7th International ISAAC Congress Volume of Abstracts

European Mathematical Society

International Mathematical Union

London Mathematical Society

7th International ISAAC Congress — Abstracts

Edited by M. Ruzhansky and J. Wirth. Prepared and typeset using LATEX. Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ

Welcoming address The ISAAC board, the Local Organising Committee and the Department of Mathematics at Imperial College London, are pleased to welcome you to the 7th International ISAAC Congress in London. The 7th International ISAAC congress continues the successful series of meetings previously held in the Delaware (USA) 1997; Fukuoka (Japan) 1999; Berlin (Germany) 2001, Toronto (Canada) 2003, Catania (Italy) 2005 and Ankara (Turkey) 2007. The success of such a series of congresses would not be possible without all the valuable contributions of all the participants. We acknowledge the financial support for this congress given by the London Mathematical Society (LMS), the International Mathematical Union (IMU), Commission on Development and Exchanges (CDE), and Developing Countries Strategy Group (DCSG), the Engineering and Physical Sciences Research Council (EPSRC), the Oxford Centre in Collaborational and Applied Mathematics (OCCAM), the Oxford Centre for Nonlinear Partial Differential Equations (OxPDE), the Bath Institute for Complex Systems (BICS), the Imperial College London, Strategic Fund, and the Department of Mathematics, Imperial College London. ISAAC Board Man Wah Wong (Toronto, Canada), President of the ISAAC Heinrich Begehr (Berlin, Germany) Alain Berlinet (Montpellier, France) Bogdan Bojarski (Warsaw,Poland) Erwin Bruning (Durban, South Africa) Victor Burenkov (Padova, Italy) Okay Celebi (Istanbul, Turkey) Robert Gilbert (Newark, Delaware, USA) Anatoly Kilbas (Minsk, Belarus) Massimo Lanza de Cristoforis (Padova, Italy) Michael Reissig (Freiberg, Germany) Luigi Rodino (Torino, Italy) Michael Ruzhansky (London, UK) John Ryan (Fayetteville, Arkansas, USA) Saburou Saitoh (Aveiro, Portugal) Bert-Wolfgang Schulze (Potsdam, Germany) Joachim Toft (V¨ axj¨ o, Sweden) Yongzhi Xu (Louisville, Kentucky, USA) Masahiro Yamamoto (Tokyo, Japan) Shangyou Zhang (Newark, Delaware, USA) Local Organising Committee Michael Ruzhansky (Chairman) Dan Crisan Brian Davies Jeroen Lamb Ari Laptev (President of the European Mathematical Society) Jens Wirth Boguslaw Zegarlinski with further support by Laura Cattaneo, Federica Dragoni, Nikki Elliott, James Inglis, Vasileios Kontis and Ioannis Papageorgiou as well as further student helpers.

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Abstracts

Plenary talks

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Sir John Ball : The Q-tensor theory of liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Louis Boutet de Monvel : Asymptotic equivariant index of Toeplitz operators and Atiyah-Weinstein conjecture Brian Davies : Non-self-adjoint spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simon Donaldson : Asymptotic analysis and complex differential geometry . . . . . . . . . . . . . . . . . . . Carlos Kenig : The global behavior of solutions to critical nonlinear dispersive and wave equations . . . . . . Vakhtang Kokilashvili : Nonlinear harmonic analysis methods in boundary value problems of analytic and harmonic functions, and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nicolas Lerner : Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems . . . . . . Paul Malliavin : Non-ergodicity of Euler deterministic fluid dynamics via stochastic analysis . . . . . . . . . Vladimir Maz’ya : Higher order elliptic problems in non-smooth domains . . . . . . . . . . . . . . . . . . . Bert-Wolfgang Schulze : Operator algebras with symbolic hierarchies on stratified spaces . . . . . . . . . . . Gunther Uhlmann : Visibility and Invisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masahiro Yamamoto : Practise of industrial mathematics related with the steel manufacturing process . . .

Public lecture

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Pierre-Louis Lions : Analysis, Models and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Sessions

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I.1. Complex variables and potential theory Tahir Aliyev Azeroˇ glo : Analytic functions in contour-solid problems . . . . . . . . . . . . . . . . . . . . . . Rauno Aulaskari : A non-α-normal function whose derivative has finite area integral of order less than 2/α Cristina Ballantine : Global mapping properties of rational functions . . . . . . . . . . . . . . . . . . . . . . Bogdan Bojarski : Beltrami equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matteo Dalal Riva : A functional analytic approach for a singularly perturbed non-linear traction problem in linearized elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Dovbush : Boundary behavior of Bloch functions and normal functions . . . . . . . . . . . . . . . . . Anatoly Golberg : Spatial quasiconformal mappings and directional dilatations . . . . . . . . . . . . . . . . Dorin Ghisa : Global mapping properties of entire and meromorphic functions . . . . . . . . . . . . . . . . . Daniyal Israfilov : Approximation in Morrey-Smirnov classes . . . . . . . . . . . . . . . . . . . . . . . . . . Dmitri Karp : Two-sided bounds for the logarithmic capacity of multiple intervals . . . . . . . . . . . . . . . Olena Karupu : On boundary smoothness of conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . Boris Kats : Structures of non-rectifiable curves and solvability of the jump problem . . . . . . . . . . . . . Gabriela Kohr : The Loewner differential equations and univalent subordination chains in several complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirela Kohr : Boundary integral equations in the study of some porous media flow problems . . . . . . . . . Massimo Lanza de Cristoforis : Singular perturbation problems in potential theory: a functional analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jamal Mamedkhanov : Classic theorems of approximation in a complex plane by rational functions . . . . Sergiy Plaksa : Commutative algebras of monogenic functions and biharmonic potentials . . . . . . . . . . . Osamu Suzuki : Fractal method for Clifford algebra and complex analysis . . . . . . . . . . . . . . . . . . . . Yunus Emre Yildirir : Approximation theorems in weighted Lorenz spaces . . . . . . . . . . . . . . . . . . . El Hassan Youssfi : Hankel operator on generalized fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . Yuriy Zelinskiy : Continues mappings between domains of manifolds . . . . . . . . . . . . . . . . . . . . . .

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I.2. Differential equations: Complex and functional analytic methods, applications Umit Aksoy : A hierarchy of polyharmonic kernel functions and the related integral operators . . . . . . . Heinrich Begehr : Boundary value problems for complex partial differential equations . . . . . . . . . . . . Peter Berglez : On some classes of bicomplex pseudoanalytic functions . . . . . . . . . . . . . . . . . . . . Carmen Bolosteanu : Boundary value problems on Klein surfaces . . . . . . . . . . . . . . . . . . . . . . . Ilya Boykov : Optimal methods for evaluation hypersingular integrals and solution of hypersingular integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Okay Celebi : Complex partial differential equations with mixed-type boundary conditions . . . . . . . . . . Natalia Chinchaladze : On a mathematical model of a cusped plate with big deflections . . . . . . . . . . .

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Jin-Yuan Du : Mixed boundary value problem with a shift for some pair of metaanalytic function and analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grigory Giorgadze : Generalized analytic functions on Riemann surfaces . . . . . . . . . . . . . . . . . . . . Sonnhard Graubner : Optimization of fixed point methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . Azhar Hussain : Generating functions of the Laguerre-Bernoulli polynomials involving bilateral series and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Kheyfits : Asymptotic behavior of subparabolic functions . . . . . . . . . . . . . . . . . . . . . . Giorgi Khimshiashvili : Elliptic Riemann-Hilbert problems for generalized Cauchy-Riemann systems . . . . Nino Manjavidze : On some qualitative issues of the elliptic systems . . . . . . . . . . . . . . . . . . . . . . Alip Mohammed : Poisson equation with the Robin boundary condition . . . . . . . . . . . . . . . . . . . . . Nusrat Rajabov : Investigation of one class of two-dimensional conjugating model and non model integral equation with fixed super-singular kernels in connection with hyperbolic equation . . . . . . . . . . . . . Lutfya Rajabova : About one class of two-dimensional Volterra type integral equation with two interior sinqular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roman Saks : Explicit global solutions of 3D rotating Navier-Stokes equations . . . . . . . . . . . . . . . . Emma Samoylova : Methods of solutions of an singular integrodifferential equation . . . . . . . . . . . . . . Tynysbek Sharipovich Kal’menov : A boundary condition of the volume potential . . . . . . . . . . . . . . . Durbudkhan Suragan : Eigenvalues and eigenfunctions of volume potential . . . . . . . . . . . . . . . . . . Zhaxylyk Tasmambetov : The ending solutions of Ince system with irregular features . . . . . . . . . . . . Ismail Taqi : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yufeng Wang : On mixed boundary-value problems of polyanalytic functions . . . . . . . . . . . . . . . . . . Oleg N. Zhdanov : An algorithm of solving the Cauchy problem and mixed problem for the two-dimensional system of quasi-linear hyperbolic partial differential equations . . . . . . . . . . . . . . . . . . . . . . . Shouguo Zhong : On solution of a kind of Riemann boundary value problem on the real axis with square roots Zhongxiang Zhang : Some Riemann boundary value problems in Clifford analysis . . . . . . . . . . . . . . .

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I.3. Complex-analytical methods for applied sciences 32 Vladimir Mityushev : R-linear problem and its applications to composites . . . . . . . . . . . . . . . . . . . 33 Michael Porter : Application of the spectral parameter power series method to conformal mapping problems 33 Sergei Rogosin : Recent results on analytic methods for 2D composite materials . . . . . . . . . . . . . . . . 33 I.4. Zeros and Gamma lines – value distributions of real and complex functions Grigor Barsegian : An universal value distribution: for arbitrary meromorphic function in a given domain Petter Branden : A generalization of the Stieltjes-Van Vleck-Bocher theorem . . . . . . . . . . . . . . . . . David Cardon : A criterion for the reality of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marios Charalambides : New properties of a class of Jacobi and generalized Laguerre polynomials . . . . . George Csordas : Meromorphic Laguerre operators and the zeros of entire functions . . . . . . . . . . . . Arturo Fern´ andez : On the logarithmic order of meromorphic functions . . . . . . . . . . . . . . . . . . . Steve Fisk : An introduction to upper (stable) polynomials in several variables . . . . . . . . . . . . . . . . Paul Gauthier : Perturbations of L-functions with or without non-trivial zeros off the critical line . . . . Rod Halburd : Tropical and number theoretic analogues of Nevanlinna theory . . . . . . . . . . . . . . . . Aimo Hinkkanen : Growth of analytic functions in unbounded open sets . . . . . . . . . . . . . . . . . . . Kazuko Kato : Zeros de la fonction holomorphe et bornee dans un polyhedre analytique de C 2 . . . . . . . Victor Katsnelson : Steiner and Weyl polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anand Prakash Singh : Spiraling Baker domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anatoliy Prykarpatsky : The algebraic Liouville integrability and the related Picard-Fuchs type equations Armen Sergeev : Quantization of universal Teichm¨ uller space: an interplay between complex analysis and quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.1 Clifford and quaternion analysis Hendrik de Bie : Clifford analysis for orthogonal, symplectic and finite reflection groups . . . . Cinzia Bisi : M¨ obius transformations and Poincar´e distance in the quaternionic setting . . . . Paula Cerejeiras : Wavelets invariant under reflection groups . . . . . . . . . . . . . . . . . . . Fabrizio Colombo : Some consequences of the quaternionic functional calculus . . . . . . . . . . Kevin Coulembier : Orthogonality of Clifford-Hermite polynomials in superspace. . . . . . . . Sirkka-Liisa Eriksson : Recent results on hyperbolic function theory . . . . . . . . . . . . . . . . Ming-Gang Fei : Symmetric properties of the Fourier transform in Clifford analysis setting . . Milton Ferreira : Factorization of M¨ obius gyrogroups - the paravector case . . . . . . . . . . . . Peter Franek : Higher spin analogues of the Dirac operator in two variables and its resolution . Ghislain R. Franssens : Cauchy kernels in ultrahyperbolic Clifford analysis – Huygens cases . . Graziano Gentili : Power series and analyticity over the quaternions . . . . . . . . . . . . . . . Anastasia Kisil : Isomorphic action of SL(2, R) on hypercomplex numbers . . . . . . . . . . . . Rolf Soeren Krausshar : Construction of 3D mappings on to the unit ball with the hypercomplex Lukas Krump : Explicit description of the resolution for 4 Dirac operators in dimension 6 . . Roman Lavicka : On polynomial solutions of Moisil-Theodoresco systems in Euclidean spaces . Matvei Libine : Quaternionic analysis, representation theory and Physics . . . . . . . . . . . .

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Maria Elena Luna-Elizarrar´ as : Hyperholomorphic functions in the sense of Moisil-Thodoresco and their different hyperderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mircea Martin : Dirac and semi-Dirac pairs of differential operators . . . . . . . . . . . . . . . . . . . . . Heikki Orelma : A differential form approach to Dirac operators on surfaces . . . . . . . . . . . . . . . . Dixan Pe˜ na Pe˜ na : CK-extension and Fischer decomposition for the inframonogenic functions . . . . . . . Alessandro Perotti : A new approach to slice-regularity on real algebras . . . . . . . . . . . . . . . . . . . . Yuying Qiao : Clifford analysis with higher order kernel over unbounded domains . . . . . . . . . . . . . . Guangbin Ren : Complex Dunkl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Ryan : p-Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irene Sabadini : Duality theorems for slice hyperholomorphic functions . . . . . . . . . . . . . . . . . . . Tomas Salac : Explicit description of operators in the resolution for the Dirac operator . . . . . . . . . . . Michael Shapiro : On the relation between the Fueter operator and the Cauchy-Riemann-type operators of Clifford analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Petr Somberg : Conformally invariant boundary valued problems for spinors and families of homomorphisms of generalized Verma modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frank Sommen : Clifford calculus in quantum variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Soucek : On relative BGG sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Caterina Stoppato : Regular Moebius transformations over the quaternions . . . . . . . . . . . . . . . . . Adrian Vajiac : Singularities of functions of one and several bicomplex variables . . . . . . . . . . . . . . Fabio Vlacci : Multiplicities of zeroes and poles of regular functions . . . . . . . . . . . . . . . . . . . . . . Zuzana Vlasakova : Gauss-Codazzi-Ricci equations in Riemannian, conformal, and CR geometry . . . . . Liesbet Van de Voorde : Compatibility conditions and higher spin Dirac operators . . . . . . . . . . . . . . II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras Swanhild Bernstein : Wavelets on spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Bock : On special monogenic power and Laurent series expansions and applications . . . . . . . Ruth Farwell : Spin gauge models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nelson Faustino : Further results in discrete Clifford analysis . . . . . . . . . . . . . . . . . . . . . . . . . Thanasis Fokas : Integrability in multidimensions, complexification and quaternions . . . . . . . . . . . . . Svetlin Georgiev : Note on the linear systems in quaternions . . . . . . . . . . . . . . . . . . . . . . . . . Jacques Helmstetter : Minimal algorithms for Lipschitzian elements and Vahlen matrices . . . . . . . . . Jeff Hogan : Clifford-Fourier transforms and hypercomplex signal processing . . . . . . . . . . . . . . . . . Uwe K¨ ahler : Discrete Clifford analysis by means of skew-Weyl relations . . . . . . . . . . . . . . . . . . . Vladimir Kisil : Hypercomplex analysis in the upper half-plane . . . . . . . . . . . . . . . . . . . . . . . . . Rolf Soeren Krausshar : Formulas for reproducing kernels of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to inhomogeneous Helmholtz equations . . . . . . . . Remi Leandre : The Ito transform for partial differential equations . . . . . . . . . . . . . . . . . . . . . . Dimitris Pinotsis : Quaternionic analysis and boundary value problems . . . . . . . . . . . . . . . . . . . Vitalii Shpakivskii : Integral theorems in a commutative three-dimensional harmonic algebra . . . . . . . . Wolfgang Spr¨ oßig : Initial boundary value problems with quaternionic analysis . . . . . . . . . . . . . . . . Tolksdorf, J¨ urgen : Real bi-graded Clifford modules, the Majorana equation and the standard model action Nelson Vieira : The regularized Schr¨ odinger semigroup acting on tensors with values in vector bundles . . III.1. Toeplitz operators and their applications Cristina Cˆ amara : On the relations between the kernel of a Toeplitz operator and the solutions to some associated Riemann-Hilbert problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luis Castro : Convolution type operators with symmetry in exterior wedge diffraction problems . . . . . . Miroslav Englis : Berezin transform on the harmonic Fock space . . . . . . . . . . . . . . . . . . . . . . . Sergey Grudsky : Inside the eigenvalues of certain Hermitian Toeplitz band matrices . . . . . . . . . . . . Turan G¨ urkanlı : Toeplitz operators of M (p, q, w)(Rd ) spaces . . . . . . . . . . . . . . . . . . . . . . . . . Oleksandr Karelin : Presentation of the kernel of a special structure matrix characteristic operator by the kernels of two operators one of them is a scalar characteristic operator . . . . . . . . . . . . . . . . . Edixon Rojas : Bounds for the kernel dimension of singular integral operators with Carleman shift . . . . Anabela Silva : Invertibility of matrix Wiener-Hopf plus Hankel operators with different Fourier symbols . Harald Upmeier : Flat Hilbert bundles and Toeplitz operators on symmetric spaces . . . . . . . . . . . . . Nikolai Vasilevski : Commutative algebras of Toeplitz operators on the unit ball . . . . . . . . . . . . . . . Kehe Zhu : Toeplitz operators on the Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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III.2. Reproducing kernels and related topics Belkacem Abdous : A general theory for kernel estimation of smooth functionals . . . . . . . . . . . . . . . Som Datt Sharma : Weighted composition operators on some spaces of analytic functions . . . . . . . . . . Keiko Fujita : Integral formulas on the boundary of some ball . . . . . . . . . . . . . . . . . . . . . . . . . . John Rowland Higgins : Paley–Wiener spaces and their reproducing formulae. . . . . . . . . . . . . . . . . . Darian Onchis : Irregular sampling in multiple-window spline-type spaces . . . . . . . . . . . . . . . . . . . Kazuo Takemura : Free boundary value problem for (−1)M (d/dx)2M and the best constant of Sobolev inequality

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III.3. Modern aspects of the theory of integral transforms Liubov Britvina : Integral transforms related to generalized convolutions and their applications to solving integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiuhui Chen : Bedrosian identity for Blaschke products in n-parameter cases . . . . . . . . . . . . . . . . Dong Hyun Cho : Evaluation formulae for analogues of conditional analytic Feynman integrals over a function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diana Doli´canin : An equation with symmetrized fractional derivatives . . . . . . . . . . . . . . . . . . . . Hiroshi Fujiwara : Numerical real inversion of the Laplace transform by reproducing kernel and multipleprecision arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anatoly Kilbas : Method of integral transforms in the theory of fractional differential equations . . . . . . Bong Jin Kim : Notes on the analytic Feynman integral over paths in abstract Wiener space . . . . . . . . Sanja Konjik : On the fractional calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Koroleva : Integral transforms with extended generalized Mittag-Leffler function . . . . . . . . . . . Ljubica Oparnica : Systems of differential equations containing distributed order fractional derivative . . . Juri M. Rappoport : Some aspects of modified Kontorovitch-Lebedev integral transforms . . . . . . . . . . Semyon Yakubovich : A new class of polynomials related to the Kontorovich-Lebedev transform . . . . . .

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III.4. Spaces of differentiable functions of several real variables and applications Alexandre Almeida : Hardy spaces with generalized parameter . . . . . . . . . . . . . . . . . . . . . . . . . . Tsegaye Gedif Ayele : Iterated norms in Nikol’ski˘ı-Besov type spaces with generalized smoothness . . . . . p(.) Ismail Aydın : Embeddings Properties of The Spaces Lw (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . Canay Aykol : On the boundedness of fractional B-maximal operators in the Lorentz spaces Lp,q,γ (Rn ) . . . Oleg Besov : Spaces of functions of fractional smoothness on an irregular domain . . . . . . . . . . . . . . . Santiago Boza : Rearrangement transformations on general measure spaces . . . . . . . . . . . . . . . . . . Maria Carro : Last developments on Rubio de Francia’s extrapolation theory . . . . . . . . . . . . . . . . . . Gurgen Dallakyan : On transformation of coordinates invariant relative to Sobolev spaces with polyhedral anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ismail Ekincioglu : The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted Lpw spaces with general weights . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Goldshtein : Composition Operators for Sobolev spaces of functions and differential forms . . . . . Vagif Guliyev : Boundedness of the fractional maximal operator and fractional integral operators in general Morrey type spaces and some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mubariz Hajibayov : Weighted estimates of generalized potentials in variable exponent Lebesque spaces . . . Ritva Hurri-Syrjanen : Our talk is on vanishing exponential integrability for Besov functions. . . . . . . . . Gennady Kalyabin : New sharp estimates for function in Sobolev spaces on finite Interval . . . . . . . . . . Leili Kusainova : On real interpolation of weighted Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . Elijah Liflyand : The Fourier transform of a radial function . . . . . . . . . . . . . . . . . . . . . . . . . . Yagub Mammadov : Necessary and sufficient conditions for the boundedness of Riesz potential in Morrey spaces associated with Dunkl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ana Moura Santos : Image normalization of Wiener-Hopf operators in diffraction problems . . . . . . . . . Bohum´ır Opic : Weighted estimates for the averaging integral operator and reverse H¨ older inequalities . . . Humberto Rafeiro : Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup Evgeniy Radkevich : On the Maxwell problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natasha Samko : Weighted potential operators in Morrey spaces. . . . . . . . . . . . . . . . . . . . . . . . . Stefan Samko : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kader Senouci : Equivalent semi-norms for Nikol’skii- Besov spaces on an interval . . . . . . . . . . . . . . Ayhan Serbetci : Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications Javier Soria : Translation-invariant bilinear operators with positive kernels . . . . . . . . . . . . . . . . . . . Sergey Tikhonov : Sharp inequalities for moduli of smoothness and K-functionals . . . . . . . . . . . . . . . Boris V. Trushin : Sobolev embedding theorems for a class of anisotropic irregular domains . . . . . . . . . Yusuf Zeren : Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.5. Analytic and harmonic function spaces Miloud Assal : Multiplier theorem in the setting of Laguerre hypergroups and applications . . . . . . . . . Boo Rim Choe : Progress on finite rank Toeplitz products . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Girela : Functions and operators in analytic Besov spaces . . . . . . . . . . . . . . . . . . . . . . . Maria Jose Gonzales : Square functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sanjiv Gupta : Convolutions of generic orbital measures in compact symmetric spaces . . . . . . . . . . . H. Turgay Kaptano˘ glu : Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Young Joo Lee : Sums of Toeplitz products on the Dirichlet space . . . . . . . . . . . . . . . . . . . . . . . Jasbir Singh Manhas : Weighted composition operators on weighted spaces of analytic functions . . . . . . Auxiliadora Marquez : Superposition operators between Qp spaces and Hardy spaces . . . . . . . . . . . . Malgorzata Michalska : Bounded Toeplitz and Hankel products on Bergman space . . . . . . . . . . . . . .

8

50 51 51 51 51 51 52 52 52 52 53 53 53 53 53 53 54 54 54 54 55 55 55 55 55 55 55 56 56 56

56 . 56 . 56 . 56 . 56 . 57 . . . . .

57 57 57 57 57

Kyesook Nam : Optimal norm estimate of the harmonic Bergman projection . . . . . . . . . . . Pekka Nieminen : Old and new on composition operators on VMOA and BMOA spaces . . . . . Maria Nowak : On Libera and Cesaro operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jordi Pau : Integration operators on weighted Bergman spaces . . . . . . . . . . . . . . . . . . . . Amol Sasane : Extension to an invertible matrix in Banach algebras of measures . . . . . . . . . Benoit F. Sehba : Multiplication operators on weighted BMOA spaces . . . . . . . . . . . . . . . . Pawel Sobolewski : Inequalities for Hardy spaces on the unit ball . . . . . . . . . . . . . . . . . . M¨ ubariz Tapdıgo˘ glu : On the Duhamel algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jari Taskinen : Toeplitz operators on Bergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . Luis Manuel Tovar : Hyperbolic weighted Bergman classes . . . . . . . . . . . . . . . . . . . . . . Dragan Vukotic : Multiplicative isometries and isometric zero-divisors . . . . . . . . . . . . . . . Zhijian Wu : Area operators on analytic function spacess . . . . . . . . . . . . . . . . . . . . . . . Hasi Wulan : Composition operators on BMOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wen Xu : Lacunary series and QK spaces on the unit ball . . . . . . . . . . . . . . . . . . . . . . Congli Yang : Some results on ϕ-Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Kehe Zhu : Holomorphic mean Lipschitz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . Nina Zorboska : Univalently induced closed range composition operators on the Bloch-type spaces

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

57 58 58 58 58 58 58 58 58 59 59 59 59 59 59 59 59

III.6. Spectral theory Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: surface potentials, equations at the boundary, and corresponding transmission problems. . . . . . . . . . . . . . . . . . . . . . Shavkat Alimov : On the spectral expansions associated with Laplace-Beltrami operator . . . . . . . . . . . . Victor Burenkov : Sharp spectral stability estimates for higher order elliptic operators . . . . . . . . . . . . Daniel Elton : Strong field asymptotics for zero modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leander Geisinger : A universal bound for the trace of the heat kernel . . . . . . . . . . . . . . . . . . . . . Tigran Harutyunyan : The eigenvalues function of the family of Sturm-Liouville operators and its applications Jan Janas : Generalized eigenvectors of some Jacobi matrices in the critical case . . . . . . . . . . . . . . . Thomas Krainer : Trace expansions for elliptic cone operators . . . . . . . . . . . . . . . . . . . . . . . . . . Pier Domenico Lamberti : Stability estimates for eigenfunctions of elliptic operators on variable domains . . Oleksii Mokhonko : Spectral theory of the normal operator with the spectra on an algebraic curve . . . . . . Jiri Neustupa : Spectral properties of operators arising from modelling of flows around rotating bodies . . . . Serge Richard : New formulae for the wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedetto Silvestri : Spectral bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Strohmaier : Scattering theory for manifolds and the scattering length . . . . . . . . . . . . . . . Yuriy Tomilov : Spectrum and wandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tomio Umeda : Eigenfunctions at the threshold energies of magnetic Dirac operators . . . . . . . . . . . . .

59

IV.1. Pseudo-differential operators Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: Dirichlet and Neumann problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chikh Bouzar : Generalized ultradistributions and their microlocal analysis . . . . . . . . . . . . . . . . . . Ernesto Buzano : Some remarks on the Sj¨ ostrand class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viorel Catana : The heat equation for the generalized Hermite and the generalized Landau operators . . . . Leon Cohen : Generalization of the Weyl rule for arbitrary operators . . . . . . . . . . . . . . . . . . . . . . Elena Cordero : Sharp results for the STFT and localization operators . . . . . . . . . . . . . . . . . . . . . Yasuo Chiba : Fuchsian mild microfunctions with fractional order and their applications to hyperbolic equations Paulo Dattori da Silva : About Gevrey semi-global solvability of a class of complex planar vector fields with degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Julio Delgado : Invertibility for a class of degenerate elliptic operators . . . . . . . . . . . . . . . . . . . . . Kenro Furutani : Heat kernel of a sub-Laplacian and Grushin type operators . . . . . . . . . . . . . . . . . . Lorenzo Galleani : Time-frequency analysis of stochastic differential equations . . . . . . . . . . . . . . . . . Gianluca Garello : Lp -microlocal regularity for pseudodifferential operators of quasi-homogeneous type . . . Claudia Garetto : Generalized Fourier integral operators methods for hyperbolic problems . . . . . . . . . . . Juan Gil : Resolvents of regular singular elliptic operators on a quantum graph . . . . . . . . . . . . . . . . Todor Gramchev : Hyperbolic systems of pseudodifferential equations with irregular symbols in t admitting superlinear growth for |x| → ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernhard Gramsch : Analytic perturbations for special Fr´echet operator algebras in the microlocal analysis . G¨ unther H¨ ormann : The Cauchy problem for a paraxial wave equation with non-smooth symbols . . . . . . Eug´enie Hunsicker : Pseudodifferential operators on locally symmetric spaces . . . . . . . . . . . . . . . . . Wataru Ichinose : On the continuity of the solutions with respect to the electromagnetic potentials to the Schr¨ odinger and the Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chisato Iwasaki : Calculus of pseudo-differential operators and a local index of Dirac operators . . . . . . . Jon Johnsen : On the theory of type 1, 1-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuryi Karlovych : Pseudo-differential operators with discontinuous symbols and their applications . . . . . . Thomas Krainer : On maximal regularity for parabolic equations on complete Riemannian manifolds . . . . Roberto de Leo : On the cohomological equation in the plane for regular vector fields . . . . . . . . . . . .

62

60 60 60 60 60 61 61 61 61 61 62 62 62 62 62 62

63 63 63 63 63 63 63 64 64 64 64 64 65 65 65 65 65 65 65 66 66 66 66 66

9

Yu Liu : Lp -boundedness and compactness of localization operators associated with Stockwell transform . . . Jean-Andr´e Marti : About transport equation with irregular coefficient and data . . . . . . . . . . . . . . . . Shahla Molahajloo : The Heat Kernel of τ -Twisted Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . Alessandro Morando : Regularity of characteristic initial-boundary value problems for symmetrizable systems David Natroshvili : Application of pseudodifferential equations in stress singularity analysis for thermoelectro-magneto-elasticity problems: a new approach for calculation of stress singularity exponents . . . Alessandro Oliaro : Wigner type transforms and pseudodifferential operators . . . . . . . . . . . . . . . . . . Michael Oberguggenberger : Local regularity of solutions to PDEs by asymptotic methods . . . . . . . . . . Nusrat Rajabov : Modern results by theory of the three dimensional Volterra type linear integral equations with singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frederic Rochon : The adiabatic limit of the Chern character . . . . . . . . . . . . . . . . . . . . . . . . . . Bert-Wolfgang Schulze : Boundary value problems as edge problems . . . . . . . . . . . . . . . . . . . . . . Elmar Schrohe : Noncommutative residues and projections associated to boundary value problems . . . . . J¨ org Seiler : On maximal regularity for mixed order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Tatyana Shaposhnikova : Dirichlet problem for higher order elliptic systems with BMO assumptions on the coefficients and the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hidetoshi Tahara : Gevrey regularities of solutions of nonlinear singular partial differential equations . . . . Nenad Teofanov : Wave-front sets and SG type operators in Fourier-Lebesgue spaces . . . . . . . . . . . . . Joachim Toft : Wave-front sets of Fourier Lebesgue types . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ville Turunen : Pseudo-differential operators and symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Vasilyev : Pseudo differential equations and boundary value problems in non-smooth domains . . Andr´ as Vasy : Diffraction at corners for the wave equation on differential forms . . . . . . . . . . . . . . . Ingo Witt : Formation of singularities near Morse points . . . . . . . . . . . . . . . . . . . . . . . . . . . . Man Wah Wong : Phases of modified Stockwell transforms and instantaneous frequencies . . . . . . . . . . Hongmei Zhu : Generalized cosine transforms in image compression . . . . . . . . . . . . . . . . . . . . . . IV.2. Dispersive equations Marcello D’Abbico : Lp –Lq estimates for hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . Q-Heung Choi : Multiple solutions for non-linear parabolic systems . . . . . . . . . . . . . . . . . . . . . . Ferruccio Colombini : Local sovability beyond condition ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniele Del Santo : Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients Marcello Ebert : On the loss of regularity for a class of weakly hyperbolic operators . . . . . . . . . . . . . Daoyuan Fang : Zakharov system in infinite energy space . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anahit Galstyan : Wave equation in Einstein-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Georgiev : Stability of solitary waves for Hartree type equation . . . . . . . . . . . . . . . . . . . . Marina Ghisi : Hyperbolic-parabolic singular perturbations for Kirchhoff-equations . . . . . . . . . . . . . . . Massimo Gobbino : Existence and uniqueness results for Kirchhoff equations in Gevrey-type spaces . . . . . Torsten Herrmann : Precise loss of derivatives for evolution type models . . . . . . . . . . . . . . . . . . . . Fumihiko Hirosawa : Wave equations with time dependent coefficients . . . . . . . . . . . . . . . . . . . . . Tacksun Jung : Critical point theory applied to a class of systems of super-quadratic wave equations . . . . Lavi Karp : On the well-posdness of the vacuum Einstein equations . . . . . . . . . . . . . . . . . . . . . . Hideo Kubo : Generalized wave operator for a system of nonlinear wave equations . . . . . . . . . . . . . . Tokio Matsuyama : Strichartz estimates for hyperbolic equations in an exterior domain . . . . . . . . . . . Kiyoshi Mochizuki : Uniform resolvent estimates and smoothing effects for magnetic Schr¨ odinger operators Hideo Nakazawa : Decay and scattering for wave equations with dissipations in layered media . . . . . . . . Tatsuo Nishitani : On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 wellposedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rainer Picard : On the structure of the material law in a linear model of poro-elasticity . . . . . . . . . . . Marco Pivetta : Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuous coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Reissig : The log-effect for 2 by 2 hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . Jun-ichi Saito : The Boussinesq equations based on the hydrostatic approximation . . . . . . . . . . . . . . . Ryuichi Suzuki : Blow-up of solutions of a quasilinear parablolic equation . . . . . . . . . . . . . . . . . . . Hiroshi Uesaka : Blow-up and a blow-up boundary for a semilinear wave equation with some convolution nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karen Yagdjian : Fundamental solutions for hyperbolic operators with variable coefficients . . . . . . . . . . Borislav Yordanov : Global existence in Sobolev spaces for a class of nonlinear Kirchhoff equations . . . . . IV.3. Control and optimisation of nonlinear evolutionary systems Lorena Bociu : Global well-posedness and long-time behavior of solutions to a wave equation . . . . . . . . Mahdi Boukrouche : Distributed optimal controls for second kind parabolic variational inequalities . . . . Muriel Boulakia : Controllability of a fluid-structure interaction problem . . . . . . . . . . . . . . . . . . Marcello Cavalcanti : Uniform decay rate estimates for the wave equation on compact surfaces and locally distributed damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moez Daoulatli : Rate of decay for non-autonomous damped wave systems . . . . . . . . . . . . . . . . . .

10

66 67 67 67 67 67 68 68 68 68 68 69 69 69 69 69 70 70 70 70 70 70 71 71 71 71 71 72 72 72 72 72 73 73 73 73 73 74 74 74 74 74 74 74 75 75 75 75 75 76

76 . 76 . 76 . 76 . .

77 77

Valeria Domingos Cavalcanti : On qualitative aspects for the damped Korteweg-de Vries and Airy type equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Eller : Optimal control of waves in anisotropic media via conservative boundary conditions . . . Genni Fragnelli : Stability for some nonlinear damped wave equations . . . . . . . . . . . . . . . . . . . . Anahit Galstyan : Global existence for the one-dimensional semilinear Tricomi-type equation . . . . . . . Catherine Lebiedzik : Optimal control of a thermoelastic structural acoustic model . . . . . . . . . . . . . Walter Littman : The Balayage method: Boundary control of a thermo-elastic plate . . . . . . . . . . . . . Paola Loreti : Hopf-Lax type formulas and Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . Vyacheslav Maksimov : Investigation of boundary control problems by on-line inversion technique . . . . . Patrick Martinez : Null controllability properties of some degenerate parabolic equations . . . . . . . . . . Maria Grazia Naso : Dissipation in contact problems: an overview and some recent results . . . . . . . . Luciano Pandolfi : Heat equations with memory: a Riesz basis approach . . . . . . . . . . . . . . . . . . . Michael Renardy : A note on a class of observability problems for PDEs . . . . . . . . . . . . . . . . . . . Roland Schnaubelt : Invariant manifolds for parabolic problems with dynamical boundary conditions . . . Ilya Shvartsman : On regularity properties of optimal control and Lagrange multipliers . . . . . . . . . . . Daniela Sforza : Evolution equations with memory terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Toundykov : Stabilization of structure-acoustics interactions for a Reissner-Mindlin plate by localized nonlinear boundary feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Julie Valein : Exponential stability of the wave equation with boundary time varying delay . . . . . . . . . Masahiro Yamamoto : State estimation for some parabolic systems . . . . . . . . . . . . . . . . . . . . . . Jean-Paul Zolesio : Euler flow and Morphing Shape Metric . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 77 77 77 78 78 78 78 78 78 79 79 79 79 79

. 79 . 80 . 80 . 80

IV.4. Nonlinear partial differential equations Piero D’Ancona : Evolution equations in nonflat waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . Mersaid Aripov : Investigation of solutions of one not divergent type . . . . . . . . . . . . . . . . . . . . . . Davide Catania : Asymptotic behavior of subparabolic functions . . . . . . . . . . . . . . . . . . . . . . . . Kuan-Ju Chen : On multiple solutions of concave and convex effects for nonlinear elliptic equation on RN Kazuyuki Doi : Nonlinear gauge invariant evolution of the plane wave . . . . . . . . . . . . . . . . . . . . . Mohammad Dehghan : New approach to solve linear parabolic problems via semigroup approximation . . . Albert Erkip : Global existence and blow-up for the nonlocal nonlinear Cauchy problem . . . . . . . . . . . . Marius Ghergu : Qualitative properties for reaction-diffusion systems modelling chemical reactions . . . . . Marco Antonio Taneco-Hern´ andez : Scattering in the zero-mass Lamb system . . . . . . . . . . . . . . . . . Soichiro Katayama : Global existence for systems of the nonlinear wave and Klein-Gordon equations in 3D Hideo Kubo : Global existence for nonlinear wave equations exterior to an obstacle in 2D . . . . . . . . . . Petr Kucera : Remark on Navier-Stokes equations with mixed boundary conditions . . . . . . . . . . . . . . Ut van Le : Contraction-Galerkin method for a semi-linear wave equation with a boundary-like antiperiodic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sandra Lucente : p − q systems of nonlinear Schrodinger equations . . . . . . . . . . . . . . . . . . . . . . . Satoshi Masaki : Semiclassical analysis for nonlinear Schrodinger equations . . . . . . . . . . . . . . . . . . Gianluca Mola : 3-D viscous Cahn-Hilliard equation with memory . . . . . . . . . . . . . . . . . . . . . . . Itir Mogultay : A symmetric error estimate for Galerkin approximations of time dependant Navier-Stokes equations in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masahito Ohta : Stability of standing waves for some systems of nonlinear Schr¨ odinger equations with three-wave interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Reissig : Decay rates for wave models with structural damping . . . . . . . . . . . . . . . . . . . . . Yoshihiro Shibata : Stability theorems in the theory of mathematical fluid mechanics . . . . . . . . . . . . . L´ aszl´ o Simon : On singular systems of parabolic functional equations . . . . . . . . . . . . . . . . . . . . . Zdenek Skalak : Survey of recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atanas Stefanov : Conditional stability theorems for Klein-Gordon type equations . . . . . . . . . . . . . . Sergio Spagnolo : A regularity result for a class of semilinear hyperbolic equations . . . . . . . . . . . . . . . Kamal Soltanov : On nonlinear equations, fixed-point theorems and their applications . . . . . . . . . . . . . Alessandro Teta : Dynamics of a quantum particle in a cloud chamber . . . . . . . . . . . . . . . . . . . . . Yoshihiro Ueda : Half space problem for the damped wave equation with a non-convex convection term . . . Nicola Visciglia : On the time-decay of solutions to a family of defocusing NLS . . . . . . . . . . . . . . . . Karen Yagdjian : The semilinear Klein-Gordon equation in de Sitter spacetime . . . . . . . . . . . . . . . .

80 80 80 81 81 81 81 81 81 82 82 82 82

IV.5. Asymptotic and multiscale analysis Natalia Babych : On the essential spectrum and singularities of solutions for Lam´e problem in cuspoidal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michel Bellieud : Torsion effects in elastic composites with high contrast . . . . . . . . . . . . . . . . . . . Yves Capdeboscq : Enhanced resolution in structured media . . . . . . . . . . . . . . . . . . . . . . . . . . Juan Casado-Diaz : Homogenization of elliptic partial differential equations with unbounded coefficients in dimension two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Cherdantsev : Two-scale Γ-convergence and its applications to homogenisation of non-linear highcontrast problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

82 83 83 83 83 83 83 84 84 84 84 84 84 84 84 85 85

. 85 . 85 . 85 .

86

.

86

11

Valentina Alekseevna Golubeva : Construction of the two-parametric generalizations of the KnizhnikZamolodchikov equations of Bn type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabricio Macia : Long-time behavior for the Wigner equation and semiclassical limits in heterogeneous media Peter Markowich : On nonlinear dispersive equations in periodic structures: Semiclassical limits and numerical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karsten Matthies : Derivation of Boltzmann-type equations from hard-sphere dynamics . . . . . . . . . . . Bernd Schmidt : Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valery Smyshlyaev : Homogenization with partial degeneracies: analytic aspects and applications . . . . . .

86 86 86 87 87 87

V.1. Inverse problems Abdellatif El Badia : An inverse conductivity problem with a single measurement . . . . . . . . . . . . . . . Fabrizio Colombo : Global in time existence and uniqueness results for some integrodifferential identification problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mourad Choulli : Stability estimate for an inverse problem for the magnetic Schr¨ odinger equation from the Dirichlet-to-Neumann map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikko Kaasalainen : Optimal combination of data modes in inverse problems: maximum compatibility estimate Christian Daveau : On an inverse problem for a linear heat conduction problem . . . . . . . . . . . . . . . Matti Lassas : Inverse problems for wave equation and a modified time reversal method . . . . . . . . . . . . Koung Hee Leem : Picard condition based regularization techniques in inverse obstacle scattering . . . . . . William Lionheart : Limited data problems in tensor tomography . . . . . . . . . . . . . . . . . . . . . . . . Marco Marletta : The finite data non-selfadjoint inverse resonance problem . . . . . . . . . . . . . . . . . . Tsutomu Matsuura : Numerical solutions of nonlinear simultaneous equations . . . . . . . . . . . . . . . . . George Pelekanos : A fixed-point algorithm for determining the regularization parameter in inverse scattering Roland Potthast : A time domain probe method for inverse scattering problems . . . . . . . . . . . . . . . . Saburou Saitoh : Explicit and direct representations of the solutions of nonlinear simultaneous equations . Vassilios Sevroglou : Direct and inverse mixed impedance problems in linear elasticity . . . . . . . . . . . . Igor Trooshin : On inverse scattering for nonsymmetric operators . . . . . . . . . . . . . . . . . . . . . . .

87 87

V.2. Stochastic analysis David Applebaum : Cylindrical Levy processes in Banach space . . . . . . . . . . . . . . . . . . . . . . . Vlad Bally : Integration by parts for locally smooth laws and applications to jump type diffusions . . . . . Dorje Brody : Information and asset pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Caruana : A (rough) pathwise approach to fully non-linear stochastic partial differential equations Dan Crisan : Solving backward stochastic differential equations using cubature methods . . . . . . . . . . . Ana Bela Cruzeiro : Some results on Lagrangian Navier-Stokes flows . . . . . . . . . . . . . . . . . . . . Alexander Davie : A uniqueness problem for SDEs and a related estimate for transition functions . . . . Mark H. A. Davis : Risk-sensitive portfolio optimization with jump-diffusion asset prices . . . . . . . . . Istvan Gyongy : Accelerated numerical schemes for nonlinear filtering . . . . . . . . . . . . . . . . . . . . Martin Hairer : Periodic homogenisation with an interface . . . . . . . . . . . . . . . . . . . . . . . . . . . Lane Hughston : Wiener chaos models for interest rates and foreign exchange . . . . . . . . . . . . . . . . Saul Jacka : Minimising the time to a decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark Kelbert : Markov process representations for polyharmonic functions . . . . . . . . . . . . . . . . . Wilfried Kendall : Networks and Poisson line patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vassili Kolokoltsov : The Levy-Khinchine type operators with variable Lipschitz continuous coefficients and stochastic differential equations driven by nonlinear Levy noise . . . . . . . . . . . . . . . . . . . . . Thomas Kurtz : Equivalence of stochastic equations and martingale problems . . . . . . . . . . . . . . . . Xue-Mei Li : Aida’s logarithmic Sobolev inequality with weights and Poincare inequalities. . . . . . . . . . Terence Lyons : Evolution equations for communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aleksandar Mijatovic : On the martingale property of certain local martingale . . . . . . . . . . . . . . . . Khairia El-Said El-Nadi : On some stochastic dynamical systems and cancer . . . . . . . . . . . . . . . . Anastasia Papavasiliou : Statistical inference for rough differential equations . . . . . . . . . . . . . . . . Martijn Pistorius : First passage for stochastic volatility models . . . . . . . . . . . . . . . . . . . . . . . . Boris Rozovsky : Unbiased random perturbations of Navier-Stokes equation . . . . . . . . . . . . . . . . . Marta Sanz-Sol´e : A Poisson equation with fractional noise . . . . . . . . . . . . . . . . . . . . . . . . . . Radu Tunaru : Constructing discrete exact approximations algorithms for financial calculus from weak convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Tretyakov : Numerical methods for parabolic SPDEs based on the averaging-over-characteristics formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elena Usoltseva : Consistent estimator in AFTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

90 90 90 90 91 91 91 91 91 91 91 92 92 92 92

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92 92 93 93 93 93 93 94 94 94

.

94

. .

94 94

V.3. Coercivity and functional inequalities Franck Barthe : Remarks on non-interacting conservative spin systems . . . . . . . . . . . . . . . . . . . Sergey Bobkov : On weak forms of Poincare-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . Messaoud Boulbrachene : L∞ -Error estimate for variational inequalities with vanishing zero order term Federica Dragoni : Convexity along vector fields and application to equations of Monge-Amp`ere type . . Ivan Gentil : Φ-entropy inequalities for diffusion semigroups . . . . . . . . . . . . . . . . . . . . . . . . .

12

. . . . .

88 88 88 88 88 88 89 89 89 89 89 89 89 90

95 . 95 . 95 . 95 . 95 . 95

Alexander Grigoryan : On positive solutions of semi-linear elliptic inequalities on manifolds Martin Hairer : Hypoellipticity in infinite dimensions . . . . . . . . . . . . . . . . . . . . . . Waldemar Hebisch : Logaritmic Sobolev inequality on nilpotent groups . . . . . . . . . . . . Nolwen Huet : Isoperimetry for spherically symmetric log-concave probability measures . . . James Inglis : Operators on the Heisenberg group with discrete spectra . . . . . . . . . . . . Mikhail Neklyudov : Liggett inequality and interacting particle systems . . . . . . . . . . . . Felix Otto : A new criterion for a covariance estimate . . . . . . . . . . . . . . . . . . . . Ioannis Papageorgiou : The Log-Sobolev inequality for non quadratic interactions . . . . . . Cyril Roberto : Isoperimetry for product probability measures . . . . . . . . . . . . . . . . .

. . . . . . . . .

95 96 96 96 96 96 96 96 96

V.4. Dynamical systems Marco Abate : Poincar´e-Bendixson theorems in holomorphic dynamics . . . . . . . . . . . . . . . . . . . . Jos´e Ferreira Alves : On the liftability of absolutely continuous ergodic expanding measures. . . . . . . . . . Flavio Abdenur : New results on stability and genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pierre Berger : Abundance of one dimensional non uniformly hyperbolic attractors for surface dynamics . . Svetlana Aleksandrovna Budochkina : First integrals in mechanics of infinite-dimensional systems . . . . . Keith Burns : Partial hyperbolicity and ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Ren´e Chazottes : On tilings, multidimensional subshifts of finite type and quasicrystals . . . . . . . . . Yi-Chiuan Chen : On topological entropy of billiard tables with small inner scatterers . . . . . . . . . . . . . Bau-Sen Du : On the nature of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Field : Mixing for flows and skew extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jorge Freitas : Rates of mixing, large deviations and recurrence times . . . . . . . . . . . . . . . . . . . . . Giovanni Forni : Limiting distributions for horocycle flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valery Gaiko : Limit cycle problems and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Jordan : Hausdorff dimension of Projections of McMullen-Bedford carpets . . . . . . . . . . . . . Jan Cees van der Meer : Fourfold 1:1 resonance, relative equilibria and moment polytopes . . . . . . . . . . Matthew Nicol : A dynamical Borel-Cantelli lemma for a class of non-uniformly hyperbolic systems . . . . Asad Niknam : Approximately inner C ∗ -dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Giovanni Panti : Dynamical systems arising in algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . Chen-chang Peng : Existence of transversal homoclinic orbits for Arneodo-Coullet-Tresser map . . . . . . . Martin Rasmussen : Bifurcations of random diffeomorphisms with bounded noise . . . . . . . . . . . . . . . Felix Sadyrbaev : Bifurcations of period annuli and solutions of nonlinear boundary value problems . . . . . J¨ org Schmeling : Large intersection properties of some invariant sets in number-theoretic dynamical systems Mike Todd : Thermodynamic formalism for unimodal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiudong Wang : Dynamics of periodically perturbed homoclinic solutions . . . . . . . . . . . . . . . . . . . .

97 97 97 97 97 97 97 98 98 98 98 98 98 98 99 99 99 99 99 99 100 100 100 100 100

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. . . . . . . . .

. . . . . . . . .

V.5. Functional differential and difference equations 100 Jarom´ır Baˇstinec : Oscillation and non-oscillation of solutions of linear second order discrete delayed equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Leonid Berezansky : New stability conditions for linear differential equations with several delays . . . . . . . 101 Aleksandr Boichuk : Boundary-value problems for differential systems with a single delay . . . . . . . . . . 101 Josef Dibl´ık : Representation of solutions of linear differential and discrete systems and their controllability 101 Alexander Domoshnitsky : Maximum principles and nonoscillation intervals in the theory of functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Marcia Federson : Averaging for impulsive functional differential equations: a new approach . . . . . . . . . 101 Yakov Goltser : Some bifurcation problems in the theory quasilinear integro differential equations . . . . . . 102 Istv´ an Gy¨ ori : Stability in Volterra type population model equations with delays . . . . . . . . . . . . . . . . 102 Ferenc Hartung : On parameter dependence in functional differential equations with state-dependent delays 102 Zeynep Kayar : Lyapunov type inequalities for nonlinear impulsive differential systems . . . . . . . . . . . . 102 Conall Kelly : Evaluating the stochastic theta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Gabor Kiss : Delay-distribution effect on stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Martina Langerov´ a : Solutions of linear impulsive differential systems bounded on the entire real axis . . . . 102 Malgorzata Migda : Oscillatory and asymptotic properties of solutions of higher-order difference equations of neutral type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 ¨ Abdullah Ozbekler : Principal and non-principal solutions of impulsive differential equations with applications103 Mihali Pituk : Nonnegative iterations with asymptotically constant coefficients . . . . . . . . . . . . . . . . . 103 Irena Rachunkova : On singular models arising in hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 103 Andrejs Reinfelds : Decoupling and simplifying of noninvertible difference equations in the neighbourhood of invariant manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 David W. Reynolds : Precise asymptotic behaviour of solutions of Volterra equations with delay . . . . . . 103 Alexandra Rodkina : On local stability of solutions of stochastic difference equations . . . . . . . . . . . . . 104 Miroslava R˚ uˇziˇckov´ a : Convergence of the solutions of a differential equation with two delayed terms . . . . 104 Vladimir Mikhailovich Savchin : Inverse problems of the calculus of variations for functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Ewa Schmeidel : Existence and nonexistence of asymptotically periodic solutions of Volterra linear difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

13

Andrei Shindiapin : Gene regulatory networks and delay equations . . . . . . . . . . . . . . . . . . . . . . . 104 Benzion Shklyar : The moment problem approach for the zero controllability of ecolution equations . . . . . 105 Svatoslav Stanek : Properties of maximal solutions of autonomous functional-differential equations with state-dependent deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Stevo Stevic : Boundedness character of some classes of difference equations . . . . . . . . . . . . . . . . . . 105 Milan Tvrdy : Continuous dependence of solutions of generalized ordinary differential equations on a parameter105 ¨ Mehmet Unal : Lyapunov type inequalities on time scales: A survey . . . . . . . . . . . . . . . . . . . . . . 105 A˘ gacık Zafer : Interval criteria for oscillation of delay dynamic equations with mixed nonlinearities . . . . . 105 V.6. Mathematical biology Robert Gilbert : Cancellous bone with a random pore structure . . . . . . . . . . . . . . . . . . . Irina Alekseevna Gainova : New computer technologies for the construction and numerical analysis ematical models for molecular genetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . Sandra Ilic : Application of the multiscale FEM in modeling the cancellous bone . . . . . . . . . Mark D. Ryser : Bone growth and destruction at the cellular level: a mathematical model . . . .

. . . . . of math. . . . . . . . . . . . . . .

105 . 106 . 106 . 106 . 106

VI. Others 106 Ruben Airapetyan : The relationship between Bezoutian matrix and Newton’s matrix of divided differences and separation of zeros of interpolation polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Hadeel Alkutubi : Bayesian shrinkage estimation of parameter exponential distribution . . . . . . . . . . . . 107 Mohammed Bokhari : Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem107 Zoubir Dahmani : The ADM method and the Tanh method for solving some non linear evolutions equations 107 Anvar Hasanov : Boundary-value problems for generalized axially-symmetric Helmholtz equation . . . . . . . 107 Maximilian Hasler : Asymptotic extension of topological modules and algebras . . . . . . . . . . . . . . . . . 107 S. Moghtada Hashemiparast : Approximation of fractional derivatives . . . . . . . . . . . . . . . . . . . . . 107 Hailiza Kamarulhaili : Discrepancy estimate for uniformly distributed sequence . . . . . . . . . . . . . . . . 108 Erdal Karapinar : Bounded linear operators on l-power series spaces . . . . . . . . . . . . . . . . . . . . . . 108 Erkinjon Karimov : On a three-dimensional elliptic equation with singular coefficients . . . . . . . . . . . . 108 Nabiullah Khan : A unified presentation of a class of generalized Humbert polynomials . . . . . . . . . . . . 108 Lixia Liu : Direct estimate for modified beta operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Eduard Marusic-Paloka : Mathematical model of an undergorund nuclear waste disposal site . . . . . . . . . 108 Abdeslam Mimouni : Compact and coprime packedness and semistar operations . . . . . . . . . . . . . . . 108 S. A. Mohiuddine : Characterization of some matrix classes involving (σ, λ)-convergence . . . . . . . . . . . 109 Mohammad Mursaleen : Sequence spaces of invariant mean and some matrix transformations . . . . . . . . 109 Ali Mussa : New convection theory for thermal plasma and NHD convection in rapidly rotating spherical configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Kourosh Nourouzi : Characterizations of Isometries on 2-modular spaces . . . . . . . . . . . . . . . . . . . . 109 Shariefuddin Pirzada : On r-imbalances in tripartite r-digraphs . . . . . . . . . . . . . . . . . . . . . . . . . 110 Hashem Parvaneh Masiha : Invariance conditions and amenability of locally compact groups . . . . . . . . . 110 Zaure Rakisheva : Motion stabilisation of a solid body with fixed point . . . . . . . . . . . . . . . . . . . . . 110 Lyazzat Sarybekova : A Lizorkin type theorem for Fourier series multipliers in regular systems . . . . . . . 110 Pedro A. Santos : Inverse-closedness problems in the stability of sequences in Banach Algebras . . . . . . . 110 Ridha Selmi : Smoothing effects for periodic NSE in critical Sobolev space . . . . . . . . . . . . . . . . . . . 110 Mariana Sibiceanu : Large deviations and almost sure convergence . . . . . . . . . . . . . . . . . . . . . . . 111 Tanfer Tanriverdi : The k- Model in Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Johnson Olaleru : The equivalence between modified Mann (with errors), Ishikawa (with errors), Noor (with errors) and modified multi-step iterations (with errors) for non-Lipschitzian strongly successively pseudo-contractive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Serap Oztop : A characterization for multipliers of weighted Banach valued Lp (G)-spaces . . . . . . . . . . 111 Karlyga Zhilisbaeva : Stationary motion of the dynamical symmetric satellite in the geomagnetic field . . . 111

Index

14

113

Plenary talks The Q-tensor theory of liquid crystals Sir John Ball Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, U.K. [email protected] The lecture will survey what is known about the mathematics of the de Gennes Q-tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lecture will describe the relation of the theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe. This is joint work with Apala Majumdar and Arghir Zarnescu. —————— Sir John Ball, FRS, is Sedleian Professor of Natural Philosophy at the University of Oxford and director of the Oxford Centre for Nonlinear PDE. He was president of the International Mathematical Union from 2003 to 2006. ————————————

Asymptotic equivariant index of Toeplitz operators and AttiyahWeinstein conjecture Louis Boutet de Monvel Universit´e Pierre et Marie Curie, Institut de Math´ematiques de Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE [email protected] The equivariant index of transversally elliptic equivariant operators was introduced by M.F. Atiyah (1974); it is a virtual trace class representation of a compact group, or equivalently the character of this representation, which is a central distribution. This does not make sense for general Toeplitz operators because the Toeplitz space where they act is only defined up to a finite dimensional space. The asymptotic index is an avatar of this, which works for Toeplitz operators : essentially it is a virtual trace class representation mod finite representations; equivalently its character is a singularity (distribution mod C ∞ ). It still is compatible with many natural operations, in particular the direct image by homogeneous symplectic maps. With E. Leichtnam, X. Tang and A. Weinstein, we have used this theory to give a new natural proof of the Atiyah-Weinstein conjecture (which was proved by C. Epstein): let X, X 0 be two compact strictly pseudoconvex boundaries (of complex domains): they carry natural cooriented contact structures. If f : X → X 0 is a contact isomorphism, we define the holomorphic pushforward Tf : u 7→ S 0 (u ◦ f −1 ) where u is the boundary value of a holomorphic function, and S 0 is the Szeg¨o projector, i.e. the orthogonal projector on the subspace of boundary values of holomorphic functions in L2 (X 0 ) (ker ∂¯b ). It is well known that Tf is a Fredholm operator; the Weinstein conjecture proposed a topological formula for its index. A particular case of this, proposed earlier by Atiyah, is the following: let V, V 0 be two smooth compact manifolds, and f a homogeneous symplectic isomorphism T ∗ V − {0} → T ∗ V 0 − {0} (equivalently a contact isomorphism between the cotangent spheres); then there exists an elliptic Fourier integral operator attached to f , whose index is given essentially by the same formula (this is a special case of the former because, if V is real analytic, the algebra of pseudodifferential operators acting on distributions is isomorphic to the algebra of Toeplitz operators acting on holomorphic boundary values on the boundary of a small tubular neighborhood of V in its complexification). One difficulty in this problem is that, since we are modifying the boundary CR structures (there are two of them), we are typically in the framework of general Toeplitz operators where the index is not well defined. Our way out was to construct a related G-elliptic operator where the index is repeated infinitely many times, but still well related geomerically to the problem, so the asymptotic index theory can be used.

15

Atiyah, M.F. Elliptic operators and compact groups. Lecture Notes in Mathematics, Vol. 401. SpringerVerlag, Berlin-New York, 1974. Boutet de Monvel, L. Asymptotic equivariant index of Toeplitz operators, RIMS Kokyuroku Bessatsu (2008). Boutet de Monvel, L.; Leichtnam E.; Tang, X. ; Weinstein A. Asymptotic equivariant index of Toeplitz operators and relative index of CR structures arXiv:0808.1365v1; to appear in the Duistermaat 65 volume, Progress in Math, Birkh¨ auser. Weinstein, A.: Some questions about the index of quantized contact transformations RIMS Kokyuroku No. 1014, pages 1-14, 1997. —————— Louis Boutet de Monvel was awarded with the 2007 Medaille Emile Picard of the French Academy of Sciences. ————————————

Non-self-adjoint spectral theory Brian Davies Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K. [email protected] Over the last twenty years there has been remarkable progress in understanding the spectral behaviour of highly non-self-adjoint operators, particularly differential operators, partly as the result of numerical experiments. The lecture will describe some of the discoveries that have been made, and theorems proved, and will contrast them with the very different spectral behaviour of self-adjoint operators. Connections with so-called pseudospectral theory, that is bounds on the norms of the resolvent operators, will be explained and illustrated. —————— Brian Davies, FRS, is Professor of Mathematics at King’s College London. In 1998 he was awarded the Senior Berwick Prize of the LMS. Brian Davies was president of the London Mathematical Society from 2007 to 2009. ————————————

Asymptotic analysis and complex differential geometry Simon Donaldson Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, U.K. [email protected] A long-standing problem in complex differential geometry is to find various preferred metrics on a complex manifold. These include K¨ ahler-Einstein, constant scalar curvature and extremal metrics. Finding such metrics comes down to solving highly nonlinear partial differential equations. For some manifolds solutions do not exist, and this is known to be related to the algebro-geometric notion of “stability”. The talk will give an overview of this area, emphasising the role of asymptotic analysis, applied to holomorphic sections of high powers of a complex line bundle. This gives a bridge between the analytical problems and algebraic geometry which is important in the general existence theory. The ideas can also be applied to construct numerical approximations to the desired metrics. —————— Simon Donaldson, FRS, holds a Royal Society Research Professorship at Imperial College London. He received a Fields Medal in 1986, was awarded with the Crafoord Prize 1994, the King Faisal International Prize in 2006 and the Nemmers Prize in Mathematics in 2008. He will receive the 2009 Shaw Prize in Mathematical Sciences. ————————————

16

The global behavior of solutions to critical nonlinear dispersive and wave equations Carlos Kenig Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, USA [email protected] In this lecture we will describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We will illustrate the method with some concrete examples and also mention other applications of these ideas. —————— Carlos Kenig is Louis Block Distinguished Service Professor of the University of Chicago. He was awarded the 2008 Bˆ ocher Memorial Prize for his contributions to harmonic analysis and non-linear dispersive partial differential equations. ————————————

Nonlinear harmonic analysis methods in boundary value problems of analytic and harmonic functions, and PDE Vakhtang Kokilashvili A. Razmadze Mathematical Institute, 1, M. Aleksidze st., 0193 Tbilisi, Georgia [email protected] The goal of our lecture is to present a survey of recent results in the nonlinear harmonic analysis operator theory and their applications in the boundary value problems for harmonic and analytic functions and related integral operators. We plan to discuss the above mentioned problems in the frame of Banach function spaces with nonstandard growth condition. For the sake of presentation, we have split the talk in the following topics: • One and two-weight norm estimates for the Cauchy singular integrals on Carleson curves in variable exponent Lebesgue spaces. • The Riemann-Hilbert problem for holomorphic functions from weighted classes of the Cauchy type integrals with densities in Lp(·) (Γ) in simply connected domains with piecewise-smooth boundaries Γ. Our aim is to give a complete solvability picture; to reveal the influence on the solvability character of the geometry of a boundary, of a weight function, and of the values of the space exponent at angular points; to give explicit formulas for solutions. • The Riemann-Hilbert-Poincar´e problem in the class of holomorphic functions whose mth order derivatives are representable by the Cauchy type integrals with densities from the variable exponent Lebesgue spaces with weights. The solvability criteria are given for the problem. The study of the problem is heavily based on the extension of I.Vekua’s integral representation of holomorphic function whose derivative is representable by the Cauchy type integral in simply connected domain with non-smooth boundary. • Baundary value problem with shift (the Hasemann BVP) for holomorphic functions in the domain with arc-chord condition. The solvability criteria and explicit formulas for solutions are established. Some part of the talk is based on joint research with V.Paatashvili. —————— Vakhtang Kokilashvili is Head of the Mathematical Analysis Department of the Razmadze Mathematical Institute. He was awarded the Razmadze Prize of the Georgian Academy of Sciences. ————————————

Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems Nicolas Lerner Institut de Math´ematiques de Jussieu, Universit´e Paris 6, 175 rue du Chevaleret, 75013 Paris, France [email protected]

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We prove that in any C-infinity neighborhood of an analytic Cauchy datum, there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first-order non-linear systems. We use a method initiated by G. M´etivier for elliptic systems based on the representation of solutions and on the FBI transform; in our case the system can be hyperbolic at initial time, but the characteristic roots leave the real line at positive times. ————————————

Non-ergodicity of Euler deterministic fluid dynamics via stochastic analysis Paul Malliavin Universit´e Pierre et Marie Curie, Institut de Math´ematiques de Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE Unitary representation associated to the motion of an incompressible fluid on the Tori. Fourier analysis of vector fields with vanishing divergence. Ergodi-city implies existence of an infinitesimal Haar measure. Randomization of Euler deterministic dynamics. Stochastic differential geometry on the group of volume preserving diffeomorphism of the Tori. Jump process describing the evolution of the repartition of the energy between modes. Non ergodicity of Euler equation via the transfert of energy towards micro scale. —————— Paul Malliavin is famous for his contributions to stochastic analysis and stochastic differential geometry. Among other distinctions he received in 1974 the Prix Gaston Julia of the French Academy of Science and is member of the Royal Swedish Academy of Sciences. ————————————

Higher order elliptic problems in non-smooth domains BICS Lecture, with an introduction by Valery Smyshlyaev Vladimir Maz’ya University of Liverpool and Linkoeping University [email protected] We discuss sharp regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions. Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron. The techniques developed in the present work allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. Then we introduce an appropriate notion of polyharmonic capacity which allows us to describe the precise correlation between the smoothness of solutions and the geometry of the domain. This is a joint work with S.Mayboroda, Perdue University. —————— His honours include the prize of the Leningrad Mathematical Society 1962, Doctor honoris causa of the University of Rostock 1990, Humbold Prize 1999, Corresponding Fellow of the Royal Society of Edinburgh 2001, Member of Royal Swedish Academy of Sciences 2002, Verdaguer Prize of the French Academy of Sciences 2003, The Celsius Gold Medal of the Royal Society of Sciences at Uppsala 2004. He is author of more than 20 books and more than 430 articles. ————————————

Operator algebras with symbolic hierarchies on stratified spaces Bert-Wolfgang Schulze Institute of Mathematics, University Potsdam, Am Neuen Palais 10, Potsdam, D-14469 Germany [email protected]

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We establish operator algebras on certain categories of stratified spaces (“corner manifolds”, or “manifolds with singularities”) that are designed to express parametrices of elliptic operators in terms of symbolic hierarchies. Our calculus contains special cases such as (pseudo-differential) boundary value problems with/without the transmission property at the boundary, and also mixed, transmission, and crack problems. The boundaries or interfaces may be smooth or have again singularities (conical points, edges, etc.). Other examples are equations on a smooth manifold, where the coefficients may have jumps or poles of a specific kind along some interfaces, smooth or singular in the above-mentioned sense, for instance, the Laplacian plus a singular interaction potential from a many-particle system. It is typical in such problems that a concrete situation (for instance, for the Laplacian in a corner domain) may generate operator-valued amplitude functions of a relatively high generality, consisting of operator functions on configurations of lower singularity order, now depending on various variables and covariables along the singular lower-dimensional strata. The calculus also contains analogues of Green functions, known from “standard” elliptic boundary value problems. In the singular case those refer again to all singular strata, operating on infinite cones. Moreover, when a stratum is of dimension zero the operator functions globally act on compact (in general singular) bases of such cones, with meromorphic dependence on a complex covariable, where non-bijectivity points (turning into poles under inversion) contribute to the asymptotics of solutions. Ellipticity in such a scenario is defined as invertibility of such operator-valued symbols. This depends on chosen weights in the respective distribution spaces. When a stratum is of dimension at least 1, this cannot be achieved in general, unless we pose extra edge conditions (analogues of boundary conditions), here of trace and potential type. The latter are possible when an analogue of the Atiyah-Bott condition for the existence of Shapiro-Lopatinskij boundary conditions is satisfied; otherwise another concept, namely, with global projection conditions may work (at least for smooth boundaries or edges, cf. the well-known work of Atiyah, Patodi, Singer, and papers of many other authors, especially, Seeley, Grubb, and also by the author, partly in joint work with J. Seiler, where corresponding operator algebras are established in a Toeplitz operator framework, unifying the structures of the Shapiro-Lopatinskij and the global projection set-up). The construction of parametrices relies on the inversion of the components of the principal symbolic hierarchy, combined with algebraic operations. Those symbols take values in spaces of operators referring to lower singularity orders. At this point, in order to express parametrices within our spaces, we need the calculus as an algebra. The analysis which is doing all this is rich in detail. Many authors contributed to the pseudo-differential methods in this framework, especially, Melrose, Mendoza, Gil, Seiler, Schrohe, Witt, and Krainer.There are several monographs of the author, a few jointly with coauthors (Rempel, Egorov, Kapanadze, Harutyunyan) containing the basics of the approach, including applications, and more references. In order to keep the calculus manageable it is important to reduce the stuctures to a few “axiomatic” principles and then to proceed in an iterative way, beginning with the pseudo-differential calculus on a smooth manifold, and then successively building up the algebras for conical, edge, corner, . . . , higher singularities. The focus of our talk is just a program of that kind. We present such an iterative process to obtain operator algebras containing the desirable (“typical”) differential operators (corner-degenerate in streched coordinates), together with the parametrices of elliptic elements, where the above-mentioned examples are covered. One of the principles to make the calculus iterative is to impose a relatively simple behaviour of the growth of norms of parameter-dependent operators when the parameters tend to infinity, then to make the parameter-dependence “edge-degenerate” at infinity of an infinite cone, and then to observe that this behaviour survives the step to the next floor of singular calculus, cf. a joint article with Abed. The general structure theory is full of new challenges and “unexpected” problems, for instance, from the point of view of index theory, or extensions to non-elliptic operators. Moreover, in concrete cases other substantial aspects remain essential, namely, to compute several data as explicitly as possible, e.g., the index of operators on infinite cones, or the number of extra edge conditions, the right weights that depend on the individual operator, the asymptotics of solutions, including iterated asymptotics, or the variable and branching behaviour connected with the above-mentioned poles when those depend on edge variables and change multipicities (cf. earlier work of Bennish, or the author, and a cycle of papers in progress jointly with Volpato). —————— Bert-Wolfgang Schulze is author of more than 240 publications and 20 books. He received the Euler Medal of the Berlin Academy of Sciences in 1984 and is doctor honoris causa of the Vekua Institute of Applied Mathematics in Tbilisi. ————————————

Visibility and Invisibility Gunther Uhlmann Department of Mathematics, C-449 Padelford Hall, Seattle, Washington 98195-4350, USA

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[email protected] We will describe the method of complex geometrical optics and its applications to find acoustic, quantum, and electromagnetic parameters of a body by making measurements at the boundary of the body. We will also survey recent results on how to make objects invisible to acoustic, quantum and electromagnetic waves. —————— Gunther Uhlmann is Walker Family Endowed Professor of Mathematics at the University of Washington. He is Fellow of the American Academy of Arts and Sciences, corresponding member of the Chilean Academy of Sciences, Fellow of the Institute of Physics and will be Clay Senior Scholar at MSRI in 2010. ————————————

Practise of industrial mathematics related with the steel manufacturing process OCCAM Lecture on Applied Mathematics, with an introduction by John Ockendon Masahiro Yamamoto University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba Meguro Tokyo 153, Japan [email protected] We will discuss several problems given by the steel industry. Those problems have originated from real working sites, are related for example with heat conduction processes and have been solved by the speaker and his research groups. Those problems can be modelled mathematically, on such a a theoretical basis, we have solved them practically as well as mathematically to satisfy demands by industry for lowering costs and improving securities. For more fruitful contribution in the industrial mathematics from the side of mathematicians, we will discuss also possible schemes. ————————————

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Public lecture Analysis, Models and Simulations OxPDE Public Lecture on Nonlinear PDE, with an introduction by Sir John Ball Pierre-Louis Lions College de France, 3 rue d’Ulm, 75005 Paris, France [email protected] In this talk, we shall first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving Partial Differential Equations and we shall briefly explain the nature, the history and the role of such equations. Then, some examples showing the importance of the mathematical analysis (i.e. understanding) of those models will be presented. And we shall conclude indicating a few trends and perspectives. —————— Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself become a renowned mathematician, making numerous important contributions to the theory of non-linear partial differential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPerna giving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Other awards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently he holds the position of Chair of Partial Differential Equations and their Applications at the prestigious Coll`ege de France in Paris. ————————————

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This is joint work with Shamil Makhmutov and Jouni R¨ atty¨ a. ———

Sessions

Global mapping properties of rational functions

I.1. Complex variables and potential theory Organisers: Tahir Aliyev, Massimo Lanza de Cristoforis, Sergiy Plaksa, Promarz Tamrazov This session is devoted to a wide range of directions of complex analysis, potential theory, their applications and related topics. —Abstracts— Analytic functions in contour-solid problems ˇ lo Tahir Aliyev Azerog Gebze Institute of Technology, Istanbul Caddesi, P.K. 141, Gebze, Kocaeli, 41400 Turkey [email protected] We generalize and strengthen certain contour-solid theorems. The generalization consists in considering finely meromorphic functions besides holomorphic, and strengthening is connected with taking into account zeroes and the multivalence of functions.

Cristina Ballantine Dept. of Mathematics, College of the Holy Cross, 1 College Street, Worcester, Massachusetts 01610 United States [email protected] The talk is based on a joint work with Dorin Ghisa. The main result is the following theorem. Every rational b = ∪n function f of degree n defines a partition C k=1 Ωk of the Riemann sphere such that the interior of every b \ Lk , where Lk is Ωk is mapped conformally by f on C part of a cut L. The mapping extends conformally to the boundary of every Ωk except forsome points b1 , b2 , ..., bj , j ≤ n, in the neighborhood of which f has one of the forms: (i) f (z) = f (bk ) + (z − bk )αk ϕk (z), or (ii) f (z) = (z − bk )−αk ϕk (z), where αk is an integer, αk ≥ 2, and ϕk is an analytic function with ϕk (bk ) 6= 0. b f ) is a branched covering Riemann surface Actually, (C, b of C having the branch points b1 , b2 , ..., bj . In the neighborhood of z = ∞ we have: (iii) f (z) = z α ϕ(z), where α ∈ Z and ϕ is analytic with lim ϕ(z) finite and non-zero. z→∞

——— A non-α-normal function whose derivative has finite area integral of order less than 2/α Rauno Aulaskari University of Joensuu Department of Physics and Mathematics Joensuu, Joensuu 80101 Finland [email protected] Let D be the unit disk {z : |z| < 1} in the complex plane. A function f , meromorphic in D, is normal, denoted by f ∈ N , if supz∈D (1 − |z|2 )f # (z) < ∞, where f # (z) = |f 0 (z)|/(1 + |f (z)|2 ). For α > 1, a meromorphic function f is called α-normal if supz∈D (1 − |z|2 )α f # (z) < ∞. H. Allen and C. Belna [J. Math. Soc. Japan, 24 (1972) 128–132] have proved that there is an analytic function f1 , defined in D, such that ZZ |f10 (z)| dxdy < ∞ D

but f1 6∈ N . S. Yamashita [Ann. Acad. Sci. Fenn. Ser. Math. 4 (1978/1979) 293–298] sharpened this result by showing that for another analytic function f2 which does not belong to N it holds ZZ |f20 (z)|p dxdy < ∞ (*)

If f is a polynomial, then every Ωk is bounded by arcs approaching asymptotically rays of the form zk (t) = tei(γ+2kπ/n) , t > 0, γ ∈ R. We will present examples of color mapping visualizations. ——— Beltrami equations Bogdan Bojarski IM PAN, Sniadeckich 8, Warsaw, 00-956 Poland [email protected] In the talk will be discussed some new approaches to the Beltrami equations and operators in the complex plane and on Riemann surfaces in connections with the general theory of quasiconformal mappings and automorphic functions. ——— A functional analytic approach for a singularly perturbed non-linear traction problem in linearized elastostatics Matteo Dalal Riva Universita’ degli Studi di Padova, Via Trieste, 63 Padova, Italy/Padova/Veneto 35121, Italy [email protected]

D

for all p, 0 < p < 2. Further, H. Wulan [Progress in analysis Vol. I,II, World Sci. Publ. 2003, 229–234] S studied more the function f2 and showed that f2 6∈ 00 Γ(αk + 1) k=0

and investigated some of its properties. This is an entire function of order 1/α. Another function having similar properties to those of Mittag-Leffler functions is given by

l+ 1

l+ 1 Jl+ 1 (µj 2 ) 2

µj 2 l+ 1 l+ 1 + (Jl− 1 (µj 2 ) − Jl+ 3 (µj 2 )) = 0. 2 2 l+1

Eα,β (z) =

∞ X k=0

——— The ending solutions of Ince system with irregular features Zhaxylyk Tasmambetov Aktobe State University after K. Zhubanov 263, Bratiev Zhubanov’s street, Aktobe city, 030000 Kazakhstan [email protected] The Ince system with irregular features: ( (0) p Zxx + p(1) q (4) Zxy + p(2) Zx + q (5) Zy + p(3) Z = 0, q (0) Zyy + p(4) q (1) Zxy + p(5) Zx + q (2) Zy + q (3) Z = 0, where coefficients p(i) = p(i) (x) and q (i) = q (i) (y) (i = 0, 5) are polynomials of (i)

p (x) =

δi X j=πi

j

pij x , q

(i)

(y) =

ζi X

qij x

j

j=ξi

type (πi , δi , ξi , ζi (i = 0, 5) - certain numbers), is studied. Let the system be collocated and let the integrability condition be executable p(0) q (0) − p(1) q (1) p(4) q (4) 6= 0. Ince established that singular curves of this system are defined by the coefficients in the case of second-order private derivatives and in the case of certain additional

zk , α > 0, β > 0. Γ(αk + β)

For β = 1, Eα,1 = Eα . Such functions arise naturally in the solution of fractional integral equations [Saxena, R., Mathai, A. and Haubold, H. (2002). On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287] and especially in the study of the fractional kinetic equation, random walks, etc. We study Mittag-Leffler type functions and derive some of their properties including integrals and recurrence relations. We also study fractional equations of the form N (t) − N0 = −c 0 Dt−1 N (t), and its generalization, where 0 Dt−ν is the RiemannLiouville operator of fractional integration. ——— On mixed boundary-value problems of polyanalytic functions Yufeng Wang School of Mathematics and Statistics, Wuhan University, Wuhan 430072 China wh [email protected] Recently, boundary value problems of higher-order complex partial differential equations have been widely investigated. For example, various kinds of boundary value problems of two-order complex partial differential equations, including the Poisson equation and the

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I.3. Complex-analytical methods for applied sciences Bitsadze equation, have been systematically discussed, and the explicit expression of solution and the condition of solvability have already been obtained. In addition, some boundary value problems of polyanalytic equation, polyharmonic equation and metaanalytic function have also been discussed. In this paper, under the appropriate decomposition of polyanalytic functions, some mixed boundary-value problems of polyanalytic functions have been discussed, and the explicit expression of solution and the condition of solvability have been obtained.

On solution of a kind of Riemann boundary value problem on the real axis with square roots Shouguo Zhong School of Mathematics and Statistics, Wuhan University, Wuhan 430072 China [email protected] Solution of the Riemann boundary value problem on the real axis X with square roots p p Ψ+ (x) = G(x) Ψ− (x) + g(x), x ∈ X

——— An algorithm of solving the Cauchy problem and mixed problem for the two-dimensional system of quasi-linear hyperbolic partial differential equations Oleg N. Zhdanov Siberian State Aerospace University “M.F. Reshetnyov”, Krasnoyarsk, Russia [email protected] Let’s consider the system of homogeneous quasilinear hyperbolic partial differential equations aij (u1 , u2 )∂x uj + bij (u1 , u2 )∂y uj = 0, i, j = 1, 2, (*) where aij , bij - smooth functions in area D. There are 3 classical boundary problems for system (*): the Cauchy problem, the Riemann problem and the anmixed problem. Earlier Cauchy and Riemann problems were solved for some particular cases using conservation laws. And now we have algorithm for the solution of the Cauchy problem of system (*) in general. Attempts to solve the mixed prolem weren’t successful for a long time. Our approach consists in applying to this system not only one conservation law, as was done in many papers, but a family of such laws with functions depending on parameters. Let’s accurately formulate the problem. Let the function u be specified on the non-characteristic curve M N in the plane C, and functions u, v be specified on a characteristic curve crossing M N . It is important that every characteristic crosses the curve M N only in one point and is not tangent to it in any point. Our aim is to find the intersections of characteristics and the values of functions u and v in these points. We reduce the mixed problem to the Cauchy problem. We choose a point on the curve M N and a point on the characteristic, and we have a system of algebraic equations - corollary fact of conservation law. Using resultant, we obtained one equation for the value of hte function v in the initial point. We find this value and repeat the procedure with another points. It allows us to find the intersections of characteristics and function values in these points with preassigned exactness using a well-known method described in [Kiryakov P. P., Senashov S. I., Yakhno A. N. Application of symmetries and conservation laws to differential equations solving. Novosibirsk, 2001., p. 170]. As application we obtained the solution of systems, describing state of plane stress of Mises‘s plastic surroundings– a problem that is interesting for mechanics for more than 100 years. ———

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for analytic function is considered, which was solved under certain assumptions on the branch points appeared. ——— Some Riemann boundary value problems in Clifford analysis Zhongxiang Zhang School of Mathematics and Statistics, Wuhan University, Wuhan 430072 China [email protected] In this paper, we mainly study the Rm (m > 0) Riemann boundary value problems for functions with values in a Clifford algebra C(V3,3 ). We firstly prove a generalized Liouville theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k-regular functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions is presented. Combining the Plemelj formula, the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m > 0) Riemann boundary value problems for regular functions, harmonic functions and biharmonic functions are solvable. The explicit solutions are given. ———

I.3. Complex-analytical methods for applied sciences Organisers: Viktor Mityushev, Sergei Rogosin The main attention will be paid to analytic-type results in complex analysis, especially those which have applications in Mathematical Physics, Mechanics, Chemistry, Biology, Medicine, Economics etc. Among the methods under consideration are: boundary value problems for holomorphic and harmonic functions and their generalizations, singular integral equations, potential analysis, conformal mappings, functional equations, entire and meromorphic functions, elliptic and doubly periodic functions etc. Applications in Fluid Mechanics, Composite Materials, Porous Media, Hydro- Aero- and Thermo-Dynamics, Elasticity, Elasto-Plasticity, will be the most considered at the session.

I.4. Zeros and Gamma lines – value distributions of real and complex functions —Abstracts— R-linear problem and its applications to composites Vladimir Mityushev Podchorazych 2 Krakow, Malopolska 30-084 Poland [email protected] We develop the method of functional equation to derive analytical approximate formulae for the effective conductivity tensor of the two–dimensional composites with elliptical inclusions. The sizes, the locations and the orientations of the ellipses can be arbitrary. The analytical formulae contains all above geometrical parameters in symbolic form.

The numbers of zeros of certain classes of meromorphic functions are studied, particularly, in the classical Nevanlinna and Ahlfors theories. Some analogous results were obtained also for the Gamma-lines of functions (i.e., preimages of curves). This enlarges the value distribution, describes not only the numbers but also the locations of a-points and, unexpectedly, leads to new distribution type phenomena for the zeros in real analysis and real algebraic geometry. Thus we are now in a stage of formation of some methods working in both real and complex analysis. The zeros (a-points, fixed-point) and Gamma-lines arising in complex analysis (particularly meromorphic functions and solutions of ODE, harmonic and polynomial mappings), real analysis, real and complex algebraic geometry will be subject of this session.

——— Application of the spectral parameter power series method to conformal mapping problems Michael Porter Department of Mathematics, CINVESTAV-IPN, Libramiento Norponiente 2000, Fracc. Real de Juriquilla Queretaro, 76230 Mexico [email protected] Many problems in conformal mapping of plane domains are determined by the Schwarzian derivative of the mapping, a third-order nonlinear differential operator, and it is well known that this can be rephrased in terms of a second-order linear differential equation y 00 + φy = 0. For many mapping problems the coefficient function φ in this equation depends on one or more real or complex parameters; a typical formulation might be y 00 + qy = λry. The global aspect of a mapping problem often translates into boundary conditions (possibly nonlinear) on a real interval and a spectral problem is thus presented. We apply the recently developed spectral parameter power series (SPPS) method for Sturm-Liouville problems to gain insight into conformal mapping problems. In particular we will calculate the complete parameter space for conformal mappings from the disk to a symmetric circular quadrilateral with right angles. ——— Recent results on analytic methods for 2D composite materials Sergei Rogosin Department of Mathematics and Mechanics, Belarusian State University, Nezavisimosti ave, 4 Minsk, BY-220030 Belarus [email protected]

—Abstracts— An universal value distribution: for arbitrary meromorphic function in a given domain Grigor Barsegian Institute of Mathematics of the National Academy of Sciences, 24-b Bagramian ave. Yerevan, 375019 Armenia [email protected] Some purely geometric results analogous to the second fundamental theorems in the classical Nevanlinna and Ahlfors theories are revealed. These analogs are valid for arbitrary analytic (meromorphic) functions in given domains unlike the classical results that are valid only for some known sub classes of functions that have “equidistributions”. The obtained results are sharp as for functions in the complex plane (the classical case) as well as for functions in a given domain. ——— A generalization of the Stieltjes-Van Vleck-Bocher theorem Petter Branden Department of Mathematics Royal Institute of Technology Stockholm, Stockholm 100 44 Sweden [email protected] A classical theorem of Stieltjes, Van Vleck and Bˆ ocher describes the polynomial solutions f (z), v(z) to the second order differential equation d Y

(z − αj )f 00 (z) +

j=1

d X j=1

βj

Y (z − αi )f 0 (z) + v(z)f (z) = 0 i6=j

It is a survey talk on the recent analytic results for 2D composite materials. Special attention will be paid to application of the boundary value problems for analytic functions, of the functional equations method and of the integral equation method.

where α1 < · · · < αd are real and β1 , . . . , βd are positive. B. Shapiro has recently developed a Heine-Stieltjes theory for linear differential operators of higher order. He conjectured a vast generalization of the Stieltjes– Van Vleck–Bˆ ocher theorem. We prove this conjecture and describe the intricate structure of the zeros of the solutions.

———

——— A criterion for the reality of zeros

I.4. Zeros and Gamma lines – value distributions of real and complex functions Organisers: Grigor Barsegian, George Csordas

David Cardon Department of Mathematics, Brigham Young University, Provo, Utah 84604 United States [email protected]

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I.4. Zeros and Gamma lines – value distributions of real and complex functions I will discuss a necessary and sufficient condition for certain real entire functions to have only real zeros. ——— New properties of a class of Jacobi and generalized Laguerre polynomials Marios Charalambides Department of Business Administration, Frederick University, 7 Yianni Frederickou street, Nicosia, Pallouriotisa 1036 Cyprus [email protected]

VA 24061-0123 United States [email protected] Polynomials with real coefficients and all real roots have many interesting and useful properties. This talk will introduce an elegant generalization to polynomials with complex coefficients in seeveral variables. These new polynomials are called upper (or stable) polynomials and are defined by their non-vanishing on the upper half plane. This is recent work of J. Borcea, P. Br¨ and en, S. Fisk, B. Shapiro, A. Sokal, and D. Wagner. ———

New properties of a class of Jacobi and generalized Laguerre polynomials are presented. The results give new classes of stable polynomials and polynomials with real negative roots. Implications of these results on the areas of geometry of polynomials and numerical analysis are also discussed. ——— Meromorphic Laguerre operators and the zeros of entire functions George Csordas Department of Mathematics University of Hawaii, Honolulu, Hawaii 96822 United States [email protected] The purpose of this lecture is to announce new results pertaining to the following open problem. Characterize the meromorphic functions, F (x), such that P∞ k F (k)a k x /k! is a transcendental entire function k=0 with only real zeros (or that the zeros all lie in the half-plane r1 } for some r1 > 0. Similarly we define a negatively oriented Baker domain. By Spiraling Baker domain, we mean either positively oriented Spiraling Baker domain or a negatively oriented spiraling Baker domain. In this paper we show the existence of Spiraling Baker domain and obtain several properties of these. ——— The algebraic Liouville integrability and the related Picard-Fuchs type equations Anatoliy Prykarpatsky The AGH-University of Science and Technology, Krakow, Poland, and Ivan Franko State Pedagogical University, Drohobych, Lviv region, Ukraine 30 Aleja Mickiewicz, N120-C Krakow, 30059 Krakow Poland [email protected] We consider a completely integrable Liouville-Arnold Hamiltonian system on a cotangent canonically symplectic manifold (T ∗ (Rn ), ω (2) ), n ∈ Z+ , possessing exactly n ∈ Z+ functionally independent and Poisson commuting algebraic polynomial invariants Hj : T ∗ (Rn ) → R, j = 1, n. Due to the Liouville-Arnold theorem this Hamiltonian system can be completely integrated by quadratures in quasi-periodic functions on its integral submanifold when taken compact. It is equivalent to the statement that this compact integral submanifold is diffeomorphic to a torus Tn , that makes it possible to integrate the system by means of searching the corresponding integral submanifold imbedding mapping. The following theorems are stated. Theorem. Every completely algebraically integrable Hamiltonian system admitting an algebraic submanifold Mhn ⊂ T ∗ (Rn ) possesses a separable canonical transformation which is described by differential algebraic Picard-Fuchs type equations whose solution is a set of some algebraic curves Theorem. Consider a completely integrable Hamiltonian system on the coadjoint manifold T ∗ (Rn ) whose integral submanifold Mhn ⊂ T ∗ (Rn ) is described by PicardFuchs type algebraic equations. The corresponding integrability embedding mapping πh : Mhn → T ∗ (Rn ) is a solution of a compatibility condition subject to the differential-algebraic relationships on the corresponding canonical transformations generating function.

Spiraling Baker domains ——— Anand Prakash Singh Department of Mathematics, University of Jammu, Jammu-180006, INDIA [email protected] Let f be a transcendental entire function. For n ∈ N, let f n denote the nth iterate of f . Fatou set F (f ) of f is defined to be the set of all points z in the complex plane C such that the family {f n }n≥1 forms a normal family in some neighbourhood of z. Julia set is defined to be the complement of Fatou set. A periodic component U of F (f ) of period m is called a Baker domain if f mn (z) → ∞ as n → ∞ for all z ∈ U . Further we define a Baker domain B as a positively oriented spiraling Baker domain if there exist positive continuous functions A(r), φ(r), ψ(r), of r all tending to ∞ as r → ∞ such that φ(r) in non decreasing, 0 < ψ(r) − φ(r) < 2π and

Quantization of universal Teichm¨ uller space: an interplay between complex analysis and quantum field theory Armen Sergeev Steklov Mathematical Institute, Gubkina 8, Moscow, 119991 Russia [email protected] Universal Teichm¨ uller space T is the quotient of the group QS(S 1 ) of quasisymmetric homeomorphisms of S 1 modulo M¨ obius transformations. It contains the quotient S of the group Diff+ (S 1 ) of diffeomorphisms of S 1 modulo M¨ obius transformations. Both groups act natu1/2 rally on Sobolev space H := H0 (S 1 , R). Quantization problem for T and S arises in string theory where these spaces are considered as phase manifolds. To solve the problem for a given phase space means to

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II.1 Clifford and quaternion analysis fix a Lie algebra of functions (observables) on it and construct its irreducible representation in a Hilbert (quantization) space. For S an algebra of observables is given by Lie algebra Vect(S 1 ) of Diff+ (S 1 ). For quantization space we take 1/2 the Fock space F (H), associated with H = H0 (S 1 , R). 1 Infinitesimal version of Diff+ (S )-action on H generates an irreducible representation of Vect(S 1 ) in F (H), yielding quantization of S. For T the situation is more subtle since QS(S 1 )-action on T is not smooth. So there is no classical Lie algebra, associated to QS(S 1 ). However, we can define a quantum Lie algebra of observables Derq (QS), generated by quantum differentials, acting on F (H). These differentials arise from integral operators dq h on H with kernels, given essentially by finite-difference derivatives of h ∈ QS(S 1 ). ———

II.1 Clifford and quaternion analysis Organisers: Irene Sabadini, Frank Sommen We call for contributions in the fields of theoretical quaternionic and Clifford analysis and, more in general, hypercomplex analysis intended as the study of the function theory related to the Dirac operator and systems of partial differential operators taking values in a Clifford algebra. All the topics varying from the study of monogenic functions, its generalisations to higher spin such as the Rarita-Schwinger system, Clifford analysis on superspace, Clifford-Radon and Fourier transforms, discrete Clifford analysis to functions with values in more general non-commutative structures are welcome. —Abstracts— Clifford analysis for orthogonal, symplectic and finite reflection groups Hendrik de Bie Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Ghent (Belgium) [email protected] In recent work we have developed a theory of Clifford analysis in superspace. This can be seen as Clifford analysis invariant under the product of the symplectic with the orthogonal group. Other authors have recently also studied Clifford analysis with respect to finite reflection groups (using Dunkl operators). In this talk we will give a general and unified framework that can be used for these different symmetries. We will also discuss some typical problems that depend on the symmetry at hand. These will include the Fischer decomposition, the Fourier transform and the Hermite polynomials. We also discuss related quantum systems. ——— M¨ obius transformations and Poincar´ e distance in the quaternionic setting Cinzia Bisi Dipartimento Matematica, Universita’ della Calabria, Cubo 30b, Ponte P.Bucci, Arcavacata di Rende Cosenza,

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Calabria 87036 Italy [email protected] In the space H of quaternions, we investigate the natural, invariant geometry of the open, unit disc ∆H and of the open half-space H+ . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M¨ obius transformations of ∆H and H+ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M¨ obius transformations, and use it to define the analogous of the Poincar´e distances and differential metrics on ∆H and H+ . As a spin-off, we directly deduce that there exists no isometry between the quaternionic Poincar´e distance of ∆H and the Kobayashi distance inherited by ∆H as a domain of C2 , in accordance with the well known classification of the non compact, rank 1, symmetric spaces. ——— Wavelets invariant under reflection groups Paula Cerejeiras Department of Mathematics, University of Aveiro, Aveiro, P-3810-193 Portugal [email protected] For signal reconstruction over a sphere, two main approaches are used: the group-theoretical one (see, for instance, Antoine/Vandergheynst or M. Ferreira) where the authors use representations over homogeneous spaces and the one using approximate identities and singular kernels (see Freeden, or Swelden). However, both rely on the Lorentz group and, therefore, are not suitable for signals with predefined symmetries which involve reflections. To overcome this problem, we consider differential-difference operators associated to specific finite reflection groups, the so-called Dunkl operators. In this setting we construct spherical Dunkl wavelets based on approximate identities and we give practical examples. ——— Some consequences of the quaternionic functional calculus Fabrizio Colombo Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9 Milano, Mi 20133 Italy [email protected] We show some of the most recent results on the quaternionic functional calculus for left and right linear quaternionic operators defined on quaternionic Banach spaces. This approach allows us to deal both with bounded and unbounded operators. In particular we use such a functional calculus to study the quaternionic evolution operator. ——— Orthogonality of Clifford-Hermite polynomials in superspace. Kevin Coulembier Department of Mathematical Analysis, Ghent University, Krijgslaan 281 Ghent 9000 Belgium [email protected]

II.1 Clifford and quaternion analysis In previous work by De Bie and Sommen, the CliffordHermite polynomials were generalized to superspace. In this talk we will construct an inner product for which these polynomials are orthogonal, using the Berezin integral. This inner product can moreover be used for quantum mechanics in superspace, as it restores the hermiticity of the anharmonic oscillator. As an application we will also derive a Mehler formula with O(m) × Sp(2n) symmetry. The Mehler formula gives an expansion of the kernel of the fractional Fourier transform in terms of the super Clifford-Hermite polynomials. This was already known in one dimension (Hermite polynomials) and formally for O(m) (CliffordHermite polynomials), but the O(m) × Sp(2n) poses some extra difficulties. ——— Recent results on hyperbolic function theory Sirkka-Liisa Eriksson Department of Mathematics, Tampere University of Technology , P.O.Box 553, Tampere 33101 Finland si[email protected] The aim of this talk is to consider the hyperbolic version of the standard Clifford analysis. The need for such a modification arises when one wants to make sure that the power function xm is included. The leading idea is that the power function is the conjugate gradient of a harmonic function, defined with respect to the hyperbolic metric of the upper half space. We present results and problems concerning power series presentation of hypermonogenic functions This work is done jointly with professor Heinz Leutwiler, University of ErlangenN¨ urnberg, Department of Mathematics, Erlangen, Germany, email: [email protected] ——— Symmetric properties of the Fourier transform in Clifford analysis setting Ming-Gang Fei Departamento de Matem´ atica, Universidade de Aveiro, Campus Universitario de Santiago Aveiro, Aveiro 3810193 Portugal [email protected] In this talk we present Fueter’s Theorem for Dunklmonogenic functions. We show that if f is a holomorphic function in one complex variable, then for any unγ +(d−1)/2 derlying space Rd1 the induced function ∆hκ f (x) is Dunkl-monogenic whenever γκ + (d − 1)/2 is a nonnegative integer, where ∆h is Dunkl Laplacian. To this end Vekua-type systems for axial Dunkl-monogenic functions are studied. ——— Factorization of M¨ obius gyrogroups - the paravector case Milton Ferreira Campus Universit´ ario de Santiago, Departamento de Matem´ atica, Universidade de Aveiro, Aveiro 3810-193 Portugal [email protected] We consider a M¨ obius gyrogroup on the unit ball of the vector space F ⊕ V, where V is a finite dimensional

vector space over the scalar field F = R or C. We will present the factorizations of the paravector unit ball by gyro-subgroups and subgroups, generalizing the case of the unit ball on Euclidean space Rn . The main differences between both cases are the replacement of the Spin group by the Spoin group and the establishment of a geometric product for the paravector case, analogous to the geometric product in the vector case. ——— Higher spin analogues of the Dirac operator in two variables and its resolution Peter Franek Mathematical Institute, Charles University Praha, Sokolovska 83 Prague, 8 186 75 Czech Republic [email protected] A resolution of the Dirac operator in two variables is well known and well understood. It consists of three invariant operators (on of those of second order) expressed using the Dirac operators in two individual variables. We shall discuss higher spin analogues of such resolutions. They are again complexes of three invariant operators acting on functions with values in more complicated representation spaces. ——— Cauchy kernels in ultrahyperbolic Clifford analysis – Huygens cases Ghislain R. Franssens Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium [email protected] ` ´ Let Rp,q , Rp+q , P , with P the canonical quadratic form of signature (p, q). Clifford Analysis (CA) over Rp,q , called Ultrahyperbolic Clifford Analysis (UCA), is a non-trivial extension of the familiar (Euclidean) CA over Rn . Essential for stating integral representation theorems in UCA is the determination of a reproducing (or Cauchy) kernel Cx0 of Rp,q , ∀p, q ∈ Z+ , for the Dirac operator ∂. Any such kernel can be obtained as Cx0 = ∂gx0 , with gx0 a fundamental distribution of the ultrahyperbolic equation p,q gx0 = δx0 , x0 ∈ Rp+q . The complexity of UCA is due to the fact that Cx0 is a rather complicated distribution, whose form profoundly depends on the parity of p and q. Iff p and q are odd is gx0 proportional to a delta distribution δ(P (x−x0 )) , having as support the null space of Rp,q relative to x0 , and then gx0 is said to satisfy Huygens’ principle. In this talk, explicit expressions for the distributions gx0 and Cx0 will be presented for the Huygens cases. We will see how δ(P (x−x0 )) arises as a pullback of the one-dimensional delta distribution δ and the matter of “regularization”, required for some of these distributions, will be carefully addressed. ——— Power series and analyticity over the quaternions Graziano Gentili Dipartimento di matematica ”U.Dini”, viale Morgagni 67/a, 50134 Firenze, Italy [email protected]

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II.1 Clifford and quaternion analysis We study power series and analyticity in the quaternionic setting. We first consider P a function f defined as the sum of a power series n∈N q n an in its domain of convergence, which is a ball B(0, R) centered at 0. At each p ∈ B(0, R), f admits expansions in terms of appropriately defined regular power series centered at P ∗n p, (q − p) bn . The expansion holds in a ball n∈N Σ(p, R − |p|) defined with respect to a (non-Euclidean) distance σ. We thus say that f is σ-analytic in B(0, R). Furthermore, we remark that Σ(p, R − |p|) is not always an Euclidean neighborhood of p; when it is, we say that f is quaternionic analytic at p. It turns out that f is quaternionic analytic in a neighborhood A of B(0, R)∩R, with A strictly contained in B(0, R) unless R is infinite. We then extend these results to the larger class of quaternionic slice regular functions, enriching their theory. Indeed, slice regularity proves equivalent to σanalyticity and slice regular functions are quaternionic analytic only in a neighborhood of the real axis.

This is joint work with D. Constales and D. Grob. ——— Explicit description of the resolution for 4 Dirac operators in dimension 6 Lukas Krump Mathematical Institute of the Charles University, Sokolovska 83, Praha 8, 186 75 Czech Republic [email protected] There are several approaches to the construction of a resolution of several Dirac operators in higher dimensions. Among them, the Penrose transform method gives satisfying results in both stable and unstable cases. Recently this method was used to determine the shape of such resolution in many cases and the next step is an explicit description of operators involved. This will be shown for the unstable case of four operators in dimension six. ———

——— Isomorphic action of SL(2, R) on hypercomplex numbers Anastasia Kisil Triniti College Cambridge, University Cambridge, Cambridgeshire CB2 1TQ, United Kingdom [email protected] We investigate the SL(2, R) invariant geodesic curves with the associated invariant distance function in parabolic geometry. Parabolic geometry naturally occurs as action of SL(2, R) on dual numbers and is placed in between the elliptic and the hyperbolic geometries (which arise from the action of SL(2, R) on complex and double numbers). Initially we attempt to use standard methods of finding geodesics but they lead to degeneracy in this set-up. Instead, by studying closely the two related hypercomplex numbers we discover a unified approach to a more exotic and less obvious dual number’s case. With aid of common invariants we describe the possible distance functions that turn out to have some unexpected, interesting properties. ——— Construction of 3D mappings on to the unit ball with the hypercomplex Szego kernel Rolf Soeren Krausshar Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan, 200-B Leuven, Vlaams Brabant, 3001 Belgium [email protected] In this talk we present a hypercomplex generalization of the Szego kernel method that allows us to construct 3D mappings from some elementary domains of R3 onto the unit sphere. More precisely, we consider an appropriately chosen line integral over the square of the hypercomplex Szego kernel. The latter one is approximated numerically by the monogenic Fueter polynomials for rectangular domains, an L-shaped domain, circular cylinders and the double cone. In all these cases the line integration provides an amazingly good mapping onto the unit sphere. We also compare the quality of results ontained with this method with the results that were obtained previously by using alternatively the Bergman kernel method.

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On polynomial solutions of Moisil-Theodoresco systems in Euclidean spaces Roman Lavicka Mathematical Institute, Charles University Sokolovska 83 Praha 8, Praha 186 75 Czech Republic [email protected] Let k be a positive integer and 0 ≤ s ≤ m. Denote by Pk the space of real-valued k-homogeneous polynomials in Rm . Moreover, Λs stands for the space of s-vectors N over Rm and Pks = Pk R Λs . We are interested in the following space Hks = {P ∈ Pks ; dP = 0, d∗ P = 0}. Here d and d∗ is the de Rham differential and its adjoint, respectively. Moreover, assume that r, p and q are non-negative integers such that p < q and r + 2q ≤ m. Putting q M (r,p,q) Pk = Pkr+2j , j=p

the space (r,p,q)

M Tk

(r,p,q)

= {P ∈ Pk

; (d + d∗ )P = 0}

is formed by all k-homogeneous polynomial solutions of the Moisil-Theodoresco system of type (r, p, q). We show that (r,p,q)

M Tk

'

q M

Hkr+2j ⊕

j=p

q−1 M

r+2j+1 Hk−1 .

j=p

Hks

Later on, the spaces are considered as SO(m)modules. We are interested in irreducibility, the highest weights and dimensions of such modules. In particular, we give a formula for the dimension of the (r,p,q) space M Tk . Moreover, we decompose the kernel of the Hodge laplacian on polynomial forms into SO(m)modules. These results were obtained jointly with R. Delanghe and V. Souˇcek. ——— Quaternionic Physics

analysis,

representation

theory

and

Matvei Libine Department of Mathematics, Indiana University, Rawles

II.1 Clifford and quaternion analysis Hall, 831 East 3rd St Bloomington, IN 47405 United States [email protected] This is a joint work with Igor Frenkel. I will describe our new developments of quaternionic analysis using representation theory of various real forms of the conformal group as a guiding principle. These developments will lead to a solution of Gelfand-Gindikin problem. Along the way we discover striking new connections between quaternionic analysis and mathematical physics. In particular, the Maxwell equations are realized as the quaternionic counterpart of the Cauchy formula for the second order pole. We also find a representation-theoretic meaning of the polarization of vacuum and one-loop Feynman integrals. This talk is partially based on the joint paper with Igor Frenkel, “Quaternionic analysis, representation theory and physics”, Advances in Mathematics 218 (2008) pp 1806-1877; also available at arXiv:0711.2699.

exterior differentiation acting on forms on Rn , and d∗ is its formal adjoint. Our goal is to prove that any Dirac and semi-Dirac pair (D, D† ) has two Cauchy-Pompeiu and two BochnerMartinelli-Koppelman type integral representation formulas. ——— A differential form approach to Dirac operators on surfaces Heikki Orelma Institute of Mathematics, Tampere University of Technology, P.O. Box 553, FI-33101 Tampere, Finland [email protected] In this talk we consider Dirac operators on surfaces. Surfaces are k-dimensional embedded submanifolds of Rm . Let F be a Clifford algebra-valued differential form and ∂x be the Dirac operator on Rm . F is called monogenic if it is a solution of the equation

——— Hyperholomorphic functions in the sense of MoisilThodoresco and their different hyperderivatives ´s Maria Elena Luna-Elizarrara ESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.Lindavista Mexico City, D.F. 07338 Mexico [email protected] Any Moisil-Th´eodoresco-hyperholomorphic function is also Fueter-hyperholomorphic, but its hyperderivative is always zero, so one could consider then that these functions are a kind of “constants” for the Fueter operator. It turns out that the skew-field of quaternions as a real linear space is wide enough, so it is possible to give another type of hyperderivatives “consistent” with the Moisil-Th´eodoresco operator. In this talk we present these notions of different hyperderivatives and the relation between them. The talk is based on a joint work with M. A. Mac´ıas Cede˜ no and M. Shapiro. The research was partially supported by CONACYT projects as well as by Instituto Polit´ecnico Nacional in the framework of COFAA and SIP programs. ——— Dirac and semi-Dirac pairs of differential operators Mircea Martin Department of Mathematics, Baker University, 8th and Grove, Baldwin City, Kansas 66006 United States [email protected] The Euclidean Dirac operator Deuc,n on Rn , n ≥ 2, is a differential operator with coefficients in the Clifford al2 gebra of Rn that has the defining property Deuc,n = −∆, where ∆ = ∆euc,n is the Laplace operator on Rn . As generalizations of this class of operators we investigate pairs (D, D† ) of differential operators on Rn with coefficients in a Banach algebra A, such that either DD† = µL ∆ and D† D = µR ∆, or DD† + D† D = µ∆, where µL , µR , or µ are some elements of A. Such pairs (D, D† ) are called Dirac or semi-Dirac pairs of differential operators. The typical examples of a Dirac or semi-Dirac pair on Rn are given by D = D† = d + d∗ , or D = d and D† = −d∗ , where d is the operator of

L∂x F = 0, where L∂x F is the Lie derivative of F with respect to ∂x . The aim of this talk is to show that if F and ∂x are restricted to the k-surface S we obtain a Dirac type equation L∂x |S F |S = 0 on S. As an application we shall consider winding numbers. This is joint work with Frank Sommen (Gent). ——— CK-extension and Fischer decomposition for the inframonogenic functions ˜ a Pen ˜a Dixan Pen Department of Mathematics, University of Aveiro, Campus Universitario de Santiago, Aveiro 3810-193, Portugal [email protected] Let ∂x denote the generalized Cauchy-Riemann operator in Rm+1 . In this communication, we will present a refinement of the biharmonic functions and at the same time an extension of the monogenic functions by considering the solutions of the sandwich equation ∂x f ∂x = 0. In this setting a CK-extension and a Fischer decomposition are studied. ——— A new approach to slice-regularity on real algebras Alessandro Perotti Dept. Mathematics, Univ. of Trento, Via Sommarive 14 Povo, Trento I-38100 Italy [email protected] We rivisit the concept of primary functions introduced by Rinehart in the ’60’s and apply it to the theory of slice regular functions introduced recently by Gentili, Struppa and other authors. (Joint work with Riccardo Ghiloni, Trento, Italy) ———

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II.1 Clifford and quaternion analysis Clifford analysis with higher order kernel over unbounded domains Yuying Qiao Yuhua east Road 113, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei Province 050016 China [email protected] In this paper we talk Clifford analysis with higher order kernel over unbounded domains. First we derive an higher order Cauchy-Pompeiu formula for the functions with rth order continuous differentiability over an unbounded domain whose complementary set contains nonempty open set. Then we obtain higher order Cauchy integral formula for k-regular functions and prove Cauchy inequality. Based on the higher order Cauchy integral formula, we define higher order Cauchytype integrals and the Plemelj formula. ——— Complex Dunkl operators Guangbin Ren Departamento de Matem´ atica - Universidade de Aveiro, Campus de Santiago, Aveiro 3810-193 Portugal ren[email protected] Complex Dunkl operators for certain Coxeter groups are introduced. These complex Dunkl operators have the commutative property, which makes it possible to establish the corresponding complex Dunkl analysis.

Purpose of this talk is to provide a characterization of the dual of the Rn -module of slice monogenic functions on a class of compact sets in the Euclidean space Rn+1 . We are able to establish a duality theorem which, since holomorphic functions are a very special case of slice monogenic functions, is the generalization of the classical K¨ othe’s theorem. The duality results are also discussed in the quaternionic setting. ——— Explicit description of operators in the resolution for the Dirac operator Tomas Salac Faculty of Mathemtarics and Physics, Sokolovsk´ a 83, Prague 8, 18675 Czech Republic [email protected] A study of Dirac operator D in several variables is a traditional part of Clifford analysis. A lot of effort was spent to find an analogue of the Dolbeaut complex, i.e. a resolution starting with the operator D. The resolution is composed (in the stable range) from operators of the first and the second order. Using representation theory, it is possible to write down an explicit form of the first order operators in the resolution. It is, however, much more difficult to compute an explicit form of second order operators. In the lecture, we shall use Casimir operators (recently introduced in study of parabolic geometries) as a new tool helping to get these explicit formulae for the second order part of the resolution. ———

——— p-Dirac equations

On the relation between the Fueter operator and the Cauchy-Riemann-type operators of Clifford analysis.

John Ryan Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72703, United States [email protected]

Michael Shapiro ESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.Lindavista Mexico City, D.F. 07338 Mexico [email protected]

Associated to Laplacians there are first order operators called Dirac operators. For instance the Dirac operator associated to the Laplacian in the complex plane is the Cauchy-Riemann operator. In euclidean space there is the euclidean Dirac operator Similar such operators exist for Laplace-Beltrami operators on Riemannian manifolds. Besides the usual Laplace equation in euclidean space there are the non-linear p-Laplace equations. These equations are covariant under M¨ obius transformations and are invariant when p = n. Here we shall introduce non-linear p-Dirac equations. We shall demonstrate their link to the p-Laplacian in euclidean space and demonstrate their covariance under M¨ obius transformations. Other basic properties of these equations will be investigated. We shall extend the p-Dirac and p-Laplace equations to spin manifolds. This is joint work with Craig A. Nolder (Florida State University). ——— Duality theorems for slice hyperholomorphic functions Irene Sabadini Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano, Mi 20133 Italy [email protected]

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The Moisil-Th´eodoresco operator has an explicitly given relation with the classic Dirac operator of Clifford analysis for Cl0,3 . It turns out that the Fueter operator does not have, as one would expect, a similar relation with the corresponding classic Cauchy-Riemann operator but a modification of the latter is necessary. The aim of the talk is to explain all this in detail thus establishing a direct relation between, on one hand, what is usually called quaternionic analysis, and, on the other hand, Clifford analysis. This is joint work with J. Bory-Reyes. M. Shapiro was partially supported by CONACYT projects as well as by Instituto Polit´ecnico Nacional in the framework of COFAA and SIP programs. ——— Conformally invariant boundary valued problems for spinors and families of homomorphisms of generalized Verma modules. Petr Somberg Mathematical Institute of Charles University, Sokolovska 83, Prague, Karlin 180 00 Czech Republic [email protected] On a conformal manifold M with boundary ∂M there is a construction associating conformally invariant non-local

II.1 Clifford and quaternion analysis operators to the boundary valued problems for conformally invariant operators on M with symbols given by power of Laplace operator. These operators belong to one parameter families of conformally invariant operators, generalizing conformal Dirichlet-to-Robin operator. We will discuss generalization towards conformally invariant boundary valued problems for the spinor representation. ——— Clifford calculus in quantum variables Frank Sommen Department of Mathematics, University of Ghent, Galglaan 2. B-9000 Gent, Belgium [email protected] Starting from the axioms of the algebra R(S) of abstract vector variables over a set S (radial algebra): z(xy + yx) = (xy + yx)z,

x, y, z ∈ S,

together with the basic q-commutation relations for coordinates: xi xj = qij xj xi we arrive at the defining relations for the q-Clifford algebra: ei ej + qji ej ei = −2gij , whereby gij is the q-metric which also consists of noncommuting parameters. The partial derivatives ∂xj satisfy the same q-relations ∂xi ∂xj = qij ∂xj ∂xi together with the q-Weyl relations: ∂xi xj = qji xj ∂xi + δij . This leads to the introduction of a reciprocal Clifford basis ej satisfying: ej ei + qji ei ej = −2δij , which is linked to the original Clifford basis by relations of the form (Einstein summation convention): ej = gjk ek .

Ez = zE,

x, z ∈ S.

However, the identities for the q-quantum lattice seem to lead (in the first approximation) to a relation of the form ∂x x + qx∂x = m + q(q + 1)E whereby Ex − q 2 xE = x.

Vladimir Soucek Sokolovska 83 Mathematical Institute, Charles University Praha, Czech Republic 186 75 Praha Czech Republic [email protected] The Penrose transform is a perfect tool for a study of generalised Dolbeault resolutions in the theory of several Clifford variables. An important notion used in the definition of the Penrose transform is the relative BGG resolution. Its construction is indicated in the book by Baston and Eastwood on the Penrose transform. They, however, deserve a better attention; their construction can be made more detailed using tools used for construction of the classical BGG sequences. ——— Regular Moebius transformations over the quaternions Caterina Stoppato Dipartimento di Matematica “U. Dini”, Universit` a di Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected] Let H denote the real algebra of quaternions. We present quaternionic transformations that are included in the class of regular quaternionic functions introduced by G. Gentili and D.C. Struppa in recent years. Regularity yields to properties that recall the complex case, although the diversity of the quaternionic setting introduces new phenomena. Specifically, the group Aut(H) of biregular functions H → H coincides with the group of regular affine transformations (namely, q 7→ qa + b with a, b ∈ H and a 6= 0). Moreover, inspired by the classical quaternionic linear fractional transformations, we define the class of regular fractional transformations. This class strictly includes the set of regular injective b = H ∪ {∞} to itself. Finally, we study functions from H regular Moebius transformations, which map the unit ball B = {q ∈ H : |q| < 1} onto itself. All regular bijections from B to itself prove to be regular Moebius transformations. ———

The vector derivative (Dirac operator) is then given by ∂x = ∂xj ej and the basic rules of Clifford calculus may be derived. On the level of radial algebra these rules are the same as for standard Clifford analysis, which indicates that the q-deformation aspect is only visible when calculations are expressed in coordinates. This raises the problem to define a kind of q-deformation on the level of abstract vector variables. This can be done by defining the Dirac operator ∂x in a suitable way as an endomorphism on R(S). This may be done by assuming the operator relation ∂x x = −qx∂x + m + 2qE, whereby m is the dimension of space and E ∈ End(R(S)) is the q-Euler operator given by the operator relations Ex − qxE = x,

——— On relative BGG sequences

Singularities of functions of one and several bicomplex variables Adrian Vajiac Chapman University, Dept of Math/CS, One University Drive, Orange, CA 92866 United States [email protected] In this talk we introduce the notion of regularity for functions of one, as well as several bicomplex variables. Moreover, using computational algebra techniques, we prove that regular functions of one bicomplex variable have the property that their compact singularities can be removed. ——— Multiplicities of zeroes and poles of regular functions Fabio Vlacci Department of Mathematics Ulisse Dini, viale Morgagni 67/a FIRENZE, FI 50134 Italy [email protected]

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II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras The aim of this talk is to give a survey on some recent results which have been obtained for the description of the zero sets (and poles) of regular functions. In particular we will focus our attention to define (and evaluate) a multiplicity for zeroes and poles of regular functions. ——— Gauss-Codazzi-Ricci equations in Riemannian, conformal, and CR geometry Zuzana Vlasakova Sokolovska 83, Faculty of Mathematics and Physics, Praha 8, 18675, Czech Republic [email protected] We will remind the Gauss-Codazzi-Ricci equations in Riemannian geometry, and the work of David Calderbank with Francis Burstal and Diemer on similar equations for conformal geometry. Then we introduce the CR geometry and explain that we can do the same thing also for this geometry (it is a complex analogue of conformal geometry). ——— Compatibility conditions and higher spin Dirac operators Liesbet Van de Voorde Department of Mathematical Analysis, Clifford Research Group, Galglaan 2, 9000 Gent, Belgium [email protected] In this talk, we investigate polynomial solutions for generalized Rarita-Schwinger operators. We will explain that there are two types of solutions, and we will explicitly construct one of them using results on compatibility conditions for systems in several Dirac operators. This is joint work with David Eelbode and Fred Brackx. ———

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras Organisers: ¨ rlebeck, Vladimir Kisil, Klaus Gu ¨ ßig Wolfgang Spro The mathematical use of above mentioned algebras reaches from hypercomplex analysis and differential geometry up to corresponding numerical methods. Therefore we call especially for contributions with applications in gauge theories, mathematical physics, image processing, robotics, cosmology, engineering sciences etc. —Abstracts— Wavelets on spheres Swanhild Bernstein Freiberg University of Mining and Technology, Institute of Applied Analysis, Pr¨ uferstr. 9, D-09596 Freiberg, Germany [email protected]

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The construction of wavelets relies on translations and dilations which are perfectly given in R. On the sphere translations can be considered as rotations but it difficult to say what are dilations. For the 2-dimensional sphere there exist two different approaches. The first concept defines wavelets by means of kernels of spherical integrals. The other approach is a purely group theoretical approach and defines dilations as dilations in the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define wavelets on the 3dimensional sphere by means of kernels of integrals and demonstrate that wavelets constructed according to the group-theoretical approach for zonal functions meet our definition. Typical examples arise quite easily from the AbelPoisson and Gauß-Weierstraß kernel. We will extend these kernels and wavelets into the Clifford-algebra setting. We specifically define spherical wavelets of order m. Theorem. The elements of {Ψρ , ρ > 0} are wavelets of order m (m ≥ 0) if the following admissibility conditions are satisfied: Z ∞ e 2ρ (k)α(ρ) dxρ = (k + 1)2 , k = m + 1, m + 2, ... Ψ 0

e ρ (k) = 0, k = 0, ..., m; ∀ρ ∈ (0, ∞) Ψ ˛ Z π ˛Z ∞ ˛ ˛ 2 (2) ˛ ˛ sin (θ) dxθ ≤ T, Ψ (θ)α(ρ) dxρ ρ ˛ ˛ 0

∀R ∈ (0, ∞),

R

(T > 0, independent of R). (2) Here, Ψρ stands for Ψρ ∗ Ψρ .Ψ1 (ρ = 1) is the mother wavelet. ——— On special monogenic power and Laurent series expansions and applications Sebastian Bock Bauhaus-University Weimar, Institute for Mathematics/Physics, Coudraystraße 13B, Weimar, 99421 Germany [email protected] The contribution focuses on some recently developed (orthogonal) monogenic power and Laurent-series expansions which are complete in the space of square integrable quaternion-valued functions and have as similar properties as the respective complex series expansions based on the well known z-powers. Starting with the Fourier series expansion we will show some structural properties of the series expansion with respect to their hypercomplex derivative and primitive. These special characteristics of the used orthonormal basis enable further the construction of a new Taylor type series expansion which can be explicitly related to the corresponding Fourier series analogously as in the complex onedimensional case. We end up by showing some orthogonality results for the exterior domain and present the corresponding Laurent series expansion for the domain of the spherical shell. These series expansions find applications in the description of the hypercomplex derivative as well as the monogenic primitive of a monogenic function which are represented as Fourier series, Taylor type and Laurent series. In this connection some further applications are presented.

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras ——— Spin gauge models Ruth Farwell Buckinghamshire New University, Queen Alexandra Road, High Wycombe, Bucks HP11 2JZ United Kingdom [email protected]

Recent progress in this direction will be reviewed. In particular integrable generalizations of KdV and NLS in 4+2 will be presented and the question of their reduction to 3+1 will be discussed. The role of quaternions for generalizing these results to higher dimensions will be investigated. ——— Note on the linear systems in quaternions

In 1999 we defined a form of spin gauge theory of particle interactions in which both standard ’left-hand’ and new ’right-hand’ interaction terms occur. In the proceedings of the 2005 Toulouse conference we reported the predictions of the value of the Weinberg angle and the mass of the Top quark based on a particular ’two-sided’ model, and we introduced the concepts of the ’quark’ and ’centroid’ representations. We also discussed new gravitational effects and the replacement of the graviton by the ’frame field quantum’. Recently, we have studied a variety of other two-sided models, and we present the predictions of another model, in which a different choice of spinor idempotent allows us to introduce a new particle interaction term. This is joint work with Roy Chisholm (Kent). ——— Further results in discrete Clifford analysis Nelson Faustino Departamento de Matem´ atica, Campus Universit´ ario de Santiago Aveiro, Aveiro 3810-193 Portugal [email protected] In this talk we will present the fundamentals of a higher dimensional discrete function theory by combining the Clifford algebra setting with the umbral calculus approach. Starting with the umbral version of Fischer decomposition, we decompose the space of umbral homogeneous polynomials in terms of umbral monogenic polynomials. This allows us to build up in a combinatorial way the theory of discrete spherical monogenics as a refinement of the theory of spherical harmonics. Furthermore, the interplay between discrete Clifford analysis and the physical model of the discrete harmonic oscillator will be explored along this talk by means of the canonical generators of Wigner Quantum Systems. ——— Integrability in multidimensions, complexification and quaternions Thanasis Fokas Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road Cambridge, Cambridgeshire CB3 0WA United Kingdom [email protected] One of the most important open problems in the area of integrable nonlinear evolution equations has been the construction of integrable equations in 3+1, i.e. in three spatial and one temporal dimensions. The celebrated KdV and NLS equations are integrable evolution equations in 1+1; the KP and DS equations are physically significant generalizations of the KdV and NLS in 2+1. Do there exist analogous equations in 3+1?

Svetlin Georgiev Sofia University, Faculty of Mathematics and Informatics, Department of Differential Equations, Blvd James Boucher 126, Sofia 1000 Bulgaria [email protected] In this talk we will discuss the linear system r X n X

s pslm xm qlm = As ,

s = 1, 2, . . . , n,

(*)

l=1 m=1 s where n, r ≥ 1 are given constants, pslm , qlm , As , l = 1, . . . , r, m = 1, . . . , n, s = 1, . . . , n, are given real quaternions, xm , m = 1, . . . , n, are unkown real quaternions. Here a propose an algorithm for finding a solution to the system (*). Also, we give necessary and sufficient condition for the solvability of the system (*) and some examples.

——— Minimal algorithms for Lipschitzian elements and Vahlen matrices Jacques Helmstetter 15 rue de l Oisans, St-Martin d’Heres, Isere 38400 France [email protected] If S is a closed algebraic manifold in a vector space V , and if d is the codimention of S in V , an algorithm that allows us to test whether an element of V belongs to S by means of only d numerical verifications, is called a minimal algorithm. If Cl(M, q) is the Clifford algebra derived from a quadratic module (M, q), the Lipschitz monoid Lip(M, q) is (in most cases but not in all cases) the monoid generated in Cl(M, q) by M . From the invariance property of Lipschitz monoids, a minimal algorithm can be deduced for the even and odd components of Lip(M, q). A minimal algorithm can also be deduced for the two components of the monoid of Vahlen matrices. ——— Clifford-Fourier transforms and hypercomplex signal processing Jeff Hogan School of Mathematical and Physical Sciences, University of Newcastle V-128, University Drive Callaghan, NSW 2308 Australia [email protected] In this talk we attempt to synthesize recent progress made in the mathematical and electrical engineering communities on topics in Clifford analysis and the processing of colour images, in particular the construction and application of Clifford-Fourier transforms designed to treat multivector-valued signals. Emphasis will be placed on the two-dimensional setting where the appropriate underlying Clifford algebra is the familiar set of

43

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras quaternions. We’ll describe some results and problems in the construction of discrete wavelet bases for colour images, and the difficulties encountered in constructing Clifford-Fourier kernels in dimensions 3 and higher. ——— Discrete Clifford analysis by means of skew-Weyl relations ¨ hler Uwe Ka Department of Mathematics, University of Aveiro, Aveiro, P-3810-193 Portugal [email protected] Recently one can observe an increased interest in higher dimensional discrete function theories. This is not only driven by the numerical application of continuous methods but also due to problems from combinatorics and quantum physics. While there is now a well-established approach in the continuous case, by means of the socalled radial algebra (F. Sommen), unfortunately, a direct translation to the discrete case is problematic. In this talk we present an alternative approach based on a recent idea of F. Sommen of replacing the Weyl relations by skew-Weyl relations. We will construct the basic ingredients for discrete Clifford analysis in this context and illustrate its applicability. ——— Hypercomplex analysis in the upper half-plane Vladimir Kisil School of Mathematics, Woodhouse Lane, University of Leeds, LS2 9JT, United Kingdom [email protected] Complex analysis seems to be the only non-trivial analytic function theory in the two dimensional case. However one can employ the group SL(2, R) and its representation theory in order to build elements of analytic functions with complex, dual and double numbers. This is a part of “Erlangen Programme at Large” approach in analysis. ——— Formulas for reproducing kernels of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to inhomogeneous Helmholtz equations Rolf Soeren Krausshar Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan, 200-B Leuven, Vlaams Brabant, 3001 Belgium [email protected] P ∂ Let D := n i=1 ∂xi ei be the Euclidean Dirac operator in n R and let P (X) = am X m + . . . + a1 X1 + a0 be a polynomial with arbitrary complex coefficients. Differential equations of the form P (D)f = 0 are called polynomial Dirac equations with complex coefficients. In this talk we consider Hilbert spaces of Clifford algebra valued functions that satisfy such a polynomial Dirac equation in annuli of the unit ball in Rn . We determine a fully explicit formula for the associated Bergman kernel for solutions of complex polynomial Dirac equations of any degree m in the annulus of radii µ and 1

44

where µ ∈]0, 1[. We further give explicit formulas for the Szeg¨ o kernel for solutions to polynomial Dirac equations of polynomial degree m < 3 in the annulus. As concrete application we give an explicit representation formula for the solutions of generalized Helmholtz and Klein-Gordon type equation inside the annulus and with prescribed data at the boundary of the annulus. The solutions are represented in terms of integral operators that involve the explicit formulas of the Bergman kernel that we computed. ——— The Ito transform for partial differential equations Remi Leandre Institut de Mathematiques. Universite de Bourgogne Bd Alain. Savary Dijon, Cote d’Or 21078. France [email protected] We give an interpretation of the celebrated Ito formula of stochastic analysis in various contexts where there is no convenient measure on a convenient path space. We begin by the case of a diffusion (the classical one), we study after the case of the heat-equation associated to an operator of order four on a torus, we continue by studying the case of the Schroedinger equation associated to a big order operator on a torus, we consider after the case on the wave equation on a torus and we finish by studying the case of a Levy type operator associated to a big fractional power of the Laplacian on the linear space. ——— Quaternionic analysis and boundary value problems Dimitris Pinotsis Department of Mathematics, University of Reading, RG6 6AX, UK [email protected] First, we will review some results appearing in the theory of quaternions. Then, we will apply these results to solve boundary value problems for linear elliptic equations in four dimensions. Further extensions of these results will also be discussed. ——— Integral theorems in a commutative three-dimensional harmonic algebra Vitalii Shpakivskii Institute of Mathematics of National Academy of Sciences of Ukraine, Tereshchenkivska str., 3, Kiev-4, 01601, Ukraine [email protected] An associative commutative three-dimensional algebra A3 with unit 1 is harmonic if in A3 there exists a harmonic basis {e1 , e2 , e3 } satisfying the conditions e21 + e22 + e23 = 0,

e2j 6= 0 for j = 1, 2, 3.

(*)

There are three harmonic algebras exactly over the field of complex numbers only, and all harmonic bases are constructed by I. Mel’nichenko. We consider a harmonic algebra A3 containing the radical with basis {ρ1 , ρ2 } and multiplication table: ρ21 = ρ2 ,

ρ22 = 0,

ρ1 ρ2 = 0.

III.1. Toeplitz operators and their applications We proved that every locally bounded function differentiable in the sense of Gateaux (such a function is monogenic) Φ(ζ) = U1 (x, y, z)e1 + U2 (x, y, z)e2 + U3 (x, y, z)e3 (here ζ = xe1 + ye2 + ze3 and x, y, z are real) has nth Gateau derivative for any n. So, the components U1 , U2 , U3 satisfy the three-dimensional Laplace equation „ 2 « ∂ ∂2 ∂2 ∆3 U := + + U (x, y, z) = 0 ∂x2 ∂y 2 ∂z 2 00

(ζ) (e21 + e22 + e23 )

owing to equality ∆3 Φ = Φ and equality (*). For monogenic functions Φ(ζ) taking values in A3 , we proved Cauchy’s theorems for surface integral and curvilinear integral. We proved also an analog of Cauchy’s formula that yields Taylor’s expansion of monogenic function. Morera’s theorem is also established. Thus, as in the complex plane, one can give different equivalent definitions of monogenic functions taking values in the algebra A3 . This is joint work with S. Plaksa. ——— Initial boundary value problems with quaternionic analysis ¨ ßig Wolfgang Spro TU Bergakademie Freiberg, Institute of Applied Analysis, Pr¨ uferstr. 9, Freiberg 09599 Germany [email protected] A quaternionic operator calculus is used to find representations of the solution of several initial boundary value problems in mathematical physics. ——— Real bi-graded Clifford modules, the Majorana equation and the standard model action ¨ rgen Tolksdorf, Ju Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22, 04105 Leipzig, Germany [email protected] The fundamental grading involution that underlies the Dirac equation is provided by parity. In contrast, the Majorana equation is based on charge conjugation. Together, these two grading involutions form what is called a Majorana module. On these modules there exist a natural class of Dirac operators encoding the action functional of the Standard Model of particle physics. ——— The regularized Schr¨ odinger semigroup acting on tensors with values in vector bundles Nelson Vieira Departamento de Matem´ atica-Universidade de Aveiro, Campus Universit´ ario de Santiago, P-3810-193 Aveiro, Portugal [email protected] In this talk we apply known techniques from semigroup theory and Clifford analysis to the homogeneous problem with initial condition of the Schr¨ odinger equation.

To do this end, we start by express the arising tensorial spaces in terms of complexified Clifford algebras and we construct a fiber bundle identification of our spaces with appropriated vector spaces of tensors and differential forms. We then establish the semi-groups for the family of regularized Schr¨ odinger operators and prove their dissipative property. We end with an application to the non-stationary Schr¨ oringer equation. ———

III.1. Toeplitz operators and their applications Organisers: Sergei Grudsky, Nikolai Vasilevski The idea of the session is to bring together the experts actively working on Toeplitz operators acting on Bergman, Fock or Hardy spaces, as well as in various related areas where Toeplitz operators play an essential role, such as asymptotic linear algebra, quantisation, approximation, singular integral and convolution type operators, financial mathematics, etc. We expect that the results presented, together with fruitful discussions, will serve as a snapshot of the current stage of the area, as well as for better understanding of the priority directions and themes of future developments. —Abstracts— On the relations between the kernel of a Toeplitz operator and the solutions to some associated RiemannHilbert problems ˆ mara Cristina Ca Departamento de Matem´ atica, Instituto Superior T´ecnico, Av. Rovisco Pais ,Lisboa, 1049-001 Portugal [email protected] It is possible, in many cases, to determine some solution to a Riemann-Hilbert problem associated to TG , of the form ± n Gh+ = h− , h± ∈ (H∞ ) . (*) Such a solution can provide important information on the properties of TG . Namely, for G ∈ (L∞ (R))n×n , with det G = 1 (or admitting a bounded canonical factorization), if h± = (h1± , h2± ) are corona pairs in C± , i.e., inf (|h1± (ξ)| + |h2± (ξ)|) > 0, (**) ξ∈C±

it can be shown that TG is invertible. In this talk, the question of what information can be obtained, as regards the kernel of TG , from a solution to (*), is considered. Several classes of symbols are studied which, if n = 2, correspond to a situation where (**) is not, or may not, be satisfied. ——— Convolution type operators with symmetry in exterior wedge diffraction problems Luis Castro Campus Universitario, Department of Mathematics, University of Aveiro, Aveiro 3810-193 Portugal [email protected]

45

III.1. Toeplitz operators and their applications We will use convolution type operators with symmetry in a Bessel potential spaces framework to analyse classes of problems of wave diffraction by a plane angular screen occupying an infinite 270 degrees wedge sector. The problems are subjected to different possible combinations of boundary conditions on the faces of the wedge. Namely, under consideration there will be boundary conditions of Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, impedanceDirichlet, and impedance-Neumann types. Existence and uniqueness results are proved for all these cases in the weak formulation. In addition, the solutions are provided within the spaces in consideration, and higher regularity of solutions are also obtained in a scale of Bessel potential spaces. The talk is based on a joint work with D. Kapanadze. ——— Berezin transform on the harmonic Fock space Miroslav Englis Mathematics Institute AS CR Zitna 25, Prague 1, Prague 11567 Czech Republic [email protected]

Lorentz space L(p, q, wdµ)(R2d ). M (p, q, w)(Rd ) is a Banach space with the norm kf kM (p,q,w) = kVg f kpq,w . In this paper we discussed the boundedness of Toeplitz operator on M (p, q, w)(Rd ) under some assumptions. We also proved that the Toeplitz operator T pg (F ) of M (2, p, w1 )(Rd ) into M (2, p, w1 )(Rd ) is S2 with the Hilbert-Schmidt norm bounded by kT pg (F )kS2 CkF k(1,t) under some condition. This is joint work with Ay¸se Sandik¸ci. ——— Presentation of the kernel of a special structure matrix characteristic operator by the kernels of two operators one of them is a scalar characteristic operator Oleksandr Karelin Advanced Research Center on Industrial Engineering, Autonomous University of the Hidalgo State, Pachuca, Hidalgo 42184 Mexico [email protected] We denote the Cauchy singular integral operator along the upper part of the unit semicircle T+ by Z ϕ(τ ) 1 (ST+ ϕ)(x) = dτ πi τ −x T+

The standard Berezin-Toeplitz quantization is based on the asymptotic expansion of the Berezin transform as the weight parameter tends to infinity. We discuss an extension of this result to the case of the harmonic SegalBargmann-Fock space on Cn .

and the identity operator on T+ by (IT+ ϕ)(t) = ϕ(t). By operator equalities, results about the integral operators with endpoint singularities are extended to matrix characteristic operators

———

DR+ = uIT+ + vST+ , DT+ ∈ [L22 (T+ )]

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

with the coefficients u, v of a special structure. The following decomposition \ ˜ ker DT+ = ker H F ker C,

Sergey Grudsky Department of Mathematics, CINVESTAV, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, 07360 Mexico [email protected] While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n goes to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. ——— Toeplitz operators of M (p, q, w)(Rd ) spaces ¨ rkanlı Turan Gu Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit Samsun, 55139 Turkey [email protected] Let g be a function in S(Rd )/0, where S(Rd ) is Schwartz space, and 1 ≤ p, q ≤ ∞. The space M (p, q, w)(Rd ) denotes the subspace of all tempered distributions f such that the Gabor transform Vg f of f is in the weighted

46

is found. Here operator C is a scalar characteristic operator, C ∈ [L2 (T+ )], operator F is invertible operator, ˜ and C are conF ∈ [L2 (T+ ), L22 (T+ )]. Operators H structed by an arbitrary nontrivial element of ker DT+ or by an arbitrary nontrivial element of the kernel of the associated operator. This is joint work with Anna Tarasenko. ——— Bounds for the kernel dimension of singular integral operators with Carleman shift Edixon Rojas Campus Universitario, Department of Mathematics, University of Aveiro, Aveiro 3810-193 Portugal [email protected] Upper bounds for the kernel dimension of singular integral operators with preserving-orientation Carleman shift are obtained. This is implemented by using some estimations which are derived with the help of certain explicit operator relations. In particular, the interplay between classes of operators with and without Carleman shifts has a preponderant importance to achieve the mentioned bounds. ——— Invertibility of matrix Wiener-Hopf plus Hankel operators with different Fourier symbols Anabela Silva Departamento de Matem´ atica - Universidade de Aveiro,

III.2. Reproducing kernels and related topics Campus de Santiago, Aveiro 3810-193 Portugal [email protected]

States [email protected]

Based on different kinds of auxiliary operators and corresponding operator relations, we will present conditions which characterize the invertibility of matrix WienerHopf plus Hankel operators having different Fourier symbols in the class of almost periodic elements.

We will discuss Toeplitz operators on the Fock space induced by positive measures. Problems considered include boundedness, compactness, and membership in the Schatten classes. ———

——— Flat Hilbert bundles and Toeplitz operators on symmetric spaces Harald Upmeier Department of Mathematics, University of Marburg, Hans-Meerwein-Strasse, Lahnberge Marburg, Hessen 35032 Germany [email protected] In generalization of the classical Fock spaces we construct a family of Hilbert spaces, viewed as a Hilbert bundle over a bounded symmetric domain (Cartan domain) B, which is equivariant under a suitable, nonholomorphic, action of the holomorphic automorphism group G of B (a semisimple Lie group). Geometrically, these Hilbert spaces live on the so-called Matsuki dual associated with the G-orbits in the boundary of B. We show that the Hilbert bundle carries a natural connection over B which is projectively flat, similar as the well-known case for the metaplectic representation on Fock space. The associated parallel transport (Bogoluybov transformations) is also determined. In the talk we emphasize relations to classical Fock spaces over real, complex and quaternion matrix spaces, although the basic construction depends mainly on the Jordan algebraic description of bounded symmetric domains. ——— Commutative algebras of Toeplitz operators on the unit ball Nikolai Vasilevski Department of Mathematics, CINVESTAV, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, 07360 Mexico nv[email protected] All known commutative C ∗ -algebras generated by Topelitz operators on the unit disk are classified as follows. Given a maximal commutative subgroup of biholomorphisms of the unit ball, the C ∗ -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. Surprisently there exist many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C ∗ -algebra, and for n = 1 all of them collapse to commutative C ∗ -algebra generated by Toeplitz operators on the unit disk. ——— Toeplitz operators on the Fock space Kehe Zhu Department of Mathematics and Statistics, 1400 Washington Ave, SUNY Albany, New York 12222 United

III.2. Reproducing kernels and related topics Organisers: Alain Berlinet, Saburu Saitoh Since the first works laying its foundations as a subfield of Complex Analysis, the theory of reproducing kernels has proved to be a powerful tool in many fields of Pure and Applied Mathematics. The aim of this session is to gather researchers interested in theoretical as well as applied modern problems involving this theory. —Abstracts— A general theory for kernel estimation of smooth functionals Belkacem Abdous Universite Laval Medecine Sociale et Preventive, Pavillon de l’Est, Quebec, Qc G1K 7P4 Canada [email protected] In this talk, we present a general framework for estimating smooth functionals of the probability distribution functions, such as the density, the hazard rate function, the mean residual time, the Lorenz curve, the spectral density, the tail index, the quantile function and many others. This framework is based on maximizing a local asymptotic pseudo-likelihood associated to the empirical distribution function. An explicit solution of this problem is obtained by means of reproducing kernels approach. Some asymptotic properties of the obtained estimators are presented as well. ——— Weighted composition operators on some spaces of analytic functions Som Datt Sharma Department of Mathematics, University of Jammu, Jammu-180006, India 66 Ashok Nagar, Canal Road, Jammu, Jammu & Kashmir 180016 India somdatt [email protected] Let D be the open unit disk in the complex plane C and H(D) be the space of holomorphic functions on D. In this article, we give a short and selective account of results known about weighted compostion operator Wψ,ϕ defined by Wψ,ϕ f (z) = ψ(z)f (ϕ(z)),

f ∈ H(D),

where ϕ is a holomorphic map of D that takes D into itself and ψ is any holomorphic map of D. Discriptions of weighted composition operators acting from Hardy spaces, weighted Bergman spaces, α-Bloch spaces and A−α -spaces into other spaces of holomorphic functions

47

III.3. Modern aspects of the theory of integral transforms have been obtained by a number of authors during recent years. We provide a unified way of treating these operators. ——— Integral formulas on the boundary of some ball Keiko Fujita Faculty of Culture and Education, Saga university, Saga 840-8502 [email protected] We have been studied integral representations for holomorphic functions and complex harmonic functions on some balls, which we call the ”Np -balls”. One of Np balls is the Lie ball. For holomorphic functions on the Lie ball we know the Cauchy-Hua integral formula, whose integral is taken over the Shilov boundary of the Lie ball. A generalization of the Cauchy-Hua integral formula was considered for holomorpic functions on subspaces of the Lie ball by M.Morimoto. Since Np -ball can be represented by a union of these subspace, the boundary of the Np -ball can be represented by a union of the boundaries of the subspaces. Considering the fact, we consider an integral representation for holomorphic functions on the Np -ball by an iterated integral. In this talk, we will review some integral formulas on holomorphic functions on the Np -ball and treat some topics.

4) Vienna, A-1090 Austria [email protected] Irregular sampling in spline-type spaces has become a vivid research area, with many contributions in the recent literature. We will describe efficient implementations of operators related to spline-type spaces with finite sets of generators on Rd , covering both the case of regular and irregular sampling. In contrast to earlier papers, which either treat the continuous setting using abstract methods (i.e. continuous Fourier transforms) or deal with the discrete case when it comes to numerical implementations, we are discussing the problem of constructively realizing the abstract concepts with methods that can be implemented on a computer, achieving a small error of reconstruction in a certain given norm. In such a situation the trade-off between realizing individual iterative steps with high precision but at high computational costs, versus the option of doing a larger number of iterations has to be analyzed. Joint work with Prof. Hans Feichtinger. ——— Free boundary value problem for (−1)M (d/dx)2M and the best constant of Sobolev inequality Kazuo Takemura Shinei 2-11-1, Narashino, Chiba 275-8576 Japan [email protected]

——— Paley–Wiener spaces and their reproducing formulae. John Rowland Higgins I.H.P., 4 rue du Bary, 11250 Montclar, France. [email protected] Classical Paley–Wiener space, denoted by PW, consists of functions that are inverse Fourier transforms of those members of L2 (R) that are null outside [−π, π]. It is well known that PW possesses two reproducing formulae; a reproducing equation and a ‘discrete’ analogue, or sampling series, and that these make a remarkable ‘concrete – discrete’ comparison. It is shown that such analogies persist in the setting of more general Paley– Wiener spaces. ‘Operator’ versions of the reproducing equation and of the sampling series will be given that are also comparable, but now in a slightly different way. The setting emerges from two sources, the approach to sampling theory via the reproducing kernel theory due to S. Saitoh, and the approach via harmonic analysis of I. Kluv´ anek, M.M. Dodson et al. The capacity for amalgamation of these two sources has gone unnoticed hitherto. The special case of multiplier operators with respect to the Fourier transform acting on Paley–Wiener space will be considered. The Hilbert transform, and in twodimensions the Riesz transforms, provide examples with possibilities of extension to higher dimensions and to further classes of operators.

Green function of free boundary value problem for (−1)M (d/dx)2M is found using Whipple’s formula. Its Green function is constructed through so-called symmetric orthogonalization method under a suitable solvability conditions. As an application, we found the best constant of Sobolev inequality for M = 1, 2, 3, 4, 5 by investigating an aspect of Green function as a reproducing kernel. For M ≥ 6, this is still open. ———

III.3. Modern aspects of the theory of integral transforms Organisers: Anatoly Kilbas, Saburu Saitoh

—Abstracts— Integral transforms related to generalized convolutions and their applications to solving integral equations Liubov Britvina Department of Theoretical and Mathematical Physics, Novgorod State University, ul.St.Petersburgskaya 41, Veliky Novgorod, Novgorod region 173003 Russia [email protected]

——— Irregular sampling in multiple-window spline-type spaces Darian Onchis Faculty of Mathematics, University of Vienna, Nordbergstrae 15 (Universit¨ ats Zentrum Althanstrae, UZA

48

The present research is devoted to some integral transforms of convolution type. The definition of polyconvolution, or generalized convolution, was first introduced by V.A. Kakichev in 1967. Let A1 , A2 and A3 be operators. The generalized convolution of function f (t) and k(t), under A1 , A2 , A3 , with weighted function α(x),

III.3. Modern aspects of the theory of integral transforms is the function h(t) denoted by

“

α

fA1 ∗ kA2

”

(t) for A3

which the following factorization property is valid: »“ ” – α (A3 h)(x) = A3 fA1 ∗ kA2 (x) A3

=

α(x)(A1 f )(x)(A2 k)(x).

Here we consider the generalized convolution for integral transforms with the Bessel functions in the kernels. Using the differential properties of these convolutions we construct some integral transforms and find their existence conditions and inverse formulas. Natural applications to the corresponding class of convolution integral equations are demonstrated. ——— Bedrosian identity for Blaschke products in n-parameter cases Qiuhui Chen Departamento de Matem´ atica - Universidade de Aveiro, Campus de Santiago, Aveiro 3810-193 Portugal [email protected] We establish a necessary and sufficient condition for the amplitude function such that a Bedrosian identity holds in the case when the phase function is determined by the boundary value of a Blaschke product with nparameters. ——— Evaluation formulae for analogues of conditional analytic Feynman integrals over a function space Dong Hyun Cho Department of Mathematics, Kyonggi University, Young-Tong-Gu Suwon, Kyonggido 443-760 South Korea [email protected] In this talk, we introduce two simple formulae for the conditional expectations over an analogue C[0, t] of the Wiener space, the space of continuous real-valued paths on the interval [0, t]. Using these formulae, we establish various formulae for analogues of the conditional analytic Wiener and Feynman integral of the functionals in a Banach algebra which corresponds to the Banach algebra on the classical Wiener space introduced by Cameron and Storvick. Finally, we evaluate the analogues of the conditional analytic Wiener and Feynman integral for the functional Z t ff exp θ(s, x(s)) dη(s) 0

which is defined on C[0, t] and is of interest in Feynman integration theories and quantum mechanics. ——— An equation with symmetrized fractional derivatives ´anin Diana Dolic Faculty of Technical Sciences, University of Priˇstina - Kosovska Mitrovica, Kneza Miloˇsa, 28000 Kosovska Mitrovica, Serbia dolicanin [email protected]

We study equation Z 1 d2 ± α ET u (t) φ(α)dα + F (t, u (t)) = 0, 0 < t < T u (t)+ dt2 0 where, ± ETα u (t) is the symmetrized Caputo fractional derivative of u, φ(α), α ∈ (0, 1), is a positive integrable function or a positive compactly supported distribution with the support in (0, 1) and F is a continuous function in [0, T ] × R and locally Lipschitz continuous with respect to the second variable. This is joint work with T. Atanackovic, S. Konjik and S. Pilipovic. ——— Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetic Hiroshi Fujiwara Kyoto University, Yoshida-Honmachi Sakyo-ku Kyoto, Kyoto 606-8501 Japan [email protected] We consider the real inversion of the Laplace transform. It appears in engineering or physics, and it is ill-posed in the sense of Hadamard. We introduce some reproducing kernel Hilbert spaces and propose an inversion algorithm employing Tikhonov regularization. The regularized equation is well-posed, and its discretization is expected to have the stability and convergence with a suitable norm. However, theoretical stability is not equivalent to the stability of computational processes. We propose the use of multiple-precision arithmetic to reduce the influence of rounding errors for reliable numerical computations. Multiple-precision arithmetic is useful for regularization as approximation. ——— Method of integral transforms in the theory of fractional differential equations Anatoly Kilbas Faculty of Mathematics and Mechanics, Belarusian State University, Independence Avenue, 4 Minsk, 220030 Belarus [email protected] Our report deals with the method of integral transforms in investigtation of differential equations with ordinary and partial fractional derivatives. First we give an overview of results in this field. Then we present application of one-dimensional Laplace, Mellin and Fourier integral transforms to solution of ordinary diferential equations with Riemann-Liouville and Caputo fracitonal derivatives. Further we give application of Laplace and Fourier integral transforms to obtain explicit solutions of Cauchy-type and Cauchy problems for the twoand multi-dimensional diffusion-wave equations with the Riemann-Liouville and Caputo partial fractional derivatives, respectively, and indicate conditions for the existence of classical solutions of these problems. Finally, we use Laplace and Fourier integral transforms to deduce explicit solutions of fractional evolution equations involving partial fractional derivatives of RiemannLiouville or Caputo with respect to time and partial Lioville fractional derivative with respect to real axis, and indicate applications.

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III.4. Spaces of differentiable functions of several real variables and applications We note that explicit solutions of the considered fractional differential equations and Cauchy-type and Cauchy problems for them are expressed in terms of special functions of Mittag-Leffler, Wright and the so-called H-functions.

where φ is ”weight” distribution with compact support and, Dγ denotes the Riemman-Liouvill fractional derivative of order γ. Differential equations of the form Z

——— 0

Notes on the analytic Feynman integral over paths in abstract Wiener space Bong Jin Kim Department of Mathematics, Daejin University, Pocheon, Kyeonggi-Do 487-711 South Korea [email protected] In this talk, we study some results about analytic Feynman integral over paths in abstract Wiener space.

2

φ1 (γ)Dγ u dγ =

Z

2

φ2 (γ)Dγ v dγ

(*)

0

are constitutive equations for viscoelastic body. We consider (*) coupled with nonlinear ordinary differential equation D2 u(t) + v = f (t, u(t))

(**)

and show existence and uniqueness of the solution to the problem (*)–(**) with initial conditions u(0) = u0 , u0 (0) = v0 in classical and mild sence.

——— On the fractional calculus of variations Sanja Konjik Faculty of Agriculture, Department of Agricultural Engineering, Trg D. Obradovica 8, Novi Sad, 21000 Serbia sanja [email protected] The purpose of this talk is to study variational principles allowing Lagrangian density to contain derivatives of arbitrary real order. We derive a necessary condition for existence of a solution to a fractional variational problem and examine invariance under the action of transformation groups. As the results we obtain the Euler-Lagrange equations, as well as infinitesimal criterion and Noether’s theorem, which in fact extend the well-known classical results. In addition, we also study the case when both function and the order of fractional derivative are varied in the minimization procedure. ——— Integral transforms with extended generalized MittagLeffler function Anna Koroleva Department of Mathematics and Mechanics, Nezavicimosti ave 4, Minsk BY-220030 Belarus [email protected] Asymptotic results for integral transforms with extended generalized Mittag-Leffler function in the kernel are discussed. Inversion formulas for such transforms in weighted spaces of integrable functions are found. ——— Systems of differential equations containing distributed order fractional derivative Ljubica Oparnica Serbian Academy of Science and Art, Kneza Mihaila 36, Belgrade, 11000 Serbia [email protected] Distributed order fractional derivatives has appeared as generalization of the finite sum of fractional derivatives and has wide applications in technical sciences. We define distributed order fractional derivative of distribution u ∈ S 0 R) supported in R+ by fomula Z h φ(γ)Dγ u dγ, ϕi = hφ, hDγ u, ϕii, ϕ ∈ S 0 , supp u

50

——— Some aspects of modified Kontorovitch-Lebedev integral transforms Juri M. Rappoport Vlasov street Building 27 apt.8 Moscow 117335 [email protected] A proof of inversion formulas of the modified Kontorovitch-Lebedev integral transforms is developed. The Parceval equations for modified KontorovitchLebedev integral transforms are proved and sufficient conditions for them are found. Some new representations and properties of these transforms are justified. The inequalities which give estimations for their kernels - the real and imaginary parts of the modified Bessel functions of the second kind Re K1/2+iτ (x) and Im K1/2+iτ (x) for all values of the variables x and τ are obtained. The applications of Kontorovitch-Lebedev transforms to the solution of some mixed boundary value problems in the wedge domains are accomplished. The solution of the appropriate dual and singular integral equations is considered. The numerical aspects of using of these transforms are elaborated in detail. ——— A new class of polynomials related to the KontorovichLebedev transform Semyon Yakubovich Department of Pure Mathematics, Faculty of Science, University of Porto, Campo Alegre str. 687, Porto 4169007 Portugal [email protected] We consider a class of polynomials related to the kernel Kiτ (x) of the Kontorovich-Lebedev transformation. Algebraic and differential properties are investigated and integral representations are derived. We draw a parallel and establish a relationship with the Bernoullis and Eulers numbers and polynomials. Finally, as an application we invert a discrete transformation with the introduced polynomials as the kernel, basing it on a decomposition of Taylors series in terms of the Kontorovich-Lebedev operator. ———

III.4. Spaces of differentiable functions of several real variables and applications

III.4. Spaces of differentiable functions of several real variables and applications Organisers: Viktor Burenkov, Stefan Samko This session intends to cover various aspects of the theory of Real Variables Function Spaces (Lebesgue, Orlich, Sobolev, Nikol’skii-Besov, Lizorkin-Triebel, Morrey, Campanato, and other spaces with zero or nonzero smoothness), such as imbedding properties, density of nice functions, weight problems, trace problems, extension theorems, duality theory etc. Various generalizations of these spaces are welcome, such as for example Orlicz-Sobolev spaces, in particular generalized Lebesgue-Sobolev spaces of variable order, Morrey-Sobolev spaces, Muiselak-Orlich spaces and their Sobolev counterparts etc. Other topics: any inequalities related to these spaces, properties of operators of real analysis acting in such spaces and also various applications to partial differential equations and integral equations. —Abstracts—

such that ϕ ~ k = {ϕk1 , ϕk2 , · · · , ϕkn } and G ⊂ Rn be an open parallelepiped with sides parallel to the coordinate planes. Then a. holds true the inclusion k

ϕ ~ Bθϕ~ (· · · Bθϕ~ (Lp (G)) · · · ) ,→ Bp,θ (G) | {z } k

b. under the additional assumption that there exists bounded extension operator ´ k` ´ k` S : Bθϕ~ G → Bθϕ~ Rn , holds true the equality of spaces k

ϕ ~ (G) Bθϕ~ (· · · Bθϕ~ (Lp (G)) · · · ) = Bp,θ | {z } k

with equivalence of norms. In these results: (1.) If we set ϕj (h) = hlj we get results which are in works of V.I. Burenkov. (2.) In case, when ϕ(h) satisfy additional condition ϕ(h) ↑ on (0, H] h leads to increment of smoothness using iterated norms. This is joint work with Abraham N. Abebe. ∃ > 0 :

Hardy spaces with generalized parameter Alexandre Almeida Department of Mathematics, University of Aveiro Aveiro, Aveiro 3810-193, Portugal [email protected] Hardy spaces with generalized parameter are introduced following the maximal characterization approach. As particular cases, they include the classical Hardy spaces H p and the Hardy-Lorentz spaces H p,q . Real interpolation results with function parameter are obtained. Based on them, the behavior of some classical operators is studied in this generalized setting. This talk is based on joint work with A. Caetano. ——— Iterated norms in Nikol’ski˘ı-Besov type spaces with generalized smoothness Tsegaye Gedif Ayele Department of Mathematics, Addis Ababa University P.O.Box 1176 Addis Ababa - Ethiopia. [email protected] In works of V.I. Burenkov iterated norms of Nikol’ski˘ıBesov type in spaces Bθl (· · · Bθl (Lp (Ω)) · · · ), k times iterated, were introduced. Using these norms, it was proved that every classical solution of the partial differential equation with constant coefficients is infinitely differentiable. In this paper we consider iterated norms of Nikol’ski˘ı-Besov type in spaces Bθϕ~ (· · · Bθϕ~ (Lp (Ω)) · · · ) with generalized smoothness ϕ ~ belonging to some class of functions Φ(~σ , θ) and with the norm kf k ϕ~ Bθ (· · · Bθϕ~ (Lp (Ω)) · · · ) | {z } k

and holds the following Theorem. Let 1 < p < ∞, 1 ≤ θ < ∞, ~σ = (σ1 , σ2 , · · · , σn ), ϕ ~ = (ϕ1 , ϕ2 , · · · , ϕn ) ∈ Φ(~σ , θ)

——— p(.)

Embeddings Properties of The Spaces Lw (Rn ) Ismail Aydın Sinop University, Faculty of Arts and Sciences, Department of Mathematics, 57000. Sinop-TURKIYE [email protected] We derive some of the basic properties of weighted varip(.) able exponent Lebesgue spaces Lw (Rn ) and investip2 (.) p (.) gate continuous embeddings Lw2 (Rn ) ,→ Lw11 (Rn ) with respect to variable exponents and weight functions under some conditions. ——— On the boundedness of fractional B-maximal operators in the Lorentz spaces Lp,q,γ (Rn ) Canay Aykol Ankara University, Faculty of Science, Department of Mathematics, Ankara, Tandogan 06100 Turkey [email protected] In this study, sharp rearrangement inequalities for the fractional B-maximal function Mα,γ f are obtained in the Lorentz spaces Lp,q,γ and by using these inequalities the boundedness conditions of the operator Mα,γ are found. Then, the conditions for the boundedness of the B-maximal operator Mγ are obtained in Lp,q,γ . ——— Spaces of functions of fractional smoothness on an irregular domain Oleg Besov Steklov Institute of Mathematics, Department of Function Theory, 8 Gubkina Str, Moscow 119991 Russia [email protected] In 1938 S.L. Sobolev proved his well-known embedding theorem Wpm (G) ⊂ Lq (G),

m ∈ N,

1 < p < q < ∞,

(*)

51

III.4. Spaces of differentiable functions of several real variables and applications m−

n n + ≥ 0, p q

(**)

for domains G ⊂ Rn satisfying the cone condition. Relationship (**) (which determines the maximum possible value of q in theorem (*)) is also a necessary condition for the embedding. Sobolev’s result has been extended to more general domains. Definition. Given σ ≥ 1, a domain G ⊂ Rn is said to satisfy the σ-cone condition if, for some T > 0 and 0 < κ0 ≤ 1, for any x ∈ G there exists a piecewise smooth path γ = γx : [0, T ] → G,

γ(0) = x,

|γ 0 | ≤ 1

a.e.,

——— On transformation of coordinates invariant relative to Sobolev spaces with polyhedral anisotropy Gurgen Dallakyan Russian-Armenian State University, Yerevan, Armenia [email protected] Let Rn be n -dimentional euclidean space of the points with real coordinates, N0n - the set of multi-indices.

such that dist (γ(t), Rn \ G) ≥ κ0 tσ

for

0 < t ≤ T.

The author established in (2001) that embedding (*) on a domain with the flexible σ-cone condition holds if m−

σ(n − 1) + 1 n + ≥ 0. p q

(***) s(m)

We construct two families of function spaces Lp,θ (G) s(m)

and Bp,θ (G) of fractional smoothness s > 0 on domains G satisfying the flexible σ-cone condition such that embeddings s(m)

s(m)

Wpm (G) ⊂ Lq,θ (G), s(m)

M f where M is the Hardy-Littlewood maximal operator and presetn several applications of it. In particular, we shall give some applications to the setting of the two weights problem for Calder´ on-Zygmund operators. This is a joint work with J. Soria and R. Torres.

Wpm (G) ⊂ Bq,θ (G),

Lp,θ (G) ⊂ Lq (G),

s(m)

Bp,θ (G) ⊂ Lq (G),

Definition. Nonempty polyhedron ℵ ⊂ Rn with vertices in N0n is said to be complete, if it has vertices at the origin and at all coordinate axes of N0n . Complete polyhedron ℵ is called completely regular (CR), if all the coordinates of outward normals of the noncoordinate (n−1)dimentional faces of ℵ are positive. Let ℵ ⊂ Rn - (CR) polyhedron with vertices e0 , e1 , e2 , e3 , ..., eM , where the vertices ej (j = 0 1, ..., n) lie on ˛ j ˛the j-th coordinate axe, e = (0, ..., 0) , l = max ˛ e ˛ . 1≤j≤n

For any domain Ω ⊂ Rn denote by Wpℵ (Ω) (1 < p < ∞) the Sobolev space with polyhedral anisotropy, i.e. the space of functions with finite norm X k u kℵ,Ω = k Dα ukLp (Ω) .

hold with the same loss of smoothness as in (***). ——— Rearrangement transformations on general measure spaces Santiago Boza EPSEVG, Avda Victor Balaguer s/n. Vilanova i Geltr´ u. 08800 (SPAIN) [email protected] For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer’s cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner’s symmetrization, the multidimensional case, and the discrete setting of trees. ——— Last developments on Rubio de Francia’s extrapolation theory Maria Carro Department of Applied Mathematics and Analysis, University of Barcelona, Gran Via 585, Barcelona 08007 Spain [email protected] Since in the early 80’s, J.L. Rubio de Francia developed his celebrated extrapolation theorem, this theory has been developed in order to cover many other situations such as boundedness of operator on rearrangement invariant spaces or multilinear operator. In this talk, we shall present a new estimate on the distribution function of T f in terms of the distribution of

52

α∈ℵ

Consider the m -dimentional manifold Γm ⊂ Ω . Definition. The piece σ ⊂ Γm we call ℵ -regular, if for some n -dimentional subdomain ω , σ ⊂ ω ⊂ Ω, there exist transformation of coordinates invariant relative to Wpℵ (Ω), mapping σ onto σ 0 ⊂ Rm . Remark. Note, that any bounded domain always has pieces of boundary, which are not ℵ -regular. Let Γ ∈ C s be m -dimentional manifold, where s ≥ r, N ˛ ˛ S (r = max ˛ej ˛), i.e., Γ = σk and each piece σk has 1≤j≤M

k=1

representation xik = ψi,k (x0 ) , 0

where x ∈ Gk , Gk ∈ R

n−m

i = 1, ..., m, , ψi,k ∈ C s (Gk ) .

Theorem. Let ℵ ⊂ Rn be (CR) polyhedron, Ω ⊂ Rn satisfies the week condition of rectangle, Γ ∈ C s -m dimentional manifold ( Γ ⊂ Ω). Let σ ⊂ Γ has the representation xi = ψi (x0 ) ,i = 1, ..., m ,x0 = (xm+1 , ..., xn ) ∈ G , ψi ∈ C s (G). Then the piece σ is ℵ -regular, if 1) s ≥ r ; 2) l ≥ r ; ˛ ˛ 3) for all i = 1, ..., m the condition ˛ei ˛ = l holds. ——— The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted Lpw spaces with general weights Ismail Ekincioglu Kutahya Dumlupinar University, Kutahya, Turkey [email protected] In this study, the boundedness of the the high order Riesz-Bessel transformations generated by generalized

III.4. Spaces of differentiable functions of several real variables and applications shift operator in weighted Lpwv-spaces with general weights is proved. ——— Composition Operators for Sobolev spaces of functions and differential forms Vladimir Goldshtein Department of Mathematics, Ben Gurion University of the Negev, P.O.Box 653, Beer Sheva, 84105 Israel [email protected] Composition operators for Sobolev spaces with first derivatives will be discussed. For such spaces composition operators are induced by mappings with bounded mean distortion. These classes of mappings represent a generalization of quasiconformal mappings. Applications to embedding theorems will be described. In the framework of so-called Lq,p -cohomology similar classes of mappings play an important role for quasiisometrical an/or Lipschitz classification of complete noncompact Riemannian manifolds with bounded geometry ———

For generalized potential operators with the kernel a[%(x,y)] on bounded measure metric space (X, µ, %) with [%(x,y)]N doubling measure µ satisfying the upper growth condition µB(x, r) ≤ CrN , N ∈ (0, ∞), we prove weighted estimates in the case of radial type power weight w = [%(x, x0 )]ν . Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the weighted variable exponent Lebesgue space Lp(·) (X, w, µ) into a certain 1

weighted Musielak-Orlicz space LΦ (X, w p(x0 ) , µ) with the N-function Φ(x, r) defined by the exponent p(x) and the function a(r). ——— Our talk is on vanishing exponential integrability for Besov functions. Ritva Hurri-Syrjanen Department of Mathematics and Statistics, University of Helsinki PL 68 (Gustaf Hallstrominkatu 2 b), Helsinki FI-00014 Finland [email protected]

Boundedness of the fractional maximal operator and fractional integral operators in general Morrey type spaces and some applications

Our talk is on vanishing exponential integrability for Besov functions.

Vagif Guliyev F. Agayev str, 7 Rasim Mukhtarov str, 10 Baku, Baku AZ 1069 Azerbaijan [email protected]

New sharp estimates for function in Sobolev spaces on finite Interval

The theory of boundedness of fractional maximal operator and fractional integral operators from one weighted Lebesgue space to another one is by now well studied. These results have good applications in the theory of partial differential equations. However, in the theory of partial differential equations, along with weighted Lebesgue spaces, general Morrey-type spaces also play an important role, but until recently there were no results, containing necessary and sufficient conditions on the weight functions ensuring boundedness of the aforementioned operators from one general Morrey-type space to another one (apart from the cases in which this follows directly from the appropriate results for weighted Lebesgue spaces). The case of power-type weights was well studied C.B. Morrey 1938, D.R. Adams 1975, F. Chiarenza and M. Frasca 1987, but for general Morreytype spaces only sufficient conditions were known (T. Mizuhara 1991, E. Nakai 1994, V.S. Guliyev 1994). In the talk a survey of results, containing necessary and sufficient conditions for boundedness of fractional maximal operator and fractional integral operators, will be given, and open problems will be discussed in detail. As applications, we establish the boundedness of some Sch¨ odinger type operators on general Morrey-type spaces related to certain nonnegative potentials belonging to the reverse H¨ older class. ——— Weighted estimates of generalized potentials in variable exponent Lebesque spaces Mubariz Hajibayov Institute of Mathematics and Mechanics of NAS of Azerbaijan, F.Agayev 9, Baku, Azerbaijan AZ1141 Azerbaijan [email protected]

———

Gennady Kalyabin Peoples Friendship University of Russia, MiklukhoMaklaya Str 6, Moscow, 117198 Russia [email protected] The smallest constants in new kind of Kolmogorov type inequalities for intermediate derivatives are obtained, namely: the quantities Ar,k (x) defined as sup{f (k) (x) : kf (r) kL2 (−1,1) ≤ 1; f

f (s) (±1) = 0,

s ∈ {0, . . . , r − 1}},

are calculated for all natural r, k ∈ {0, . . . , r − 1} and x ∈ (−1, 1). In particular A2r,0 (x) =

(1 − x2 )2r−1 . − 1)

Γ2 (r)22r−1 (2r

As a Corollary it is established that for the first eigenvalue of the boundary problem (−D2 )r y(x) = λy(x), y (s) (±1) = 0, s ∈ {0, . . . , r − 1}, the asymptotic formula √ λ1,r ≈ 2π 2r(2r e)2 r, r → ∞, holds. ——— On real interpolation of weighted Sobolev spaces Leili Kusainova L.N. Gumilev Eurasian National University Astana, Munaitpasov 5, Akmola 010008 Kazakhstan [email protected]

53

III.4. Spaces of differentiable functions of several real variables and applications Let 1 < p < ∞, m ∈ N, Ω ∈ Rn an open set, and let υ be a non-negative function locally integrable in Ω. We denote by Wpm (υ) the weighted Sobolev space with the finite norm

maximal operator Mβ and Dunkl type fractional integral operator Iβ on the Dunkl-type Morrey spaces Lp,λ,α (R), 1 ≤ p < ∞.

|u; Wpm (υ)| = |∇m u; Lp (Ω)| + |u; Lp (Ω; υ)|.

Image normalization of Wiener-Hopf operators in diffraction problems

In talk we describe Peetre interpolation spaces ` this ´ Wpm0 (υ0 ), Wpm1 (υ 1 θ,p for weights υi , which allow introducing local variable average characteristics. Here 0 ≤ m1 < m0 , 1 ≤ p < ∞, mi p 6= n (i = 0, 1). Let d(x) be a positive bounded function in Ω such that for some a > 1 and all x ∈ Ω Qad(x) (x) ⊂ Ω, where Qd (x) = {y ∈ Rn : |yi − xi | < d/2, i = 1, 2, ..., n}. Let Bps (υ) denotes Besov space with the finite norm (s > 0) |u; Bps (υ)| = |u; bsp | + |u; Lp (Ω; υ)|. For certain class of weights υi (i = 0, 1) we have obtained the equality of type ` m0 ´ −sp Wp (υ0 ), Wpm1 (υ1 ) θ,p = Bps ((υ ∗ ) ), where 0 < θ < 1, s = (1 − θ)m0 + θm1 , υ ∗ = max υi∗ (x) i=0,1

and

Ana Moura Santos Dept. de Matematica, IST, Av. Rovisco Pais 1, Lisbon, 1049-001 Portugal [email protected] In this work we discuss the normalization problem for Wiener-Hopf Operators (WHO), which arrives in certain ill-posed boundary-transmission value problems on halfplanes. We first consider a wave diffraction problem by a junction of two infinite half-planes, and different combinations of normal and oblique derivatives on the planes. Then a generalization for higher order derivatives follows. For all studied diffraction problems, which are associated with not normally solvable WHO, we solve the normalization problem based on the image normalization technique previously developed for one half-plane. ———

0 < υi∗ (x) = sup{d : dmi p−n υi (Qd (x)) ≤ 1} ≤ d(x). d>0

——— The Fourier transform of a radial function Elijah Liflyand Department of Mathematics Bar-Ilan University RamatGan, Gush-Dan 52900 Israel [email protected] This talk naturally consists of two parts. In the first one we survey the known results on representation of the Fourier transform of a radial function as the onedimensional Fourier transform of a related function. One of such results, due to Leray, gave an impact to obtaining a series of new such formulas. We discuss those already obtained in a joint work with S. Samko as well as tentative formulas. Correspondingly, already obtained applications are given and certain conjectures are posed. ——— Necessary and sufficient conditions for the boundedness of Riesz potential in Morrey spaces associated with Dunkl operator Yagub Mammadov Institute of Mathematics and Mechanics, Rasim Mukhtarov str. 10, Narimanov area Baku, AZ 1141 Azerbaijan [email protected] The maximal function, fractional maximal function and fractional integrals associated with the Dunkl operator were studied extensively in Lebesgue spaces on R. We study the fractional maximal function (Dunkl-type fractional maximal function) and fractional integrals (Dunkl-type fractional integrals) associated with the Dunkl operator in the Dunkl-type Morrey space Lp,λ,α (R) and Dunkl-type Besov-Morrey spaces s Bpθ,λ,α (R). We obtain the necessary and sufficient conditions for the boundedness of Dunkl-type fractional

54

———

Weighted estimates for the averaging integral operator and reverse H¨ older inequalities Bohum´ır Opic Institute of Mathematics, AS CR ˇ a 25 Zitn´ 11567 Praha 1 Czech Republic [email protected] Let 1 < p < +∞ and let v be a weight on (0, +∞) satisfying v(x)xρ is equivalent to a non-decreasing function on (0, +∞) for some ρ ≥ 0. Let A R x be the averaging operator given by (Af )(x) := x1 0 f (t) dt, x ∈ (0, +∞), and let Lp (v) denote the weighted Lebesgue space of all measurable functions f on (0, +∞) for which ”1/p “R +∞ p |f (x)| v(x) dx < +∞. 0 First, we prove that the following statements are equivalent: (i) A is bounded on Lp (v); (ii) A is bounded on Lp−ε (v) for some ε ∈ (0, p − 1); (iii) A is bounded on Lp (v 1+ε ) for some ε > 0; (iv) A is bounded on Lp (v(x)xε ) for some ε > 0. Moreover, if A is bounded on Lp (v), then A is bounded on Lq (v) for all q ∈ [p, +∞). Second, we show that the boundedness of the averaging operator A on the space Lp (v) implies that, for all r > 0, 0 the weight v 1−p satisfies the reverse H¨ older inequality over the interval (0, r) with respect to the measure dt, while the weight v satisfies the reverse H¨ older inequality over the interval (r, +∞) with respect to the measure t−p dt. Third, assuming moreover that p ≤ q < +∞ and that w is a weight on (0, +∞) such that [w(x)x]1/q ≈ [v(x)x]1/p

for all x ∈ (0, +∞),

we prove that the operator A is bounded on Lp (v) if and only if the operator A : Lp (v) → Lq (w) is bounded. ———

III.4. Spaces of differentiable functions of several real variables and applications Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup Humberto Rafeiro Universidade do Algarve, Dep. Matem´ atica, Campus de Gambelas Faro, Faro 8005-139 Portugal [email protected] In this talk we give a characterization of the variable exponent Bessel potential space in terms of the convergence of the Poisson semigroup. We show that Grunwald-Letnikov construction with respect to the Poisson semigroup coincides with the Riesz fractional differentiation under some natural restrictions on the exponent p(x). ——— On the Maxwell problem Evgeniy Radkevich Mathematics Department, Moscow State University, Vorobievy Gori, Moscow 119992 Russia [email protected]

We admit variable complex valued orders α(x), where 0, k ∈ N, k > l, −∞ ≤ a 6 b 6 ∞. Recall thatf ∈ blp,θ (a, b), if f is measurable on (a, b) and 0 kf kbl

p,θ

(a,b)

B [email protected]

b−a

Zk “

1 θ1 ”θ dh C h−l k∆kh f kLp (a,b−kh) A h

0

is finite.

TBA ———

Theorem. 1 < p, θ 6 ∞, l > 0,k ∈ N, 0 < l < k,α1 ≥ 0, α2 ≥ k. Then for an arbitrary interval (a, b)

Weighted potential operators in Morrey spaces. 0 Natasha Samko Department of Mathematics, University of Algarve, Campus de Gambelas, 8005-139 Faro, Portugal [email protected] We study the weighted (p, λ)-(q, λ)-boundedness of Hardy and potential operators. We show that the weighted boundedness of potential operators is reduced to the boundedness of weighted Hardy operators. In case of power weights or oscillating weights from the BaryStechkin class we find conditions for weighted Hardy operators to be bounded in Morrey spaces. ——— Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces Stefan Samko Universidade do Algarve Campus de Gambelas Faro, Algarve 8005-139 Portugal [email protected] We consider non-standard H¨ older spaces H λ(·) (X) of functions f on a metric measure space (X, d, µ), whose H¨ older exponent λ(x) is variable, depending on x ∈ X. We establish theorems on mapping properties of potential operators of variable order α(x), from such a variable exponent H¨ older space with the exponent λ(x) to another one with a ‘better’ exponent λ(x) + α(x), and similar mapping properties of hypersingular integrals of variable order α(x) from such a space into the space with the ‘worse’ exponent λ(x)−α(x) in the case α(x) < λ(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spaces H λ(·) (X), but also the generalized H¨ older spaces H w(·,·) (X) of functions whose continuity modulus is dominated by a given function w(x, h), x ∈ X, h > 0.

kf kbl

p,θ

(a,b)

B ∼B @

1 θ1

b−a α1 +α2

Z

“

h−l k∆kh f kLp (a+α1 h,b−α2 h)

”θ dh C C , hA

0

where the equivalence constants are independent of a and b. ——— Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications Ayhan Serbetci Department of Mathematics, Ankara University, Tandogan, Ankara 06100 Turkey [email protected] In this study we consider the fractional integral operator Iα on any Carnot group G (i.e., nilpotent stratified Lie group) in the weighted Lebesgue spaces Lp,ρ(x)β (G). We establish Stein-Weiss inequalities for Iα , and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator Iα from the spaces Lp,ρ(x)β (G) to Lq,ρ(x)−γ (G), and from the spaces L1,ρ(x)β (G) to the weak spaces W Lq,ρ(x)−γ (G) by using the Stein-Weiss inequalities. Q In the limiting case p = α−β−γ , we prove that the modified fractional integral operator Ieα is bounded from the space Lp,ρ(x)β (G) to the weighted BMO space BM Oρ(x)−γ (G), where Q is the homogeneous dimension of G. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two SobolevStein embedding theorems on weighted Lebesgue and weighted Besov spaces in the Carnot group setting. As an another application, we prove the boundedness of Iα s s from the weighted Besov spaces Bpθ,β (G) to Bqθ,−γ (G). ———

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III.5. Analytic and harmonic function spaces Translation-invariant bilinear operators with positive kernels Javier Soria Department of Applied Mathematics and Analysis, University of Barcelona, Gran Via 585, Barcelona 08007 Spain [email protected] We study the boundedness of bilinear convolutions operators with positive kernels. We prove both necessary and sufficient conditions and, by means of several counterexamples we show that near the endpoints the behavior of positive translation-invariant bilinear operators can be quite different than that of positive linear ones. ——— Sharp inequalities for moduli of smoothness and Kfunctionals Sergey Tikhonov Centre de Recerca Matem` atica, Facultat de Ci`encies, UAB, Bellaterra, Barcelona 08193 Spain [email protected] We discuss the (p − p) and (p − q) sharp inequalities (Jackson-type, Marchaud-type, Ulyanov-type, etc) for moduli of smoothness/K-functionals. Corresponding embedding theorems are studied. ——— Sobolev embedding theorems for a class of anisotropic irregular domains Boris V. Trushin MIAN (Departament of Function Theory), ul. Gubkina, d. 8, Moscow 119991 Russia [email protected] Sufficient conditions for the embedding of a Sobolev space in Lebesgue spaces and the space of continuous functions on a domain depend on the integrability and smoothness parameters of the spaces and on the geometric features of the domain. In our talk, Sobolev embedding theorems will obtaine for a class of domains with irregular boundary. This new class includes the well-known classes of σ-John domains, domains with the flexible cone condition, and their anisotropic analogs. The results can be extended to weighted spaces with power weights. ——— Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces Yusuf Zeren Department of Mathematics of Harran University, Campus of Osmanbey, SanliUrfa, Region 6300 Turkey [email protected] We obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator Mα , and the Riesz potential operator e p,λ (Rn ) to the Iα from the modified Morrey spaces L n e spaces Lq,λ (R ), 1 < p < q < ∞, and from the e 1,λ (Rn ) to the weak modified Morrey spaces spaces L e W Lq,λ (Rn ), 1 < q < ∞. ———

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III.5. Analytic and harmonic function spaces Organisers: Rauno Aulaskari, Turgay Kaptanoglu, ¨ ttya ¨ Jouni Ra Anticipated topics are normal families, complex valued function spaces and classes, function spaces and local theory of complex differential equations, composition operators between function spaces, boundary behaviour etc.; Hardy, Bergman, Bloch, Besov, Lipschitz, Fock, Qp spaces of one and several holomorphic or harmonic variables, Toeplitz, Hankel, composition, Volterra, multiplication operators, C ∗ or other algebras of such operators, Toeplitz algebras, reproducing kernel Hilbert spaces of holomorphic or harmonic functions, and other similar topics. —Abstracts— Multiplier theorem in the setting of Laguerre hypergroups and applications Miloud Assal Department of Mathematics, Faculty of Sciences of Bizerte Zarzouna, Bizerte 7021, Tunisia [email protected] In this work we study a multiplier theorem in the setting of Laguerre hypergroups and their applications to estimate the solution of Schrdinger equation in Hardy spaces. ——— Progress on finite rank Toeplitz products Boo Rim Choe Department of Mathematics, Korea University, Anamdong 5 ga 1, Seongbuk-gu, Seoul 136-713 South Korea [email protected] It has been conjectured that a product of Toeplitz operators with function symbols, either on Hardy space or Bergman space, has finite rank, then one of the factor must be the zero operator. In this talk we survey recent results towards the conjecture as well as related results. ——— Functions and operators in analytic Besov spaces Daniel Girela Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Campus de Teatinos, Universidad de M´ alaga, M´ alaga 29071 Spain [email protected] In this talk we shall focus on structural and geometric properties of the functions in analytic Besov, primarily on the univalent functions in such spaces, and in operators acting on them. ——— Square functions Maria Jose Gonzales Department of Mathematics, Casem Rio San Pedro

III.5. Analytic and harmonic function spaces Puerto Real, Cadiz 11560 Spain [email protected] We will study multiplicative versions of the usual martingale square function and of the Lusin area of a harmonic function. ——— Convolutions of generic orbital measures in compact symmetric spaces Sanjiv Gupta DOMAS, PO BOX-36 Al-Khodh-123 Sultan Qaboos University Muscat, Oman gupta s [email protected] We prove that in any compact symmetric space, G/K, there is a dense set of a1 , a2 ∈ G such that if µj = mK ∗ δaj ∗ mk is the K-bi-invariant measure supported on Kaj K, then µ1 ∗ µ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1 Ka2 K, has non-empty interior in G. ——— Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman projections ˘ lu H. Turgay Kaptanog Department of Mathematics, Bilkent University, Ankara 06800, Turkey [email protected] Weighted Bergman spaces bpq are well-known spaces of harmonic functions for which q > −1 and 1 ≤ p < ∞. Besov spaces, also denoted bpq , generalize them to all q ∈ R. Our Besov spaces consist of harmonic functions on the unit ball B of Rn so that their sufficiently highorder (t) derivatives are in Bergman spaces (q+pt > −1). We compute the reproducing kernels Rq (x, y) of the Besov spaces b2q with q ≤ −1. The kernels turn out to be weighted infinite sums of zonal harmonics, and also radial fractional derivatives of the Poisson kernel. The new kernels give rise to R generalized Bergman projections by way of Qs ϕ(x) = B ϕ(y) Rs (x, y) (1−|x|2 )s dν(y), where s ∈ R. We prove that Qs : Lpq → bpq are bounded if and only if q + 1 < p(s + 1). This requires new estimates on the integral growth of Bergman kernels near ∂B. We obtain various applications of the Qs . This is joint work with Se¸cil Gerg¨ un and A. Ersin ¨ ¨ ITAK ˙ Ureyen. The work is supported by TUB under Research Project Grant 108T329. ——— Sums of Toeplitz products on the Dirichlet space Young Joo Lee Department of Mathematics, Chonnam National University, Gwangju, Yongbongdong 500-757, South Korea [email protected] In this talk, we will consider a class of operators which contains finite sums of products of two Toeplitz operators with harmonic symbols on the Dirichlet space of the unit disk. We will give characterizations of when an operators in that class is zero or compact. Also, we solve the zero product problem for products of finitely many Toeplitz operators with harmonic symbols.

——— Weighted composition operators on weighted spaces of analytic functions Jasbir Singh Manhas Sultan Qaboos University, Department of Mathematics & Statistics, College of Science, P.O. Box 36, Al-Khod Muscat, Muscat 123 Oman [email protected] Let V be an arbitrary system of weights on an open connected subset G of CN (N ≥ 1). Let HV0 (G) and HVb (G) be the weighted locally convex spaces of analytic functions with topology defined by seminorms which are weighted analogues of the supremum norm. Let Hv0 (G) and Hvb (G) be the weighted Banach spaces of analytic functions defined by a single weight v. In this talk besides presenting the characterizations of weighted composition operators on HV0 (G) ( Hv0 (G) )and HVb (G) ( Hvb (G) ), we shall present some results pertaining to topological structures ( e.g. component structure, Isolated points, compact differences ) of weighted composition operators on the spaces H ∞ (D) and Hv0 (D) ( Hvb (D) ). ——— Superposition operators between Qp spaces and Hardy spaces Auxiliadora Marquez Departamento de Analisis Matematico, Facultad de Ciencias, Campus de Teatinos, Malaga 29071 Spain [email protected] For any pair of numbers (s, p) with 0 ≤ s < ∞ and 0 < p ≤ ∞ we characterize the superposition operators which apply the conformally invariant Qs space into the Hardy space H p and, also, those which apply H p into Qs . ——— Bounded Toeplitz and Hankel products on Bergman space Malgorzata Michalska Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] We improve the sufficient condition for boundedness of products of Toeplitz operators Tf Tg¯ on the Bergman space obtained by K. Stroethoff and D. Zheng in 1999. Using our result we give a short proof of the sufficient and necessary condition for the boundedness of Tf T1/f¯ obtained also by Stroethoff and Zheng in 2002. We consider also the products of Hankel operators Hf Hg∗ . ——— Optimal norm estimate of the harmonic Bergman projection Kyesook Nam Department of Mathematics, Hanshin University, Osansi, Gyeonggi-do 447-791 South Korea [email protected] On the unit ball of the Euclidean n-spaces, we give an optimal norm estimate for one-parameter family of operators associated with the weighted harmonic Bergman

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III.5. Analytic and harmonic function spaces projections. Using this result, we obtain an optimal norm estimate for the weighted harmonic Bergman projections. This is the joint work with Boo Rim Choe and Hyungwoon Koo. ——— Old and new on composition operators on VMOA and BMOA spaces Pekka Nieminen Dept. of Mathematics and Statistics, Univ of Helsinki, PO Box 68, Helsinki, 00014 Finland [email protected] We review various compactness characterizations for analytic composition operators acting on the spaces VMOA and BMOA, and give some new formulations. We also discuss the equivalence of weak compactness and (norm) compactness for these operators. Joint work with Jussi Laitila, Eero Saksman and Hans-Olav Tylli (Helsinki). ———

Houghton Street London, WC2A 2AE United Kingdom [email protected] We will address the question of whether a left invertible matrix with entries in certain convolution Banach algebras of measures supported in [0, +∞) can be completed to an invertible matrix with entries from the same Banach algebra. The Banach algebras we consider arise naturally in control theory as classes of inverse Laplace transforms of stable transfer functions, and the relevance of the problem of completion to an isomorphism in control theory will also be explained. ——— Multiplication operators on weighted BMOA spaces Benoit F. Sehba Department of Mathematics, University of Glasgow, G12 8QW, Glasgow, UK [email protected] We give some (test function) criteria for symbols of bounded multiplication operators for a special familly of weighted BMOA spaces in the unit ball.

On Libera and Cesaro operators Maria Nowak Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] Let H(D) denote the class of functions holomorphic in the unit disk D. The Ces` aro operator C is defined on P∞ “ 1 Pn ˆ ” n H(D) by Cf (z) = n=0 n+1 k=0 f (k) z , where P∞ ˆ n f (z) = operator L, defined n=0 f (n)z“ . The Libera P∞ P∞ fˆ(k) ” n z , can be considered by Lf (z) = n=0 k=n k+1 as an extension of the conjugate operator C ∗ defined on H(D) - the space of holomorphic functions defined in a neighborhood of D. We obtain results on Libera operator acting on known spaces of holomorphic functions in the unit disk. (Joint work with Miroslav Pavlovic) ——— Integration operators on weighted Bergman spaces Jordi Pau Departament de Matem` atica Aplicada i An` alisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, Barcelona, 08007 Spain [email protected] For an analytic function g on the unit disc, we consider the operators Z z Jg f (z) = f (ζ)g 0 (ζ)dζ. 0

We describe the boundedness and compactness of Jg on Bergman spaces with exponential weights, answering an open question posed by Aleman and Siskakis in 1997. ——— Extension to an invertible matrix in Banach algebras of measures Amol Sasane Mathematics Department, London School of Economics,

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——— Inequalities for Hardy spaces on the unit ball Pawel Sobolewski Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] In 1988 (TAMS 103(3)) D. Luecking obtained the following results for Hardy spaces H p in the unit disk D ⊂ C. The inequality Z |h(z)|p−s |h0 (z)|s (1 − |z|)s−1 dA(z) ≤ CkhkpH p D

holds for h ∈ H p , p > 0 if and only if 2 ≤ s < p + 2. We obtain analogous results for the Hardy spaces on the unit ball of Cn , n ≥ 2. ——— On the Duhamel algebras ¨ bariz Tapdıgog ˘ lu Mu Isparta Vocational School, Suleyman Demirel University, Dogu Campus Isparta, Cunur 32260 Turkey [email protected] We introduce the notion of Duhamel algebra. We prove that under some natural conditions any Banach space of analytic functions in the unit disc D is the Duhamel algebra and describe its all closed ideals. In particular, we improve some results of Wigley. ——— Toeplitz operators on Bergman spaces Jari Taskinen P.O.Box 68, Department of Mathematics and Statistics University of Helsinki Helsinki, Helsinki FI-00014 Finland [email protected] We give sufficient conditions for boundedness and compactness of Toeplitz operators in the Bergman spaces on the unit disc of the complex plane. We consider both the cases 1 < p < ∞ and p = 1. The conditions concern a

III.6. Spectral theory kind of averages of the symbol on hypebolic rectangles. The sufficient condition is also necessary in the case of positive symbols, and it thus coincides with known results in this case. An approach to the Fredholm properties of Toeplitz operators is also given. ——— Hyperbolic weighted Bergman classes Luis Manuel Tovar Department of Mathematics, Esc. Sup. de Fsica y Mat. I.P.N., Edificio 9, Unidad Prof. A.L.M. Zacatenco del I.P.N., Mexico City, 07738 Mexico [email protected] A new class of like-hyperbolic Bergman class of analytic functions in the unit complex disk is introduced, which has several interesting properties and relationships with several classical weighted spaces, like Bloch, Dirichlet and Qp . ——— Multiplicative isometries and isometric zero-divisors Dragan Vukotic Departamento de Matematicas & ICMAT, Modulo CXV, Universidad Autonoma de Madrid, Madrid, 28049 Spain [email protected] For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlettype spaces), we show that their isometric pointwise multipliers are necessarily unimodular constants. As a consequence, it follows that none of those spaces have isometric zero-divisors. We also investigate the isometric coefficient multipliers. ——— Area operators on analytic function spacess Zhijian Wu Department of Mathematics, The University of Alabama, Tuscaloosa, Alabama 35487 United States [email protected] We characterize non-negative measures µ on the unit disk D for which the area operator Aµ is bounded or compact on Hardy and Bergman spaces. ——— Composition operators on BMOA Hasi Wulan Department of Mathematics, Shantou University Shantou, Guangdong 515063 China [email protected] We give a new and simple compactness criterion for composition operators Cϕ on BMOA and the Bloch space in terms of the norms of ϕn in the respective spaces.

joensuu 80100 Finland [email protected] In this paper, we give a necessary and sufficient condition for a kind of lacunary series on the unit ball to be in Qp spaces for (m − 1)/m < p ≤ 1. The necessity is extended to more general QK spaces. This is a generalization of the result of Aulaskari, Xiao and Zhao for that on the unit disk. ——— Some results on ϕ-Bloch functions Congli Yang Yliopistokatu 7 Metria Building (Y6), Joensuu 80101 Finland [email protected] Let ϕ : [0, 1)→(0, ∞) be an increasing function, such that ϕ(r)(1 − r) → ∞, as r → 1− . An analytic function f (z) in the unit disc is said to be ϕ-Bloch function if it’s derivative satisfies |f 0 (z)| = O(ϕ(|z|)) as |z| → 1− . This paper is devoted to the study of analytic ϕ-Bloch functions. we obtain some new characterizations for ϕBloch functions are established under certain regularity conditions on ϕ. ——— Holomorphic mean Lipschitz spaces Kehe Zhu Department of Mathematics and Statistics, 1400 Washington Ave, SUNY Albany, New York 12222 United States [email protected] I will talk about the connections between holomorphic mean Lipschitz spaces and several other classes of function spaces, including Bergman spaces, Besov spaces, and Bloch type spaces. The setting is the open unit ball in C n . ——— Univalently induced closed range composition operators on the Bloch-type spaces Nina Zorboska Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2 Canada [email protected] We will show that if the closed range composition operator is univalently induced, then the inducing function has to be a disk automorphism, whenever the underlying space is a Bloch-type space B α with alpha not equal to one. The proof uses a combination of methods and results from operator theory, complex analysis and the pseudohyperbolic geometry on the unit disk. ———

——— Lacunary series and QK spaces on the unit ball

III.6. Spectral theory

Wen Xu Yliopistokatu 7 Metria Building (Y6) 3rd floor Joensuu,

Organisers: Brian Davies, Ari Laptev, Yuri Safarov

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III.6. Spectral theory Anticipated topics are: Spectral theory of differential operators. Spectra of non-self-adjoint operators. Spectral asymptotics. Scattering theory. General spectral theory and related topics.

and the coefficients Aαβ are real-valued Lipschitz continuous functions satisfying Aαβ = Aβα and the uniform ellipticity condition X Aαβ (x)ξα ξβ ≥ θ|ξ|2 |α|=|β|=m

—Abstracts— Strongly elliptic second-order systems in Lipschitz domains: surface potentials, equations at the boundary, and corresponding transmission problems. Mikhael Agranovich [email protected]

for all x ∈ Ω and for all ξα ∈ R, |α| = m, where θ > 0 is the ellipticity constant. We consider open sets Ω for which the spectrum is discrete and can be represented by means of a non-decreasing sequence of non-negative eigenvalues of finite multiplicity λ1 [Ω] ≤ λ2 [Ω] ≤ · · · ≤ λn [Ω] ≤ . . . Here each eigenvalue is repeated as many times as its multiplicity and lim λn [Ω] = ∞ . n→∞

We consider a strongly elliptic second-order system in a bounded Lipschitz domain Ω. For convenience, we assume that Ω = Ω+ lies in the standard torus T = T n and consider the system in the domain Ω− = T \ Ω too. Assuming that the Dirichlet and Neumann problems in the variational setting in Ω± are uniquely solvable in some spaces Hpσ or Bpσ , we describe properties of the surface potentials. We define these operators and derive corresponding formulas following Costabel and McLean, without using properties of the fundamental solution (but do not assume that p = 2 and that the coefficients are smooth). Main results: boundedness of the surface potentials, their invertibility at the boundary (in particular, of the single layer and hypersingular operators) in Besov spaces, and a description of their spectral properties in these spaces (including the case p 6= 2). We also describe applications to the corresponding transmission problems, general and spectral.

The aim is sharp estimates for the variation |λn [Ω1 ] − λn [Ω2 ]| of the eigenvalues corresponding to two open sets Ω1 , Ω2 with continuous boundaries, described by means of the same fixed atlas A. Three types of estimates will be under discussion: for each n ∈ N for some cn > 0 depending only on n, A, m, θ and the Lipschitz constant L of the coefficients Aαβ |λn [Ω1 ] − λn [Ω2 ]| ≤ cn dA (Ω1 , Ω2 ), where dA (Ω1 , Ω2 ) is the so-called atlas distance of Ω1 to Ω2 , |λn [Ω1 ] − λn [Ω2 ]| ≤ cn ω(dHP (∂Ω1 , ∂Ω2 )), where dHP (∂Ω1 , ∂Ω2 ) is the so-called lower HausdoffPompeiu deviation of the boundaries ∂Ω1 and ∂Ω2 and ω is the common modulus of continuity of ∂Ω1 and ∂Ω2 , and, under certain regularity assumptions on ∂Ω1 and ∂Ω2 ,

———

|λn [Ω1 ] − λn [Ω2 ]| ≤ cn meas (Ω1 ∆Ω2 ) ,

On the spectral expansions associated with LaplaceBeltrami operator

where Ω1 ∆Ω2 is the symmetric difference of Ω1 and Ω2 . Joint work with Dr P. D. Lamberti.

Shavkat Alimov Vuzgorodok National University Tashkent, 100174 Uzbekistan shavkat [email protected]

——— of

Uzbekistan,

The eigenfunction expansions associated with LaplaceBeltrami operator on n-dimensional symmetrical manifold Ω of rank 1 is considered. If the eigenfunction expansion of the piecewise smooth function, which depends on the geodesic distance from some point, converges at this point, then considered function belongs to C (n−3)/2 (Ω). This result is the generalization of the result, which was proved by M. Pinsky and W. O. Bray for geodesic ball. ——— Sharp spectral stability estimates for higher order elliptic operators Victor Burenkov Via Trieste 63, Padova University, Padova, 35121, Italy [email protected], [email protected] We consider the eigenvalue problem for the operator “ ” X Dα Aαβ (x)Dβ u , x ∈ Ω, Hu = (−1)m |α|=|β|=m

subject to homogeneous Dirichlet or Neumann boundary conditions, where m ∈ N, Ω is a bounded open set in RN

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Strong field asymptotics for zero modes Daniel Elton Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF United Kingdom [email protected] Given a magnetic potential A one can consider the existence of zero modes (or zero-energy L2 eigenfunctions) of the Weyl-Dirac operator σ.(−i∇ − tA) on R3 ; here t is a positive parameter, with the limit t → ∞ corresponding to the strong field (or, equivalently, semi-classical) regime. General O(t3 ) bounds on the number of zero modes can be obtained. These bounds can be refined to O(t2 ) asymptotics for a special class of potentials A that are constructed from potentials on R2 ; a key step involves localising the Aharonov-Casher theorem to obtain good estimates for the number of “approximate zero modes” for two-dimensional Pauli operators. ——— A universal bound for the trace of the heat kernel Leander Geisinger Universit¨ at Stuttgart, Fakult¨ at Mathematik und Physik, IADM, Pfaffenwaldring 57, Stuttgart 70569, Germany [email protected]

III.6. Spectral theory We derive a unviersal P upper bound for the trace of the heat kernel Z(t) = k e−λk t , where (λk )k∈N denote the eigenvalues of the Dirichlet Laplace Operator in an open set Ω ⊂ R2 with finite volume. The result improves an inequality of Kac and holds true without further assumptions on Ω. The proof is based on improved Berezin-LiYau inequalities with a remainder term. ——— The eigenvalues function of the family of SturmLiouville operators and its applications Tigran Harutyunyan Faculty of Math. and Mechanics of Yerevan State University, Alek Manukyan 1, Yerevan 0049 Armenia [email protected] In order to study the dependence of the eigenvalues of the Sturm-Liouville problem on parameters, defining the boundary conditions, we introduce the concept of the eigenvalues function of the family of Sturm-Liouville operators. We find the necessary and sufficient conditions for some function (of two variables) to be the eigenvalues function. Actually, we solve the direct and inverse Sturm-Liouville problems. This solution particularly includes: a) the new (more precise) asymptotic formulae for the eigenvalues and normalized constants, b) some new uniqueness theorems in the inverse problems, c) the constructive solution of the inverse problems in known and some new statements. Also we introduce the concept of the eigenvalues function of the family of Dirac operators and solve similar problems for that case. ——— Generalized eigenvectors of some Jacobi matrices in the critical case Jan Janas Sniadeckich 8 Warsaw, Warsaw 00-956 Poland [email protected] The talk will be concerned with asymptotic behavior of generalized eigenvectors of a class of Hermitian Jacobi matrices J in the critical case. The last means that the fraction qn /λn generated by the diagonal entries qn of J and its subdiagonal elements λn has the limit ±2. In other word, the limit transfer matrix as n → ∞ contains a Jordan box (=double root in terms of BirkhoffAdams theory). This is the situation where the asymptotic Levinson theorem does not work and one has to elaborate more special methods for asymptotic analysis. ——— Trace expansions for elliptic cone operators Thomas Krainer Penn State Altoona 3000 Ivyside Park Altoona, Pennsylvania 16601 United States [email protected]

I plan to report on recent joint work with Juan Gil and Gerardo Mendoza on the expansion of the resolvent trace and the heat kernel for (nonselfadjoint) elliptic operators on manifolds with conical singularities. Our approach allows for the treatment of elliptic operators A of general form without simplifying assumptions on the coefficients or the geometry near the singularities, and we achieve results for a wide range of closed extensions of A in the appropriate metric L2 -space. In particular, we obtain results for selfadjoint and nonselfadjoint extensions of Hodge-Laplacians in the presence of warped conical singularities where conventional methods that are based on separation of variables and special functions fail. ——— Stability estimates for eigenfunctions of elliptic operators on variable domains Pier Domenico Lamberti Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 Padova, Padova 35121, Italy [email protected] We prove stability estimates for the variation of resolvents and eigenfunctions of second order uniformly elliptic operators subject to homogeneous boundary conditions upon variation of the domain. We consider classes ˜ parametrized by suitable bi-Lipschitz of open sets Ω homeomorphisms φ˜ defined on a fixed reference domain Ω. We obtain estimates expressed in terms of k∇φ˜ − IkLp (Ω) for finite values of p. We apply these estimates in order to control the variation of the eigenfunctions via the measure of the symmetric difference ˜ We also discuss an application to the stability Ω M Ω. of the solutions to the Poisson problem. This is joint work with G. Barbatis and V.I. Burenkov. ——— Spectral theory of the normal operator with the spectra on an algebraic curve Oleksii Mokhonko Kyiv National Taras Shevchenko University, Volodymyrska street, 01033 Kyiv, Ukraine [email protected]

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The Jacobi (three-diagonal) structure of self-adjoint multiplication operator is well-known. Berezansky Yu.M. and Dudkin M.E. proved that similar Jacobi structure is typical not only for self-adjoint operators but also for arbitrary unitary and even for any bounded normal operators for which a cyclic vector exists. This leads to numerous applications of these objects just in the same way as it is for the classical Jacobi matrices, e.g. application to non-abelian difference-differential lattices generated by Lax equation (Golinskii L.B., Mokhonko O.A.). The following results will be presented. 1. Block Jacobi matrix of a bounded normal operator J acts in C1 ⊕C2 ⊕C3 ⊕C4 ⊕· · · . If one knows that the spectrum of J is a subset of a curve {z ∈ C : p(z, z¯) = 0}, p ∈ C[x, y] then its structure can be simplified: it acts over C1 ⊕C2 ⊕· · ·⊕Cn ⊕Cn ⊕· · · (dimension stabilization phenomenon). E.g. if a normal operator is in fact the unitary one then it acts over C1 ⊕ C2 ⊕ C2 ⊕ · · · (CMV matrix structure) and if it is self-adjoint then it acts over `2 ' C1 ⊕ C1 ⊕ · · · .

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IV.1. Pseudo-differential operators 2. The Direct Spectral Problem (the generalized eigenvalue expansion theorem) and the Inverse Spectral Problem will be presented for this type of normal operators. ——— Spectral properties of operators arising from modelling of flows around rotating bodies Jiri Neustupa Mathematical Institute of the Czech Academy of Sciences Zitna 25 Prague 1, Czech Republic 115 67 Czech Republic [email protected] We give a description of the spectrum of a Stokes-type or an Oseen-type operator which appears in mathematical models of flows of a viscous incompressible fluid around rotating bodies. The special attention is paid to the essential spectrum. The operator is considered in an Lq space. ——— New formulae for the wave operators Serge Richard Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB United Kingdom [email protected] We review some new formulae recently obtained for the wave operators of various scattering systems. Different applications of these formulae will be presented. ——— Spectral bundles Benedetto Silvestri Dipartimento di Matematica Pura ed Applicata, Universit di Padova, Via Trieste 63 Padova, 35121 Italy [email protected] In this talk I will construct certain bundles hM, ρ, Xi and hB, η, Xi of Hausdorff locally convex spaces associated to a given Banach bundle hE, π, Xi. Then I will present conditions Q ensuring the existence Q of bounded selections U ∈ M and P ∈ x x∈X x∈X Bx both continuous at a point x∞ ∈ X, such that U(x) is a C0 −semigroup on Ex and P(x) is a spectral projector of the infinitesimal generator of the semigroup U(x), for every x ∈ X. ——— Scattering theory for manifolds and the scattering length Alexander Strohmaier Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom [email protected] We define the so-called scattering length for Riemannian manifolds with cylindrical ends as the time delay that waves experience when scattered in the manifold. We show that this scattering length can be estimated

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by geometric quantities. For vector valued wave equations our estimates depend on quite involved geometric quantities like lengths of homological systoles. ——— Spectrum and wandering Yuriy Tomilov Chopina Str. 12/18 Department of Mathematics and Informatics, Nicholas Copernicus University, Torun and Institute of Mathematics, PAN, Warsaw Torun, Torun 87-100, Poland [email protected] Let T be a bounded linear operator on a Hilbert space H. A vector x ∈ H is called weakly wandering for T if there is an increasing sequence (nk ) such that the vectors T nk x are mutually orthogonal. By a well-known result due to Krengel, every unitary operator on H without point spectrum has a dense subset of weakly wandering vectors. We will present several far-reaching extensions of the Krengel result. In particular, we will show that if T is a power bounded operator on H with infinite peripheral spectrum and with empty peripheral point spectrum then the set of weakly wandering vectors for T is dense in H. Our spectral assumptions on T are in a sense best possible. This is joint work with V. M¨ uller (Prague). ——— Eigenfunctions at the threshold energies of magnetic Dirac operators Tomio Umeda Department of Mathematical Science University of Hyoto, Shosha 2167 Himeji, Hyogo 671-2201 Japan [email protected] This talk will be devoted to investigation of the eigenfunctions at the threshold` energies ±m of ´ the magnetic Dirac operator H = α · − i∇x − A(x) + mβ, where α = (α1 , α2 , α3 ) and β are Dirac matrices and m is a positive constant. It will be considered three different cases of the vector potential A to decay at infinity. In all the cases, it will be` shown that zero ´ modes of the WeylDirac operator σ · − i∇x − A(x) play crucial roles in the analysis of the eigenfunctions at the threshold of H. Here σ = (σ1 , σ2 , σ3 ) denotes Pauli matrices. It turns out that many existing works on the Weyl-Dirac operator can be utilized. Accordingly, various results on the threshold eigenfunctions of the magnetic Dirac operator H are obtained. This talk is based on joint work with Yoshimi Sait¯ o, University of Alabama at Birmingham, U.S.A. ———

IV.1. Pseudo-differential operators Organisers: Luigi Rodino, Man Wah Wong Topics related to pseudo-differential operators such as PDE, geometry, quantisation, wavelet transforms, localisation operators on groups and symmetric domains, mathematical physics, signal and image processing, among others, are the embodiment of the special session.

IV.1. Pseudo-differential operators —Abstracts— Strongly elliptic second-order systems in Lipschitz domains: Dirichlet and Neumann problems. Mikhael Agranovich [email protected] This is a survey talk. We consider a strongly elliptic second-order system in a bounded Lipschitz domain. The coefficients have minimized smoothness. The aim of the talk is to describe the investigation of the Dirichlet and (under natural additional assumptions) Neumann problems in the variational setting in the spaces Hpσ and Bpσ . The main case: the principal symbol is Hermitian. Then we can use the Savar´e approach to the analysis of the smoothness of solutions and combine it with some tools of the interpolation theory, in particular, with Shneiberg’s results on the extrapolation of the invertibility of operators. The main results: conditions for the unique solvability of the problems and some spectral results (including the case p 6= 2) for the corresponding operators. Applications to Neumann-to-Dirichlet operators. Some results are also true for general strongly elliptic systems. We compare this approach with the deep approach based on the investigation of the surface potentials and corresponding equations at the boundary (Calder´ on, Jerisson, Kenig, Verchota and many other mathematicians) in terms of the non-tangential convergence, maximal functions, Rellich-type identities, etc. ——— Generalized ultradistributions and their microlocal analysis Chikh Bouzar Department of Mathematics, Oran-Essenia University B. P. 1925 EL MNAOUER Oran, Oran 31003 Algeria [email protected] We first introduce new algebras of generalized functions containing ultradistributions. We then develop a microlocal analysis suitable for these algebras. Finally, we give an application through an extension of the wellknown H¨ ormander’s theorem on the wave front of the product of two distributions.

Following Wong’s point of view (Wong M.W., The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. Journal, vol. 34 (2005), 393404), we give a formula for the one-parameter strongly λ continuous semigroup e−tL , t > 0, generated by the generalized Hermite operator Lλ , for a fixed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L2 estimate for the solution of the initial value problem for the heat equation governed by Lλ , in terms of the Lp norm, 1 ≤ p ≤??, of the initial data. Similar results have also been derived for the generalized Landau operator A˜ which was firstly introduced by M.A. De Gosson (M.A. De Gosson, Spectral Properties of a class of generalized Landau operators, Comm. Part. Diff. Equ., 33 (2008), 2096–2104), who has studied its spectral properties. ——— Generalization of the Weyl rule for arbitrary operators Leon Cohen City University-Hunter College, 695 Park. Ave, New York, 10471 United States [email protected] The Weyl rule generally deals with two operators whose commutator is a c-number. The generalization to arbitrary operators is of importance and offers interesting and challenging mathematical issues. We review the basic ideas, present new results and discuss the unsolved problems. We also show how our generalization leads to the consideration of quasi-probability distributions for arbitrary variables. In addition to the Weyl rule we consider other rules of association between operators and symbols. ——— Sharp results for the STFT and localization operators Elena Cordero Departimento di Matematica, Universita di Torino, via Carlo Alberto 10 Torino, TO 10123 Italy [email protected]

We show that the bi-dual of the closure of C0∞ in M ∞,1 is an extension of M ∞,1 as a subalgebra of the algebra of bounded operators on L2 .

We completely characterize the boundedness on Lp spaces and on Wiener amalgam spaces of the shorttime Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a well-known STFT boundedness result on Lp spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W (Lp , Lq ) are given and their sharpness is shown. Localization operators are treated similarly. Using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for localization operators on Lp spaces and prove the optimality of our results. More generally, we prove sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.

———

———

The heat equation for the generalized Hermite and the generalized Landau operators

Fuchsian mild microfunctions with fractional order and their applications to hyperbolic equations

Viorel Catana University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest 060042 Romania catana [email protected]

Yasuo Chiba 1404-1, Katakura-cho Hachioji, Tokyo 1920982 Japan [email protected]

——— Some remarks on the Sj¨ ostrand class Ernesto Buzano Dipartimento di Matematica, Universit` a di Torino, Via Carlo Alberto 10, Torino 10123 Italy [email protected]

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IV.1. Pseudo-differential operators Kataoka introduced a concept of mildness in boundary value problems. He defined mild microfunctions with boundary values. This theory has effective results in propagation of singularities of diffraction. Furthermore, Oaku introduced F-mild microfunctions and applied them to Fuchsian partial differential equations. Based on these theories, we introduce Fuchsian mild microfunctions with fractional order. We show the properties of such microfunctions and their applications to partial differential equations of hyperbolic type. By using a fractional coordinate transform and a quantized Legendre transform, degenerate hyperbolic equations are transformed into equations with derivatives of fractional order. We present a correspondence between solutions for the hyperbolic equations and those for the transformed equations. ——— About Gevrey semi-global solvability of a class of complex planar vector fields with degeneracies Paulo Dattori da Silva Faculdade de Filosofia, Ciˆencias e Letras de Ribeir˜ ao Preto - Departamento de Fsica e Matem´ atica, Avenida dos Bandeirantes, 3900 - Monte Alegre Ribeirao Preto, Sao Paulo 14040-901 Brazil [email protected]

Heat kernel of a sub-Laplacian and Grushin type operators Kenro Furutani Department of Mathematics, Science University of Tokyo, 2641 Yamazaki Noda, Chiba 278-8510 Japan [email protected] First, I will introduce a framework of a sub-Riemannian structure which is compatible with a submersion and define Grushin type operators. My purpose is to construct heat kernel for various Grushin type operators from known heat kernel in an explicit integral form. As a typical example, I explain the original Grushin operator and its heat kernel constucted from the heat kernel on three dimensional Heisenberg group. Then as a generalization to dimension three, I define Grushin type operators on R3 , R4 and R5 from a sub-Laplacian on the 6−dimensional free nilpotent Lie group, and give their heat kernels in terms of fiber integration. Also in the case that the submersion is a covering map from the Heisenberg group to Heisenberg manifolds, I will determine the spectral zeta function for a subLaplacian on them in terms of Riemann zeta function. If possible, I will also show a heat kernel for a spherical Grushin operator on S 2 and CP 3 which come from a sub-Laplacian on S 3 or S 7 , respectively. ———

Let Ω = (−, ) × S 1 , where > 0 and S 1 is the unit circle. Let L = ∂/∂t + (a(x) + ib(x))∂/∂x,

b 6≡ 0,

(*)

be a complex vector field defined on Ω , where a and b are real-valued s-Gevrey functions on (−, ), and s ≥ 1. We will assume that Σ = {0} × S 1 is the characteristic set of L and that L is tangent to Σ. In particular, L is elliptic on Ω \Σ and (a+ib)(0) = 0. Hence, we may write (a + ib)(x) = xn a0 (x) + ixm b0 (x) in Ω , with m, n ≥ 1, and a0 , b0 smooth. In this talk we shall present results about Gevrey solvability of L, given by (*), in a neighborhood of Σ, in the following sense: there exists s0 > 1 such that for any f belonging to a subspace of finite codimension of Gs (Ω ) 0 there exists a solution, u ∈ Gs , to the equation Lu = f in a neighborhood of Σ. We will see that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability. Moreover, lost of regularity occurs. This is a joint work with Adalberto P. Bergamasco (ICMC/USP) and Marcelo R. Ebert (FFCLRP/USP). ——— Invertibility for a class of degenerate elliptic operators Julio Delgado Cra 82 Bis 49-03 Ciudad Real Cali, Valle 9999 Colombia [email protected] In this work we study fundamental solutions for a class of degenerate elliptic operators. The type of operator considered is obtained as a sum of operators of the form 2 Dx2i + x2k i Dxj . The invertibility for an operator of type 2 2k 2 Dx1 +x1 Dx2 on R2 is known, here we extend this result to higher dimensions. ———

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Time-frequency analysis of stochastic differential equations Lorenzo Galleani Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, TO 10129 Italy [email protected] Most of the stochastic processes used to model physical systems are nonstationary, and yet most of the theoretical results on stochastic processes are related to the stationary case. We consider a nonstationary random process defined as the solution of a stochastic differential equation. We first transform the stochastic equation to the Wigner spectrum domain, where we obtain a deterministic differential equation. Then, by applying the Laplace transform, we obtain the exact solution of the deterministic equation. Finally, we rewrite the general solution in a form which clarifies the structure of the nonstationary stochastic process, and which highlights the connection to the classical results obtained by Fourier analysis. ——— p

L -microlocal regularity for pseudodifferential operators of quasi-homogeneous type Gianluca Garello Universit` a di Torino, Department of Mathematics, Via Carlo Alberto 10 Torino, Torino I-107123 Italy [email protected] Pseudodifferential operators whose symbols have decay at infinitive of quasi-homogeneous are considered and their behavior on the wave front set of distributions in weighted Zygmund-H¨ older spaces and weighted Sobolev spaces in Lp framework is studied. Then microlocal properties for solutions of linear partial differential equations with coefficients in weighted ZygmundH¨ older spaces are obtained.

IV.1. Pseudo-differential operators ——— Generalized Fourier integral operators methods for hyperbolic problems Claudia Garetto Arbeitsbereich f¨ ur technische Mathematik, Universit¨ at Innsbruck Technikerstrasse 13 Innsbruck, Austria 6020 Austria [email protected] The past decade has seen the emergence of a differentialalgebraic theory of generalized functions that answered a wealth of questions on solutions to partial differential equations involving non-smooth coefficients and strongly singular data. In such cases, the theory of distributions does not provide a general framework in which solutions exist due to inherent constraints in dealing with nonlinear operations. An alternative framework is provided by the theory of Colombeau algebras of generalized functions. In this talk we solve hyperbolic equations, generated by highly singular coefficients and data, by means of generalized FIO techniques developed in the Colombeau context. Finally, we provide a careful microlocal investigation of the solution by studying the microlocal mapping properties of these operators. ——— Resolvents of regular singular elliptic operators on a quantum graph Juan Gil Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601 United States [email protected] We will discuss the pseudodifferential structure of the resolvent of regular singular differential operators on a graph. For second order operators, we give a simple, explicit, sufficient condition for the existence of a sector of minimal growth. In particular, we will discuss operators with a singular potential of Coulomb type. Our analysis is based on the theory of elliptic cone operators. ——— Hyperbolic systems of pseudodifferential equations with irregular symbols in t admitting superlinear growth for |x| → ∞. Todor Gramchev Dipartimento di Matematica e Informatica, Universit` a di Cagliari, via Ospedale 72, 09124 Cagliari [email protected] We consider hyperbolic systems of pseudodifferential equations with irregular symbols with respect to the time variable t and admitting superlinear growth for |x| → ∞. We investigate the global well-posedness of the Cauchy problem for such systems in the framework of weighted spaces which generalize the Cordes type spaces H s1 ,s2 (Rn ). ——— Analytic perturbations for special Fr´ echet operator algebras in the microlocal analysis Bernhard Gramsch Institut f¨ ur Mathematik, Universit¨ at Mainz, Staudingerweg 9, Mainz, 55099 Germany [email protected]

The symmetric Hoermander class of type (1, 1) (interesting for paradifferential operators) is included in the theory of holomorphic Fredholm functions in connection with the Oka principle. This class is known to be not spectrally invariant. But commutator methods lead to the submultiplicativity of this symmetric Fr´echet algebra. Some relations to operator algebras on singular and stratified spaces are given. Stochastic PDE lead to holomorphic operator functions on infinite dimensional domains in DFN - spaces with basis such as the distribution space S 0 of Schwartz. A series of open problems is mentioned for Fr´echet operator algebras connected to parameter dependent equations on singular rep. ramified manifolds. ——— The Cauchy problem for a paraxial wave equation with non-smooth symbols ¨ nther Ho ¨ rmann Gu Nordbergstraße 15 Fakult¨ at f¨ ur Mathematik Wien, Wien A-1090 Austria [email protected] We discuss evolution systems in L2 for Schroedingertype pseudodifferential equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. Solutions with low regularity in the operator symbol will provide a basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data. ——— Pseudodifferential operators on locally symmetric spaces ´nie Hunsicker Euge Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU United Kingdom [email protected] I will discuss recent work with D. Grieser of U. Oldenburg on the first stages of the construction of a pseudodifferential operator calculus tailored to locally symmetric spaces and other noncompact spaces with similar structures. ——— On the continuity of the solutions with respect to the electromagnetic potentials to the Schr¨ odinger and the Dirac equations Wataru Ichinose Department of Mathematical Science, Shinshu University Matsumoto, Nagano 390-8621 Japan [email protected] The initial problem to families of the Schr¨ odinger equations and the Dirac equations with the electromagnetic

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IV.1. Pseudo-differential operators potentials are studied, respectively. Assume that the solutions have the same initial data and that the electromagnetic potentials converge. Then, it is proved that the solutions of Schr¨ odinger equations and the Dirac equations with the corresponding electromagnetic potentials also converge, respectively. The proof follows from the uniqueness and the boundedness of the solutions, and the functional method, ex. the abstract Ascoli-Arzel` a theorem, which will be seen to be applied to nonlinear equations. ——— Calculus of pseudo-differential operators and a local index of Dirac operators Chisato Iwasaki Department of Mathematical Sciences, Shosha 2167 Himeji, Hyogo 671-2201 Japan [email protected] I will show a method to obtain a local index of Dirac operators. This method depends on construction of the fundamental solution to the Cauchy problem for heat equations by introducing a weight for symbols of pseudodifferential operators. ——— On the theory of type 1, 1-operators Jon Johnsen Mathematics Department, Aalborg University, Fredrik Bajers Vej 7G, Dk-9220 Aalborg Øst, Denmark [email protected] After an introduction with a brief review of celebrated contributions on type 1, 1-operators of G. Bourdaud (1983,1988) and L. H¨ ormander (1988–89), their results will be set in relation to the general definition of type 1, 1-operators, which was introduced at the ISAAC 2007 congress. Progress in the area will be described as time permits. ——— Pseudo-differential operators with discontinuous symbols and their applications Yuryi Karlovych Universidad Aut´ onoma del Estado de Morelos, Facultad de Ciencias, Av. Universidad 1001, Cuernavaca, Morelos 62209 Mexico [email protected] Applying a weighted analogue of the Litllewood-Paley theorem and the boundedness of the maximal singular integral operator S∗ related to the Carleson-Hunt theorem on almost everywhere convergence on all weighted Lebesgue spaces Lp (R, w), where 1 < p < ∞ and w ∈ Ap (R), we study the boundedness and compactness of pseudo-differential operators a(x, D) with non-regular symbols in the classes L∞ (R, V (R)) and Λγ (R, Vd (R)) on the spaces Lp (R, w). The Banach algebra L∞ (R, V (R)) consists of all bounded measurable V (R)-valued functions on R where V (R) is the Banach algebra of all functions on R of bounded total variation, and the Banach algebra Λγ (R, Vd (R)) consists of all Lipschitz Vd (R)valued functions of exponent γ ∈ (0, 1) on R where Vd (R) is the Banach algebra of all functions on R of

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bounded variation on dyadic shells. For some Banach algebras of pseudo-differential operators acting on the space Lp (R, w) and having symbols discontinuous with respect to spatial and dual variables, we construct a noncommutative Fredholm symbol calculi and give Fredholm criteria and index formulas for the operators in these algebras. Applications to algebras of generalized singular integral operators with shifts are considered. ——— On maximal regularity for parabolic equations on complete Riemannian manifolds Thomas Krainer Penn State Altoona 3000 Ivyside Park Altoona, Pennsylvania 16601 United States [email protected] In this talk I plan to demonstrate how recent advances in the theory of pseudodifferential operators lead to a method to effectively establish optimal Lp –Lq a priori estimates for solutions to parabolic equations on certain complete Riemannian manifolds. The approach is based on Weis’ functional analytic characterization of maximal regularity in terms of the Rboundedness of the resolvent. In recent work, partly in collaboration with Robert Denk (Univ. of Constance, Germany), we have shown that the approximation of resolvents of elliptic operators by parameter-dependent parametrices in suitable classes of pseudodifferential operators readily leads to the desired R-boundedness, thus to maximal regularity. In my talk I plan to survey our results and the basic underlying principles of the method. ——— On the cohomological equation in the plane for regular vector fields Roberto de Leo INFN, Complesso Universitario di Monserrato Monserrato (CA), Sardegna 09042 Italy [email protected] In this talk we present our recent results about the solvability of the equation Xf = g, where X is a vector field on the plane without zeros, in the cases when X is generic and when it is Hamiltonian with respect to some symplectic form. This work slightly generalizes a recent result of S.P. Novikov, which showed recently that a generic vector field on a compact surface, seen as a 1st order operator on the set of smooth functions, has an infinite-dimensional cokernel. Our study is also related to aspects of pseudo-differential operators on the plane. ——— p

L -boundedness and compactness of localization operators associated with Stockwell transform Yu Liu Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario M3J1P3 Canada [email protected] Localization operators associated with the Stockwell transform, with respect to the filter symbol and the windows, are a class of operators defined on Lp (R). Under

IV.1. Pseudo-differential operators suitable conditions for the symbol and the windows, the localization operators turn to be bounded and compact. ——— About transport equation with irregular coefficient and data ´ Marti Jean-Andre Campus de Schoelcher, Laboratoire CEREGMIA, Universit´e des Antilles et de la Guyane Schoelcher, Martinique B.P. 7209-97275 France [email protected] We are interested in the study of the Cauchy problem for transport equation in the formally simplified case where the coefficients α and β are discontinuous and even distributions. For the data u0 , we suppose it is a distribution and even a more singular object like δxp ⊗ δyq we will give later a generalized meaning. Then the problem is formally written as 8 ∂ ∂ < ∂ u + α ⊗ 1xy u + β ⊗ 1xy u = 0, (Pf orm ) ∂t ∂x ∂y : u| = u (= δ p ⊗ δ q .) 0

{t=0}

x

y

We remark that the product and the restriction written above are generally not defined in a distributional sense. Consequently we begin in associating to (Pf orm ) a generalized one (P`gen )´well formulated in a convenient (C, E, P) algebra A R3 and recall the definition and main properties of this generalized multiparametric factor algebra. In our case, we construct such an algebra by means of independant regularizations involving three independant parameters and obtain 8 ∂ ∂ < ∂ u+F u + G u = 0, (Pgen ) ∂t ∂x ∂y : u| = H. {t=0}

` 3´ where F` and ´ G (resp.H) are the classes in A R 2 (resp.A R ) of the families regularizing the coefficients (resp. the data). First we solve (Pgen ) and examine the existence of a solution. To study more pecisely its singularities, we refer to a generalization of the asymptotic singular spectrum defined previously and adapted here to the threeparametric case. The so-called ”(a, D0 )-singular spectrum” of u ∈ A(R3 ) propose a spectral analysis of the singularities: by means of an ”analyzing” function a we can see where and why u is not locally (associated to a section of) D0 .The localization of such singularities of u is always the ”D0 -singular support” of u, and the asymptotic causis is described by a fiber ΣX (u) (above each X = (t, x, y) ∈ R3 ) which is the complement in R3+ of a conic subset of R3+ . In our case, the D0 -singularities of the data propagate along the ”regularized characteristic Γ of the problem (Pgen )” on which the fiber ΣX (u) remains constant. This joint work of V. Dvou, M. Hasler and J.-A. Marti of Universit Antilles-Guyane. ——— The Heat Kernel of τ -Twisted Laplacian Shahla Molahajloo Department of Mathematics and statistics, York University 4700 Keele street, Toronto, Ontario M3J1P3

Canada [email protected] For a family of τ -twisted Laplacians that includes the usual twisted Laplacian when τ = 1/2, we compute the heat kernel for each τ -twisted Laplacian for [0, 1]. ——— Regularity of characteristic initial-boundary value problems for symmetrizable systems Alessandro Morando Department of Mathematics - University of Brescia, Via Valotti, 9, I-25133, Brescia, Italy [email protected] We study the initial-boundary value problem for a linear Friedrichs symmetrizable system, with characteristic boundary of constant rank. We assume the existence of the strong L2 solution satisfying a suitable energy estimate, but we do not assume any structural assumption sufficient for existence, such as the fact that the boundary conditions are maximally dissipative or the KreissLopatinski condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth. ——— Application of pseudodifferential equations in stress singularity analysis for thermo-electro-magneto-elasticity problems: a new approach for calculation of stress singularity exponents David Natroshvili Georgian Technical University 77 M.Kostava st. Tbilisi, Tbilisi 0175 Georgia [email protected] We apply the potential method and the pseudodifferential equations technique to the mathematical model of the thermo-electro-magneto-elasticity theory. We study mixed and crack type boundary value problems. Along with the existence and uniqueness questions our main goal is a detailed theoretical investigation of singularities of the thermo-mechanical and electro-magnetic fields near the crack edges and the curves where the boundary conditions change their type. In particular, the most important question is description of the dependence of the stress singularity exponents on the material parameters. We reduce the three-dimensional mixed and crack type boundary value problems of the thermo-electromagneto-elasticity to the equivalent system of pseudodifferential equations which live on proper parts of the boundary of the elastic body under consideration. We show that with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators it is possible to determine explicitly the singularity exponents for physical fields. We give an efficient method for computation of these exponents. Moreover, we establish that these exponents essentially depend on the material parameters, in general. ——— Wigner type transforms and pseudodifferential operators Alessandro Oliaro Department of Mathematics, University of Torino, Via

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IV.1. Pseudo-differential operators Carlo Alberto, 10 Torino, TO I-10123 Italy [email protected] We present some modifications of the Wigner transform (Wig), suggested by the connections of Wig with pseudodifferential operators. We analyze some properties of these representations, in particular the positivity and the behaviour with respect to the cross terms. ——— Local regularity of solutions to PDEs by asymptotic methods Michael Oberguggenberger Unit for Engineering Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria [email protected] In the nonlinear theory of generalized functions, algebras of generalized functions are commonly constructed by means of nets of smooth functions (uε )ε∈E depending on one or more parameters. Typically, these nets do not converge as ε → 0, say, but exhibit a certain asymptotic behavior. This behavior not only determines the algebras to which such an object belongs to, but may describe local regularity properties. This type of regularity theory has become increasingly important in applications to partial- and pseudodifferential operators. This presentation is devoted to a general framework – the so-called asymptotic spectrum – for measuring the asymptotic behavior in algebras of generalized functions, using asymptotic scales and various topologies. It has been developed in joint work with A. Delcroix and J.-A. Marti [Asymptotic Analysis 59(2008), 169 – 199] and forms a nonlinear alternative to the wave front set approach. Various applications to propagation of singularities as well as to regularity in Colombeau algebras and to jump discontinuities in hyperbolic systems will be given. Modern results by theory of the three dimensional Volterra type linear integral equations with singularity Nusrat Rajabov Tajik National University Rudaki Av. 17 Dushanbe, Dushanbe 734025 Tajikistan [email protected] Let Ω denote the parallelepiped Ω = {(x, y, z) : a < x < a0 , b < y < b0 , c < z < c0 }, D1 = {(x, y) : a < x < a0 , b < y < b0 , z = c}, D2 = {(x, z) : a < x < a0 , y = b, c < z < c0 }, D3 = {(y, z) : x = a, b < y < b0 , c < z < c0 }. In the domain Ω we consider the following integral equation Z x Z b Φ(t, y, z) Φ(x, s, z) Φ(x, y, z) + A dt + B ds t − a s−b a y Z z Z x Z y Φ(x, y, τ ) Φ(t, s, z) dt +E dτ + A1 ds τ −c s−b c a t−a b Z x Z z Φ(t, y, τ ) dt + B1 dτ τ −c a t−a c Z y Z z Φ(x, s, τ ) ds + C1 dτ τ −c b s−b c Z x Z y Z z Φ(t, s, τ ) dt ds +D dτ t − a s − b τ −s b c a

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——— The adiabatic limit of the Chern character Frederic Rochon Department of Mathematics, 40 St. Toronto, Ontario M5S 2E4 Canada [email protected]

George Street,

Certain spaces of pseudo-differential operators can be used as classifying spaces for K-theory. In this context, Bott periodicity can be realized by taking a certain adiabatic limit. In this talk, we will indicate how natural forms representing the universal Chern chararcter on these spaces behave under such an adiabatic limit. This a joint work with Richard Melrose. ——— Boundary value problems as edge problems

———

= f (x, y, z),

where A, B, E, A1 , B1 , C1 , D are constants, f (x, y, z) – is a given function in Ω, Φ(x, y, z) is the desired function. Some cases equation (*) investigated N. Rajabov [Ac. of Sciences Dokl. V. 409, No 6, 2006, pp.749-753]. In this lecture the general solution of the integral equation (*) is constructed, using the connection equation (*) with one dimensional integral equation of the type (*). In the case, when A1 = AB, B1 = AE, D = AC1 , then the problem is determination general solution equation (*) redused, to problems found general solution single one-dimensional integral equation and single two– dimensional integral equation of the type (*). In this basis in the case when C1 = EB and A < 0, B < 0, E < 0, find general solution equation (*) by three arbitrary functions two variables. In the case when C1 6= EB, find the solution equation (*) by means of one arbitrary function two variabe and infinity number arbitrary function one variabe. Select the cases, when equation (*) has unique solution.

(*)

Bert-Wolfgang Schulze Institute of Mathematics, University Potsdam, Am Neuen Palais 10, Potsdam, D-14469 Germany [email protected] The calculus of pseudo-differential operators on a manifold with edges can be established in such a way that standard boundary value problems (BVPs) with the transmission property at the boundary appear as a special case (up to some simple modifications). Also the case without the transmission property can be formulated as a special case of the edge calculus (as is shown in a joint paper of the author with J. Seiler, 2009). The remarkable fact here is that the symbols of the respective (classical) pseudo-differential operators are not required to be of edge-degenerate form but are only smooth up to the boundary in the usual sense. In our talk we illustrate the specific properties of that theory for the case of symbols with the anti-transmission property (recently singlet out by the author to investigate specific asymptotics of solutions). Symbols with the transmission property together with those with the anti-transmission prope rty span the full space of symbols that are smooth up to the boundary. ———

IV.1. Pseudo-differential operators Noncommutative residues and projections associated to boundary value problems

7, Chiyoda-ku Tokyo, Tokyo 102-8554 Japan [email protected]

Elmar Schrohe Institut f¨ ur Analysis, Leibniz Universit¨ at Hannover, Welfengarten 1, 30167 Hannover [email protected]

In this talk, I will consider the regularity of the solution of a nonlinear singular partial differential equation (E):

On a compact manifold X with boundary we consider the realization B = PT of an elliptic boundary problem, consisting of a differential operator P and a differential boundary condition T . We assume that B is parameterelliptic in small sectors around two rays in the complex plane, say arg λ = φ and arg λ = θ. Associated to the cuts along the rays one can then define two zeta function ζφ and ζθ for B. Both extend to meromorphic functions on the plane; the origin is a regular point. We relate the difference of the values at the origin to the associated spectral projection Πθ,φ (B) defined by Z i Πθ,φ u = λ−1 B(B − λ)−1 u dλ, u ∈ dom(B), 2π Γθ,φ where Γθ,φ is the contour which runs on the first ray from infinity to r0 eiφ for some r0 > 0, then clockwise about the origin on the circle of radius r0 to r0 eiθ and back to infinity along the second ray. ——— On maximal regularity for mixed order systems ¨ rg Seiler Jo School of Mathematics Loughborough University Loughborough, Leicestershire LE113TU United Kingdom [email protected] I will discuss some results on maximal Lp -regularity for parabolic mixed order systems based on the so-called H∞ -calculus as well as on a calculus of Volterra pseudodifferential operators. This is a joint work with R. Denk and J. Saal. ——— Dirichlet problem for higher order elliptic systems with BMO assumptions on the coefficients and the boundary Tatyana Shaposhnikova Department of Mathematics, Linkoeping Universitym Linkoeping, Ostergotland SE-58183 Sweden [email protected] Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients. The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO) which guarantee that the solution operator associated with this problem is an isomorphism. ——— Gevrey regularities of solutions of nonlinear singular partial differential equations Hidetoshi Tahara Department of Mathematics, Sophia University, Kioicho

(t∂/∂t)m u = F (t, x, {(t∂/∂t)j (∂/∂x)α u}j+|α|≤m,j 0, for t ≥ t0 . Let U be the solution of the Cauchy Problem; then kU (t, ·)kLq ≤ C(n, p) (1 + t)

1 − 1 )+s − n−1 (p 0 2 q

——— Multiple solutions for non-linear parabolic systems Q-Heung Choi Dept. of Mathematics San 68 Miryong Dong Kunsan National University , Kunsan 573-701 South Korea [email protected] We have a concern with the existence of solutions (ξ, η) for perturbations of the parabolic system with Dirichlet boundary condition ξt = −Lξ + µg(3ξ + η) − sφ1 − h1 (x, t)

in Ω × (0, 2π),

ηt = −Lη + νg(3ξ + η) − sφ1 − h2 (x, t)

in Ω × (0, 2π).

We prove the uniqueness theorem when the nonlinearity does not cross eigenvalues. We also investigate multiple solutions (ξ(x, t), η(x, t)) for perturbations of the parabolic system with Dirichlet boundary condition when the nonlinearity f 0 is bounded and f 0 (−∞) < λ1 , λn < (3µ + ν)f 0 (+∞) < λn+1 . This is joint work with Tacksun Jung. ——— Local sovability beyond condition ψ Ferruccio Colombini Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 Pisa, PI 56127 Italy [email protected]

—Abstracts—

Dt U =

This is a joint work with Sandra Lucente and Giovanni Taglialatela from University of Bari.

kU0 kH Np ,p ,

for some s0 > 0, where 1 = p−1 + q −1 , 1 < p ≤ 2 and Np ≥ n(1/p − 1/q). One can take s0 = 0 if γ = 0 and s0 = for any > 0 if γ ∈ (0, 1) and C = C(n, p, ε).

It is well known that condition ψ (PSI) is necessary and sufficient in order to have local solvability for differential (pseudo-differential) operators of principal type with coefficients sufficiently regular. We study some cases when such conditions are not satisfied. These are two joint papers with Ludovico Pernazza and Fran¸cois Treves and with Paulo Cordaro and Ludovico Pernazza. ——— Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients Daniele Del Santo Dipartimento di Matematica e Informatica, Via Valerio 12/1, Trieste, 34127 Italy [email protected] We consider the following backward parabolic equation ∂t u +

X

∂xi (ai,j (t, x)∂xj u)

i,j

+

X

bj (t, x)∂xj u + c(t, x)u = 0

(*)

j

on the strip [0, T ] × Rn 3 (t, x). We suppose that • for all (t, x) ∈ [0, T ] × Rn and for all i, j = 1 . . . n, ai,j (t, x) = aj,i (t, x);

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IV.2. Dispersive equations • there exists k > 0 such that, for all (t, x, ξ) ∈ [0, T ] × Rn × Rn , X k|ξ|2 ≤ ai,j (t, x)ξi ξj ≤ k−1 |ξ|2 ; i,j

• for all i, j = 1, . . . , n, ai,j ∈ LL([0, T ], L∞ (Rn )) ∩ L∞ ([0, T ], Cb2 (Rn )) and bj , c ∈ L∞ ([0, T ], Cb2 (Rn )), (where a ∈ LL([0, T ], L∞ (Rn )) means that the function a is Log–Lipschitz–continuous with respect to time with values in L∞ , i.e. sup t,s∈[0,T ], 0 0 there exist ρ0 , M 0 , N 0 , δ 0 > 0 such that if u ∈ E is a solution of the equation (*) with supt∈[0,T ] ku(t, ·)kL2 ≤ D and ku(0, ·)kL2 ≤ ρ0 , then sup ku(t, ·)kL2 ≤ M 0 e−N

0

| log ku(0,·)kL2 |δ

.

(joint work with Martino Prizzi, Trieste University) ——— On the loss of regularity for a class of weakly hyperbolic operators Marcello Ebert Universidade de S˜ ao Paulo, Faculdade de Filosofia, Ciˆencias e Letras, Dept. de Fisica e Matem´ atica, Av. dos Bandeirantes, 3900 Ribeir˜ ao Preto, S˜ ao Paulo 14040-901 Brazil [email protected] In this work we consider the Cauchy problem n X

aij (t)∂x2i xj u +λ(t)

i,j=1

n X

2 ci (t)∂tx u i

i=1

0

= f (x, t, u, ∂t u, λ (t)∇x u), u(x, 0) = u0 (x), ∂t u(x, 0) = u1 (x)

(*) (**)

where P is weakly hyperbolic in a neighborhood of {t = 0}, that is, the roots of p(x, t, ξ, τ ) in τ are real;

Daoyuan Fang Zhejiang University, Hangzhou, China [email protected] We consider the Zakharov system in space dimension two. We will show a L2 -concentration result for the data without finite energy, when blow-up of the solution happens, and a low regularity global well-posedness result. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given. This talk is based on some joint works with Sijia Zhong and Hartmut Pecher. ——— Wave equation in Einstein-de Sitter spacetime Anahit Galstyan Department of Mathematics, University of Texas-Pan American, 1201 West University Drive, Edinburg, Texas 78539 United States [email protected]

0

t∈[0,T 0 ]

P u = ∂t2 u−λ2 (t)

Zakharov system in infinite energy space

In this talk we introduce the fundamental solutions of the wave equation in the Einstein-de Sitter spacetime. The last one describes the simplest non-empty expanding model of the universe. The covariant d’Alembert’s operator in the Einstein-de Sitter spacetime belongs to the family of the non-Fuchsian partial differential operators. In this talk we investigate initial value problem for this equation and give the explicit representation formulas for the solutions. The equation is strictly hyperbolic in the domain with positive time. On the initial hypersurface its coefficients have singularities that make difficulties in studying of the initial value problem. In particular, one cannot anticipate the well-posedness in the Cauchy problem for the wave equation in the Einstein-de Sitter spacetime. The initial conditions must be modified to so-called weighted initial conditions in order to adjust them to the equation. We also show the Lp − Lq estimates for solutions. Thus, we have prepared all necessary tools in order to study the solvability of semilinear wave equation in the Einstein-de Sitter spacetime. This is a joint work with Tamotu Kinoshita (University of Tsukuba, Japan) and Karen Yagdjian (UTPA, U.S.A.). ———

(***)

Stability of solitary waves for Hartree type equation here p = p(x, t, ξ, τ ) is the principal symbol of P . Examples show that, differently of the hyperbolic case, under (*), (**) and (***) the solution might not exist. In addition to condition (***), various authors presented sufficient conditions, usually called Levi conditions, for the Cauchy problem to be well posed in Sobolev spaces. Those type of conditions relate p with lower order terms of P . In this work, we narrowed the bounds for the optimal Sobolevs loss of regularity under some sharp Levi conditions. This work was done in collaboration with Rafael A. dos Santos Kapp and Jos Ruidival dos Santos Filho, both from UFSCar(Brazil). ———

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Vladimir Georgiev Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 Pisa, PI 56127 Italy [email protected] We prove the stability of solitary manifold associated with the solitary solutions of Hatree type equation with external Coulomb type potential. ——— Hyperbolic-parabolic Kirchhoff-equations

singular

perturbations

for

Marina Ghisi Department of Mathematics, University of Pisa, Largo

IV.2. Dispersive equations Pontecorvo 5 Pisa, Pi 56127 Italy [email protected] We consider the second order Cauchy problem u00 + g(t)u0 + m(|A1/2 u|2 )Au = 0, u(0) = u0 , u0 (0) = u1 where > 0, g is a positive function, m is a nonnegative C 1 function, A is a self-adjoint non-negative operator with dense domain D(A) in a Hilbert space, and (u0 , u1 ) ∈ D(A) × D(A1/2 ). We prove the global solvability of the Cauchy problem under different conditions on the functions m and g, including the case where m(0) = 0, and the case where g(t) tends to 0 as t tends to +infinity (weak dissipation). We also consider the behavior of solutions as t tends to +infinity (decay estimates), and as tends to 0. ——— Existence and uniqueness results for Kirchhoff equations in Gevrey-type spaces Massimo Gobbino Dipartimento di Matematica Applicata, via Filippo Buonarroti 1c, Pisa, PI 56127 Italy [email protected] We consider the second order Cauchy problem 00

1/2

u + m(|A

2

u| )Au = 0,

u(0) = u0 ,

——— Wave equations with time dependent coefficients Fumihiko Hirosawa Department of Mathematics, Yamaguchi University, 753-8512, Japan [email protected] The total energy of the wave equation is conserved with respect to time if the propagation speed is a constant, but it is not true in general for time dependent propagation speeds. Indeed, it is considered in [F. Hirosawa, Math. Ann. 339 (2007), 819-839] that the following properties of the propagation speed are crucial for the estimates of the total energy: oscillating speed, difference from the mean, and the smoothness in C m category. The main purpose of our talk is to derive a benefit of a further smoothness of the propagation speed in the Gevrey class for the energy estimates. ———

0

u (0) = u1 ,

where m : [0, +∞) → [0, +∞) is a continuous function, and A is a self-adjoint nonnegative operator with dense domain on a Hilbert space. In this conference we present three results. • The first result is local existence for initial data in suitable spaces depending on the continuity modulus of the nonlinear term m. This spaces are a natural generalization of Gevrey spaces to the abstract setting. We also show that solutions with less regular data may exhibit an instantaneous derivative loss. • The second result concerns uniqueness in the case where the nonlinear term is not Lipschitz continuous. • The last result concerns the global solvability. Roughly speaking, we show that every initial datum in the spaces where local solutions exist is the sum of two initial data for which the solution is actually global. ——— Precise loss of derivatives for evolution type models Torsten Herrmann Faculty 1, TU Bergakademie Freiberg, Pr¨ uferstr. Freiberg, 09596 Germany [email protected]

derive results for well-posedness with a (possible) loss of regularity. On the other hand we discuss strategies how to show optimality of the results and sharpness of the assumptions. Here Floquet theory and instability arguments form the core of our strategies. We distinguish between optimality for the leading coefficients of the principal part and for coefficients of the remaining principal part.

9,

The goal of this talk is to present statements about well-posedness for Cauchy problems for degenerate pevolution equations with time-dependent coefficients. Degeneracy means that the p-evolution operators may have characteristics of variable multiplicity. On the one hand we are interested to apply phase space analysis to

Critical point theory applied to a class of systems of super-quadratic wave equations Tacksun Jung Dept. of Mathematics San 68 Miryong Dong Kunsan National University , Kunsan 573-701 South Korea [email protected] We show the existence of a nontrivial solution for a class of the systems of the super-quadratic nonlinear wave equations with Dirichlet boundary conditions and periodic conditions with super-quadratic nonlinear terms at infinity which have continuous derivatives. We approach the variational method and use the critical point theory which is the Linking Theorem for the strongly indefinite corresponding functional. This is joint work with Q-Heung Choi. ——— On the well-posdness of the vacuum Einstein equations Lavi Karp P.O. Box 78 Karmiel, Galilee 21982 Israel [email protected] The Cauchy problem of the vacuum Einstein’s equations determines a semi-metric gαβ of a spacetime with vanishing Ricci curvature Rα,β and prescribe initial data. under harmonic gauge condition, the equations Rα,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric hab and a second fundamental form Kab . However, these data must satisfy Einstein constraint equations and therefore the pair (hab , Kab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in

73

IV.2. Dispersive equations one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The aim of our work is to resolve this incompatibility and to show well-posedness of the reduced Einstein vacuum equations in one type of Sobolev spaces. ——— Generalized wave operator for a system of nonlinear wave equations Hideo Kubo Graduate School of Information Sciences, Tohoku University 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai , Miyagi 980-8579 Japan [email protected] In this talk we discuss the asymptotic behavior of solutions to a system of nonlinear wave equations whose decaying rate is actually slower than that of the free solutions. Desipte of that fact, we are able to construct wave operators in a generalized sense. The proof is done by finding a nice approximation and introducing a suitable metric (that is not a norm in fact). Moreover, the sacttering operators are defined in a generarized sence.

with Dirichlet boundary conditions w(x, 0, t) = w(x, π, t) = 0,

(x, t) ∈ RN × (0, ∞).

For long-range type of dissipations, e.g., b0 (1 + |x|)−1 ≤ b(x, y) ≤ b1 in RN × [0, π] for some b0 , b1 > 0, the total energy decays as t goes to infinity. For short-range type of dissipations, e.g., 0 ≤ b(x, y) ≤ b2 (1 + |x|)−1−δ in RN × [0, π] for some b2 > 0 and δ > 0, scattering solution exists. Although the proof for scattering is based on Kato’s smooth perturbation theory, the singular points called thresholds in the spectrum cause to difficulty. To eliminate this, density argument using some approximate operators are employed. This is joint work with Mitsuteru Kadowaki (Ehime University) and Kazuo Watanabe (Gskushuin University). ——— On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 well-posedness Tatsuo Nishitani Machikaneyama-cho 1-1 Toyonaka, Osaka 560-0043 Japan [email protected]

———

In this talk we will present Strichartz estimates for higher oder hyperbolic equations in an exterior domain outside a star-shaped obstacle.

The Cauchy problem for non-effectively hyperbolic operators is discussed in the Gevrey classes. Our operators belong to the class of non-effectively hyperbolic operators with symbols vanishing of order 2 on a smooth submanifold of codimension 3 on which the canonical symplectic 2-form has a constant rank. Assuming that there is no null bicharacteristic issuing from a simple characteristic point and landing tangentialy on the double characteristic manifold, we prove that the Cauchy problem is Gevrey s well-posed for any lower order term whenever 1 ≤ s < 4.

———

———

Uniform resolvent estimates and smoothing effects for magnetic Schr¨ odinger operators

On the structure of the material law in a linear model of poro-elasticity

Kiyoshi Mochizuki Department of Mathematics, Chuo University, Kasuga, Bunnkyo 1-13-27, Tokyo 112-8551 Japan [email protected]

Rainer Picard Institut f¨ ur Analysis, FB Mathematik,TU Dresden, 01062 Dresden, Germany [email protected]

Uniform resolvent estimates for magnetic Schr¨ odinger operators in an exterior domain are obtained under smallness conditions on the magnetic fields and scalar potentials. The results are then used to obtain spacetime L2 -estimates for the corresponding Schr¨ odinger, Klein-Gordon and wave equations.

A modification of the material law associated with the well-known Biot system first investigated by R.E. Showalter is re-considered in the light of a new approach to a comprehensive class of evolutionary problems.The particular material law is of the form

Strichartz estimates for hyperbolic equations in an exterior domain Tokio Matsuyama Tokai University 1117 Kitakaname Hiratsuka, Kanagawa 259-1292 Japan [email protected]

T = (C + trace∗ λ trace ∂0 ) E − trace∗ α p

——— Decay and scattering for wave equations with dissipations in layered media Hideo Nakazawa Chiba Institute of Technology, Narashino, Chiba 275-0023 Japan [email protected]

Shibazono

2-1-1

We consider wave equations with linear dissipations in some layered regions;

connecting the stress tensor T with strain tensor E and fluid pressure p via parameters λ, α and C as the isotropic elasticity tensor. Here ∂0 denotes the time derivative and trace the matrix trace operation with trace∗ as its adjoint. This model is generalized to anisotropic media and well-posedness of the generalized model is shown. ——— Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuous coefficients

wtt (x, y, t) − ∆w(x, y, t) + b(x, y)wt (x, y, t) = 0, (x, y, t) ∈ RN × [0, π] × (0, ∞)

74

Marco Pivetta Dipartimento di Matematica e Informatica, Via Valerio

IV.2. Dispersive equations 12/1, Trieste, Italy 34127 Italy [email protected]

Japan [email protected]

Using the Carleman estimates developed by Koch and Lasiecka [Functional analysis and evolution equations, 389-403, Birkh¨ auser, Basel, 2008] together with an approximation technique in the phase space, a uniqueness result for the backward Cauchy problem is proved for the system of themoelastic waves having coefficients which are in a class of log-Lipschitz-continuous functions.

We consider nonnegative solutions of the Cauchy problem for quasilinear parabolic equations

———

where m > 1 and f (ξ) is a positive function R ∞in ξ > 0 satisfying f (0) = 0 and a blow-up condition 1 1/f (ξ) dξ < ∞. We study under what conditions on f (ξ) all nontrivial solutions blow up. ———

The log-effect for 2 by 2 hyperbolic systems Michael Reissig Faculty 1, TU Bergakademie Freiberg, Pr¨ uferstr. Freiberg, 09596 Germany [email protected]

ut = ∆um + f (u),

9,

In the talk we are interested to explain how to extend the Log-effect from wave equations with timedependent coefficients to 2 by 2 strictly hyperbolic systems ∂t U − A(t)∂x U = 0. From wave models we know that besides oscillations in the coefficients a possible interaction of oscillations has a strong influence on H ∞ well- or ill-posedness. Moreover, the precise loss of derivatives can be proved. In the case of systems the situation is more complicate. Besides the effects of oscillating entries of the matrix A = A(t) and interactions between the entries of A we have to take into consideration the system character itself. We will prove by using tools from phase space analysis results about H ∞ well- or ill-posedness. The precise loss of regularity is of interest. Moreover, we discuss the question if the loss of derivatives does really appear. These considerations base on the interplay between the Ljapunov and energy functional. Finally, we discuss the cone of dependence property for solutions to 2 by 2 systems. This is a joint talk with T.Kinoshita (Tsukuba). ——— The Boussinesq equations based on the hydrostatic approximation

Blow-up and a blow-up boundary for a semilinear wave equation with some convolution nonlinearity Hiroshi Uesaka Department of Mathematics, College of Science and Technology, Nihon University, Tokyo 101-8308, Chiyodaku Kanda Surugadai 1-8, Japan [email protected] We consider the Cauchy problem with a convolution nonlinearity, (∂t2 − 4)u = uq (V ∗ up ), in R3 × (0, T ), (0.1) u(x, 0) = f (x), ∂t u(x, 0) = g(x) in R3 , R p (y,t) where uq (V ∗ up )(x, t) = uq (x, t)( R3 u|x−y| γ dy) with p, q > 1, 0 < γ < 3. The blow-up boundary is defined by Γ = ∂{u < ∞} ∩ {t > 0}. We can give several suitable conditions to initial data to show that 1. the Cauchy problem has a classical positive realvalued local solution u, 2. u is monotone increasing in t for any fixed x and moreover satisfies ∂t u ≥ |∇u|, 3. there exists a positive T (x) for any x such that u keeps its regularity in (0, T (x)) and limt%T (x) u(x, t) = ∞ .

Jun-ichi Saito Minamisenju 8-17-1 Arakawa-ku, Tokyo 116-0003 Japan j [email protected]

Then the blow-up boundary Γ exists and T (x) satisfies |T (x) − T (y)| ≤ |x − y|.

The Boussinesq equations is studied in the field of dynamic meteorology. Atmospheric flow in meteorology are described by the Boussinesq equations. Due to the fact that the aspect ratio

Fundamental solutions for hyperbolic operators with variable coefficients

ε=

characteristic depth characteristic width

is very small in most geophysical domains, asymptotic models have been used. One of the models is the hydrostatic approximation of the Boussinesq equations. We consider the Boussinesq equations in the domains with very small aspect ratio and prove the convergence theorem for this model. ——— Blow-up of solutions of a quasilinear parablolic equation Ryuichi Suzuki School of Science and Engineering, Kokushikan University, 4-28-1 Setagaya, Setagaya-ku Tokyo, 154-8515

———

Karen Yagdjian Department of Mathematics, University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78541-2999, USA [email protected] The goal of this talk is to give a survey of a new approach in the constructing of fundamental solutions for the partial differential operators with variable coefficients and of some resent results obtaining by that approach. This new approach appeals neither to the Fourier transform, nor to the Microlocal Analysis, nor to the WKBapproximation. More precisely, the new integral transformation is suggested which transforms the family of the fundamental solutions of the Cauchy problem for the operator with the constant coefficients to the fundamental solutions for the operators with variable coefficients.

75

IV.3. Control and optimisation of nonlinear evolutionary systems The kernel of that transformation contains Gauss’s hypergeometric function. This approach was applied by the author and his coauthors, T.Kinoshita (University of Tsukuba) and A.Galstyan (University of Texas-Pan American), to investigate in the unified way several equations such as the linear and semilinear Tricomi and Tricomi-type equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the KleinGordon equations in the de Sitter and anti-de Sitter spacetimes. The listed equations play important role in the gas dynamics, elementary particle physics, quantum field theory in the curved spaces, and cosmology. In particular, for all above mentioned equations, we have obtained representation formulas for the initial-value problem, the Lp − Lq -estimates, local and global solutions for the semilinear equations, blow up phenomena, selfsimilar solutions and number of other results.

cavities, control of turbulence), geophysics (reconstruction of seismic data) and others. Recent years have witnessed rapid development of new mathematical tools in both analysis and geometry that allow to obtain various PDE estimates of inverse type. These are enabling to establish properties such as controllability, reconstruction of the data, stabilisation or optimal feedback control.

———

The model under consideration is the semilinear wave equation with supercritical nonlinear sources and damping terms and the aim is to discuss the wellposedness of the system on finite energy space and the long-time behavior of solutions. A distinct feature of the equation is the presence of the double interaction of source and damping, both in the interior of the domain and on the boundary. Moreover, the nonlinear boundary sources are driven by Neumann boundary conditions. Since Lopatinski condition fails to hold for dimension greater or equal than 2, the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves strong interaction between the source and the damping. I will provide positive answers to the questions of local existence and uniqueness of weak solutions and moreover give complete and sharp description of parameters corresponding to global existence and blow-up of solutions in finite time. I will also discuss asymptotic energy-decay rates and blow-up of solutions originating in a potential well.

Global existence in Sobolev spaces for a class of nonlinear Kirchhoff equations Borislav Yordanov Borislav Yordanov, 226 Swain Ct, Belle Mead, NJ 085024239 United States [email protected] The nonlinear Kirchhoff equation utt − m(k∇uk2L2 )∆u = 0 is studied for initial data (u, ut )t=0 = (u0 , u1 ) in the Sobolev spaces H s (Rn ) × H s−1 (Rn ) with s ≥ 2 and for smooth perturbations m(ρ) of the Pokhozhaev function m0 (ρ) = (k1 ρ + k0 )−2 with k0 , k1 > 0. Global existence is shown when ku1 kL2 is large and m is close to m0 in a suitable metric. Moreover, the asymptotic behavior of solutions is found as t → ±∞. It turns out that the norms k∇ukL2 grow like |t|, so the propagation speeds decrease like t−2 and the waves remain trapped in bounded regions. This is joint work with Lubin Vulkov. ———

IV.3. Control and optimisation of nonlinear evolutionary systems Organisers: Francesca Bucci, Irena Lasiecka The session is focused on new developments in the area of well-posedness, optimisation, and control of systems described by evolutionary partial differential equations. These include: non-linear wave and plate equations, Navier-Stokes and Euler equations, non-linear thermoelasticity, viscoelasticity and electromagnetism. Of particular interest to the session are interacting systems that involve PDE’s of different type describing the dynamics on two (or more) separate regions with a coupling on an interface between these regions. Particular examples of such systemsare: structural acoustic interactions, fluid structure interactions, magnetostructure interactions. These have a wide range of applications that include medicine (diagnostic imaging such as MRI, ultrasound), engineering (noise reduction in an acoustic

76

—Abstracts— Global well-posedness and long-time behavior of solutions to a wave equation Lorena Bociu University of Nebraska-Lincoln, Department of Mathematics, 203 Avery Hall Lincoln, NE 68588 United States [email protected]

——— Distributed optimal controls for second kind parabolic variational inequalities Mahdi Boukrouche LaMUSE Saint-Etienne University, 23 Rue Dr Paul Michelon, Saint-Etienne, 42023, France [email protected] Let ugi be the unique solution of a second kind parabolic variational inequality with second member gi (i = 1, 2). We establish, in the general case, the error estimate between u3 (µ) = µug1 + (1 − µ)ug2 and u4 (µ) = uµg1 +(1−µ)g2 for µ ∈ [0, 1], and prove a monotony property between u3 (µ) and u4 (µ) using a regularization method. For a given constant M > 0, and the cost functional we establish the existence of solutions for a family of control problems, over the the external force g for each parameter h > 0. Using the monotony property between u3 (µ) and u4 (µ), we establish the uniqueness of the solution for each control problem of the above family. We prove also the convergence of the optimalcontrols and states associated to this family of control problems governed by a second kind parabolic variational inequalities. ———

IV.3. Control and optimisation of nonlinear evolutionary systems Controllability of a fluid-structure interaction problem Muriel Boulakia 175 rue du Chevaleret, 75013 Paris, France [email protected]

In addition, we investigate the existence of uniform decay rates for both, the Airy type equation ut + uxxx + g(u) = 0, in [0, L] × (0, +∞),

We are interested by the controllability of a fluidstructure interaction problem. The fluid in governed by the incompressible Navier-Stokes equations and a rigid structure is immersed inside. The control acts on a fixed subset of the fluid domain. For small initial data, we prove that this system is null controllable i.e. that the system can be driven at rest. This result is obtained with the help of a Carleman inequality proven for the adjoint linearized system. ——— Uniform decay rate estimates for the wave equation on compact surfaces and locally distributed damping Marcello Cavalcanti Department of Mathematics - State University of Maringa, Av. Colombo 5790, Maringa, PR 87020-900 Brazil [email protected] In this talk we present new contributions concerning uniform decay rates of the energy associated with the wave equation on compact surfaces subject to a dissipation locally distributed. We present a method that gives us a sharp result in what concerns of reducing arbitrarily the area where the dissipative effect lies. ——— Rate of decay for non-autonomous damped wave systems Moez Daoulatli ISSATS, University of Sousse (& LAMSIN) Cit´e Taffala (Ibn Khaldoun), Sousse 4003 Tunisia [email protected] We study the rate of decay of solutions of the wave systems with time dependent nonlinear damping which is localized on a subset of the domain. We prove that the asymptotic decay rates of the energy functional are obtained by solving nonlinear non-autonomous ODE. ——— On qualitative aspects for the damped Korteweg-de Vries and Airy type equations

posed in a bounded interval [0, L] and supplemented by a nonlinear damping g(u). By considering suitable assumptions on g and on the initial data, general decay rates are proved in L2 − level as well as exponential decay rates are established in H 1 −level. ——— Optimal control of waves in anisotropic media via conservative boundary conditions Matthias Eller Department of Mathematics, Georgetown University, Washington, DC 20057 United States [email protected] An optimal boundary control problem for symmetric hyperbolic systems is considered. The quadratic cost functional is of tracking type and the control acts through a conservative boundary condition. Some loss of regularity is associated with these boundary conditions. This results in certain choices for the underlying function spaces in the cost functional. The loss of regularity occurs only near the boundary and it may be attributed to the occurrence of surface waves. As examples we consider the anisotropic Maxwell equations as well as the anisotropic equations of elasticity. ——— Stability for some nonlinear damped wave equations Genni Fragnelli Dipartimento di Ingegneria dell’Informazione, Universit` a degli Studi di Siena, via Roma 56, c.a.p. 53100 [email protected] We prove stability results for a large class of abstract nonlinear damped wave equations, whose prototype is the usual wave equation 8 in (0, +∞) × Ω, < utt + h(t)ut = ∆u + f (u) u(t, x) = 0 in (0, +∞) × ∂Ω, : u(0, x) = u0 (x), ut (0, x) = u1 (x) x ∈ Ω, where Ω is a bounded and smooth domain of RN , N ≥ 1, u0 ∈ H01 (Ω), u1 ∈ L2 (Ω) and f : R → R. At first, the damping is nonnegative, but it is allowed to be zero either on negligible sets or even in a sequence of intervals. Then, also the case of a positive–negative damping is treated. ———

Valeria Domingos Cavalcanti Department of Mathematics - State University of Maringa, Av. Colombo 5790, Maringa, PR 87020-900 Brazil [email protected] This talk is concerned with the study of the damped Korteweg-de Vries equation posed in whole real line ut + uxxx + uux + λ u = 0, in R × [0, +∞),

λ > 0.

We establish two invariant subsets of H 1 (R) where just one of the following statements holds: (i) solutions decay exponentially in H 1 − level or (ii) solutions do not decay to zero in H 1 − level as t goes to infinity.

Global existence for the one-dimensional semilinear Tricomi-type equation Anahit Galstyan Department of Mathematics, University of Texas-Pan American, 1201 W. University Dr., Edinburg 78541, TX, U.S.A. [email protected] In this talk the issue of global existence of the solutions of the Cauchy problem for one-dimensional semilinear weakly hyperbolic equations, appearing in the boundary value problems of gas dynamics is investigated. We solve the Cauchy problem trough integral equation and give some sufficient conditions for the existence of the global

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IV.3. Control and optimisation of nonlinear evolutionary systems weak solutions. The necessary condition for the existence of the similarity solutions for the one-dimensional semilinear Tricomi-type equation will be presented as well. Our approach is based on the fundamental solution of the operator and the Lp − Lq estimates for the linear Tricomi equation. ——— Optimal control of a thermoelastic structural acoustic model Catherine Lebiedzik Department of Mathematics, Wayne State University, 656 W Kirby Detroit, MI 48202 United States [email protected] We consider point control of a structural acoustic model with thermoelastic effects. The key feature of this paper is that the two-dimensional plate modeling the active wall of the acoustic chamber has clamped boundary conditions. For this case a new optimal regularity result has recently become available. Using this new result for the plate alone, we derive a sharp regularity result for the overall coupled system of wave and thermoelastic plate equations. This allows for the study of optimal control of the coupled system. ——— The Balayage method: Boundary control of a thermoelastic plate Walter Littman University of Minnesota, School of Mathematics, 206 Church Street, Southeast Minneapolis, Minnesota 55455 United States [email protected] We discuss the null boundary controllablity thermo-elastic plate. The method employs ing property of the system of PDEs which boundary controls to be calculated directly two Cauchy problems.

of a linear a smoothallows the by solving

——— Hopf-Lax type formulas and Hamilton-Jacobi equations Paola Loreti Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, via Antonio Scarpa n.16, 00161 Roma, Italia. [email protected] Here we discuss Hopf-Lax type formulas related to the class of Hamilton-Jacobi equations ut (x, t) + αxDu(x, t) + H(Du(x, t)) = 0, N

in R × (0, +∞) with initial condition u(x, 0) = u0 in Rn , with α a positive, real number. The talk is based on some joint works with A. Avantaggiati. ——— Investigation of boundary control problems by on-line inversion technique Vyacheslav Maksimov S.Kovalevskaya 16 Ekaterinburg, Sverdlovsk 620219

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Russia [email protected] A number of applied studies address such fundamental issues as (i) reconstruction of uncertain parameters of multidimensional dynamical systems and (ii) control of uncertain dynamical systems. We discuss a technical approach intended to help solve problems of this kind. The approach employs the on-line inversion theory adjoining theory of closed-loop control and theory of ill-posed problems. On-line inversion algorithms involve artificially designed dynamical models whose parameters track non-observable parameters of the system; it is important that the tracking quality is insensitive to perturbations in the observation channels. In combination with appropriate closed-loop regulators, on-line parameter tracking algorithms give raise to robust observer-controller patterns allowing one to guide the uncertain system close to the trajectories designed via an optimal feedback to a complete set of observed signals. The goal of this report is to demonstrate the essence and abilities of the on-line inversion technique; for this purpose we consider three types of problems, namely, a problem of etalon motion tracking, a problem of game control, and a problem of dynamical input identification for a parabolic equation with the Neumann and Dirichlet boundary condition. ——— Null controllability properties of some degenerate parabolic equations Patrick Martinez Universit´e Paul Sabatier Toulouse III, Institut de Math´ematiques, 118 route de Narbonne, Toulouse, 31062 France [email protected] Motivated by several problems in fluid dynamics, biology, or economics, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. After considering the one dimensional case, this talk will mainly focus on the N-dimensional case: ut − div(A(x)∇u) = f (x, t)χω (x),

x ∈ Ω, t > 0

where ω ⊂ Ω and the matrix A(x) is definite positive for all x ∈ Ω, and but has at least one eigenvalue equal to 0 for all x ∈ ∂Ω. Mainly, we assume that - the least eigenvalue of the matrix A(x) behaves as d(x, ∂Ω)α , where α ≥ 0, - the degeneracy occurs in the normal direction: when x ∈ ∂Ω, the associated eigenvector is the unit outward vector. When α ∈ [0, 2), we prove the null controllability via new Carleman estimates for the adjoint degenerate parabolic equation. When α ∈ [2, +∞), we prove that the problem is not null controllable, using earlier results of Micu ˇ ak related to nondegenZuazua, Escauriaza-Seregin-Sver´ erate parabolic equations in unbounded domains. These results were obtained in collaboration with P. Cannarsa (Univ Tor Vergata, Roma 2), and J. Vancostenoble (Univ Toulouse 3). ———

IV.3. Control and optimisation of nonlinear evolutionary systems Dissipation in contact problems: an overview and some recent results Maria Grazia Naso Dipartimento di Matematica, Universit` a degli Studi di Brescia, Via Valotti, 9 Brescia, BS 25133 Italy [email protected] In this talk we investigate the longtime behaviour of a dynamic unilateral contact problem between two thermoelastic beams. Under suitable mechanical and thermal boundary conditions the evolution problem is shown to possess an energy decaying exponentially to zero, as time goes to infinity. ——— Heat equations with memory: a Riesz basis approach Luciano Pandolfi Politecnico di Torino, Dipartimento di Matematica, C.so Duca degli Abruzzi 24, Torino, 10129 Italy [email protected]

and on its boundary combined with a nonlinear coupled boundary condition. Such problems arise from free boundary value problems as the Stefan problem with surface tension after a suitable transformation. Besides local well posedness and smoothing properties, we focus on the qualitative behavior near an equilibrium. To that purpose we construct locally invariant manifolds and establish their main properties. ——— On regularity properties of optimal control and Lagrange multipliers Ilya Shvartsman Dept. of Mathematics and Computer Science, 777 W. Harrisburg Pike Middletown, PA 17110 United States [email protected] In this talk we will go over classical and recent results on regularity properties (such as continuity, Holder and Lipschitz continuity) of optimal controls and Lagrange multipliers. ———

In this talk we present recent results concerning a Riesz basis approach to the heat equation with memory Z t θt (x, t) = N (t − s) [∆θ(x, s) − q(x)θ(x, s)] ds 0

(x ∈ [0, π]) and square integrable initial conditions. We shall construct a special sequence {zn (t)} associated to this equations and we shall prove that it is a Riesz sequence on a suitable interval [0, T ], using Bari Theroem. These results are applied to the study of control/observability problems. ——— A note on a class of observability problems for PDEs Michael Renardy Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123 United States [email protected] The question of observability arises naturally in the analysis of control problems. If the solution of a PDE initialboundary value problem is known to be zero in a part of the domain, does this guarantee it is zero everywhere? The most popular techniques to establish such results are based on local unique continuation results (Holmgren’s theorem) or Carleman estimates. The lecture will draw attention to a class of problems where the observed region is bounded by characteristics, and local unique continuation fails. Nevertheless, observability may hold. A problem of this nature arose in recent work by the author on control of viscoelastic flows. ——— Invariant manifolds for parabolic problems with dynamical boundary conditions Roland Schnaubelt University of Karlsruhe, Department of Mathematics, Kaiserstrasse 89, Karlsruhe, 76128 Germany [email protected] We study a class of nonlinear parabolic systems described by coupled evolution equations on a domain

Evolution equations with memory terms Daniela Sforza Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, via Antonio Scarpa 16, Roma, 00161 Italy [email protected] The purpose of the talk is to show some results concerning control problems for integro-differential equations of hyperbolic type. More precisely, we consider non-linear equations in Hilbert spaces with integral convolution terms and assume the corresponding kernels to exhibit a polynomial or exponential decay. We show that the solutions have the same decay behaviour as the kernel. Our main tool is the multipliers method and we succeed in finding suitable multipliers which work even in the presence of integral terms. Besides, we provide a reachability result for a class of linear integro-differential problems. Our strategy is founded on the so-called Reachability Hilbert Uniqueness Method, introduced by Lagnese - Lions, which amounts to proving Ingham type inequalities for the Fourier series expansion of the solution of the adjoint problem. To conclude, we observe that our abstract results may be used to treat some problems arising in the study of viscoelastic systems. ——— Stabilization of structure-acoustics interactions for a Reissner-Mindlin plate by localized nonlinear boundary feedbacks Daniel Toundykov University of Nebraska-Lincoln Department of Mathematics, 203 Avery Hall Lincoln, NE 68588 United States [email protected] This work addresses observability and energy decay for a structural-acoustics model comprised of a wave equation coupled with a Reissner-Mindlin plate. Both components of the dynamics are subject to localized boundary damping: the acoustic dissipative feedback is restricted to the flexible boundary and only a portion of the rigid

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IV.4. Nonlinear partial differential equations wall; the plate is likewise damped on a segment of its boundary. The derivation of the “coupled” stabilization/observability inequalities requires weighted energy multipliers related to the geometry of the domain, and special tangential trace estimates for the displacement and the filament angles of the Reissner-Mindlin plate model. The behavior of the energy at infinity can be quantified by a solution to an explicitly constructed nonlinear ODE. The nonlinearities in the feedbacks may include sub- and super-linear growth at infinity, in which case the decay scheme presents a trade-off between the regularity of solutions and attainable uniform decay rates of the finite-energy. ——— Exponential stability of the wave equation with boundary time varying delay Julie Valein Universit´e de Valenciennes et du Hainaut-Cambr´esis LAMAV - ISTV2 - Le Mont-Houy Valenciennes, NordPas de Calais 59313 France [email protected] We consider the wave equation with a time-varying delay term in the boundary condition in a bounded domain Ω ⊂ Rn with a boundary Γ of class C 2 . We assume Γ = ΓD ∪ ΓN , with ΓD ∩ ΓN = ∅, ΓD 6= ∅, and we consider 8 > > utt (x, t) − ∆u(x, t) = 0 in Ω × (0, +∞) > > u(x, t) = 0 on ΓD × (0, +∞) > > > > (x, t) = −µ1 ut (x, t) − µ2 ut (x, t − τ (t)) < ∂u ∂ν on ΓN × (0, +∞) > > > u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω > > > ut (x, t − τ (0)) = f0 (x, t − τ (0)) > > : in ΓN × (0, τ (0)), (*) where τ (t) is the delay, µ1 , µ2 > 0. We assume 0 ≤ τ (t) ≤ τ , ∀t>0

0

τ (t) ≤ d < 1, and τ ∈ W 2,∞ ([0, T ]),

Under µ2

0.

1 − dµ1 ,

we prove the existence and uniqueness results of (*) by using the variable norm technique of Kato and we show the exponential decay of an appropriate energy. Due to the time-dependence of the delay, we can not use an observability estimate since the system is not invariant by translation in time. Hence we introduce a Lyapunov functional. We extend this result to a nonlinear version of the model. This is a joint work with Serge Nicaise and Cristina Pignotti. ——— State estimation for some parabolic systems Masahiro Yamamoto University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba Meguro Tokyo 153, Japan [email protected]

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For some types of parabolic systems, we consider inequalities of Carleman’s type and prove conditional stability estimates for state determination problems such as a backward problem. ——— Euler flow and Morphing Shape Metric Jean-Paul Zolesio CNRS-INLN nd INRIA RTE Lucioles 1361 and 2004 Sophia Antipolis, France 06565 France [email protected] We extend the so-called ”Courant metric” into a new ”Tube metric” beetwen measurable sets and characterize the geodesic as variational solution to incompressible Euler flow. Such geodesic can modelise topological changes. We make use of a new ”Sobolev perimeter”, and Sobolev Mean curvature for the moving boundary which turns to be Shape differentiable under smooth transverse perturbation. Then working with L2 speed vector fields (we don’t use any renormalization benefit) we succed in the existence of connecting tubes and in variational solution to the usual incompressible Euler flow under surface tension associated to the Sobolev perimeter. This technic applies to several situations in [Shape-Morphing Metric by Variational Formulation for Incompressible Euler Flow.J. of Control and Cybernetics, vol 38 (2009), No. 4], [Control of Moving Domains...,. Int.Ser.Num.Math., vol. 155, 329-382, Birkhauser Verlag Basel,2007]. ———

IV.4. Nonlinear partial differential equations Organisers: Vladimir Georgiev, Tohru Ozawa The Session intends to discuss various nonlinear partial differential equations in mathematical physics. Among possible arguments the following ones shall be discussed: existence and qualitative properties of the solutions, existence of wave operators and scattering for these problems, stability of solitary waves and other special solutions. —Abstracts— Evolution equations in nonflat waveguides Piero D’Ancona Sapienza - Universit` a di Roma - Dipartimento di Matematica P. Moro, 2 Roma, RM 00185 Italy [email protected] In a joint work with Reinhard Racke (Konstanz) we prove smoothing and Strichartz estimates for evolution equations of Schroedinger or wave type on waveguides which are deformations, in a suitable sense, of flat waveguides of the form O × Rk , O a bounded open set in Rm . For the proof, new weighted estimates for fractional powers of Schroedinger operators are required. ———

IV.4. Nonlinear partial differential equations Investigation of solutions of one not divergent type Mersaid Aripov Mech.,Math, National University of Uzbekistan, Universitet 1, Tashkent, Tashkent 100174 Uzbekistan [email protected] The properties of the weak solution of problem Cauchy and the first boundary value problem for one parabolic equation of not divergent type double nonlinearity and with lower members are investigated. The researched equation is the best combination of forms of the equation of nonlinear diffusion, fast diffusion, the equation to very fast diffusion and p-Laplace heat conductivity equation. This equation describes various processes of nonlinear diffusion, heat conductivity, a filtration, magnetic rheology, etc. The method of investigation of the qualitative properties having physical sense weak solution on the basis of a method of a nonlinear splitting and a method of the standard equations is offered. Two side estimations of the solutions and free boundary, a condition of existence of global solutions (including case of critical value of parameter and exponent) generalizing of known results of H. Fujite, A.A., Samarskii, S.P. Kurdyumov, A.P. Mikhajlov, V.A. Galaktionov, H.Vaskes, S. A. Posashkov are received. On the basis of the analysis of properties of solutions the numerical modeling and visualization of solutions carried out. ——— Asymptotic behavior of subparabolic functions Davide Catania Universit` a di Brescia - Dip. Matematica Via Valotti, n. 9 Brescia, BS 25133 Italy [email protected] We consider an MHD-α model with regularized velocity for an incompressible fluid in two space dimension. Such a model is introduced in analogy with the Navier– Stokes equation to study the turbulent behavior of fluids in presence of a magnetic field, since this problem is otherwise difficult to study, both analitically and numerically. We prove local and global existence for the related Cauchy problem, where the velocity field is viscous, while we have not any magnetic diffusivity. ——— On multiple solutions of concave and convex effects for nonlinear elliptic equation on RN Kuan-Ju Chen Department of Applied Science, Naval Academy, P.O.BOX 90175 Zuoying, Taiwan, R.O.C. [email protected] In this paper we consider the existence of multiple solutions of the elliptic equation on RN with concave and convex nonlinearities. ——— Nonlinear gauge invariant evolution of the plane wave Kazuyuki Doi Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai,

Miyagi 980-8579 Japan [email protected] We consider nonlinear gauge invariant evolution of the plane wave. In this talk, we deal with the power and logarithmic type nonlinearities. Although the plane wave does not decay at infinity, by an elementary and simple argument we find an extremely smooth solution which has an explicit expression. Additionally, we study the global behavior of the solution from its representation. ——— New approach to solve linear parabolic problems via semigroup approximation Mohammad Dehghan Ferdowsi University of Mashhhad Azadi Square Mashhad, Khorasan-e-Razavi 9177948974 Iran [email protected] We consider Linear Parabolic Problems (LPPs) whose solutions can be expressed via semigroups. Computing the solutions of these LPPs depends on existing explicit formulas for the corresponding semigroups. However, in general explicit formulas are not available. The proposed approach defines a sequence of linear problems which are semidiscrete approximations of the considered LPP. The solutions of approximant linear problems can be expressed via corresponding semigroups which have explicit formulas. These solutions converge uniformly to the solution of LPP. So the corresponding semigroup of LPP can be approximated by semigroups which have explicit formulas. The approximant linear problems are defined on the finite dimensional subspaces of the LPP solution space, via a hybrid finite-differenceprojection method. The accuracy of approximations, order of convergency and their relations to the proposed hybrid method are discussed and some examples are presented. ——— Global existence and blow-up for the nonlocal nonlinear Cauchy problem Albert Erkip Sabanci University, Faculty of Engineering and Natural Sciences, Orhanli, Tuzla Istanbul / 3495 Turkey [email protected] We study the Cauchy problem utt

=

(β ∗ (u + g (u)))xx

u (x, 0)

=

φ (x) ,

x ∈ R, t > 0

ut (x, 0) = ψ (x)

x ∈ R,

for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function β whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided. ———

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IV.4. Nonlinear partial differential equations Qualitative properties for reaction-diffusion systems modelling chemical reactions Marius Ghergu School of Mathematical Sciences University College Dublin Belfield, , Dublin 4 Dublin Ireland [email protected] In 1952 the British mathematician Alan M. Turing published the foundation of reaction-diffusion theory for morphogenesis, the development of form and shape in biological systems. Since then, many Turing-type models described by coupled reaction-diffusion equations have been proposed for generating patterns in both organic and inorganic systems. In this talk we present a qualitative study for reactiondiffusion systems of the type ut − d1 ∆u = a + bu + f (u)v

in Ω × (0, ∞),

vt − d2 ∆v = c + du − f (u)v

in Ω × (0, ∞),

u(x, 0) = u0 (x), v(x, 0) = v0 (x) ∂u ∂u (x, t) = (x, t) = 0 ∂ν ∂ν

on Ω,

on ∂Ω × (0, ∞).

Here Ω ⊂ RN (N ≥ 1) is a bounded domain, a, b, c, d, d1 , d2 ∈ R, u0 , v0 ∈ C(Ω) are non-negative and f ∈ C[0, ∞) ∩ C 1 (0, ∞) is a non-negative and nondecreasing such that f (0) = 0 and f > 0 in (0, ∞). The system encompasses two well known chemical models: the Brusselator and the Schnackenberg models which are a rich source of varied spatio-temporal patterns. We present several existence and stability results. A particular attention is paid to the associated steady-state system where the crucial role played by the diffusion coefficients d1 , d2 and the behavior of the nonlinearity f is emphasized. The proofs rely on a-priori estimates combined with analytical and topological methods. ——— Scattering in the zero-mass Lamb system ´ ndez Marco Antonio Taneco-Herna Instituto de F´ısica y Matem´ aticas, Universidad Michoacana de San Nicol´ as de Hidalgo, Edificio C-3, Ciudad Universitaria Av., Francisco J. Mujica s/n, Colonia Felicitas del Rio Morelia, Michoac´ an 58040, Mexico [email protected] We consider nonlinear conservative Lamb system, which is the wave equation coupled with a particle of zero mass: u ¨(x, t) = u00 (x, t), F (y(t)) + u0 (0+, t) − u0 (0−, t),

y(t) = u(0, t),

with x ∈ R \ {0}, t ∈ R. Here u˙ := ∂u , u0 := ∂u ∂t ∂x and so on. The solutions u(x, t) take the values in Rd with d ≥ 1 and F := −∇V with V : Rd → R is a potential force field. For the first time we establish long time asymptotics in global energy norm for all finite energy solutions. Namely, under some Ginzburg-Landau type conditions to V , each solution from some functional space decays to a sum of a stationary state, outgoing wave and the rest which tends to zero in global energy norm as t → +∞.

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The outgoing wave is a solution to the free wave equation with some asymptotics states as initial data. We introduce corresponding nonlinear scattering operator, and obtain a necessary condition for the asymptotic states. Also we prove the asymptotics completeness in the Lamb system. This is joint work with A.E. Merzon and A.I. Komech. ——— Global existence for systems of the nonlinear wave and Klein-Gordon equations in 3D Soichiro Katayama Department of Mathematics, Wakayama University, 930 Sakaedani Wakayama, Wakayama 640-8510 Japan [email protected] We consider the Cauchy problem for coupled systems of the nonlinear wave and Klein-Gordon equations in three space dimensions. We present a sufficient condition for global existence of small amplitude solutions to such systems. Our condition is much weaker than the strong null condition for this kind of coupled system, and our result is a natural extension of the global existence theorem for the nonlinear wave equations under the null condition, as well as that for the Klein-Gordon equations with quadratic nonlinearities. Our result is applicable to a certain kind of model equation in physics, such as the Klein-Gordon-Dirac equations, the Klein-GordonZakharov equations, and the Dirac-Proca equations. ——— Global existence for nonlinear wave equations exterior to an obstacle in 2D Hideo Kubo Graduate School of Information Sciences, Tohoku University 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai , Miyagi 980-8579 Japan [email protected] In this talk we discuss the global existence for the exterior problem of nonlinear wave equations in two space dimensions. The obstacle is assumed to be a star-shaped, so that the decay of the local energy is available. The main difficulty compared with the three space dimensional case is the weaker decay of solutions in 2D, as well as the lack of the sharp Huygens principle. However, we are able to show the global existence for small initial data, provided the nonlinearity is of the cubic order and fulfills the so-called null condition. ——— Remark on Navier-Stokes equations with mixed boundary conditions Petr Kucera Czech Technical University, Fac. of Civil Engineering, Dept. of Math., Thakurova 7, Prague 166 29 Czech Republic [email protected] We solve a system of the Navier-Stokes equations for incompressible heat conducting fluid with mixed boundary conditions (of the Dirichlet or non-Dirichlet type on different parts of the boundary). We suppose that the viscosity of the fluid depends on temperarure. ———

IV.4. Nonlinear partial differential equations Contraction-Galerkin method for a semi-linear wave equation with a boundary-like antiperiodic condition Ut van Le Department of Mathematical Sciences, P.O. Box 3000, Oulu FI-90014 Finland [email protected] We consider the unique solvability of initial-boundary value problems of semi-linear wave equations with spacetime dependent coefficients and special mixed nonhomogeneous boundary values which make the so-called boundary-like antiperiodic condition. The procedure in this project is the combination of the Galerkin method and a contraction. ——— p − q systems of nonlinear Schrodinger equations Sandra Lucente Dipartimento di Matematica, Via Orabona 4, Bari 70124, Italy [email protected]

A symmetric error estimate for Galerkin approximations of time dependant Navier-Stokes equations in two dimensions Itir Mogultay Department of Mathematics, Yeditepe University, 26 Agustos Yerlesimi Kayisdagi Caddesi Kayisdagi Istanbul, 81120 Turkey [email protected] A symmetric error estimate for Galerkin approximation of solutions of the Navier-Stokes equations in two space dimensions plus time is given. The finite dimensional function spaces are taken to be divergence free, and time is left continuous. The estimate is similar to known results for scalar parabolic equations. An application of the result is given for mixed method formulations. A short discussion of examples is included. Finally, there are some remarks about a partial expansion to three space dimensions. Note: This is a joint work with Prof. Todd F. Dupont at the University of Chicago. ———

In a joint work with L. Fanelli and E. Montefusco, we consider coupled nonlinear Schr¨ odinger equations

Stability of standing waves for some systems of nonlinear Schr¨ odinger equations with three-wave interactions

iut + ∆u ± N1 (u, v) = 0, ivt + ∆v ± N2 (u, v) = 0,

Masahito Ohta Department of Mathematics, Saitama University, 255 Shimo-Ohkubo, Saitama, 338-8570 Japan [email protected]

with suitable semilinear terms N1 (u, v) and N2 (u, v) having polynomial growth. We investigate on local and global existence critical exponents and describe the corresponding solutions.

We discuss orbital stability and instability of several types of standing waves for some three-component systems of nonlinear Schr¨ odinger equations.

——— ——— Semiclassical analysis for nonlinear Schrodinger equations

Decay rates for wave models with structural damping

Satoshi Masaki 6-3-09 Aza-aoba Aramki Aoba-ku Sendai, Miyagi 9808579 Japan [email protected]

Michael Reissig Faculty 1, TU Bergakademie Freiberg, Pr¨ uferstr. 9, Freiberg, 09596 Germany [email protected]

We consider the semiclassical limit of the nonlinear Schrodinger equations. We approximate the solution by a function of phase-amplitude form, called WKB analysis. We mainly treat the nonlocal nonlinearites. ——— 3-D viscous Cahn-Hilliard equation with memory Gianluca Mola Universit` a di Milano, Dipartimento di Matamatica, via Saldini 50 Milano, MI 20133 Italy [email protected] We deal with the memory relaxation of the viscous Cahn-Hilliard equation in 3-D, covering the well–known hyperbolic version of the model. We study the longterm dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. ———

In this talk, we will present results on the behavior of higher order energies of solutions to the following Cauchy problem for a wave model with structural damping: utt − ∆u + b(t)(−∆)σ ut = 0, u(0, x) = u0 (x), σ ∈ (0, 1],

ut (0, x) = u1 (x),

b(t) = µ(1 + t)δ ,

µ > 0, δ ∈ [−1, 1].

We are interested in the influence of the structural dissipation (between external and visco-elastic damping) b(t)(−∆)σ ut on L2 − L2 estimates. Our main goal is to study under which conditions do we have a parabolic effect for the solutions, that is, the decay rates depend on the order of energy. In the talk we will explain how hyperbolic or elliptic WKB analysis comes in. The main tools are a correct division of the extended phase space into zones, diagonalization procedures, construction of fundamental solutions and a gluing procedure. Some open problems complete the talk. This is joint work with Xiaojun Lu (Hangzhou). ———

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IV.4. Nonlinear partial differential equations Stability theorems in the theory of mathematical fluid mechanics

towards the stable manifold etc. Several outstanding open questions will be discussed as well.

Yoshihiro Shibata Department of Mathematics, Waseda University, Ohkubo 3-4-1 Shinjuku-ku Tokyo, Tokyo 169-8555 Japan [email protected]

———

I would like to talk about some stability theorem of stationary solutions of incompressible fluid flow with initial disturbance. ——— On singular systems of parabolic functional equations ´ szlo ´ Simon La P´ azm´ any P. s´et´ any 1/C, L. E¨ otv¨ os University, Institute of Mathematics Budapest, Hungary H-1117 Hungary [email protected] We shall consider initial-boundary value problems for a system consisting of a quasilinear parabolic functional equation and an ordinary differential equation with functional terms. The parabolic equation may contain the gradient with respect to the space variable of the unknown function in the ODE. It will be proved global existence of weak solutions, by using the theory of monotone type operators and Schauder’s fixed point theorem. Such problems are motivated by models describing reaction-mineralogy-porosity changes in porous media and polymer diffusion. ——— Survey of recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations Zdenek Skalak Thakurova 7, Czech Technical University, Prague, 16629 Czech Republic [email protected] We present some recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations. For example, if w is such a solution satisfying the strong energy inequality then there exists a ≥ 0 such that lim ||Eλ w(t)||/||w(t)|| = 1 t→∞

for every λ > a, where {Eλ ; λ ≥ 0} denotes the resolution of identity of the Stokes operator. ——— Conditional stability theorems for Klein-Gordon type equations Atanas Stefanov 1460, Jayhawk Blvd., Department of Mathematics, University of Kansas, Lawrence, KS 66049, USA [email protected] We consider unstable ground state solutions of the KleinGordon equation with various power nonlinearities. The main result is a fairly precise construction of a stable manifold in a close vicinity of the ground state. In particular, we provide an asymptotic formula for the asymptotic phase, an estimate of the rate of the convergence

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A regularity result for a class of semilinear hyperbolic equations Sergio Spagnolo Department of Mathematics, University of Pisa, Largo Pontecorvo 5 Pisa, 56127 Italy [email protected] We first recall a former result of global wellposedness in C-infinity (resp., in each Gevrey class) for a special kind of homogeneous, linear hyperbolic equations with analytic (resp., C-infinity) coefficients depending only on time. Then, we add to these equations an analytic semilinear term, and we prove that the resulting equations enjoy the following regularity property; each solution which is real analytic at the initial time together with all its time derivatives, remains analytic as long as it is bounded in C-infinity (resp., in some Gevrey class). ——— On nonlinear equations, fixed-point theorems and their applications Kamal Soltanov Department of Mathematics, Faculty of Sciences, Hacettepe University, Beytepe Campus Ankara, Cankaya TR-06532 Turkey [email protected] In this work we investigated some class of the nonlinear operators and a nonlinear equations with such type operators in a Banach spaces. Here we obtained some new results on the solvability of the nonlinear equations, and also a fixed-point theorems for continuous mappings. With use of the obtained here results we studied various boundary value problems (BVP) (and mixed problems) for the different nonlinear differential equations. ——— Dynamics of a quantum particle in a cloud chamber Alessandro Teta Dipartimento di matematica pura e applicata, Universita’ di L’Aquila via Vetoio - loc. Coppito L’Aquila, Abruzzo 67100 Italy [email protected] We consider the Schroedinger equation for a system composed by a particle (the α-particle) interacting with two other particles (the atoms) subject to attractive potentials centered in a1 , a2 ∈ R3 . At time zero the α-particle is described by a diverging spherical wave centered in the origin and the atoms are in their ground state. The aim is to show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless a2 lies on the line joining the origin with a1 . The work (in collaboration with G. Dell’Antonio and R. Figari) is a fully time-dependent version of the original analysis performed by Mott in 1929. ———

IV.5. Asymptotic and multiscale analysis Half space problem for the damped wave equation with a non-convex convection term Yoshihiro Ueda Graduate School of Sciences, Tohoku University 6-3 Aramaki-Aza-Aoba, Aoba-ku Sendai, Miyagi 980-8578 Japan [email protected] We consider the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space. In the case where the flux is convex, it had already known that the solution tends to the corresponding stationary wave. In this talk, we show that even for a quite wide class of flux functions which are not necessarily convex, such the stationary wave is asymptotically stable. The proof is given by a technical weighted energy method. ——— On the time-decay of solutions to a family of defocusing NLS Nicola Visciglia Dipartimento di Matematica, Universita di Pisa, Via F. Buonarroti 2, Pisa 56127 Italy [email protected] Let u(t, x) be any solution to the defocusing NLS with 4 pure power nonlinearity u|u|α , where 0 < α < n−2 , and 1 n with initial condition u(0, x) ∈ H (R ). Then the Lp norm of u(t, x) goes to zero as t → ∞ provided that 2n 2 < p < n−2 . In particular we extend previous result due to Ginibre and Velo who have shown the property 4 above under the extra assumption n4 < α < n−2 . ——— The semilinear Klein-Gordon equation in de Sitter spacetime Karen Yagdjian Department of Mathematics, University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78541-2999, USA [email protected] In this talk we present the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation g φ − m2 φ = −|φ|p with the small mass m ≤ n/2 in de Sitter spacetime with the metric g. We prove that for every p > 1 large energy solutions blow up, while for the small energy solutions we give a borderline p = p(m, n) for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of Kato’s lemma. ———

IV.5. Asymptotic and multiscale analysis Organisers: Ilia Kamotski, Valery Smyshlyaev BICS Mini-Symposium The minisymposium will focus on fundamental analytical issues associated with differential equations (linear

and nonlinear, partial or ordinary) with a small parameter and/or multiple scales, and relevant applications. This includes singularly perturbed problems, problems in thin domain or with singular boundaries, homogenization. The applications may include propagation and localization of waves, blow-up phenomena, metamaterials, etc. The relevant analytic issues are convergence and relevant functional spaces, compactness and propagation of oscillations, asymptotic expansions with error bounds, etc. —Abstracts— On the essential spectrum and singularities of solutions for Lam´ e problem in cuspoidal domain Natalia Babych Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY United Kingdom [email protected] Within a Lam´e problem of linear elasticity, we investigate singularities of solutions in the vicinity of an outward cusp at the boundary. In case of a sharp cusp (the H¨ older constant is less or equal 12 ), we describe the essential spectrum that consists of a certain real ray accessing +∞. We analyse all possible local singularities of solutions and construct radiation conditions defining suitable spaces that guarantee a Fredholm type solvability for the problem. We demonstrate that the sharp outward cusp at the boundary is somewhat similar to infinity for unbounded domains. This is joint work with Dr. I. Kamotski. ——— Torsion effects in elastic composites with high contrast Michel Bellieud Universit´e de Perpignan, 52 avenue Paul Alduy, Perpignan, 66860 France [email protected] In the context of linearized elasticity, we analyze as ε → 0 a vibration problem for a two-phase medium whereby an ε-periodic set of ”stiff” elastic fibers of elastic moduli of the order 1 is embedded in a ”soft” elastic matrix of elastic moduli of the order ε2 . We show that torsional vibrations take place at an infinitesimal scale. ——— Enhanced resolution in structured media Yves Capdeboscq OxPDE Centre for Nonlinear Partial Differential Equations, University of Oxford, Mathematical Institute, Oxford, OX1 3LB United Kingdom [email protected] In this talk, we show that it is possible to achieve a resolution enhancement in detecting a target inclusion if it is surrounded by an appropriate structured medium. This work is motivated by the advances in physics concerning the so-called super resolution, or resolution beyond the diffraction limit. We first revisit the notion of resolution and focal spot, and then show that in a structured medium, the resolution is conditioned by effective parameters. This is a joint work with Habib Ammari & Eric Bonnetier

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IV.5. Asymptotic and multiscale analysis

This is work in collaboration with Marc Briane, where we study the asymptotic behaviour of a given sequence of diffusion energies in L2 (Ω) for a bounded open subset Ω of R2 . The corresponding diffusion matrices are assumed to be coercive but any upper bound is considered. We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of Fn is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. These results are based on the uniform convergence satisfied by some minimizers of the equicoercive sequence Fn , which is specific to the dimension two.

The Knizhnik-Zamolodchikov equation associated with the root system Bn is investigated. This root system has two orbits with respect Weyl group. By this reason KZ equation naturally contains two parameters. Singular locus of this equation consists from hyperplanes xi − xj = 0, xi + xj = 0, xk = 0, i, j, k = 1, 2, . . . , n, x = (x1 , . . . , xn ) ∈ Cn . The following inverse problem of Riemann-Hilbert type is considered: given a representation of a fundamental group of complement to the singular locus in Cn to the orthogonal group of odd order. To define the coefficients of the two-parametric differential KZ equation as elements of tensor power of universal envelopping algebra for odd orthogonal Lie algebra. Oneparametric case was investigated by A.Leibman. For coefficients were used Casimir elements of second order. In two-parametric case the coefficients are defined by using the families of Casimir elements of higher order described by A. Molev. For construction of these elements are used Capelli operators which permit to describe the centre of corresponding universal enveloping algebra. The invariants used in explicit form of coefficients for the case o(5) are expressed by means of Pfaffian for matrix defined using the root system.

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Two-scale Γ-convergence and its applications to homogenisation of non-linear high-contrast problems

Long-time behavior for the Wigner equation and semiclassical limits in heterogeneous media

Mikhail Cherdantsev School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG United Kingdom [email protected]

Fabricio Macia Universidad Politecnica de Madrid, DEBIN ETSI Navales, Avda. Arco de la Victoria, Madrid 28040 Spain [email protected]

It is a resent results of Bouchitte, Felbacq, Zhikov and others that passing to the limit in high-contrast elliptic PDEs may lead to non-classical effects, which are due to the two-scale nature of the limit problem. These have so far been studied in the linear setting, or under the assumption of convexity of the stored energy function. It seems of practical interest however to investigate the effect of high-contrast in the general non-linear case, such as of finite elasticity. With this aim in mind, we develop a new tool to study non-linear high-contrast problems, which may be thought of as a “hybrid” of the classical Γ-convergence (De Giorgi, Dal Maso, Braides) and two-scale convergence (Allaire, Briane, Zhikov). We demonstrate the need for such a tool by showing that in the high-contrast case the minimising sequences may be non-compact in Lp space and the corresponding minima may not converge to the minimum of the usual Γ-limit. We prove a compactness principle for high-contrast functionals with respect to the two-scale Γ-convergence, which in particular implies convergence of their minima. We briefly discuss possible applications of this new technique in the mechanics of composites. (This is a joint work with K.D. Cherednichenko.)

We study the semiclassical limit for a class of linear Schr¨ odinger equations in an heterogeneous medium (for instance, a Riemannian manifold) at time scales tending to infinity as the characteristic frequencies of the initial data tend to zero. We are interested, in particular, in dealing with time scales larger than the Eherenfest time, for which the high frequency behavior is completely characterized by classical mechanics via Egorov’s theorem. Our analysis is performed by studying the highfrequency behavior of Wigner functions corresponding to solutions to the Schrodinger equation at very long times. We give a complete characterization of their structure for systems arising as the quantization of a completely integrable classical Hamiltonian flow. In particular, we prove that in such systems the asymptotic behavior of Wigner functions for times larger than Ehrenfest’s might no longer be determined by the classical flow. This is due to effects caused by resonances, that have to be studied via a new object, the resonant Wigner distribution.

——— Homogenization of elliptic partial differential equations with unbounded coefficients in dimension two Juan Casado-Diaz Dpto. de Ecuaciones Diferenciales y Analisis Numerico, Facultad de Matematicas, C. Tarfia s/n Sevilla, Sevilla 41012 Spain [email protected]

——— Construction of the two-parametric generalizations of the Knizhnik-Zamolodchikov equations of Bn type Valentina Alekseevna Golubeva Steklov Mathematical Institute, Gubkina 8, Moscow 119991 Russia [email protected]

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——— On nonlinear dispersive equations in periodic structures: Semiclassical limits and numerical schemes Peter Markowich DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA United Kingdom [email protected] We discuss (nonlinear) dispersive equations, such as the Schrdinger equation, the Gross-Pitaevskii equation mod- eling Bose-Einstein condensation, the MaxwellDirac system and semilinear wave equations. Semiclassical limits are analysed using WKB and Wigner tech-

V.1. Inverse problems niques, in particular for periodic structures, and connections to classical homogenisation problems for HamiltonJacobi equations and hyperbolic conservation laws are established. We present a new numerical technique for such PDE problems, based on Bloch decomposition, and show applications in semiconductor modelling, BoseEinstein condensation and Anderson localisation for random wave equations. ——— Derivation of Boltzmann-type equations from hardsphere dynamics Karsten Matthies Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY United Kingdom [email protected] The derivation of the continuum models from deterministic atomistic descriptions is a longstanding and fundamental challenge. In particular the emergence of irreversible macroscopic evolution from reversible deterministic microscopic evolution is still not fully understood. We study a classic system: N balls that interact with each other via a hard-core potential and show rigorously that in the case of kinetic annihilation (particles annihilate each other upon collision) the asymptotic behavior as N tends to infinity is correctly described by the Boltzmann equation without gain-term for non-concentrated initial distributions. The mean-field description fails, when there are concentrations in the space or the velocity coordinates. This is joint work with Florian Theil. ——— Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape Bernd Schmidt Zentrum Mathematik, TU Muenchen, Boltzmannstr. 3, Garching b. Muenchen, 85747 Germany [email protected]

in the (tensorial) coefficients. The employed tools are those of ”non-classical” (high contrast type) homogenisation. This leads to interesting effect physically, for example allowing ”directional localisation”, with no wave propagation in certain directions, and mathematically allows treating form a unified perspective ”classical”, high-contrast homogenizations and intermediate cases. We discuss some related analytic issues, including the need to develop appropriate versions of two-scale convergence and of the theory of compensated compactness. ———

V.1. Inverse problems Organisers: Yaroslav Kurylev, Masahiro Yamamoto Inverse problems is a multidisciplinary subject having its firm origin in application of mathematics to such problems as search for oil, gas and other mineral resources, medical imaging, process monitoring in micro-biological, chemical and other industries, non-destructive testing of materials, to mention just few. Its mathematical underpinning stretches from discrete mathematics, to geometry, to computational methods with, however, the principal background being in analysis. In particular, the use of analytic methods makes it possible to address such issues of IP as their strongly non-linear nature and severe ill-posdnesss. In recent years, these relations have made it possible to solve a number of long-standing inverse problems, including those with data on a part of the boundary, with significantly reduced requirements on regularity and the number of measurements, etc. These were based on the advancing and employing such topics in analysis as Carleman estimates for PDE’s, harmonic and quasiconformal analysis, global and geometric analysis, microlocal calculus and stochastic/probabilistic methods. In this section we intend to represent those progress by inviting the leading people in the area to give relevant talks. —Abstracts—

We investigate ground state configurations of atomic systems in two dimensions as the number of atoms tends to infinity for suitable pair interaction models. Suitably rescaled, these configurations are shown to crystallize on a triangular lattice and to converge to a macroscopic Wulff shape which is obtained from an anisotropic surface energy induced by the microscopic atomic lattice. Moreover, sharp estimates on the microscopic fluctuations about the limiting Wullf shape are obtained. (Joined work with Y. Au Yeung and G. Friesecke.) ——— Homogenization with partial degeneracies: analytic aspects and applications Valery Smyshlyaev Department of Mathematical Sciences, University of Bath, Claverton Down Bath, BA2 7AY United Kingdom [email protected] We consider homogenization problems for a generic class of (scalar or vector) operators with ”partial” degeneracy

An inverse conductivity problem with a single measurement Abdellatif El Badia LMAC, University of Compiegne, Compiegne, Oise 60200 France [email protected] We revisit in this paper the inverse boundary value problem of Calderon for a coated domain, where the conductivity is constant in each subdomain. This geometric distribution of conductivity corresponds to the well accepted model of heads in ElectroEncephaloGraphy (EEG). For instance, the inmost interior domain is occupied by the brain, and it is surrounded by the skull and the scalp. The so-called spherical model, where these regions are concentric spherical layers, is also frequently used. We show for this distribution of conductivity that the inverse problem is completely solved with only one suitably chosen Cauchy data, instead of the whole Dirichlet-to-Neumann operator. The criterion of choice for these Cauchy data is completely set up in

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V.1. Inverse problems the spherical model, using spherical harmonics. Also, a stability result is established. As for the numerical method to compute the conductivity, we propose a least square procedure with a Kohn-Vogelius functional, and a boundary integral method for the direct problem. This is joint work with T. Ha-Duong. ——— Global in time existence and uniqueness results for some integrodifferential identification problems Fabrizio Colombo Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9 Milano, Mi 20133 Italy [email protected] We show some results on the identification of memory kernels in some nonlinear equations such as the heat equation with memory, the strongly damped wave equation with memory, the beam equation with memory and a peculiar model in the theory of combustion. An additional restriction on the state variable is given to determine both the state variable and the memory kernels. We prove global in time uniqueness results and for suitable nonlinearities we prove existence and uniqueness results for the solution of the identification problems associated to the models mentioned above. ——— Stability estimate for an inverse problem for the magnetic Schr¨ odinger equation from the Dirichlet-toNeumann map Mourad Choulli Department of Mathematics, Metz University, Ile du Saulcy Metz, Lorraine 57000 France [email protected] In this talk we consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schr¨ odinger equation in a bounded smooth domain of Rn with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the solutions of the magnetic Schr¨ odinger equation. We prove in dimension n ≥ 2 that the knowledge of the Dirichletto-Neumann map for the magnetic Schr¨ odinger equation measured on the boundary determines uniquely the magnetic field and we prove a H¨ older-type stability in determining the magnetic field induced by the magnetic potential. ——— Optimal combination of data modes in inverse problems: maximum compatibility estimate Mikko Kaasalainen Department of Mathematics and Statistics, PO Box 68, Helsinki, FI-00014 Finland [email protected] We present an optimal strategy for weighting various data modes in inverse problems. The solution, maximum compatibility estimate, corresponds to the maximum likelihood estimate of the single-mode case (with,

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e.g., regularization functions included). We illustrate the method by showing that one can reconstruct a body with sparse data of the boundary curves (profiles) and volumes (brightnesses) of its generalized projections. ——— On an inverse problem for a linear heat conduction problem Christian Daveau CNRS (UMR 8088) and Department of Mathematics, University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. [email protected] In this talk, a boundary integral method is used to solve an inverse linear heat conduction problem in twodimensional bounded domain. An inverse problem of measuring the heat flux from partial (on a part of the boundary) dynamic boundary measurements is considered. This talk presents joint work with A. Khelifi. ——— Inverse problems for wave equation and a modified time reversal method Matti Lassas Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hallstromin katu 2b), Helsinki, University of Helsinki 00014 Finland [email protected] A novel method to solve inverse problems for the wave equation is introduced. Suppose that we can send waves from the boundary into an unknown body with spatially varying wave speed c(x). Using a combination of the boundary control method and an iterative time reversal scheme, we show how to focus waves near a point x0 inside the medium and simultaneously recover c(x0 ) if the wave speed is isotropic. In the anisotropic case we can reconstruct the wave speed up to a change of coordinates. These results are obtained in collaboration with Kenrick Bingham, Yaroslav Kurylev, and Samuli Siltanen. Also, we will disucss how the energy of a wave can be focused near a single point in an unknown medium. These results are done in collaboration with Matias Dahl and Anna Kirpichnikova. ——— Picard condition based regularization techniques in inverse obstacle scattering Koung Hee Leem Dept. of Mathematics & Statistics, Southern Illinois University, Edwardsville, IL 62026 United States [email protected] The problem of determining the shape of an obstacle from far-field measurements is considered. It is well known that linear sampling methods have been widely used for shape reconstructions obtained via the singular system of an ill conditioned discretized far-field operator. For our reconstructions we assume that the far field data are noisy and we present two novel regularization methods that are based on the Picard Condition and do not require a priori knowledge of the noise level. Both

V.1. Inverse problems approaches yield results comparable to the ones obtained via the L-curve method and the discrepancy principle. ——— Limited data problems in tensor tomography William Lionheart School of Mathematic, University of Manchester, Oxford Rd, Manchester, M13 9PL United Kingdom [email protected] n photoelastic tomography one seeks to recover a trace free symmetric second rank tensor from its truncated transverse ray transform. We present constructive uniqueness results in the case where realistic subsets of data are known and numerical reconstruction methods. This is joint work with V Sharafutdinov and D Szotten.

The factorization method is a fast inversion technique for visualizing the profile of a scatterer from measurements of the far-field pattern. The mathematical basis of this method is given by the far-field equation, which is a Fredholm integral equation of the first kind in which the data function is a known analytic function and the integral kernel is the measured (and therefore noisy) far field pattern. We present a Tikhonov parameter choice approach based on a fast fixed-point method developed by Bazan. The method determines a Tikhonov parameter associated with a point near the corner of the L-curve in log-log scale and it works well even for cases where the L-curve exhibits more than one convex corner. The performance of the method is evaluated by comparing our reconstructions with those obtained via the L-curve method.

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The finite data non-selfadjoint inverse resonance problem

A time domain probe method for inverse scattering problems

Marco Marletta Cardiff School of Mathematics Senghennydd Road Cardiff, Wales CF24 4AG United Kingdom [email protected]

Roland Potthast Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire, RG6 6AX, UK [email protected]

We consider Schr¨ odinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L1 [0, ∞). It is known (at least in the case of a realvalued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignicant for the reconstruction of the potential from the data. We prove the validity of this statement, i.e., we show conditional stability for nite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials. This is joint work with S. Naboko, S. Shterenberg and R, Weikard. ——— Numerical solutions of nonlinear simultaneous equations Tsutomu Matsuura Graduate School of Engineering, Gunma University 1-51 Tenjintyo Kiryu, Gunma 376-8515 Japan [email protected] In this paper we shall give practical and numerical representations of inverse mappings of 2-dimensional mappings (of the solutions of 2-nonlinear simultaneous equations) and show their numerical experiments by using computers. We derive a concrete formula from a very general idea for the representation of the inverse function ——— A fixed-point algorithm for determining the regularization parameter in inverse scattering George Pelekanos Dept. of Mathematics & Statistics, Southern Illinois University, Edwardsville, IL 62026 United States [email protected]

The goal of the talk is to discuss the development of probe methods for inverse scattering problems in the time-domain. We will study wave scattering by threedimensional rough surface problems. Both the mathematics of these problems as well as the algorithmical solution of direct and inverse problems and the numerical analysis of algorithms provide a sincere challenges since the methods developed for bounded objects cannot be directly translated into the setting of unbounded scatterers. We survey recent results on the direct and inverse problems by Burkard, Chandler-Wilde, Heinemeyer, Lindner and the speaker. With the multi-section method we present a numerical scheme for which convergence both for direct and inverse scattering (using a multi-section Kirsch-Kress approach) can be shown. The time-domain probe method is then formulated and discussed. Convergence for the reconstruction of surfaces can be shown and numerical examples are presented. ——— Explicit and direct representations of the solutions of nonlinear simultaneous equations Saburou Saitoh Department of Mathematics, University of Aveiro, 3810193 Aveiro, Portugal [email protected] We shall present our recent results with Dr. Masato Yamada on practical, numerical and explicit representations of inverse mappings of n-dimensional mappings (of the solutions of n-nonlinear simultaneous equations) and show their numerical experiments by using computers. We derive those concrete formulas from very general ideas for the representation of the inverse functions. ———

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V.2. Stochastic analysis Direct and inverse mixed impedance problems in linear elasticity Vassilios Sevroglou University of Piraeus, Department of Statistics and Insurance Science, 80 Karaoli & Dimitriou Str., Piraeus, Athens 18534, Greece [email protected] Direct and inverse scattering problems with mixed boundary conditions in linear elasticity are considered. We formulate the direct scattering problem for a partially coated obstacle as well as the mathematical setting for the inverse one. Uniqueness theorems are presented and an inversion algorithm for the determination of the scattering obstacle is established. In particular, a linear integral equation due to the linear sampling method which arises from an application of the reciprocity gap functional and the fundamental solution, connected with the appoximate solution of the inverse problem, is investigated. Finally, a discussion about the validity of our method for mixed boundary value problems in elastic scattering theory is presented. ——— On inverse scattering for nonsymmetric operators Igor Trooshin Institute of Problems of Precise Mechanics and Control, Russian Academy of Sciences, Rabochaya 24, Saratov, 410028 Russia [email protected] We consider a nonsymmetric operator AP {L2 (0, ∞)}2 . defined by differential expression (AP u)(x) = Bu0 (x) + P (x)u(x),

in

0 0. Second, we study a class of two-dimensional maps (or called Mira map) and prove that there exist snapback repellers for the map near its anti-integrable limits. Finally, combining the above two results, we establish the existence of transversal homoclinic orbits in family of Arneodo-Coullet-Tresser map near singularities. ——— Bifurcations of random diffeomorphisms with bounded noise Martin Rasmussen Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom [email protected] We discuss iterates of random diffeomorphisms with identically distributed and bounded noise. In this context, minimal forward invariant sets play an important role, since they support stationary measures, and when the noise is interpreted as external control, minimal forward invariant sets coincide with invariant control sets. Discontinuous bifurcations of minimal forward invariant sets are analysed, and a numerical method to approximate these sets is presented. The results are applied to study a bifurcation of the randomly perturbed Henon map. This talk is based on joint work with Jeroen Lamb (Imperial College) and Christian Rodrigues (University of Aberdeen). ——— Bifurcations of period annuli and solutions of nonlinear boundary value problems Felix Sadyrbaev Institute of Mathematics and Computer Science, Rainis boul. 29 Riga, Latvia LV-1459 Latvia [email protected]

be generalized to non-linear and also non-integer expansions of a real number. This talk is based on joined work with T. Persson and D. F¨ arm. ——— Thermodynamic formalism for unimodal maps Mike Todd Departamento de Matem´ atica Pura, Rua do Campo Alegre, 687 Porto, 4169-007 Portugal [email protected] Notions from thermodynamic formalism such as pressure, equilibrium states and large deviations can give a rich qualitative description of a dynamical system. Recently there has been a lot of activity in the development of thermodynamic formalism applied to non-uniformly hyperbolic dynamical systems. These systems have been shown to exhibit a wide variety of phenomena, most interestingly critical phenomena such as phase transitions. In this talk I will give a fairly complete description of the possible behaviour of the class of unimodal interval maps, including the relation between phase transitions and the existence of a natural measure for the system. ——— Dynamics of periodically perturbed homoclinic solutions Qiudong Wang Department of Mathematics, University of Arizona, Tucson, Arizona 85721 United States [email protected] We study the dynamics of homoclinic tangles in periodically perturbed second order equations. Let µ be the size of the perturbation and Λµ be the homoclinic tangles. We prove that (i) for infinitely many µ, Λµ contain nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many µ, Λµ contain nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure set of µ so that Λµ admits strange attractors with Sinai-Ruelle-Bowen measure. ———

Differential equations of the type x00 + λf (x) = 0 are considered, where f (x) are polynomials. First bifurcations of period annuli (continua of periodic solutions) are studied under the change of coefficients of f (x). Secondly, bifurcations of solutions to the Dirichlet problem x(a) = 0, x(b) = 0 are investigated under the change of λ. ——— Large intersection properties of some invariant sets in number-theoretic dynamical systems ¨ rg Schmeling Jo Center of Mathematical Sciences, LTH, Box 118, Slvegatan 18 Lund, 22100 Sweden [email protected] In this talk we consider sets of real numbers that have a given approximation property by rationals with denominators g n . We prove that these sets have large intersection properties and are winning in a modified (α, β) game or belong to Falconers s-class. This result will

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V.5. Functional differential and difference equations Organisers: ˘ acık Zafer Leonid Berezansky, Josef Dibl´ık, Ag Scope of the session: Qualitative theory of functional differential and difference equations: stability, boundedness, oscillation, asymptotic behaviour, positive solutions, dynamic equations on time scales, applications to population dynamics. —Abstracts— Oscillation and non-oscillation of solutions of linear second order discrete delayed equations Jarom´ır Baˇ stinec Department of Mathematics, The Faculty of Electrical Engineering and Communication, Brno University

V.5. Functional differential and difference equations of Technology, Technick´ a 8, 616 00 Brno, Czech Republic [email protected] The phenomenon of the existence of a positive solution of difference equations is often encountered when analysing mathematical models describing various processes. This is a motivation for an intensive study of the conditions for the existence of positive solutions of difference equations. Such analysis is related to an investigation of the case of all solutions being oscillating. In the talk, conditions for the existence of a positive solution are given for a class of linear delayed discrete equations ∆x(n) = −p(n)x(n − 1) where n ∈ Za∞ := {a, a + 1, . . . }, a ∈ N is fixed, ∆x(n) = x(n + 1) − x(n), p : Za∞ → (0, ∞). For the same class of equations, also conditions are given for all the solutions being oscillating. The results obtained indicate sharp sufficient conditions for the existence of a positive solution or for the case of all solutions being oscillating. The investigation was supported by the grant 201/07/0145 of the Czech Grant Agency (Prague) and by the Councils of Czech Government MSM 0021630529 and by MSM 00216 30503. This is joint work with Josef Diblik. ——— New stability conditions for linear differential equations with several delays Leonid Berezansky Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, Negev 84105 Israel [email protected] New explicit conditions of asymptotic and exponential stability are obtained for the general scalar nonautonomous linear delay differential equation with measurable delays and coefficients. These results are compared to known stability tests. ——— Boundary-value problems for differential systems with a single delay Aleksandr Boichuk ˇ ˇ Faculty of Science, Zilina University, Zilina, 01 026 Slovakia [email protected] Conditions are derived of the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay. Utilizing a delayed matrix exponential and a method of pseudo-inverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay. This work was supported by the grant 1/0771/08 of the Grant Agency of Slovak Republic (VEGA) and by the

project APVV-0700-07 of Slovak Research and Development Agency.This is joint work with J. Dibl´ık, D. Khusainov, M. R˚ uˇziˇckov´ a. ——— Representation of solutions of linear differential and discrete systems and their controllability Josef Dibl´ık Brno University of Technology, Brno, Czech Republic, Kiev State University, Kiev, Ukraine [email protected] We study discrete controlled systems ∆x(k) = Bx(k − m) + bu(k), q where m ≥ 1 is a fixed integer, k ∈ Z∞ 0 , Zs := {s, s + n 1, . . . , q}, B is a constant n × n matrix, x : Z∞ −m → R is unknown solution, b ∈ Rn is given nonzero vector and u : Z∞ 0 → R is input scalar function. Moreover, we consider the system of delayed linear differential equations of second order

y 00 (t) + Ω2 y(t − τ ) = bu(t) and an initial problem y(t) = ϕ(t), y 0 (t) = ϕ0 (t), t ∈ [−τ, 0] where τ > 0 and ϕ : [−τ, 0] → Rn is twice differentiable. Special matrix functions are defined: the delayed matrix sine and the delayed matrix cosine. These matrix functions are applied to obtain explicit formulas for the solution of the initial problem and a controllability criterion. The investigation was supported by the grant 201/08/0469 of the Czech Grant Agency (Prague), by the Councils of Czech Government MSM 0021630519 and MSM 00216 30503 and by the project M/34-2008 of Ukrainian Ministry of Education. This is joint work with Denys Khusainov, Blanka Mor´ avkov´ a. ——— Maximum principles and nonoscillation intervals in the theory of functional differential equations Alexander Domoshnitsky Ariel University Center, Department of Mathematics and Computer Science, Ariel, 44837 Israel [email protected] Many classical topics in the theory of functional differential equations, such as nonoscillation, differential inequalities and stability, were historically studied without any connection between them. As a result, assertions associated with maximum principles for such equations in contrast with the cases of ordinary and even partial differential equations do not add so much in problems of existence and uniqueness of solutions to boundary value problems and stability for functional differential equations. One of the goals of this talk is to present a concept of the maximum principles for functional differential equations. New results on existence and uniqueness of solutions of boundary value problems are proposed. Assertions about positivity og Green’s functions are formulated. Tests of the exponential stability are obtained on the basis of nonoscillation and positivity of the Cauchy function. ———

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V.5. Functional differential and difference equations Averaging for impulsive functional differential equations: a new approach

Lyapunov type inequalities for nonlinear impulsive differential systems

Marcia Federson Av. Trabalhador Sao-carlense 400, CP 668, Sao Carlos, SP 13560-970 Brazil [email protected]

Zeynep Kayar Middle East Technical University, Department of Mathematics, Ankara, Cankaya 06531, Turkey [email protected]

We consider a large class of functional differential equations subject to impulse effects and state an averaging result by means of the techniques of the theory of generalized ordinary differential equations introduced by J. Kurzweil. ——— Some bifurcation problems in the theory quasilinear integro differential equations Yakov Goltser Department of Computer Sciences and Mathematics, Ariel University Center of Samaria, Ariel, 44837 Isser Natanzon, 27/7, Pisgat Zeev, 97877 Jerusalem, Israel [email protected] Our goal is to study parametrical perturbed nonlinear quasiperiodic systems of differential and integrodifferential equations.Study bifurcation problems similarly Hopf bifurcation, Bogdanov-Takkens bifurcation and bifurcation of invariant torus,based on the normal form theory and the truncated method for countable systems of ordinary differential equations. ——— Stability in Volterra type population model equations with delays ´ n Gyo ¨ ri Istva Egyetem u. 10 Department of Mathematics, University of Pannonia Veszprem, Veszprem County H-8200 Hungary [email protected] In this talk some delay dependent and delay independent stability conditions will be given for differential equations arising in population dynamics. The proofs are based on the construction of a Lyapunov functional and some monotone techniques for nonautonomous systems. At the end of the talk we shall formulate some open problems and conjectures. ——— On parameter dependence in functional differential equations with state-dependent delays Ferenc Hartung University of Pannonia Egyetem str 10 Veszprem, H8200 Hungary [email protected] In this talk we study smooth dependence on parameters of solutions of several classes of functional differential equations with state-dependent delays. As an application of our results, we discuss the parameter estimation problems for FDEs with state-dependent delays using a quasilinearization method. ———

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We obtain Lyapunov-type inequalities for systems of nonlinear impulsive differential equations. In particular, these sytems contain the Emden-Fowler-type systems and half linear systems in the special cases. In addition, as an application we make use of these inequalities to derive some boundedness and disconjugacy criteria and sufficient conditions for the asymptotic behaviour of solutions. ——— Evaluating the stochastic theta method Conall Kelly Department of Mathematics, University of the West Indies, Mona Kingston, Sn.Andrew 7, Jamaica [email protected] When a numerical method is applied to a differential equation, the result is a difference equation. Ideally the dynamics of the difference equation should reflect those of the original as closely as possible, but in general this can be difficult to check. It is therefore useful to perform a linear stability analysis: applying the method of interest to a linear test equation possessed of an equilibrium solution with known stability properties, and determining the asymptotic properties of the resultant difference equation for comparison. We examine the issues that arise for this kind of analysis in the context of stochastic differential equations, and review the relevant literature. These issues have yet to be adequately addressed. We propose a new approach and demonstrate its usage for the class of θ-Maruyama methods with constant step-size. ——— Delay-distribution effect on stability Gabor Kiss Department of Engineering Mathematics, University of Bristol Queen’s Building Bristol, South West England BS8 1TR United Kingdom [email protected] We consider the effect of delay distribution on retarted functional differential equations with one delay. More specifically, we study the effect of delay distribution on the stability of solutions of first- and second-order equations by comparing the stability regions of the respective equation with a single delay with that of the equation with distributed delays. ——— Solutions of linear impulsive differential systems bounded on the entire real axis ´ Martina Langerova Dept. of Mathematics, Faculty of Science, University of ˇ ˇ Zilina, Univerzitn´ a 1, 010 26 Zilina, Slovakia [email protected] We consider the problem of existence and structure of solutions bounded on the entire real axis of the linear

V.5. Functional differential and difference equations differential system with impulsive action at fixed points of time x˙ = A(t)x + f (t), ˛ ˛ ∆x˛

= ai , t=τi

t, τi ∈ R,

t 6= τi , i ∈ Z,

ai ∈ R n .

Under the assumption that the corresponding homogeneous system is exponentially dichotomous on the semiaxes R+ and R− and by using the results of the wellknown Palmer lemma and the theory of pseudoinverse matrices we establish necessary and sufficient conditions for the indicated problem. Co-authors: Oleksandr ˇ ıkov´ Boichuk, Jaroslava Skor´ a. This research was supported by the Grants 1/0771/08, 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and APVV 0700-07. ——— Oscillatory and asymptotic properties of solutions of higher-order difference equations of neutral type Malgorzata Migda Institute of Mathematics, Poznan University of Technology, ul. Piotrowo 3A, Poznan, 60-965 Poland [email protected] We consider higher-order linear difference equations with delayed and advanced terms ∆m (xn − pxn−τ ) = qn xn−σ + hn xn+η where p is a nonnegative number, τ, σ, η are positive integers and (qn ), (hn ) are sequences of nonnegative real numbers. We give sufficient conditions under which all nonoscillatory solutions of the delayed part of the equation are unbounded and under which all nonoscillatory solutions of the advanced part tend to zero as n → ∞. We establish also sufficient conditions for the oscillation of all solutions of the full equation.

We shall discuss some results on the asymptotic behaviour of the nonnegative solutions of systems of linear difference equations with asymptotically constant coefficients. The main result describes the relationship between the nonnegative solutions of the perturbed system and the positive eigenvalues and the corresponding nonnegative eigenvectors of the limiting system. The proofs are based on Pringsheim’s Theorem and the Extended Liouville Theorem from complex analysis. ——— On singular models arising in hydrodynamics Irena Rachunkova Palacky University, Fakulty of Science, Dept. of Mathematics, Tomkova 40, Olomouc, 77900 Czech Republic [email protected] We investigate models arising in hydrodynamics. These models have the form of the singular second order differential equation (p(t)u0 (t))0 = p(t)f (u(t)) on the half-line. Here f is locally Lipsichtz on R and changes its sign and p is continuous on [0, ∞) and p(0) = 0. A discrete formulation of this equation is investigated as well. We are interested in strictly increasing solutions and homoclinic solutions and provide conditions for p and f which guarantee the existence of such solutions. In particular cases a homoclinic solution determines an increasing mass density in centrally symmetric gas bubbles which are surrounded by an external liquid. ——— Decoupling and simplifying of noninvertible difference equations in the neighbourhood of invariant manifold Andrejs Reinfelds University of Latvia, Institute of Mathematics and Computer Science; Rai¸ na bulv¯ aris 29, LV-1459, R¯ıga, Latvia

——— Principal and non-principal solutions of impulsive differential equations with applications ¨ Abdullah Ozbekler Atılım University, Department of Mathematics, ˙ Kızılca¸sar K¨ oy¨ u, Incek G¨ olba¸sı, Ankara 06836 Turkey [email protected] In this work we first prove a theorem on the existence of principal and nonprincipal solutions for second order differential equations having fixed moments of impulse actions. Next, by means of nonprincipal solution we give new oscillation criteria for related impulsive differential equations. Examples are provided with numerical simulations to illustrate the importance of the study. ——— Nonnegative iterations with asymptotically constant coefficients Mihali Pituk Egyetem u. 10 Department of Mathematics, University of Pannonia Veszprem, Veszprem County H-8200 Hungary [email protected]

[email protected] In Banach space X×E the system of difference equations x(t + 1) = g(x(t)) + G(x(t), p(t)), p(t + 1) = A(x(t))p(t) + Φ(x(t), p(t))

(*)

is considered. Sufficient conditions under which there is an local Lipschitzian invariant manifold u : X → E are obtained. Using this result we find sufficient conditions of partial decoupling and simplifying of the system of noninvertible difference equations (*). ——— Precise asymptotic behaviour of solutions of Volterra equations with delay David W. Reynolds School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland. [email protected] This talk considers the rates at which solutions of Volterra equations with delay converge to asymptotic equilibria. It is found that these convergence rates depend delicately on prescribed data. The results are established using admissibility techniques. This work is motivated by logistic equations with infinite delay.

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V.5. Functional differential and difference equations ———

———

On local stability of solutions of stochastic difference equations

Existence and nonexistence of asymptotically periodic solutions of Volterra linear difference equations

Alexandra Rodkina Department of Mathematics, University of the West Indies, Mona Kingston, Sn.Andrew 7, Jamaica [email protected]

Ewa Schmeidel Instytute of Mathematics, ul. Piotrowo 3A, Poznan, Wielkopolska 60-965 Poland [email protected]

We present results on the local stability of solutions of a stochastic difference equation with polynomial coefficients. Two cases are considered: when stochastic perturbation are a state-independent and asymptotically fading and when stochastic perturbation are a statedependent.

In this talk we investigate Volterra difference equation of the form

——— Convergence of the solutions of a differential equation with two delayed terms ˇkova ´ Miroslava R˚ uˇ zic ˇ Faculty of science, University of Zilina, Slovak Republic [email protected] In this contribution we deal with asymptotic behavior of solutions to a linear homogeneous differential equation containing two discrete delays y(t) ˙ = β(t)[y(t − δ) − y(t − τ )]

(*)

for t → ∞. We assume δ, τ ∈ R+ := (0, +∞), τ > δ, β : I−1 → R+ is a continuous function, I−1 := [t0 − τ, ∞), t0 ∈ R. Denote I := [t0 , ∞) and the symbol “ ˙ ” denotes (at least) the right-hand derivative. Similarly, if necessary, the value of a function at a point of I−1 is understood (at least) as value of the corresponding limit from the right. The main results concern the asymptotic convergence of all solutions of Eq. (*). Especially we deal with so called critical case with respect to the function β. When the function β is the constant function than this critical case is represented with the value β := (τ − σ)−1 . The proof of results is, except other, based on comparison of solutions of Eq. (*) with solutions of an auxiliary inequality which formally copies Eq. (*). This research was supported the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA), by the project APVV-0700-07 of Slovak Research and Development Agency and by the Slovak-Ukrainian project SK-UA-0028-07 (Ukrainian-Slovak project M/34 MOH Ukraine 27.03.2008). This is joint work with Josef Dibl´ık. ——— Inverse problems of the calculus of variations for functional differential equations Vladimir Mikhailovich Savchin Peoples Friendship University of Russia, MikluxoMaklaya street 6, Moscow, 117198 Russia [email protected] The problem of existense of solutions of inverse problems of the calculus of variations for partial differencial difference operators is investigated. Necessary and sufficient conditions of potentiality for such operators are obtained. Methods of construction of variational multiplies are suggested.

104

x(n + 1) = a(n) + b(n)x(n) +

n X

K(n, i)x(i)

i=0

where n ∈ N = {0, 1, 2, . . . }, a, b, x : N → R and K : N × N → R, the special case of this equation is Volterra difference equation of convolution type x(n + 1) = Ax(n) +

n X

K(n − i)x(i).

i=0

This equation may be considered as a discrete analogue of famous Volterra integrodifferential equation Z t b(t − s)x(s)ds. x0 (t) = Ax(t) + 0

Such equation has been widely used as a mathematical model in population dynamics. Both discrete equations represents a system in which the future state x(n + 1) does not depend only on the present state x(n) but also on all past states x(n − 1), x(n − 2), . . . , x(0). These system are sometimes called hereditary. Given the initial condition x(0) = x0 , one can easy generate the solution x(n, x0 ). Sufficient conditions for the existence of asymptotically periodic solutions of Volterra difference equation are presented. In addition we present sufficient conditions for non-existence of an asymptotically periodic solution satisfying some auxiliary conditions. The results are illustrated by examples. ——— Gene regulatory networks and delay equations Andrei Shindiapin Eduardo Mondlane University, Maputo, Mozambique [email protected] Gene regulatory networks consist of differential equations with smooth but steep nonlinearities (”sigmoids”). As the number of genes may be rather large, any theoretical or computer-based analysis of such networks can be complicated. That is why a simplified approach based on replacing sigmoids with step functions is widely used. However, this leads to some mathematical challenges, as for instance analysis of stationary points belonging to the discontinuity set of the system (thresholds) cannot be done directly. Additional problems occur if one tries to incorporate time delays into the network. The delay effects naturally arise from the time required to complete transcription, translation and diffusion to the place of action of a protein. We offer an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. This algorithm is combined with a method to study delay systems by replacing them with an equivalent system of ordinary differential equations, commonly known as the linear chain trick. However, a direct application of this ”trick” is not

V.6. Mathematical biology possible in our case, so that we suggest a modification of it based on the general framework of representing delay equations as ordinary differential equations using the integral transforms”. This is joint work with Arcady Ponosov. ——— The moment problem approach for the zero controllability of ecolution equations Benzion Shklyar 52 Golomb St., P.O.B. 305, Dept. of Appl. Math, Holon Institute of Technology, Holon, 58102 Israel shk [email protected] The exact controllability to the origin for linear evolution control equation is considered.The problem is investigated by its transformation to infinite linear moment problem. Controllability conditions for linear evolution control equations have been obtained. The obtained results are applied to the zero controllability for partial differential and functional differential equations. ——— Properties of maximal solutions of autonomous functional-differential equations with state-dependent deviations Svatoslav Stanek Palacky University, Fakulty of Science, Dept. of Mathematics, Tomkova 40, Olomouc, 77900 Czech Republic [email protected] Equations of the type x00 +x(t−kx)) = 0 are considered. Here k is a positive parameter. It is described (i) the set of all periodic solutions x satisfying x0 < 1/k and (ii) the set of all maximal solutions x (that is, solutions which have no extension) satisfying x0 ≥ 1/k. ——— Boundedness character of some classes of difference equations Stevo Stevic Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000 Serbia [email protected] Some results on the boundedness character of the positive solutions of the following two classes of difference equations xn+1 = A +

xpn , q xn−1 xrn−2

xn+1 = max A,

ff xpn , xqn−1 xrn−2

n ∈ N0 ; n ∈ N0 ,

Republic [email protected]

This contribution deals with systems of linear generalized linear differential equations of the form t

Z x(t) = x e+

d[A(s)] x(s) + g(t) − g(a),

t ∈ [a, b], (*)

a

where −∞ < a < b < ∞, A is an n × n-complex matrix valued function, g is an n-complex vector valued function, A has a bounded variation on [a, b] and g is regulated on [a, b]. The integrals are understood in the Kurzweil-Stieltjes sense. Our aim is to present some new results on continuous dependence of solutions to linear generalized differential equations (*) on parameters and initial data. ——— Lyapunov type inequalities on time scales: A survey ¨ Mehmet Unal Bahcesehir University, C ¸ ıra˘ gan Caddesi, Osmanpa¸sa Mektebi Sokak No. 4–6, Be¸sikta¸s, Istanbul 34353 Turkey [email protected]

We survey Lyapunov type inequalities for linear and nonlinear dynamic equations on time scales. The inequalities contain the well-known classical Lyapunov inequalities as special cases. We also give some applications to illustrate the importance of such inequalities. ——— Interval criteria for oscillation of delay dynamic equations with mixed nonlinearities ˘ acık Zafer Ag Department of Mathematics Middle East Technical University Cankaya, Ankara 06531 Turkey [email protected]

We obtain interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities on an arbitrary time scale T. All results are new even for T = R and T = Z. Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second order delay dynamic equations regardless of the choice of delta (∆) or nabla (∇) derivatives. ———

where the parameters A, p, q and r are positive numbers, are presented. ——— Continuous dependence of solutions of generalized ordinary differential equations on a parameter Milan Tvrdy Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, Praha 1, CZ 115 67 Czech

V.6. Mathematical biology Organisers: Robert Gilbert

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VI. Others —Abstracts— Cancellous bone with a random pore structure Robert Gilbert Department of Mathematics, University of Delaware, 317 Ewing Hall, Newark, DE 19716 United States [email protected] We continue the study of acoustic wave propagation for an elastic medium that is randomly fissured. Moreover, the fissures are assumed to be statistically homogeneous. Although the underlying stochastic process does not necessarily have to be ergodic, we assume for simplicity of exposition that it is. This allows us to obtain an explicit and computationaly easier auxillary problem in a Representative Elementary Volume. In a later work we intend to study the more general case. This is joint work with Ana Vasilic. ——— New computer technologies for the construction and numerical analysis of mathematical models for molecular genetic systems Irina Alekseevna Gainova Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences Acad. Koptyug avenue, 4 Novosibirsk, Novosibirsk region 630090 Russia [email protected] We have created an integrative computer system, which includes three program modules (Institute of Cytology and Genetics, SB RAS): GeneNet, MGSgenerator, MGSmodeller, and the software package STEP+ (Sobolev Institute of Mathematics, SB RAS). The system is used to construct and numerically analyze models describing dynamics of the molecular genetic systems (MGS) functioning in pro- and eukaryotes. Using module GeneNet we can reconstruct structure functional organization of gene networks. We use MGSgenerator as an intermediate module in generation of mathematical models based on gene networks reconstructed in GeneNet. Moreover, in the module MGSgenerator we represent obtained mathematical models in the input format of STEP+. Module MGSmodeller contains tools for the gene network models to be developed and numerically analyzed. Package STEP+ is intended for the numerical analysis of mathematical models represented by autonomous systems ODEs. We have tested our integrated system on the MGS model for intracellular auxin metabolism in a plant cell. This work has been partially supported by the Siberian Branch of the Russian Academy of Sciences (Interdisciplinary integration project Post-genomic bioinformatics: computer analysis and modeling of the molecular genetic systems, No. 119). ———

Due to the presence of the fluid and solid phase, the modeling of cancellous bone represents a complex, extensive task where the dynamic investigation and viscosity effects must be taken into consideration. The already established approach for the investigation of this material type is Biots method, originally developed for simulating saturated porous materials. In this contribution we present the homogenization multiscale FEM as an alternative to Biots method. The motivation for this choice is decreasing the extent of the necessary laboratory investigations. According to the multiscale FEM, the bone is understood as the homogenized medium whose effective material parameters are obtained by the analysis of an appropriate representative volume element (RVE). This is also the main topic of the presentation: a comparison of the effective values obtained by studying different types of RVEs where the particular attention is paid to the numerical values for Youngs modulus and attenuation coefficient. The distinction between the models pertains to the geometry of the solid frame of the RVE, the type of the applied elements as well as the type of the coupling conditions on the interface of the phases. ——— Bone growth and destruction at the cellular level: a mathematical model Mark D. Ryser McGill University, W. Burnside Hall, Room 1005, 805 Sherbrooke Street, Montral, Quebec H3A 2K6 Canada [email protected] The process of bone destruction and subsequent growth is continually occurring in healthy bone tissue. This process is referred to as ’remodeling’ and plays a key role in many pathologies such as osteoporosis and osteoarthritis. We describe remodeling at the cellular level and discuss the cells and biochemical pathways involved. We then develop a mathematical model for remodeling, consisting of a system of coupled nonlinear PDEs. We discuss how physiological parameters may be obtained through scaling of the equations and we comment on their mathematical properties. Numerical experiments validating the model will be presented. This is joint work with Nilima Nigam (SFU) and Svetlana Komarova (McGill). ———

VI. Others Organisers: local organising committee

—Abstracts—

Application of the multiscale FEM in modeling the cancellous bone

The relationship between Bezoutian matrix and Newton’s matrix of divided differences and separation of zeros of interpolation polynomials

Sandra Ilic Institute of Mechanics, Ruhr-University of Bochum, Bochum, 44780 Germany [email protected]

Ruben Airapetyan Kettering University, 1700 W Third Ave. Flint, Michigan 48504, United States [email protected]

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VI. Others Let x1 , . . . , xn be real numbers, Pn (x) = an (x − x1 ) · · · (x−xn ). Denote by Dn g the matrix of generalized divided differences of function g in Newton’s interpolation formula with nodes x1 , . . . , xn and by Gn (x) the Newton’s interpolation polynomial of function g. Denote by B = B(Pn (x), Gn (x)) the Bezoutian matrix of Pn and Gn . The relationship between the corresponding principal minors of the matrices Dn g and B counted from the left lower corner is establish. Then, it follows that if these principal minors of the matrix of divided differences are positive or have alternating signs then the roots of the interpolation polynomial are real and separated by the nodes of interpolation. ——— Bayesian shrinkage estimation of parameter exponential distribution Hadeel Alkutubi B-23-1 , The Heritage, JLN SB, Dagang Mines Resort City, Seri Kembangan, Serdang, 43300 Malaysia [email protected]

In our work, we use the ADM method for solving some nonlinear evolution equations with time and space fractional derivative. Then we use the Extended Tanh method to formally derive traveling wave solutions for some evolution equations. The obtained solutions include, also, kink soltuions. ——— Boundary-value problems for symmetric Helmholtz equation

generalized

axially-

Anvar Hasanov 34 Durmon yoli, Tashkent branch of the Russian State University of oil and gas named after Gubkin, Tashkent, Tashkent 100125, Uzbekistan [email protected] In this talk several main boundary-value problems such the Dirichlet, Neumann problem and other problems will be considered. The unique solvability of afore-mentioned problems will be proved. ———

In this paper, we would like to test the best estimator (smallest MSE and MPE) of shrinkage estimator of parameter exponential distribution . To do this , we derived this estimators depend on Bayesian method with Jeffreys prior information and square error loss function . To compared between estimators we used MSE and MPE with respect of simulation study. We found the shrinkage estimator between Bayes estimators under different loss function is the best estimator.

Asymptotic extension of topological modules and algebras

———

Given a topological R-module or algebra E and an asymptotic scale M ⊂ RΛ , we exhibit a natural M extended topology on the sequence space E Λ , and define the M -extension of E as the Hausdorff space associated with the subspace of nets for which multiplication is continuous with respect to this topology. Commonly used spaces of generalized functions are obtained as special cases, but this new approach applies in many different situations. It also allows the iteration of the construction, which is not possible with previously existing theories. We use only the topology, i.e. neighbourhoods of zero, but not its explicit definition in terms of seminorms, inductive or projective limits etc., which is particularly convenient in non-metrizable spaces. Many ideas commonly used in the context of generalized functions (functoriality, association, sheaf structure, algebraic analysis, . . . ) can be applied to a large extent. Reasoning on a category-theoretic level allows to establish several results so far only known for particular cases, for the whole class of such spaces.

Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem Mohammed Bokhari Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia [email protected] An elegant result due to Erdos and Turan states that the sequence of Lagrange interpolants to a given continuous function f at the zeros of orthogonal polynomials over a closed interval converges to f in the mean square sense. We introduce certain sequences of polynomials which preserve both interpolation as well as convergence properties of Erdos-Teran Theorem. In addition, they interpolate f at a finite number of pre-assigned points lying outside the underlying open interval. We shall introduce a method to construct the suggested polynomials and also investigate their properties. Computational aspects will also be discussed. ——— The ADM method and the Tanh method for solving some non linear evolutions equations Zoubir Dahmani Department of Mathematics, Faculty of Sciences, University of Mostaganem Les HLM, 21 street les HLM mostaganem, mostaganem 27000 Algeria [email protected]

Maximilian Hasler Laboratoire AOC, Universit Antilles-Guyane, B.P. 7209, campus de Schoelcher, Schoelcher, Martinique 97275 France [email protected]

——— Approximation of fractional derivatives S. Moghtada Hashemiparast Mathematics and Statistics, Jolfa Ave, Seyed Khandan, Tehran 193953358 Iran ha[email protected] Series represantations are presented to approximate the fractional derivatives which have extensive application in ordinary,partial difrential equations and specilly the stable probability distributions.The convergence of the

107

VI. Others series are considered and are applied to solving the equations,finally toillustrate the accuracy of the apprpximations examples are solved. ——— Discrepancy estimate for uniformly distributed sequence Hailiza Kamarulhaili School of Mathematical Sciences Universiti Sains Malaysia Minden, Penang 11800 Malaysia [email protected] A general metrical result of discrepancy estimate related to uniform distribution of a sequence is proved . This work extends result of R.C. Baker where the sequence can be assumed to be real. The lighter version of this theorem will also be discussed in this talk. ——— Bounded linear operators on l-power series spaces Erdal Karapinar ATILIM University, Department of Mathematics, Kizilcasar Koyu, INCEK ANKARA, 06836 Turkey [email protected] Let A be the class of Banach space ` of scalar sequences with a norm k · k` such that (i) a = (ai ) ∈ l∞ , x = (ξi ) ∈ ` ⇒ ax = (ai ξi ) ∈ `, kaxk` ≤ kakl∞ kxk` , (ii) kei k` = 1, ∀i ∈ N where ei = (δij )j∈N .

Theorem. For Fr´echet space E and ` ∈ A, (E, K l1 (A)) ∈ B ⇒ (E, K ` (A)) ∈ B ⇒ (E, K l∞ (A)) ∈ B. and `

∈

A,

——— On a three-dimensional elliptic equation with singular coefficients Erkinjon Karimov Durmon yuli street 29, Akademgorodok Tashkent, Tashkent 100125 Uzbekistan [email protected] In this talk some questions such as finding fundamental solutions, investigations of main boundary-value problems for an equation uxx + uyy + uzz +

108

2α 2β 2ζ ux + uy + uz x y z

——— A unified presentation of a class of generalized Humbert polynomials Nabiullah Khan Department of Applied Mathematics, Z.H. College of Engineering and Technology, Aligarh Muslim University, Aligarh 202002 India nabi [email protected] The principal object of this paper is to present a natural further step toward the unified presentation of a class of Humbert’s polynomials which generalizes the wellknown class of Gegenbauer, Humbert, Legendre, Tchebycheff, Pincherle, Horadam, Dave, Kinnsy, Sinha, Shreshtha, Horadam-Pethe, Djordjevie, Gould, Milovanovi´c and Djordjevi´c, Pathan and Khan polynomials and many not so wellknown polynomials. We shall give some basic relations involving the generalized Humbert polynomials and then take up several generating functions, hypergeometric representations and expansions in series of some relatively more familier polynomials of Legendre, Gegenbauer, Rice, Hermite, Jacobi, Laguerre Fasenmyer Sister M. Celine, Bateman, Rainville and Khandekar. We also show that our results provide useful extensions of known results of Dilcher, Horadam, Sinha, Shreshtha, Milovanovi´c-Djordjevi´c, Pathan and Khan. ——— Direct estimate for modified beta operators

For a given ` ∈ A and a K¨ othe matrix A, we define `-K¨ othe space K ` (ai,n ) as a Fr´echet space of all scalar sequences x = (ξi ) such that (ξi ai,n ) ∈ ` for each n, endowed with the topology of Fr echet space, determined by the canonical system of norms {kxkn = k (ξi ai,n ) k` , n ∈ N }. We write (E, F ) ∈ B, if every continuous linear map from E to F is bounded. In 1983, D.Vogt has characterized those Fr´echet spaces E for which (E, K l∞ (A)) ∈ B holds. This gives also a characterization of (E, K c0 (A)) ∈ B. We extend this results and prove that

Theorem. For Fr´echet space F (K ` (A), F ) ∈ B ⇒ (K l1 (A), F ) ∈ B.

will be discussed. Here α, β, zeta are constants, moreover 0 < 2α, 2β, ζ < 1.

Lixia Liu Yuhua east Road 113, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei Province 050016 China [email protected] 2 Use the Ditzian modulous of smoothness ωϕ λ (f, t), (0 ≤ λ ≤ 1), to study the pointwise direct results for modified Beta operators, which extend the approximation result for Beta operators.

——— Mathematical model of an undergorund nuclear waste disposal site Eduard Marusic-Paloka Department of Mathematics, University of Zagreb, Bijenicka 30, Zagreb, 10000 Croatia [email protected] The goal of our research is to find an accurate model for numerical simulations of the nuclear waste disposal site. The purpose of such model is to perform safety analysis of the site and find out its possible impact on the biosphere. Due to the large dimension of the site and very long lifetime of radioelements, realistic experiments are not possible. Thus, predictions based on numerical simulations are all we have. Starting from the microscopic model given by the reaction-diffusion-convection equation, using the asymptotic analysis and homogenization, we derive a macroscopic model and discuss ity accuracy. ———

VI. Others Compact and coprime packedness and semistar operations Abdeslam Mimouni Department of Mathematics and Statistics King Fahd University of Petoleum and Minerals Dhahran, Estern 31261 Saudi Arabia [email protected] In this talk we will present new developments on the study of compact and coprime packedness of an integral domain with respect to a star operation of finite character. Let R be an integral domain with quotient field K and let ∗ be a star operation of finite type on R. A ∗-ideal I is said to be ∗-compaclty S (respectively ∗-coprimely) packed if whenever I ⊆ α∈Ω Pα , where {Pα }α∈Ω is a family of ∗-prime ideals of R, I is actually contained in Pα (resp. (I + Pα )∗ ( R) for some α ∈ Ω; and R is said to be ∗-compactly (resp. ∗-coprimely) packed if every ∗-ideal of R is ∗-compactly (resp. ∗coprimely) packed. In the particular case where ∗ = d is the trivial operation, we obtain the so-called compactly and coprimely packed domains. Our objectives is to study some ring-theoretic aspects of these notions in different classes of integral domains, paying particular attention to the the t-operation as the largest and well-known operation. ——— Characterization of some matrix classes involving (σ, λ)convergence S. A. Mohiuddine Department of Mathematics Aligarh Muslim University Aligarh, Uttar Pradesh 202002 India [email protected] Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional ϕ on the space `∞ of bounded single sequences is said to be an invariant mean or σ-mean if and only if (i) ϕ(x) ≥ 0 if x ≥ 0 (i.e. xk ≥ 0 for all k); (ii) ϕ(e) = 1, where e = (1, 1, 1, · · · ); (iii) ϕ(x) = ϕ((xσ(k) )) for all x ∈ `∞ . Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1,

λ1 = 0.

In this paper, first we define (σ, λ)-convergence and show that Vσλ is a Banach space with kxk = supm,n |tmn (x)|, where Vσλ is the set of all (σ, λ)-convergent sequences x = (xk ). We also define and characterize (σ, λ)conservative, (σ, λ)-regular and (σ, λ)-coercive matrices. Further, we characterize the class (`1 , Vσλ ), where `1 is the space of all absolutely convergent series.

if and only if (i) φ(x) ≥ 0 when the sequence x = (xk ) has xk ≥ 0 for all k, (ii) φ(e) = 1, where e = (1, 1, 1, · · · ), and (iii) φ(x) = φ((xσ(k) )) for all x ∈ `∞ . Throughout this paper we consider the mapping σ which has no finite orbits, that is, σ p (k) 6= k for all integer k ≥ 0 and p ≥ 1, where σ p (k) denotes the pth iterate of σ at k. Note that, a σ-mean extends the limit functional on the space c in the sense that φ(x) = lim x for all x ∈ c. In this paper we define a new sequence space Vσ∞ (λ) which is related to the concept of σ-mean and the sequence λ = (λn ) described as above and characterize the matrix classes (`∞ , Vσ∞ (λ)) and (`1 , Vσ∞ (λ)). Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 0. Then we define the following sequence space and show that it is a BK-space: Vσ∞ (λ) := {x ∈ `∞ : sup |τmn (x)| ≤ ∞}, m,n

where X

τmn (x) = (1/(λm ))

xσj (n) .

j∈`m

——— New convection theory for thermal plasma and NHD convection in rapidly rotating spherical configurations Ali Mussa King Abdulaziz City for Science and Technology Building # 2 King Abdullah Bin Abdulaziz Street Riyadh, Riyadh 6086/11442 Saudi Arabia [email protected] We extend Jones-Soward-Mussa (JSM) theory (2000): “analytic and computational solution for E → 0 and P r/E → ∞”. We also make use of Zhang (2001) ansatz for: “E 1 arbitrary but fixed and 0 ≤ P r < ∞” the so-called enhanced Nearly Geostrophic Inertial Wave (NGIW) approach. Such extension represented as a construction of a new MHD plasma convection and magnetoconvection force theory. The flow field confinement in the study assumed to be in spherical geometry configuration and our investigation is made under the basis of magnetic balance and scaling theory. Furthermore, strong inertial turbulence can be achieved in presence of high Reynolds number so strong forces govern the flow fields have to be sufficiently understood. Indeed, strong rotation and strong magnetic field for the flow field inside the spherical rotating geometry; take into consideration the effect of the anticipated vigorous convection and magnetoconvection in the flow field confinement. ——— Characterizations of Isometries on 2-modular spaces

Sequence spaces of invariant mean and some matrix transformations

Kourosh Nourouzi Department of Mathematics, K.N. Toosi University of Technology Tehran, Tehran 16315-1618 Iran [email protected]

Mohammad Mursaleen Department of Mathematics Aligarh Muslim University Aligarh, UP 202002 India [email protected]

Let X be a real vector space of dimension greater than one. A real valued function ρ(·, ·) on X 2 satisfying the following properties is called a 2-modular on X, for all x, y, z ∈ X:

———

Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional φ on the space `∞ is said to be an invariant mean or a σ-mean

1. ρ(x, y) = 0 if and only if x, y are linearly dependent, 2. ρ(x, y) = ρ(y, x),

109

VI. Others 3. ρ(−x, y) = ρ(x, y), 4. ρ(x, αy + βz) ≤ ρ(x, y) + ρ(x, z), for any nonnegative real numbers α, β with α + β = 1. In this talk, we discuss on the characterization of isometries defined on 2-modular spaces. ——— On r-imbalances in tripartite r-digraphs Shariefuddin Pirzada King Fahd University of Petroleum and Minerals, Dhahran, 31261 Saudi Arabia [email protected] A tripartite r-digraph(r ≥ 1) is an orientation of a tripartite multigraph that is without loops and contains atmost r edges between any pair of vertices from distinct parts. For any vertex x in a tripartite r-digraph − D(U, V, W ), let d+ x and dx denote the outdegree and − indegree respectively of x. Define aui = d+ ui − dui , + − + − bvj = dvj − dvj and cwk = dwk − dwk as the r-imbalances of the vertices ui in U , vj in V and wk in W respectively. In this paper, we characterize r-imbalances in tripartite r-digraphs and obtain some results. ——— Invariance conditions and amenability of locally compact groups Hashem Parvaneh Masiha Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology. No. 41, Kavian St., Seyyed Khandan Bridge (N.), Shariati Ave., Tehran, Tehran 16315-1613 Iran [email protected] Adler and Hamilton showed that a semigroup S is left amenable if and only if it satisfies the following invariance condition. For any subsets A1 , A2 , · · · , Ak of S and any s1 , s2 , · · · , sk ∈ S, there exists a nonempty finite subset E of S such that n(s−1 i Ai ∩ E) = n(Ai ∩ E), for i = 1, 2, · · · , k, where s−1 A = {t ∈ S : st ∈ A} and n(A) is the number of elements in A. In this talk, we shall prove an analogous result for locally compact groups. More precisely, we show that amenability of a locally compact group G is equivalent to: For any λ-measurable subsets A1 , A2 , · · · , Ak of G, any g1 , g2 , · · · , gk ∈ G and any ε > 0, there exists a compact subset K of G such that |λ(gi−1 Ai ∩ K) − λ(Ai ∩ K)| < ελ(K), for i = 1, 2, · · · , k, where λ(A) denotes the left Haar measure of A. In this paper, we suppose that G be a locally compact group and λ a fixed left Haar measure on G. We let X = {K ⊂ G : K is compact and R λ(K) > 0}. For 1 f ∈ L∞ (G), we define f¯(K) = λ(K) f dλ, K ∈ X, K then f¯ : X → R is well defined. ———

of one of the major classical problems of theoretical mechanics, dynamics of a solid body with one fixed point in a gravity field. Motion of a solid body with one fixed point is described by the well-known system of Euler and Poisson equations. It is known the general solution exists if one considers two first terms of force function expansion into a series. By original change of variables the system is reduced to the normal form with the first integral of norm type. The solution of this system is considered as the non-perturbed motion and it is investigated on stability. The procedure is offered for obtaining of asymptotically steady motion in general case. The controlling force nature was defined. This method was applied to three cases with special restrictions on the bodys inertia moments, so-called generalized classical cases of Euler, Lagrange and Kovalevskaya. The numerical solution for the problem in Euler case was constructed. ——— A Lizorkin type theorem for Fourier series multipliers in regular systems Lyazzat Sarybekova Munaitpasov 7, Astana, 010010 Kazakhstan [email protected] A new Fourier series multiplier theorem of Lizorkin type is proved for the case 1 < q < p < ∞. The result is given for a general regular system and, in particular, for the trigonometrical system it implies an analogy of the original Lizorkin theorem. ——— Inverse-closedness problems in the stability of sequences in Banach Algebras Pedro A. Santos Departamento de Matem´ atica, Instituto Superior T´ecnico, Av Rovisco Pais, Lisboa, 1049-001 Portugal [email protected] We are concerned with the applicability of the finite sections method to operators belonging to the closed subalgebra of L(Lp (R)), 1 < p < ∞, generated by operators of multiplication by piecewise continuous functions in ˙ and operators of convolution by piecewise continuous R Fourier multipliers. The usual technique is to introduce a larger algebra of sequences, which contains the special sequences we are interested and the usual operator algebra generated by the operators of multiplication and convolution. There is a direct relationship between the applicability of the finite section method for a given operator and invertibility of the corresponding sequence in this algebra. But, contrarily to the C ∗ case and Banach analogue for Toeplitz operators, in our case several inverse-closedness problems must be solved. ———

Motion stabilisation of a solid body with fixed point

Smoothing effects for periodic NSE in critical Sobolev space

Zaure Rakisheva Al-Faraby Kazak National University, Almaty, Kazakstan zaure [email protected]

Ridha Selmi Department of Mathematics, Faculty of Sciences of Gabes, 6072, TUNISIA [email protected]

The problem of a solid body dynamics in the central Newton field of forces is considered. It is generalization

We prove smoothing effects for 3D incompressible Navier Stokes Equation for initial data belonging to critical

110

VI. Others 1

Sobolev Space H 2 (T3 ). Asymptotic behavior of the global solution when the time goes to +∞ is studied. ——— Large deviations and almost sure convergence Mariana Sibiceanu Gh.Mihoc-C.Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie Nr.13, Sector 5 Bucharest, 050711 Romania [email protected] In our setup, the Large Deviation Principle for a sequence P (n) of probabilities on a separable Banach space E, with a convex good rate function I is assumed, also the existence of the finite limits g(w) of the associated logarithmic moment generating function. We establish precise upper and lower bounds of the values that P (n) assigns almost sure in the weak and strong topology of E, respectively, determined by the amounts of the canonical dual product on E 0 × N , N being the nullifying set of the rate function. Also, we reveal the significance of the derivative of the function g(tw) of real t for the almost sure convergence, in the situation when g is Gateaux differentiable on (E 0 , t(E 0 , E)). ———

on the recent results of Z.Y.Huang [Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively pseudocontractive mappings without Lipschitzian assumptions, J.Math.Anal.Appl. 329(2007),935-947], Z.Y. Huang, F.W. Bu, M.A. Noor [On the equivalence of the convergence criteria between modified Mann-Ishikawa and multistep iteration with errors for strongly pseudocontractive operators, Appl. Math. Compt. 181(2006), 641-647], B.E.Rhoades, S.M.Soltuz [The equivalence between the convergences of Ishikawa and Mann iteration for an asymptotically non-expansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289(2004), 266-278] and B.E.Rhoades, S.M.Soltuz [The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal.58(2004),219-228] among others. ——— A characterization for multipliers of weighted Banach valued Lp (G)-spaces Serap Oztop Istanbul University, Faculty of Sciences, Istanbul, Vezneciler 34134 Turkey [email protected] Let G be a locally compact group, 1 < p < ∞. The aim of this paper is to characterize the multipliers of the weighted Banach valued intersection Lp (G) spaces as the space of multipliers of a certain Banach algebra.

The k- Model in Turbulence ——— Tanfer Tanriverdi Harran University, Faculty of Arts and Sciences, Department of Mathematics, Sanlurfa 63300, Turkey [email protected] We prove analytically the existence of self-similar solutions for the k- model arising in the evolution of turbulent bursts by employing the topological shooting technique where α > β with the some other conditions. The first author was supported by the Scientific and ¨ Technological Research Council of Turkey (TUBITAK). He is also thankful to the Oxford Center for Nonlinear PDE, and to the Mathematical Institute of the University of Oxford, for the hospitality they offered him during his visit. This is joint work with Bryce McLeod (Oxford). ——— The equivalence between modified Mann (with errors), Ishikawa (with errors), Noor (with errors) and modified multi-step iterations (with errors) for non-Lipschitzian strongly successively pseudo-contractive operators Johnson Olaleru Mathematics Department, University of Lagos, University of Lagos Road, Yaba, Lagos, Nigeria [email protected]

Stationary motion of the dynamical symmetric satellite in the geomagnetic field Karlyga Zhilisbaeva Al-Faraby Kazak National University, Almaty, Kazakstan [email protected] Stationary solutions of the system of the satellite’s motion equations are of special interest for the various problems of space researches, and first of all for the satellite’s magnetic stabilization. In the paper stationary motions of the equatorial magnetized dynamically symmetric satellite round the centre of mass on a circular orbit are considered. Strong magnets are placed on the satellite’s board. Perturbations are taken into account, caused by insignificant deviation of a satellite’s axis of dynamic symmetry and by magnetization of its cover. The equations of the satellite’s perturbed motion in Euler’s canonical variables are obtained. Conditions of stationary motion existence are defined, necessary and sufficient conditions of their stability are found with taking into account of small perturbations. ———

In this paper, the equivalence of the convergence between modified Mann(with errors), Ishikawa(with errors), Noor(with errors) and modified multistep iteration(with errors) is proved for generalized strongly successively pseudocontractive mapping without Lipschitzian assumption. Our results generalise and improve

111

Index Abate, M., 97 D’Abbico, M., 71 Abdenur, F., 97 Abdous, B., 47 Agranovich, M., 60, 63 Airapetyan, R., 106 Aksoy, U., 27 Alimov, S., 60 Aliyev, T., 23 Alkutubi, H., 107 Almeida, A., 51 Alves, J.F., 97 D’Ancona, P., 80 Applebaum, D., 90 Aripov, M., 81 Assal, M., 56 Aulaskari, R., 23, 56 Aydin, I., 51 Aykol, C., 51 Babych, N., 85 El Badia, A., 87 Bakry, D., 95 Ball, J., 15 Ballantine, C., 23 Bally, V., 90 Barsegian, G., 33 Barthe, F., 95 Baˇstinec, J., 101 Begehr, H., 3, 26, 27 Bellieud, M., 85 Berezansky, L., 100, 101 Berger, P., 97 Berglez, P., 27 Berlinet, A., 3, 47 Bernstein, S., 42 Besov, O., 51 de Bie, H., 36 Bisi, C., 36 Bobkov, S., 95 Bociu, L., 76 Bock, S., 42 Boichuk, A., 101 Bojarski, B., 3, 23 Bokhari, M., 107 Bolosteanu, C., 27 Boukrouche, M., 76 Boulakia, M., 77 Boulbrachene, M., 95 Boutet de Monvel, L., 15 Bouzar, C., 63 Boykov, I., 27 Boza, S., 52 Branden, P., 33 Britvina, L., 48 Brody, D., 90 Bruning, E., 3 Bucci, F., 76

Budochkina, S.A., 97 Burenkov, V., 3, 51, 60 Burns, K., 98 Buzano, E., 63

Dovbush, P., 24 Dragoni, F., 3, 95 Du Bau-Sen, 98 Du Jinyuan, 26, 28

Cˆ amara, C., 45 Capdeboscq, Y., 85 Cardon, D., 33 Carro, M., 52 Caruana, M., 91 Casado-Diaz, J., 86 Castro, L., 45 Catana, V., 63 Catania, D., 81 Cattaneo, L., 3 Cavalcanti, M., 77 Celebi, O., 3, 28 Cerejeiras, P., 36 Charalambides, M., 34 Chazottes, J.-R., 98 Chen Kuan-Ju, 81 Chen Qiuhui, 49 Chen Yi-Chiuan, 98 Cherdantsev, M., 86 Chiba, Y., 63 Chinchaladze, N., 28 Cho, D.H., 49 Choe, B.R., 56 Choi, Q-H., 71 Choulli, M., 88 Cohen, L., 63 Colombini, F., 71 Colombo, F., 36, 88 Cordero, E., 63 Coulembier, K, 37 Crisan, D., 3, 90, 91 Cruzeira, A.B., 91 Csordas, G., 33, 34

Ebert, M., 72 Ekincioglu, I., 52 El-Nadi, K., 93 Eller, M., 77 Elliott, N., 3 Elton, D., 60 Englis, M., 46 Eriksson, S.-L., 37 Erkip, A., 81

Dahmani, Z., 107 Dai Daoquin, 26 Dalla Riva, M., 23 Dallakyan, G., 52 Daoulatli, M., 77 Datt Sharma, S., 47 Dattori da Silva, P., 64 Daveau, C., 88 Davie, A., 91 Davies, B., 3, 16, 59 Davis, M., 91 Dehgan, M., 81 Del Santo, D., 71 Delgado, J., 64 Diblik, J., 100, 101 Doi, K., 81 Doli´canin, D., 49 Domingos Cavalcanti, V., 77 Domoshnitzky, A., 101 Donaldson, S., 16

Fang Daoyuan, 72 Farwell, R., 43 Faustino, N., 43 Federson, M., 102 Fei, M.-G., 37 Fern´ andez, A., 34 Ferreira, M., 37 Field, M., 98 Fisk, S., 34 Fokas, T., 43 Forni, G., 98 Fragnelli, G., 77 Franek, P., 37 Franssens, G.R., 37 Freitas, J., 98 Fujita, K., 48 Fujiwara, H., 49 Furutani, K., 64 Gaiko, V., 98 Gainova, I.A., 106 Galleani, L., 64 Galstyan, A., 72, 77 Garello, G., 64 Garetto, C., 65 Gauthier, P., 34 Gedif Ayele, T., 51 Geisinger, L., 60 Gentil, I., 95 Gentili, G., 37 Georgiev, S., 43 Georgiev, V., 72, 80 Ghergu, M., 82 Ghisa, D., 24 Ghisi, M., 73 Gil, J., 65 Gilbert, R., 3, 105, 106 Giorgadze, G., 28 Girela, D., 56 Gobbino, M., 73 Golberg, A., 24 Goldshtein, V., 53 Goltser, Y., 102 Golubeva, V.A., 86

113

Index Gonzalez, M.J., 57 Gramchev, T., 65 Gramsch, B., 65 Graubner, S., 28 Grigoryan, A., 95 Grudsky, S., 45, 46 Guliyev, V., 53 Gupta, S., 57 G¨ urlebeck, K., 42 G¨ urkanlı, A.T., 46 Gyongy, I., 91 Gy¨ ori, I., 102 Hairer, M., 92, 96 Hajibayov, M., 53 Halburd, R., 34 Hartung, F., 102 Harutyunyan, T., 61 Hasanov, A., 107 Hashemiparast, S.M., 107 Hasler, M., 107 Hebisch, W., 96 Helmstetter, J., 43 Herrmann, T., 73 Higgins, J.R., 48 Hinkkanen, A., 34 Hirosawa, F., 71, 73 Hogan, J., 43 H¨ ormann, G., 65 Huet, N., 96 Hughston, L., 92 Hunsicker, E., 65 Hurri-Syrvanen, H., 53 Hussain, A., 28 Ichinose, W., 65 Ilic, S., 106 Inglis, J., 3, 96 Israfilov, D., 24 Iwasaki, C., 66 Jacka, S., 92 Janas, J., 61 Johnson, J., 66 Jordan, T., 99 Jung, T., 73 Kaasalainen, M., 88 K¨ ahler, U., 44 Kalmenov, T.S., 30 Kalyabin, G., 53 Kamarulhaili, H., 108 Kamotski, I., 85 Kaptanoglu, T., 56, 57 Karapinar, E., 108 Karelin, O., 46 Karimov, E., 108 Karlovych, Y., 66 Karp, D., 24 Karp, L., 73 Karupu, O., 25 Katayama, S., 82 Kato, K., 35 Kats, B., 25 Katsnelson, V., 35 Kayar, Z., 102 Kelbert, M., 92

114

Kelly, C., 102 Kendall, W., 92 Kenig, C., 17 Khan, N., 108 Kheyfits, A., 29 Khimshiashvili, G., 29 Kilbas, A., 3, 48, 49 Kim, B.J., 50 Kisil, A., 38 Kisil, V., 42, 44 Kiss, G., 102 Kohr, G., 25 Kohr, M., 25 Kokilashvili, V., 17 Kolokoltsov, V., 92 Konjik, S., 50 Kontis, V., 3 Koroleva, A., 50 Krainer, T., 61, 66 Krausshar, R.S., 38, 44 Krump, L., 38 Kubo, H., 74, 82 Kucera, P., 82 Kurtz, T., 93 Kurylev, Y., 87 Kusainova, L., 54 Lamb, J., 3, 97 Lamberti, P., 61 Langerov´ a, M., 102 Lanza de Cristoforis, M., 3, 23, 25 Laptev, A., 3, 59 Lasiecka, I., 76 Lassas, M., 88 Lavicka, R., 38 Le, U., 83 Leandre, R., 44 Lebiedzik, C., 78 Lee, Y.L., 57 Leem, K.H., 88 de Leo, R., 66 Lerner, N., 17 Li Xue-Mei, 93 Libine, M., 39 Liflyand, E., 54 Lionheart, W., 89 Lions, P.-L., 21 Littman, W., 78 Liu Lixia, 108 Liu Yu, 66 Loreti, P., 78 Lucente, S., 83 Luna-Elizarrar´ as, M.E., 39 Luzzatto, S., 97 Lyons, T., 90, 93 Macia, F., 86 Maksimov, V., 78 Malliavin, P, 18 Mamedkhanov, J., 25 Mammadov, Y., 54 Manhas, J.S., 57 Manjavidze, N., 29 Markowich, P., 86 Marletta, M., 89

Marquez, A., 57 Marti, J.-A., 67 Martin, M., 39 Martinez, P., 78 Marusic-Paloka, E., 108 Masaki, S., 83 Matsuura, T., 89 Matsuyama, T., 74 Matthies, K., 87 Maz’ya, V., 18 van der Meer, J. C., 99 Michalska, M., 57 Migda, M., 103 Mijatovic, A., 93 Mimoiuni, A., 109 Mityushev, V., 32, 33 Mochizuki, K., 74 Mogultay, I., 83 Mohammed, A., 29 Mohiuddine, S.A., 109 Mokhonko, A., 61 Mola, G., 83 Molahajloo, S., 67 Morando, A., 67 Moura Santos, A., 54 Mursaleen, M., 109 Mussa, A., 109 Nakazawa, H., 74 Nam, K., 57 Naso, M.G., 79 Natroshvili, D., 67 Neklyudov, M., 96 Neustupa, J., 62 Nicol, M., 99 Nieminen, P., 58 Niknam, A., 99 Nishitani, T., 74 Nourouzi, K., 109 Nowak, M., 58 Oberguggenberger, A., 68 Ockendon, J., 20 Ohta, M., 83 Olaleru, J., 111 Oliaro, A., 68 Onchis, D., 48 Oparnica, L., 50 Opic, B., 54 Orelma, H., 39 Otto, F., 96 Ozawa, T., 80 ¨ Ozbekler, A., 103 Oztop, S., 111 Pandolfi, L., 79 Panti, G., 99 Papageorgiou, I., 3, 96 Papavasiliou, A., 93 Parvaneh Masoha, H., 110 Pau, J., 58 Pelekanos, G., 89 Pe˜ na Pe˜ na, D., 39 Peng, C., 99 Perotti, A., 39 Picard, R., 74 Pinotsis, D., 44

Index Pirzada, S., 110 Pistorius, M., 94 Pituk, M., 103 Pivetta, M., 75 Plaksa, S., 23, 26 Porter, M., 33 Potthast, R., 89 Prakash Sing, A., 35 Prykarpatsky, A., 35 Quiao Yuying, 40 R¨ atty¨ a, J., 56 Rachunkova, I., 103 Radkevich, E., 55 Rafeiro, H., 55 Rajabov, N., 29, 68 Rajabova, L., 30 Rakisheva, Z., 110 Rappoport, J., 50 Rasmussen, M., 100 Reinfelds, A., 103 Reissig, M., 3, 71, 75, 83 Ren Guangbin, 40 Renardy, M., 79 Reynolds, D., 103 Richard, S., 62 Roberto, C., 96 Rochon, F., 68 Rodino, L., 3, 62 Rodkina, A., 104 Rogosin, S., 32, 33 Rojas, E., 46 Rozovsky, B., 94 R˚ uˇziˇckov´ a , M., 104 Ruzhansky, M., 3 Ryan, J., 40 Ryan, M., 3 Ryser, M.D., 106 Sabadini, I., 36, 40 Sadyrbaev, F., 100 Safarov, Y., 59 Saito, J., 75 Saitoh, S., 3, 47, 48, 89 Saks, R., 30 Salac, T., 40 Samko, N., 55 Samko, S., 51, 55 Samoylova, E., 30 Santos, P.A., 110 Sanz-Sol´e, M., 94 Sarybekova, L., 110 Sasane, A., 58 Savchin, V.M., 104 Schmeidel, E., 104 Schmeling, J., 100 Schmidt, B., 87 Schnaubelt, R., 79

Schrohe, E., 69 Schulze, B.-W., 3, 18, 68 Sehba, B.F., 58 Seiler, J., 69 Selmi, R., 110 Senouci, K., 55 Serbetci, A., 55 Sergeev, A., 35 Sevroglou, V., 90 Sforza, D., 79 Shapiro, M., 40 Shaposhnikova, T., 69 Shibata, Y., 84 Shindiapin, A., 104 Shklyar, B., 105 Shpakivskii, V., 44 Shvartsman, I., 79 Sibiceanu, M., 111 Silva, A., 47 Silvestri, B., 62 Simon, L., 84 Skalak, Z., 84 Smyshlyaev, V., 85, 87 Sobolewski, P., 58 Soltanov, K., 84 Somberg, P., 40 Sommen, F., 36, 41 Soria, J., 56 Soucek, V., 41 Spagnolo, S., 84 Spr¨ oßig, W., 42, 45 Stanek, S., 105 Stefanov, A., 84 Stevic, S., 105 Stoppato, C., 41 Strohmaier, A., 62 Suragan, D., 30 Suzuki, O., 26 Suzuki, R., 75 Tahara, H., 69 Takemura, K., 48 Tamrazov, P., 23 Taneco-Hern´ andez, M.A., 82 Tanriverdi, T., 111 Tapdıgo˘ glu, M., 58 Taqi, I., 31 Taskinen, J., 58 Tasmambetov, Zh., 31 Teofanov, N., 69 Teta, A., 84 Tikhonov, S., 56 Todd, M., 100 Toft, J., 3, 69 Tolksdorf, J., 45 Tomilov, Y., 62 Toundykov, D., 79 Tovar, L.M., 59

Tretyakov, M., 94 Trooshin, I., 90 Trushin, B.V., 56 Tunaru, R., 94 Turunen, V., 70 Tvrdy, M., 105 Ueda, Y., 85 Uesaka, H., 75 Uhlmann, G., 20 Umeda, T., 62 ¨ Unal, M., 105 Upmeier, H., 47 Usoltseva, E., 94 Vajiac, A., 41 Valein, J., 80 Vasilevski, N., 45, 47 Vasilyev, V., 70 Vasy, A., 70 Vieira, N., 45 Visciglia, N., 85 Vlacci, F., 42 Vlasakova, Z., 42 van de Voorde, L., 42 Vukotic, D., 59 Wang Qiudong, 100 Wang Yufeng, 31 Wirth, J., 3 Witt, I., 70 Wong, M.W., 3, 62, 70 Wu Zhijian, 59 Wulan Hasi, 59 Xu Wen, 59 Xu Yongzhi, 3 Yagdjian, K., 75, 85 Yakubovich, S., 50 Yamamoto, M., 3, 20, 80, 87 Yang Congli, 59 Yildirir, Y.E., 26 Yordanov, B., 76 Youssfi, E.H., 26 Zafer, A., 100, 105 Zegarlinski, B., 3, 95 Zelinskiy, Y., 26 Zeren, Y., 56 Zhang Shangyou, 3 Zhang Zhongxiang, 32 Zhdanov, O.N., 32 Zhilisbaeva. K., 111 Zhong Shouguo, 32 Zhu Hongmei, 71 Zhu Kehe, 47, 59 Zolesio, J.-P., 80 Zorboska, N., 59

115

European Mathematical Society

International Mathematical Union

London Mathematical Society

7th International ISAAC Congress — Abstracts

Edited by M. Ruzhansky and J. Wirth. Prepared and typeset using LATEX. Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ

Welcoming address The ISAAC board, the Local Organising Committee and the Department of Mathematics at Imperial College London, are pleased to welcome you to the 7th International ISAAC Congress in London. The 7th International ISAAC congress continues the successful series of meetings previously held in the Delaware (USA) 1997; Fukuoka (Japan) 1999; Berlin (Germany) 2001, Toronto (Canada) 2003, Catania (Italy) 2005 and Ankara (Turkey) 2007. The success of such a series of congresses would not be possible without all the valuable contributions of all the participants. We acknowledge the financial support for this congress given by the London Mathematical Society (LMS), the International Mathematical Union (IMU), Commission on Development and Exchanges (CDE), and Developing Countries Strategy Group (DCSG), the Engineering and Physical Sciences Research Council (EPSRC), the Oxford Centre in Collaborational and Applied Mathematics (OCCAM), the Oxford Centre for Nonlinear Partial Differential Equations (OxPDE), the Bath Institute for Complex Systems (BICS), the Imperial College London, Strategic Fund, and the Department of Mathematics, Imperial College London. ISAAC Board Man Wah Wong (Toronto, Canada), President of the ISAAC Heinrich Begehr (Berlin, Germany) Alain Berlinet (Montpellier, France) Bogdan Bojarski (Warsaw,Poland) Erwin Bruning (Durban, South Africa) Victor Burenkov (Padova, Italy) Okay Celebi (Istanbul, Turkey) Robert Gilbert (Newark, Delaware, USA) Anatoly Kilbas (Minsk, Belarus) Massimo Lanza de Cristoforis (Padova, Italy) Michael Reissig (Freiberg, Germany) Luigi Rodino (Torino, Italy) Michael Ruzhansky (London, UK) John Ryan (Fayetteville, Arkansas, USA) Saburou Saitoh (Aveiro, Portugal) Bert-Wolfgang Schulze (Potsdam, Germany) Joachim Toft (V¨ axj¨ o, Sweden) Yongzhi Xu (Louisville, Kentucky, USA) Masahiro Yamamoto (Tokyo, Japan) Shangyou Zhang (Newark, Delaware, USA) Local Organising Committee Michael Ruzhansky (Chairman) Dan Crisan Brian Davies Jeroen Lamb Ari Laptev (President of the European Mathematical Society) Jens Wirth Boguslaw Zegarlinski with further support by Laura Cattaneo, Federica Dragoni, Nikki Elliott, James Inglis, Vasileios Kontis and Ioannis Papageorgiou as well as further student helpers.

3

Abstracts

Plenary talks

15

Sir John Ball : The Q-tensor theory of liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Louis Boutet de Monvel : Asymptotic equivariant index of Toeplitz operators and Atiyah-Weinstein conjecture Brian Davies : Non-self-adjoint spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simon Donaldson : Asymptotic analysis and complex differential geometry . . . . . . . . . . . . . . . . . . . Carlos Kenig : The global behavior of solutions to critical nonlinear dispersive and wave equations . . . . . . Vakhtang Kokilashvili : Nonlinear harmonic analysis methods in boundary value problems of analytic and harmonic functions, and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nicolas Lerner : Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems . . . . . . Paul Malliavin : Non-ergodicity of Euler deterministic fluid dynamics via stochastic analysis . . . . . . . . . Vladimir Maz’ya : Higher order elliptic problems in non-smooth domains . . . . . . . . . . . . . . . . . . . Bert-Wolfgang Schulze : Operator algebras with symbolic hierarchies on stratified spaces . . . . . . . . . . . Gunther Uhlmann : Visibility and Invisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masahiro Yamamoto : Practise of industrial mathematics related with the steel manufacturing process . . .

Public lecture

15 15 16 16 16 17 17 18 18 18 19 20

21

Pierre-Louis Lions : Analysis, Models and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Sessions

23

I.1. Complex variables and potential theory Tahir Aliyev Azeroˇ glo : Analytic functions in contour-solid problems . . . . . . . . . . . . . . . . . . . . . . Rauno Aulaskari : A non-α-normal function whose derivative has finite area integral of order less than 2/α Cristina Ballantine : Global mapping properties of rational functions . . . . . . . . . . . . . . . . . . . . . . Bogdan Bojarski : Beltrami equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matteo Dalal Riva : A functional analytic approach for a singularly perturbed non-linear traction problem in linearized elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Dovbush : Boundary behavior of Bloch functions and normal functions . . . . . . . . . . . . . . . . . Anatoly Golberg : Spatial quasiconformal mappings and directional dilatations . . . . . . . . . . . . . . . . Dorin Ghisa : Global mapping properties of entire and meromorphic functions . . . . . . . . . . . . . . . . . Daniyal Israfilov : Approximation in Morrey-Smirnov classes . . . . . . . . . . . . . . . . . . . . . . . . . . Dmitri Karp : Two-sided bounds for the logarithmic capacity of multiple intervals . . . . . . . . . . . . . . . Olena Karupu : On boundary smoothness of conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . Boris Kats : Structures of non-rectifiable curves and solvability of the jump problem . . . . . . . . . . . . . Gabriela Kohr : The Loewner differential equations and univalent subordination chains in several complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirela Kohr : Boundary integral equations in the study of some porous media flow problems . . . . . . . . . Massimo Lanza de Cristoforis : Singular perturbation problems in potential theory: a functional analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jamal Mamedkhanov : Classic theorems of approximation in a complex plane by rational functions . . . . Sergiy Plaksa : Commutative algebras of monogenic functions and biharmonic potentials . . . . . . . . . . . Osamu Suzuki : Fractal method for Clifford algebra and complex analysis . . . . . . . . . . . . . . . . . . . . Yunus Emre Yildirir : Approximation theorems in weighted Lorenz spaces . . . . . . . . . . . . . . . . . . . El Hassan Youssfi : Hankel operator on generalized fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . Yuriy Zelinskiy : Continues mappings between domains of manifolds . . . . . . . . . . . . . . . . . . . . . .

23 23 23 23 23

I.2. Differential equations: Complex and functional analytic methods, applications Umit Aksoy : A hierarchy of polyharmonic kernel functions and the related integral operators . . . . . . . Heinrich Begehr : Boundary value problems for complex partial differential equations . . . . . . . . . . . . Peter Berglez : On some classes of bicomplex pseudoanalytic functions . . . . . . . . . . . . . . . . . . . . Carmen Bolosteanu : Boundary value problems on Klein surfaces . . . . . . . . . . . . . . . . . . . . . . . Ilya Boykov : Optimal methods for evaluation hypersingular integrals and solution of hypersingular integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Okay Celebi : Complex partial differential equations with mixed-type boundary conditions . . . . . . . . . . Natalia Chinchaladze : On a mathematical model of a cusped plate with big deflections . . . . . . . . . . .

26 27 27 27 27

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23 23 24 24 24 24 24 25 25 25 25 25 26 26 26 26 26

. 27 . 28 . 28

5

Jin-Yuan Du : Mixed boundary value problem with a shift for some pair of metaanalytic function and analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grigory Giorgadze : Generalized analytic functions on Riemann surfaces . . . . . . . . . . . . . . . . . . . . Sonnhard Graubner : Optimization of fixed point methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . Azhar Hussain : Generating functions of the Laguerre-Bernoulli polynomials involving bilateral series and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Kheyfits : Asymptotic behavior of subparabolic functions . . . . . . . . . . . . . . . . . . . . . . Giorgi Khimshiashvili : Elliptic Riemann-Hilbert problems for generalized Cauchy-Riemann systems . . . . Nino Manjavidze : On some qualitative issues of the elliptic systems . . . . . . . . . . . . . . . . . . . . . . Alip Mohammed : Poisson equation with the Robin boundary condition . . . . . . . . . . . . . . . . . . . . . Nusrat Rajabov : Investigation of one class of two-dimensional conjugating model and non model integral equation with fixed super-singular kernels in connection with hyperbolic equation . . . . . . . . . . . . . Lutfya Rajabova : About one class of two-dimensional Volterra type integral equation with two interior sinqular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roman Saks : Explicit global solutions of 3D rotating Navier-Stokes equations . . . . . . . . . . . . . . . . Emma Samoylova : Methods of solutions of an singular integrodifferential equation . . . . . . . . . . . . . . Tynysbek Sharipovich Kal’menov : A boundary condition of the volume potential . . . . . . . . . . . . . . . Durbudkhan Suragan : Eigenvalues and eigenfunctions of volume potential . . . . . . . . . . . . . . . . . . Zhaxylyk Tasmambetov : The ending solutions of Ince system with irregular features . . . . . . . . . . . . Ismail Taqi : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yufeng Wang : On mixed boundary-value problems of polyanalytic functions . . . . . . . . . . . . . . . . . . Oleg N. Zhdanov : An algorithm of solving the Cauchy problem and mixed problem for the two-dimensional system of quasi-linear hyperbolic partial differential equations . . . . . . . . . . . . . . . . . . . . . . . Shouguo Zhong : On solution of a kind of Riemann boundary value problem on the real axis with square roots Zhongxiang Zhang : Some Riemann boundary value problems in Clifford analysis . . . . . . . . . . . . . . .

28 28 28 28 29 29 29 29 29 30 30 30 30 30 31 31 31 32 32 32

I.3. Complex-analytical methods for applied sciences 32 Vladimir Mityushev : R-linear problem and its applications to composites . . . . . . . . . . . . . . . . . . . 33 Michael Porter : Application of the spectral parameter power series method to conformal mapping problems 33 Sergei Rogosin : Recent results on analytic methods for 2D composite materials . . . . . . . . . . . . . . . . 33 I.4. Zeros and Gamma lines – value distributions of real and complex functions Grigor Barsegian : An universal value distribution: for arbitrary meromorphic function in a given domain Petter Branden : A generalization of the Stieltjes-Van Vleck-Bocher theorem . . . . . . . . . . . . . . . . . David Cardon : A criterion for the reality of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marios Charalambides : New properties of a class of Jacobi and generalized Laguerre polynomials . . . . . George Csordas : Meromorphic Laguerre operators and the zeros of entire functions . . . . . . . . . . . . Arturo Fern´ andez : On the logarithmic order of meromorphic functions . . . . . . . . . . . . . . . . . . . Steve Fisk : An introduction to upper (stable) polynomials in several variables . . . . . . . . . . . . . . . . Paul Gauthier : Perturbations of L-functions with or without non-trivial zeros off the critical line . . . . Rod Halburd : Tropical and number theoretic analogues of Nevanlinna theory . . . . . . . . . . . . . . . . Aimo Hinkkanen : Growth of analytic functions in unbounded open sets . . . . . . . . . . . . . . . . . . . Kazuko Kato : Zeros de la fonction holomorphe et bornee dans un polyhedre analytique de C 2 . . . . . . . Victor Katsnelson : Steiner and Weyl polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anand Prakash Singh : Spiraling Baker domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anatoliy Prykarpatsky : The algebraic Liouville integrability and the related Picard-Fuchs type equations Armen Sergeev : Quantization of universal Teichm¨ uller space: an interplay between complex analysis and quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.1 Clifford and quaternion analysis Hendrik de Bie : Clifford analysis for orthogonal, symplectic and finite reflection groups . . . . Cinzia Bisi : M¨ obius transformations and Poincar´e distance in the quaternionic setting . . . . Paula Cerejeiras : Wavelets invariant under reflection groups . . . . . . . . . . . . . . . . . . . Fabrizio Colombo : Some consequences of the quaternionic functional calculus . . . . . . . . . . Kevin Coulembier : Orthogonality of Clifford-Hermite polynomials in superspace. . . . . . . . Sirkka-Liisa Eriksson : Recent results on hyperbolic function theory . . . . . . . . . . . . . . . . Ming-Gang Fei : Symmetric properties of the Fourier transform in Clifford analysis setting . . Milton Ferreira : Factorization of M¨ obius gyrogroups - the paravector case . . . . . . . . . . . . Peter Franek : Higher spin analogues of the Dirac operator in two variables and its resolution . Ghislain R. Franssens : Cauchy kernels in ultrahyperbolic Clifford analysis – Huygens cases . . Graziano Gentili : Power series and analyticity over the quaternions . . . . . . . . . . . . . . . Anastasia Kisil : Isomorphic action of SL(2, R) on hypercomplex numbers . . . . . . . . . . . . Rolf Soeren Krausshar : Construction of 3D mappings on to the unit ball with the hypercomplex Lukas Krump : Explicit description of the resolution for 4 Dirac operators in dimension 6 . . Roman Lavicka : On polynomial solutions of Moisil-Theodoresco systems in Euclidean spaces . Matvei Libine : Quaternionic analysis, representation theory and Physics . . . . . . . . . . . .

6

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33 33 33 33 34 34 34 34 34 34 34 35 35 35 35

.

35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Szego kernel . . . . . . . . . . . . . . . . . . . . .

36 36 36 36 36 36 37 37 37 37 37 37 38 38 38 38 38

Maria Elena Luna-Elizarrar´ as : Hyperholomorphic functions in the sense of Moisil-Thodoresco and their different hyperderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mircea Martin : Dirac and semi-Dirac pairs of differential operators . . . . . . . . . . . . . . . . . . . . . Heikki Orelma : A differential form approach to Dirac operators on surfaces . . . . . . . . . . . . . . . . Dixan Pe˜ na Pe˜ na : CK-extension and Fischer decomposition for the inframonogenic functions . . . . . . . Alessandro Perotti : A new approach to slice-regularity on real algebras . . . . . . . . . . . . . . . . . . . . Yuying Qiao : Clifford analysis with higher order kernel over unbounded domains . . . . . . . . . . . . . . Guangbin Ren : Complex Dunkl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Ryan : p-Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irene Sabadini : Duality theorems for slice hyperholomorphic functions . . . . . . . . . . . . . . . . . . . Tomas Salac : Explicit description of operators in the resolution for the Dirac operator . . . . . . . . . . . Michael Shapiro : On the relation between the Fueter operator and the Cauchy-Riemann-type operators of Clifford analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Petr Somberg : Conformally invariant boundary valued problems for spinors and families of homomorphisms of generalized Verma modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frank Sommen : Clifford calculus in quantum variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Soucek : On relative BGG sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Caterina Stoppato : Regular Moebius transformations over the quaternions . . . . . . . . . . . . . . . . . Adrian Vajiac : Singularities of functions of one and several bicomplex variables . . . . . . . . . . . . . . Fabio Vlacci : Multiplicities of zeroes and poles of regular functions . . . . . . . . . . . . . . . . . . . . . . Zuzana Vlasakova : Gauss-Codazzi-Ricci equations in Riemannian, conformal, and CR geometry . . . . . Liesbet Van de Voorde : Compatibility conditions and higher spin Dirac operators . . . . . . . . . . . . . . II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras Swanhild Bernstein : Wavelets on spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Bock : On special monogenic power and Laurent series expansions and applications . . . . . . . Ruth Farwell : Spin gauge models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nelson Faustino : Further results in discrete Clifford analysis . . . . . . . . . . . . . . . . . . . . . . . . . Thanasis Fokas : Integrability in multidimensions, complexification and quaternions . . . . . . . . . . . . . Svetlin Georgiev : Note on the linear systems in quaternions . . . . . . . . . . . . . . . . . . . . . . . . . Jacques Helmstetter : Minimal algorithms for Lipschitzian elements and Vahlen matrices . . . . . . . . . Jeff Hogan : Clifford-Fourier transforms and hypercomplex signal processing . . . . . . . . . . . . . . . . . Uwe K¨ ahler : Discrete Clifford analysis by means of skew-Weyl relations . . . . . . . . . . . . . . . . . . . Vladimir Kisil : Hypercomplex analysis in the upper half-plane . . . . . . . . . . . . . . . . . . . . . . . . . Rolf Soeren Krausshar : Formulas for reproducing kernels of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to inhomogeneous Helmholtz equations . . . . . . . . Remi Leandre : The Ito transform for partial differential equations . . . . . . . . . . . . . . . . . . . . . . Dimitris Pinotsis : Quaternionic analysis and boundary value problems . . . . . . . . . . . . . . . . . . . Vitalii Shpakivskii : Integral theorems in a commutative three-dimensional harmonic algebra . . . . . . . . Wolfgang Spr¨ oßig : Initial boundary value problems with quaternionic analysis . . . . . . . . . . . . . . . . Tolksdorf, J¨ urgen : Real bi-graded Clifford modules, the Majorana equation and the standard model action Nelson Vieira : The regularized Schr¨ odinger semigroup acting on tensors with values in vector bundles . . III.1. Toeplitz operators and their applications Cristina Cˆ amara : On the relations between the kernel of a Toeplitz operator and the solutions to some associated Riemann-Hilbert problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luis Castro : Convolution type operators with symmetry in exterior wedge diffraction problems . . . . . . Miroslav Englis : Berezin transform on the harmonic Fock space . . . . . . . . . . . . . . . . . . . . . . . Sergey Grudsky : Inside the eigenvalues of certain Hermitian Toeplitz band matrices . . . . . . . . . . . . Turan G¨ urkanlı : Toeplitz operators of M (p, q, w)(Rd ) spaces . . . . . . . . . . . . . . . . . . . . . . . . . Oleksandr Karelin : Presentation of the kernel of a special structure matrix characteristic operator by the kernels of two operators one of them is a scalar characteristic operator . . . . . . . . . . . . . . . . . Edixon Rojas : Bounds for the kernel dimension of singular integral operators with Carleman shift . . . . Anabela Silva : Invertibility of matrix Wiener-Hopf plus Hankel operators with different Fourier symbols . Harald Upmeier : Flat Hilbert bundles and Toeplitz operators on symmetric spaces . . . . . . . . . . . . . Nikolai Vasilevski : Commutative algebras of Toeplitz operators on the unit ball . . . . . . . . . . . . . . . Kehe Zhu : Toeplitz operators on the Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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39 39 39 39 39 39 40 40 40 40

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40

. 40 . 41 . 41 . 41 . 41 . 41 . 42 . 42

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42 42 42 43 43 43 43 43 43 44 44

. 44 . 44 . 44 . 44 . 45 . 45 . 45 45 . 45 . 45 . 46 . 46 . 46 . 46 . 46 . 46 . 47 . 47 . 47

III.2. Reproducing kernels and related topics Belkacem Abdous : A general theory for kernel estimation of smooth functionals . . . . . . . . . . . . . . . Som Datt Sharma : Weighted composition operators on some spaces of analytic functions . . . . . . . . . . Keiko Fujita : Integral formulas on the boundary of some ball . . . . . . . . . . . . . . . . . . . . . . . . . . John Rowland Higgins : Paley–Wiener spaces and their reproducing formulae. . . . . . . . . . . . . . . . . . Darian Onchis : Irregular sampling in multiple-window spline-type spaces . . . . . . . . . . . . . . . . . . . Kazuo Takemura : Free boundary value problem for (−1)M (d/dx)2M and the best constant of Sobolev inequality

47 47 47 48 48 48 48

7

III.3. Modern aspects of the theory of integral transforms Liubov Britvina : Integral transforms related to generalized convolutions and their applications to solving integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiuhui Chen : Bedrosian identity for Blaschke products in n-parameter cases . . . . . . . . . . . . . . . . Dong Hyun Cho : Evaluation formulae for analogues of conditional analytic Feynman integrals over a function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diana Doli´canin : An equation with symmetrized fractional derivatives . . . . . . . . . . . . . . . . . . . . Hiroshi Fujiwara : Numerical real inversion of the Laplace transform by reproducing kernel and multipleprecision arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anatoly Kilbas : Method of integral transforms in the theory of fractional differential equations . . . . . . Bong Jin Kim : Notes on the analytic Feynman integral over paths in abstract Wiener space . . . . . . . . Sanja Konjik : On the fractional calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Koroleva : Integral transforms with extended generalized Mittag-Leffler function . . . . . . . . . . . Ljubica Oparnica : Systems of differential equations containing distributed order fractional derivative . . . Juri M. Rappoport : Some aspects of modified Kontorovitch-Lebedev integral transforms . . . . . . . . . . Semyon Yakubovich : A new class of polynomials related to the Kontorovich-Lebedev transform . . . . . .

48 . .

48 49

. .

49 49

. 49 . 49 . 50 . 50 . 50 . 50 . 50 . 50

III.4. Spaces of differentiable functions of several real variables and applications Alexandre Almeida : Hardy spaces with generalized parameter . . . . . . . . . . . . . . . . . . . . . . . . . . Tsegaye Gedif Ayele : Iterated norms in Nikol’ski˘ı-Besov type spaces with generalized smoothness . . . . . p(.) Ismail Aydın : Embeddings Properties of The Spaces Lw (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . Canay Aykol : On the boundedness of fractional B-maximal operators in the Lorentz spaces Lp,q,γ (Rn ) . . . Oleg Besov : Spaces of functions of fractional smoothness on an irregular domain . . . . . . . . . . . . . . . Santiago Boza : Rearrangement transformations on general measure spaces . . . . . . . . . . . . . . . . . . Maria Carro : Last developments on Rubio de Francia’s extrapolation theory . . . . . . . . . . . . . . . . . . Gurgen Dallakyan : On transformation of coordinates invariant relative to Sobolev spaces with polyhedral anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ismail Ekincioglu : The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted Lpw spaces with general weights . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Goldshtein : Composition Operators for Sobolev spaces of functions and differential forms . . . . . Vagif Guliyev : Boundedness of the fractional maximal operator and fractional integral operators in general Morrey type spaces and some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mubariz Hajibayov : Weighted estimates of generalized potentials in variable exponent Lebesque spaces . . . Ritva Hurri-Syrjanen : Our talk is on vanishing exponential integrability for Besov functions. . . . . . . . . Gennady Kalyabin : New sharp estimates for function in Sobolev spaces on finite Interval . . . . . . . . . . Leili Kusainova : On real interpolation of weighted Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . Elijah Liflyand : The Fourier transform of a radial function . . . . . . . . . . . . . . . . . . . . . . . . . . Yagub Mammadov : Necessary and sufficient conditions for the boundedness of Riesz potential in Morrey spaces associated with Dunkl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ana Moura Santos : Image normalization of Wiener-Hopf operators in diffraction problems . . . . . . . . . Bohum´ır Opic : Weighted estimates for the averaging integral operator and reverse H¨ older inequalities . . . Humberto Rafeiro : Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup Evgeniy Radkevich : On the Maxwell problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natasha Samko : Weighted potential operators in Morrey spaces. . . . . . . . . . . . . . . . . . . . . . . . . Stefan Samko : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kader Senouci : Equivalent semi-norms for Nikol’skii- Besov spaces on an interval . . . . . . . . . . . . . . Ayhan Serbetci : Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications Javier Soria : Translation-invariant bilinear operators with positive kernels . . . . . . . . . . . . . . . . . . . Sergey Tikhonov : Sharp inequalities for moduli of smoothness and K-functionals . . . . . . . . . . . . . . . Boris V. Trushin : Sobolev embedding theorems for a class of anisotropic irregular domains . . . . . . . . . Yusuf Zeren : Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.5. Analytic and harmonic function spaces Miloud Assal : Multiplier theorem in the setting of Laguerre hypergroups and applications . . . . . . . . . Boo Rim Choe : Progress on finite rank Toeplitz products . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Girela : Functions and operators in analytic Besov spaces . . . . . . . . . . . . . . . . . . . . . . . Maria Jose Gonzales : Square functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sanjiv Gupta : Convolutions of generic orbital measures in compact symmetric spaces . . . . . . . . . . . H. Turgay Kaptano˘ glu : Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Young Joo Lee : Sums of Toeplitz products on the Dirichlet space . . . . . . . . . . . . . . . . . . . . . . . Jasbir Singh Manhas : Weighted composition operators on weighted spaces of analytic functions . . . . . . Auxiliadora Marquez : Superposition operators between Qp spaces and Hardy spaces . . . . . . . . . . . . Malgorzata Michalska : Bounded Toeplitz and Hankel products on Bergman space . . . . . . . . . . . . . .

8

50 51 51 51 51 51 52 52 52 52 53 53 53 53 53 53 54 54 54 54 55 55 55 55 55 55 55 56 56 56

56 . 56 . 56 . 56 . 56 . 57 . . . . .

57 57 57 57 57

Kyesook Nam : Optimal norm estimate of the harmonic Bergman projection . . . . . . . . . . . Pekka Nieminen : Old and new on composition operators on VMOA and BMOA spaces . . . . . Maria Nowak : On Libera and Cesaro operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jordi Pau : Integration operators on weighted Bergman spaces . . . . . . . . . . . . . . . . . . . . Amol Sasane : Extension to an invertible matrix in Banach algebras of measures . . . . . . . . . Benoit F. Sehba : Multiplication operators on weighted BMOA spaces . . . . . . . . . . . . . . . . Pawel Sobolewski : Inequalities for Hardy spaces on the unit ball . . . . . . . . . . . . . . . . . . M¨ ubariz Tapdıgo˘ glu : On the Duhamel algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jari Taskinen : Toeplitz operators on Bergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . Luis Manuel Tovar : Hyperbolic weighted Bergman classes . . . . . . . . . . . . . . . . . . . . . . Dragan Vukotic : Multiplicative isometries and isometric zero-divisors . . . . . . . . . . . . . . . Zhijian Wu : Area operators on analytic function spacess . . . . . . . . . . . . . . . . . . . . . . . Hasi Wulan : Composition operators on BMOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wen Xu : Lacunary series and QK spaces on the unit ball . . . . . . . . . . . . . . . . . . . . . . Congli Yang : Some results on ϕ-Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Kehe Zhu : Holomorphic mean Lipschitz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . Nina Zorboska : Univalently induced closed range composition operators on the Bloch-type spaces

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. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

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57 58 58 58 58 58 58 58 58 59 59 59 59 59 59 59 59

III.6. Spectral theory Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: surface potentials, equations at the boundary, and corresponding transmission problems. . . . . . . . . . . . . . . . . . . . . . Shavkat Alimov : On the spectral expansions associated with Laplace-Beltrami operator . . . . . . . . . . . . Victor Burenkov : Sharp spectral stability estimates for higher order elliptic operators . . . . . . . . . . . . Daniel Elton : Strong field asymptotics for zero modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leander Geisinger : A universal bound for the trace of the heat kernel . . . . . . . . . . . . . . . . . . . . . Tigran Harutyunyan : The eigenvalues function of the family of Sturm-Liouville operators and its applications Jan Janas : Generalized eigenvectors of some Jacobi matrices in the critical case . . . . . . . . . . . . . . . Thomas Krainer : Trace expansions for elliptic cone operators . . . . . . . . . . . . . . . . . . . . . . . . . . Pier Domenico Lamberti : Stability estimates for eigenfunctions of elliptic operators on variable domains . . Oleksii Mokhonko : Spectral theory of the normal operator with the spectra on an algebraic curve . . . . . . Jiri Neustupa : Spectral properties of operators arising from modelling of flows around rotating bodies . . . . Serge Richard : New formulae for the wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedetto Silvestri : Spectral bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Strohmaier : Scattering theory for manifolds and the scattering length . . . . . . . . . . . . . . . Yuriy Tomilov : Spectrum and wandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tomio Umeda : Eigenfunctions at the threshold energies of magnetic Dirac operators . . . . . . . . . . . . .

59

IV.1. Pseudo-differential operators Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: Dirichlet and Neumann problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chikh Bouzar : Generalized ultradistributions and their microlocal analysis . . . . . . . . . . . . . . . . . . Ernesto Buzano : Some remarks on the Sj¨ ostrand class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viorel Catana : The heat equation for the generalized Hermite and the generalized Landau operators . . . . Leon Cohen : Generalization of the Weyl rule for arbitrary operators . . . . . . . . . . . . . . . . . . . . . . Elena Cordero : Sharp results for the STFT and localization operators . . . . . . . . . . . . . . . . . . . . . Yasuo Chiba : Fuchsian mild microfunctions with fractional order and their applications to hyperbolic equations Paulo Dattori da Silva : About Gevrey semi-global solvability of a class of complex planar vector fields with degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Julio Delgado : Invertibility for a class of degenerate elliptic operators . . . . . . . . . . . . . . . . . . . . . Kenro Furutani : Heat kernel of a sub-Laplacian and Grushin type operators . . . . . . . . . . . . . . . . . . Lorenzo Galleani : Time-frequency analysis of stochastic differential equations . . . . . . . . . . . . . . . . . Gianluca Garello : Lp -microlocal regularity for pseudodifferential operators of quasi-homogeneous type . . . Claudia Garetto : Generalized Fourier integral operators methods for hyperbolic problems . . . . . . . . . . . Juan Gil : Resolvents of regular singular elliptic operators on a quantum graph . . . . . . . . . . . . . . . . Todor Gramchev : Hyperbolic systems of pseudodifferential equations with irregular symbols in t admitting superlinear growth for |x| → ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernhard Gramsch : Analytic perturbations for special Fr´echet operator algebras in the microlocal analysis . G¨ unther H¨ ormann : The Cauchy problem for a paraxial wave equation with non-smooth symbols . . . . . . Eug´enie Hunsicker : Pseudodifferential operators on locally symmetric spaces . . . . . . . . . . . . . . . . . Wataru Ichinose : On the continuity of the solutions with respect to the electromagnetic potentials to the Schr¨ odinger and the Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chisato Iwasaki : Calculus of pseudo-differential operators and a local index of Dirac operators . . . . . . . Jon Johnsen : On the theory of type 1, 1-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuryi Karlovych : Pseudo-differential operators with discontinuous symbols and their applications . . . . . . Thomas Krainer : On maximal regularity for parabolic equations on complete Riemannian manifolds . . . . Roberto de Leo : On the cohomological equation in the plane for regular vector fields . . . . . . . . . . . .

62

60 60 60 60 60 61 61 61 61 61 62 62 62 62 62 62

63 63 63 63 63 63 63 64 64 64 64 64 65 65 65 65 65 65 65 66 66 66 66 66

9

Yu Liu : Lp -boundedness and compactness of localization operators associated with Stockwell transform . . . Jean-Andr´e Marti : About transport equation with irregular coefficient and data . . . . . . . . . . . . . . . . Shahla Molahajloo : The Heat Kernel of τ -Twisted Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . Alessandro Morando : Regularity of characteristic initial-boundary value problems for symmetrizable systems David Natroshvili : Application of pseudodifferential equations in stress singularity analysis for thermoelectro-magneto-elasticity problems: a new approach for calculation of stress singularity exponents . . . Alessandro Oliaro : Wigner type transforms and pseudodifferential operators . . . . . . . . . . . . . . . . . . Michael Oberguggenberger : Local regularity of solutions to PDEs by asymptotic methods . . . . . . . . . . Nusrat Rajabov : Modern results by theory of the three dimensional Volterra type linear integral equations with singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frederic Rochon : The adiabatic limit of the Chern character . . . . . . . . . . . . . . . . . . . . . . . . . . Bert-Wolfgang Schulze : Boundary value problems as edge problems . . . . . . . . . . . . . . . . . . . . . . Elmar Schrohe : Noncommutative residues and projections associated to boundary value problems . . . . . J¨ org Seiler : On maximal regularity for mixed order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Tatyana Shaposhnikova : Dirichlet problem for higher order elliptic systems with BMO assumptions on the coefficients and the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hidetoshi Tahara : Gevrey regularities of solutions of nonlinear singular partial differential equations . . . . Nenad Teofanov : Wave-front sets and SG type operators in Fourier-Lebesgue spaces . . . . . . . . . . . . . Joachim Toft : Wave-front sets of Fourier Lebesgue types . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ville Turunen : Pseudo-differential operators and symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Vasilyev : Pseudo differential equations and boundary value problems in non-smooth domains . . Andr´ as Vasy : Diffraction at corners for the wave equation on differential forms . . . . . . . . . . . . . . . Ingo Witt : Formation of singularities near Morse points . . . . . . . . . . . . . . . . . . . . . . . . . . . . Man Wah Wong : Phases of modified Stockwell transforms and instantaneous frequencies . . . . . . . . . . Hongmei Zhu : Generalized cosine transforms in image compression . . . . . . . . . . . . . . . . . . . . . . IV.2. Dispersive equations Marcello D’Abbico : Lp –Lq estimates for hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . Q-Heung Choi : Multiple solutions for non-linear parabolic systems . . . . . . . . . . . . . . . . . . . . . . Ferruccio Colombini : Local sovability beyond condition ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniele Del Santo : Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients Marcello Ebert : On the loss of regularity for a class of weakly hyperbolic operators . . . . . . . . . . . . . Daoyuan Fang : Zakharov system in infinite energy space . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anahit Galstyan : Wave equation in Einstein-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . Vladimir Georgiev : Stability of solitary waves for Hartree type equation . . . . . . . . . . . . . . . . . . . . Marina Ghisi : Hyperbolic-parabolic singular perturbations for Kirchhoff-equations . . . . . . . . . . . . . . . Massimo Gobbino : Existence and uniqueness results for Kirchhoff equations in Gevrey-type spaces . . . . . Torsten Herrmann : Precise loss of derivatives for evolution type models . . . . . . . . . . . . . . . . . . . . Fumihiko Hirosawa : Wave equations with time dependent coefficients . . . . . . . . . . . . . . . . . . . . . Tacksun Jung : Critical point theory applied to a class of systems of super-quadratic wave equations . . . . Lavi Karp : On the well-posdness of the vacuum Einstein equations . . . . . . . . . . . . . . . . . . . . . . Hideo Kubo : Generalized wave operator for a system of nonlinear wave equations . . . . . . . . . . . . . . Tokio Matsuyama : Strichartz estimates for hyperbolic equations in an exterior domain . . . . . . . . . . . Kiyoshi Mochizuki : Uniform resolvent estimates and smoothing effects for magnetic Schr¨ odinger operators Hideo Nakazawa : Decay and scattering for wave equations with dissipations in layered media . . . . . . . . Tatsuo Nishitani : On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 wellposedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rainer Picard : On the structure of the material law in a linear model of poro-elasticity . . . . . . . . . . . Marco Pivetta : Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuous coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Reissig : The log-effect for 2 by 2 hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . Jun-ichi Saito : The Boussinesq equations based on the hydrostatic approximation . . . . . . . . . . . . . . . Ryuichi Suzuki : Blow-up of solutions of a quasilinear parablolic equation . . . . . . . . . . . . . . . . . . . Hiroshi Uesaka : Blow-up and a blow-up boundary for a semilinear wave equation with some convolution nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karen Yagdjian : Fundamental solutions for hyperbolic operators with variable coefficients . . . . . . . . . . Borislav Yordanov : Global existence in Sobolev spaces for a class of nonlinear Kirchhoff equations . . . . . IV.3. Control and optimisation of nonlinear evolutionary systems Lorena Bociu : Global well-posedness and long-time behavior of solutions to a wave equation . . . . . . . . Mahdi Boukrouche : Distributed optimal controls for second kind parabolic variational inequalities . . . . Muriel Boulakia : Controllability of a fluid-structure interaction problem . . . . . . . . . . . . . . . . . . Marcello Cavalcanti : Uniform decay rate estimates for the wave equation on compact surfaces and locally distributed damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moez Daoulatli : Rate of decay for non-autonomous damped wave systems . . . . . . . . . . . . . . . . . .

10

66 67 67 67 67 67 68 68 68 68 68 69 69 69 69 69 70 70 70 70 70 70 71 71 71 71 71 72 72 72 72 72 73 73 73 73 73 74 74 74 74 74 74 74 75 75 75 75 75 76

76 . 76 . 76 . 76 . .

77 77

Valeria Domingos Cavalcanti : On qualitative aspects for the damped Korteweg-de Vries and Airy type equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Eller : Optimal control of waves in anisotropic media via conservative boundary conditions . . . Genni Fragnelli : Stability for some nonlinear damped wave equations . . . . . . . . . . . . . . . . . . . . Anahit Galstyan : Global existence for the one-dimensional semilinear Tricomi-type equation . . . . . . . Catherine Lebiedzik : Optimal control of a thermoelastic structural acoustic model . . . . . . . . . . . . . Walter Littman : The Balayage method: Boundary control of a thermo-elastic plate . . . . . . . . . . . . . Paola Loreti : Hopf-Lax type formulas and Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . Vyacheslav Maksimov : Investigation of boundary control problems by on-line inversion technique . . . . . Patrick Martinez : Null controllability properties of some degenerate parabolic equations . . . . . . . . . . Maria Grazia Naso : Dissipation in contact problems: an overview and some recent results . . . . . . . . Luciano Pandolfi : Heat equations with memory: a Riesz basis approach . . . . . . . . . . . . . . . . . . . Michael Renardy : A note on a class of observability problems for PDEs . . . . . . . . . . . . . . . . . . . Roland Schnaubelt : Invariant manifolds for parabolic problems with dynamical boundary conditions . . . Ilya Shvartsman : On regularity properties of optimal control and Lagrange multipliers . . . . . . . . . . . Daniela Sforza : Evolution equations with memory terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Toundykov : Stabilization of structure-acoustics interactions for a Reissner-Mindlin plate by localized nonlinear boundary feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Julie Valein : Exponential stability of the wave equation with boundary time varying delay . . . . . . . . . Masahiro Yamamoto : State estimation for some parabolic systems . . . . . . . . . . . . . . . . . . . . . . Jean-Paul Zolesio : Euler flow and Morphing Shape Metric . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 77 77 77 78 78 78 78 78 78 79 79 79 79 79

. 79 . 80 . 80 . 80

IV.4. Nonlinear partial differential equations Piero D’Ancona : Evolution equations in nonflat waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . Mersaid Aripov : Investigation of solutions of one not divergent type . . . . . . . . . . . . . . . . . . . . . . Davide Catania : Asymptotic behavior of subparabolic functions . . . . . . . . . . . . . . . . . . . . . . . . Kuan-Ju Chen : On multiple solutions of concave and convex effects for nonlinear elliptic equation on RN Kazuyuki Doi : Nonlinear gauge invariant evolution of the plane wave . . . . . . . . . . . . . . . . . . . . . Mohammad Dehghan : New approach to solve linear parabolic problems via semigroup approximation . . . Albert Erkip : Global existence and blow-up for the nonlocal nonlinear Cauchy problem . . . . . . . . . . . . Marius Ghergu : Qualitative properties for reaction-diffusion systems modelling chemical reactions . . . . . Marco Antonio Taneco-Hern´ andez : Scattering in the zero-mass Lamb system . . . . . . . . . . . . . . . . . Soichiro Katayama : Global existence for systems of the nonlinear wave and Klein-Gordon equations in 3D Hideo Kubo : Global existence for nonlinear wave equations exterior to an obstacle in 2D . . . . . . . . . . Petr Kucera : Remark on Navier-Stokes equations with mixed boundary conditions . . . . . . . . . . . . . . Ut van Le : Contraction-Galerkin method for a semi-linear wave equation with a boundary-like antiperiodic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sandra Lucente : p − q systems of nonlinear Schrodinger equations . . . . . . . . . . . . . . . . . . . . . . . Satoshi Masaki : Semiclassical analysis for nonlinear Schrodinger equations . . . . . . . . . . . . . . . . . . Gianluca Mola : 3-D viscous Cahn-Hilliard equation with memory . . . . . . . . . . . . . . . . . . . . . . . Itir Mogultay : A symmetric error estimate for Galerkin approximations of time dependant Navier-Stokes equations in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masahito Ohta : Stability of standing waves for some systems of nonlinear Schr¨ odinger equations with three-wave interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Reissig : Decay rates for wave models with structural damping . . . . . . . . . . . . . . . . . . . . . Yoshihiro Shibata : Stability theorems in the theory of mathematical fluid mechanics . . . . . . . . . . . . . L´ aszl´ o Simon : On singular systems of parabolic functional equations . . . . . . . . . . . . . . . . . . . . . Zdenek Skalak : Survey of recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atanas Stefanov : Conditional stability theorems for Klein-Gordon type equations . . . . . . . . . . . . . . Sergio Spagnolo : A regularity result for a class of semilinear hyperbolic equations . . . . . . . . . . . . . . . Kamal Soltanov : On nonlinear equations, fixed-point theorems and their applications . . . . . . . . . . . . . Alessandro Teta : Dynamics of a quantum particle in a cloud chamber . . . . . . . . . . . . . . . . . . . . . Yoshihiro Ueda : Half space problem for the damped wave equation with a non-convex convection term . . . Nicola Visciglia : On the time-decay of solutions to a family of defocusing NLS . . . . . . . . . . . . . . . . Karen Yagdjian : The semilinear Klein-Gordon equation in de Sitter spacetime . . . . . . . . . . . . . . . .

80 80 80 81 81 81 81 81 81 82 82 82 82

IV.5. Asymptotic and multiscale analysis Natalia Babych : On the essential spectrum and singularities of solutions for Lam´e problem in cuspoidal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michel Bellieud : Torsion effects in elastic composites with high contrast . . . . . . . . . . . . . . . . . . . Yves Capdeboscq : Enhanced resolution in structured media . . . . . . . . . . . . . . . . . . . . . . . . . . Juan Casado-Diaz : Homogenization of elliptic partial differential equations with unbounded coefficients in dimension two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Cherdantsev : Two-scale Γ-convergence and its applications to homogenisation of non-linear highcontrast problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

82 83 83 83 83 83 83 84 84 84 84 84 84 84 84 85 85

. 85 . 85 . 85 .

86

.

86

11

Valentina Alekseevna Golubeva : Construction of the two-parametric generalizations of the KnizhnikZamolodchikov equations of Bn type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabricio Macia : Long-time behavior for the Wigner equation and semiclassical limits in heterogeneous media Peter Markowich : On nonlinear dispersive equations in periodic structures: Semiclassical limits and numerical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karsten Matthies : Derivation of Boltzmann-type equations from hard-sphere dynamics . . . . . . . . . . . Bernd Schmidt : Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valery Smyshlyaev : Homogenization with partial degeneracies: analytic aspects and applications . . . . . .

86 86 86 87 87 87

V.1. Inverse problems Abdellatif El Badia : An inverse conductivity problem with a single measurement . . . . . . . . . . . . . . . Fabrizio Colombo : Global in time existence and uniqueness results for some integrodifferential identification problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mourad Choulli : Stability estimate for an inverse problem for the magnetic Schr¨ odinger equation from the Dirichlet-to-Neumann map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikko Kaasalainen : Optimal combination of data modes in inverse problems: maximum compatibility estimate Christian Daveau : On an inverse problem for a linear heat conduction problem . . . . . . . . . . . . . . . Matti Lassas : Inverse problems for wave equation and a modified time reversal method . . . . . . . . . . . . Koung Hee Leem : Picard condition based regularization techniques in inverse obstacle scattering . . . . . . William Lionheart : Limited data problems in tensor tomography . . . . . . . . . . . . . . . . . . . . . . . . Marco Marletta : The finite data non-selfadjoint inverse resonance problem . . . . . . . . . . . . . . . . . . Tsutomu Matsuura : Numerical solutions of nonlinear simultaneous equations . . . . . . . . . . . . . . . . . George Pelekanos : A fixed-point algorithm for determining the regularization parameter in inverse scattering Roland Potthast : A time domain probe method for inverse scattering problems . . . . . . . . . . . . . . . . Saburou Saitoh : Explicit and direct representations of the solutions of nonlinear simultaneous equations . Vassilios Sevroglou : Direct and inverse mixed impedance problems in linear elasticity . . . . . . . . . . . . Igor Trooshin : On inverse scattering for nonsymmetric operators . . . . . . . . . . . . . . . . . . . . . . .

87 87

V.2. Stochastic analysis David Applebaum : Cylindrical Levy processes in Banach space . . . . . . . . . . . . . . . . . . . . . . . Vlad Bally : Integration by parts for locally smooth laws and applications to jump type diffusions . . . . . Dorje Brody : Information and asset pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Caruana : A (rough) pathwise approach to fully non-linear stochastic partial differential equations Dan Crisan : Solving backward stochastic differential equations using cubature methods . . . . . . . . . . . Ana Bela Cruzeiro : Some results on Lagrangian Navier-Stokes flows . . . . . . . . . . . . . . . . . . . . Alexander Davie : A uniqueness problem for SDEs and a related estimate for transition functions . . . . Mark H. A. Davis : Risk-sensitive portfolio optimization with jump-diffusion asset prices . . . . . . . . . Istvan Gyongy : Accelerated numerical schemes for nonlinear filtering . . . . . . . . . . . . . . . . . . . . Martin Hairer : Periodic homogenisation with an interface . . . . . . . . . . . . . . . . . . . . . . . . . . . Lane Hughston : Wiener chaos models for interest rates and foreign exchange . . . . . . . . . . . . . . . . Saul Jacka : Minimising the time to a decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark Kelbert : Markov process representations for polyharmonic functions . . . . . . . . . . . . . . . . . Wilfried Kendall : Networks and Poisson line patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vassili Kolokoltsov : The Levy-Khinchine type operators with variable Lipschitz continuous coefficients and stochastic differential equations driven by nonlinear Levy noise . . . . . . . . . . . . . . . . . . . . . Thomas Kurtz : Equivalence of stochastic equations and martingale problems . . . . . . . . . . . . . . . . Xue-Mei Li : Aida’s logarithmic Sobolev inequality with weights and Poincare inequalities. . . . . . . . . . Terence Lyons : Evolution equations for communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aleksandar Mijatovic : On the martingale property of certain local martingale . . . . . . . . . . . . . . . . Khairia El-Said El-Nadi : On some stochastic dynamical systems and cancer . . . . . . . . . . . . . . . . Anastasia Papavasiliou : Statistical inference for rough differential equations . . . . . . . . . . . . . . . . Martijn Pistorius : First passage for stochastic volatility models . . . . . . . . . . . . . . . . . . . . . . . . Boris Rozovsky : Unbiased random perturbations of Navier-Stokes equation . . . . . . . . . . . . . . . . . Marta Sanz-Sol´e : A Poisson equation with fractional noise . . . . . . . . . . . . . . . . . . . . . . . . . . Radu Tunaru : Constructing discrete exact approximations algorithms for financial calculus from weak convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Tretyakov : Numerical methods for parabolic SPDEs based on the averaging-over-characteristics formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elena Usoltseva : Consistent estimator in AFTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

90 90 90 90 91 91 91 91 91 91 91 92 92 92 92

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92 92 93 93 93 93 93 94 94 94

.

94

. .

94 94

V.3. Coercivity and functional inequalities Franck Barthe : Remarks on non-interacting conservative spin systems . . . . . . . . . . . . . . . . . . . Sergey Bobkov : On weak forms of Poincare-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . Messaoud Boulbrachene : L∞ -Error estimate for variational inequalities with vanishing zero order term Federica Dragoni : Convexity along vector fields and application to equations of Monge-Amp`ere type . . Ivan Gentil : Φ-entropy inequalities for diffusion semigroups . . . . . . . . . . . . . . . . . . . . . . . . .

12

. . . . .

88 88 88 88 88 88 89 89 89 89 89 89 89 90

95 . 95 . 95 . 95 . 95 . 95

Alexander Grigoryan : On positive solutions of semi-linear elliptic inequalities on manifolds Martin Hairer : Hypoellipticity in infinite dimensions . . . . . . . . . . . . . . . . . . . . . . Waldemar Hebisch : Logaritmic Sobolev inequality on nilpotent groups . . . . . . . . . . . . Nolwen Huet : Isoperimetry for spherically symmetric log-concave probability measures . . . James Inglis : Operators on the Heisenberg group with discrete spectra . . . . . . . . . . . . Mikhail Neklyudov : Liggett inequality and interacting particle systems . . . . . . . . . . . . Felix Otto : A new criterion for a covariance estimate . . . . . . . . . . . . . . . . . . . . Ioannis Papageorgiou : The Log-Sobolev inequality for non quadratic interactions . . . . . . Cyril Roberto : Isoperimetry for product probability measures . . . . . . . . . . . . . . . . .

. . . . . . . . .

95 96 96 96 96 96 96 96 96

V.4. Dynamical systems Marco Abate : Poincar´e-Bendixson theorems in holomorphic dynamics . . . . . . . . . . . . . . . . . . . . Jos´e Ferreira Alves : On the liftability of absolutely continuous ergodic expanding measures. . . . . . . . . . Flavio Abdenur : New results on stability and genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pierre Berger : Abundance of one dimensional non uniformly hyperbolic attractors for surface dynamics . . Svetlana Aleksandrovna Budochkina : First integrals in mechanics of infinite-dimensional systems . . . . . Keith Burns : Partial hyperbolicity and ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Ren´e Chazottes : On tilings, multidimensional subshifts of finite type and quasicrystals . . . . . . . . . Yi-Chiuan Chen : On topological entropy of billiard tables with small inner scatterers . . . . . . . . . . . . . Bau-Sen Du : On the nature of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Field : Mixing for flows and skew extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jorge Freitas : Rates of mixing, large deviations and recurrence times . . . . . . . . . . . . . . . . . . . . . Giovanni Forni : Limiting distributions for horocycle flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valery Gaiko : Limit cycle problems and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Jordan : Hausdorff dimension of Projections of McMullen-Bedford carpets . . . . . . . . . . . . . Jan Cees van der Meer : Fourfold 1:1 resonance, relative equilibria and moment polytopes . . . . . . . . . . Matthew Nicol : A dynamical Borel-Cantelli lemma for a class of non-uniformly hyperbolic systems . . . . Asad Niknam : Approximately inner C ∗ -dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Giovanni Panti : Dynamical systems arising in algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . Chen-chang Peng : Existence of transversal homoclinic orbits for Arneodo-Coullet-Tresser map . . . . . . . Martin Rasmussen : Bifurcations of random diffeomorphisms with bounded noise . . . . . . . . . . . . . . . Felix Sadyrbaev : Bifurcations of period annuli and solutions of nonlinear boundary value problems . . . . . J¨ org Schmeling : Large intersection properties of some invariant sets in number-theoretic dynamical systems Mike Todd : Thermodynamic formalism for unimodal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiudong Wang : Dynamics of periodically perturbed homoclinic solutions . . . . . . . . . . . . . . . . . . . .

97 97 97 97 97 97 97 98 98 98 98 98 98 98 99 99 99 99 99 99 100 100 100 100 100

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. . . . . . . . .

V.5. Functional differential and difference equations 100 Jarom´ır Baˇstinec : Oscillation and non-oscillation of solutions of linear second order discrete delayed equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Leonid Berezansky : New stability conditions for linear differential equations with several delays . . . . . . . 101 Aleksandr Boichuk : Boundary-value problems for differential systems with a single delay . . . . . . . . . . 101 Josef Dibl´ık : Representation of solutions of linear differential and discrete systems and their controllability 101 Alexander Domoshnitsky : Maximum principles and nonoscillation intervals in the theory of functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Marcia Federson : Averaging for impulsive functional differential equations: a new approach . . . . . . . . . 101 Yakov Goltser : Some bifurcation problems in the theory quasilinear integro differential equations . . . . . . 102 Istv´ an Gy¨ ori : Stability in Volterra type population model equations with delays . . . . . . . . . . . . . . . . 102 Ferenc Hartung : On parameter dependence in functional differential equations with state-dependent delays 102 Zeynep Kayar : Lyapunov type inequalities for nonlinear impulsive differential systems . . . . . . . . . . . . 102 Conall Kelly : Evaluating the stochastic theta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Gabor Kiss : Delay-distribution effect on stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Martina Langerov´ a : Solutions of linear impulsive differential systems bounded on the entire real axis . . . . 102 Malgorzata Migda : Oscillatory and asymptotic properties of solutions of higher-order difference equations of neutral type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 ¨ Abdullah Ozbekler : Principal and non-principal solutions of impulsive differential equations with applications103 Mihali Pituk : Nonnegative iterations with asymptotically constant coefficients . . . . . . . . . . . . . . . . . 103 Irena Rachunkova : On singular models arising in hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 103 Andrejs Reinfelds : Decoupling and simplifying of noninvertible difference equations in the neighbourhood of invariant manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 David W. Reynolds : Precise asymptotic behaviour of solutions of Volterra equations with delay . . . . . . 103 Alexandra Rodkina : On local stability of solutions of stochastic difference equations . . . . . . . . . . . . . 104 Miroslava R˚ uˇziˇckov´ a : Convergence of the solutions of a differential equation with two delayed terms . . . . 104 Vladimir Mikhailovich Savchin : Inverse problems of the calculus of variations for functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Ewa Schmeidel : Existence and nonexistence of asymptotically periodic solutions of Volterra linear difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

13

Andrei Shindiapin : Gene regulatory networks and delay equations . . . . . . . . . . . . . . . . . . . . . . . 104 Benzion Shklyar : The moment problem approach for the zero controllability of ecolution equations . . . . . 105 Svatoslav Stanek : Properties of maximal solutions of autonomous functional-differential equations with state-dependent deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Stevo Stevic : Boundedness character of some classes of difference equations . . . . . . . . . . . . . . . . . . 105 Milan Tvrdy : Continuous dependence of solutions of generalized ordinary differential equations on a parameter105 ¨ Mehmet Unal : Lyapunov type inequalities on time scales: A survey . . . . . . . . . . . . . . . . . . . . . . 105 A˘ gacık Zafer : Interval criteria for oscillation of delay dynamic equations with mixed nonlinearities . . . . . 105 V.6. Mathematical biology Robert Gilbert : Cancellous bone with a random pore structure . . . . . . . . . . . . . . . . . . . Irina Alekseevna Gainova : New computer technologies for the construction and numerical analysis ematical models for molecular genetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . Sandra Ilic : Application of the multiscale FEM in modeling the cancellous bone . . . . . . . . . Mark D. Ryser : Bone growth and destruction at the cellular level: a mathematical model . . . .

. . . . . of math. . . . . . . . . . . . . . .

105 . 106 . 106 . 106 . 106

VI. Others 106 Ruben Airapetyan : The relationship between Bezoutian matrix and Newton’s matrix of divided differences and separation of zeros of interpolation polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Hadeel Alkutubi : Bayesian shrinkage estimation of parameter exponential distribution . . . . . . . . . . . . 107 Mohammed Bokhari : Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem107 Zoubir Dahmani : The ADM method and the Tanh method for solving some non linear evolutions equations 107 Anvar Hasanov : Boundary-value problems for generalized axially-symmetric Helmholtz equation . . . . . . . 107 Maximilian Hasler : Asymptotic extension of topological modules and algebras . . . . . . . . . . . . . . . . . 107 S. Moghtada Hashemiparast : Approximation of fractional derivatives . . . . . . . . . . . . . . . . . . . . . 107 Hailiza Kamarulhaili : Discrepancy estimate for uniformly distributed sequence . . . . . . . . . . . . . . . . 108 Erdal Karapinar : Bounded linear operators on l-power series spaces . . . . . . . . . . . . . . . . . . . . . . 108 Erkinjon Karimov : On a three-dimensional elliptic equation with singular coefficients . . . . . . . . . . . . 108 Nabiullah Khan : A unified presentation of a class of generalized Humbert polynomials . . . . . . . . . . . . 108 Lixia Liu : Direct estimate for modified beta operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Eduard Marusic-Paloka : Mathematical model of an undergorund nuclear waste disposal site . . . . . . . . . 108 Abdeslam Mimouni : Compact and coprime packedness and semistar operations . . . . . . . . . . . . . . . 108 S. A. Mohiuddine : Characterization of some matrix classes involving (σ, λ)-convergence . . . . . . . . . . . 109 Mohammad Mursaleen : Sequence spaces of invariant mean and some matrix transformations . . . . . . . . 109 Ali Mussa : New convection theory for thermal plasma and NHD convection in rapidly rotating spherical configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Kourosh Nourouzi : Characterizations of Isometries on 2-modular spaces . . . . . . . . . . . . . . . . . . . . 109 Shariefuddin Pirzada : On r-imbalances in tripartite r-digraphs . . . . . . . . . . . . . . . . . . . . . . . . . 110 Hashem Parvaneh Masiha : Invariance conditions and amenability of locally compact groups . . . . . . . . . 110 Zaure Rakisheva : Motion stabilisation of a solid body with fixed point . . . . . . . . . . . . . . . . . . . . . 110 Lyazzat Sarybekova : A Lizorkin type theorem for Fourier series multipliers in regular systems . . . . . . . 110 Pedro A. Santos : Inverse-closedness problems in the stability of sequences in Banach Algebras . . . . . . . 110 Ridha Selmi : Smoothing effects for periodic NSE in critical Sobolev space . . . . . . . . . . . . . . . . . . . 110 Mariana Sibiceanu : Large deviations and almost sure convergence . . . . . . . . . . . . . . . . . . . . . . . 111 Tanfer Tanriverdi : The k- Model in Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Johnson Olaleru : The equivalence between modified Mann (with errors), Ishikawa (with errors), Noor (with errors) and modified multi-step iterations (with errors) for non-Lipschitzian strongly successively pseudo-contractive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Serap Oztop : A characterization for multipliers of weighted Banach valued Lp (G)-spaces . . . . . . . . . . 111 Karlyga Zhilisbaeva : Stationary motion of the dynamical symmetric satellite in the geomagnetic field . . . 111

Index

14

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Plenary talks The Q-tensor theory of liquid crystals Sir John Ball Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, U.K. [email protected] The lecture will survey what is known about the mathematics of the de Gennes Q-tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lecture will describe the relation of the theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe. This is joint work with Apala Majumdar and Arghir Zarnescu. —————— Sir John Ball, FRS, is Sedleian Professor of Natural Philosophy at the University of Oxford and director of the Oxford Centre for Nonlinear PDE. He was president of the International Mathematical Union from 2003 to 2006. ————————————

Asymptotic equivariant index of Toeplitz operators and AttiyahWeinstein conjecture Louis Boutet de Monvel Universit´e Pierre et Marie Curie, Institut de Math´ematiques de Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE [email protected] The equivariant index of transversally elliptic equivariant operators was introduced by M.F. Atiyah (1974); it is a virtual trace class representation of a compact group, or equivalently the character of this representation, which is a central distribution. This does not make sense for general Toeplitz operators because the Toeplitz space where they act is only defined up to a finite dimensional space. The asymptotic index is an avatar of this, which works for Toeplitz operators : essentially it is a virtual trace class representation mod finite representations; equivalently its character is a singularity (distribution mod C ∞ ). It still is compatible with many natural operations, in particular the direct image by homogeneous symplectic maps. With E. Leichtnam, X. Tang and A. Weinstein, we have used this theory to give a new natural proof of the Atiyah-Weinstein conjecture (which was proved by C. Epstein): let X, X 0 be two compact strictly pseudoconvex boundaries (of complex domains): they carry natural cooriented contact structures. If f : X → X 0 is a contact isomorphism, we define the holomorphic pushforward Tf : u 7→ S 0 (u ◦ f −1 ) where u is the boundary value of a holomorphic function, and S 0 is the Szeg¨o projector, i.e. the orthogonal projector on the subspace of boundary values of holomorphic functions in L2 (X 0 ) (ker ∂¯b ). It is well known that Tf is a Fredholm operator; the Weinstein conjecture proposed a topological formula for its index. A particular case of this, proposed earlier by Atiyah, is the following: let V, V 0 be two smooth compact manifolds, and f a homogeneous symplectic isomorphism T ∗ V − {0} → T ∗ V 0 − {0} (equivalently a contact isomorphism between the cotangent spheres); then there exists an elliptic Fourier integral operator attached to f , whose index is given essentially by the same formula (this is a special case of the former because, if V is real analytic, the algebra of pseudodifferential operators acting on distributions is isomorphic to the algebra of Toeplitz operators acting on holomorphic boundary values on the boundary of a small tubular neighborhood of V in its complexification). One difficulty in this problem is that, since we are modifying the boundary CR structures (there are two of them), we are typically in the framework of general Toeplitz operators where the index is not well defined. Our way out was to construct a related G-elliptic operator where the index is repeated infinitely many times, but still well related geomerically to the problem, so the asymptotic index theory can be used.

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Atiyah, M.F. Elliptic operators and compact groups. Lecture Notes in Mathematics, Vol. 401. SpringerVerlag, Berlin-New York, 1974. Boutet de Monvel, L. Asymptotic equivariant index of Toeplitz operators, RIMS Kokyuroku Bessatsu (2008). Boutet de Monvel, L.; Leichtnam E.; Tang, X. ; Weinstein A. Asymptotic equivariant index of Toeplitz operators and relative index of CR structures arXiv:0808.1365v1; to appear in the Duistermaat 65 volume, Progress in Math, Birkh¨ auser. Weinstein, A.: Some questions about the index of quantized contact transformations RIMS Kokyuroku No. 1014, pages 1-14, 1997. —————— Louis Boutet de Monvel was awarded with the 2007 Medaille Emile Picard of the French Academy of Sciences. ————————————

Non-self-adjoint spectral theory Brian Davies Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K. [email protected] Over the last twenty years there has been remarkable progress in understanding the spectral behaviour of highly non-self-adjoint operators, particularly differential operators, partly as the result of numerical experiments. The lecture will describe some of the discoveries that have been made, and theorems proved, and will contrast them with the very different spectral behaviour of self-adjoint operators. Connections with so-called pseudospectral theory, that is bounds on the norms of the resolvent operators, will be explained and illustrated. —————— Brian Davies, FRS, is Professor of Mathematics at King’s College London. In 1998 he was awarded the Senior Berwick Prize of the LMS. Brian Davies was president of the London Mathematical Society from 2007 to 2009. ————————————

Asymptotic analysis and complex differential geometry Simon Donaldson Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, U.K. [email protected] A long-standing problem in complex differential geometry is to find various preferred metrics on a complex manifold. These include K¨ ahler-Einstein, constant scalar curvature and extremal metrics. Finding such metrics comes down to solving highly nonlinear partial differential equations. For some manifolds solutions do not exist, and this is known to be related to the algebro-geometric notion of “stability”. The talk will give an overview of this area, emphasising the role of asymptotic analysis, applied to holomorphic sections of high powers of a complex line bundle. This gives a bridge between the analytical problems and algebraic geometry which is important in the general existence theory. The ideas can also be applied to construct numerical approximations to the desired metrics. —————— Simon Donaldson, FRS, holds a Royal Society Research Professorship at Imperial College London. He received a Fields Medal in 1986, was awarded with the Crafoord Prize 1994, the King Faisal International Prize in 2006 and the Nemmers Prize in Mathematics in 2008. He will receive the 2009 Shaw Prize in Mathematical Sciences. ————————————

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The global behavior of solutions to critical nonlinear dispersive and wave equations Carlos Kenig Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, USA [email protected] In this lecture we will describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We will illustrate the method with some concrete examples and also mention other applications of these ideas. —————— Carlos Kenig is Louis Block Distinguished Service Professor of the University of Chicago. He was awarded the 2008 Bˆ ocher Memorial Prize for his contributions to harmonic analysis and non-linear dispersive partial differential equations. ————————————

Nonlinear harmonic analysis methods in boundary value problems of analytic and harmonic functions, and PDE Vakhtang Kokilashvili A. Razmadze Mathematical Institute, 1, M. Aleksidze st., 0193 Tbilisi, Georgia [email protected] The goal of our lecture is to present a survey of recent results in the nonlinear harmonic analysis operator theory and their applications in the boundary value problems for harmonic and analytic functions and related integral operators. We plan to discuss the above mentioned problems in the frame of Banach function spaces with nonstandard growth condition. For the sake of presentation, we have split the talk in the following topics: • One and two-weight norm estimates for the Cauchy singular integrals on Carleson curves in variable exponent Lebesgue spaces. • The Riemann-Hilbert problem for holomorphic functions from weighted classes of the Cauchy type integrals with densities in Lp(·) (Γ) in simply connected domains with piecewise-smooth boundaries Γ. Our aim is to give a complete solvability picture; to reveal the influence on the solvability character of the geometry of a boundary, of a weight function, and of the values of the space exponent at angular points; to give explicit formulas for solutions. • The Riemann-Hilbert-Poincar´e problem in the class of holomorphic functions whose mth order derivatives are representable by the Cauchy type integrals with densities from the variable exponent Lebesgue spaces with weights. The solvability criteria are given for the problem. The study of the problem is heavily based on the extension of I.Vekua’s integral representation of holomorphic function whose derivative is representable by the Cauchy type integral in simply connected domain with non-smooth boundary. • Baundary value problem with shift (the Hasemann BVP) for holomorphic functions in the domain with arc-chord condition. The solvability criteria and explicit formulas for solutions are established. Some part of the talk is based on joint research with V.Paatashvili. —————— Vakhtang Kokilashvili is Head of the Mathematical Analysis Department of the Razmadze Mathematical Institute. He was awarded the Razmadze Prize of the Georgian Academy of Sciences. ————————————

Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems Nicolas Lerner Institut de Math´ematiques de Jussieu, Universit´e Paris 6, 175 rue du Chevaleret, 75013 Paris, France [email protected]

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We prove that in any C-infinity neighborhood of an analytic Cauchy datum, there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first-order non-linear systems. We use a method initiated by G. M´etivier for elliptic systems based on the representation of solutions and on the FBI transform; in our case the system can be hyperbolic at initial time, but the characteristic roots leave the real line at positive times. ————————————

Non-ergodicity of Euler deterministic fluid dynamics via stochastic analysis Paul Malliavin Universit´e Pierre et Marie Curie, Institut de Math´ematiques de Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE Unitary representation associated to the motion of an incompressible fluid on the Tori. Fourier analysis of vector fields with vanishing divergence. Ergodi-city implies existence of an infinitesimal Haar measure. Randomization of Euler deterministic dynamics. Stochastic differential geometry on the group of volume preserving diffeomorphism of the Tori. Jump process describing the evolution of the repartition of the energy between modes. Non ergodicity of Euler equation via the transfert of energy towards micro scale. —————— Paul Malliavin is famous for his contributions to stochastic analysis and stochastic differential geometry. Among other distinctions he received in 1974 the Prix Gaston Julia of the French Academy of Science and is member of the Royal Swedish Academy of Sciences. ————————————

Higher order elliptic problems in non-smooth domains BICS Lecture, with an introduction by Valery Smyshlyaev Vladimir Maz’ya University of Liverpool and Linkoeping University [email protected] We discuss sharp regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions. Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron. The techniques developed in the present work allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. Then we introduce an appropriate notion of polyharmonic capacity which allows us to describe the precise correlation between the smoothness of solutions and the geometry of the domain. This is a joint work with S.Mayboroda, Perdue University. —————— His honours include the prize of the Leningrad Mathematical Society 1962, Doctor honoris causa of the University of Rostock 1990, Humbold Prize 1999, Corresponding Fellow of the Royal Society of Edinburgh 2001, Member of Royal Swedish Academy of Sciences 2002, Verdaguer Prize of the French Academy of Sciences 2003, The Celsius Gold Medal of the Royal Society of Sciences at Uppsala 2004. He is author of more than 20 books and more than 430 articles. ————————————

Operator algebras with symbolic hierarchies on stratified spaces Bert-Wolfgang Schulze Institute of Mathematics, University Potsdam, Am Neuen Palais 10, Potsdam, D-14469 Germany [email protected]

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We establish operator algebras on certain categories of stratified spaces (“corner manifolds”, or “manifolds with singularities”) that are designed to express parametrices of elliptic operators in terms of symbolic hierarchies. Our calculus contains special cases such as (pseudo-differential) boundary value problems with/without the transmission property at the boundary, and also mixed, transmission, and crack problems. The boundaries or interfaces may be smooth or have again singularities (conical points, edges, etc.). Other examples are equations on a smooth manifold, where the coefficients may have jumps or poles of a specific kind along some interfaces, smooth or singular in the above-mentioned sense, for instance, the Laplacian plus a singular interaction potential from a many-particle system. It is typical in such problems that a concrete situation (for instance, for the Laplacian in a corner domain) may generate operator-valued amplitude functions of a relatively high generality, consisting of operator functions on configurations of lower singularity order, now depending on various variables and covariables along the singular lower-dimensional strata. The calculus also contains analogues of Green functions, known from “standard” elliptic boundary value problems. In the singular case those refer again to all singular strata, operating on infinite cones. Moreover, when a stratum is of dimension zero the operator functions globally act on compact (in general singular) bases of such cones, with meromorphic dependence on a complex covariable, where non-bijectivity points (turning into poles under inversion) contribute to the asymptotics of solutions. Ellipticity in such a scenario is defined as invertibility of such operator-valued symbols. This depends on chosen weights in the respective distribution spaces. When a stratum is of dimension at least 1, this cannot be achieved in general, unless we pose extra edge conditions (analogues of boundary conditions), here of trace and potential type. The latter are possible when an analogue of the Atiyah-Bott condition for the existence of Shapiro-Lopatinskij boundary conditions is satisfied; otherwise another concept, namely, with global projection conditions may work (at least for smooth boundaries or edges, cf. the well-known work of Atiyah, Patodi, Singer, and papers of many other authors, especially, Seeley, Grubb, and also by the author, partly in joint work with J. Seiler, where corresponding operator algebras are established in a Toeplitz operator framework, unifying the structures of the Shapiro-Lopatinskij and the global projection set-up). The construction of parametrices relies on the inversion of the components of the principal symbolic hierarchy, combined with algebraic operations. Those symbols take values in spaces of operators referring to lower singularity orders. At this point, in order to express parametrices within our spaces, we need the calculus as an algebra. The analysis which is doing all this is rich in detail. Many authors contributed to the pseudo-differential methods in this framework, especially, Melrose, Mendoza, Gil, Seiler, Schrohe, Witt, and Krainer.There are several monographs of the author, a few jointly with coauthors (Rempel, Egorov, Kapanadze, Harutyunyan) containing the basics of the approach, including applications, and more references. In order to keep the calculus manageable it is important to reduce the stuctures to a few “axiomatic” principles and then to proceed in an iterative way, beginning with the pseudo-differential calculus on a smooth manifold, and then successively building up the algebras for conical, edge, corner, . . . , higher singularities. The focus of our talk is just a program of that kind. We present such an iterative process to obtain operator algebras containing the desirable (“typical”) differential operators (corner-degenerate in streched coordinates), together with the parametrices of elliptic elements, where the above-mentioned examples are covered. One of the principles to make the calculus iterative is to impose a relatively simple behaviour of the growth of norms of parameter-dependent operators when the parameters tend to infinity, then to make the parameter-dependence “edge-degenerate” at infinity of an infinite cone, and then to observe that this behaviour survives the step to the next floor of singular calculus, cf. a joint article with Abed. The general structure theory is full of new challenges and “unexpected” problems, for instance, from the point of view of index theory, or extensions to non-elliptic operators. Moreover, in concrete cases other substantial aspects remain essential, namely, to compute several data as explicitly as possible, e.g., the index of operators on infinite cones, or the number of extra edge conditions, the right weights that depend on the individual operator, the asymptotics of solutions, including iterated asymptotics, or the variable and branching behaviour connected with the above-mentioned poles when those depend on edge variables and change multipicities (cf. earlier work of Bennish, or the author, and a cycle of papers in progress jointly with Volpato). —————— Bert-Wolfgang Schulze is author of more than 240 publications and 20 books. He received the Euler Medal of the Berlin Academy of Sciences in 1984 and is doctor honoris causa of the Vekua Institute of Applied Mathematics in Tbilisi. ————————————

Visibility and Invisibility Gunther Uhlmann Department of Mathematics, C-449 Padelford Hall, Seattle, Washington 98195-4350, USA

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[email protected] We will describe the method of complex geometrical optics and its applications to find acoustic, quantum, and electromagnetic parameters of a body by making measurements at the boundary of the body. We will also survey recent results on how to make objects invisible to acoustic, quantum and electromagnetic waves. —————— Gunther Uhlmann is Walker Family Endowed Professor of Mathematics at the University of Washington. He is Fellow of the American Academy of Arts and Sciences, corresponding member of the Chilean Academy of Sciences, Fellow of the Institute of Physics and will be Clay Senior Scholar at MSRI in 2010. ————————————

Practise of industrial mathematics related with the steel manufacturing process OCCAM Lecture on Applied Mathematics, with an introduction by John Ockendon Masahiro Yamamoto University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba Meguro Tokyo 153, Japan [email protected] We will discuss several problems given by the steel industry. Those problems have originated from real working sites, are related for example with heat conduction processes and have been solved by the speaker and his research groups. Those problems can be modelled mathematically, on such a a theoretical basis, we have solved them practically as well as mathematically to satisfy demands by industry for lowering costs and improving securities. For more fruitful contribution in the industrial mathematics from the side of mathematicians, we will discuss also possible schemes. ————————————

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Public lecture Analysis, Models and Simulations OxPDE Public Lecture on Nonlinear PDE, with an introduction by Sir John Ball Pierre-Louis Lions College de France, 3 rue d’Ulm, 75005 Paris, France [email protected] In this talk, we shall first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving Partial Differential Equations and we shall briefly explain the nature, the history and the role of such equations. Then, some examples showing the importance of the mathematical analysis (i.e. understanding) of those models will be presented. And we shall conclude indicating a few trends and perspectives. —————— Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself become a renowned mathematician, making numerous important contributions to the theory of non-linear partial differential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPerna giving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Other awards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently he holds the position of Chair of Partial Differential Equations and their Applications at the prestigious Coll`ege de France in Paris. ————————————

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This is joint work with Shamil Makhmutov and Jouni R¨ atty¨ a. ———

Sessions

Global mapping properties of rational functions

I.1. Complex variables and potential theory Organisers: Tahir Aliyev, Massimo Lanza de Cristoforis, Sergiy Plaksa, Promarz Tamrazov This session is devoted to a wide range of directions of complex analysis, potential theory, their applications and related topics. —Abstracts— Analytic functions in contour-solid problems ˇ lo Tahir Aliyev Azerog Gebze Institute of Technology, Istanbul Caddesi, P.K. 141, Gebze, Kocaeli, 41400 Turkey [email protected] We generalize and strengthen certain contour-solid theorems. The generalization consists in considering finely meromorphic functions besides holomorphic, and strengthening is connected with taking into account zeroes and the multivalence of functions.

Cristina Ballantine Dept. of Mathematics, College of the Holy Cross, 1 College Street, Worcester, Massachusetts 01610 United States [email protected] The talk is based on a joint work with Dorin Ghisa. The main result is the following theorem. Every rational b = ∪n function f of degree n defines a partition C k=1 Ωk of the Riemann sphere such that the interior of every b \ Lk , where Lk is Ωk is mapped conformally by f on C part of a cut L. The mapping extends conformally to the boundary of every Ωk except forsome points b1 , b2 , ..., bj , j ≤ n, in the neighborhood of which f has one of the forms: (i) f (z) = f (bk ) + (z − bk )αk ϕk (z), or (ii) f (z) = (z − bk )−αk ϕk (z), where αk is an integer, αk ≥ 2, and ϕk is an analytic function with ϕk (bk ) 6= 0. b f ) is a branched covering Riemann surface Actually, (C, b of C having the branch points b1 , b2 , ..., bj . In the neighborhood of z = ∞ we have: (iii) f (z) = z α ϕ(z), where α ∈ Z and ϕ is analytic with lim ϕ(z) finite and non-zero. z→∞

——— A non-α-normal function whose derivative has finite area integral of order less than 2/α Rauno Aulaskari University of Joensuu Department of Physics and Mathematics Joensuu, Joensuu 80101 Finland [email protected] Let D be the unit disk {z : |z| < 1} in the complex plane. A function f , meromorphic in D, is normal, denoted by f ∈ N , if supz∈D (1 − |z|2 )f # (z) < ∞, where f # (z) = |f 0 (z)|/(1 + |f (z)|2 ). For α > 1, a meromorphic function f is called α-normal if supz∈D (1 − |z|2 )α f # (z) < ∞. H. Allen and C. Belna [J. Math. Soc. Japan, 24 (1972) 128–132] have proved that there is an analytic function f1 , defined in D, such that ZZ |f10 (z)| dxdy < ∞ D

but f1 6∈ N . S. Yamashita [Ann. Acad. Sci. Fenn. Ser. Math. 4 (1978/1979) 293–298] sharpened this result by showing that for another analytic function f2 which does not belong to N it holds ZZ |f20 (z)|p dxdy < ∞ (*)

If f is a polynomial, then every Ωk is bounded by arcs approaching asymptotically rays of the form zk (t) = tei(γ+2kπ/n) , t > 0, γ ∈ R. We will present examples of color mapping visualizations. ——— Beltrami equations Bogdan Bojarski IM PAN, Sniadeckich 8, Warsaw, 00-956 Poland [email protected] In the talk will be discussed some new approaches to the Beltrami equations and operators in the complex plane and on Riemann surfaces in connections with the general theory of quasiconformal mappings and automorphic functions. ——— A functional analytic approach for a singularly perturbed non-linear traction problem in linearized elastostatics Matteo Dalal Riva Universita’ degli Studi di Padova, Via Trieste, 63 Padova, Italy/Padova/Veneto 35121, Italy [email protected]

D

for all p, 0 < p < 2. Further, H. Wulan [Progress in analysis Vol. I,II, World Sci. Publ. 2003, 229–234] S studied more the function f2 and showed that f2 6∈ 00 Γ(αk + 1) k=0

and investigated some of its properties. This is an entire function of order 1/α. Another function having similar properties to those of Mittag-Leffler functions is given by

l+ 1

l+ 1 Jl+ 1 (µj 2 ) 2

µj 2 l+ 1 l+ 1 + (Jl− 1 (µj 2 ) − Jl+ 3 (µj 2 )) = 0. 2 2 l+1

Eα,β (z) =

∞ X k=0

——— The ending solutions of Ince system with irregular features Zhaxylyk Tasmambetov Aktobe State University after K. Zhubanov 263, Bratiev Zhubanov’s street, Aktobe city, 030000 Kazakhstan [email protected] The Ince system with irregular features: ( (0) p Zxx + p(1) q (4) Zxy + p(2) Zx + q (5) Zy + p(3) Z = 0, q (0) Zyy + p(4) q (1) Zxy + p(5) Zx + q (2) Zy + q (3) Z = 0, where coefficients p(i) = p(i) (x) and q (i) = q (i) (y) (i = 0, 5) are polynomials of (i)

p (x) =

δi X j=πi

j

pij x , q

(i)

(y) =

ζi X

qij x

j

j=ξi

type (πi , δi , ξi , ζi (i = 0, 5) - certain numbers), is studied. Let the system be collocated and let the integrability condition be executable p(0) q (0) − p(1) q (1) p(4) q (4) 6= 0. Ince established that singular curves of this system are defined by the coefficients in the case of second-order private derivatives and in the case of certain additional

zk , α > 0, β > 0. Γ(αk + β)

For β = 1, Eα,1 = Eα . Such functions arise naturally in the solution of fractional integral equations [Saxena, R., Mathai, A. and Haubold, H. (2002). On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287] and especially in the study of the fractional kinetic equation, random walks, etc. We study Mittag-Leffler type functions and derive some of their properties including integrals and recurrence relations. We also study fractional equations of the form N (t) − N0 = −c 0 Dt−1 N (t), and its generalization, where 0 Dt−ν is the RiemannLiouville operator of fractional integration. ——— On mixed boundary-value problems of polyanalytic functions Yufeng Wang School of Mathematics and Statistics, Wuhan University, Wuhan 430072 China wh [email protected] Recently, boundary value problems of higher-order complex partial differential equations have been widely investigated. For example, various kinds of boundary value problems of two-order complex partial differential equations, including the Poisson equation and the

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I.3. Complex-analytical methods for applied sciences Bitsadze equation, have been systematically discussed, and the explicit expression of solution and the condition of solvability have already been obtained. In addition, some boundary value problems of polyanalytic equation, polyharmonic equation and metaanalytic function have also been discussed. In this paper, under the appropriate decomposition of polyanalytic functions, some mixed boundary-value problems of polyanalytic functions have been discussed, and the explicit expression of solution and the condition of solvability have been obtained.

On solution of a kind of Riemann boundary value problem on the real axis with square roots Shouguo Zhong School of Mathematics and Statistics, Wuhan University, Wuhan 430072 China [email protected] Solution of the Riemann boundary value problem on the real axis X with square roots p p Ψ+ (x) = G(x) Ψ− (x) + g(x), x ∈ X

——— An algorithm of solving the Cauchy problem and mixed problem for the two-dimensional system of quasi-linear hyperbolic partial differential equations Oleg N. Zhdanov Siberian State Aerospace University “M.F. Reshetnyov”, Krasnoyarsk, Russia [email protected] Let’s consider the system of homogeneous quasilinear hyperbolic partial differential equations aij (u1 , u2 )∂x uj + bij (u1 , u2 )∂y uj = 0, i, j = 1, 2, (*) where aij , bij - smooth functions in area D. There are 3 classical boundary problems for system (*): the Cauchy problem, the Riemann problem and the anmixed problem. Earlier Cauchy and Riemann problems were solved for some particular cases using conservation laws. And now we have algorithm for the solution of the Cauchy problem of system (*) in general. Attempts to solve the mixed prolem weren’t successful for a long time. Our approach consists in applying to this system not only one conservation law, as was done in many papers, but a family of such laws with functions depending on parameters. Let’s accurately formulate the problem. Let the function u be specified on the non-characteristic curve M N in the plane C, and functions u, v be specified on a characteristic curve crossing M N . It is important that every characteristic crosses the curve M N only in one point and is not tangent to it in any point. Our aim is to find the intersections of characteristics and the values of functions u and v in these points. We reduce the mixed problem to the Cauchy problem. We choose a point on the curve M N and a point on the characteristic, and we have a system of algebraic equations - corollary fact of conservation law. Using resultant, we obtained one equation for the value of hte function v in the initial point. We find this value and repeat the procedure with another points. It allows us to find the intersections of characteristics and function values in these points with preassigned exactness using a well-known method described in [Kiryakov P. P., Senashov S. I., Yakhno A. N. Application of symmetries and conservation laws to differential equations solving. Novosibirsk, 2001., p. 170]. As application we obtained the solution of systems, describing state of plane stress of Mises‘s plastic surroundings– a problem that is interesting for mechanics for more than 100 years. ———

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for analytic function is considered, which was solved under certain assumptions on the branch points appeared. ——— Some Riemann boundary value problems in Clifford analysis Zhongxiang Zhang School of Mathematics and Statistics, Wuhan University, Wuhan 430072 China [email protected] In this paper, we mainly study the Rm (m > 0) Riemann boundary value problems for functions with values in a Clifford algebra C(V3,3 ). We firstly prove a generalized Liouville theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k-regular functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions is presented. Combining the Plemelj formula, the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m > 0) Riemann boundary value problems for regular functions, harmonic functions and biharmonic functions are solvable. The explicit solutions are given. ———

I.3. Complex-analytical methods for applied sciences Organisers: Viktor Mityushev, Sergei Rogosin The main attention will be paid to analytic-type results in complex analysis, especially those which have applications in Mathematical Physics, Mechanics, Chemistry, Biology, Medicine, Economics etc. Among the methods under consideration are: boundary value problems for holomorphic and harmonic functions and their generalizations, singular integral equations, potential analysis, conformal mappings, functional equations, entire and meromorphic functions, elliptic and doubly periodic functions etc. Applications in Fluid Mechanics, Composite Materials, Porous Media, Hydro- Aero- and Thermo-Dynamics, Elasticity, Elasto-Plasticity, will be the most considered at the session.

I.4. Zeros and Gamma lines – value distributions of real and complex functions —Abstracts— R-linear problem and its applications to composites Vladimir Mityushev Podchorazych 2 Krakow, Malopolska 30-084 Poland [email protected] We develop the method of functional equation to derive analytical approximate formulae for the effective conductivity tensor of the two–dimensional composites with elliptical inclusions. The sizes, the locations and the orientations of the ellipses can be arbitrary. The analytical formulae contains all above geometrical parameters in symbolic form.

The numbers of zeros of certain classes of meromorphic functions are studied, particularly, in the classical Nevanlinna and Ahlfors theories. Some analogous results were obtained also for the Gamma-lines of functions (i.e., preimages of curves). This enlarges the value distribution, describes not only the numbers but also the locations of a-points and, unexpectedly, leads to new distribution type phenomena for the zeros in real analysis and real algebraic geometry. Thus we are now in a stage of formation of some methods working in both real and complex analysis. The zeros (a-points, fixed-point) and Gamma-lines arising in complex analysis (particularly meromorphic functions and solutions of ODE, harmonic and polynomial mappings), real analysis, real and complex algebraic geometry will be subject of this session.

——— Application of the spectral parameter power series method to conformal mapping problems Michael Porter Department of Mathematics, CINVESTAV-IPN, Libramiento Norponiente 2000, Fracc. Real de Juriquilla Queretaro, 76230 Mexico [email protected] Many problems in conformal mapping of plane domains are determined by the Schwarzian derivative of the mapping, a third-order nonlinear differential operator, and it is well known that this can be rephrased in terms of a second-order linear differential equation y 00 + φy = 0. For many mapping problems the coefficient function φ in this equation depends on one or more real or complex parameters; a typical formulation might be y 00 + qy = λry. The global aspect of a mapping problem often translates into boundary conditions (possibly nonlinear) on a real interval and a spectral problem is thus presented. We apply the recently developed spectral parameter power series (SPPS) method for Sturm-Liouville problems to gain insight into conformal mapping problems. In particular we will calculate the complete parameter space for conformal mappings from the disk to a symmetric circular quadrilateral with right angles. ——— Recent results on analytic methods for 2D composite materials Sergei Rogosin Department of Mathematics and Mechanics, Belarusian State University, Nezavisimosti ave, 4 Minsk, BY-220030 Belarus [email protected]

—Abstracts— An universal value distribution: for arbitrary meromorphic function in a given domain Grigor Barsegian Institute of Mathematics of the National Academy of Sciences, 24-b Bagramian ave. Yerevan, 375019 Armenia [email protected] Some purely geometric results analogous to the second fundamental theorems in the classical Nevanlinna and Ahlfors theories are revealed. These analogs are valid for arbitrary analytic (meromorphic) functions in given domains unlike the classical results that are valid only for some known sub classes of functions that have “equidistributions”. The obtained results are sharp as for functions in the complex plane (the classical case) as well as for functions in a given domain. ——— A generalization of the Stieltjes-Van Vleck-Bocher theorem Petter Branden Department of Mathematics Royal Institute of Technology Stockholm, Stockholm 100 44 Sweden [email protected] A classical theorem of Stieltjes, Van Vleck and Bˆ ocher describes the polynomial solutions f (z), v(z) to the second order differential equation d Y

(z − αj )f 00 (z) +

j=1

d X j=1

βj

Y (z − αi )f 0 (z) + v(z)f (z) = 0 i6=j

It is a survey talk on the recent analytic results for 2D composite materials. Special attention will be paid to application of the boundary value problems for analytic functions, of the functional equations method and of the integral equation method.

where α1 < · · · < αd are real and β1 , . . . , βd are positive. B. Shapiro has recently developed a Heine-Stieltjes theory for linear differential operators of higher order. He conjectured a vast generalization of the Stieltjes– Van Vleck–Bˆ ocher theorem. We prove this conjecture and describe the intricate structure of the zeros of the solutions.

———

——— A criterion for the reality of zeros

I.4. Zeros and Gamma lines – value distributions of real and complex functions Organisers: Grigor Barsegian, George Csordas

David Cardon Department of Mathematics, Brigham Young University, Provo, Utah 84604 United States [email protected]

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I.4. Zeros and Gamma lines – value distributions of real and complex functions I will discuss a necessary and sufficient condition for certain real entire functions to have only real zeros. ——— New properties of a class of Jacobi and generalized Laguerre polynomials Marios Charalambides Department of Business Administration, Frederick University, 7 Yianni Frederickou street, Nicosia, Pallouriotisa 1036 Cyprus [email protected]

VA 24061-0123 United States [email protected] Polynomials with real coefficients and all real roots have many interesting and useful properties. This talk will introduce an elegant generalization to polynomials with complex coefficients in seeveral variables. These new polynomials are called upper (or stable) polynomials and are defined by their non-vanishing on the upper half plane. This is recent work of J. Borcea, P. Br¨ and en, S. Fisk, B. Shapiro, A. Sokal, and D. Wagner. ———

New properties of a class of Jacobi and generalized Laguerre polynomials are presented. The results give new classes of stable polynomials and polynomials with real negative roots. Implications of these results on the areas of geometry of polynomials and numerical analysis are also discussed. ——— Meromorphic Laguerre operators and the zeros of entire functions George Csordas Department of Mathematics University of Hawaii, Honolulu, Hawaii 96822 United States [email protected] The purpose of this lecture is to announce new results pertaining to the following open problem. Characterize the meromorphic functions, F (x), such that P∞ k F (k)a k x /k! is a transcendental entire function k=0 with only real zeros (or that the zeros all lie in the half-plane r1 } for some r1 > 0. Similarly we define a negatively oriented Baker domain. By Spiraling Baker domain, we mean either positively oriented Spiraling Baker domain or a negatively oriented spiraling Baker domain. In this paper we show the existence of Spiraling Baker domain and obtain several properties of these. ——— The algebraic Liouville integrability and the related Picard-Fuchs type equations Anatoliy Prykarpatsky The AGH-University of Science and Technology, Krakow, Poland, and Ivan Franko State Pedagogical University, Drohobych, Lviv region, Ukraine 30 Aleja Mickiewicz, N120-C Krakow, 30059 Krakow Poland [email protected] We consider a completely integrable Liouville-Arnold Hamiltonian system on a cotangent canonically symplectic manifold (T ∗ (Rn ), ω (2) ), n ∈ Z+ , possessing exactly n ∈ Z+ functionally independent and Poisson commuting algebraic polynomial invariants Hj : T ∗ (Rn ) → R, j = 1, n. Due to the Liouville-Arnold theorem this Hamiltonian system can be completely integrated by quadratures in quasi-periodic functions on its integral submanifold when taken compact. It is equivalent to the statement that this compact integral submanifold is diffeomorphic to a torus Tn , that makes it possible to integrate the system by means of searching the corresponding integral submanifold imbedding mapping. The following theorems are stated. Theorem. Every completely algebraically integrable Hamiltonian system admitting an algebraic submanifold Mhn ⊂ T ∗ (Rn ) possesses a separable canonical transformation which is described by differential algebraic Picard-Fuchs type equations whose solution is a set of some algebraic curves Theorem. Consider a completely integrable Hamiltonian system on the coadjoint manifold T ∗ (Rn ) whose integral submanifold Mhn ⊂ T ∗ (Rn ) is described by PicardFuchs type algebraic equations. The corresponding integrability embedding mapping πh : Mhn → T ∗ (Rn ) is a solution of a compatibility condition subject to the differential-algebraic relationships on the corresponding canonical transformations generating function.

Spiraling Baker domains ——— Anand Prakash Singh Department of Mathematics, University of Jammu, Jammu-180006, INDIA [email protected] Let f be a transcendental entire function. For n ∈ N, let f n denote the nth iterate of f . Fatou set F (f ) of f is defined to be the set of all points z in the complex plane C such that the family {f n }n≥1 forms a normal family in some neighbourhood of z. Julia set is defined to be the complement of Fatou set. A periodic component U of F (f ) of period m is called a Baker domain if f mn (z) → ∞ as n → ∞ for all z ∈ U . Further we define a Baker domain B as a positively oriented spiraling Baker domain if there exist positive continuous functions A(r), φ(r), ψ(r), of r all tending to ∞ as r → ∞ such that φ(r) in non decreasing, 0 < ψ(r) − φ(r) < 2π and

Quantization of universal Teichm¨ uller space: an interplay between complex analysis and quantum field theory Armen Sergeev Steklov Mathematical Institute, Gubkina 8, Moscow, 119991 Russia [email protected] Universal Teichm¨ uller space T is the quotient of the group QS(S 1 ) of quasisymmetric homeomorphisms of S 1 modulo M¨ obius transformations. It contains the quotient S of the group Diff+ (S 1 ) of diffeomorphisms of S 1 modulo M¨ obius transformations. Both groups act natu1/2 rally on Sobolev space H := H0 (S 1 , R). Quantization problem for T and S arises in string theory where these spaces are considered as phase manifolds. To solve the problem for a given phase space means to

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II.1 Clifford and quaternion analysis fix a Lie algebra of functions (observables) on it and construct its irreducible representation in a Hilbert (quantization) space. For S an algebra of observables is given by Lie algebra Vect(S 1 ) of Diff+ (S 1 ). For quantization space we take 1/2 the Fock space F (H), associated with H = H0 (S 1 , R). 1 Infinitesimal version of Diff+ (S )-action on H generates an irreducible representation of Vect(S 1 ) in F (H), yielding quantization of S. For T the situation is more subtle since QS(S 1 )-action on T is not smooth. So there is no classical Lie algebra, associated to QS(S 1 ). However, we can define a quantum Lie algebra of observables Derq (QS), generated by quantum differentials, acting on F (H). These differentials arise from integral operators dq h on H with kernels, given essentially by finite-difference derivatives of h ∈ QS(S 1 ). ———

II.1 Clifford and quaternion analysis Organisers: Irene Sabadini, Frank Sommen We call for contributions in the fields of theoretical quaternionic and Clifford analysis and, more in general, hypercomplex analysis intended as the study of the function theory related to the Dirac operator and systems of partial differential operators taking values in a Clifford algebra. All the topics varying from the study of monogenic functions, its generalisations to higher spin such as the Rarita-Schwinger system, Clifford analysis on superspace, Clifford-Radon and Fourier transforms, discrete Clifford analysis to functions with values in more general non-commutative structures are welcome. —Abstracts— Clifford analysis for orthogonal, symplectic and finite reflection groups Hendrik de Bie Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Ghent (Belgium) [email protected] In recent work we have developed a theory of Clifford analysis in superspace. This can be seen as Clifford analysis invariant under the product of the symplectic with the orthogonal group. Other authors have recently also studied Clifford analysis with respect to finite reflection groups (using Dunkl operators). In this talk we will give a general and unified framework that can be used for these different symmetries. We will also discuss some typical problems that depend on the symmetry at hand. These will include the Fischer decomposition, the Fourier transform and the Hermite polynomials. We also discuss related quantum systems. ——— M¨ obius transformations and Poincar´ e distance in the quaternionic setting Cinzia Bisi Dipartimento Matematica, Universita’ della Calabria, Cubo 30b, Ponte P.Bucci, Arcavacata di Rende Cosenza,

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Calabria 87036 Italy [email protected] In the space H of quaternions, we investigate the natural, invariant geometry of the open, unit disc ∆H and of the open half-space H+ . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M¨ obius transformations of ∆H and H+ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M¨ obius transformations, and use it to define the analogous of the Poincar´e distances and differential metrics on ∆H and H+ . As a spin-off, we directly deduce that there exists no isometry between the quaternionic Poincar´e distance of ∆H and the Kobayashi distance inherited by ∆H as a domain of C2 , in accordance with the well known classification of the non compact, rank 1, symmetric spaces. ——— Wavelets invariant under reflection groups Paula Cerejeiras Department of Mathematics, University of Aveiro, Aveiro, P-3810-193 Portugal [email protected] For signal reconstruction over a sphere, two main approaches are used: the group-theoretical one (see, for instance, Antoine/Vandergheynst or M. Ferreira) where the authors use representations over homogeneous spaces and the one using approximate identities and singular kernels (see Freeden, or Swelden). However, both rely on the Lorentz group and, therefore, are not suitable for signals with predefined symmetries which involve reflections. To overcome this problem, we consider differential-difference operators associated to specific finite reflection groups, the so-called Dunkl operators. In this setting we construct spherical Dunkl wavelets based on approximate identities and we give practical examples. ——— Some consequences of the quaternionic functional calculus Fabrizio Colombo Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9 Milano, Mi 20133 Italy [email protected] We show some of the most recent results on the quaternionic functional calculus for left and right linear quaternionic operators defined on quaternionic Banach spaces. This approach allows us to deal both with bounded and unbounded operators. In particular we use such a functional calculus to study the quaternionic evolution operator. ——— Orthogonality of Clifford-Hermite polynomials in superspace. Kevin Coulembier Department of Mathematical Analysis, Ghent University, Krijgslaan 281 Ghent 9000 Belgium [email protected]

II.1 Clifford and quaternion analysis In previous work by De Bie and Sommen, the CliffordHermite polynomials were generalized to superspace. In this talk we will construct an inner product for which these polynomials are orthogonal, using the Berezin integral. This inner product can moreover be used for quantum mechanics in superspace, as it restores the hermiticity of the anharmonic oscillator. As an application we will also derive a Mehler formula with O(m) × Sp(2n) symmetry. The Mehler formula gives an expansion of the kernel of the fractional Fourier transform in terms of the super Clifford-Hermite polynomials. This was already known in one dimension (Hermite polynomials) and formally for O(m) (CliffordHermite polynomials), but the O(m) × Sp(2n) poses some extra difficulties. ——— Recent results on hyperbolic function theory Sirkka-Liisa Eriksson Department of Mathematics, Tampere University of Technology , P.O.Box 553, Tampere 33101 Finland si[email protected] The aim of this talk is to consider the hyperbolic version of the standard Clifford analysis. The need for such a modification arises when one wants to make sure that the power function xm is included. The leading idea is that the power function is the conjugate gradient of a harmonic function, defined with respect to the hyperbolic metric of the upper half space. We present results and problems concerning power series presentation of hypermonogenic functions This work is done jointly with professor Heinz Leutwiler, University of ErlangenN¨ urnberg, Department of Mathematics, Erlangen, Germany, email: [email protected] ——— Symmetric properties of the Fourier transform in Clifford analysis setting Ming-Gang Fei Departamento de Matem´ atica, Universidade de Aveiro, Campus Universitario de Santiago Aveiro, Aveiro 3810193 Portugal [email protected] In this talk we present Fueter’s Theorem for Dunklmonogenic functions. We show that if f is a holomorphic function in one complex variable, then for any unγ +(d−1)/2 derlying space Rd1 the induced function ∆hκ f (x) is Dunkl-monogenic whenever γκ + (d − 1)/2 is a nonnegative integer, where ∆h is Dunkl Laplacian. To this end Vekua-type systems for axial Dunkl-monogenic functions are studied. ——— Factorization of M¨ obius gyrogroups - the paravector case Milton Ferreira Campus Universit´ ario de Santiago, Departamento de Matem´ atica, Universidade de Aveiro, Aveiro 3810-193 Portugal [email protected] We consider a M¨ obius gyrogroup on the unit ball of the vector space F ⊕ V, where V is a finite dimensional

vector space over the scalar field F = R or C. We will present the factorizations of the paravector unit ball by gyro-subgroups and subgroups, generalizing the case of the unit ball on Euclidean space Rn . The main differences between both cases are the replacement of the Spin group by the Spoin group and the establishment of a geometric product for the paravector case, analogous to the geometric product in the vector case. ——— Higher spin analogues of the Dirac operator in two variables and its resolution Peter Franek Mathematical Institute, Charles University Praha, Sokolovska 83 Prague, 8 186 75 Czech Republic [email protected] A resolution of the Dirac operator in two variables is well known and well understood. It consists of three invariant operators (on of those of second order) expressed using the Dirac operators in two individual variables. We shall discuss higher spin analogues of such resolutions. They are again complexes of three invariant operators acting on functions with values in more complicated representation spaces. ——— Cauchy kernels in ultrahyperbolic Clifford analysis – Huygens cases Ghislain R. Franssens Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium [email protected] ` ´ Let Rp,q , Rp+q , P , with P the canonical quadratic form of signature (p, q). Clifford Analysis (CA) over Rp,q , called Ultrahyperbolic Clifford Analysis (UCA), is a non-trivial extension of the familiar (Euclidean) CA over Rn . Essential for stating integral representation theorems in UCA is the determination of a reproducing (or Cauchy) kernel Cx0 of Rp,q , ∀p, q ∈ Z+ , for the Dirac operator ∂. Any such kernel can be obtained as Cx0 = ∂gx0 , with gx0 a fundamental distribution of the ultrahyperbolic equation p,q gx0 = δx0 , x0 ∈ Rp+q . The complexity of UCA is due to the fact that Cx0 is a rather complicated distribution, whose form profoundly depends on the parity of p and q. Iff p and q are odd is gx0 proportional to a delta distribution δ(P (x−x0 )) , having as support the null space of Rp,q relative to x0 , and then gx0 is said to satisfy Huygens’ principle. In this talk, explicit expressions for the distributions gx0 and Cx0 will be presented for the Huygens cases. We will see how δ(P (x−x0 )) arises as a pullback of the one-dimensional delta distribution δ and the matter of “regularization”, required for some of these distributions, will be carefully addressed. ——— Power series and analyticity over the quaternions Graziano Gentili Dipartimento di matematica ”U.Dini”, viale Morgagni 67/a, 50134 Firenze, Italy [email protected]

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II.1 Clifford and quaternion analysis We study power series and analyticity in the quaternionic setting. We first consider P a function f defined as the sum of a power series n∈N q n an in its domain of convergence, which is a ball B(0, R) centered at 0. At each p ∈ B(0, R), f admits expansions in terms of appropriately defined regular power series centered at P ∗n p, (q − p) bn . The expansion holds in a ball n∈N Σ(p, R − |p|) defined with respect to a (non-Euclidean) distance σ. We thus say that f is σ-analytic in B(0, R). Furthermore, we remark that Σ(p, R − |p|) is not always an Euclidean neighborhood of p; when it is, we say that f is quaternionic analytic at p. It turns out that f is quaternionic analytic in a neighborhood A of B(0, R)∩R, with A strictly contained in B(0, R) unless R is infinite. We then extend these results to the larger class of quaternionic slice regular functions, enriching their theory. Indeed, slice regularity proves equivalent to σanalyticity and slice regular functions are quaternionic analytic only in a neighborhood of the real axis.

This is joint work with D. Constales and D. Grob. ——— Explicit description of the resolution for 4 Dirac operators in dimension 6 Lukas Krump Mathematical Institute of the Charles University, Sokolovska 83, Praha 8, 186 75 Czech Republic [email protected] There are several approaches to the construction of a resolution of several Dirac operators in higher dimensions. Among them, the Penrose transform method gives satisfying results in both stable and unstable cases. Recently this method was used to determine the shape of such resolution in many cases and the next step is an explicit description of operators involved. This will be shown for the unstable case of four operators in dimension six. ———

——— Isomorphic action of SL(2, R) on hypercomplex numbers Anastasia Kisil Triniti College Cambridge, University Cambridge, Cambridgeshire CB2 1TQ, United Kingdom [email protected] We investigate the SL(2, R) invariant geodesic curves with the associated invariant distance function in parabolic geometry. Parabolic geometry naturally occurs as action of SL(2, R) on dual numbers and is placed in between the elliptic and the hyperbolic geometries (which arise from the action of SL(2, R) on complex and double numbers). Initially we attempt to use standard methods of finding geodesics but they lead to degeneracy in this set-up. Instead, by studying closely the two related hypercomplex numbers we discover a unified approach to a more exotic and less obvious dual number’s case. With aid of common invariants we describe the possible distance functions that turn out to have some unexpected, interesting properties. ——— Construction of 3D mappings on to the unit ball with the hypercomplex Szego kernel Rolf Soeren Krausshar Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan, 200-B Leuven, Vlaams Brabant, 3001 Belgium [email protected] In this talk we present a hypercomplex generalization of the Szego kernel method that allows us to construct 3D mappings from some elementary domains of R3 onto the unit sphere. More precisely, we consider an appropriately chosen line integral over the square of the hypercomplex Szego kernel. The latter one is approximated numerically by the monogenic Fueter polynomials for rectangular domains, an L-shaped domain, circular cylinders and the double cone. In all these cases the line integration provides an amazingly good mapping onto the unit sphere. We also compare the quality of results ontained with this method with the results that were obtained previously by using alternatively the Bergman kernel method.

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On polynomial solutions of Moisil-Theodoresco systems in Euclidean spaces Roman Lavicka Mathematical Institute, Charles University Sokolovska 83 Praha 8, Praha 186 75 Czech Republic [email protected] Let k be a positive integer and 0 ≤ s ≤ m. Denote by Pk the space of real-valued k-homogeneous polynomials in Rm . Moreover, Λs stands for the space of s-vectors N over Rm and Pks = Pk R Λs . We are interested in the following space Hks = {P ∈ Pks ; dP = 0, d∗ P = 0}. Here d and d∗ is the de Rham differential and its adjoint, respectively. Moreover, assume that r, p and q are non-negative integers such that p < q and r + 2q ≤ m. Putting q M (r,p,q) Pk = Pkr+2j , j=p

the space (r,p,q)

M Tk

(r,p,q)

= {P ∈ Pk

; (d + d∗ )P = 0}

is formed by all k-homogeneous polynomial solutions of the Moisil-Theodoresco system of type (r, p, q). We show that (r,p,q)

M Tk

'

q M

Hkr+2j ⊕

j=p

q−1 M

r+2j+1 Hk−1 .

j=p

Hks

Later on, the spaces are considered as SO(m)modules. We are interested in irreducibility, the highest weights and dimensions of such modules. In particular, we give a formula for the dimension of the (r,p,q) space M Tk . Moreover, we decompose the kernel of the Hodge laplacian on polynomial forms into SO(m)modules. These results were obtained jointly with R. Delanghe and V. Souˇcek. ——— Quaternionic Physics

analysis,

representation

theory

and

Matvei Libine Department of Mathematics, Indiana University, Rawles

II.1 Clifford and quaternion analysis Hall, 831 East 3rd St Bloomington, IN 47405 United States [email protected] This is a joint work with Igor Frenkel. I will describe our new developments of quaternionic analysis using representation theory of various real forms of the conformal group as a guiding principle. These developments will lead to a solution of Gelfand-Gindikin problem. Along the way we discover striking new connections between quaternionic analysis and mathematical physics. In particular, the Maxwell equations are realized as the quaternionic counterpart of the Cauchy formula for the second order pole. We also find a representation-theoretic meaning of the polarization of vacuum and one-loop Feynman integrals. This talk is partially based on the joint paper with Igor Frenkel, “Quaternionic analysis, representation theory and physics”, Advances in Mathematics 218 (2008) pp 1806-1877; also available at arXiv:0711.2699.

exterior differentiation acting on forms on Rn , and d∗ is its formal adjoint. Our goal is to prove that any Dirac and semi-Dirac pair (D, D† ) has two Cauchy-Pompeiu and two BochnerMartinelli-Koppelman type integral representation formulas. ——— A differential form approach to Dirac operators on surfaces Heikki Orelma Institute of Mathematics, Tampere University of Technology, P.O. Box 553, FI-33101 Tampere, Finland [email protected] In this talk we consider Dirac operators on surfaces. Surfaces are k-dimensional embedded submanifolds of Rm . Let F be a Clifford algebra-valued differential form and ∂x be the Dirac operator on Rm . F is called monogenic if it is a solution of the equation

——— Hyperholomorphic functions in the sense of MoisilThodoresco and their different hyperderivatives ´s Maria Elena Luna-Elizarrara ESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.Lindavista Mexico City, D.F. 07338 Mexico [email protected] Any Moisil-Th´eodoresco-hyperholomorphic function is also Fueter-hyperholomorphic, but its hyperderivative is always zero, so one could consider then that these functions are a kind of “constants” for the Fueter operator. It turns out that the skew-field of quaternions as a real linear space is wide enough, so it is possible to give another type of hyperderivatives “consistent” with the Moisil-Th´eodoresco operator. In this talk we present these notions of different hyperderivatives and the relation between them. The talk is based on a joint work with M. A. Mac´ıas Cede˜ no and M. Shapiro. The research was partially supported by CONACYT projects as well as by Instituto Polit´ecnico Nacional in the framework of COFAA and SIP programs. ——— Dirac and semi-Dirac pairs of differential operators Mircea Martin Department of Mathematics, Baker University, 8th and Grove, Baldwin City, Kansas 66006 United States [email protected] The Euclidean Dirac operator Deuc,n on Rn , n ≥ 2, is a differential operator with coefficients in the Clifford al2 gebra of Rn that has the defining property Deuc,n = −∆, where ∆ = ∆euc,n is the Laplace operator on Rn . As generalizations of this class of operators we investigate pairs (D, D† ) of differential operators on Rn with coefficients in a Banach algebra A, such that either DD† = µL ∆ and D† D = µR ∆, or DD† + D† D = µ∆, where µL , µR , or µ are some elements of A. Such pairs (D, D† ) are called Dirac or semi-Dirac pairs of differential operators. The typical examples of a Dirac or semi-Dirac pair on Rn are given by D = D† = d + d∗ , or D = d and D† = −d∗ , where d is the operator of

L∂x F = 0, where L∂x F is the Lie derivative of F with respect to ∂x . The aim of this talk is to show that if F and ∂x are restricted to the k-surface S we obtain a Dirac type equation L∂x |S F |S = 0 on S. As an application we shall consider winding numbers. This is joint work with Frank Sommen (Gent). ——— CK-extension and Fischer decomposition for the inframonogenic functions ˜ a Pen ˜a Dixan Pen Department of Mathematics, University of Aveiro, Campus Universitario de Santiago, Aveiro 3810-193, Portugal [email protected] Let ∂x denote the generalized Cauchy-Riemann operator in Rm+1 . In this communication, we will present a refinement of the biharmonic functions and at the same time an extension of the monogenic functions by considering the solutions of the sandwich equation ∂x f ∂x = 0. In this setting a CK-extension and a Fischer decomposition are studied. ——— A new approach to slice-regularity on real algebras Alessandro Perotti Dept. Mathematics, Univ. of Trento, Via Sommarive 14 Povo, Trento I-38100 Italy [email protected] We rivisit the concept of primary functions introduced by Rinehart in the ’60’s and apply it to the theory of slice regular functions introduced recently by Gentili, Struppa and other authors. (Joint work with Riccardo Ghiloni, Trento, Italy) ———

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II.1 Clifford and quaternion analysis Clifford analysis with higher order kernel over unbounded domains Yuying Qiao Yuhua east Road 113, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei Province 050016 China [email protected] In this paper we talk Clifford analysis with higher order kernel over unbounded domains. First we derive an higher order Cauchy-Pompeiu formula for the functions with rth order continuous differentiability over an unbounded domain whose complementary set contains nonempty open set. Then we obtain higher order Cauchy integral formula for k-regular functions and prove Cauchy inequality. Based on the higher order Cauchy integral formula, we define higher order Cauchytype integrals and the Plemelj formula. ——— Complex Dunkl operators Guangbin Ren Departamento de Matem´ atica - Universidade de Aveiro, Campus de Santiago, Aveiro 3810-193 Portugal ren[email protected] Complex Dunkl operators for certain Coxeter groups are introduced. These complex Dunkl operators have the commutative property, which makes it possible to establish the corresponding complex Dunkl analysis.

Purpose of this talk is to provide a characterization of the dual of the Rn -module of slice monogenic functions on a class of compact sets in the Euclidean space Rn+1 . We are able to establish a duality theorem which, since holomorphic functions are a very special case of slice monogenic functions, is the generalization of the classical K¨ othe’s theorem. The duality results are also discussed in the quaternionic setting. ——— Explicit description of operators in the resolution for the Dirac operator Tomas Salac Faculty of Mathemtarics and Physics, Sokolovsk´ a 83, Prague 8, 18675 Czech Republic [email protected] A study of Dirac operator D in several variables is a traditional part of Clifford analysis. A lot of effort was spent to find an analogue of the Dolbeaut complex, i.e. a resolution starting with the operator D. The resolution is composed (in the stable range) from operators of the first and the second order. Using representation theory, it is possible to write down an explicit form of the first order operators in the resolution. It is, however, much more difficult to compute an explicit form of second order operators. In the lecture, we shall use Casimir operators (recently introduced in study of parabolic geometries) as a new tool helping to get these explicit formulae for the second order part of the resolution. ———

——— p-Dirac equations

On the relation between the Fueter operator and the Cauchy-Riemann-type operators of Clifford analysis.

John Ryan Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72703, United States [email protected]

Michael Shapiro ESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.Lindavista Mexico City, D.F. 07338 Mexico [email protected]

Associated to Laplacians there are first order operators called Dirac operators. For instance the Dirac operator associated to the Laplacian in the complex plane is the Cauchy-Riemann operator. In euclidean space there is the euclidean Dirac operator Similar such operators exist for Laplace-Beltrami operators on Riemannian manifolds. Besides the usual Laplace equation in euclidean space there are the non-linear p-Laplace equations. These equations are covariant under M¨ obius transformations and are invariant when p = n. Here we shall introduce non-linear p-Dirac equations. We shall demonstrate their link to the p-Laplacian in euclidean space and demonstrate their covariance under M¨ obius transformations. Other basic properties of these equations will be investigated. We shall extend the p-Dirac and p-Laplace equations to spin manifolds. This is joint work with Craig A. Nolder (Florida State University). ——— Duality theorems for slice hyperholomorphic functions Irene Sabadini Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano, Mi 20133 Italy [email protected]

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The Moisil-Th´eodoresco operator has an explicitly given relation with the classic Dirac operator of Clifford analysis for Cl0,3 . It turns out that the Fueter operator does not have, as one would expect, a similar relation with the corresponding classic Cauchy-Riemann operator but a modification of the latter is necessary. The aim of the talk is to explain all this in detail thus establishing a direct relation between, on one hand, what is usually called quaternionic analysis, and, on the other hand, Clifford analysis. This is joint work with J. Bory-Reyes. M. Shapiro was partially supported by CONACYT projects as well as by Instituto Polit´ecnico Nacional in the framework of COFAA and SIP programs. ——— Conformally invariant boundary valued problems for spinors and families of homomorphisms of generalized Verma modules. Petr Somberg Mathematical Institute of Charles University, Sokolovska 83, Prague, Karlin 180 00 Czech Republic [email protected] On a conformal manifold M with boundary ∂M there is a construction associating conformally invariant non-local

II.1 Clifford and quaternion analysis operators to the boundary valued problems for conformally invariant operators on M with symbols given by power of Laplace operator. These operators belong to one parameter families of conformally invariant operators, generalizing conformal Dirichlet-to-Robin operator. We will discuss generalization towards conformally invariant boundary valued problems for the spinor representation. ——— Clifford calculus in quantum variables Frank Sommen Department of Mathematics, University of Ghent, Galglaan 2. B-9000 Gent, Belgium [email protected] Starting from the axioms of the algebra R(S) of abstract vector variables over a set S (radial algebra): z(xy + yx) = (xy + yx)z,

x, y, z ∈ S,

together with the basic q-commutation relations for coordinates: xi xj = qij xj xi we arrive at the defining relations for the q-Clifford algebra: ei ej + qji ej ei = −2gij , whereby gij is the q-metric which also consists of noncommuting parameters. The partial derivatives ∂xj satisfy the same q-relations ∂xi ∂xj = qij ∂xj ∂xi together with the q-Weyl relations: ∂xi xj = qji xj ∂xi + δij . This leads to the introduction of a reciprocal Clifford basis ej satisfying: ej ei + qji ei ej = −2δij , which is linked to the original Clifford basis by relations of the form (Einstein summation convention): ej = gjk ek .

Ez = zE,

x, z ∈ S.

However, the identities for the q-quantum lattice seem to lead (in the first approximation) to a relation of the form ∂x x + qx∂x = m + q(q + 1)E whereby Ex − q 2 xE = x.

Vladimir Soucek Sokolovska 83 Mathematical Institute, Charles University Praha, Czech Republic 186 75 Praha Czech Republic [email protected] The Penrose transform is a perfect tool for a study of generalised Dolbeault resolutions in the theory of several Clifford variables. An important notion used in the definition of the Penrose transform is the relative BGG resolution. Its construction is indicated in the book by Baston and Eastwood on the Penrose transform. They, however, deserve a better attention; their construction can be made more detailed using tools used for construction of the classical BGG sequences. ——— Regular Moebius transformations over the quaternions Caterina Stoppato Dipartimento di Matematica “U. Dini”, Universit` a di Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected] Let H denote the real algebra of quaternions. We present quaternionic transformations that are included in the class of regular quaternionic functions introduced by G. Gentili and D.C. Struppa in recent years. Regularity yields to properties that recall the complex case, although the diversity of the quaternionic setting introduces new phenomena. Specifically, the group Aut(H) of biregular functions H → H coincides with the group of regular affine transformations (namely, q 7→ qa + b with a, b ∈ H and a 6= 0). Moreover, inspired by the classical quaternionic linear fractional transformations, we define the class of regular fractional transformations. This class strictly includes the set of regular injective b = H ∪ {∞} to itself. Finally, we study functions from H regular Moebius transformations, which map the unit ball B = {q ∈ H : |q| < 1} onto itself. All regular bijections from B to itself prove to be regular Moebius transformations. ———

The vector derivative (Dirac operator) is then given by ∂x = ∂xj ej and the basic rules of Clifford calculus may be derived. On the level of radial algebra these rules are the same as for standard Clifford analysis, which indicates that the q-deformation aspect is only visible when calculations are expressed in coordinates. This raises the problem to define a kind of q-deformation on the level of abstract vector variables. This can be done by defining the Dirac operator ∂x in a suitable way as an endomorphism on R(S). This may be done by assuming the operator relation ∂x x = −qx∂x + m + 2qE, whereby m is the dimension of space and E ∈ End(R(S)) is the q-Euler operator given by the operator relations Ex − qxE = x,

——— On relative BGG sequences

Singularities of functions of one and several bicomplex variables Adrian Vajiac Chapman University, Dept of Math/CS, One University Drive, Orange, CA 92866 United States [email protected] In this talk we introduce the notion of regularity for functions of one, as well as several bicomplex variables. Moreover, using computational algebra techniques, we prove that regular functions of one bicomplex variable have the property that their compact singularities can be removed. ——— Multiplicities of zeroes and poles of regular functions Fabio Vlacci Department of Mathematics Ulisse Dini, viale Morgagni 67/a FIRENZE, FI 50134 Italy [email protected]

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II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras The aim of this talk is to give a survey on some recent results which have been obtained for the description of the zero sets (and poles) of regular functions. In particular we will focus our attention to define (and evaluate) a multiplicity for zeroes and poles of regular functions. ——— Gauss-Codazzi-Ricci equations in Riemannian, conformal, and CR geometry Zuzana Vlasakova Sokolovska 83, Faculty of Mathematics and Physics, Praha 8, 18675, Czech Republic [email protected] We will remind the Gauss-Codazzi-Ricci equations in Riemannian geometry, and the work of David Calderbank with Francis Burstal and Diemer on similar equations for conformal geometry. Then we introduce the CR geometry and explain that we can do the same thing also for this geometry (it is a complex analogue of conformal geometry). ——— Compatibility conditions and higher spin Dirac operators Liesbet Van de Voorde Department of Mathematical Analysis, Clifford Research Group, Galglaan 2, 9000 Gent, Belgium [email protected] In this talk, we investigate polynomial solutions for generalized Rarita-Schwinger operators. We will explain that there are two types of solutions, and we will explicitly construct one of them using results on compatibility conditions for systems in several Dirac operators. This is joint work with David Eelbode and Fred Brackx. ———

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras Organisers: ¨ rlebeck, Vladimir Kisil, Klaus Gu ¨ ßig Wolfgang Spro The mathematical use of above mentioned algebras reaches from hypercomplex analysis and differential geometry up to corresponding numerical methods. Therefore we call especially for contributions with applications in gauge theories, mathematical physics, image processing, robotics, cosmology, engineering sciences etc. —Abstracts— Wavelets on spheres Swanhild Bernstein Freiberg University of Mining and Technology, Institute of Applied Analysis, Pr¨ uferstr. 9, D-09596 Freiberg, Germany [email protected]

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The construction of wavelets relies on translations and dilations which are perfectly given in R. On the sphere translations can be considered as rotations but it difficult to say what are dilations. For the 2-dimensional sphere there exist two different approaches. The first concept defines wavelets by means of kernels of spherical integrals. The other approach is a purely group theoretical approach and defines dilations as dilations in the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define wavelets on the 3dimensional sphere by means of kernels of integrals and demonstrate that wavelets constructed according to the group-theoretical approach for zonal functions meet our definition. Typical examples arise quite easily from the AbelPoisson and Gauß-Weierstraß kernel. We will extend these kernels and wavelets into the Clifford-algebra setting. We specifically define spherical wavelets of order m. Theorem. The elements of {Ψρ , ρ > 0} are wavelets of order m (m ≥ 0) if the following admissibility conditions are satisfied: Z ∞ e 2ρ (k)α(ρ) dxρ = (k + 1)2 , k = m + 1, m + 2, ... Ψ 0

e ρ (k) = 0, k = 0, ..., m; ∀ρ ∈ (0, ∞) Ψ ˛ Z π ˛Z ∞ ˛ ˛ 2 (2) ˛ ˛ sin (θ) dxθ ≤ T, Ψ (θ)α(ρ) dxρ ρ ˛ ˛ 0

∀R ∈ (0, ∞),

R

(T > 0, independent of R). (2) Here, Ψρ stands for Ψρ ∗ Ψρ .Ψ1 (ρ = 1) is the mother wavelet. ——— On special monogenic power and Laurent series expansions and applications Sebastian Bock Bauhaus-University Weimar, Institute for Mathematics/Physics, Coudraystraße 13B, Weimar, 99421 Germany [email protected] The contribution focuses on some recently developed (orthogonal) monogenic power and Laurent-series expansions which are complete in the space of square integrable quaternion-valued functions and have as similar properties as the respective complex series expansions based on the well known z-powers. Starting with the Fourier series expansion we will show some structural properties of the series expansion with respect to their hypercomplex derivative and primitive. These special characteristics of the used orthonormal basis enable further the construction of a new Taylor type series expansion which can be explicitly related to the corresponding Fourier series analogously as in the complex onedimensional case. We end up by showing some orthogonality results for the exterior domain and present the corresponding Laurent series expansion for the domain of the spherical shell. These series expansions find applications in the description of the hypercomplex derivative as well as the monogenic primitive of a monogenic function which are represented as Fourier series, Taylor type and Laurent series. In this connection some further applications are presented.

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras ——— Spin gauge models Ruth Farwell Buckinghamshire New University, Queen Alexandra Road, High Wycombe, Bucks HP11 2JZ United Kingdom [email protected]

Recent progress in this direction will be reviewed. In particular integrable generalizations of KdV and NLS in 4+2 will be presented and the question of their reduction to 3+1 will be discussed. The role of quaternions for generalizing these results to higher dimensions will be investigated. ——— Note on the linear systems in quaternions

In 1999 we defined a form of spin gauge theory of particle interactions in which both standard ’left-hand’ and new ’right-hand’ interaction terms occur. In the proceedings of the 2005 Toulouse conference we reported the predictions of the value of the Weinberg angle and the mass of the Top quark based on a particular ’two-sided’ model, and we introduced the concepts of the ’quark’ and ’centroid’ representations. We also discussed new gravitational effects and the replacement of the graviton by the ’frame field quantum’. Recently, we have studied a variety of other two-sided models, and we present the predictions of another model, in which a different choice of spinor idempotent allows us to introduce a new particle interaction term. This is joint work with Roy Chisholm (Kent). ——— Further results in discrete Clifford analysis Nelson Faustino Departamento de Matem´ atica, Campus Universit´ ario de Santiago Aveiro, Aveiro 3810-193 Portugal [email protected] In this talk we will present the fundamentals of a higher dimensional discrete function theory by combining the Clifford algebra setting with the umbral calculus approach. Starting with the umbral version of Fischer decomposition, we decompose the space of umbral homogeneous polynomials in terms of umbral monogenic polynomials. This allows us to build up in a combinatorial way the theory of discrete spherical monogenics as a refinement of the theory of spherical harmonics. Furthermore, the interplay between discrete Clifford analysis and the physical model of the discrete harmonic oscillator will be explored along this talk by means of the canonical generators of Wigner Quantum Systems. ——— Integrability in multidimensions, complexification and quaternions Thanasis Fokas Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road Cambridge, Cambridgeshire CB3 0WA United Kingdom [email protected] One of the most important open problems in the area of integrable nonlinear evolution equations has been the construction of integrable equations in 3+1, i.e. in three spatial and one temporal dimensions. The celebrated KdV and NLS equations are integrable evolution equations in 1+1; the KP and DS equations are physically significant generalizations of the KdV and NLS in 2+1. Do there exist analogous equations in 3+1?

Svetlin Georgiev Sofia University, Faculty of Mathematics and Informatics, Department of Differential Equations, Blvd James Boucher 126, Sofia 1000 Bulgaria [email protected] In this talk we will discuss the linear system r X n X

s pslm xm qlm = As ,

s = 1, 2, . . . , n,

(*)

l=1 m=1 s where n, r ≥ 1 are given constants, pslm , qlm , As , l = 1, . . . , r, m = 1, . . . , n, s = 1, . . . , n, are given real quaternions, xm , m = 1, . . . , n, are unkown real quaternions. Here a propose an algorithm for finding a solution to the system (*). Also, we give necessary and sufficient condition for the solvability of the system (*) and some examples.

——— Minimal algorithms for Lipschitzian elements and Vahlen matrices Jacques Helmstetter 15 rue de l Oisans, St-Martin d’Heres, Isere 38400 France [email protected] If S is a closed algebraic manifold in a vector space V , and if d is the codimention of S in V , an algorithm that allows us to test whether an element of V belongs to S by means of only d numerical verifications, is called a minimal algorithm. If Cl(M, q) is the Clifford algebra derived from a quadratic module (M, q), the Lipschitz monoid Lip(M, q) is (in most cases but not in all cases) the monoid generated in Cl(M, q) by M . From the invariance property of Lipschitz monoids, a minimal algorithm can be deduced for the even and odd components of Lip(M, q). A minimal algorithm can also be deduced for the two components of the monoid of Vahlen matrices. ——— Clifford-Fourier transforms and hypercomplex signal processing Jeff Hogan School of Mathematical and Physical Sciences, University of Newcastle V-128, University Drive Callaghan, NSW 2308 Australia [email protected] In this talk we attempt to synthesize recent progress made in the mathematical and electrical engineering communities on topics in Clifford analysis and the processing of colour images, in particular the construction and application of Clifford-Fourier transforms designed to treat multivector-valued signals. Emphasis will be placed on the two-dimensional setting where the appropriate underlying Clifford algebra is the familiar set of

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II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras quaternions. We’ll describe some results and problems in the construction of discrete wavelet bases for colour images, and the difficulties encountered in constructing Clifford-Fourier kernels in dimensions 3 and higher. ——— Discrete Clifford analysis by means of skew-Weyl relations ¨ hler Uwe Ka Department of Mathematics, University of Aveiro, Aveiro, P-3810-193 Portugal [email protected] Recently one can observe an increased interest in higher dimensional discrete function theories. This is not only driven by the numerical application of continuous methods but also due to problems from combinatorics and quantum physics. While there is now a well-established approach in the continuous case, by means of the socalled radial algebra (F. Sommen), unfortunately, a direct translation to the discrete case is problematic. In this talk we present an alternative approach based on a recent idea of F. Sommen of replacing the Weyl relations by skew-Weyl relations. We will construct the basic ingredients for discrete Clifford analysis in this context and illustrate its applicability. ——— Hypercomplex analysis in the upper half-plane Vladimir Kisil School of Mathematics, Woodhouse Lane, University of Leeds, LS2 9JT, United Kingdom [email protected] Complex analysis seems to be the only non-trivial analytic function theory in the two dimensional case. However one can employ the group SL(2, R) and its representation theory in order to build elements of analytic functions with complex, dual and double numbers. This is a part of “Erlangen Programme at Large” approach in analysis. ——— Formulas for reproducing kernels of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to inhomogeneous Helmholtz equations Rolf Soeren Krausshar Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan, 200-B Leuven, Vlaams Brabant, 3001 Belgium [email protected] P ∂ Let D := n i=1 ∂xi ei be the Euclidean Dirac operator in n R and let P (X) = am X m + . . . + a1 X1 + a0 be a polynomial with arbitrary complex coefficients. Differential equations of the form P (D)f = 0 are called polynomial Dirac equations with complex coefficients. In this talk we consider Hilbert spaces of Clifford algebra valued functions that satisfy such a polynomial Dirac equation in annuli of the unit ball in Rn . We determine a fully explicit formula for the associated Bergman kernel for solutions of complex polynomial Dirac equations of any degree m in the annulus of radii µ and 1

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where µ ∈]0, 1[. We further give explicit formulas for the Szeg¨ o kernel for solutions to polynomial Dirac equations of polynomial degree m < 3 in the annulus. As concrete application we give an explicit representation formula for the solutions of generalized Helmholtz and Klein-Gordon type equation inside the annulus and with prescribed data at the boundary of the annulus. The solutions are represented in terms of integral operators that involve the explicit formulas of the Bergman kernel that we computed. ——— The Ito transform for partial differential equations Remi Leandre Institut de Mathematiques. Universite de Bourgogne Bd Alain. Savary Dijon, Cote d’Or 21078. France [email protected] We give an interpretation of the celebrated Ito formula of stochastic analysis in various contexts where there is no convenient measure on a convenient path space. We begin by the case of a diffusion (the classical one), we study after the case of the heat-equation associated to an operator of order four on a torus, we continue by studying the case of the Schroedinger equation associated to a big order operator on a torus, we consider after the case on the wave equation on a torus and we finish by studying the case of a Levy type operator associated to a big fractional power of the Laplacian on the linear space. ——— Quaternionic analysis and boundary value problems Dimitris Pinotsis Department of Mathematics, University of Reading, RG6 6AX, UK [email protected] First, we will review some results appearing in the theory of quaternions. Then, we will apply these results to solve boundary value problems for linear elliptic equations in four dimensions. Further extensions of these results will also be discussed. ——— Integral theorems in a commutative three-dimensional harmonic algebra Vitalii Shpakivskii Institute of Mathematics of National Academy of Sciences of Ukraine, Tereshchenkivska str., 3, Kiev-4, 01601, Ukraine [email protected] An associative commutative three-dimensional algebra A3 with unit 1 is harmonic if in A3 there exists a harmonic basis {e1 , e2 , e3 } satisfying the conditions e21 + e22 + e23 = 0,

e2j 6= 0 for j = 1, 2, 3.

(*)

There are three harmonic algebras exactly over the field of complex numbers only, and all harmonic bases are constructed by I. Mel’nichenko. We consider a harmonic algebra A3 containing the radical with basis {ρ1 , ρ2 } and multiplication table: ρ21 = ρ2 ,

ρ22 = 0,

ρ1 ρ2 = 0.

III.1. Toeplitz operators and their applications We proved that every locally bounded function differentiable in the sense of Gateaux (such a function is monogenic) Φ(ζ) = U1 (x, y, z)e1 + U2 (x, y, z)e2 + U3 (x, y, z)e3 (here ζ = xe1 + ye2 + ze3 and x, y, z are real) has nth Gateau derivative for any n. So, the components U1 , U2 , U3 satisfy the three-dimensional Laplace equation „ 2 « ∂ ∂2 ∂2 ∆3 U := + + U (x, y, z) = 0 ∂x2 ∂y 2 ∂z 2 00

(ζ) (e21 + e22 + e23 )

owing to equality ∆3 Φ = Φ and equality (*). For monogenic functions Φ(ζ) taking values in A3 , we proved Cauchy’s theorems for surface integral and curvilinear integral. We proved also an analog of Cauchy’s formula that yields Taylor’s expansion of monogenic function. Morera’s theorem is also established. Thus, as in the complex plane, one can give different equivalent definitions of monogenic functions taking values in the algebra A3 . This is joint work with S. Plaksa. ——— Initial boundary value problems with quaternionic analysis ¨ ßig Wolfgang Spro TU Bergakademie Freiberg, Institute of Applied Analysis, Pr¨ uferstr. 9, Freiberg 09599 Germany [email protected] A quaternionic operator calculus is used to find representations of the solution of several initial boundary value problems in mathematical physics. ——— Real bi-graded Clifford modules, the Majorana equation and the standard model action ¨ rgen Tolksdorf, Ju Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22, 04105 Leipzig, Germany [email protected] The fundamental grading involution that underlies the Dirac equation is provided by parity. In contrast, the Majorana equation is based on charge conjugation. Together, these two grading involutions form what is called a Majorana module. On these modules there exist a natural class of Dirac operators encoding the action functional of the Standard Model of particle physics. ——— The regularized Schr¨ odinger semigroup acting on tensors with values in vector bundles Nelson Vieira Departamento de Matem´ atica-Universidade de Aveiro, Campus Universit´ ario de Santiago, P-3810-193 Aveiro, Portugal [email protected] In this talk we apply known techniques from semigroup theory and Clifford analysis to the homogeneous problem with initial condition of the Schr¨ odinger equation.

To do this end, we start by express the arising tensorial spaces in terms of complexified Clifford algebras and we construct a fiber bundle identification of our spaces with appropriated vector spaces of tensors and differential forms. We then establish the semi-groups for the family of regularized Schr¨ odinger operators and prove their dissipative property. We end with an application to the non-stationary Schr¨ oringer equation. ———

III.1. Toeplitz operators and their applications Organisers: Sergei Grudsky, Nikolai Vasilevski The idea of the session is to bring together the experts actively working on Toeplitz operators acting on Bergman, Fock or Hardy spaces, as well as in various related areas where Toeplitz operators play an essential role, such as asymptotic linear algebra, quantisation, approximation, singular integral and convolution type operators, financial mathematics, etc. We expect that the results presented, together with fruitful discussions, will serve as a snapshot of the current stage of the area, as well as for better understanding of the priority directions and themes of future developments. —Abstracts— On the relations between the kernel of a Toeplitz operator and the solutions to some associated RiemannHilbert problems ˆ mara Cristina Ca Departamento de Matem´ atica, Instituto Superior T´ecnico, Av. Rovisco Pais ,Lisboa, 1049-001 Portugal [email protected] It is possible, in many cases, to determine some solution to a Riemann-Hilbert problem associated to TG , of the form ± n Gh+ = h− , h± ∈ (H∞ ) . (*) Such a solution can provide important information on the properties of TG . Namely, for G ∈ (L∞ (R))n×n , with det G = 1 (or admitting a bounded canonical factorization), if h± = (h1± , h2± ) are corona pairs in C± , i.e., inf (|h1± (ξ)| + |h2± (ξ)|) > 0, (**) ξ∈C±

it can be shown that TG is invertible. In this talk, the question of what information can be obtained, as regards the kernel of TG , from a solution to (*), is considered. Several classes of symbols are studied which, if n = 2, correspond to a situation where (**) is not, or may not, be satisfied. ——— Convolution type operators with symmetry in exterior wedge diffraction problems Luis Castro Campus Universitario, Department of Mathematics, University of Aveiro, Aveiro 3810-193 Portugal [email protected]

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III.1. Toeplitz operators and their applications We will use convolution type operators with symmetry in a Bessel potential spaces framework to analyse classes of problems of wave diffraction by a plane angular screen occupying an infinite 270 degrees wedge sector. The problems are subjected to different possible combinations of boundary conditions on the faces of the wedge. Namely, under consideration there will be boundary conditions of Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, impedanceDirichlet, and impedance-Neumann types. Existence and uniqueness results are proved for all these cases in the weak formulation. In addition, the solutions are provided within the spaces in consideration, and higher regularity of solutions are also obtained in a scale of Bessel potential spaces. The talk is based on a joint work with D. Kapanadze. ——— Berezin transform on the harmonic Fock space Miroslav Englis Mathematics Institute AS CR Zitna 25, Prague 1, Prague 11567 Czech Republic [email protected]

Lorentz space L(p, q, wdµ)(R2d ). M (p, q, w)(Rd ) is a Banach space with the norm kf kM (p,q,w) = kVg f kpq,w . In this paper we discussed the boundedness of Toeplitz operator on M (p, q, w)(Rd ) under some assumptions. We also proved that the Toeplitz operator T pg (F ) of M (2, p, w1 )(Rd ) into M (2, p, w1 )(Rd ) is S2 with the Hilbert-Schmidt norm bounded by kT pg (F )kS2 CkF k(1,t) under some condition. This is joint work with Ay¸se Sandik¸ci. ——— Presentation of the kernel of a special structure matrix characteristic operator by the kernels of two operators one of them is a scalar characteristic operator Oleksandr Karelin Advanced Research Center on Industrial Engineering, Autonomous University of the Hidalgo State, Pachuca, Hidalgo 42184 Mexico [email protected] We denote the Cauchy singular integral operator along the upper part of the unit semicircle T+ by Z ϕ(τ ) 1 (ST+ ϕ)(x) = dτ πi τ −x T+

The standard Berezin-Toeplitz quantization is based on the asymptotic expansion of the Berezin transform as the weight parameter tends to infinity. We discuss an extension of this result to the case of the harmonic SegalBargmann-Fock space on Cn .

and the identity operator on T+ by (IT+ ϕ)(t) = ϕ(t). By operator equalities, results about the integral operators with endpoint singularities are extended to matrix characteristic operators

———

DR+ = uIT+ + vST+ , DT+ ∈ [L22 (T+ )]

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

with the coefficients u, v of a special structure. The following decomposition \ ˜ ker DT+ = ker H F ker C,

Sergey Grudsky Department of Mathematics, CINVESTAV, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, 07360 Mexico [email protected] While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n goes to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. ——— Toeplitz operators of M (p, q, w)(Rd ) spaces ¨ rkanlı Turan Gu Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit Samsun, 55139 Turkey [email protected] Let g be a function in S(Rd )/0, where S(Rd ) is Schwartz space, and 1 ≤ p, q ≤ ∞. The space M (p, q, w)(Rd ) denotes the subspace of all tempered distributions f such that the Gabor transform Vg f of f is in the weighted

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is found. Here operator C is a scalar characteristic operator, C ∈ [L2 (T+ )], operator F is invertible operator, ˜ and C are conF ∈ [L2 (T+ ), L22 (T+ )]. Operators H structed by an arbitrary nontrivial element of ker DT+ or by an arbitrary nontrivial element of the kernel of the associated operator. This is joint work with Anna Tarasenko. ——— Bounds for the kernel dimension of singular integral operators with Carleman shift Edixon Rojas Campus Universitario, Department of Mathematics, University of Aveiro, Aveiro 3810-193 Portugal [email protected] Upper bounds for the kernel dimension of singular integral operators with preserving-orientation Carleman shift are obtained. This is implemented by using some estimations which are derived with the help of certain explicit operator relations. In particular, the interplay between classes of operators with and without Carleman shifts has a preponderant importance to achieve the mentioned bounds. ——— Invertibility of matrix Wiener-Hopf plus Hankel operators with different Fourier symbols Anabela Silva Departamento de Matem´ atica - Universidade de Aveiro,

III.2. Reproducing kernels and related topics Campus de Santiago, Aveiro 3810-193 Portugal [email protected]

States [email protected]

Based on different kinds of auxiliary operators and corresponding operator relations, we will present conditions which characterize the invertibility of matrix WienerHopf plus Hankel operators having different Fourier symbols in the class of almost periodic elements.

We will discuss Toeplitz operators on the Fock space induced by positive measures. Problems considered include boundedness, compactness, and membership in the Schatten classes. ———

——— Flat Hilbert bundles and Toeplitz operators on symmetric spaces Harald Upmeier Department of Mathematics, University of Marburg, Hans-Meerwein-Strasse, Lahnberge Marburg, Hessen 35032 Germany [email protected] In generalization of the classical Fock spaces we construct a family of Hilbert spaces, viewed as a Hilbert bundle over a bounded symmetric domain (Cartan domain) B, which is equivariant under a suitable, nonholomorphic, action of the holomorphic automorphism group G of B (a semisimple Lie group). Geometrically, these Hilbert spaces live on the so-called Matsuki dual associated with the G-orbits in the boundary of B. We show that the Hilbert bundle carries a natural connection over B which is projectively flat, similar as the well-known case for the metaplectic representation on Fock space. The associated parallel transport (Bogoluybov transformations) is also determined. In the talk we emphasize relations to classical Fock spaces over real, complex and quaternion matrix spaces, although the basic construction depends mainly on the Jordan algebraic description of bounded symmetric domains. ——— Commutative algebras of Toeplitz operators on the unit ball Nikolai Vasilevski Department of Mathematics, CINVESTAV, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, 07360 Mexico nv[email protected] All known commutative C ∗ -algebras generated by Topelitz operators on the unit disk are classified as follows. Given a maximal commutative subgroup of biholomorphisms of the unit ball, the C ∗ -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. Surprisently there exist many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C ∗ -algebra, and for n = 1 all of them collapse to commutative C ∗ -algebra generated by Toeplitz operators on the unit disk. ——— Toeplitz operators on the Fock space Kehe Zhu Department of Mathematics and Statistics, 1400 Washington Ave, SUNY Albany, New York 12222 United

III.2. Reproducing kernels and related topics Organisers: Alain Berlinet, Saburu Saitoh Since the first works laying its foundations as a subfield of Complex Analysis, the theory of reproducing kernels has proved to be a powerful tool in many fields of Pure and Applied Mathematics. The aim of this session is to gather researchers interested in theoretical as well as applied modern problems involving this theory. —Abstracts— A general theory for kernel estimation of smooth functionals Belkacem Abdous Universite Laval Medecine Sociale et Preventive, Pavillon de l’Est, Quebec, Qc G1K 7P4 Canada [email protected] In this talk, we present a general framework for estimating smooth functionals of the probability distribution functions, such as the density, the hazard rate function, the mean residual time, the Lorenz curve, the spectral density, the tail index, the quantile function and many others. This framework is based on maximizing a local asymptotic pseudo-likelihood associated to the empirical distribution function. An explicit solution of this problem is obtained by means of reproducing kernels approach. Some asymptotic properties of the obtained estimators are presented as well. ——— Weighted composition operators on some spaces of analytic functions Som Datt Sharma Department of Mathematics, University of Jammu, Jammu-180006, India 66 Ashok Nagar, Canal Road, Jammu, Jammu & Kashmir 180016 India somdatt [email protected] Let D be the open unit disk in the complex plane C and H(D) be the space of holomorphic functions on D. In this article, we give a short and selective account of results known about weighted compostion operator Wψ,ϕ defined by Wψ,ϕ f (z) = ψ(z)f (ϕ(z)),

f ∈ H(D),

where ϕ is a holomorphic map of D that takes D into itself and ψ is any holomorphic map of D. Discriptions of weighted composition operators acting from Hardy spaces, weighted Bergman spaces, α-Bloch spaces and A−α -spaces into other spaces of holomorphic functions

47

III.3. Modern aspects of the theory of integral transforms have been obtained by a number of authors during recent years. We provide a unified way of treating these operators. ——— Integral formulas on the boundary of some ball Keiko Fujita Faculty of Culture and Education, Saga university, Saga 840-8502 [email protected] We have been studied integral representations for holomorphic functions and complex harmonic functions on some balls, which we call the ”Np -balls”. One of Np balls is the Lie ball. For holomorphic functions on the Lie ball we know the Cauchy-Hua integral formula, whose integral is taken over the Shilov boundary of the Lie ball. A generalization of the Cauchy-Hua integral formula was considered for holomorpic functions on subspaces of the Lie ball by M.Morimoto. Since Np -ball can be represented by a union of these subspace, the boundary of the Np -ball can be represented by a union of the boundaries of the subspaces. Considering the fact, we consider an integral representation for holomorphic functions on the Np -ball by an iterated integral. In this talk, we will review some integral formulas on holomorphic functions on the Np -ball and treat some topics.

4) Vienna, A-1090 Austria [email protected] Irregular sampling in spline-type spaces has become a vivid research area, with many contributions in the recent literature. We will describe efficient implementations of operators related to spline-type spaces with finite sets of generators on Rd , covering both the case of regular and irregular sampling. In contrast to earlier papers, which either treat the continuous setting using abstract methods (i.e. continuous Fourier transforms) or deal with the discrete case when it comes to numerical implementations, we are discussing the problem of constructively realizing the abstract concepts with methods that can be implemented on a computer, achieving a small error of reconstruction in a certain given norm. In such a situation the trade-off between realizing individual iterative steps with high precision but at high computational costs, versus the option of doing a larger number of iterations has to be analyzed. Joint work with Prof. Hans Feichtinger. ——— Free boundary value problem for (−1)M (d/dx)2M and the best constant of Sobolev inequality Kazuo Takemura Shinei 2-11-1, Narashino, Chiba 275-8576 Japan [email protected]

——— Paley–Wiener spaces and their reproducing formulae. John Rowland Higgins I.H.P., 4 rue du Bary, 11250 Montclar, France. [email protected] Classical Paley–Wiener space, denoted by PW, consists of functions that are inverse Fourier transforms of those members of L2 (R) that are null outside [−π, π]. It is well known that PW possesses two reproducing formulae; a reproducing equation and a ‘discrete’ analogue, or sampling series, and that these make a remarkable ‘concrete – discrete’ comparison. It is shown that such analogies persist in the setting of more general Paley– Wiener spaces. ‘Operator’ versions of the reproducing equation and of the sampling series will be given that are also comparable, but now in a slightly different way. The setting emerges from two sources, the approach to sampling theory via the reproducing kernel theory due to S. Saitoh, and the approach via harmonic analysis of I. Kluv´ anek, M.M. Dodson et al. The capacity for amalgamation of these two sources has gone unnoticed hitherto. The special case of multiplier operators with respect to the Fourier transform acting on Paley–Wiener space will be considered. The Hilbert transform, and in twodimensions the Riesz transforms, provide examples with possibilities of extension to higher dimensions and to further classes of operators.

Green function of free boundary value problem for (−1)M (d/dx)2M is found using Whipple’s formula. Its Green function is constructed through so-called symmetric orthogonalization method under a suitable solvability conditions. As an application, we found the best constant of Sobolev inequality for M = 1, 2, 3, 4, 5 by investigating an aspect of Green function as a reproducing kernel. For M ≥ 6, this is still open. ———

III.3. Modern aspects of the theory of integral transforms Organisers: Anatoly Kilbas, Saburu Saitoh

—Abstracts— Integral transforms related to generalized convolutions and their applications to solving integral equations Liubov Britvina Department of Theoretical and Mathematical Physics, Novgorod State University, ul.St.Petersburgskaya 41, Veliky Novgorod, Novgorod region 173003 Russia [email protected]

——— Irregular sampling in multiple-window spline-type spaces Darian Onchis Faculty of Mathematics, University of Vienna, Nordbergstrae 15 (Universit¨ ats Zentrum Althanstrae, UZA

48

The present research is devoted to some integral transforms of convolution type. The definition of polyconvolution, or generalized convolution, was first introduced by V.A. Kakichev in 1967. Let A1 , A2 and A3 be operators. The generalized convolution of function f (t) and k(t), under A1 , A2 , A3 , with weighted function α(x),

III.3. Modern aspects of the theory of integral transforms is the function h(t) denoted by

“

α

fA1 ∗ kA2

”

(t) for A3

which the following factorization property is valid: »“ ” – α (A3 h)(x) = A3 fA1 ∗ kA2 (x) A3

=

α(x)(A1 f )(x)(A2 k)(x).

Here we consider the generalized convolution for integral transforms with the Bessel functions in the kernels. Using the differential properties of these convolutions we construct some integral transforms and find their existence conditions and inverse formulas. Natural applications to the corresponding class of convolution integral equations are demonstrated. ——— Bedrosian identity for Blaschke products in n-parameter cases Qiuhui Chen Departamento de Matem´ atica - Universidade de Aveiro, Campus de Santiago, Aveiro 3810-193 Portugal [email protected] We establish a necessary and sufficient condition for the amplitude function such that a Bedrosian identity holds in the case when the phase function is determined by the boundary value of a Blaschke product with nparameters. ——— Evaluation formulae for analogues of conditional analytic Feynman integrals over a function space Dong Hyun Cho Department of Mathematics, Kyonggi University, Young-Tong-Gu Suwon, Kyonggido 443-760 South Korea [email protected] In this talk, we introduce two simple formulae for the conditional expectations over an analogue C[0, t] of the Wiener space, the space of continuous real-valued paths on the interval [0, t]. Using these formulae, we establish various formulae for analogues of the conditional analytic Wiener and Feynman integral of the functionals in a Banach algebra which corresponds to the Banach algebra on the classical Wiener space introduced by Cameron and Storvick. Finally, we evaluate the analogues of the conditional analytic Wiener and Feynman integral for the functional Z t ff exp θ(s, x(s)) dη(s) 0

which is defined on C[0, t] and is of interest in Feynman integration theories and quantum mechanics. ——— An equation with symmetrized fractional derivatives ´anin Diana Dolic Faculty of Technical Sciences, University of Priˇstina - Kosovska Mitrovica, Kneza Miloˇsa, 28000 Kosovska Mitrovica, Serbia dolicanin [email protected]

We study equation Z 1 d2 ± α ET u (t) φ(α)dα + F (t, u (t)) = 0, 0 < t < T u (t)+ dt2 0 where, ± ETα u (t) is the symmetrized Caputo fractional derivative of u, φ(α), α ∈ (0, 1), is a positive integrable function or a positive compactly supported distribution with the support in (0, 1) and F is a continuous function in [0, T ] × R and locally Lipschitz continuous with respect to the second variable. This is joint work with T. Atanackovic, S. Konjik and S. Pilipovic. ——— Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetic Hiroshi Fujiwara Kyoto University, Yoshida-Honmachi Sakyo-ku Kyoto, Kyoto 606-8501 Japan [email protected] We consider the real inversion of the Laplace transform. It appears in engineering or physics, and it is ill-posed in the sense of Hadamard. We introduce some reproducing kernel Hilbert spaces and propose an inversion algorithm employing Tikhonov regularization. The regularized equation is well-posed, and its discretization is expected to have the stability and convergence with a suitable norm. However, theoretical stability is not equivalent to the stability of computational processes. We propose the use of multiple-precision arithmetic to reduce the influence of rounding errors for reliable numerical computations. Multiple-precision arithmetic is useful for regularization as approximation. ——— Method of integral transforms in the theory of fractional differential equations Anatoly Kilbas Faculty of Mathematics and Mechanics, Belarusian State University, Independence Avenue, 4 Minsk, 220030 Belarus [email protected] Our report deals with the method of integral transforms in investigtation of differential equations with ordinary and partial fractional derivatives. First we give an overview of results in this field. Then we present application of one-dimensional Laplace, Mellin and Fourier integral transforms to solution of ordinary diferential equations with Riemann-Liouville and Caputo fracitonal derivatives. Further we give application of Laplace and Fourier integral transforms to obtain explicit solutions of Cauchy-type and Cauchy problems for the twoand multi-dimensional diffusion-wave equations with the Riemann-Liouville and Caputo partial fractional derivatives, respectively, and indicate conditions for the existence of classical solutions of these problems. Finally, we use Laplace and Fourier integral transforms to deduce explicit solutions of fractional evolution equations involving partial fractional derivatives of RiemannLiouville or Caputo with respect to time and partial Lioville fractional derivative with respect to real axis, and indicate applications.

49

III.4. Spaces of differentiable functions of several real variables and applications We note that explicit solutions of the considered fractional differential equations and Cauchy-type and Cauchy problems for them are expressed in terms of special functions of Mittag-Leffler, Wright and the so-called H-functions.

where φ is ”weight” distribution with compact support and, Dγ denotes the Riemman-Liouvill fractional derivative of order γ. Differential equations of the form Z

——— 0

Notes on the analytic Feynman integral over paths in abstract Wiener space Bong Jin Kim Department of Mathematics, Daejin University, Pocheon, Kyeonggi-Do 487-711 South Korea [email protected] In this talk, we study some results about analytic Feynman integral over paths in abstract Wiener space.

2

φ1 (γ)Dγ u dγ =

Z

2

φ2 (γ)Dγ v dγ

(*)

0

are constitutive equations for viscoelastic body. We consider (*) coupled with nonlinear ordinary differential equation D2 u(t) + v = f (t, u(t))

(**)

and show existence and uniqueness of the solution to the problem (*)–(**) with initial conditions u(0) = u0 , u0 (0) = v0 in classical and mild sence.

——— On the fractional calculus of variations Sanja Konjik Faculty of Agriculture, Department of Agricultural Engineering, Trg D. Obradovica 8, Novi Sad, 21000 Serbia sanja [email protected] The purpose of this talk is to study variational principles allowing Lagrangian density to contain derivatives of arbitrary real order. We derive a necessary condition for existence of a solution to a fractional variational problem and examine invariance under the action of transformation groups. As the results we obtain the Euler-Lagrange equations, as well as infinitesimal criterion and Noether’s theorem, which in fact extend the well-known classical results. In addition, we also study the case when both function and the order of fractional derivative are varied in the minimization procedure. ——— Integral transforms with extended generalized MittagLeffler function Anna Koroleva Department of Mathematics and Mechanics, Nezavicimosti ave 4, Minsk BY-220030 Belarus [email protected] Asymptotic results for integral transforms with extended generalized Mittag-Leffler function in the kernel are discussed. Inversion formulas for such transforms in weighted spaces of integrable functions are found. ——— Systems of differential equations containing distributed order fractional derivative Ljubica Oparnica Serbian Academy of Science and Art, Kneza Mihaila 36, Belgrade, 11000 Serbia [email protected] Distributed order fractional derivatives has appeared as generalization of the finite sum of fractional derivatives and has wide applications in technical sciences. We define distributed order fractional derivative of distribution u ∈ S 0 R) supported in R+ by fomula Z h φ(γ)Dγ u dγ, ϕi = hφ, hDγ u, ϕii, ϕ ∈ S 0 , supp u

50

——— Some aspects of modified Kontorovitch-Lebedev integral transforms Juri M. Rappoport Vlasov street Building 27 apt.8 Moscow 117335 [email protected] A proof of inversion formulas of the modified Kontorovitch-Lebedev integral transforms is developed. The Parceval equations for modified KontorovitchLebedev integral transforms are proved and sufficient conditions for them are found. Some new representations and properties of these transforms are justified. The inequalities which give estimations for their kernels - the real and imaginary parts of the modified Bessel functions of the second kind Re K1/2+iτ (x) and Im K1/2+iτ (x) for all values of the variables x and τ are obtained. The applications of Kontorovitch-Lebedev transforms to the solution of some mixed boundary value problems in the wedge domains are accomplished. The solution of the appropriate dual and singular integral equations is considered. The numerical aspects of using of these transforms are elaborated in detail. ——— A new class of polynomials related to the KontorovichLebedev transform Semyon Yakubovich Department of Pure Mathematics, Faculty of Science, University of Porto, Campo Alegre str. 687, Porto 4169007 Portugal [email protected] We consider a class of polynomials related to the kernel Kiτ (x) of the Kontorovich-Lebedev transformation. Algebraic and differential properties are investigated and integral representations are derived. We draw a parallel and establish a relationship with the Bernoullis and Eulers numbers and polynomials. Finally, as an application we invert a discrete transformation with the introduced polynomials as the kernel, basing it on a decomposition of Taylors series in terms of the Kontorovich-Lebedev operator. ———

III.4. Spaces of differentiable functions of several real variables and applications

III.4. Spaces of differentiable functions of several real variables and applications Organisers: Viktor Burenkov, Stefan Samko This session intends to cover various aspects of the theory of Real Variables Function Spaces (Lebesgue, Orlich, Sobolev, Nikol’skii-Besov, Lizorkin-Triebel, Morrey, Campanato, and other spaces with zero or nonzero smoothness), such as imbedding properties, density of nice functions, weight problems, trace problems, extension theorems, duality theory etc. Various generalizations of these spaces are welcome, such as for example Orlicz-Sobolev spaces, in particular generalized Lebesgue-Sobolev spaces of variable order, Morrey-Sobolev spaces, Muiselak-Orlich spaces and their Sobolev counterparts etc. Other topics: any inequalities related to these spaces, properties of operators of real analysis acting in such spaces and also various applications to partial differential equations and integral equations. —Abstracts—

such that ϕ ~ k = {ϕk1 , ϕk2 , · · · , ϕkn } and G ⊂ Rn be an open parallelepiped with sides parallel to the coordinate planes. Then a. holds true the inclusion k

ϕ ~ Bθϕ~ (· · · Bθϕ~ (Lp (G)) · · · ) ,→ Bp,θ (G) | {z } k

b. under the additional assumption that there exists bounded extension operator ´ k` ´ k` S : Bθϕ~ G → Bθϕ~ Rn , holds true the equality of spaces k

ϕ ~ (G) Bθϕ~ (· · · Bθϕ~ (Lp (G)) · · · ) = Bp,θ | {z } k

with equivalence of norms. In these results: (1.) If we set ϕj (h) = hlj we get results which are in works of V.I. Burenkov. (2.) In case, when ϕ(h) satisfy additional condition ϕ(h) ↑ on (0, H] h leads to increment of smoothness using iterated norms. This is joint work with Abraham N. Abebe. ∃ > 0 :

Hardy spaces with generalized parameter Alexandre Almeida Department of Mathematics, University of Aveiro Aveiro, Aveiro 3810-193, Portugal [email protected] Hardy spaces with generalized parameter are introduced following the maximal characterization approach. As particular cases, they include the classical Hardy spaces H p and the Hardy-Lorentz spaces H p,q . Real interpolation results with function parameter are obtained. Based on them, the behavior of some classical operators is studied in this generalized setting. This talk is based on joint work with A. Caetano. ——— Iterated norms in Nikol’ski˘ı-Besov type spaces with generalized smoothness Tsegaye Gedif Ayele Department of Mathematics, Addis Ababa University P.O.Box 1176 Addis Ababa - Ethiopia. [email protected] In works of V.I. Burenkov iterated norms of Nikol’ski˘ıBesov type in spaces Bθl (· · · Bθl (Lp (Ω)) · · · ), k times iterated, were introduced. Using these norms, it was proved that every classical solution of the partial differential equation with constant coefficients is infinitely differentiable. In this paper we consider iterated norms of Nikol’ski˘ı-Besov type in spaces Bθϕ~ (· · · Bθϕ~ (Lp (Ω)) · · · ) with generalized smoothness ϕ ~ belonging to some class of functions Φ(~σ , θ) and with the norm kf k ϕ~ Bθ (· · · Bθϕ~ (Lp (Ω)) · · · ) | {z } k

and holds the following Theorem. Let 1 < p < ∞, 1 ≤ θ < ∞, ~σ = (σ1 , σ2 , · · · , σn ), ϕ ~ = (ϕ1 , ϕ2 , · · · , ϕn ) ∈ Φ(~σ , θ)

——— p(.)

Embeddings Properties of The Spaces Lw (Rn ) Ismail Aydın Sinop University, Faculty of Arts and Sciences, Department of Mathematics, 57000. Sinop-TURKIYE [email protected] We derive some of the basic properties of weighted varip(.) able exponent Lebesgue spaces Lw (Rn ) and investip2 (.) p (.) gate continuous embeddings Lw2 (Rn ) ,→ Lw11 (Rn ) with respect to variable exponents and weight functions under some conditions. ——— On the boundedness of fractional B-maximal operators in the Lorentz spaces Lp,q,γ (Rn ) Canay Aykol Ankara University, Faculty of Science, Department of Mathematics, Ankara, Tandogan 06100 Turkey [email protected] In this study, sharp rearrangement inequalities for the fractional B-maximal function Mα,γ f are obtained in the Lorentz spaces Lp,q,γ and by using these inequalities the boundedness conditions of the operator Mα,γ are found. Then, the conditions for the boundedness of the B-maximal operator Mγ are obtained in Lp,q,γ . ——— Spaces of functions of fractional smoothness on an irregular domain Oleg Besov Steklov Institute of Mathematics, Department of Function Theory, 8 Gubkina Str, Moscow 119991 Russia [email protected] In 1938 S.L. Sobolev proved his well-known embedding theorem Wpm (G) ⊂ Lq (G),

m ∈ N,

1 < p < q < ∞,

(*)

51

III.4. Spaces of differentiable functions of several real variables and applications m−

n n + ≥ 0, p q

(**)

for domains G ⊂ Rn satisfying the cone condition. Relationship (**) (which determines the maximum possible value of q in theorem (*)) is also a necessary condition for the embedding. Sobolev’s result has been extended to more general domains. Definition. Given σ ≥ 1, a domain G ⊂ Rn is said to satisfy the σ-cone condition if, for some T > 0 and 0 < κ0 ≤ 1, for any x ∈ G there exists a piecewise smooth path γ = γx : [0, T ] → G,

γ(0) = x,

|γ 0 | ≤ 1

a.e.,

——— On transformation of coordinates invariant relative to Sobolev spaces with polyhedral anisotropy Gurgen Dallakyan Russian-Armenian State University, Yerevan, Armenia [email protected] Let Rn be n -dimentional euclidean space of the points with real coordinates, N0n - the set of multi-indices.

such that dist (γ(t), Rn \ G) ≥ κ0 tσ

for

0 < t ≤ T.

The author established in (2001) that embedding (*) on a domain with the flexible σ-cone condition holds if m−

σ(n − 1) + 1 n + ≥ 0. p q

(***) s(m)

We construct two families of function spaces Lp,θ (G) s(m)

and Bp,θ (G) of fractional smoothness s > 0 on domains G satisfying the flexible σ-cone condition such that embeddings s(m)

s(m)

Wpm (G) ⊂ Lq,θ (G), s(m)

M f where M is the Hardy-Littlewood maximal operator and presetn several applications of it. In particular, we shall give some applications to the setting of the two weights problem for Calder´ on-Zygmund operators. This is a joint work with J. Soria and R. Torres.

Wpm (G) ⊂ Bq,θ (G),

Lp,θ (G) ⊂ Lq (G),

s(m)

Bp,θ (G) ⊂ Lq (G),

Definition. Nonempty polyhedron ℵ ⊂ Rn with vertices in N0n is said to be complete, if it has vertices at the origin and at all coordinate axes of N0n . Complete polyhedron ℵ is called completely regular (CR), if all the coordinates of outward normals of the noncoordinate (n−1)dimentional faces of ℵ are positive. Let ℵ ⊂ Rn - (CR) polyhedron with vertices e0 , e1 , e2 , e3 , ..., eM , where the vertices ej (j = 0 1, ..., n) lie on ˛ j ˛the j-th coordinate axe, e = (0, ..., 0) , l = max ˛ e ˛ . 1≤j≤n

For any domain Ω ⊂ Rn denote by Wpℵ (Ω) (1 < p < ∞) the Sobolev space with polyhedral anisotropy, i.e. the space of functions with finite norm X k u kℵ,Ω = k Dα ukLp (Ω) .

hold with the same loss of smoothness as in (***). ——— Rearrangement transformations on general measure spaces Santiago Boza EPSEVG, Avda Victor Balaguer s/n. Vilanova i Geltr´ u. 08800 (SPAIN) [email protected] For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer’s cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner’s symmetrization, the multidimensional case, and the discrete setting of trees. ——— Last developments on Rubio de Francia’s extrapolation theory Maria Carro Department of Applied Mathematics and Analysis, University of Barcelona, Gran Via 585, Barcelona 08007 Spain [email protected] Since in the early 80’s, J.L. Rubio de Francia developed his celebrated extrapolation theorem, this theory has been developed in order to cover many other situations such as boundedness of operator on rearrangement invariant spaces or multilinear operator. In this talk, we shall present a new estimate on the distribution function of T f in terms of the distribution of

52

α∈ℵ

Consider the m -dimentional manifold Γm ⊂ Ω . Definition. The piece σ ⊂ Γm we call ℵ -regular, if for some n -dimentional subdomain ω , σ ⊂ ω ⊂ Ω, there exist transformation of coordinates invariant relative to Wpℵ (Ω), mapping σ onto σ 0 ⊂ Rm . Remark. Note, that any bounded domain always has pieces of boundary, which are not ℵ -regular. Let Γ ∈ C s be m -dimentional manifold, where s ≥ r, N ˛ ˛ S (r = max ˛ej ˛), i.e., Γ = σk and each piece σk has 1≤j≤M

k=1

representation xik = ψi,k (x0 ) , 0

where x ∈ Gk , Gk ∈ R

n−m

i = 1, ..., m, , ψi,k ∈ C s (Gk ) .

Theorem. Let ℵ ⊂ Rn be (CR) polyhedron, Ω ⊂ Rn satisfies the week condition of rectangle, Γ ∈ C s -m dimentional manifold ( Γ ⊂ Ω). Let σ ⊂ Γ has the representation xi = ψi (x0 ) ,i = 1, ..., m ,x0 = (xm+1 , ..., xn ) ∈ G , ψi ∈ C s (G). Then the piece σ is ℵ -regular, if 1) s ≥ r ; 2) l ≥ r ; ˛ ˛ 3) for all i = 1, ..., m the condition ˛ei ˛ = l holds. ——— The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted Lpw spaces with general weights Ismail Ekincioglu Kutahya Dumlupinar University, Kutahya, Turkey [email protected] In this study, the boundedness of the the high order Riesz-Bessel transformations generated by generalized

III.4. Spaces of differentiable functions of several real variables and applications shift operator in weighted Lpwv-spaces with general weights is proved. ——— Composition Operators for Sobolev spaces of functions and differential forms Vladimir Goldshtein Department of Mathematics, Ben Gurion University of the Negev, P.O.Box 653, Beer Sheva, 84105 Israel [email protected] Composition operators for Sobolev spaces with first derivatives will be discussed. For such spaces composition operators are induced by mappings with bounded mean distortion. These classes of mappings represent a generalization of quasiconformal mappings. Applications to embedding theorems will be described. In the framework of so-called Lq,p -cohomology similar classes of mappings play an important role for quasiisometrical an/or Lipschitz classification of complete noncompact Riemannian manifolds with bounded geometry ———

For generalized potential operators with the kernel a[%(x,y)] on bounded measure metric space (X, µ, %) with [%(x,y)]N doubling measure µ satisfying the upper growth condition µB(x, r) ≤ CrN , N ∈ (0, ∞), we prove weighted estimates in the case of radial type power weight w = [%(x, x0 )]ν . Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the weighted variable exponent Lebesgue space Lp(·) (X, w, µ) into a certain 1

weighted Musielak-Orlicz space LΦ (X, w p(x0 ) , µ) with the N-function Φ(x, r) defined by the exponent p(x) and the function a(r). ——— Our talk is on vanishing exponential integrability for Besov functions. Ritva Hurri-Syrjanen Department of Mathematics and Statistics, University of Helsinki PL 68 (Gustaf Hallstrominkatu 2 b), Helsinki FI-00014 Finland [email protected]

Boundedness of the fractional maximal operator and fractional integral operators in general Morrey type spaces and some applications

Our talk is on vanishing exponential integrability for Besov functions.

Vagif Guliyev F. Agayev str, 7 Rasim Mukhtarov str, 10 Baku, Baku AZ 1069 Azerbaijan [email protected]

New sharp estimates for function in Sobolev spaces on finite Interval

The theory of boundedness of fractional maximal operator and fractional integral operators from one weighted Lebesgue space to another one is by now well studied. These results have good applications in the theory of partial differential equations. However, in the theory of partial differential equations, along with weighted Lebesgue spaces, general Morrey-type spaces also play an important role, but until recently there were no results, containing necessary and sufficient conditions on the weight functions ensuring boundedness of the aforementioned operators from one general Morrey-type space to another one (apart from the cases in which this follows directly from the appropriate results for weighted Lebesgue spaces). The case of power-type weights was well studied C.B. Morrey 1938, D.R. Adams 1975, F. Chiarenza and M. Frasca 1987, but for general Morreytype spaces only sufficient conditions were known (T. Mizuhara 1991, E. Nakai 1994, V.S. Guliyev 1994). In the talk a survey of results, containing necessary and sufficient conditions for boundedness of fractional maximal operator and fractional integral operators, will be given, and open problems will be discussed in detail. As applications, we establish the boundedness of some Sch¨ odinger type operators on general Morrey-type spaces related to certain nonnegative potentials belonging to the reverse H¨ older class. ——— Weighted estimates of generalized potentials in variable exponent Lebesque spaces Mubariz Hajibayov Institute of Mathematics and Mechanics of NAS of Azerbaijan, F.Agayev 9, Baku, Azerbaijan AZ1141 Azerbaijan [email protected]

———

Gennady Kalyabin Peoples Friendship University of Russia, MiklukhoMaklaya Str 6, Moscow, 117198 Russia [email protected] The smallest constants in new kind of Kolmogorov type inequalities for intermediate derivatives are obtained, namely: the quantities Ar,k (x) defined as sup{f (k) (x) : kf (r) kL2 (−1,1) ≤ 1; f

f (s) (±1) = 0,

s ∈ {0, . . . , r − 1}},

are calculated for all natural r, k ∈ {0, . . . , r − 1} and x ∈ (−1, 1). In particular A2r,0 (x) =

(1 − x2 )2r−1 . − 1)

Γ2 (r)22r−1 (2r

As a Corollary it is established that for the first eigenvalue of the boundary problem (−D2 )r y(x) = λy(x), y (s) (±1) = 0, s ∈ {0, . . . , r − 1}, the asymptotic formula √ λ1,r ≈ 2π 2r(2r e)2 r, r → ∞, holds. ——— On real interpolation of weighted Sobolev spaces Leili Kusainova L.N. Gumilev Eurasian National University Astana, Munaitpasov 5, Akmola 010008 Kazakhstan [email protected]

53

III.4. Spaces of differentiable functions of several real variables and applications Let 1 < p < ∞, m ∈ N, Ω ∈ Rn an open set, and let υ be a non-negative function locally integrable in Ω. We denote by Wpm (υ) the weighted Sobolev space with the finite norm

maximal operator Mβ and Dunkl type fractional integral operator Iβ on the Dunkl-type Morrey spaces Lp,λ,α (R), 1 ≤ p < ∞.

|u; Wpm (υ)| = |∇m u; Lp (Ω)| + |u; Lp (Ω; υ)|.

Image normalization of Wiener-Hopf operators in diffraction problems

In talk we describe Peetre interpolation spaces ` this ´ Wpm0 (υ0 ), Wpm1 (υ 1 θ,p for weights υi , which allow introducing local variable average characteristics. Here 0 ≤ m1 < m0 , 1 ≤ p < ∞, mi p 6= n (i = 0, 1). Let d(x) be a positive bounded function in Ω such that for some a > 1 and all x ∈ Ω Qad(x) (x) ⊂ Ω, where Qd (x) = {y ∈ Rn : |yi − xi | < d/2, i = 1, 2, ..., n}. Let Bps (υ) denotes Besov space with the finite norm (s > 0) |u; Bps (υ)| = |u; bsp | + |u; Lp (Ω; υ)|. For certain class of weights υi (i = 0, 1) we have obtained the equality of type ` m0 ´ −sp Wp (υ0 ), Wpm1 (υ1 ) θ,p = Bps ((υ ∗ ) ), where 0 < θ < 1, s = (1 − θ)m0 + θm1 , υ ∗ = max υi∗ (x) i=0,1

and

Ana Moura Santos Dept. de Matematica, IST, Av. Rovisco Pais 1, Lisbon, 1049-001 Portugal [email protected] In this work we discuss the normalization problem for Wiener-Hopf Operators (WHO), which arrives in certain ill-posed boundary-transmission value problems on halfplanes. We first consider a wave diffraction problem by a junction of two infinite half-planes, and different combinations of normal and oblique derivatives on the planes. Then a generalization for higher order derivatives follows. For all studied diffraction problems, which are associated with not normally solvable WHO, we solve the normalization problem based on the image normalization technique previously developed for one half-plane. ———

0 < υi∗ (x) = sup{d : dmi p−n υi (Qd (x)) ≤ 1} ≤ d(x). d>0

——— The Fourier transform of a radial function Elijah Liflyand Department of Mathematics Bar-Ilan University RamatGan, Gush-Dan 52900 Israel [email protected] This talk naturally consists of two parts. In the first one we survey the known results on representation of the Fourier transform of a radial function as the onedimensional Fourier transform of a related function. One of such results, due to Leray, gave an impact to obtaining a series of new such formulas. We discuss those already obtained in a joint work with S. Samko as well as tentative formulas. Correspondingly, already obtained applications are given and certain conjectures are posed. ——— Necessary and sufficient conditions for the boundedness of Riesz potential in Morrey spaces associated with Dunkl operator Yagub Mammadov Institute of Mathematics and Mechanics, Rasim Mukhtarov str. 10, Narimanov area Baku, AZ 1141 Azerbaijan [email protected] The maximal function, fractional maximal function and fractional integrals associated with the Dunkl operator were studied extensively in Lebesgue spaces on R. We study the fractional maximal function (Dunkl-type fractional maximal function) and fractional integrals (Dunkl-type fractional integrals) associated with the Dunkl operator in the Dunkl-type Morrey space Lp,λ,α (R) and Dunkl-type Besov-Morrey spaces s Bpθ,λ,α (R). We obtain the necessary and sufficient conditions for the boundedness of Dunkl-type fractional

54

———

Weighted estimates for the averaging integral operator and reverse H¨ older inequalities Bohum´ır Opic Institute of Mathematics, AS CR ˇ a 25 Zitn´ 11567 Praha 1 Czech Republic [email protected] Let 1 < p < +∞ and let v be a weight on (0, +∞) satisfying v(x)xρ is equivalent to a non-decreasing function on (0, +∞) for some ρ ≥ 0. Let A R x be the averaging operator given by (Af )(x) := x1 0 f (t) dt, x ∈ (0, +∞), and let Lp (v) denote the weighted Lebesgue space of all measurable functions f on (0, +∞) for which ”1/p “R +∞ p |f (x)| v(x) dx < +∞. 0 First, we prove that the following statements are equivalent: (i) A is bounded on Lp (v); (ii) A is bounded on Lp−ε (v) for some ε ∈ (0, p − 1); (iii) A is bounded on Lp (v 1+ε ) for some ε > 0; (iv) A is bounded on Lp (v(x)xε ) for some ε > 0. Moreover, if A is bounded on Lp (v), then A is bounded on Lq (v) for all q ∈ [p, +∞). Second, we show that the boundedness of the averaging operator A on the space Lp (v) implies that, for all r > 0, 0 the weight v 1−p satisfies the reverse H¨ older inequality over the interval (0, r) with respect to the measure dt, while the weight v satisfies the reverse H¨ older inequality over the interval (r, +∞) with respect to the measure t−p dt. Third, assuming moreover that p ≤ q < +∞ and that w is a weight on (0, +∞) such that [w(x)x]1/q ≈ [v(x)x]1/p

for all x ∈ (0, +∞),

we prove that the operator A is bounded on Lp (v) if and only if the operator A : Lp (v) → Lq (w) is bounded. ———

III.4. Spaces of differentiable functions of several real variables and applications Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup Humberto Rafeiro Universidade do Algarve, Dep. Matem´ atica, Campus de Gambelas Faro, Faro 8005-139 Portugal [email protected] In this talk we give a characterization of the variable exponent Bessel potential space in terms of the convergence of the Poisson semigroup. We show that Grunwald-Letnikov construction with respect to the Poisson semigroup coincides with the Riesz fractional differentiation under some natural restrictions on the exponent p(x). ——— On the Maxwell problem Evgeniy Radkevich Mathematics Department, Moscow State University, Vorobievy Gori, Moscow 119992 Russia [email protected]

We admit variable complex valued orders α(x), where 0, k ∈ N, k > l, −∞ ≤ a 6 b 6 ∞. Recall thatf ∈ blp,θ (a, b), if f is measurable on (a, b) and 0 kf kbl

p,θ

(a,b)

B [email protected]

b−a

Zk “

1 θ1 ”θ dh C h−l k∆kh f kLp (a,b−kh) A h

0

is finite.

TBA ———

Theorem. 1 < p, θ 6 ∞, l > 0,k ∈ N, 0 < l < k,α1 ≥ 0, α2 ≥ k. Then for an arbitrary interval (a, b)

Weighted potential operators in Morrey spaces. 0 Natasha Samko Department of Mathematics, University of Algarve, Campus de Gambelas, 8005-139 Faro, Portugal [email protected] We study the weighted (p, λ)-(q, λ)-boundedness of Hardy and potential operators. We show that the weighted boundedness of potential operators is reduced to the boundedness of weighted Hardy operators. In case of power weights or oscillating weights from the BaryStechkin class we find conditions for weighted Hardy operators to be bounded in Morrey spaces. ——— Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces Stefan Samko Universidade do Algarve Campus de Gambelas Faro, Algarve 8005-139 Portugal [email protected] We consider non-standard H¨ older spaces H λ(·) (X) of functions f on a metric measure space (X, d, µ), whose H¨ older exponent λ(x) is variable, depending on x ∈ X. We establish theorems on mapping properties of potential operators of variable order α(x), from such a variable exponent H¨ older space with the exponent λ(x) to another one with a ‘better’ exponent λ(x) + α(x), and similar mapping properties of hypersingular integrals of variable order α(x) from such a space into the space with the ‘worse’ exponent λ(x)−α(x) in the case α(x) < λ(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spaces H λ(·) (X), but also the generalized H¨ older spaces H w(·,·) (X) of functions whose continuity modulus is dominated by a given function w(x, h), x ∈ X, h > 0.

kf kbl

p,θ

(a,b)

B ∼B @

1 θ1

b−a α1 +α2

Z

“

h−l k∆kh f kLp (a+α1 h,b−α2 h)

”θ dh C C , hA

0

where the equivalence constants are independent of a and b. ——— Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications Ayhan Serbetci Department of Mathematics, Ankara University, Tandogan, Ankara 06100 Turkey [email protected] In this study we consider the fractional integral operator Iα on any Carnot group G (i.e., nilpotent stratified Lie group) in the weighted Lebesgue spaces Lp,ρ(x)β (G). We establish Stein-Weiss inequalities for Iα , and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator Iα from the spaces Lp,ρ(x)β (G) to Lq,ρ(x)−γ (G), and from the spaces L1,ρ(x)β (G) to the weak spaces W Lq,ρ(x)−γ (G) by using the Stein-Weiss inequalities. Q In the limiting case p = α−β−γ , we prove that the modified fractional integral operator Ieα is bounded from the space Lp,ρ(x)β (G) to the weighted BMO space BM Oρ(x)−γ (G), where Q is the homogeneous dimension of G. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two SobolevStein embedding theorems on weighted Lebesgue and weighted Besov spaces in the Carnot group setting. As an another application, we prove the boundedness of Iα s s from the weighted Besov spaces Bpθ,β (G) to Bqθ,−γ (G). ———

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III.5. Analytic and harmonic function spaces Translation-invariant bilinear operators with positive kernels Javier Soria Department of Applied Mathematics and Analysis, University of Barcelona, Gran Via 585, Barcelona 08007 Spain [email protected] We study the boundedness of bilinear convolutions operators with positive kernels. We prove both necessary and sufficient conditions and, by means of several counterexamples we show that near the endpoints the behavior of positive translation-invariant bilinear operators can be quite different than that of positive linear ones. ——— Sharp inequalities for moduli of smoothness and Kfunctionals Sergey Tikhonov Centre de Recerca Matem` atica, Facultat de Ci`encies, UAB, Bellaterra, Barcelona 08193 Spain [email protected] We discuss the (p − p) and (p − q) sharp inequalities (Jackson-type, Marchaud-type, Ulyanov-type, etc) for moduli of smoothness/K-functionals. Corresponding embedding theorems are studied. ——— Sobolev embedding theorems for a class of anisotropic irregular domains Boris V. Trushin MIAN (Departament of Function Theory), ul. Gubkina, d. 8, Moscow 119991 Russia [email protected] Sufficient conditions for the embedding of a Sobolev space in Lebesgue spaces and the space of continuous functions on a domain depend on the integrability and smoothness parameters of the spaces and on the geometric features of the domain. In our talk, Sobolev embedding theorems will obtaine for a class of domains with irregular boundary. This new class includes the well-known classes of σ-John domains, domains with the flexible cone condition, and their anisotropic analogs. The results can be extended to weighted spaces with power weights. ——— Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces Yusuf Zeren Department of Mathematics of Harran University, Campus of Osmanbey, SanliUrfa, Region 6300 Turkey [email protected] We obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator Mα , and the Riesz potential operator e p,λ (Rn ) to the Iα from the modified Morrey spaces L n e spaces Lq,λ (R ), 1 < p < q < ∞, and from the e 1,λ (Rn ) to the weak modified Morrey spaces spaces L e W Lq,λ (Rn ), 1 < q < ∞. ———

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III.5. Analytic and harmonic function spaces Organisers: Rauno Aulaskari, Turgay Kaptanoglu, ¨ ttya ¨ Jouni Ra Anticipated topics are normal families, complex valued function spaces and classes, function spaces and local theory of complex differential equations, composition operators between function spaces, boundary behaviour etc.; Hardy, Bergman, Bloch, Besov, Lipschitz, Fock, Qp spaces of one and several holomorphic or harmonic variables, Toeplitz, Hankel, composition, Volterra, multiplication operators, C ∗ or other algebras of such operators, Toeplitz algebras, reproducing kernel Hilbert spaces of holomorphic or harmonic functions, and other similar topics. —Abstracts— Multiplier theorem in the setting of Laguerre hypergroups and applications Miloud Assal Department of Mathematics, Faculty of Sciences of Bizerte Zarzouna, Bizerte 7021, Tunisia [email protected] In this work we study a multiplier theorem in the setting of Laguerre hypergroups and their applications to estimate the solution of Schrdinger equation in Hardy spaces. ——— Progress on finite rank Toeplitz products Boo Rim Choe Department of Mathematics, Korea University, Anamdong 5 ga 1, Seongbuk-gu, Seoul 136-713 South Korea [email protected] It has been conjectured that a product of Toeplitz operators with function symbols, either on Hardy space or Bergman space, has finite rank, then one of the factor must be the zero operator. In this talk we survey recent results towards the conjecture as well as related results. ——— Functions and operators in analytic Besov spaces Daniel Girela Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Campus de Teatinos, Universidad de M´ alaga, M´ alaga 29071 Spain [email protected] In this talk we shall focus on structural and geometric properties of the functions in analytic Besov, primarily on the univalent functions in such spaces, and in operators acting on them. ——— Square functions Maria Jose Gonzales Department of Mathematics, Casem Rio San Pedro

III.5. Analytic and harmonic function spaces Puerto Real, Cadiz 11560 Spain [email protected] We will study multiplicative versions of the usual martingale square function and of the Lusin area of a harmonic function. ——— Convolutions of generic orbital measures in compact symmetric spaces Sanjiv Gupta DOMAS, PO BOX-36 Al-Khodh-123 Sultan Qaboos University Muscat, Oman gupta s [email protected] We prove that in any compact symmetric space, G/K, there is a dense set of a1 , a2 ∈ G such that if µj = mK ∗ δaj ∗ mk is the K-bi-invariant measure supported on Kaj K, then µ1 ∗ µ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1 Ka2 K, has non-empty interior in G. ——— Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman projections ˘ lu H. Turgay Kaptanog Department of Mathematics, Bilkent University, Ankara 06800, Turkey [email protected] Weighted Bergman spaces bpq are well-known spaces of harmonic functions for which q > −1 and 1 ≤ p < ∞. Besov spaces, also denoted bpq , generalize them to all q ∈ R. Our Besov spaces consist of harmonic functions on the unit ball B of Rn so that their sufficiently highorder (t) derivatives are in Bergman spaces (q+pt > −1). We compute the reproducing kernels Rq (x, y) of the Besov spaces b2q with q ≤ −1. The kernels turn out to be weighted infinite sums of zonal harmonics, and also radial fractional derivatives of the Poisson kernel. The new kernels give rise to R generalized Bergman projections by way of Qs ϕ(x) = B ϕ(y) Rs (x, y) (1−|x|2 )s dν(y), where s ∈ R. We prove that Qs : Lpq → bpq are bounded if and only if q + 1 < p(s + 1). This requires new estimates on the integral growth of Bergman kernels near ∂B. We obtain various applications of the Qs . This is joint work with Se¸cil Gerg¨ un and A. Ersin ¨ ¨ ITAK ˙ Ureyen. The work is supported by TUB under Research Project Grant 108T329. ——— Sums of Toeplitz products on the Dirichlet space Young Joo Lee Department of Mathematics, Chonnam National University, Gwangju, Yongbongdong 500-757, South Korea [email protected] In this talk, we will consider a class of operators which contains finite sums of products of two Toeplitz operators with harmonic symbols on the Dirichlet space of the unit disk. We will give characterizations of when an operators in that class is zero or compact. Also, we solve the zero product problem for products of finitely many Toeplitz operators with harmonic symbols.

——— Weighted composition operators on weighted spaces of analytic functions Jasbir Singh Manhas Sultan Qaboos University, Department of Mathematics & Statistics, College of Science, P.O. Box 36, Al-Khod Muscat, Muscat 123 Oman [email protected] Let V be an arbitrary system of weights on an open connected subset G of CN (N ≥ 1). Let HV0 (G) and HVb (G) be the weighted locally convex spaces of analytic functions with topology defined by seminorms which are weighted analogues of the supremum norm. Let Hv0 (G) and Hvb (G) be the weighted Banach spaces of analytic functions defined by a single weight v. In this talk besides presenting the characterizations of weighted composition operators on HV0 (G) ( Hv0 (G) )and HVb (G) ( Hvb (G) ), we shall present some results pertaining to topological structures ( e.g. component structure, Isolated points, compact differences ) of weighted composition operators on the spaces H ∞ (D) and Hv0 (D) ( Hvb (D) ). ——— Superposition operators between Qp spaces and Hardy spaces Auxiliadora Marquez Departamento de Analisis Matematico, Facultad de Ciencias, Campus de Teatinos, Malaga 29071 Spain [email protected] For any pair of numbers (s, p) with 0 ≤ s < ∞ and 0 < p ≤ ∞ we characterize the superposition operators which apply the conformally invariant Qs space into the Hardy space H p and, also, those which apply H p into Qs . ——— Bounded Toeplitz and Hankel products on Bergman space Malgorzata Michalska Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] We improve the sufficient condition for boundedness of products of Toeplitz operators Tf Tg¯ on the Bergman space obtained by K. Stroethoff and D. Zheng in 1999. Using our result we give a short proof of the sufficient and necessary condition for the boundedness of Tf T1/f¯ obtained also by Stroethoff and Zheng in 2002. We consider also the products of Hankel operators Hf Hg∗ . ——— Optimal norm estimate of the harmonic Bergman projection Kyesook Nam Department of Mathematics, Hanshin University, Osansi, Gyeonggi-do 447-791 South Korea [email protected] On the unit ball of the Euclidean n-spaces, we give an optimal norm estimate for one-parameter family of operators associated with the weighted harmonic Bergman

57

III.5. Analytic and harmonic function spaces projections. Using this result, we obtain an optimal norm estimate for the weighted harmonic Bergman projections. This is the joint work with Boo Rim Choe and Hyungwoon Koo. ——— Old and new on composition operators on VMOA and BMOA spaces Pekka Nieminen Dept. of Mathematics and Statistics, Univ of Helsinki, PO Box 68, Helsinki, 00014 Finland [email protected] We review various compactness characterizations for analytic composition operators acting on the spaces VMOA and BMOA, and give some new formulations. We also discuss the equivalence of weak compactness and (norm) compactness for these operators. Joint work with Jussi Laitila, Eero Saksman and Hans-Olav Tylli (Helsinki). ———

Houghton Street London, WC2A 2AE United Kingdom [email protected] We will address the question of whether a left invertible matrix with entries in certain convolution Banach algebras of measures supported in [0, +∞) can be completed to an invertible matrix with entries from the same Banach algebra. The Banach algebras we consider arise naturally in control theory as classes of inverse Laplace transforms of stable transfer functions, and the relevance of the problem of completion to an isomorphism in control theory will also be explained. ——— Multiplication operators on weighted BMOA spaces Benoit F. Sehba Department of Mathematics, University of Glasgow, G12 8QW, Glasgow, UK [email protected] We give some (test function) criteria for symbols of bounded multiplication operators for a special familly of weighted BMOA spaces in the unit ball.

On Libera and Cesaro operators Maria Nowak Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] Let H(D) denote the class of functions holomorphic in the unit disk D. The Ces` aro operator C is defined on P∞ “ 1 Pn ˆ ” n H(D) by Cf (z) = n=0 n+1 k=0 f (k) z , where P∞ ˆ n f (z) = operator L, defined n=0 f (n)z“ . The Libera P∞ P∞ fˆ(k) ” n z , can be considered by Lf (z) = n=0 k=n k+1 as an extension of the conjugate operator C ∗ defined on H(D) - the space of holomorphic functions defined in a neighborhood of D. We obtain results on Libera operator acting on known spaces of holomorphic functions in the unit disk. (Joint work with Miroslav Pavlovic) ——— Integration operators on weighted Bergman spaces Jordi Pau Departament de Matem` atica Aplicada i An` alisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, Barcelona, 08007 Spain [email protected] For an analytic function g on the unit disc, we consider the operators Z z Jg f (z) = f (ζ)g 0 (ζ)dζ. 0

We describe the boundedness and compactness of Jg on Bergman spaces with exponential weights, answering an open question posed by Aleman and Siskakis in 1997. ——— Extension to an invertible matrix in Banach algebras of measures Amol Sasane Mathematics Department, London School of Economics,

58

——— Inequalities for Hardy spaces on the unit ball Pawel Sobolewski Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] In 1988 (TAMS 103(3)) D. Luecking obtained the following results for Hardy spaces H p in the unit disk D ⊂ C. The inequality Z |h(z)|p−s |h0 (z)|s (1 − |z|)s−1 dA(z) ≤ CkhkpH p D

holds for h ∈ H p , p > 0 if and only if 2 ≤ s < p + 2. We obtain analogous results for the Hardy spaces on the unit ball of Cn , n ≥ 2. ——— On the Duhamel algebras ¨ bariz Tapdıgog ˘ lu Mu Isparta Vocational School, Suleyman Demirel University, Dogu Campus Isparta, Cunur 32260 Turkey [email protected] We introduce the notion of Duhamel algebra. We prove that under some natural conditions any Banach space of analytic functions in the unit disc D is the Duhamel algebra and describe its all closed ideals. In particular, we improve some results of Wigley. ——— Toeplitz operators on Bergman spaces Jari Taskinen P.O.Box 68, Department of Mathematics and Statistics University of Helsinki Helsinki, Helsinki FI-00014 Finland [email protected] We give sufficient conditions for boundedness and compactness of Toeplitz operators in the Bergman spaces on the unit disc of the complex plane. We consider both the cases 1 < p < ∞ and p = 1. The conditions concern a

III.6. Spectral theory kind of averages of the symbol on hypebolic rectangles. The sufficient condition is also necessary in the case of positive symbols, and it thus coincides with known results in this case. An approach to the Fredholm properties of Toeplitz operators is also given. ——— Hyperbolic weighted Bergman classes Luis Manuel Tovar Department of Mathematics, Esc. Sup. de Fsica y Mat. I.P.N., Edificio 9, Unidad Prof. A.L.M. Zacatenco del I.P.N., Mexico City, 07738 Mexico [email protected] A new class of like-hyperbolic Bergman class of analytic functions in the unit complex disk is introduced, which has several interesting properties and relationships with several classical weighted spaces, like Bloch, Dirichlet and Qp . ——— Multiplicative isometries and isometric zero-divisors Dragan Vukotic Departamento de Matematicas & ICMAT, Modulo CXV, Universidad Autonoma de Madrid, Madrid, 28049 Spain [email protected] For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlettype spaces), we show that their isometric pointwise multipliers are necessarily unimodular constants. As a consequence, it follows that none of those spaces have isometric zero-divisors. We also investigate the isometric coefficient multipliers. ——— Area operators on analytic function spacess Zhijian Wu Department of Mathematics, The University of Alabama, Tuscaloosa, Alabama 35487 United States [email protected] We characterize non-negative measures µ on the unit disk D for which the area operator Aµ is bounded or compact on Hardy and Bergman spaces. ——— Composition operators on BMOA Hasi Wulan Department of Mathematics, Shantou University Shantou, Guangdong 515063 China [email protected] We give a new and simple compactness criterion for composition operators Cϕ on BMOA and the Bloch space in terms of the norms of ϕn in the respective spaces.

joensuu 80100 Finland [email protected] In this paper, we give a necessary and sufficient condition for a kind of lacunary series on the unit ball to be in Qp spaces for (m − 1)/m < p ≤ 1. The necessity is extended to more general QK spaces. This is a generalization of the result of Aulaskari, Xiao and Zhao for that on the unit disk. ——— Some results on ϕ-Bloch functions Congli Yang Yliopistokatu 7 Metria Building (Y6), Joensuu 80101 Finland [email protected] Let ϕ : [0, 1)→(0, ∞) be an increasing function, such that ϕ(r)(1 − r) → ∞, as r → 1− . An analytic function f (z) in the unit disc is said to be ϕ-Bloch function if it’s derivative satisfies |f 0 (z)| = O(ϕ(|z|)) as |z| → 1− . This paper is devoted to the study of analytic ϕ-Bloch functions. we obtain some new characterizations for ϕBloch functions are established under certain regularity conditions on ϕ. ——— Holomorphic mean Lipschitz spaces Kehe Zhu Department of Mathematics and Statistics, 1400 Washington Ave, SUNY Albany, New York 12222 United States [email protected] I will talk about the connections between holomorphic mean Lipschitz spaces and several other classes of function spaces, including Bergman spaces, Besov spaces, and Bloch type spaces. The setting is the open unit ball in C n . ——— Univalently induced closed range composition operators on the Bloch-type spaces Nina Zorboska Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2 Canada [email protected] We will show that if the closed range composition operator is univalently induced, then the inducing function has to be a disk automorphism, whenever the underlying space is a Bloch-type space B α with alpha not equal to one. The proof uses a combination of methods and results from operator theory, complex analysis and the pseudohyperbolic geometry on the unit disk. ———

——— Lacunary series and QK spaces on the unit ball

III.6. Spectral theory

Wen Xu Yliopistokatu 7 Metria Building (Y6) 3rd floor Joensuu,

Organisers: Brian Davies, Ari Laptev, Yuri Safarov

59

III.6. Spectral theory Anticipated topics are: Spectral theory of differential operators. Spectra of non-self-adjoint operators. Spectral asymptotics. Scattering theory. General spectral theory and related topics.

and the coefficients Aαβ are real-valued Lipschitz continuous functions satisfying Aαβ = Aβα and the uniform ellipticity condition X Aαβ (x)ξα ξβ ≥ θ|ξ|2 |α|=|β|=m

—Abstracts— Strongly elliptic second-order systems in Lipschitz domains: surface potentials, equations at the boundary, and corresponding transmission problems. Mikhael Agranovich [email protected]

for all x ∈ Ω and for all ξα ∈ R, |α| = m, where θ > 0 is the ellipticity constant. We consider open sets Ω for which the spectrum is discrete and can be represented by means of a non-decreasing sequence of non-negative eigenvalues of finite multiplicity λ1 [Ω] ≤ λ2 [Ω] ≤ · · · ≤ λn [Ω] ≤ . . . Here each eigenvalue is repeated as many times as its multiplicity and lim λn [Ω] = ∞ . n→∞

We consider a strongly elliptic second-order system in a bounded Lipschitz domain Ω. For convenience, we assume that Ω = Ω+ lies in the standard torus T = T n and consider the system in the domain Ω− = T \ Ω too. Assuming that the Dirichlet and Neumann problems in the variational setting in Ω± are uniquely solvable in some spaces Hpσ or Bpσ , we describe properties of the surface potentials. We define these operators and derive corresponding formulas following Costabel and McLean, without using properties of the fundamental solution (but do not assume that p = 2 and that the coefficients are smooth). Main results: boundedness of the surface potentials, their invertibility at the boundary (in particular, of the single layer and hypersingular operators) in Besov spaces, and a description of their spectral properties in these spaces (including the case p 6= 2). We also describe applications to the corresponding transmission problems, general and spectral.

The aim is sharp estimates for the variation |λn [Ω1 ] − λn [Ω2 ]| of the eigenvalues corresponding to two open sets Ω1 , Ω2 with continuous boundaries, described by means of the same fixed atlas A. Three types of estimates will be under discussion: for each n ∈ N for some cn > 0 depending only on n, A, m, θ and the Lipschitz constant L of the coefficients Aαβ |λn [Ω1 ] − λn [Ω2 ]| ≤ cn dA (Ω1 , Ω2 ), where dA (Ω1 , Ω2 ) is the so-called atlas distance of Ω1 to Ω2 , |λn [Ω1 ] − λn [Ω2 ]| ≤ cn ω(dHP (∂Ω1 , ∂Ω2 )), where dHP (∂Ω1 , ∂Ω2 ) is the so-called lower HausdoffPompeiu deviation of the boundaries ∂Ω1 and ∂Ω2 and ω is the common modulus of continuity of ∂Ω1 and ∂Ω2 , and, under certain regularity assumptions on ∂Ω1 and ∂Ω2 ,

———

|λn [Ω1 ] − λn [Ω2 ]| ≤ cn meas (Ω1 ∆Ω2 ) ,

On the spectral expansions associated with LaplaceBeltrami operator

where Ω1 ∆Ω2 is the symmetric difference of Ω1 and Ω2 . Joint work with Dr P. D. Lamberti.

Shavkat Alimov Vuzgorodok National University Tashkent, 100174 Uzbekistan shavkat [email protected]

——— of

Uzbekistan,

The eigenfunction expansions associated with LaplaceBeltrami operator on n-dimensional symmetrical manifold Ω of rank 1 is considered. If the eigenfunction expansion of the piecewise smooth function, which depends on the geodesic distance from some point, converges at this point, then considered function belongs to C (n−3)/2 (Ω). This result is the generalization of the result, which was proved by M. Pinsky and W. O. Bray for geodesic ball. ——— Sharp spectral stability estimates for higher order elliptic operators Victor Burenkov Via Trieste 63, Padova University, Padova, 35121, Italy [email protected], [email protected] We consider the eigenvalue problem for the operator “ ” X Dα Aαβ (x)Dβ u , x ∈ Ω, Hu = (−1)m |α|=|β|=m

subject to homogeneous Dirichlet or Neumann boundary conditions, where m ∈ N, Ω is a bounded open set in RN

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Strong field asymptotics for zero modes Daniel Elton Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF United Kingdom [email protected] Given a magnetic potential A one can consider the existence of zero modes (or zero-energy L2 eigenfunctions) of the Weyl-Dirac operator σ.(−i∇ − tA) on R3 ; here t is a positive parameter, with the limit t → ∞ corresponding to the strong field (or, equivalently, semi-classical) regime. General O(t3 ) bounds on the number of zero modes can be obtained. These bounds can be refined to O(t2 ) asymptotics for a special class of potentials A that are constructed from potentials on R2 ; a key step involves localising the Aharonov-Casher theorem to obtain good estimates for the number of “approximate zero modes” for two-dimensional Pauli operators. ——— A universal bound for the trace of the heat kernel Leander Geisinger Universit¨ at Stuttgart, Fakult¨ at Mathematik und Physik, IADM, Pfaffenwaldring 57, Stuttgart 70569, Germany [email protected]

III.6. Spectral theory We derive a unviersal P upper bound for the trace of the heat kernel Z(t) = k e−λk t , where (λk )k∈N denote the eigenvalues of the Dirichlet Laplace Operator in an open set Ω ⊂ R2 with finite volume. The result improves an inequality of Kac and holds true without further assumptions on Ω. The proof is based on improved Berezin-LiYau inequalities with a remainder term. ——— The eigenvalues function of the family of SturmLiouville operators and its applications Tigran Harutyunyan Faculty of Math. and Mechanics of Yerevan State University, Alek Manukyan 1, Yerevan 0049 Armenia [email protected] In order to study the dependence of the eigenvalues of the Sturm-Liouville problem on parameters, defining the boundary conditions, we introduce the concept of the eigenvalues function of the family of Sturm-Liouville operators. We find the necessary and sufficient conditions for some function (of two variables) to be the eigenvalues function. Actually, we solve the direct and inverse Sturm-Liouville problems. This solution particularly includes: a) the new (more precise) asymptotic formulae for the eigenvalues and normalized constants, b) some new uniqueness theorems in the inverse problems, c) the constructive solution of the inverse problems in known and some new statements. Also we introduce the concept of the eigenvalues function of the family of Dirac operators and solve similar problems for that case. ——— Generalized eigenvectors of some Jacobi matrices in the critical case Jan Janas Sniadeckich 8 Warsaw, Warsaw 00-956 Poland [email protected] The talk will be concerned with asymptotic behavior of generalized eigenvectors of a class of Hermitian Jacobi matrices J in the critical case. The last means that the fraction qn /λn generated by the diagonal entries qn of J and its subdiagonal elements λn has the limit ±2. In other word, the limit transfer matrix as n → ∞ contains a Jordan box (=double root in terms of BirkhoffAdams theory). This is the situation where the asymptotic Levinson theorem does not work and one has to elaborate more special methods for asymptotic analysis. ——— Trace expansions for elliptic cone operators Thomas Krainer Penn State Altoona 3000 Ivyside Park Altoona, Pennsylvania 16601 United States [email protected]

I plan to report on recent joint work with Juan Gil and Gerardo Mendoza on the expansion of the resolvent trace and the heat kernel for (nonselfadjoint) elliptic operators on manifolds with conical singularities. Our approach allows for the treatment of elliptic operators A of general form without simplifying assumptions on the coefficients or the geometry near the singularities, and we achieve results for a wide range of closed extensions of A in the appropriate metric L2 -space. In particular, we obtain results for selfadjoint and nonselfadjoint extensions of Hodge-Laplacians in the presence of warped conical singularities where conventional methods that are based on separation of variables and special functions fail. ——— Stability estimates for eigenfunctions of elliptic operators on variable domains Pier Domenico Lamberti Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 Padova, Padova 35121, Italy [email protected] We prove stability estimates for the variation of resolvents and eigenfunctions of second order uniformly elliptic operators subject to homogeneous boundary conditions upon variation of the domain. We consider classes ˜ parametrized by suitable bi-Lipschitz of open sets Ω homeomorphisms φ˜ defined on a fixed reference domain Ω. We obtain estimates expressed in terms of k∇φ˜ − IkLp (Ω) for finite values of p. We apply these estimates in order to control the variation of the eigenfunctions via the measure of the symmetric difference ˜ We also discuss an application to the stability Ω M Ω. of the solutions to the Poisson problem. This is joint work with G. Barbatis and V.I. Burenkov. ——— Spectral theory of the normal operator with the spectra on an algebraic curve Oleksii Mokhonko Kyiv National Taras Shevchenko University, Volodymyrska street, 01033 Kyiv, Ukraine [email protected]

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The Jacobi (three-diagonal) structure of self-adjoint multiplication operator is well-known. Berezansky Yu.M. and Dudkin M.E. proved that similar Jacobi structure is typical not only for self-adjoint operators but also for arbitrary unitary and even for any bounded normal operators for which a cyclic vector exists. This leads to numerous applications of these objects just in the same way as it is for the classical Jacobi matrices, e.g. application to non-abelian difference-differential lattices generated by Lax equation (Golinskii L.B., Mokhonko O.A.). The following results will be presented. 1. Block Jacobi matrix of a bounded normal operator J acts in C1 ⊕C2 ⊕C3 ⊕C4 ⊕· · · . If one knows that the spectrum of J is a subset of a curve {z ∈ C : p(z, z¯) = 0}, p ∈ C[x, y] then its structure can be simplified: it acts over C1 ⊕C2 ⊕· · ·⊕Cn ⊕Cn ⊕· · · (dimension stabilization phenomenon). E.g. if a normal operator is in fact the unitary one then it acts over C1 ⊕ C2 ⊕ C2 ⊕ · · · (CMV matrix structure) and if it is self-adjoint then it acts over `2 ' C1 ⊕ C1 ⊕ · · · .

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IV.1. Pseudo-differential operators 2. The Direct Spectral Problem (the generalized eigenvalue expansion theorem) and the Inverse Spectral Problem will be presented for this type of normal operators. ——— Spectral properties of operators arising from modelling of flows around rotating bodies Jiri Neustupa Mathematical Institute of the Czech Academy of Sciences Zitna 25 Prague 1, Czech Republic 115 67 Czech Republic [email protected] We give a description of the spectrum of a Stokes-type or an Oseen-type operator which appears in mathematical models of flows of a viscous incompressible fluid around rotating bodies. The special attention is paid to the essential spectrum. The operator is considered in an Lq space. ——— New formulae for the wave operators Serge Richard Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB United Kingdom [email protected] We review some new formulae recently obtained for the wave operators of various scattering systems. Different applications of these formulae will be presented. ——— Spectral bundles Benedetto Silvestri Dipartimento di Matematica Pura ed Applicata, Universit di Padova, Via Trieste 63 Padova, 35121 Italy [email protected] In this talk I will construct certain bundles hM, ρ, Xi and hB, η, Xi of Hausdorff locally convex spaces associated to a given Banach bundle hE, π, Xi. Then I will present conditions Q ensuring the existence Q of bounded selections U ∈ M and P ∈ x x∈X x∈X Bx both continuous at a point x∞ ∈ X, such that U(x) is a C0 −semigroup on Ex and P(x) is a spectral projector of the infinitesimal generator of the semigroup U(x), for every x ∈ X. ——— Scattering theory for manifolds and the scattering length Alexander Strohmaier Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom [email protected] We define the so-called scattering length for Riemannian manifolds with cylindrical ends as the time delay that waves experience when scattered in the manifold. We show that this scattering length can be estimated

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by geometric quantities. For vector valued wave equations our estimates depend on quite involved geometric quantities like lengths of homological systoles. ——— Spectrum and wandering Yuriy Tomilov Chopina Str. 12/18 Department of Mathematics and Informatics, Nicholas Copernicus University, Torun and Institute of Mathematics, PAN, Warsaw Torun, Torun 87-100, Poland [email protected] Let T be a bounded linear operator on a Hilbert space H. A vector x ∈ H is called weakly wandering for T if there is an increasing sequence (nk ) such that the vectors T nk x are mutually orthogonal. By a well-known result due to Krengel, every unitary operator on H without point spectrum has a dense subset of weakly wandering vectors. We will present several far-reaching extensions of the Krengel result. In particular, we will show that if T is a power bounded operator on H with infinite peripheral spectrum and with empty peripheral point spectrum then the set of weakly wandering vectors for T is dense in H. Our spectral assumptions on T are in a sense best possible. This is joint work with V. M¨ uller (Prague). ——— Eigenfunctions at the threshold energies of magnetic Dirac operators Tomio Umeda Department of Mathematical Science University of Hyoto, Shosha 2167 Himeji, Hyogo 671-2201 Japan [email protected] This talk will be devoted to investigation of the eigenfunctions at the threshold` energies ±m of ´ the magnetic Dirac operator H = α · − i∇x − A(x) + mβ, where α = (α1 , α2 , α3 ) and β are Dirac matrices and m is a positive constant. It will be considered three different cases of the vector potential A to decay at infinity. In all the cases, it will be` shown that zero ´ modes of the WeylDirac operator σ · − i∇x − A(x) play crucial roles in the analysis of the eigenfunctions at the threshold of H. Here σ = (σ1 , σ2 , σ3 ) denotes Pauli matrices. It turns out that many existing works on the Weyl-Dirac operator can be utilized. Accordingly, various results on the threshold eigenfunctions of the magnetic Dirac operator H are obtained. This talk is based on joint work with Yoshimi Sait¯ o, University of Alabama at Birmingham, U.S.A. ———

IV.1. Pseudo-differential operators Organisers: Luigi Rodino, Man Wah Wong Topics related to pseudo-differential operators such as PDE, geometry, quantisation, wavelet transforms, localisation operators on groups and symmetric domains, mathematical physics, signal and image processing, among others, are the embodiment of the special session.

IV.1. Pseudo-differential operators —Abstracts— Strongly elliptic second-order systems in Lipschitz domains: Dirichlet and Neumann problems. Mikhael Agranovich [email protected] This is a survey talk. We consider a strongly elliptic second-order system in a bounded Lipschitz domain. The coefficients have minimized smoothness. The aim of the talk is to describe the investigation of the Dirichlet and (under natural additional assumptions) Neumann problems in the variational setting in the spaces Hpσ and Bpσ . The main case: the principal symbol is Hermitian. Then we can use the Savar´e approach to the analysis of the smoothness of solutions and combine it with some tools of the interpolation theory, in particular, with Shneiberg’s results on the extrapolation of the invertibility of operators. The main results: conditions for the unique solvability of the problems and some spectral results (including the case p 6= 2) for the corresponding operators. Applications to Neumann-to-Dirichlet operators. Some results are also true for general strongly elliptic systems. We compare this approach with the deep approach based on the investigation of the surface potentials and corresponding equations at the boundary (Calder´ on, Jerisson, Kenig, Verchota and many other mathematicians) in terms of the non-tangential convergence, maximal functions, Rellich-type identities, etc. ——— Generalized ultradistributions and their microlocal analysis Chikh Bouzar Department of Mathematics, Oran-Essenia University B. P. 1925 EL MNAOUER Oran, Oran 31003 Algeria [email protected] We first introduce new algebras of generalized functions containing ultradistributions. We then develop a microlocal analysis suitable for these algebras. Finally, we give an application through an extension of the wellknown H¨ ormander’s theorem on the wave front of the product of two distributions.

Following Wong’s point of view (Wong M.W., The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. Journal, vol. 34 (2005), 393404), we give a formula for the one-parameter strongly λ continuous semigroup e−tL , t > 0, generated by the generalized Hermite operator Lλ , for a fixed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L2 estimate for the solution of the initial value problem for the heat equation governed by Lλ , in terms of the Lp norm, 1 ≤ p ≤??, of the initial data. Similar results have also been derived for the generalized Landau operator A˜ which was firstly introduced by M.A. De Gosson (M.A. De Gosson, Spectral Properties of a class of generalized Landau operators, Comm. Part. Diff. Equ., 33 (2008), 2096–2104), who has studied its spectral properties. ——— Generalization of the Weyl rule for arbitrary operators Leon Cohen City University-Hunter College, 695 Park. Ave, New York, 10471 United States [email protected] The Weyl rule generally deals with two operators whose commutator is a c-number. The generalization to arbitrary operators is of importance and offers interesting and challenging mathematical issues. We review the basic ideas, present new results and discuss the unsolved problems. We also show how our generalization leads to the consideration of quasi-probability distributions for arbitrary variables. In addition to the Weyl rule we consider other rules of association between operators and symbols. ——— Sharp results for the STFT and localization operators Elena Cordero Departimento di Matematica, Universita di Torino, via Carlo Alberto 10 Torino, TO 10123 Italy [email protected]

We show that the bi-dual of the closure of C0∞ in M ∞,1 is an extension of M ∞,1 as a subalgebra of the algebra of bounded operators on L2 .

We completely characterize the boundedness on Lp spaces and on Wiener amalgam spaces of the shorttime Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a well-known STFT boundedness result on Lp spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W (Lp , Lq ) are given and their sharpness is shown. Localization operators are treated similarly. Using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for localization operators on Lp spaces and prove the optimality of our results. More generally, we prove sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.

———

———

The heat equation for the generalized Hermite and the generalized Landau operators

Fuchsian mild microfunctions with fractional order and their applications to hyperbolic equations

Viorel Catana University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest 060042 Romania catana [email protected]

Yasuo Chiba 1404-1, Katakura-cho Hachioji, Tokyo 1920982 Japan [email protected]

——— Some remarks on the Sj¨ ostrand class Ernesto Buzano Dipartimento di Matematica, Universit` a di Torino, Via Carlo Alberto 10, Torino 10123 Italy [email protected]

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IV.1. Pseudo-differential operators Kataoka introduced a concept of mildness in boundary value problems. He defined mild microfunctions with boundary values. This theory has effective results in propagation of singularities of diffraction. Furthermore, Oaku introduced F-mild microfunctions and applied them to Fuchsian partial differential equations. Based on these theories, we introduce Fuchsian mild microfunctions with fractional order. We show the properties of such microfunctions and their applications to partial differential equations of hyperbolic type. By using a fractional coordinate transform and a quantized Legendre transform, degenerate hyperbolic equations are transformed into equations with derivatives of fractional order. We present a correspondence between solutions for the hyperbolic equations and those for the transformed equations. ——— About Gevrey semi-global solvability of a class of complex planar vector fields with degeneracies Paulo Dattori da Silva Faculdade de Filosofia, Ciˆencias e Letras de Ribeir˜ ao Preto - Departamento de Fsica e Matem´ atica, Avenida dos Bandeirantes, 3900 - Monte Alegre Ribeirao Preto, Sao Paulo 14040-901 Brazil [email protected]

Heat kernel of a sub-Laplacian and Grushin type operators Kenro Furutani Department of Mathematics, Science University of Tokyo, 2641 Yamazaki Noda, Chiba 278-8510 Japan [email protected] First, I will introduce a framework of a sub-Riemannian structure which is compatible with a submersion and define Grushin type operators. My purpose is to construct heat kernel for various Grushin type operators from known heat kernel in an explicit integral form. As a typical example, I explain the original Grushin operator and its heat kernel constucted from the heat kernel on three dimensional Heisenberg group. Then as a generalization to dimension three, I define Grushin type operators on R3 , R4 and R5 from a sub-Laplacian on the 6−dimensional free nilpotent Lie group, and give their heat kernels in terms of fiber integration. Also in the case that the submersion is a covering map from the Heisenberg group to Heisenberg manifolds, I will determine the spectral zeta function for a subLaplacian on them in terms of Riemann zeta function. If possible, I will also show a heat kernel for a spherical Grushin operator on S 2 and CP 3 which come from a sub-Laplacian on S 3 or S 7 , respectively. ———

Let Ω = (−, ) × S 1 , where > 0 and S 1 is the unit circle. Let L = ∂/∂t + (a(x) + ib(x))∂/∂x,

b 6≡ 0,

(*)

be a complex vector field defined on Ω , where a and b are real-valued s-Gevrey functions on (−, ), and s ≥ 1. We will assume that Σ = {0} × S 1 is the characteristic set of L and that L is tangent to Σ. In particular, L is elliptic on Ω \Σ and (a+ib)(0) = 0. Hence, we may write (a + ib)(x) = xn a0 (x) + ixm b0 (x) in Ω , with m, n ≥ 1, and a0 , b0 smooth. In this talk we shall present results about Gevrey solvability of L, given by (*), in a neighborhood of Σ, in the following sense: there exists s0 > 1 such that for any f belonging to a subspace of finite codimension of Gs (Ω ) 0 there exists a solution, u ∈ Gs , to the equation Lu = f in a neighborhood of Σ. We will see that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability. Moreover, lost of regularity occurs. This is a joint work with Adalberto P. Bergamasco (ICMC/USP) and Marcelo R. Ebert (FFCLRP/USP). ——— Invertibility for a class of degenerate elliptic operators Julio Delgado Cra 82 Bis 49-03 Ciudad Real Cali, Valle 9999 Colombia [email protected] In this work we study fundamental solutions for a class of degenerate elliptic operators. The type of operator considered is obtained as a sum of operators of the form 2 Dx2i + x2k i Dxj . The invertibility for an operator of type 2 2k 2 Dx1 +x1 Dx2 on R2 is known, here we extend this result to higher dimensions. ———

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Time-frequency analysis of stochastic differential equations Lorenzo Galleani Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, TO 10129 Italy [email protected] Most of the stochastic processes used to model physical systems are nonstationary, and yet most of the theoretical results on stochastic processes are related to the stationary case. We consider a nonstationary random process defined as the solution of a stochastic differential equation. We first transform the stochastic equation to the Wigner spectrum domain, where we obtain a deterministic differential equation. Then, by applying the Laplace transform, we obtain the exact solution of the deterministic equation. Finally, we rewrite the general solution in a form which clarifies the structure of the nonstationary stochastic process, and which highlights the connection to the classical results obtained by Fourier analysis. ——— p

L -microlocal regularity for pseudodifferential operators of quasi-homogeneous type Gianluca Garello Universit` a di Torino, Department of Mathematics, Via Carlo Alberto 10 Torino, Torino I-107123 Italy [email protected] Pseudodifferential operators whose symbols have decay at infinitive of quasi-homogeneous are considered and their behavior on the wave front set of distributions in weighted Zygmund-H¨ older spaces and weighted Sobolev spaces in Lp framework is studied. Then microlocal properties for solutions of linear partial differential equations with coefficients in weighted ZygmundH¨ older spaces are obtained.

IV.1. Pseudo-differential operators ——— Generalized Fourier integral operators methods for hyperbolic problems Claudia Garetto Arbeitsbereich f¨ ur technische Mathematik, Universit¨ at Innsbruck Technikerstrasse 13 Innsbruck, Austria 6020 Austria [email protected] The past decade has seen the emergence of a differentialalgebraic theory of generalized functions that answered a wealth of questions on solutions to partial differential equations involving non-smooth coefficients and strongly singular data. In such cases, the theory of distributions does not provide a general framework in which solutions exist due to inherent constraints in dealing with nonlinear operations. An alternative framework is provided by the theory of Colombeau algebras of generalized functions. In this talk we solve hyperbolic equations, generated by highly singular coefficients and data, by means of generalized FIO techniques developed in the Colombeau context. Finally, we provide a careful microlocal investigation of the solution by studying the microlocal mapping properties of these operators. ——— Resolvents of regular singular elliptic operators on a quantum graph Juan Gil Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601 United States [email protected] We will discuss the pseudodifferential structure of the resolvent of regular singular differential operators on a graph. For second order operators, we give a simple, explicit, sufficient condition for the existence of a sector of minimal growth. In particular, we will discuss operators with a singular potential of Coulomb type. Our analysis is based on the theory of elliptic cone operators. ——— Hyperbolic systems of pseudodifferential equations with irregular symbols in t admitting superlinear growth for |x| → ∞. Todor Gramchev Dipartimento di Matematica e Informatica, Universit` a di Cagliari, via Ospedale 72, 09124 Cagliari [email protected] We consider hyperbolic systems of pseudodifferential equations with irregular symbols with respect to the time variable t and admitting superlinear growth for |x| → ∞. We investigate the global well-posedness of the Cauchy problem for such systems in the framework of weighted spaces which generalize the Cordes type spaces H s1 ,s2 (Rn ). ——— Analytic perturbations for special Fr´ echet operator algebras in the microlocal analysis Bernhard Gramsch Institut f¨ ur Mathematik, Universit¨ at Mainz, Staudingerweg 9, Mainz, 55099 Germany [email protected]

The symmetric Hoermander class of type (1, 1) (interesting for paradifferential operators) is included in the theory of holomorphic Fredholm functions in connection with the Oka principle. This class is known to be not spectrally invariant. But commutator methods lead to the submultiplicativity of this symmetric Fr´echet algebra. Some relations to operator algebras on singular and stratified spaces are given. Stochastic PDE lead to holomorphic operator functions on infinite dimensional domains in DFN - spaces with basis such as the distribution space S 0 of Schwartz. A series of open problems is mentioned for Fr´echet operator algebras connected to parameter dependent equations on singular rep. ramified manifolds. ——— The Cauchy problem for a paraxial wave equation with non-smooth symbols ¨ nther Ho ¨ rmann Gu Nordbergstraße 15 Fakult¨ at f¨ ur Mathematik Wien, Wien A-1090 Austria [email protected] We discuss evolution systems in L2 for Schroedingertype pseudodifferential equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. Solutions with low regularity in the operator symbol will provide a basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data. ——— Pseudodifferential operators on locally symmetric spaces ´nie Hunsicker Euge Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU United Kingdom [email protected] I will discuss recent work with D. Grieser of U. Oldenburg on the first stages of the construction of a pseudodifferential operator calculus tailored to locally symmetric spaces and other noncompact spaces with similar structures. ——— On the continuity of the solutions with respect to the electromagnetic potentials to the Schr¨ odinger and the Dirac equations Wataru Ichinose Department of Mathematical Science, Shinshu University Matsumoto, Nagano 390-8621 Japan [email protected] The initial problem to families of the Schr¨ odinger equations and the Dirac equations with the electromagnetic

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IV.1. Pseudo-differential operators potentials are studied, respectively. Assume that the solutions have the same initial data and that the electromagnetic potentials converge. Then, it is proved that the solutions of Schr¨ odinger equations and the Dirac equations with the corresponding electromagnetic potentials also converge, respectively. The proof follows from the uniqueness and the boundedness of the solutions, and the functional method, ex. the abstract Ascoli-Arzel` a theorem, which will be seen to be applied to nonlinear equations. ——— Calculus of pseudo-differential operators and a local index of Dirac operators Chisato Iwasaki Department of Mathematical Sciences, Shosha 2167 Himeji, Hyogo 671-2201 Japan [email protected] I will show a method to obtain a local index of Dirac operators. This method depends on construction of the fundamental solution to the Cauchy problem for heat equations by introducing a weight for symbols of pseudodifferential operators. ——— On the theory of type 1, 1-operators Jon Johnsen Mathematics Department, Aalborg University, Fredrik Bajers Vej 7G, Dk-9220 Aalborg Øst, Denmark [email protected] After an introduction with a brief review of celebrated contributions on type 1, 1-operators of G. Bourdaud (1983,1988) and L. H¨ ormander (1988–89), their results will be set in relation to the general definition of type 1, 1-operators, which was introduced at the ISAAC 2007 congress. Progress in the area will be described as time permits. ——— Pseudo-differential operators with discontinuous symbols and their applications Yuryi Karlovych Universidad Aut´ onoma del Estado de Morelos, Facultad de Ciencias, Av. Universidad 1001, Cuernavaca, Morelos 62209 Mexico [email protected] Applying a weighted analogue of the Litllewood-Paley theorem and the boundedness of the maximal singular integral operator S∗ related to the Carleson-Hunt theorem on almost everywhere convergence on all weighted Lebesgue spaces Lp (R, w), where 1 < p < ∞ and w ∈ Ap (R), we study the boundedness and compactness of pseudo-differential operators a(x, D) with non-regular symbols in the classes L∞ (R, V (R)) and Λγ (R, Vd (R)) on the spaces Lp (R, w). The Banach algebra L∞ (R, V (R)) consists of all bounded measurable V (R)-valued functions on R where V (R) is the Banach algebra of all functions on R of bounded total variation, and the Banach algebra Λγ (R, Vd (R)) consists of all Lipschitz Vd (R)valued functions of exponent γ ∈ (0, 1) on R where Vd (R) is the Banach algebra of all functions on R of

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bounded variation on dyadic shells. For some Banach algebras of pseudo-differential operators acting on the space Lp (R, w) and having symbols discontinuous with respect to spatial and dual variables, we construct a noncommutative Fredholm symbol calculi and give Fredholm criteria and index formulas for the operators in these algebras. Applications to algebras of generalized singular integral operators with shifts are considered. ——— On maximal regularity for parabolic equations on complete Riemannian manifolds Thomas Krainer Penn State Altoona 3000 Ivyside Park Altoona, Pennsylvania 16601 United States [email protected] In this talk I plan to demonstrate how recent advances in the theory of pseudodifferential operators lead to a method to effectively establish optimal Lp –Lq a priori estimates for solutions to parabolic equations on certain complete Riemannian manifolds. The approach is based on Weis’ functional analytic characterization of maximal regularity in terms of the Rboundedness of the resolvent. In recent work, partly in collaboration with Robert Denk (Univ. of Constance, Germany), we have shown that the approximation of resolvents of elliptic operators by parameter-dependent parametrices in suitable classes of pseudodifferential operators readily leads to the desired R-boundedness, thus to maximal regularity. In my talk I plan to survey our results and the basic underlying principles of the method. ——— On the cohomological equation in the plane for regular vector fields Roberto de Leo INFN, Complesso Universitario di Monserrato Monserrato (CA), Sardegna 09042 Italy [email protected] In this talk we present our recent results about the solvability of the equation Xf = g, where X is a vector field on the plane without zeros, in the cases when X is generic and when it is Hamiltonian with respect to some symplectic form. This work slightly generalizes a recent result of S.P. Novikov, which showed recently that a generic vector field on a compact surface, seen as a 1st order operator on the set of smooth functions, has an infinite-dimensional cokernel. Our study is also related to aspects of pseudo-differential operators on the plane. ——— p

L -boundedness and compactness of localization operators associated with Stockwell transform Yu Liu Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario M3J1P3 Canada [email protected] Localization operators associated with the Stockwell transform, with respect to the filter symbol and the windows, are a class of operators defined on Lp (R). Under

IV.1. Pseudo-differential operators suitable conditions for the symbol and the windows, the localization operators turn to be bounded and compact. ——— About transport equation with irregular coefficient and data ´ Marti Jean-Andre Campus de Schoelcher, Laboratoire CEREGMIA, Universit´e des Antilles et de la Guyane Schoelcher, Martinique B.P. 7209-97275 France [email protected] We are interested in the study of the Cauchy problem for transport equation in the formally simplified case where the coefficients α and β are discontinuous and even distributions. For the data u0 , we suppose it is a distribution and even a more singular object like δxp ⊗ δyq we will give later a generalized meaning. Then the problem is formally written as 8 ∂ ∂ < ∂ u + α ⊗ 1xy u + β ⊗ 1xy u = 0, (Pf orm ) ∂t ∂x ∂y : u| = u (= δ p ⊗ δ q .) 0

{t=0}

x

y

We remark that the product and the restriction written above are generally not defined in a distributional sense. Consequently we begin in associating to (Pf orm ) a generalized one (P`gen )´well formulated in a convenient (C, E, P) algebra A R3 and recall the definition and main properties of this generalized multiparametric factor algebra. In our case, we construct such an algebra by means of independant regularizations involving three independant parameters and obtain 8 ∂ ∂ < ∂ u+F u + G u = 0, (Pgen ) ∂t ∂x ∂y : u| = H. {t=0}

` 3´ where F` and ´ G (resp.H) are the classes in A R 2 (resp.A R ) of the families regularizing the coefficients (resp. the data). First we solve (Pgen ) and examine the existence of a solution. To study more pecisely its singularities, we refer to a generalization of the asymptotic singular spectrum defined previously and adapted here to the threeparametric case. The so-called ”(a, D0 )-singular spectrum” of u ∈ A(R3 ) propose a spectral analysis of the singularities: by means of an ”analyzing” function a we can see where and why u is not locally (associated to a section of) D0 .The localization of such singularities of u is always the ”D0 -singular support” of u, and the asymptotic causis is described by a fiber ΣX (u) (above each X = (t, x, y) ∈ R3 ) which is the complement in R3+ of a conic subset of R3+ . In our case, the D0 -singularities of the data propagate along the ”regularized characteristic Γ of the problem (Pgen )” on which the fiber ΣX (u) remains constant. This joint work of V. Dvou, M. Hasler and J.-A. Marti of Universit Antilles-Guyane. ——— The Heat Kernel of τ -Twisted Laplacian Shahla Molahajloo Department of Mathematics and statistics, York University 4700 Keele street, Toronto, Ontario M3J1P3

Canada [email protected] For a family of τ -twisted Laplacians that includes the usual twisted Laplacian when τ = 1/2, we compute the heat kernel for each τ -twisted Laplacian for [0, 1]. ——— Regularity of characteristic initial-boundary value problems for symmetrizable systems Alessandro Morando Department of Mathematics - University of Brescia, Via Valotti, 9, I-25133, Brescia, Italy [email protected] We study the initial-boundary value problem for a linear Friedrichs symmetrizable system, with characteristic boundary of constant rank. We assume the existence of the strong L2 solution satisfying a suitable energy estimate, but we do not assume any structural assumption sufficient for existence, such as the fact that the boundary conditions are maximally dissipative or the KreissLopatinski condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth. ——— Application of pseudodifferential equations in stress singularity analysis for thermo-electro-magneto-elasticity problems: a new approach for calculation of stress singularity exponents David Natroshvili Georgian Technical University 77 M.Kostava st. Tbilisi, Tbilisi 0175 Georgia [email protected] We apply the potential method and the pseudodifferential equations technique to the mathematical model of the thermo-electro-magneto-elasticity theory. We study mixed and crack type boundary value problems. Along with the existence and uniqueness questions our main goal is a detailed theoretical investigation of singularities of the thermo-mechanical and electro-magnetic fields near the crack edges and the curves where the boundary conditions change their type. In particular, the most important question is description of the dependence of the stress singularity exponents on the material parameters. We reduce the three-dimensional mixed and crack type boundary value problems of the thermo-electromagneto-elasticity to the equivalent system of pseudodifferential equations which live on proper parts of the boundary of the elastic body under consideration. We show that with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators it is possible to determine explicitly the singularity exponents for physical fields. We give an efficient method for computation of these exponents. Moreover, we establish that these exponents essentially depend on the material parameters, in general. ——— Wigner type transforms and pseudodifferential operators Alessandro Oliaro Department of Mathematics, University of Torino, Via

67

IV.1. Pseudo-differential operators Carlo Alberto, 10 Torino, TO I-10123 Italy [email protected] We present some modifications of the Wigner transform (Wig), suggested by the connections of Wig with pseudodifferential operators. We analyze some properties of these representations, in particular the positivity and the behaviour with respect to the cross terms. ——— Local regularity of solutions to PDEs by asymptotic methods Michael Oberguggenberger Unit for Engineering Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria [email protected] In the nonlinear theory of generalized functions, algebras of generalized functions are commonly constructed by means of nets of smooth functions (uε )ε∈E depending on one or more parameters. Typically, these nets do not converge as ε → 0, say, but exhibit a certain asymptotic behavior. This behavior not only determines the algebras to which such an object belongs to, but may describe local regularity properties. This type of regularity theory has become increasingly important in applications to partial- and pseudodifferential operators. This presentation is devoted to a general framework – the so-called asymptotic spectrum – for measuring the asymptotic behavior in algebras of generalized functions, using asymptotic scales and various topologies. It has been developed in joint work with A. Delcroix and J.-A. Marti [Asymptotic Analysis 59(2008), 169 – 199] and forms a nonlinear alternative to the wave front set approach. Various applications to propagation of singularities as well as to regularity in Colombeau algebras and to jump discontinuities in hyperbolic systems will be given. Modern results by theory of the three dimensional Volterra type linear integral equations with singularity Nusrat Rajabov Tajik National University Rudaki Av. 17 Dushanbe, Dushanbe 734025 Tajikistan [email protected] Let Ω denote the parallelepiped Ω = {(x, y, z) : a < x < a0 , b < y < b0 , c < z < c0 }, D1 = {(x, y) : a < x < a0 , b < y < b0 , z = c}, D2 = {(x, z) : a < x < a0 , y = b, c < z < c0 }, D3 = {(y, z) : x = a, b < y < b0 , c < z < c0 }. In the domain Ω we consider the following integral equation Z x Z b Φ(t, y, z) Φ(x, s, z) Φ(x, y, z) + A dt + B ds t − a s−b a y Z z Z x Z y Φ(x, y, τ ) Φ(t, s, z) dt +E dτ + A1 ds τ −c s−b c a t−a b Z x Z z Φ(t, y, τ ) dt + B1 dτ τ −c a t−a c Z y Z z Φ(x, s, τ ) ds + C1 dτ τ −c b s−b c Z x Z y Z z Φ(t, s, τ ) dt ds +D dτ t − a s − b τ −s b c a

68

——— The adiabatic limit of the Chern character Frederic Rochon Department of Mathematics, 40 St. Toronto, Ontario M5S 2E4 Canada [email protected]

George Street,

Certain spaces of pseudo-differential operators can be used as classifying spaces for K-theory. In this context, Bott periodicity can be realized by taking a certain adiabatic limit. In this talk, we will indicate how natural forms representing the universal Chern chararcter on these spaces behave under such an adiabatic limit. This a joint work with Richard Melrose. ——— Boundary value problems as edge problems

———

= f (x, y, z),

where A, B, E, A1 , B1 , C1 , D are constants, f (x, y, z) – is a given function in Ω, Φ(x, y, z) is the desired function. Some cases equation (*) investigated N. Rajabov [Ac. of Sciences Dokl. V. 409, No 6, 2006, pp.749-753]. In this lecture the general solution of the integral equation (*) is constructed, using the connection equation (*) with one dimensional integral equation of the type (*). In the case, when A1 = AB, B1 = AE, D = AC1 , then the problem is determination general solution equation (*) redused, to problems found general solution single one-dimensional integral equation and single two– dimensional integral equation of the type (*). In this basis in the case when C1 = EB and A < 0, B < 0, E < 0, find general solution equation (*) by three arbitrary functions two variables. In the case when C1 6= EB, find the solution equation (*) by means of one arbitrary function two variabe and infinity number arbitrary function one variabe. Select the cases, when equation (*) has unique solution.

(*)

Bert-Wolfgang Schulze Institute of Mathematics, University Potsdam, Am Neuen Palais 10, Potsdam, D-14469 Germany [email protected] The calculus of pseudo-differential operators on a manifold with edges can be established in such a way that standard boundary value problems (BVPs) with the transmission property at the boundary appear as a special case (up to some simple modifications). Also the case without the transmission property can be formulated as a special case of the edge calculus (as is shown in a joint paper of the author with J. Seiler, 2009). The remarkable fact here is that the symbols of the respective (classical) pseudo-differential operators are not required to be of edge-degenerate form but are only smooth up to the boundary in the usual sense. In our talk we illustrate the specific properties of that theory for the case of symbols with the anti-transmission property (recently singlet out by the author to investigate specific asymptotics of solutions). Symbols with the transmission property together with those with the anti-transmission prope rty span the full space of symbols that are smooth up to the boundary. ———

IV.1. Pseudo-differential operators Noncommutative residues and projections associated to boundary value problems

7, Chiyoda-ku Tokyo, Tokyo 102-8554 Japan [email protected]

Elmar Schrohe Institut f¨ ur Analysis, Leibniz Universit¨ at Hannover, Welfengarten 1, 30167 Hannover [email protected]

In this talk, I will consider the regularity of the solution of a nonlinear singular partial differential equation (E):

On a compact manifold X with boundary we consider the realization B = PT of an elliptic boundary problem, consisting of a differential operator P and a differential boundary condition T . We assume that B is parameterelliptic in small sectors around two rays in the complex plane, say arg λ = φ and arg λ = θ. Associated to the cuts along the rays one can then define two zeta function ζφ and ζθ for B. Both extend to meromorphic functions on the plane; the origin is a regular point. We relate the difference of the values at the origin to the associated spectral projection Πθ,φ (B) defined by Z i Πθ,φ u = λ−1 B(B − λ)−1 u dλ, u ∈ dom(B), 2π Γθ,φ where Γθ,φ is the contour which runs on the first ray from infinity to r0 eiφ for some r0 > 0, then clockwise about the origin on the circle of radius r0 to r0 eiθ and back to infinity along the second ray. ——— On maximal regularity for mixed order systems ¨ rg Seiler Jo School of Mathematics Loughborough University Loughborough, Leicestershire LE113TU United Kingdom [email protected] I will discuss some results on maximal Lp -regularity for parabolic mixed order systems based on the so-called H∞ -calculus as well as on a calculus of Volterra pseudodifferential operators. This is a joint work with R. Denk and J. Saal. ——— Dirichlet problem for higher order elliptic systems with BMO assumptions on the coefficients and the boundary Tatyana Shaposhnikova Department of Mathematics, Linkoeping Universitym Linkoeping, Ostergotland SE-58183 Sweden [email protected] Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients. The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO) which guarantee that the solution operator associated with this problem is an isomorphism. ——— Gevrey regularities of solutions of nonlinear singular partial differential equations Hidetoshi Tahara Department of Mathematics, Sophia University, Kioicho

(t∂/∂t)m u = F (t, x, {(t∂/∂t)j (∂/∂x)α u}j+|α|≤m,j 0, for t ≥ t0 . Let U be the solution of the Cauchy Problem; then kU (t, ·)kLq ≤ C(n, p) (1 + t)

1 − 1 )+s − n−1 (p 0 2 q

——— Multiple solutions for non-linear parabolic systems Q-Heung Choi Dept. of Mathematics San 68 Miryong Dong Kunsan National University , Kunsan 573-701 South Korea [email protected] We have a concern with the existence of solutions (ξ, η) for perturbations of the parabolic system with Dirichlet boundary condition ξt = −Lξ + µg(3ξ + η) − sφ1 − h1 (x, t)

in Ω × (0, 2π),

ηt = −Lη + νg(3ξ + η) − sφ1 − h2 (x, t)

in Ω × (0, 2π).

We prove the uniqueness theorem when the nonlinearity does not cross eigenvalues. We also investigate multiple solutions (ξ(x, t), η(x, t)) for perturbations of the parabolic system with Dirichlet boundary condition when the nonlinearity f 0 is bounded and f 0 (−∞) < λ1 , λn < (3µ + ν)f 0 (+∞) < λn+1 . This is joint work with Tacksun Jung. ——— Local sovability beyond condition ψ Ferruccio Colombini Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 Pisa, PI 56127 Italy [email protected]

—Abstracts—

Dt U =

This is a joint work with Sandra Lucente and Giovanni Taglialatela from University of Bari.

kU0 kH Np ,p ,

for some s0 > 0, where 1 = p−1 + q −1 , 1 < p ≤ 2 and Np ≥ n(1/p − 1/q). One can take s0 = 0 if γ = 0 and s0 = for any > 0 if γ ∈ (0, 1) and C = C(n, p, ε).

It is well known that condition ψ (PSI) is necessary and sufficient in order to have local solvability for differential (pseudo-differential) operators of principal type with coefficients sufficiently regular. We study some cases when such conditions are not satisfied. These are two joint papers with Ludovico Pernazza and Fran¸cois Treves and with Paulo Cordaro and Ludovico Pernazza. ——— Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients Daniele Del Santo Dipartimento di Matematica e Informatica, Via Valerio 12/1, Trieste, 34127 Italy [email protected] We consider the following backward parabolic equation ∂t u +

X

∂xi (ai,j (t, x)∂xj u)

i,j

+

X

bj (t, x)∂xj u + c(t, x)u = 0

(*)

j

on the strip [0, T ] × Rn 3 (t, x). We suppose that • for all (t, x) ∈ [0, T ] × Rn and for all i, j = 1 . . . n, ai,j (t, x) = aj,i (t, x);

71

IV.2. Dispersive equations • there exists k > 0 such that, for all (t, x, ξ) ∈ [0, T ] × Rn × Rn , X k|ξ|2 ≤ ai,j (t, x)ξi ξj ≤ k−1 |ξ|2 ; i,j

• for all i, j = 1, . . . , n, ai,j ∈ LL([0, T ], L∞ (Rn )) ∩ L∞ ([0, T ], Cb2 (Rn )) and bj , c ∈ L∞ ([0, T ], Cb2 (Rn )), (where a ∈ LL([0, T ], L∞ (Rn )) means that the function a is Log–Lipschitz–continuous with respect to time with values in L∞ , i.e. sup t,s∈[0,T ], 0 0 there exist ρ0 , M 0 , N 0 , δ 0 > 0 such that if u ∈ E is a solution of the equation (*) with supt∈[0,T ] ku(t, ·)kL2 ≤ D and ku(0, ·)kL2 ≤ ρ0 , then sup ku(t, ·)kL2 ≤ M 0 e−N

0

| log ku(0,·)kL2 |δ

.

(joint work with Martino Prizzi, Trieste University) ——— On the loss of regularity for a class of weakly hyperbolic operators Marcello Ebert Universidade de S˜ ao Paulo, Faculdade de Filosofia, Ciˆencias e Letras, Dept. de Fisica e Matem´ atica, Av. dos Bandeirantes, 3900 Ribeir˜ ao Preto, S˜ ao Paulo 14040-901 Brazil [email protected] In this work we consider the Cauchy problem n X

aij (t)∂x2i xj u +λ(t)

i,j=1

n X

2 ci (t)∂tx u i

i=1

0

= f (x, t, u, ∂t u, λ (t)∇x u), u(x, 0) = u0 (x), ∂t u(x, 0) = u1 (x)

(*) (**)

where P is weakly hyperbolic in a neighborhood of {t = 0}, that is, the roots of p(x, t, ξ, τ ) in τ are real;

Daoyuan Fang Zhejiang University, Hangzhou, China [email protected] We consider the Zakharov system in space dimension two. We will show a L2 -concentration result for the data without finite energy, when blow-up of the solution happens, and a low regularity global well-posedness result. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given. This talk is based on some joint works with Sijia Zhong and Hartmut Pecher. ——— Wave equation in Einstein-de Sitter spacetime Anahit Galstyan Department of Mathematics, University of Texas-Pan American, 1201 West University Drive, Edinburg, Texas 78539 United States [email protected]

0

t∈[0,T 0 ]

P u = ∂t2 u−λ2 (t)

Zakharov system in infinite energy space

In this talk we introduce the fundamental solutions of the wave equation in the Einstein-de Sitter spacetime. The last one describes the simplest non-empty expanding model of the universe. The covariant d’Alembert’s operator in the Einstein-de Sitter spacetime belongs to the family of the non-Fuchsian partial differential operators. In this talk we investigate initial value problem for this equation and give the explicit representation formulas for the solutions. The equation is strictly hyperbolic in the domain with positive time. On the initial hypersurface its coefficients have singularities that make difficulties in studying of the initial value problem. In particular, one cannot anticipate the well-posedness in the Cauchy problem for the wave equation in the Einstein-de Sitter spacetime. The initial conditions must be modified to so-called weighted initial conditions in order to adjust them to the equation. We also show the Lp − Lq estimates for solutions. Thus, we have prepared all necessary tools in order to study the solvability of semilinear wave equation in the Einstein-de Sitter spacetime. This is a joint work with Tamotu Kinoshita (University of Tsukuba, Japan) and Karen Yagdjian (UTPA, U.S.A.). ———

(***)

Stability of solitary waves for Hartree type equation here p = p(x, t, ξ, τ ) is the principal symbol of P . Examples show that, differently of the hyperbolic case, under (*), (**) and (***) the solution might not exist. In addition to condition (***), various authors presented sufficient conditions, usually called Levi conditions, for the Cauchy problem to be well posed in Sobolev spaces. Those type of conditions relate p with lower order terms of P . In this work, we narrowed the bounds for the optimal Sobolevs loss of regularity under some sharp Levi conditions. This work was done in collaboration with Rafael A. dos Santos Kapp and Jos Ruidival dos Santos Filho, both from UFSCar(Brazil). ———

72

Vladimir Georgiev Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 Pisa, PI 56127 Italy [email protected] We prove the stability of solitary manifold associated with the solitary solutions of Hatree type equation with external Coulomb type potential. ——— Hyperbolic-parabolic Kirchhoff-equations

singular

perturbations

for

Marina Ghisi Department of Mathematics, University of Pisa, Largo

IV.2. Dispersive equations Pontecorvo 5 Pisa, Pi 56127 Italy [email protected] We consider the second order Cauchy problem u00 + g(t)u0 + m(|A1/2 u|2 )Au = 0, u(0) = u0 , u0 (0) = u1 where > 0, g is a positive function, m is a nonnegative C 1 function, A is a self-adjoint non-negative operator with dense domain D(A) in a Hilbert space, and (u0 , u1 ) ∈ D(A) × D(A1/2 ). We prove the global solvability of the Cauchy problem under different conditions on the functions m and g, including the case where m(0) = 0, and the case where g(t) tends to 0 as t tends to +infinity (weak dissipation). We also consider the behavior of solutions as t tends to +infinity (decay estimates), and as tends to 0. ——— Existence and uniqueness results for Kirchhoff equations in Gevrey-type spaces Massimo Gobbino Dipartimento di Matematica Applicata, via Filippo Buonarroti 1c, Pisa, PI 56127 Italy [email protected] We consider the second order Cauchy problem 00

1/2

u + m(|A

2

u| )Au = 0,

u(0) = u0 ,

——— Wave equations with time dependent coefficients Fumihiko Hirosawa Department of Mathematics, Yamaguchi University, 753-8512, Japan [email protected] The total energy of the wave equation is conserved with respect to time if the propagation speed is a constant, but it is not true in general for time dependent propagation speeds. Indeed, it is considered in [F. Hirosawa, Math. Ann. 339 (2007), 819-839] that the following properties of the propagation speed are crucial for the estimates of the total energy: oscillating speed, difference from the mean, and the smoothness in C m category. The main purpose of our talk is to derive a benefit of a further smoothness of the propagation speed in the Gevrey class for the energy estimates. ———

0

u (0) = u1 ,

where m : [0, +∞) → [0, +∞) is a continuous function, and A is a self-adjoint nonnegative operator with dense domain on a Hilbert space. In this conference we present three results. • The first result is local existence for initial data in suitable spaces depending on the continuity modulus of the nonlinear term m. This spaces are a natural generalization of Gevrey spaces to the abstract setting. We also show that solutions with less regular data may exhibit an instantaneous derivative loss. • The second result concerns uniqueness in the case where the nonlinear term is not Lipschitz continuous. • The last result concerns the global solvability. Roughly speaking, we show that every initial datum in the spaces where local solutions exist is the sum of two initial data for which the solution is actually global. ——— Precise loss of derivatives for evolution type models Torsten Herrmann Faculty 1, TU Bergakademie Freiberg, Pr¨ uferstr. Freiberg, 09596 Germany [email protected]

derive results for well-posedness with a (possible) loss of regularity. On the other hand we discuss strategies how to show optimality of the results and sharpness of the assumptions. Here Floquet theory and instability arguments form the core of our strategies. We distinguish between optimality for the leading coefficients of the principal part and for coefficients of the remaining principal part.

9,

The goal of this talk is to present statements about well-posedness for Cauchy problems for degenerate pevolution equations with time-dependent coefficients. Degeneracy means that the p-evolution operators may have characteristics of variable multiplicity. On the one hand we are interested to apply phase space analysis to

Critical point theory applied to a class of systems of super-quadratic wave equations Tacksun Jung Dept. of Mathematics San 68 Miryong Dong Kunsan National University , Kunsan 573-701 South Korea [email protected] We show the existence of a nontrivial solution for a class of the systems of the super-quadratic nonlinear wave equations with Dirichlet boundary conditions and periodic conditions with super-quadratic nonlinear terms at infinity which have continuous derivatives. We approach the variational method and use the critical point theory which is the Linking Theorem for the strongly indefinite corresponding functional. This is joint work with Q-Heung Choi. ——— On the well-posdness of the vacuum Einstein equations Lavi Karp P.O. Box 78 Karmiel, Galilee 21982 Israel [email protected] The Cauchy problem of the vacuum Einstein’s equations determines a semi-metric gαβ of a spacetime with vanishing Ricci curvature Rα,β and prescribe initial data. under harmonic gauge condition, the equations Rα,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric hab and a second fundamental form Kab . However, these data must satisfy Einstein constraint equations and therefore the pair (hab , Kab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in

73

IV.2. Dispersive equations one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The aim of our work is to resolve this incompatibility and to show well-posedness of the reduced Einstein vacuum equations in one type of Sobolev spaces. ——— Generalized wave operator for a system of nonlinear wave equations Hideo Kubo Graduate School of Information Sciences, Tohoku University 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai , Miyagi 980-8579 Japan [email protected] In this talk we discuss the asymptotic behavior of solutions to a system of nonlinear wave equations whose decaying rate is actually slower than that of the free solutions. Desipte of that fact, we are able to construct wave operators in a generalized sense. The proof is done by finding a nice approximation and introducing a suitable metric (that is not a norm in fact). Moreover, the sacttering operators are defined in a generarized sence.

with Dirichlet boundary conditions w(x, 0, t) = w(x, π, t) = 0,

(x, t) ∈ RN × (0, ∞).

For long-range type of dissipations, e.g., b0 (1 + |x|)−1 ≤ b(x, y) ≤ b1 in RN × [0, π] for some b0 , b1 > 0, the total energy decays as t goes to infinity. For short-range type of dissipations, e.g., 0 ≤ b(x, y) ≤ b2 (1 + |x|)−1−δ in RN × [0, π] for some b2 > 0 and δ > 0, scattering solution exists. Although the proof for scattering is based on Kato’s smooth perturbation theory, the singular points called thresholds in the spectrum cause to difficulty. To eliminate this, density argument using some approximate operators are employed. This is joint work with Mitsuteru Kadowaki (Ehime University) and Kazuo Watanabe (Gskushuin University). ——— On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 well-posedness Tatsuo Nishitani Machikaneyama-cho 1-1 Toyonaka, Osaka 560-0043 Japan [email protected]

———

In this talk we will present Strichartz estimates for higher oder hyperbolic equations in an exterior domain outside a star-shaped obstacle.

The Cauchy problem for non-effectively hyperbolic operators is discussed in the Gevrey classes. Our operators belong to the class of non-effectively hyperbolic operators with symbols vanishing of order 2 on a smooth submanifold of codimension 3 on which the canonical symplectic 2-form has a constant rank. Assuming that there is no null bicharacteristic issuing from a simple characteristic point and landing tangentialy on the double characteristic manifold, we prove that the Cauchy problem is Gevrey s well-posed for any lower order term whenever 1 ≤ s < 4.

———

———

Uniform resolvent estimates and smoothing effects for magnetic Schr¨ odinger operators

On the structure of the material law in a linear model of poro-elasticity

Kiyoshi Mochizuki Department of Mathematics, Chuo University, Kasuga, Bunnkyo 1-13-27, Tokyo 112-8551 Japan [email protected]

Rainer Picard Institut f¨ ur Analysis, FB Mathematik,TU Dresden, 01062 Dresden, Germany [email protected]

Uniform resolvent estimates for magnetic Schr¨ odinger operators in an exterior domain are obtained under smallness conditions on the magnetic fields and scalar potentials. The results are then used to obtain spacetime L2 -estimates for the corresponding Schr¨ odinger, Klein-Gordon and wave equations.

A modification of the material law associated with the well-known Biot system first investigated by R.E. Showalter is re-considered in the light of a new approach to a comprehensive class of evolutionary problems.The particular material law is of the form

Strichartz estimates for hyperbolic equations in an exterior domain Tokio Matsuyama Tokai University 1117 Kitakaname Hiratsuka, Kanagawa 259-1292 Japan [email protected]

T = (C + trace∗ λ trace ∂0 ) E − trace∗ α p

——— Decay and scattering for wave equations with dissipations in layered media Hideo Nakazawa Chiba Institute of Technology, Narashino, Chiba 275-0023 Japan [email protected]

Shibazono

2-1-1

We consider wave equations with linear dissipations in some layered regions;

connecting the stress tensor T with strain tensor E and fluid pressure p via parameters λ, α and C as the isotropic elasticity tensor. Here ∂0 denotes the time derivative and trace the matrix trace operation with trace∗ as its adjoint. This model is generalized to anisotropic media and well-posedness of the generalized model is shown. ——— Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuous coefficients

wtt (x, y, t) − ∆w(x, y, t) + b(x, y)wt (x, y, t) = 0, (x, y, t) ∈ RN × [0, π] × (0, ∞)

74

Marco Pivetta Dipartimento di Matematica e Informatica, Via Valerio

IV.2. Dispersive equations 12/1, Trieste, Italy 34127 Italy [email protected]

Japan [email protected]

Using the Carleman estimates developed by Koch and Lasiecka [Functional analysis and evolution equations, 389-403, Birkh¨ auser, Basel, 2008] together with an approximation technique in the phase space, a uniqueness result for the backward Cauchy problem is proved for the system of themoelastic waves having coefficients which are in a class of log-Lipschitz-continuous functions.

We consider nonnegative solutions of the Cauchy problem for quasilinear parabolic equations

———

where m > 1 and f (ξ) is a positive function R ∞in ξ > 0 satisfying f (0) = 0 and a blow-up condition 1 1/f (ξ) dξ < ∞. We study under what conditions on f (ξ) all nontrivial solutions blow up. ———

The log-effect for 2 by 2 hyperbolic systems Michael Reissig Faculty 1, TU Bergakademie Freiberg, Pr¨ uferstr. Freiberg, 09596 Germany [email protected]

ut = ∆um + f (u),

9,

In the talk we are interested to explain how to extend the Log-effect from wave equations with timedependent coefficients to 2 by 2 strictly hyperbolic systems ∂t U − A(t)∂x U = 0. From wave models we know that besides oscillations in the coefficients a possible interaction of oscillations has a strong influence on H ∞ well- or ill-posedness. Moreover, the precise loss of derivatives can be proved. In the case of systems the situation is more complicate. Besides the effects of oscillating entries of the matrix A = A(t) and interactions between the entries of A we have to take into consideration the system character itself. We will prove by using tools from phase space analysis results about H ∞ well- or ill-posedness. The precise loss of regularity is of interest. Moreover, we discuss the question if the loss of derivatives does really appear. These considerations base on the interplay between the Ljapunov and energy functional. Finally, we discuss the cone of dependence property for solutions to 2 by 2 systems. This is a joint talk with T.Kinoshita (Tsukuba). ——— The Boussinesq equations based on the hydrostatic approximation

Blow-up and a blow-up boundary for a semilinear wave equation with some convolution nonlinearity Hiroshi Uesaka Department of Mathematics, College of Science and Technology, Nihon University, Tokyo 101-8308, Chiyodaku Kanda Surugadai 1-8, Japan [email protected] We consider the Cauchy problem with a convolution nonlinearity, (∂t2 − 4)u = uq (V ∗ up ), in R3 × (0, T ), (0.1) u(x, 0) = f (x), ∂t u(x, 0) = g(x) in R3 , R p (y,t) where uq (V ∗ up )(x, t) = uq (x, t)( R3 u|x−y| γ dy) with p, q > 1, 0 < γ < 3. The blow-up boundary is defined by Γ = ∂{u < ∞} ∩ {t > 0}. We can give several suitable conditions to initial data to show that 1. the Cauchy problem has a classical positive realvalued local solution u, 2. u is monotone increasing in t for any fixed x and moreover satisfies ∂t u ≥ |∇u|, 3. there exists a positive T (x) for any x such that u keeps its regularity in (0, T (x)) and limt%T (x) u(x, t) = ∞ .

Jun-ichi Saito Minamisenju 8-17-1 Arakawa-ku, Tokyo 116-0003 Japan j [email protected]

Then the blow-up boundary Γ exists and T (x) satisfies |T (x) − T (y)| ≤ |x − y|.

The Boussinesq equations is studied in the field of dynamic meteorology. Atmospheric flow in meteorology are described by the Boussinesq equations. Due to the fact that the aspect ratio

Fundamental solutions for hyperbolic operators with variable coefficients

ε=

characteristic depth characteristic width

is very small in most geophysical domains, asymptotic models have been used. One of the models is the hydrostatic approximation of the Boussinesq equations. We consider the Boussinesq equations in the domains with very small aspect ratio and prove the convergence theorem for this model. ——— Blow-up of solutions of a quasilinear parablolic equation Ryuichi Suzuki School of Science and Engineering, Kokushikan University, 4-28-1 Setagaya, Setagaya-ku Tokyo, 154-8515

———

Karen Yagdjian Department of Mathematics, University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78541-2999, USA [email protected] The goal of this talk is to give a survey of a new approach in the constructing of fundamental solutions for the partial differential operators with variable coefficients and of some resent results obtaining by that approach. This new approach appeals neither to the Fourier transform, nor to the Microlocal Analysis, nor to the WKBapproximation. More precisely, the new integral transformation is suggested which transforms the family of the fundamental solutions of the Cauchy problem for the operator with the constant coefficients to the fundamental solutions for the operators with variable coefficients.

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IV.3. Control and optimisation of nonlinear evolutionary systems The kernel of that transformation contains Gauss’s hypergeometric function. This approach was applied by the author and his coauthors, T.Kinoshita (University of Tsukuba) and A.Galstyan (University of Texas-Pan American), to investigate in the unified way several equations such as the linear and semilinear Tricomi and Tricomi-type equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the KleinGordon equations in the de Sitter and anti-de Sitter spacetimes. The listed equations play important role in the gas dynamics, elementary particle physics, quantum field theory in the curved spaces, and cosmology. In particular, for all above mentioned equations, we have obtained representation formulas for the initial-value problem, the Lp − Lq -estimates, local and global solutions for the semilinear equations, blow up phenomena, selfsimilar solutions and number of other results.

cavities, control of turbulence), geophysics (reconstruction of seismic data) and others. Recent years have witnessed rapid development of new mathematical tools in both analysis and geometry that allow to obtain various PDE estimates of inverse type. These are enabling to establish properties such as controllability, reconstruction of the data, stabilisation or optimal feedback control.

———

The model under consideration is the semilinear wave equation with supercritical nonlinear sources and damping terms and the aim is to discuss the wellposedness of the system on finite energy space and the long-time behavior of solutions. A distinct feature of the equation is the presence of the double interaction of source and damping, both in the interior of the domain and on the boundary. Moreover, the nonlinear boundary sources are driven by Neumann boundary conditions. Since Lopatinski condition fails to hold for dimension greater or equal than 2, the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves strong interaction between the source and the damping. I will provide positive answers to the questions of local existence and uniqueness of weak solutions and moreover give complete and sharp description of parameters corresponding to global existence and blow-up of solutions in finite time. I will also discuss asymptotic energy-decay rates and blow-up of solutions originating in a potential well.

Global existence in Sobolev spaces for a class of nonlinear Kirchhoff equations Borislav Yordanov Borislav Yordanov, 226 Swain Ct, Belle Mead, NJ 085024239 United States [email protected] The nonlinear Kirchhoff equation utt − m(k∇uk2L2 )∆u = 0 is studied for initial data (u, ut )t=0 = (u0 , u1 ) in the Sobolev spaces H s (Rn ) × H s−1 (Rn ) with s ≥ 2 and for smooth perturbations m(ρ) of the Pokhozhaev function m0 (ρ) = (k1 ρ + k0 )−2 with k0 , k1 > 0. Global existence is shown when ku1 kL2 is large and m is close to m0 in a suitable metric. Moreover, the asymptotic behavior of solutions is found as t → ±∞. It turns out that the norms k∇ukL2 grow like |t|, so the propagation speeds decrease like t−2 and the waves remain trapped in bounded regions. This is joint work with Lubin Vulkov. ———

IV.3. Control and optimisation of nonlinear evolutionary systems Organisers: Francesca Bucci, Irena Lasiecka The session is focused on new developments in the area of well-posedness, optimisation, and control of systems described by evolutionary partial differential equations. These include: non-linear wave and plate equations, Navier-Stokes and Euler equations, non-linear thermoelasticity, viscoelasticity and electromagnetism. Of particular interest to the session are interacting systems that involve PDE’s of different type describing the dynamics on two (or more) separate regions with a coupling on an interface between these regions. Particular examples of such systemsare: structural acoustic interactions, fluid structure interactions, magnetostructure interactions. These have a wide range of applications that include medicine (diagnostic imaging such as MRI, ultrasound), engineering (noise reduction in an acoustic

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—Abstracts— Global well-posedness and long-time behavior of solutions to a wave equation Lorena Bociu University of Nebraska-Lincoln, Department of Mathematics, 203 Avery Hall Lincoln, NE 68588 United States [email protected]

——— Distributed optimal controls for second kind parabolic variational inequalities Mahdi Boukrouche LaMUSE Saint-Etienne University, 23 Rue Dr Paul Michelon, Saint-Etienne, 42023, France [email protected] Let ugi be the unique solution of a second kind parabolic variational inequality with second member gi (i = 1, 2). We establish, in the general case, the error estimate between u3 (µ) = µug1 + (1 − µ)ug2 and u4 (µ) = uµg1 +(1−µ)g2 for µ ∈ [0, 1], and prove a monotony property between u3 (µ) and u4 (µ) using a regularization method. For a given constant M > 0, and the cost functional we establish the existence of solutions for a family of control problems, over the the external force g for each parameter h > 0. Using the monotony property between u3 (µ) and u4 (µ), we establish the uniqueness of the solution for each control problem of the above family. We prove also the convergence of the optimalcontrols and states associated to this family of control problems governed by a second kind parabolic variational inequalities. ———

IV.3. Control and optimisation of nonlinear evolutionary systems Controllability of a fluid-structure interaction problem Muriel Boulakia 175 rue du Chevaleret, 75013 Paris, France [email protected]

In addition, we investigate the existence of uniform decay rates for both, the Airy type equation ut + uxxx + g(u) = 0, in [0, L] × (0, +∞),

We are interested by the controllability of a fluidstructure interaction problem. The fluid in governed by the incompressible Navier-Stokes equations and a rigid structure is immersed inside. The control acts on a fixed subset of the fluid domain. For small initial data, we prove that this system is null controllable i.e. that the system can be driven at rest. This result is obtained with the help of a Carleman inequality proven for the adjoint linearized system. ——— Uniform decay rate estimates for the wave equation on compact surfaces and locally distributed damping Marcello Cavalcanti Department of Mathematics - State University of Maringa, Av. Colombo 5790, Maringa, PR 87020-900 Brazil [email protected] In this talk we present new contributions concerning uniform decay rates of the energy associated with the wave equation on compact surfaces subject to a dissipation locally distributed. We present a method that gives us a sharp result in what concerns of reducing arbitrarily the area where the dissipative effect lies. ——— Rate of decay for non-autonomous damped wave systems Moez Daoulatli ISSATS, University of Sousse (& LAMSIN) Cit´e Taffala (Ibn Khaldoun), Sousse 4003 Tunisia [email protected] We study the rate of decay of solutions of the wave systems with time dependent nonlinear damping which is localized on a subset of the domain. We prove that the asymptotic decay rates of the energy functional are obtained by solving nonlinear non-autonomous ODE. ——— On qualitative aspects for the damped Korteweg-de Vries and Airy type equations

posed in a bounded interval [0, L] and supplemented by a nonlinear damping g(u). By considering suitable assumptions on g and on the initial data, general decay rates are proved in L2 − level as well as exponential decay rates are established in H 1 −level. ——— Optimal control of waves in anisotropic media via conservative boundary conditions Matthias Eller Department of Mathematics, Georgetown University, Washington, DC 20057 United States [email protected] An optimal boundary control problem for symmetric hyperbolic systems is considered. The quadratic cost functional is of tracking type and the control acts through a conservative boundary condition. Some loss of regularity is associated with these boundary conditions. This results in certain choices for the underlying function spaces in the cost functional. The loss of regularity occurs only near the boundary and it may be attributed to the occurrence of surface waves. As examples we consider the anisotropic Maxwell equations as well as the anisotropic equations of elasticity. ——— Stability for some nonlinear damped wave equations Genni Fragnelli Dipartimento di Ingegneria dell’Informazione, Universit` a degli Studi di Siena, via Roma 56, c.a.p. 53100 [email protected] We prove stability results for a large class of abstract nonlinear damped wave equations, whose prototype is the usual wave equation 8 in (0, +∞) × Ω, < utt + h(t)ut = ∆u + f (u) u(t, x) = 0 in (0, +∞) × ∂Ω, : u(0, x) = u0 (x), ut (0, x) = u1 (x) x ∈ Ω, where Ω is a bounded and smooth domain of RN , N ≥ 1, u0 ∈ H01 (Ω), u1 ∈ L2 (Ω) and f : R → R. At first, the damping is nonnegative, but it is allowed to be zero either on negligible sets or even in a sequence of intervals. Then, also the case of a positive–negative damping is treated. ———

Valeria Domingos Cavalcanti Department of Mathematics - State University of Maringa, Av. Colombo 5790, Maringa, PR 87020-900 Brazil [email protected] This talk is concerned with the study of the damped Korteweg-de Vries equation posed in whole real line ut + uxxx + uux + λ u = 0, in R × [0, +∞),

λ > 0.

We establish two invariant subsets of H 1 (R) where just one of the following statements holds: (i) solutions decay exponentially in H 1 − level or (ii) solutions do not decay to zero in H 1 − level as t goes to infinity.

Global existence for the one-dimensional semilinear Tricomi-type equation Anahit Galstyan Department of Mathematics, University of Texas-Pan American, 1201 W. University Dr., Edinburg 78541, TX, U.S.A. [email protected] In this talk the issue of global existence of the solutions of the Cauchy problem for one-dimensional semilinear weakly hyperbolic equations, appearing in the boundary value problems of gas dynamics is investigated. We solve the Cauchy problem trough integral equation and give some sufficient conditions for the existence of the global

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IV.3. Control and optimisation of nonlinear evolutionary systems weak solutions. The necessary condition for the existence of the similarity solutions for the one-dimensional semilinear Tricomi-type equation will be presented as well. Our approach is based on the fundamental solution of the operator and the Lp − Lq estimates for the linear Tricomi equation. ——— Optimal control of a thermoelastic structural acoustic model Catherine Lebiedzik Department of Mathematics, Wayne State University, 656 W Kirby Detroit, MI 48202 United States [email protected] We consider point control of a structural acoustic model with thermoelastic effects. The key feature of this paper is that the two-dimensional plate modeling the active wall of the acoustic chamber has clamped boundary conditions. For this case a new optimal regularity result has recently become available. Using this new result for the plate alone, we derive a sharp regularity result for the overall coupled system of wave and thermoelastic plate equations. This allows for the study of optimal control of the coupled system. ——— The Balayage method: Boundary control of a thermoelastic plate Walter Littman University of Minnesota, School of Mathematics, 206 Church Street, Southeast Minneapolis, Minnesota 55455 United States [email protected] We discuss the null boundary controllablity thermo-elastic plate. The method employs ing property of the system of PDEs which boundary controls to be calculated directly two Cauchy problems.

of a linear a smoothallows the by solving

——— Hopf-Lax type formulas and Hamilton-Jacobi equations Paola Loreti Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, via Antonio Scarpa n.16, 00161 Roma, Italia. [email protected] Here we discuss Hopf-Lax type formulas related to the class of Hamilton-Jacobi equations ut (x, t) + αxDu(x, t) + H(Du(x, t)) = 0, N

in R × (0, +∞) with initial condition u(x, 0) = u0 in Rn , with α a positive, real number. The talk is based on some joint works with A. Avantaggiati. ——— Investigation of boundary control problems by on-line inversion technique Vyacheslav Maksimov S.Kovalevskaya 16 Ekaterinburg, Sverdlovsk 620219

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Russia [email protected] A number of applied studies address such fundamental issues as (i) reconstruction of uncertain parameters of multidimensional dynamical systems and (ii) control of uncertain dynamical systems. We discuss a technical approach intended to help solve problems of this kind. The approach employs the on-line inversion theory adjoining theory of closed-loop control and theory of ill-posed problems. On-line inversion algorithms involve artificially designed dynamical models whose parameters track non-observable parameters of the system; it is important that the tracking quality is insensitive to perturbations in the observation channels. In combination with appropriate closed-loop regulators, on-line parameter tracking algorithms give raise to robust observer-controller patterns allowing one to guide the uncertain system close to the trajectories designed via an optimal feedback to a complete set of observed signals. The goal of this report is to demonstrate the essence and abilities of the on-line inversion technique; for this purpose we consider three types of problems, namely, a problem of etalon motion tracking, a problem of game control, and a problem of dynamical input identification for a parabolic equation with the Neumann and Dirichlet boundary condition. ——— Null controllability properties of some degenerate parabolic equations Patrick Martinez Universit´e Paul Sabatier Toulouse III, Institut de Math´ematiques, 118 route de Narbonne, Toulouse, 31062 France [email protected] Motivated by several problems in fluid dynamics, biology, or economics, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. After considering the one dimensional case, this talk will mainly focus on the N-dimensional case: ut − div(A(x)∇u) = f (x, t)χω (x),

x ∈ Ω, t > 0

where ω ⊂ Ω and the matrix A(x) is definite positive for all x ∈ Ω, and but has at least one eigenvalue equal to 0 for all x ∈ ∂Ω. Mainly, we assume that - the least eigenvalue of the matrix A(x) behaves as d(x, ∂Ω)α , where α ≥ 0, - the degeneracy occurs in the normal direction: when x ∈ ∂Ω, the associated eigenvector is the unit outward vector. When α ∈ [0, 2), we prove the null controllability via new Carleman estimates for the adjoint degenerate parabolic equation. When α ∈ [2, +∞), we prove that the problem is not null controllable, using earlier results of Micu ˇ ak related to nondegenZuazua, Escauriaza-Seregin-Sver´ erate parabolic equations in unbounded domains. These results were obtained in collaboration with P. Cannarsa (Univ Tor Vergata, Roma 2), and J. Vancostenoble (Univ Toulouse 3). ———

IV.3. Control and optimisation of nonlinear evolutionary systems Dissipation in contact problems: an overview and some recent results Maria Grazia Naso Dipartimento di Matematica, Universit` a degli Studi di Brescia, Via Valotti, 9 Brescia, BS 25133 Italy [email protected] In this talk we investigate the longtime behaviour of a dynamic unilateral contact problem between two thermoelastic beams. Under suitable mechanical and thermal boundary conditions the evolution problem is shown to possess an energy decaying exponentially to zero, as time goes to infinity. ——— Heat equations with memory: a Riesz basis approach Luciano Pandolfi Politecnico di Torino, Dipartimento di Matematica, C.so Duca degli Abruzzi 24, Torino, 10129 Italy [email protected]

and on its boundary combined with a nonlinear coupled boundary condition. Such problems arise from free boundary value problems as the Stefan problem with surface tension after a suitable transformation. Besides local well posedness and smoothing properties, we focus on the qualitative behavior near an equilibrium. To that purpose we construct locally invariant manifolds and establish their main properties. ——— On regularity properties of optimal control and Lagrange multipliers Ilya Shvartsman Dept. of Mathematics and Computer Science, 777 W. Harrisburg Pike Middletown, PA 17110 United States [email protected] In this talk we will go over classical and recent results on regularity properties (such as continuity, Holder and Lipschitz continuity) of optimal controls and Lagrange multipliers. ———

In this talk we present recent results concerning a Riesz basis approach to the heat equation with memory Z t θt (x, t) = N (t − s) [∆θ(x, s) − q(x)θ(x, s)] ds 0

(x ∈ [0, π]) and square integrable initial conditions. We shall construct a special sequence {zn (t)} associated to this equations and we shall prove that it is a Riesz sequence on a suitable interval [0, T ], using Bari Theroem. These results are applied to the study of control/observability problems. ——— A note on a class of observability problems for PDEs Michael Renardy Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123 United States [email protected] The question of observability arises naturally in the analysis of control problems. If the solution of a PDE initialboundary value problem is known to be zero in a part of the domain, does this guarantee it is zero everywhere? The most popular techniques to establish such results are based on local unique continuation results (Holmgren’s theorem) or Carleman estimates. The lecture will draw attention to a class of problems where the observed region is bounded by characteristics, and local unique continuation fails. Nevertheless, observability may hold. A problem of this nature arose in recent work by the author on control of viscoelastic flows. ——— Invariant manifolds for parabolic problems with dynamical boundary conditions Roland Schnaubelt University of Karlsruhe, Department of Mathematics, Kaiserstrasse 89, Karlsruhe, 76128 Germany [email protected] We study a class of nonlinear parabolic systems described by coupled evolution equations on a domain

Evolution equations with memory terms Daniela Sforza Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, via Antonio Scarpa 16, Roma, 00161 Italy [email protected] The purpose of the talk is to show some results concerning control problems for integro-differential equations of hyperbolic type. More precisely, we consider non-linear equations in Hilbert spaces with integral convolution terms and assume the corresponding kernels to exhibit a polynomial or exponential decay. We show that the solutions have the same decay behaviour as the kernel. Our main tool is the multipliers method and we succeed in finding suitable multipliers which work even in the presence of integral terms. Besides, we provide a reachability result for a class of linear integro-differential problems. Our strategy is founded on the so-called Reachability Hilbert Uniqueness Method, introduced by Lagnese - Lions, which amounts to proving Ingham type inequalities for the Fourier series expansion of the solution of the adjoint problem. To conclude, we observe that our abstract results may be used to treat some problems arising in the study of viscoelastic systems. ——— Stabilization of structure-acoustics interactions for a Reissner-Mindlin plate by localized nonlinear boundary feedbacks Daniel Toundykov University of Nebraska-Lincoln Department of Mathematics, 203 Avery Hall Lincoln, NE 68588 United States [email protected] This work addresses observability and energy decay for a structural-acoustics model comprised of a wave equation coupled with a Reissner-Mindlin plate. Both components of the dynamics are subject to localized boundary damping: the acoustic dissipative feedback is restricted to the flexible boundary and only a portion of the rigid

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IV.4. Nonlinear partial differential equations wall; the plate is likewise damped on a segment of its boundary. The derivation of the “coupled” stabilization/observability inequalities requires weighted energy multipliers related to the geometry of the domain, and special tangential trace estimates for the displacement and the filament angles of the Reissner-Mindlin plate model. The behavior of the energy at infinity can be quantified by a solution to an explicitly constructed nonlinear ODE. The nonlinearities in the feedbacks may include sub- and super-linear growth at infinity, in which case the decay scheme presents a trade-off between the regularity of solutions and attainable uniform decay rates of the finite-energy. ——— Exponential stability of the wave equation with boundary time varying delay Julie Valein Universit´e de Valenciennes et du Hainaut-Cambr´esis LAMAV - ISTV2 - Le Mont-Houy Valenciennes, NordPas de Calais 59313 France [email protected] We consider the wave equation with a time-varying delay term in the boundary condition in a bounded domain Ω ⊂ Rn with a boundary Γ of class C 2 . We assume Γ = ΓD ∪ ΓN , with ΓD ∩ ΓN = ∅, ΓD 6= ∅, and we consider 8 > > utt (x, t) − ∆u(x, t) = 0 in Ω × (0, +∞) > > u(x, t) = 0 on ΓD × (0, +∞) > > > > (x, t) = −µ1 ut (x, t) − µ2 ut (x, t − τ (t)) < ∂u ∂ν on ΓN × (0, +∞) > > > u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω > > > ut (x, t − τ (0)) = f0 (x, t − τ (0)) > > : in ΓN × (0, τ (0)), (*) where τ (t) is the delay, µ1 , µ2 > 0. We assume 0 ≤ τ (t) ≤ τ , ∀t>0

0

τ (t) ≤ d < 1, and τ ∈ W 2,∞ ([0, T ]),

Under µ2

0.

1 − dµ1 ,

we prove the existence and uniqueness results of (*) by using the variable norm technique of Kato and we show the exponential decay of an appropriate energy. Due to the time-dependence of the delay, we can not use an observability estimate since the system is not invariant by translation in time. Hence we introduce a Lyapunov functional. We extend this result to a nonlinear version of the model. This is a joint work with Serge Nicaise and Cristina Pignotti. ——— State estimation for some parabolic systems Masahiro Yamamoto University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba Meguro Tokyo 153, Japan [email protected]

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For some types of parabolic systems, we consider inequalities of Carleman’s type and prove conditional stability estimates for state determination problems such as a backward problem. ——— Euler flow and Morphing Shape Metric Jean-Paul Zolesio CNRS-INLN nd INRIA RTE Lucioles 1361 and 2004 Sophia Antipolis, France 06565 France [email protected] We extend the so-called ”Courant metric” into a new ”Tube metric” beetwen measurable sets and characterize the geodesic as variational solution to incompressible Euler flow. Such geodesic can modelise topological changes. We make use of a new ”Sobolev perimeter”, and Sobolev Mean curvature for the moving boundary which turns to be Shape differentiable under smooth transverse perturbation. Then working with L2 speed vector fields (we don’t use any renormalization benefit) we succed in the existence of connecting tubes and in variational solution to the usual incompressible Euler flow under surface tension associated to the Sobolev perimeter. This technic applies to several situations in [Shape-Morphing Metric by Variational Formulation for Incompressible Euler Flow.J. of Control and Cybernetics, vol 38 (2009), No. 4], [Control of Moving Domains...,. Int.Ser.Num.Math., vol. 155, 329-382, Birkhauser Verlag Basel,2007]. ———

IV.4. Nonlinear partial differential equations Organisers: Vladimir Georgiev, Tohru Ozawa The Session intends to discuss various nonlinear partial differential equations in mathematical physics. Among possible arguments the following ones shall be discussed: existence and qualitative properties of the solutions, existence of wave operators and scattering for these problems, stability of solitary waves and other special solutions. —Abstracts— Evolution equations in nonflat waveguides Piero D’Ancona Sapienza - Universit` a di Roma - Dipartimento di Matematica P. Moro, 2 Roma, RM 00185 Italy [email protected] In a joint work with Reinhard Racke (Konstanz) we prove smoothing and Strichartz estimates for evolution equations of Schroedinger or wave type on waveguides which are deformations, in a suitable sense, of flat waveguides of the form O × Rk , O a bounded open set in Rm . For the proof, new weighted estimates for fractional powers of Schroedinger operators are required. ———

IV.4. Nonlinear partial differential equations Investigation of solutions of one not divergent type Mersaid Aripov Mech.,Math, National University of Uzbekistan, Universitet 1, Tashkent, Tashkent 100174 Uzbekistan [email protected] The properties of the weak solution of problem Cauchy and the first boundary value problem for one parabolic equation of not divergent type double nonlinearity and with lower members are investigated. The researched equation is the best combination of forms of the equation of nonlinear diffusion, fast diffusion, the equation to very fast diffusion and p-Laplace heat conductivity equation. This equation describes various processes of nonlinear diffusion, heat conductivity, a filtration, magnetic rheology, etc. The method of investigation of the qualitative properties having physical sense weak solution on the basis of a method of a nonlinear splitting and a method of the standard equations is offered. Two side estimations of the solutions and free boundary, a condition of existence of global solutions (including case of critical value of parameter and exponent) generalizing of known results of H. Fujite, A.A., Samarskii, S.P. Kurdyumov, A.P. Mikhajlov, V.A. Galaktionov, H.Vaskes, S. A. Posashkov are received. On the basis of the analysis of properties of solutions the numerical modeling and visualization of solutions carried out. ——— Asymptotic behavior of subparabolic functions Davide Catania Universit` a di Brescia - Dip. Matematica Via Valotti, n. 9 Brescia, BS 25133 Italy [email protected] We consider an MHD-α model with regularized velocity for an incompressible fluid in two space dimension. Such a model is introduced in analogy with the Navier– Stokes equation to study the turbulent behavior of fluids in presence of a magnetic field, since this problem is otherwise difficult to study, both analitically and numerically. We prove local and global existence for the related Cauchy problem, where the velocity field is viscous, while we have not any magnetic diffusivity. ——— On multiple solutions of concave and convex effects for nonlinear elliptic equation on RN Kuan-Ju Chen Department of Applied Science, Naval Academy, P.O.BOX 90175 Zuoying, Taiwan, R.O.C. [email protected] In this paper we consider the existence of multiple solutions of the elliptic equation on RN with concave and convex nonlinearities. ——— Nonlinear gauge invariant evolution of the plane wave Kazuyuki Doi Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai,

Miyagi 980-8579 Japan [email protected] We consider nonlinear gauge invariant evolution of the plane wave. In this talk, we deal with the power and logarithmic type nonlinearities. Although the plane wave does not decay at infinity, by an elementary and simple argument we find an extremely smooth solution which has an explicit expression. Additionally, we study the global behavior of the solution from its representation. ——— New approach to solve linear parabolic problems via semigroup approximation Mohammad Dehghan Ferdowsi University of Mashhhad Azadi Square Mashhad, Khorasan-e-Razavi 9177948974 Iran [email protected] We consider Linear Parabolic Problems (LPPs) whose solutions can be expressed via semigroups. Computing the solutions of these LPPs depends on existing explicit formulas for the corresponding semigroups. However, in general explicit formulas are not available. The proposed approach defines a sequence of linear problems which are semidiscrete approximations of the considered LPP. The solutions of approximant linear problems can be expressed via corresponding semigroups which have explicit formulas. These solutions converge uniformly to the solution of LPP. So the corresponding semigroup of LPP can be approximated by semigroups which have explicit formulas. The approximant linear problems are defined on the finite dimensional subspaces of the LPP solution space, via a hybrid finite-differenceprojection method. The accuracy of approximations, order of convergency and their relations to the proposed hybrid method are discussed and some examples are presented. ——— Global existence and blow-up for the nonlocal nonlinear Cauchy problem Albert Erkip Sabanci University, Faculty of Engineering and Natural Sciences, Orhanli, Tuzla Istanbul / 3495 Turkey [email protected] We study the Cauchy problem utt

=

(β ∗ (u + g (u)))xx

u (x, 0)

=

φ (x) ,

x ∈ R, t > 0

ut (x, 0) = ψ (x)

x ∈ R,

for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function β whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided. ———

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IV.4. Nonlinear partial differential equations Qualitative properties for reaction-diffusion systems modelling chemical reactions Marius Ghergu School of Mathematical Sciences University College Dublin Belfield, , Dublin 4 Dublin Ireland [email protected] In 1952 the British mathematician Alan M. Turing published the foundation of reaction-diffusion theory for morphogenesis, the development of form and shape in biological systems. Since then, many Turing-type models described by coupled reaction-diffusion equations have been proposed for generating patterns in both organic and inorganic systems. In this talk we present a qualitative study for reactiondiffusion systems of the type ut − d1 ∆u = a + bu + f (u)v

in Ω × (0, ∞),

vt − d2 ∆v = c + du − f (u)v

in Ω × (0, ∞),

u(x, 0) = u0 (x), v(x, 0) = v0 (x) ∂u ∂u (x, t) = (x, t) = 0 ∂ν ∂ν

on Ω,

on ∂Ω × (0, ∞).

Here Ω ⊂ RN (N ≥ 1) is a bounded domain, a, b, c, d, d1 , d2 ∈ R, u0 , v0 ∈ C(Ω) are non-negative and f ∈ C[0, ∞) ∩ C 1 (0, ∞) is a non-negative and nondecreasing such that f (0) = 0 and f > 0 in (0, ∞). The system encompasses two well known chemical models: the Brusselator and the Schnackenberg models which are a rich source of varied spatio-temporal patterns. We present several existence and stability results. A particular attention is paid to the associated steady-state system where the crucial role played by the diffusion coefficients d1 , d2 and the behavior of the nonlinearity f is emphasized. The proofs rely on a-priori estimates combined with analytical and topological methods. ——— Scattering in the zero-mass Lamb system ´ ndez Marco Antonio Taneco-Herna Instituto de F´ısica y Matem´ aticas, Universidad Michoacana de San Nicol´ as de Hidalgo, Edificio C-3, Ciudad Universitaria Av., Francisco J. Mujica s/n, Colonia Felicitas del Rio Morelia, Michoac´ an 58040, Mexico [email protected] We consider nonlinear conservative Lamb system, which is the wave equation coupled with a particle of zero mass: u ¨(x, t) = u00 (x, t), F (y(t)) + u0 (0+, t) − u0 (0−, t),

y(t) = u(0, t),

with x ∈ R \ {0}, t ∈ R. Here u˙ := ∂u , u0 := ∂u ∂t ∂x and so on. The solutions u(x, t) take the values in Rd with d ≥ 1 and F := −∇V with V : Rd → R is a potential force field. For the first time we establish long time asymptotics in global energy norm for all finite energy solutions. Namely, under some Ginzburg-Landau type conditions to V , each solution from some functional space decays to a sum of a stationary state, outgoing wave and the rest which tends to zero in global energy norm as t → +∞.

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The outgoing wave is a solution to the free wave equation with some asymptotics states as initial data. We introduce corresponding nonlinear scattering operator, and obtain a necessary condition for the asymptotic states. Also we prove the asymptotics completeness in the Lamb system. This is joint work with A.E. Merzon and A.I. Komech. ——— Global existence for systems of the nonlinear wave and Klein-Gordon equations in 3D Soichiro Katayama Department of Mathematics, Wakayama University, 930 Sakaedani Wakayama, Wakayama 640-8510 Japan [email protected] We consider the Cauchy problem for coupled systems of the nonlinear wave and Klein-Gordon equations in three space dimensions. We present a sufficient condition for global existence of small amplitude solutions to such systems. Our condition is much weaker than the strong null condition for this kind of coupled system, and our result is a natural extension of the global existence theorem for the nonlinear wave equations under the null condition, as well as that for the Klein-Gordon equations with quadratic nonlinearities. Our result is applicable to a certain kind of model equation in physics, such as the Klein-Gordon-Dirac equations, the Klein-GordonZakharov equations, and the Dirac-Proca equations. ——— Global existence for nonlinear wave equations exterior to an obstacle in 2D Hideo Kubo Graduate School of Information Sciences, Tohoku University 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai , Miyagi 980-8579 Japan [email protected] In this talk we discuss the global existence for the exterior problem of nonlinear wave equations in two space dimensions. The obstacle is assumed to be a star-shaped, so that the decay of the local energy is available. The main difficulty compared with the three space dimensional case is the weaker decay of solutions in 2D, as well as the lack of the sharp Huygens principle. However, we are able to show the global existence for small initial data, provided the nonlinearity is of the cubic order and fulfills the so-called null condition. ——— Remark on Navier-Stokes equations with mixed boundary conditions Petr Kucera Czech Technical University, Fac. of Civil Engineering, Dept. of Math., Thakurova 7, Prague 166 29 Czech Republic [email protected] We solve a system of the Navier-Stokes equations for incompressible heat conducting fluid with mixed boundary conditions (of the Dirichlet or non-Dirichlet type on different parts of the boundary). We suppose that the viscosity of the fluid depends on temperarure. ———

IV.4. Nonlinear partial differential equations Contraction-Galerkin method for a semi-linear wave equation with a boundary-like antiperiodic condition Ut van Le Department of Mathematical Sciences, P.O. Box 3000, Oulu FI-90014 Finland [email protected] We consider the unique solvability of initial-boundary value problems of semi-linear wave equations with spacetime dependent coefficients and special mixed nonhomogeneous boundary values which make the so-called boundary-like antiperiodic condition. The procedure in this project is the combination of the Galerkin method and a contraction. ——— p − q systems of nonlinear Schrodinger equations Sandra Lucente Dipartimento di Matematica, Via Orabona 4, Bari 70124, Italy [email protected]

A symmetric error estimate for Galerkin approximations of time dependant Navier-Stokes equations in two dimensions Itir Mogultay Department of Mathematics, Yeditepe University, 26 Agustos Yerlesimi Kayisdagi Caddesi Kayisdagi Istanbul, 81120 Turkey [email protected] A symmetric error estimate for Galerkin approximation of solutions of the Navier-Stokes equations in two space dimensions plus time is given. The finite dimensional function spaces are taken to be divergence free, and time is left continuous. The estimate is similar to known results for scalar parabolic equations. An application of the result is given for mixed method formulations. A short discussion of examples is included. Finally, there are some remarks about a partial expansion to three space dimensions. Note: This is a joint work with Prof. Todd F. Dupont at the University of Chicago. ———

In a joint work with L. Fanelli and E. Montefusco, we consider coupled nonlinear Schr¨ odinger equations

Stability of standing waves for some systems of nonlinear Schr¨ odinger equations with three-wave interactions

iut + ∆u ± N1 (u, v) = 0, ivt + ∆v ± N2 (u, v) = 0,

Masahito Ohta Department of Mathematics, Saitama University, 255 Shimo-Ohkubo, Saitama, 338-8570 Japan [email protected]

with suitable semilinear terms N1 (u, v) and N2 (u, v) having polynomial growth. We investigate on local and global existence critical exponents and describe the corresponding solutions.

We discuss orbital stability and instability of several types of standing waves for some three-component systems of nonlinear Schr¨ odinger equations.

——— ——— Semiclassical analysis for nonlinear Schrodinger equations

Decay rates for wave models with structural damping

Satoshi Masaki 6-3-09 Aza-aoba Aramki Aoba-ku Sendai, Miyagi 9808579 Japan [email protected]

Michael Reissig Faculty 1, TU Bergakademie Freiberg, Pr¨ uferstr. 9, Freiberg, 09596 Germany [email protected]

We consider the semiclassical limit of the nonlinear Schrodinger equations. We approximate the solution by a function of phase-amplitude form, called WKB analysis. We mainly treat the nonlocal nonlinearites. ——— 3-D viscous Cahn-Hilliard equation with memory Gianluca Mola Universit` a di Milano, Dipartimento di Matamatica, via Saldini 50 Milano, MI 20133 Italy [email protected] We deal with the memory relaxation of the viscous Cahn-Hilliard equation in 3-D, covering the well–known hyperbolic version of the model. We study the longterm dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. ———

In this talk, we will present results on the behavior of higher order energies of solutions to the following Cauchy problem for a wave model with structural damping: utt − ∆u + b(t)(−∆)σ ut = 0, u(0, x) = u0 (x), σ ∈ (0, 1],

ut (0, x) = u1 (x),

b(t) = µ(1 + t)δ ,

µ > 0, δ ∈ [−1, 1].

We are interested in the influence of the structural dissipation (between external and visco-elastic damping) b(t)(−∆)σ ut on L2 − L2 estimates. Our main goal is to study under which conditions do we have a parabolic effect for the solutions, that is, the decay rates depend on the order of energy. In the talk we will explain how hyperbolic or elliptic WKB analysis comes in. The main tools are a correct division of the extended phase space into zones, diagonalization procedures, construction of fundamental solutions and a gluing procedure. Some open problems complete the talk. This is joint work with Xiaojun Lu (Hangzhou). ———

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IV.4. Nonlinear partial differential equations Stability theorems in the theory of mathematical fluid mechanics

towards the stable manifold etc. Several outstanding open questions will be discussed as well.

Yoshihiro Shibata Department of Mathematics, Waseda University, Ohkubo 3-4-1 Shinjuku-ku Tokyo, Tokyo 169-8555 Japan [email protected]

———

I would like to talk about some stability theorem of stationary solutions of incompressible fluid flow with initial disturbance. ——— On singular systems of parabolic functional equations ´ szlo ´ Simon La P´ azm´ any P. s´et´ any 1/C, L. E¨ otv¨ os University, Institute of Mathematics Budapest, Hungary H-1117 Hungary [email protected] We shall consider initial-boundary value problems for a system consisting of a quasilinear parabolic functional equation and an ordinary differential equation with functional terms. The parabolic equation may contain the gradient with respect to the space variable of the unknown function in the ODE. It will be proved global existence of weak solutions, by using the theory of monotone type operators and Schauder’s fixed point theorem. Such problems are motivated by models describing reaction-mineralogy-porosity changes in porous media and polymer diffusion. ——— Survey of recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations Zdenek Skalak Thakurova 7, Czech Technical University, Prague, 16629 Czech Republic [email protected] We present some recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations. For example, if w is such a solution satisfying the strong energy inequality then there exists a ≥ 0 such that lim ||Eλ w(t)||/||w(t)|| = 1 t→∞

for every λ > a, where {Eλ ; λ ≥ 0} denotes the resolution of identity of the Stokes operator. ——— Conditional stability theorems for Klein-Gordon type equations Atanas Stefanov 1460, Jayhawk Blvd., Department of Mathematics, University of Kansas, Lawrence, KS 66049, USA [email protected] We consider unstable ground state solutions of the KleinGordon equation with various power nonlinearities. The main result is a fairly precise construction of a stable manifold in a close vicinity of the ground state. In particular, we provide an asymptotic formula for the asymptotic phase, an estimate of the rate of the convergence

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A regularity result for a class of semilinear hyperbolic equations Sergio Spagnolo Department of Mathematics, University of Pisa, Largo Pontecorvo 5 Pisa, 56127 Italy [email protected] We first recall a former result of global wellposedness in C-infinity (resp., in each Gevrey class) for a special kind of homogeneous, linear hyperbolic equations with analytic (resp., C-infinity) coefficients depending only on time. Then, we add to these equations an analytic semilinear term, and we prove that the resulting equations enjoy the following regularity property; each solution which is real analytic at the initial time together with all its time derivatives, remains analytic as long as it is bounded in C-infinity (resp., in some Gevrey class). ——— On nonlinear equations, fixed-point theorems and their applications Kamal Soltanov Department of Mathematics, Faculty of Sciences, Hacettepe University, Beytepe Campus Ankara, Cankaya TR-06532 Turkey [email protected] In this work we investigated some class of the nonlinear operators and a nonlinear equations with such type operators in a Banach spaces. Here we obtained some new results on the solvability of the nonlinear equations, and also a fixed-point theorems for continuous mappings. With use of the obtained here results we studied various boundary value problems (BVP) (and mixed problems) for the different nonlinear differential equations. ——— Dynamics of a quantum particle in a cloud chamber Alessandro Teta Dipartimento di matematica pura e applicata, Universita’ di L’Aquila via Vetoio - loc. Coppito L’Aquila, Abruzzo 67100 Italy [email protected] We consider the Schroedinger equation for a system composed by a particle (the α-particle) interacting with two other particles (the atoms) subject to attractive potentials centered in a1 , a2 ∈ R3 . At time zero the α-particle is described by a diverging spherical wave centered in the origin and the atoms are in their ground state. The aim is to show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless a2 lies on the line joining the origin with a1 . The work (in collaboration with G. Dell’Antonio and R. Figari) is a fully time-dependent version of the original analysis performed by Mott in 1929. ———

IV.5. Asymptotic and multiscale analysis Half space problem for the damped wave equation with a non-convex convection term Yoshihiro Ueda Graduate School of Sciences, Tohoku University 6-3 Aramaki-Aza-Aoba, Aoba-ku Sendai, Miyagi 980-8578 Japan [email protected] We consider the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space. In the case where the flux is convex, it had already known that the solution tends to the corresponding stationary wave. In this talk, we show that even for a quite wide class of flux functions which are not necessarily convex, such the stationary wave is asymptotically stable. The proof is given by a technical weighted energy method. ——— On the time-decay of solutions to a family of defocusing NLS Nicola Visciglia Dipartimento di Matematica, Universita di Pisa, Via F. Buonarroti 2, Pisa 56127 Italy [email protected] Let u(t, x) be any solution to the defocusing NLS with 4 pure power nonlinearity u|u|α , where 0 < α < n−2 , and 1 n with initial condition u(0, x) ∈ H (R ). Then the Lp norm of u(t, x) goes to zero as t → ∞ provided that 2n 2 < p < n−2 . In particular we extend previous result due to Ginibre and Velo who have shown the property 4 above under the extra assumption n4 < α < n−2 . ——— The semilinear Klein-Gordon equation in de Sitter spacetime Karen Yagdjian Department of Mathematics, University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78541-2999, USA [email protected] In this talk we present the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation g φ − m2 φ = −|φ|p with the small mass m ≤ n/2 in de Sitter spacetime with the metric g. We prove that for every p > 1 large energy solutions blow up, while for the small energy solutions we give a borderline p = p(m, n) for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of Kato’s lemma. ———

IV.5. Asymptotic and multiscale analysis Organisers: Ilia Kamotski, Valery Smyshlyaev BICS Mini-Symposium The minisymposium will focus on fundamental analytical issues associated with differential equations (linear

and nonlinear, partial or ordinary) with a small parameter and/or multiple scales, and relevant applications. This includes singularly perturbed problems, problems in thin domain or with singular boundaries, homogenization. The applications may include propagation and localization of waves, blow-up phenomena, metamaterials, etc. The relevant analytic issues are convergence and relevant functional spaces, compactness and propagation of oscillations, asymptotic expansions with error bounds, etc. —Abstracts— On the essential spectrum and singularities of solutions for Lam´ e problem in cuspoidal domain Natalia Babych Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY United Kingdom [email protected] Within a Lam´e problem of linear elasticity, we investigate singularities of solutions in the vicinity of an outward cusp at the boundary. In case of a sharp cusp (the H¨ older constant is less or equal 12 ), we describe the essential spectrum that consists of a certain real ray accessing +∞. We analyse all possible local singularities of solutions and construct radiation conditions defining suitable spaces that guarantee a Fredholm type solvability for the problem. We demonstrate that the sharp outward cusp at the boundary is somewhat similar to infinity for unbounded domains. This is joint work with Dr. I. Kamotski. ——— Torsion effects in elastic composites with high contrast Michel Bellieud Universit´e de Perpignan, 52 avenue Paul Alduy, Perpignan, 66860 France [email protected] In the context of linearized elasticity, we analyze as ε → 0 a vibration problem for a two-phase medium whereby an ε-periodic set of ”stiff” elastic fibers of elastic moduli of the order 1 is embedded in a ”soft” elastic matrix of elastic moduli of the order ε2 . We show that torsional vibrations take place at an infinitesimal scale. ——— Enhanced resolution in structured media Yves Capdeboscq OxPDE Centre for Nonlinear Partial Differential Equations, University of Oxford, Mathematical Institute, Oxford, OX1 3LB United Kingdom [email protected] In this talk, we show that it is possible to achieve a resolution enhancement in detecting a target inclusion if it is surrounded by an appropriate structured medium. This work is motivated by the advances in physics concerning the so-called super resolution, or resolution beyond the diffraction limit. We first revisit the notion of resolution and focal spot, and then show that in a structured medium, the resolution is conditioned by effective parameters. This is a joint work with Habib Ammari & Eric Bonnetier

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IV.5. Asymptotic and multiscale analysis

This is work in collaboration with Marc Briane, where we study the asymptotic behaviour of a given sequence of diffusion energies in L2 (Ω) for a bounded open subset Ω of R2 . The corresponding diffusion matrices are assumed to be coercive but any upper bound is considered. We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of Fn is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. These results are based on the uniform convergence satisfied by some minimizers of the equicoercive sequence Fn , which is specific to the dimension two.

The Knizhnik-Zamolodchikov equation associated with the root system Bn is investigated. This root system has two orbits with respect Weyl group. By this reason KZ equation naturally contains two parameters. Singular locus of this equation consists from hyperplanes xi − xj = 0, xi + xj = 0, xk = 0, i, j, k = 1, 2, . . . , n, x = (x1 , . . . , xn ) ∈ Cn . The following inverse problem of Riemann-Hilbert type is considered: given a representation of a fundamental group of complement to the singular locus in Cn to the orthogonal group of odd order. To define the coefficients of the two-parametric differential KZ equation as elements of tensor power of universal envelopping algebra for odd orthogonal Lie algebra. Oneparametric case was investigated by A.Leibman. For coefficients were used Casimir elements of second order. In two-parametric case the coefficients are defined by using the families of Casimir elements of higher order described by A. Molev. For construction of these elements are used Capelli operators which permit to describe the centre of corresponding universal enveloping algebra. The invariants used in explicit form of coefficients for the case o(5) are expressed by means of Pfaffian for matrix defined using the root system.

———

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Two-scale Γ-convergence and its applications to homogenisation of non-linear high-contrast problems

Long-time behavior for the Wigner equation and semiclassical limits in heterogeneous media

Mikhail Cherdantsev School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG United Kingdom [email protected]

Fabricio Macia Universidad Politecnica de Madrid, DEBIN ETSI Navales, Avda. Arco de la Victoria, Madrid 28040 Spain [email protected]

It is a resent results of Bouchitte, Felbacq, Zhikov and others that passing to the limit in high-contrast elliptic PDEs may lead to non-classical effects, which are due to the two-scale nature of the limit problem. These have so far been studied in the linear setting, or under the assumption of convexity of the stored energy function. It seems of practical interest however to investigate the effect of high-contrast in the general non-linear case, such as of finite elasticity. With this aim in mind, we develop a new tool to study non-linear high-contrast problems, which may be thought of as a “hybrid” of the classical Γ-convergence (De Giorgi, Dal Maso, Braides) and two-scale convergence (Allaire, Briane, Zhikov). We demonstrate the need for such a tool by showing that in the high-contrast case the minimising sequences may be non-compact in Lp space and the corresponding minima may not converge to the minimum of the usual Γ-limit. We prove a compactness principle for high-contrast functionals with respect to the two-scale Γ-convergence, which in particular implies convergence of their minima. We briefly discuss possible applications of this new technique in the mechanics of composites. (This is a joint work with K.D. Cherednichenko.)

We study the semiclassical limit for a class of linear Schr¨ odinger equations in an heterogeneous medium (for instance, a Riemannian manifold) at time scales tending to infinity as the characteristic frequencies of the initial data tend to zero. We are interested, in particular, in dealing with time scales larger than the Eherenfest time, for which the high frequency behavior is completely characterized by classical mechanics via Egorov’s theorem. Our analysis is performed by studying the highfrequency behavior of Wigner functions corresponding to solutions to the Schrodinger equation at very long times. We give a complete characterization of their structure for systems arising as the quantization of a completely integrable classical Hamiltonian flow. In particular, we prove that in such systems the asymptotic behavior of Wigner functions for times larger than Ehrenfest’s might no longer be determined by the classical flow. This is due to effects caused by resonances, that have to be studied via a new object, the resonant Wigner distribution.

——— Homogenization of elliptic partial differential equations with unbounded coefficients in dimension two Juan Casado-Diaz Dpto. de Ecuaciones Diferenciales y Analisis Numerico, Facultad de Matematicas, C. Tarfia s/n Sevilla, Sevilla 41012 Spain [email protected]

——— Construction of the two-parametric generalizations of the Knizhnik-Zamolodchikov equations of Bn type Valentina Alekseevna Golubeva Steklov Mathematical Institute, Gubkina 8, Moscow 119991 Russia [email protected]

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——— On nonlinear dispersive equations in periodic structures: Semiclassical limits and numerical schemes Peter Markowich DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA United Kingdom [email protected] We discuss (nonlinear) dispersive equations, such as the Schrdinger equation, the Gross-Pitaevskii equation mod- eling Bose-Einstein condensation, the MaxwellDirac system and semilinear wave equations. Semiclassical limits are analysed using WKB and Wigner tech-

V.1. Inverse problems niques, in particular for periodic structures, and connections to classical homogenisation problems for HamiltonJacobi equations and hyperbolic conservation laws are established. We present a new numerical technique for such PDE problems, based on Bloch decomposition, and show applications in semiconductor modelling, BoseEinstein condensation and Anderson localisation for random wave equations. ——— Derivation of Boltzmann-type equations from hardsphere dynamics Karsten Matthies Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY United Kingdom [email protected] The derivation of the continuum models from deterministic atomistic descriptions is a longstanding and fundamental challenge. In particular the emergence of irreversible macroscopic evolution from reversible deterministic microscopic evolution is still not fully understood. We study a classic system: N balls that interact with each other via a hard-core potential and show rigorously that in the case of kinetic annihilation (particles annihilate each other upon collision) the asymptotic behavior as N tends to infinity is correctly described by the Boltzmann equation without gain-term for non-concentrated initial distributions. The mean-field description fails, when there are concentrations in the space or the velocity coordinates. This is joint work with Florian Theil. ——— Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape Bernd Schmidt Zentrum Mathematik, TU Muenchen, Boltzmannstr. 3, Garching b. Muenchen, 85747 Germany [email protected]

in the (tensorial) coefficients. The employed tools are those of ”non-classical” (high contrast type) homogenisation. This leads to interesting effect physically, for example allowing ”directional localisation”, with no wave propagation in certain directions, and mathematically allows treating form a unified perspective ”classical”, high-contrast homogenizations and intermediate cases. We discuss some related analytic issues, including the need to develop appropriate versions of two-scale convergence and of the theory of compensated compactness. ———

V.1. Inverse problems Organisers: Yaroslav Kurylev, Masahiro Yamamoto Inverse problems is a multidisciplinary subject having its firm origin in application of mathematics to such problems as search for oil, gas and other mineral resources, medical imaging, process monitoring in micro-biological, chemical and other industries, non-destructive testing of materials, to mention just few. Its mathematical underpinning stretches from discrete mathematics, to geometry, to computational methods with, however, the principal background being in analysis. In particular, the use of analytic methods makes it possible to address such issues of IP as their strongly non-linear nature and severe ill-posdnesss. In recent years, these relations have made it possible to solve a number of long-standing inverse problems, including those with data on a part of the boundary, with significantly reduced requirements on regularity and the number of measurements, etc. These were based on the advancing and employing such topics in analysis as Carleman estimates for PDE’s, harmonic and quasiconformal analysis, global and geometric analysis, microlocal calculus and stochastic/probabilistic methods. In this section we intend to represent those progress by inviting the leading people in the area to give relevant talks. —Abstracts—

We investigate ground state configurations of atomic systems in two dimensions as the number of atoms tends to infinity for suitable pair interaction models. Suitably rescaled, these configurations are shown to crystallize on a triangular lattice and to converge to a macroscopic Wulff shape which is obtained from an anisotropic surface energy induced by the microscopic atomic lattice. Moreover, sharp estimates on the microscopic fluctuations about the limiting Wullf shape are obtained. (Joined work with Y. Au Yeung and G. Friesecke.) ——— Homogenization with partial degeneracies: analytic aspects and applications Valery Smyshlyaev Department of Mathematical Sciences, University of Bath, Claverton Down Bath, BA2 7AY United Kingdom [email protected] We consider homogenization problems for a generic class of (scalar or vector) operators with ”partial” degeneracy

An inverse conductivity problem with a single measurement Abdellatif El Badia LMAC, University of Compiegne, Compiegne, Oise 60200 France [email protected] We revisit in this paper the inverse boundary value problem of Calderon for a coated domain, where the conductivity is constant in each subdomain. This geometric distribution of conductivity corresponds to the well accepted model of heads in ElectroEncephaloGraphy (EEG). For instance, the inmost interior domain is occupied by the brain, and it is surrounded by the skull and the scalp. The so-called spherical model, where these regions are concentric spherical layers, is also frequently used. We show for this distribution of conductivity that the inverse problem is completely solved with only one suitably chosen Cauchy data, instead of the whole Dirichlet-to-Neumann operator. The criterion of choice for these Cauchy data is completely set up in

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V.1. Inverse problems the spherical model, using spherical harmonics. Also, a stability result is established. As for the numerical method to compute the conductivity, we propose a least square procedure with a Kohn-Vogelius functional, and a boundary integral method for the direct problem. This is joint work with T. Ha-Duong. ——— Global in time existence and uniqueness results for some integrodifferential identification problems Fabrizio Colombo Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9 Milano, Mi 20133 Italy [email protected] We show some results on the identification of memory kernels in some nonlinear equations such as the heat equation with memory, the strongly damped wave equation with memory, the beam equation with memory and a peculiar model in the theory of combustion. An additional restriction on the state variable is given to determine both the state variable and the memory kernels. We prove global in time uniqueness results and for suitable nonlinearities we prove existence and uniqueness results for the solution of the identification problems associated to the models mentioned above. ——— Stability estimate for an inverse problem for the magnetic Schr¨ odinger equation from the Dirichlet-toNeumann map Mourad Choulli Department of Mathematics, Metz University, Ile du Saulcy Metz, Lorraine 57000 France [email protected] In this talk we consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schr¨ odinger equation in a bounded smooth domain of Rn with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the solutions of the magnetic Schr¨ odinger equation. We prove in dimension n ≥ 2 that the knowledge of the Dirichletto-Neumann map for the magnetic Schr¨ odinger equation measured on the boundary determines uniquely the magnetic field and we prove a H¨ older-type stability in determining the magnetic field induced by the magnetic potential. ——— Optimal combination of data modes in inverse problems: maximum compatibility estimate Mikko Kaasalainen Department of Mathematics and Statistics, PO Box 68, Helsinki, FI-00014 Finland [email protected] We present an optimal strategy for weighting various data modes in inverse problems. The solution, maximum compatibility estimate, corresponds to the maximum likelihood estimate of the single-mode case (with,

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e.g., regularization functions included). We illustrate the method by showing that one can reconstruct a body with sparse data of the boundary curves (profiles) and volumes (brightnesses) of its generalized projections. ——— On an inverse problem for a linear heat conduction problem Christian Daveau CNRS (UMR 8088) and Department of Mathematics, University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. [email protected] In this talk, a boundary integral method is used to solve an inverse linear heat conduction problem in twodimensional bounded domain. An inverse problem of measuring the heat flux from partial (on a part of the boundary) dynamic boundary measurements is considered. This talk presents joint work with A. Khelifi. ——— Inverse problems for wave equation and a modified time reversal method Matti Lassas Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hallstromin katu 2b), Helsinki, University of Helsinki 00014 Finland [email protected] A novel method to solve inverse problems for the wave equation is introduced. Suppose that we can send waves from the boundary into an unknown body with spatially varying wave speed c(x). Using a combination of the boundary control method and an iterative time reversal scheme, we show how to focus waves near a point x0 inside the medium and simultaneously recover c(x0 ) if the wave speed is isotropic. In the anisotropic case we can reconstruct the wave speed up to a change of coordinates. These results are obtained in collaboration with Kenrick Bingham, Yaroslav Kurylev, and Samuli Siltanen. Also, we will disucss how the energy of a wave can be focused near a single point in an unknown medium. These results are done in collaboration with Matias Dahl and Anna Kirpichnikova. ——— Picard condition based regularization techniques in inverse obstacle scattering Koung Hee Leem Dept. of Mathematics & Statistics, Southern Illinois University, Edwardsville, IL 62026 United States [email protected] The problem of determining the shape of an obstacle from far-field measurements is considered. It is well known that linear sampling methods have been widely used for shape reconstructions obtained via the singular system of an ill conditioned discretized far-field operator. For our reconstructions we assume that the far field data are noisy and we present two novel regularization methods that are based on the Picard Condition and do not require a priori knowledge of the noise level. Both

V.1. Inverse problems approaches yield results comparable to the ones obtained via the L-curve method and the discrepancy principle. ——— Limited data problems in tensor tomography William Lionheart School of Mathematic, University of Manchester, Oxford Rd, Manchester, M13 9PL United Kingdom [email protected] n photoelastic tomography one seeks to recover a trace free symmetric second rank tensor from its truncated transverse ray transform. We present constructive uniqueness results in the case where realistic subsets of data are known and numerical reconstruction methods. This is joint work with V Sharafutdinov and D Szotten.

The factorization method is a fast inversion technique for visualizing the profile of a scatterer from measurements of the far-field pattern. The mathematical basis of this method is given by the far-field equation, which is a Fredholm integral equation of the first kind in which the data function is a known analytic function and the integral kernel is the measured (and therefore noisy) far field pattern. We present a Tikhonov parameter choice approach based on a fast fixed-point method developed by Bazan. The method determines a Tikhonov parameter associated with a point near the corner of the L-curve in log-log scale and it works well even for cases where the L-curve exhibits more than one convex corner. The performance of the method is evaluated by comparing our reconstructions with those obtained via the L-curve method.

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The finite data non-selfadjoint inverse resonance problem

A time domain probe method for inverse scattering problems

Marco Marletta Cardiff School of Mathematics Senghennydd Road Cardiff, Wales CF24 4AG United Kingdom [email protected]

Roland Potthast Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire, RG6 6AX, UK [email protected]

We consider Schr¨ odinger operators on [0, ∞) with compactly supported, possibly complex-valued potentials in L1 [0, ∞). It is known (at least in the case of a realvalued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignicant for the reconstruction of the potential from the data. We prove the validity of this statement, i.e., we show conditional stability for nite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials. This is joint work with S. Naboko, S. Shterenberg and R, Weikard. ——— Numerical solutions of nonlinear simultaneous equations Tsutomu Matsuura Graduate School of Engineering, Gunma University 1-51 Tenjintyo Kiryu, Gunma 376-8515 Japan [email protected] In this paper we shall give practical and numerical representations of inverse mappings of 2-dimensional mappings (of the solutions of 2-nonlinear simultaneous equations) and show their numerical experiments by using computers. We derive a concrete formula from a very general idea for the representation of the inverse function ——— A fixed-point algorithm for determining the regularization parameter in inverse scattering George Pelekanos Dept. of Mathematics & Statistics, Southern Illinois University, Edwardsville, IL 62026 United States [email protected]

The goal of the talk is to discuss the development of probe methods for inverse scattering problems in the time-domain. We will study wave scattering by threedimensional rough surface problems. Both the mathematics of these problems as well as the algorithmical solution of direct and inverse problems and the numerical analysis of algorithms provide a sincere challenges since the methods developed for bounded objects cannot be directly translated into the setting of unbounded scatterers. We survey recent results on the direct and inverse problems by Burkard, Chandler-Wilde, Heinemeyer, Lindner and the speaker. With the multi-section method we present a numerical scheme for which convergence both for direct and inverse scattering (using a multi-section Kirsch-Kress approach) can be shown. The time-domain probe method is then formulated and discussed. Convergence for the reconstruction of surfaces can be shown and numerical examples are presented. ——— Explicit and direct representations of the solutions of nonlinear simultaneous equations Saburou Saitoh Department of Mathematics, University of Aveiro, 3810193 Aveiro, Portugal [email protected] We shall present our recent results with Dr. Masato Yamada on practical, numerical and explicit representations of inverse mappings of n-dimensional mappings (of the solutions of n-nonlinear simultaneous equations) and show their numerical experiments by using computers. We derive those concrete formulas from very general ideas for the representation of the inverse functions. ———

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V.2. Stochastic analysis Direct and inverse mixed impedance problems in linear elasticity Vassilios Sevroglou University of Piraeus, Department of Statistics and Insurance Science, 80 Karaoli & Dimitriou Str., Piraeus, Athens 18534, Greece [email protected] Direct and inverse scattering problems with mixed boundary conditions in linear elasticity are considered. We formulate the direct scattering problem for a partially coated obstacle as well as the mathematical setting for the inverse one. Uniqueness theorems are presented and an inversion algorithm for the determination of the scattering obstacle is established. In particular, a linear integral equation due to the linear sampling method which arises from an application of the reciprocity gap functional and the fundamental solution, connected with the appoximate solution of the inverse problem, is investigated. Finally, a discussion about the validity of our method for mixed boundary value problems in elastic scattering theory is presented. ——— On inverse scattering for nonsymmetric operators Igor Trooshin Institute of Problems of Precise Mechanics and Control, Russian Academy of Sciences, Rabochaya 24, Saratov, 410028 Russia [email protected] We consider a nonsymmetric operator AP {L2 (0, ∞)}2 . defined by differential expression (AP u)(x) = Bu0 (x) + P (x)u(x),

in

0 0. Second, we study a class of two-dimensional maps (or called Mira map) and prove that there exist snapback repellers for the map near its anti-integrable limits. Finally, combining the above two results, we establish the existence of transversal homoclinic orbits in family of Arneodo-Coullet-Tresser map near singularities. ——— Bifurcations of random diffeomorphisms with bounded noise Martin Rasmussen Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom [email protected] We discuss iterates of random diffeomorphisms with identically distributed and bounded noise. In this context, minimal forward invariant sets play an important role, since they support stationary measures, and when the noise is interpreted as external control, minimal forward invariant sets coincide with invariant control sets. Discontinuous bifurcations of minimal forward invariant sets are analysed, and a numerical method to approximate these sets is presented. The results are applied to study a bifurcation of the randomly perturbed Henon map. This talk is based on joint work with Jeroen Lamb (Imperial College) and Christian Rodrigues (University of Aberdeen). ——— Bifurcations of period annuli and solutions of nonlinear boundary value problems Felix Sadyrbaev Institute of Mathematics and Computer Science, Rainis boul. 29 Riga, Latvia LV-1459 Latvia [email protected]

be generalized to non-linear and also non-integer expansions of a real number. This talk is based on joined work with T. Persson and D. F¨ arm. ——— Thermodynamic formalism for unimodal maps Mike Todd Departamento de Matem´ atica Pura, Rua do Campo Alegre, 687 Porto, 4169-007 Portugal [email protected] Notions from thermodynamic formalism such as pressure, equilibrium states and large deviations can give a rich qualitative description of a dynamical system. Recently there has been a lot of activity in the development of thermodynamic formalism applied to non-uniformly hyperbolic dynamical systems. These systems have been shown to exhibit a wide variety of phenomena, most interestingly critical phenomena such as phase transitions. In this talk I will give a fairly complete description of the possible behaviour of the class of unimodal interval maps, including the relation between phase transitions and the existence of a natural measure for the system. ——— Dynamics of periodically perturbed homoclinic solutions Qiudong Wang Department of Mathematics, University of Arizona, Tucson, Arizona 85721 United States [email protected] We study the dynamics of homoclinic tangles in periodically perturbed second order equations. Let µ be the size of the perturbation and Λµ be the homoclinic tangles. We prove that (i) for infinitely many µ, Λµ contain nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many µ, Λµ contain nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure set of µ so that Λµ admits strange attractors with Sinai-Ruelle-Bowen measure. ———

Differential equations of the type x00 + λf (x) = 0 are considered, where f (x) are polynomials. First bifurcations of period annuli (continua of periodic solutions) are studied under the change of coefficients of f (x). Secondly, bifurcations of solutions to the Dirichlet problem x(a) = 0, x(b) = 0 are investigated under the change of λ. ——— Large intersection properties of some invariant sets in number-theoretic dynamical systems ¨ rg Schmeling Jo Center of Mathematical Sciences, LTH, Box 118, Slvegatan 18 Lund, 22100 Sweden [email protected] In this talk we consider sets of real numbers that have a given approximation property by rationals with denominators g n . We prove that these sets have large intersection properties and are winning in a modified (α, β) game or belong to Falconers s-class. This result will

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V.5. Functional differential and difference equations Organisers: ˘ acık Zafer Leonid Berezansky, Josef Dibl´ık, Ag Scope of the session: Qualitative theory of functional differential and difference equations: stability, boundedness, oscillation, asymptotic behaviour, positive solutions, dynamic equations on time scales, applications to population dynamics. —Abstracts— Oscillation and non-oscillation of solutions of linear second order discrete delayed equations Jarom´ır Baˇ stinec Department of Mathematics, The Faculty of Electrical Engineering and Communication, Brno University

V.5. Functional differential and difference equations of Technology, Technick´ a 8, 616 00 Brno, Czech Republic [email protected] The phenomenon of the existence of a positive solution of difference equations is often encountered when analysing mathematical models describing various processes. This is a motivation for an intensive study of the conditions for the existence of positive solutions of difference equations. Such analysis is related to an investigation of the case of all solutions being oscillating. In the talk, conditions for the existence of a positive solution are given for a class of linear delayed discrete equations ∆x(n) = −p(n)x(n − 1) where n ∈ Za∞ := {a, a + 1, . . . }, a ∈ N is fixed, ∆x(n) = x(n + 1) − x(n), p : Za∞ → (0, ∞). For the same class of equations, also conditions are given for all the solutions being oscillating. The results obtained indicate sharp sufficient conditions for the existence of a positive solution or for the case of all solutions being oscillating. The investigation was supported by the grant 201/07/0145 of the Czech Grant Agency (Prague) and by the Councils of Czech Government MSM 0021630529 and by MSM 00216 30503. This is joint work with Josef Diblik. ——— New stability conditions for linear differential equations with several delays Leonid Berezansky Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, Negev 84105 Israel [email protected] New explicit conditions of asymptotic and exponential stability are obtained for the general scalar nonautonomous linear delay differential equation with measurable delays and coefficients. These results are compared to known stability tests. ——— Boundary-value problems for differential systems with a single delay Aleksandr Boichuk ˇ ˇ Faculty of Science, Zilina University, Zilina, 01 026 Slovakia [email protected] Conditions are derived of the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay. Utilizing a delayed matrix exponential and a method of pseudo-inverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay. This work was supported by the grant 1/0771/08 of the Grant Agency of Slovak Republic (VEGA) and by the

project APVV-0700-07 of Slovak Research and Development Agency.This is joint work with J. Dibl´ık, D. Khusainov, M. R˚ uˇziˇckov´ a. ——— Representation of solutions of linear differential and discrete systems and their controllability Josef Dibl´ık Brno University of Technology, Brno, Czech Republic, Kiev State University, Kiev, Ukraine [email protected] We study discrete controlled systems ∆x(k) = Bx(k − m) + bu(k), q where m ≥ 1 is a fixed integer, k ∈ Z∞ 0 , Zs := {s, s + n 1, . . . , q}, B is a constant n × n matrix, x : Z∞ −m → R is unknown solution, b ∈ Rn is given nonzero vector and u : Z∞ 0 → R is input scalar function. Moreover, we consider the system of delayed linear differential equations of second order

y 00 (t) + Ω2 y(t − τ ) = bu(t) and an initial problem y(t) = ϕ(t), y 0 (t) = ϕ0 (t), t ∈ [−τ, 0] where τ > 0 and ϕ : [−τ, 0] → Rn is twice differentiable. Special matrix functions are defined: the delayed matrix sine and the delayed matrix cosine. These matrix functions are applied to obtain explicit formulas for the solution of the initial problem and a controllability criterion. The investigation was supported by the grant 201/08/0469 of the Czech Grant Agency (Prague), by the Councils of Czech Government MSM 0021630519 and MSM 00216 30503 and by the project M/34-2008 of Ukrainian Ministry of Education. This is joint work with Denys Khusainov, Blanka Mor´ avkov´ a. ——— Maximum principles and nonoscillation intervals in the theory of functional differential equations Alexander Domoshnitsky Ariel University Center, Department of Mathematics and Computer Science, Ariel, 44837 Israel [email protected] Many classical topics in the theory of functional differential equations, such as nonoscillation, differential inequalities and stability, were historically studied without any connection between them. As a result, assertions associated with maximum principles for such equations in contrast with the cases of ordinary and even partial differential equations do not add so much in problems of existence and uniqueness of solutions to boundary value problems and stability for functional differential equations. One of the goals of this talk is to present a concept of the maximum principles for functional differential equations. New results on existence and uniqueness of solutions of boundary value problems are proposed. Assertions about positivity og Green’s functions are formulated. Tests of the exponential stability are obtained on the basis of nonoscillation and positivity of the Cauchy function. ———

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V.5. Functional differential and difference equations Averaging for impulsive functional differential equations: a new approach

Lyapunov type inequalities for nonlinear impulsive differential systems

Marcia Federson Av. Trabalhador Sao-carlense 400, CP 668, Sao Carlos, SP 13560-970 Brazil [email protected]

Zeynep Kayar Middle East Technical University, Department of Mathematics, Ankara, Cankaya 06531, Turkey [email protected]

We consider a large class of functional differential equations subject to impulse effects and state an averaging result by means of the techniques of the theory of generalized ordinary differential equations introduced by J. Kurzweil. ——— Some bifurcation problems in the theory quasilinear integro differential equations Yakov Goltser Department of Computer Sciences and Mathematics, Ariel University Center of Samaria, Ariel, 44837 Isser Natanzon, 27/7, Pisgat Zeev, 97877 Jerusalem, Israel [email protected] Our goal is to study parametrical perturbed nonlinear quasiperiodic systems of differential and integrodifferential equations.Study bifurcation problems similarly Hopf bifurcation, Bogdanov-Takkens bifurcation and bifurcation of invariant torus,based on the normal form theory and the truncated method for countable systems of ordinary differential equations. ——— Stability in Volterra type population model equations with delays ´ n Gyo ¨ ri Istva Egyetem u. 10 Department of Mathematics, University of Pannonia Veszprem, Veszprem County H-8200 Hungary [email protected] In this talk some delay dependent and delay independent stability conditions will be given for differential equations arising in population dynamics. The proofs are based on the construction of a Lyapunov functional and some monotone techniques for nonautonomous systems. At the end of the talk we shall formulate some open problems and conjectures. ——— On parameter dependence in functional differential equations with state-dependent delays Ferenc Hartung University of Pannonia Egyetem str 10 Veszprem, H8200 Hungary [email protected] In this talk we study smooth dependence on parameters of solutions of several classes of functional differential equations with state-dependent delays. As an application of our results, we discuss the parameter estimation problems for FDEs with state-dependent delays using a quasilinearization method. ———

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We obtain Lyapunov-type inequalities for systems of nonlinear impulsive differential equations. In particular, these sytems contain the Emden-Fowler-type systems and half linear systems in the special cases. In addition, as an application we make use of these inequalities to derive some boundedness and disconjugacy criteria and sufficient conditions for the asymptotic behaviour of solutions. ——— Evaluating the stochastic theta method Conall Kelly Department of Mathematics, University of the West Indies, Mona Kingston, Sn.Andrew 7, Jamaica [email protected] When a numerical method is applied to a differential equation, the result is a difference equation. Ideally the dynamics of the difference equation should reflect those of the original as closely as possible, but in general this can be difficult to check. It is therefore useful to perform a linear stability analysis: applying the method of interest to a linear test equation possessed of an equilibrium solution with known stability properties, and determining the asymptotic properties of the resultant difference equation for comparison. We examine the issues that arise for this kind of analysis in the context of stochastic differential equations, and review the relevant literature. These issues have yet to be adequately addressed. We propose a new approach and demonstrate its usage for the class of θ-Maruyama methods with constant step-size. ——— Delay-distribution effect on stability Gabor Kiss Department of Engineering Mathematics, University of Bristol Queen’s Building Bristol, South West England BS8 1TR United Kingdom [email protected] We consider the effect of delay distribution on retarted functional differential equations with one delay. More specifically, we study the effect of delay distribution on the stability of solutions of first- and second-order equations by comparing the stability regions of the respective equation with a single delay with that of the equation with distributed delays. ——— Solutions of linear impulsive differential systems bounded on the entire real axis ´ Martina Langerova Dept. of Mathematics, Faculty of Science, University of ˇ ˇ Zilina, Univerzitn´ a 1, 010 26 Zilina, Slovakia [email protected] We consider the problem of existence and structure of solutions bounded on the entire real axis of the linear

V.5. Functional differential and difference equations differential system with impulsive action at fixed points of time x˙ = A(t)x + f (t), ˛ ˛ ∆x˛

= ai , t=τi

t, τi ∈ R,

t 6= τi , i ∈ Z,

ai ∈ R n .

Under the assumption that the corresponding homogeneous system is exponentially dichotomous on the semiaxes R+ and R− and by using the results of the wellknown Palmer lemma and the theory of pseudoinverse matrices we establish necessary and sufficient conditions for the indicated problem. Co-authors: Oleksandr ˇ ıkov´ Boichuk, Jaroslava Skor´ a. This research was supported by the Grants 1/0771/08, 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and APVV 0700-07. ——— Oscillatory and asymptotic properties of solutions of higher-order difference equations of neutral type Malgorzata Migda Institute of Mathematics, Poznan University of Technology, ul. Piotrowo 3A, Poznan, 60-965 Poland [email protected] We consider higher-order linear difference equations with delayed and advanced terms ∆m (xn − pxn−τ ) = qn xn−σ + hn xn+η where p is a nonnegative number, τ, σ, η are positive integers and (qn ), (hn ) are sequences of nonnegative real numbers. We give sufficient conditions under which all nonoscillatory solutions of the delayed part of the equation are unbounded and under which all nonoscillatory solutions of the advanced part tend to zero as n → ∞. We establish also sufficient conditions for the oscillation of all solutions of the full equation.

We shall discuss some results on the asymptotic behaviour of the nonnegative solutions of systems of linear difference equations with asymptotically constant coefficients. The main result describes the relationship between the nonnegative solutions of the perturbed system and the positive eigenvalues and the corresponding nonnegative eigenvectors of the limiting system. The proofs are based on Pringsheim’s Theorem and the Extended Liouville Theorem from complex analysis. ——— On singular models arising in hydrodynamics Irena Rachunkova Palacky University, Fakulty of Science, Dept. of Mathematics, Tomkova 40, Olomouc, 77900 Czech Republic [email protected] We investigate models arising in hydrodynamics. These models have the form of the singular second order differential equation (p(t)u0 (t))0 = p(t)f (u(t)) on the half-line. Here f is locally Lipsichtz on R and changes its sign and p is continuous on [0, ∞) and p(0) = 0. A discrete formulation of this equation is investigated as well. We are interested in strictly increasing solutions and homoclinic solutions and provide conditions for p and f which guarantee the existence of such solutions. In particular cases a homoclinic solution determines an increasing mass density in centrally symmetric gas bubbles which are surrounded by an external liquid. ——— Decoupling and simplifying of noninvertible difference equations in the neighbourhood of invariant manifold Andrejs Reinfelds University of Latvia, Institute of Mathematics and Computer Science; Rai¸ na bulv¯ aris 29, LV-1459, R¯ıga, Latvia

——— Principal and non-principal solutions of impulsive differential equations with applications ¨ Abdullah Ozbekler Atılım University, Department of Mathematics, ˙ Kızılca¸sar K¨ oy¨ u, Incek G¨ olba¸sı, Ankara 06836 Turkey [email protected] In this work we first prove a theorem on the existence of principal and nonprincipal solutions for second order differential equations having fixed moments of impulse actions. Next, by means of nonprincipal solution we give new oscillation criteria for related impulsive differential equations. Examples are provided with numerical simulations to illustrate the importance of the study. ——— Nonnegative iterations with asymptotically constant coefficients Mihali Pituk Egyetem u. 10 Department of Mathematics, University of Pannonia Veszprem, Veszprem County H-8200 Hungary [email protected]

[email protected] In Banach space X×E the system of difference equations x(t + 1) = g(x(t)) + G(x(t), p(t)), p(t + 1) = A(x(t))p(t) + Φ(x(t), p(t))

(*)

is considered. Sufficient conditions under which there is an local Lipschitzian invariant manifold u : X → E are obtained. Using this result we find sufficient conditions of partial decoupling and simplifying of the system of noninvertible difference equations (*). ——— Precise asymptotic behaviour of solutions of Volterra equations with delay David W. Reynolds School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland. [email protected] This talk considers the rates at which solutions of Volterra equations with delay converge to asymptotic equilibria. It is found that these convergence rates depend delicately on prescribed data. The results are established using admissibility techniques. This work is motivated by logistic equations with infinite delay.

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V.5. Functional differential and difference equations ———

———

On local stability of solutions of stochastic difference equations

Existence and nonexistence of asymptotically periodic solutions of Volterra linear difference equations

Alexandra Rodkina Department of Mathematics, University of the West Indies, Mona Kingston, Sn.Andrew 7, Jamaica [email protected]

Ewa Schmeidel Instytute of Mathematics, ul. Piotrowo 3A, Poznan, Wielkopolska 60-965 Poland [email protected]

We present results on the local stability of solutions of a stochastic difference equation with polynomial coefficients. Two cases are considered: when stochastic perturbation are a state-independent and asymptotically fading and when stochastic perturbation are a statedependent.

In this talk we investigate Volterra difference equation of the form

——— Convergence of the solutions of a differential equation with two delayed terms ˇkova ´ Miroslava R˚ uˇ zic ˇ Faculty of science, University of Zilina, Slovak Republic [email protected] In this contribution we deal with asymptotic behavior of solutions to a linear homogeneous differential equation containing two discrete delays y(t) ˙ = β(t)[y(t − δ) − y(t − τ )]

(*)

for t → ∞. We assume δ, τ ∈ R+ := (0, +∞), τ > δ, β : I−1 → R+ is a continuous function, I−1 := [t0 − τ, ∞), t0 ∈ R. Denote I := [t0 , ∞) and the symbol “ ˙ ” denotes (at least) the right-hand derivative. Similarly, if necessary, the value of a function at a point of I−1 is understood (at least) as value of the corresponding limit from the right. The main results concern the asymptotic convergence of all solutions of Eq. (*). Especially we deal with so called critical case with respect to the function β. When the function β is the constant function than this critical case is represented with the value β := (τ − σ)−1 . The proof of results is, except other, based on comparison of solutions of Eq. (*) with solutions of an auxiliary inequality which formally copies Eq. (*). This research was supported the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA), by the project APVV-0700-07 of Slovak Research and Development Agency and by the Slovak-Ukrainian project SK-UA-0028-07 (Ukrainian-Slovak project M/34 MOH Ukraine 27.03.2008). This is joint work with Josef Dibl´ık. ——— Inverse problems of the calculus of variations for functional differential equations Vladimir Mikhailovich Savchin Peoples Friendship University of Russia, MikluxoMaklaya street 6, Moscow, 117198 Russia [email protected] The problem of existense of solutions of inverse problems of the calculus of variations for partial differencial difference operators is investigated. Necessary and sufficient conditions of potentiality for such operators are obtained. Methods of construction of variational multiplies are suggested.

104

x(n + 1) = a(n) + b(n)x(n) +

n X

K(n, i)x(i)

i=0

where n ∈ N = {0, 1, 2, . . . }, a, b, x : N → R and K : N × N → R, the special case of this equation is Volterra difference equation of convolution type x(n + 1) = Ax(n) +

n X

K(n − i)x(i).

i=0

This equation may be considered as a discrete analogue of famous Volterra integrodifferential equation Z t b(t − s)x(s)ds. x0 (t) = Ax(t) + 0

Such equation has been widely used as a mathematical model in population dynamics. Both discrete equations represents a system in which the future state x(n + 1) does not depend only on the present state x(n) but also on all past states x(n − 1), x(n − 2), . . . , x(0). These system are sometimes called hereditary. Given the initial condition x(0) = x0 , one can easy generate the solution x(n, x0 ). Sufficient conditions for the existence of asymptotically periodic solutions of Volterra difference equation are presented. In addition we present sufficient conditions for non-existence of an asymptotically periodic solution satisfying some auxiliary conditions. The results are illustrated by examples. ——— Gene regulatory networks and delay equations Andrei Shindiapin Eduardo Mondlane University, Maputo, Mozambique [email protected] Gene regulatory networks consist of differential equations with smooth but steep nonlinearities (”sigmoids”). As the number of genes may be rather large, any theoretical or computer-based analysis of such networks can be complicated. That is why a simplified approach based on replacing sigmoids with step functions is widely used. However, this leads to some mathematical challenges, as for instance analysis of stationary points belonging to the discontinuity set of the system (thresholds) cannot be done directly. Additional problems occur if one tries to incorporate time delays into the network. The delay effects naturally arise from the time required to complete transcription, translation and diffusion to the place of action of a protein. We offer an algorithm of localizing stationary points in the presence of delays as well as stability analysis around such points. This algorithm is combined with a method to study delay systems by replacing them with an equivalent system of ordinary differential equations, commonly known as the linear chain trick. However, a direct application of this ”trick” is not

V.6. Mathematical biology possible in our case, so that we suggest a modification of it based on the general framework of representing delay equations as ordinary differential equations using the integral transforms”. This is joint work with Arcady Ponosov. ——— The moment problem approach for the zero controllability of ecolution equations Benzion Shklyar 52 Golomb St., P.O.B. 305, Dept. of Appl. Math, Holon Institute of Technology, Holon, 58102 Israel shk [email protected] The exact controllability to the origin for linear evolution control equation is considered.The problem is investigated by its transformation to infinite linear moment problem. Controllability conditions for linear evolution control equations have been obtained. The obtained results are applied to the zero controllability for partial differential and functional differential equations. ——— Properties of maximal solutions of autonomous functional-differential equations with state-dependent deviations Svatoslav Stanek Palacky University, Fakulty of Science, Dept. of Mathematics, Tomkova 40, Olomouc, 77900 Czech Republic [email protected] Equations of the type x00 +x(t−kx)) = 0 are considered. Here k is a positive parameter. It is described (i) the set of all periodic solutions x satisfying x0 < 1/k and (ii) the set of all maximal solutions x (that is, solutions which have no extension) satisfying x0 ≥ 1/k. ——— Boundedness character of some classes of difference equations Stevo Stevic Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000 Serbia [email protected] Some results on the boundedness character of the positive solutions of the following two classes of difference equations xn+1 = A +

xpn , q xn−1 xrn−2

xn+1 = max A,

ff xpn , xqn−1 xrn−2

n ∈ N0 ; n ∈ N0 ,

Republic [email protected]

This contribution deals with systems of linear generalized linear differential equations of the form t

Z x(t) = x e+

d[A(s)] x(s) + g(t) − g(a),

t ∈ [a, b], (*)

a

where −∞ < a < b < ∞, A is an n × n-complex matrix valued function, g is an n-complex vector valued function, A has a bounded variation on [a, b] and g is regulated on [a, b]. The integrals are understood in the Kurzweil-Stieltjes sense. Our aim is to present some new results on continuous dependence of solutions to linear generalized differential equations (*) on parameters and initial data. ——— Lyapunov type inequalities on time scales: A survey ¨ Mehmet Unal Bahcesehir University, C ¸ ıra˘ gan Caddesi, Osmanpa¸sa Mektebi Sokak No. 4–6, Be¸sikta¸s, Istanbul 34353 Turkey [email protected]

We survey Lyapunov type inequalities for linear and nonlinear dynamic equations on time scales. The inequalities contain the well-known classical Lyapunov inequalities as special cases. We also give some applications to illustrate the importance of such inequalities. ——— Interval criteria for oscillation of delay dynamic equations with mixed nonlinearities ˘ acık Zafer Ag Department of Mathematics Middle East Technical University Cankaya, Ankara 06531 Turkey [email protected]

We obtain interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities on an arbitrary time scale T. All results are new even for T = R and T = Z. Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second order delay dynamic equations regardless of the choice of delta (∆) or nabla (∇) derivatives. ———

where the parameters A, p, q and r are positive numbers, are presented. ——— Continuous dependence of solutions of generalized ordinary differential equations on a parameter Milan Tvrdy Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, Praha 1, CZ 115 67 Czech

V.6. Mathematical biology Organisers: Robert Gilbert

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VI. Others —Abstracts— Cancellous bone with a random pore structure Robert Gilbert Department of Mathematics, University of Delaware, 317 Ewing Hall, Newark, DE 19716 United States [email protected] We continue the study of acoustic wave propagation for an elastic medium that is randomly fissured. Moreover, the fissures are assumed to be statistically homogeneous. Although the underlying stochastic process does not necessarily have to be ergodic, we assume for simplicity of exposition that it is. This allows us to obtain an explicit and computationaly easier auxillary problem in a Representative Elementary Volume. In a later work we intend to study the more general case. This is joint work with Ana Vasilic. ——— New computer technologies for the construction and numerical analysis of mathematical models for molecular genetic systems Irina Alekseevna Gainova Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences Acad. Koptyug avenue, 4 Novosibirsk, Novosibirsk region 630090 Russia [email protected] We have created an integrative computer system, which includes three program modules (Institute of Cytology and Genetics, SB RAS): GeneNet, MGSgenerator, MGSmodeller, and the software package STEP+ (Sobolev Institute of Mathematics, SB RAS). The system is used to construct and numerically analyze models describing dynamics of the molecular genetic systems (MGS) functioning in pro- and eukaryotes. Using module GeneNet we can reconstruct structure functional organization of gene networks. We use MGSgenerator as an intermediate module in generation of mathematical models based on gene networks reconstructed in GeneNet. Moreover, in the module MGSgenerator we represent obtained mathematical models in the input format of STEP+. Module MGSmodeller contains tools for the gene network models to be developed and numerically analyzed. Package STEP+ is intended for the numerical analysis of mathematical models represented by autonomous systems ODEs. We have tested our integrated system on the MGS model for intracellular auxin metabolism in a plant cell. This work has been partially supported by the Siberian Branch of the Russian Academy of Sciences (Interdisciplinary integration project Post-genomic bioinformatics: computer analysis and modeling of the molecular genetic systems, No. 119). ———

Due to the presence of the fluid and solid phase, the modeling of cancellous bone represents a complex, extensive task where the dynamic investigation and viscosity effects must be taken into consideration. The already established approach for the investigation of this material type is Biots method, originally developed for simulating saturated porous materials. In this contribution we present the homogenization multiscale FEM as an alternative to Biots method. The motivation for this choice is decreasing the extent of the necessary laboratory investigations. According to the multiscale FEM, the bone is understood as the homogenized medium whose effective material parameters are obtained by the analysis of an appropriate representative volume element (RVE). This is also the main topic of the presentation: a comparison of the effective values obtained by studying different types of RVEs where the particular attention is paid to the numerical values for Youngs modulus and attenuation coefficient. The distinction between the models pertains to the geometry of the solid frame of the RVE, the type of the applied elements as well as the type of the coupling conditions on the interface of the phases. ——— Bone growth and destruction at the cellular level: a mathematical model Mark D. Ryser McGill University, W. Burnside Hall, Room 1005, 805 Sherbrooke Street, Montral, Quebec H3A 2K6 Canada [email protected] The process of bone destruction and subsequent growth is continually occurring in healthy bone tissue. This process is referred to as ’remodeling’ and plays a key role in many pathologies such as osteoporosis and osteoarthritis. We describe remodeling at the cellular level and discuss the cells and biochemical pathways involved. We then develop a mathematical model for remodeling, consisting of a system of coupled nonlinear PDEs. We discuss how physiological parameters may be obtained through scaling of the equations and we comment on their mathematical properties. Numerical experiments validating the model will be presented. This is joint work with Nilima Nigam (SFU) and Svetlana Komarova (McGill). ———

VI. Others Organisers: local organising committee

—Abstracts—

Application of the multiscale FEM in modeling the cancellous bone

The relationship between Bezoutian matrix and Newton’s matrix of divided differences and separation of zeros of interpolation polynomials

Sandra Ilic Institute of Mechanics, Ruhr-University of Bochum, Bochum, 44780 Germany [email protected]

Ruben Airapetyan Kettering University, 1700 W Third Ave. Flint, Michigan 48504, United States [email protected]

106

VI. Others Let x1 , . . . , xn be real numbers, Pn (x) = an (x − x1 ) · · · (x−xn ). Denote by Dn g the matrix of generalized divided differences of function g in Newton’s interpolation formula with nodes x1 , . . . , xn and by Gn (x) the Newton’s interpolation polynomial of function g. Denote by B = B(Pn (x), Gn (x)) the Bezoutian matrix of Pn and Gn . The relationship between the corresponding principal minors of the matrices Dn g and B counted from the left lower corner is establish. Then, it follows that if these principal minors of the matrix of divided differences are positive or have alternating signs then the roots of the interpolation polynomial are real and separated by the nodes of interpolation. ——— Bayesian shrinkage estimation of parameter exponential distribution Hadeel Alkutubi B-23-1 , The Heritage, JLN SB, Dagang Mines Resort City, Seri Kembangan, Serdang, 43300 Malaysia [email protected]

In our work, we use the ADM method for solving some nonlinear evolution equations with time and space fractional derivative. Then we use the Extended Tanh method to formally derive traveling wave solutions for some evolution equations. The obtained solutions include, also, kink soltuions. ——— Boundary-value problems for symmetric Helmholtz equation

generalized

axially-

Anvar Hasanov 34 Durmon yoli, Tashkent branch of the Russian State University of oil and gas named after Gubkin, Tashkent, Tashkent 100125, Uzbekistan [email protected] In this talk several main boundary-value problems such the Dirichlet, Neumann problem and other problems will be considered. The unique solvability of afore-mentioned problems will be proved. ———

In this paper, we would like to test the best estimator (smallest MSE and MPE) of shrinkage estimator of parameter exponential distribution . To do this , we derived this estimators depend on Bayesian method with Jeffreys prior information and square error loss function . To compared between estimators we used MSE and MPE with respect of simulation study. We found the shrinkage estimator between Bayes estimators under different loss function is the best estimator.

Asymptotic extension of topological modules and algebras

———

Given a topological R-module or algebra E and an asymptotic scale M ⊂ RΛ , we exhibit a natural M extended topology on the sequence space E Λ , and define the M -extension of E as the Hausdorff space associated with the subspace of nets for which multiplication is continuous with respect to this topology. Commonly used spaces of generalized functions are obtained as special cases, but this new approach applies in many different situations. It also allows the iteration of the construction, which is not possible with previously existing theories. We use only the topology, i.e. neighbourhoods of zero, but not its explicit definition in terms of seminorms, inductive or projective limits etc., which is particularly convenient in non-metrizable spaces. Many ideas commonly used in the context of generalized functions (functoriality, association, sheaf structure, algebraic analysis, . . . ) can be applied to a large extent. Reasoning on a category-theoretic level allows to establish several results so far only known for particular cases, for the whole class of such spaces.

Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem Mohammed Bokhari Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia [email protected] An elegant result due to Erdos and Turan states that the sequence of Lagrange interpolants to a given continuous function f at the zeros of orthogonal polynomials over a closed interval converges to f in the mean square sense. We introduce certain sequences of polynomials which preserve both interpolation as well as convergence properties of Erdos-Teran Theorem. In addition, they interpolate f at a finite number of pre-assigned points lying outside the underlying open interval. We shall introduce a method to construct the suggested polynomials and also investigate their properties. Computational aspects will also be discussed. ——— The ADM method and the Tanh method for solving some non linear evolutions equations Zoubir Dahmani Department of Mathematics, Faculty of Sciences, University of Mostaganem Les HLM, 21 street les HLM mostaganem, mostaganem 27000 Algeria [email protected]

Maximilian Hasler Laboratoire AOC, Universit Antilles-Guyane, B.P. 7209, campus de Schoelcher, Schoelcher, Martinique 97275 France [email protected]

——— Approximation of fractional derivatives S. Moghtada Hashemiparast Mathematics and Statistics, Jolfa Ave, Seyed Khandan, Tehran 193953358 Iran ha[email protected] Series represantations are presented to approximate the fractional derivatives which have extensive application in ordinary,partial difrential equations and specilly the stable probability distributions.The convergence of the

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VI. Others series are considered and are applied to solving the equations,finally toillustrate the accuracy of the apprpximations examples are solved. ——— Discrepancy estimate for uniformly distributed sequence Hailiza Kamarulhaili School of Mathematical Sciences Universiti Sains Malaysia Minden, Penang 11800 Malaysia [email protected] A general metrical result of discrepancy estimate related to uniform distribution of a sequence is proved . This work extends result of R.C. Baker where the sequence can be assumed to be real. The lighter version of this theorem will also be discussed in this talk. ——— Bounded linear operators on l-power series spaces Erdal Karapinar ATILIM University, Department of Mathematics, Kizilcasar Koyu, INCEK ANKARA, 06836 Turkey [email protected] Let A be the class of Banach space ` of scalar sequences with a norm k · k` such that (i) a = (ai ) ∈ l∞ , x = (ξi ) ∈ ` ⇒ ax = (ai ξi ) ∈ `, kaxk` ≤ kakl∞ kxk` , (ii) kei k` = 1, ∀i ∈ N where ei = (δij )j∈N .

Theorem. For Fr´echet space E and ` ∈ A, (E, K l1 (A)) ∈ B ⇒ (E, K ` (A)) ∈ B ⇒ (E, K l∞ (A)) ∈ B. and `

∈

A,

——— On a three-dimensional elliptic equation with singular coefficients Erkinjon Karimov Durmon yuli street 29, Akademgorodok Tashkent, Tashkent 100125 Uzbekistan [email protected] In this talk some questions such as finding fundamental solutions, investigations of main boundary-value problems for an equation uxx + uyy + uzz +

108

2α 2β 2ζ ux + uy + uz x y z

——— A unified presentation of a class of generalized Humbert polynomials Nabiullah Khan Department of Applied Mathematics, Z.H. College of Engineering and Technology, Aligarh Muslim University, Aligarh 202002 India nabi [email protected] The principal object of this paper is to present a natural further step toward the unified presentation of a class of Humbert’s polynomials which generalizes the wellknown class of Gegenbauer, Humbert, Legendre, Tchebycheff, Pincherle, Horadam, Dave, Kinnsy, Sinha, Shreshtha, Horadam-Pethe, Djordjevie, Gould, Milovanovi´c and Djordjevi´c, Pathan and Khan polynomials and many not so wellknown polynomials. We shall give some basic relations involving the generalized Humbert polynomials and then take up several generating functions, hypergeometric representations and expansions in series of some relatively more familier polynomials of Legendre, Gegenbauer, Rice, Hermite, Jacobi, Laguerre Fasenmyer Sister M. Celine, Bateman, Rainville and Khandekar. We also show that our results provide useful extensions of known results of Dilcher, Horadam, Sinha, Shreshtha, Milovanovi´c-Djordjevi´c, Pathan and Khan. ——— Direct estimate for modified beta operators

For a given ` ∈ A and a K¨ othe matrix A, we define `-K¨ othe space K ` (ai,n ) as a Fr´echet space of all scalar sequences x = (ξi ) such that (ξi ai,n ) ∈ ` for each n, endowed with the topology of Fr echet space, determined by the canonical system of norms {kxkn = k (ξi ai,n ) k` , n ∈ N }. We write (E, F ) ∈ B, if every continuous linear map from E to F is bounded. In 1983, D.Vogt has characterized those Fr´echet spaces E for which (E, K l∞ (A)) ∈ B holds. This gives also a characterization of (E, K c0 (A)) ∈ B. We extend this results and prove that

Theorem. For Fr´echet space F (K ` (A), F ) ∈ B ⇒ (K l1 (A), F ) ∈ B.

will be discussed. Here α, β, zeta are constants, moreover 0 < 2α, 2β, ζ < 1.

Lixia Liu Yuhua east Road 113, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei Province 050016 China [email protected] 2 Use the Ditzian modulous of smoothness ωϕ λ (f, t), (0 ≤ λ ≤ 1), to study the pointwise direct results for modified Beta operators, which extend the approximation result for Beta operators.

——— Mathematical model of an undergorund nuclear waste disposal site Eduard Marusic-Paloka Department of Mathematics, University of Zagreb, Bijenicka 30, Zagreb, 10000 Croatia [email protected] The goal of our research is to find an accurate model for numerical simulations of the nuclear waste disposal site. The purpose of such model is to perform safety analysis of the site and find out its possible impact on the biosphere. Due to the large dimension of the site and very long lifetime of radioelements, realistic experiments are not possible. Thus, predictions based on numerical simulations are all we have. Starting from the microscopic model given by the reaction-diffusion-convection equation, using the asymptotic analysis and homogenization, we derive a macroscopic model and discuss ity accuracy. ———

VI. Others Compact and coprime packedness and semistar operations Abdeslam Mimouni Department of Mathematics and Statistics King Fahd University of Petoleum and Minerals Dhahran, Estern 31261 Saudi Arabia [email protected] In this talk we will present new developments on the study of compact and coprime packedness of an integral domain with respect to a star operation of finite character. Let R be an integral domain with quotient field K and let ∗ be a star operation of finite type on R. A ∗-ideal I is said to be ∗-compaclty S (respectively ∗-coprimely) packed if whenever I ⊆ α∈Ω Pα , where {Pα }α∈Ω is a family of ∗-prime ideals of R, I is actually contained in Pα (resp. (I + Pα )∗ ( R) for some α ∈ Ω; and R is said to be ∗-compactly (resp. ∗-coprimely) packed if every ∗-ideal of R is ∗-compactly (resp. ∗coprimely) packed. In the particular case where ∗ = d is the trivial operation, we obtain the so-called compactly and coprimely packed domains. Our objectives is to study some ring-theoretic aspects of these notions in different classes of integral domains, paying particular attention to the the t-operation as the largest and well-known operation. ——— Characterization of some matrix classes involving (σ, λ)convergence S. A. Mohiuddine Department of Mathematics Aligarh Muslim University Aligarh, Uttar Pradesh 202002 India [email protected] Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional ϕ on the space `∞ of bounded single sequences is said to be an invariant mean or σ-mean if and only if (i) ϕ(x) ≥ 0 if x ≥ 0 (i.e. xk ≥ 0 for all k); (ii) ϕ(e) = 1, where e = (1, 1, 1, · · · ); (iii) ϕ(x) = ϕ((xσ(k) )) for all x ∈ `∞ . Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1,

λ1 = 0.

In this paper, first we define (σ, λ)-convergence and show that Vσλ is a Banach space with kxk = supm,n |tmn (x)|, where Vσλ is the set of all (σ, λ)-convergent sequences x = (xk ). We also define and characterize (σ, λ)conservative, (σ, λ)-regular and (σ, λ)-coercive matrices. Further, we characterize the class (`1 , Vσλ ), where `1 is the space of all absolutely convergent series.

if and only if (i) φ(x) ≥ 0 when the sequence x = (xk ) has xk ≥ 0 for all k, (ii) φ(e) = 1, where e = (1, 1, 1, · · · ), and (iii) φ(x) = φ((xσ(k) )) for all x ∈ `∞ . Throughout this paper we consider the mapping σ which has no finite orbits, that is, σ p (k) 6= k for all integer k ≥ 0 and p ≥ 1, where σ p (k) denotes the pth iterate of σ at k. Note that, a σ-mean extends the limit functional on the space c in the sense that φ(x) = lim x for all x ∈ c. In this paper we define a new sequence space Vσ∞ (λ) which is related to the concept of σ-mean and the sequence λ = (λn ) described as above and characterize the matrix classes (`∞ , Vσ∞ (λ)) and (`1 , Vσ∞ (λ)). Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 0. Then we define the following sequence space and show that it is a BK-space: Vσ∞ (λ) := {x ∈ `∞ : sup |τmn (x)| ≤ ∞}, m,n

where X

τmn (x) = (1/(λm ))

xσj (n) .

j∈`m

——— New convection theory for thermal plasma and NHD convection in rapidly rotating spherical configurations Ali Mussa King Abdulaziz City for Science and Technology Building # 2 King Abdullah Bin Abdulaziz Street Riyadh, Riyadh 6086/11442 Saudi Arabia [email protected] We extend Jones-Soward-Mussa (JSM) theory (2000): “analytic and computational solution for E → 0 and P r/E → ∞”. We also make use of Zhang (2001) ansatz for: “E 1 arbitrary but fixed and 0 ≤ P r < ∞” the so-called enhanced Nearly Geostrophic Inertial Wave (NGIW) approach. Such extension represented as a construction of a new MHD plasma convection and magnetoconvection force theory. The flow field confinement in the study assumed to be in spherical geometry configuration and our investigation is made under the basis of magnetic balance and scaling theory. Furthermore, strong inertial turbulence can be achieved in presence of high Reynolds number so strong forces govern the flow fields have to be sufficiently understood. Indeed, strong rotation and strong magnetic field for the flow field inside the spherical rotating geometry; take into consideration the effect of the anticipated vigorous convection and magnetoconvection in the flow field confinement. ——— Characterizations of Isometries on 2-modular spaces

Sequence spaces of invariant mean and some matrix transformations

Kourosh Nourouzi Department of Mathematics, K.N. Toosi University of Technology Tehran, Tehran 16315-1618 Iran [email protected]

Mohammad Mursaleen Department of Mathematics Aligarh Muslim University Aligarh, UP 202002 India [email protected]

Let X be a real vector space of dimension greater than one. A real valued function ρ(·, ·) on X 2 satisfying the following properties is called a 2-modular on X, for all x, y, z ∈ X:

———

Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional φ on the space `∞ is said to be an invariant mean or a σ-mean

1. ρ(x, y) = 0 if and only if x, y are linearly dependent, 2. ρ(x, y) = ρ(y, x),

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VI. Others 3. ρ(−x, y) = ρ(x, y), 4. ρ(x, αy + βz) ≤ ρ(x, y) + ρ(x, z), for any nonnegative real numbers α, β with α + β = 1. In this talk, we discuss on the characterization of isometries defined on 2-modular spaces. ——— On r-imbalances in tripartite r-digraphs Shariefuddin Pirzada King Fahd University of Petroleum and Minerals, Dhahran, 31261 Saudi Arabia [email protected] A tripartite r-digraph(r ≥ 1) is an orientation of a tripartite multigraph that is without loops and contains atmost r edges between any pair of vertices from distinct parts. For any vertex x in a tripartite r-digraph − D(U, V, W ), let d+ x and dx denote the outdegree and − indegree respectively of x. Define aui = d+ ui − dui , + − + − bvj = dvj − dvj and cwk = dwk − dwk as the r-imbalances of the vertices ui in U , vj in V and wk in W respectively. In this paper, we characterize r-imbalances in tripartite r-digraphs and obtain some results. ——— Invariance conditions and amenability of locally compact groups Hashem Parvaneh Masiha Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology. No. 41, Kavian St., Seyyed Khandan Bridge (N.), Shariati Ave., Tehran, Tehran 16315-1613 Iran [email protected] Adler and Hamilton showed that a semigroup S is left amenable if and only if it satisfies the following invariance condition. For any subsets A1 , A2 , · · · , Ak of S and any s1 , s2 , · · · , sk ∈ S, there exists a nonempty finite subset E of S such that n(s−1 i Ai ∩ E) = n(Ai ∩ E), for i = 1, 2, · · · , k, where s−1 A = {t ∈ S : st ∈ A} and n(A) is the number of elements in A. In this talk, we shall prove an analogous result for locally compact groups. More precisely, we show that amenability of a locally compact group G is equivalent to: For any λ-measurable subsets A1 , A2 , · · · , Ak of G, any g1 , g2 , · · · , gk ∈ G and any ε > 0, there exists a compact subset K of G such that |λ(gi−1 Ai ∩ K) − λ(Ai ∩ K)| < ελ(K), for i = 1, 2, · · · , k, where λ(A) denotes the left Haar measure of A. In this paper, we suppose that G be a locally compact group and λ a fixed left Haar measure on G. We let X = {K ⊂ G : K is compact and R λ(K) > 0}. For 1 f ∈ L∞ (G), we define f¯(K) = λ(K) f dλ, K ∈ X, K then f¯ : X → R is well defined. ———

of one of the major classical problems of theoretical mechanics, dynamics of a solid body with one fixed point in a gravity field. Motion of a solid body with one fixed point is described by the well-known system of Euler and Poisson equations. It is known the general solution exists if one considers two first terms of force function expansion into a series. By original change of variables the system is reduced to the normal form with the first integral of norm type. The solution of this system is considered as the non-perturbed motion and it is investigated on stability. The procedure is offered for obtaining of asymptotically steady motion in general case. The controlling force nature was defined. This method was applied to three cases with special restrictions on the bodys inertia moments, so-called generalized classical cases of Euler, Lagrange and Kovalevskaya. The numerical solution for the problem in Euler case was constructed. ——— A Lizorkin type theorem for Fourier series multipliers in regular systems Lyazzat Sarybekova Munaitpasov 7, Astana, 010010 Kazakhstan [email protected] A new Fourier series multiplier theorem of Lizorkin type is proved for the case 1 < q < p < ∞. The result is given for a general regular system and, in particular, for the trigonometrical system it implies an analogy of the original Lizorkin theorem. ——— Inverse-closedness problems in the stability of sequences in Banach Algebras Pedro A. Santos Departamento de Matem´ atica, Instituto Superior T´ecnico, Av Rovisco Pais, Lisboa, 1049-001 Portugal [email protected] We are concerned with the applicability of the finite sections method to operators belonging to the closed subalgebra of L(Lp (R)), 1 < p < ∞, generated by operators of multiplication by piecewise continuous functions in ˙ and operators of convolution by piecewise continuous R Fourier multipliers. The usual technique is to introduce a larger algebra of sequences, which contains the special sequences we are interested and the usual operator algebra generated by the operators of multiplication and convolution. There is a direct relationship between the applicability of the finite section method for a given operator and invertibility of the corresponding sequence in this algebra. But, contrarily to the C ∗ case and Banach analogue for Toeplitz operators, in our case several inverse-closedness problems must be solved. ———

Motion stabilisation of a solid body with fixed point

Smoothing effects for periodic NSE in critical Sobolev space

Zaure Rakisheva Al-Faraby Kazak National University, Almaty, Kazakstan zaure [email protected]

Ridha Selmi Department of Mathematics, Faculty of Sciences of Gabes, 6072, TUNISIA [email protected]

The problem of a solid body dynamics in the central Newton field of forces is considered. It is generalization

We prove smoothing effects for 3D incompressible Navier Stokes Equation for initial data belonging to critical

110

VI. Others 1

Sobolev Space H 2 (T3 ). Asymptotic behavior of the global solution when the time goes to +∞ is studied. ——— Large deviations and almost sure convergence Mariana Sibiceanu Gh.Mihoc-C.Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie Nr.13, Sector 5 Bucharest, 050711 Romania [email protected] In our setup, the Large Deviation Principle for a sequence P (n) of probabilities on a separable Banach space E, with a convex good rate function I is assumed, also the existence of the finite limits g(w) of the associated logarithmic moment generating function. We establish precise upper and lower bounds of the values that P (n) assigns almost sure in the weak and strong topology of E, respectively, determined by the amounts of the canonical dual product on E 0 × N , N being the nullifying set of the rate function. Also, we reveal the significance of the derivative of the function g(tw) of real t for the almost sure convergence, in the situation when g is Gateaux differentiable on (E 0 , t(E 0 , E)). ———

on the recent results of Z.Y.Huang [Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively pseudocontractive mappings without Lipschitzian assumptions, J.Math.Anal.Appl. 329(2007),935-947], Z.Y. Huang, F.W. Bu, M.A. Noor [On the equivalence of the convergence criteria between modified Mann-Ishikawa and multistep iteration with errors for strongly pseudocontractive operators, Appl. Math. Compt. 181(2006), 641-647], B.E.Rhoades, S.M.Soltuz [The equivalence between the convergences of Ishikawa and Mann iteration for an asymptotically non-expansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289(2004), 266-278] and B.E.Rhoades, S.M.Soltuz [The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal.58(2004),219-228] among others. ——— A characterization for multipliers of weighted Banach valued Lp (G)-spaces Serap Oztop Istanbul University, Faculty of Sciences, Istanbul, Vezneciler 34134 Turkey [email protected] Let G be a locally compact group, 1 < p < ∞. The aim of this paper is to characterize the multipliers of the weighted Banach valued intersection Lp (G) spaces as the space of multipliers of a certain Banach algebra.

The k- Model in Turbulence ——— Tanfer Tanriverdi Harran University, Faculty of Arts and Sciences, Department of Mathematics, Sanlurfa 63300, Turkey [email protected] We prove analytically the existence of self-similar solutions for the k- model arising in the evolution of turbulent bursts by employing the topological shooting technique where α > β with the some other conditions. The first author was supported by the Scientific and ¨ Technological Research Council of Turkey (TUBITAK). He is also thankful to the Oxford Center for Nonlinear PDE, and to the Mathematical Institute of the University of Oxford, for the hospitality they offered him during his visit. This is joint work with Bryce McLeod (Oxford). ——— The equivalence between modified Mann (with errors), Ishikawa (with errors), Noor (with errors) and modified multi-step iterations (with errors) for non-Lipschitzian strongly successively pseudo-contractive operators Johnson Olaleru Mathematics Department, University of Lagos, University of Lagos Road, Yaba, Lagos, Nigeria [email protected]

Stationary motion of the dynamical symmetric satellite in the geomagnetic field Karlyga Zhilisbaeva Al-Faraby Kazak National University, Almaty, Kazakstan [email protected] Stationary solutions of the system of the satellite’s motion equations are of special interest for the various problems of space researches, and first of all for the satellite’s magnetic stabilization. In the paper stationary motions of the equatorial magnetized dynamically symmetric satellite round the centre of mass on a circular orbit are considered. Strong magnets are placed on the satellite’s board. Perturbations are taken into account, caused by insignificant deviation of a satellite’s axis of dynamic symmetry and by magnetization of its cover. The equations of the satellite’s perturbed motion in Euler’s canonical variables are obtained. Conditions of stationary motion existence are defined, necessary and sufficient conditions of their stability are found with taking into account of small perturbations. ———

In this paper, the equivalence of the convergence between modified Mann(with errors), Ishikawa(with errors), Noor(with errors) and modified multistep iteration(with errors) is proved for generalized strongly successively pseudocontractive mapping without Lipschitzian assumption. Our results generalise and improve

111

Index Abate, M., 97 D’Abbico, M., 71 Abdenur, F., 97 Abdous, B., 47 Agranovich, M., 60, 63 Airapetyan, R., 106 Aksoy, U., 27 Alimov, S., 60 Aliyev, T., 23 Alkutubi, H., 107 Almeida, A., 51 Alves, J.F., 97 D’Ancona, P., 80 Applebaum, D., 90 Aripov, M., 81 Assal, M., 56 Aulaskari, R., 23, 56 Aydin, I., 51 Aykol, C., 51 Babych, N., 85 El Badia, A., 87 Bakry, D., 95 Ball, J., 15 Ballantine, C., 23 Bally, V., 90 Barsegian, G., 33 Barthe, F., 95 Baˇstinec, J., 101 Begehr, H., 3, 26, 27 Bellieud, M., 85 Berezansky, L., 100, 101 Berger, P., 97 Berglez, P., 27 Berlinet, A., 3, 47 Bernstein, S., 42 Besov, O., 51 de Bie, H., 36 Bisi, C., 36 Bobkov, S., 95 Bociu, L., 76 Bock, S., 42 Boichuk, A., 101 Bojarski, B., 3, 23 Bokhari, M., 107 Bolosteanu, C., 27 Boukrouche, M., 76 Boulakia, M., 77 Boulbrachene, M., 95 Boutet de Monvel, L., 15 Bouzar, C., 63 Boykov, I., 27 Boza, S., 52 Branden, P., 33 Britvina, L., 48 Brody, D., 90 Bruning, E., 3 Bucci, F., 76

Budochkina, S.A., 97 Burenkov, V., 3, 51, 60 Burns, K., 98 Buzano, E., 63

Dovbush, P., 24 Dragoni, F., 3, 95 Du Bau-Sen, 98 Du Jinyuan, 26, 28

Cˆ amara, C., 45 Capdeboscq, Y., 85 Cardon, D., 33 Carro, M., 52 Caruana, M., 91 Casado-Diaz, J., 86 Castro, L., 45 Catana, V., 63 Catania, D., 81 Cattaneo, L., 3 Cavalcanti, M., 77 Celebi, O., 3, 28 Cerejeiras, P., 36 Charalambides, M., 34 Chazottes, J.-R., 98 Chen Kuan-Ju, 81 Chen Qiuhui, 49 Chen Yi-Chiuan, 98 Cherdantsev, M., 86 Chiba, Y., 63 Chinchaladze, N., 28 Cho, D.H., 49 Choe, B.R., 56 Choi, Q-H., 71 Choulli, M., 88 Cohen, L., 63 Colombini, F., 71 Colombo, F., 36, 88 Cordero, E., 63 Coulembier, K, 37 Crisan, D., 3, 90, 91 Cruzeira, A.B., 91 Csordas, G., 33, 34

Ebert, M., 72 Ekincioglu, I., 52 El-Nadi, K., 93 Eller, M., 77 Elliott, N., 3 Elton, D., 60 Englis, M., 46 Eriksson, S.-L., 37 Erkip, A., 81

Dahmani, Z., 107 Dai Daoquin, 26 Dalla Riva, M., 23 Dallakyan, G., 52 Daoulatli, M., 77 Datt Sharma, S., 47 Dattori da Silva, P., 64 Daveau, C., 88 Davie, A., 91 Davies, B., 3, 16, 59 Davis, M., 91 Dehgan, M., 81 Del Santo, D., 71 Delgado, J., 64 Diblik, J., 100, 101 Doi, K., 81 Doli´canin, D., 49 Domingos Cavalcanti, V., 77 Domoshnitzky, A., 101 Donaldson, S., 16

Fang Daoyuan, 72 Farwell, R., 43 Faustino, N., 43 Federson, M., 102 Fei, M.-G., 37 Fern´ andez, A., 34 Ferreira, M., 37 Field, M., 98 Fisk, S., 34 Fokas, T., 43 Forni, G., 98 Fragnelli, G., 77 Franek, P., 37 Franssens, G.R., 37 Freitas, J., 98 Fujita, K., 48 Fujiwara, H., 49 Furutani, K., 64 Gaiko, V., 98 Gainova, I.A., 106 Galleani, L., 64 Galstyan, A., 72, 77 Garello, G., 64 Garetto, C., 65 Gauthier, P., 34 Gedif Ayele, T., 51 Geisinger, L., 60 Gentil, I., 95 Gentili, G., 37 Georgiev, S., 43 Georgiev, V., 72, 80 Ghergu, M., 82 Ghisa, D., 24 Ghisi, M., 73 Gil, J., 65 Gilbert, R., 3, 105, 106 Giorgadze, G., 28 Girela, D., 56 Gobbino, M., 73 Golberg, A., 24 Goldshtein, V., 53 Goltser, Y., 102 Golubeva, V.A., 86

113

Index Gonzalez, M.J., 57 Gramchev, T., 65 Gramsch, B., 65 Graubner, S., 28 Grigoryan, A., 95 Grudsky, S., 45, 46 Guliyev, V., 53 Gupta, S., 57 G¨ urlebeck, K., 42 G¨ urkanlı, A.T., 46 Gyongy, I., 91 Gy¨ ori, I., 102 Hairer, M., 92, 96 Hajibayov, M., 53 Halburd, R., 34 Hartung, F., 102 Harutyunyan, T., 61 Hasanov, A., 107 Hashemiparast, S.M., 107 Hasler, M., 107 Hebisch, W., 96 Helmstetter, J., 43 Herrmann, T., 73 Higgins, J.R., 48 Hinkkanen, A., 34 Hirosawa, F., 71, 73 Hogan, J., 43 H¨ ormann, G., 65 Huet, N., 96 Hughston, L., 92 Hunsicker, E., 65 Hurri-Syrvanen, H., 53 Hussain, A., 28 Ichinose, W., 65 Ilic, S., 106 Inglis, J., 3, 96 Israfilov, D., 24 Iwasaki, C., 66 Jacka, S., 92 Janas, J., 61 Johnson, J., 66 Jordan, T., 99 Jung, T., 73 Kaasalainen, M., 88 K¨ ahler, U., 44 Kalmenov, T.S., 30 Kalyabin, G., 53 Kamarulhaili, H., 108 Kamotski, I., 85 Kaptanoglu, T., 56, 57 Karapinar, E., 108 Karelin, O., 46 Karimov, E., 108 Karlovych, Y., 66 Karp, D., 24 Karp, L., 73 Karupu, O., 25 Katayama, S., 82 Kato, K., 35 Kats, B., 25 Katsnelson, V., 35 Kayar, Z., 102 Kelbert, M., 92

114

Kelly, C., 102 Kendall, W., 92 Kenig, C., 17 Khan, N., 108 Kheyfits, A., 29 Khimshiashvili, G., 29 Kilbas, A., 3, 48, 49 Kim, B.J., 50 Kisil, A., 38 Kisil, V., 42, 44 Kiss, G., 102 Kohr, G., 25 Kohr, M., 25 Kokilashvili, V., 17 Kolokoltsov, V., 92 Konjik, S., 50 Kontis, V., 3 Koroleva, A., 50 Krainer, T., 61, 66 Krausshar, R.S., 38, 44 Krump, L., 38 Kubo, H., 74, 82 Kucera, P., 82 Kurtz, T., 93 Kurylev, Y., 87 Kusainova, L., 54 Lamb, J., 3, 97 Lamberti, P., 61 Langerov´ a, M., 102 Lanza de Cristoforis, M., 3, 23, 25 Laptev, A., 3, 59 Lasiecka, I., 76 Lassas, M., 88 Lavicka, R., 38 Le, U., 83 Leandre, R., 44 Lebiedzik, C., 78 Lee, Y.L., 57 Leem, K.H., 88 de Leo, R., 66 Lerner, N., 17 Li Xue-Mei, 93 Libine, M., 39 Liflyand, E., 54 Lionheart, W., 89 Lions, P.-L., 21 Littman, W., 78 Liu Lixia, 108 Liu Yu, 66 Loreti, P., 78 Lucente, S., 83 Luna-Elizarrar´ as, M.E., 39 Luzzatto, S., 97 Lyons, T., 90, 93 Macia, F., 86 Maksimov, V., 78 Malliavin, P, 18 Mamedkhanov, J., 25 Mammadov, Y., 54 Manhas, J.S., 57 Manjavidze, N., 29 Markowich, P., 86 Marletta, M., 89

Marquez, A., 57 Marti, J.-A., 67 Martin, M., 39 Martinez, P., 78 Marusic-Paloka, E., 108 Masaki, S., 83 Matsuura, T., 89 Matsuyama, T., 74 Matthies, K., 87 Maz’ya, V., 18 van der Meer, J. C., 99 Michalska, M., 57 Migda, M., 103 Mijatovic, A., 93 Mimoiuni, A., 109 Mityushev, V., 32, 33 Mochizuki, K., 74 Mogultay, I., 83 Mohammed, A., 29 Mohiuddine, S.A., 109 Mokhonko, A., 61 Mola, G., 83 Molahajloo, S., 67 Morando, A., 67 Moura Santos, A., 54 Mursaleen, M., 109 Mussa, A., 109 Nakazawa, H., 74 Nam, K., 57 Naso, M.G., 79 Natroshvili, D., 67 Neklyudov, M., 96 Neustupa, J., 62 Nicol, M., 99 Nieminen, P., 58 Niknam, A., 99 Nishitani, T., 74 Nourouzi, K., 109 Nowak, M., 58 Oberguggenberger, A., 68 Ockendon, J., 20 Ohta, M., 83 Olaleru, J., 111 Oliaro, A., 68 Onchis, D., 48 Oparnica, L., 50 Opic, B., 54 Orelma, H., 39 Otto, F., 96 Ozawa, T., 80 ¨ Ozbekler, A., 103 Oztop, S., 111 Pandolfi, L., 79 Panti, G., 99 Papageorgiou, I., 3, 96 Papavasiliou, A., 93 Parvaneh Masoha, H., 110 Pau, J., 58 Pelekanos, G., 89 Pe˜ na Pe˜ na, D., 39 Peng, C., 99 Perotti, A., 39 Picard, R., 74 Pinotsis, D., 44

Index Pirzada, S., 110 Pistorius, M., 94 Pituk, M., 103 Pivetta, M., 75 Plaksa, S., 23, 26 Porter, M., 33 Potthast, R., 89 Prakash Sing, A., 35 Prykarpatsky, A., 35 Quiao Yuying, 40 R¨ atty¨ a, J., 56 Rachunkova, I., 103 Radkevich, E., 55 Rafeiro, H., 55 Rajabov, N., 29, 68 Rajabova, L., 30 Rakisheva, Z., 110 Rappoport, J., 50 Rasmussen, M., 100 Reinfelds, A., 103 Reissig, M., 3, 71, 75, 83 Ren Guangbin, 40 Renardy, M., 79 Reynolds, D., 103 Richard, S., 62 Roberto, C., 96 Rochon, F., 68 Rodino, L., 3, 62 Rodkina, A., 104 Rogosin, S., 32, 33 Rojas, E., 46 Rozovsky, B., 94 R˚ uˇziˇckov´ a , M., 104 Ruzhansky, M., 3 Ryan, J., 40 Ryan, M., 3 Ryser, M.D., 106 Sabadini, I., 36, 40 Sadyrbaev, F., 100 Safarov, Y., 59 Saito, J., 75 Saitoh, S., 3, 47, 48, 89 Saks, R., 30 Salac, T., 40 Samko, N., 55 Samko, S., 51, 55 Samoylova, E., 30 Santos, P.A., 110 Sanz-Sol´e, M., 94 Sarybekova, L., 110 Sasane, A., 58 Savchin, V.M., 104 Schmeidel, E., 104 Schmeling, J., 100 Schmidt, B., 87 Schnaubelt, R., 79

Schrohe, E., 69 Schulze, B.-W., 3, 18, 68 Sehba, B.F., 58 Seiler, J., 69 Selmi, R., 110 Senouci, K., 55 Serbetci, A., 55 Sergeev, A., 35 Sevroglou, V., 90 Sforza, D., 79 Shapiro, M., 40 Shaposhnikova, T., 69 Shibata, Y., 84 Shindiapin, A., 104 Shklyar, B., 105 Shpakivskii, V., 44 Shvartsman, I., 79 Sibiceanu, M., 111 Silva, A., 47 Silvestri, B., 62 Simon, L., 84 Skalak, Z., 84 Smyshlyaev, V., 85, 87 Sobolewski, P., 58 Soltanov, K., 84 Somberg, P., 40 Sommen, F., 36, 41 Soria, J., 56 Soucek, V., 41 Spagnolo, S., 84 Spr¨ oßig, W., 42, 45 Stanek, S., 105 Stefanov, A., 84 Stevic, S., 105 Stoppato, C., 41 Strohmaier, A., 62 Suragan, D., 30 Suzuki, O., 26 Suzuki, R., 75 Tahara, H., 69 Takemura, K., 48 Tamrazov, P., 23 Taneco-Hern´ andez, M.A., 82 Tanriverdi, T., 111 Tapdıgo˘ glu, M., 58 Taqi, I., 31 Taskinen, J., 58 Tasmambetov, Zh., 31 Teofanov, N., 69 Teta, A., 84 Tikhonov, S., 56 Todd, M., 100 Toft, J., 3, 69 Tolksdorf, J., 45 Tomilov, Y., 62 Toundykov, D., 79 Tovar, L.M., 59

Tretyakov, M., 94 Trooshin, I., 90 Trushin, B.V., 56 Tunaru, R., 94 Turunen, V., 70 Tvrdy, M., 105 Ueda, Y., 85 Uesaka, H., 75 Uhlmann, G., 20 Umeda, T., 62 ¨ Unal, M., 105 Upmeier, H., 47 Usoltseva, E., 94 Vajiac, A., 41 Valein, J., 80 Vasilevski, N., 45, 47 Vasilyev, V., 70 Vasy, A., 70 Vieira, N., 45 Visciglia, N., 85 Vlacci, F., 42 Vlasakova, Z., 42 van de Voorde, L., 42 Vukotic, D., 59 Wang Qiudong, 100 Wang Yufeng, 31 Wirth, J., 3 Witt, I., 70 Wong, M.W., 3, 62, 70 Wu Zhijian, 59 Wulan Hasi, 59 Xu Wen, 59 Xu Yongzhi, 3 Yagdjian, K., 75, 85 Yakubovich, S., 50 Yamamoto, M., 3, 20, 80, 87 Yang Congli, 59 Yildirir, Y.E., 26 Yordanov, B., 76 Youssfi, E.H., 26 Zafer, A., 100, 105 Zegarlinski, B., 3, 95 Zelinskiy, Y., 26 Zeren, Y., 56 Zhang Shangyou, 3 Zhang Zhongxiang, 32 Zhdanov, O.N., 32 Zhilisbaeva. K., 111 Zhong Shouguo, 32 Zhu Hongmei, 71 Zhu Kehe, 47, 59 Zolesio, J.-P., 80 Zorboska, N., 59

115