IJCMI-VOL2-NO1 _2010_ final

INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN: 0974-8652 Volume 2 / Number 1 / January 2010 --------------------------------------------------------------------------------------------------------------------

CONTENTS Research Papers

Page No

n in which The Least Part is k

6-12

Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature

13-21

A Note on r - partitions of K.Hanuma Reddy

C.N.B.Rao, V. Lakshmi Prasannam , T.Raja Rani

On Completely Prime And Completely Semi-Prime Ideals In Γ-Near-Rings

22-27

Satyanarayana Bhavanari, Pradeep Kumar T.V, Sreenadh Sridharamalle, Eswaraiah Setty Sriramula

A Unified Frame Work For Searching Digital Libraries Using Document Clustering

28-32

Shaik Sagar Imambi, Thatimakula Sudha

Reducibility For The Fiorini-Wilson-Fisk Conjecture

33-42

S.Satyanarayana, J.Venkateswara Rao, V.Amarendra Babu

Perceiving Plagiarism Using Weighted Window Approach- Performance Analysis

43-47

Bobba Veeramallu, T. Pavan Kumar, Prof.V.Srikanth, Prof.K.Rajasekhara Rao

System Representation For Software Architecture Recovery

48-55

Shaheda Akthar, Sk.MD.Rafi

Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15 and 318.15k. Narendra K, Narayanamurthy P & Srinivasu Ch

55-59

INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN: 0974-8652 Volume 2 / Number 2 / August 2010 --------------------------------------------------------------------------------------------------------------------

CONTENTS Research Papers Over view to Implementation of robotics with Voice recognition

Page No 60-64

Ande Stanly Kumar, Dr.K.Mallikarjuna Rao, Dr.A.Bala Krishna, B.Venkatesh,

Novelty of Extreme Programming

65-72

Ch.V.Phani Krishna, S.Satyanarayana, K.Rajasekhara Rao

Radiation Effects On Mhd Free Convection Flow Past A Semi-Infinite Moving Vertical Porous Plate With Soret And Dufour Effect

73-81

G.Venkata Ramana Reddy and Dr. A.Rami Reddy

Pareto Distribution - Some Methods Of Estimation R. Subba Rao, R.R.L. Kantam, G.Srinivasa Rao

82-92

INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI) ISSN: 0974-8652 www.ijcmi.webs.com Dr.S.Satyanarayana EDITOR-IN-CHIEF –IJCMI E-Mail: [email protected],[email protected] S.Rajeswara Rao MANAGING EDITOR S.Satish Babu CONSULTING EDITOR S.Srinivasara Rao PRODUCTION EDITOR HONARARY EDITORIAL TEAM 1. Prof.Frederick P. Brooks, Jr (U.S.A) 2. Prof. John Thompson(U.S.A) 3. Prof.S.R.Srinivasa Varadhan(U.S.A) 4. Prof.EDMUND M. CLARKE (U.S.A) 5. Prof. Dr Bhavanari Satyanarayana(INDIA) 6. Prof. Dr.J.Venkateswara Rao(INDIA) 7. Prof.Curtis T McMullen(U.K) 8. Prof.Terence Tao(U.S.A) 9. Dr.Kuncham Syam Prasad(INDIA) 10. Prof. Dr T V Pradeep Kumar(INDIA) 11. Prof.S.Pallam Setty(INDIA) 12.Prof.Dr.J.Saibabu(INDIA) 13.Dr.D.Naga Raju(INDIA) 14.Dr.B.V.Appa Rao(INDIA) 15.Prof.V.Kalyan Raju(INDIA) 16. Prof.Dr.G.Murugusundaramoorthy(INDIA) 17. T.Madhu Mohan(INDIA) 18. A.Sri Krishna Chaitanya(INDIA) 19. Dr. Sumanta Kumar Tripathy (INDIA) 20.Prof.Andreea S.Calude(NZ) 21.Dr.A.Rami Reddy (INDIA) 22.Dr.K.Srinivasa Rao (INDIA) 23.Dr.P.Srinivasa Rao (INDIA) About this Journal The INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI) is a refereed Mathematics &Computer science and Engineering journal devoted to publication of original research papers, research notes, and review articles, with emphasis on unsolved problems and open questions in mathematics &Computer science and Engineering. All areas listed on the cover of Mathematical Reviews, such as pure and applied mathematics. The INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI) is an international research journal, which publishes top-level work on computational aspects of

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Editorial We are pleased to announce the launch Fourth & Fifth issue of the INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI).The IJCMI is a refereed Mathematics &Computer science and Engineering journal devoted to publication of original research papers, research notes, and review articles, with emphasis on unsolved problems and open questions in mathematics &Computer science and Engineering. All areas listed on the cover of Mathematical Reviews, such as pure and applied mathematics & Computer science and Engineering. The IJCMI is an international research journal, which publishes top-level work on computational aspects of mathematics interface between applied mathematics, numerical computation, and applications of systems oriented ideas to the physical, biological, social, and behavioral sciences. It includes but not limited to areas of mathematics such as algebra (especially group theory), combinatorics (especially graph theory), geometry, number theory and numerical analysis; computational complexity; cryptology; symbolic and algebraic computation; optimization; the mathematical aspects of: models of computation; automata theory; categories and logic in computer science; proof theory; type theory; semantics of programming languages; process algebra and concurrent systems; specification and verification; databases; rewriting; neural nets and genetic algorithms; computational learning theory; theorem proving, Applied Physics, Solid State Physics, Nuclear Physics Theoretical Physics and more... IJCMI Journal publishes research articles and reviews within the whole field of Mathematical Sciences& Computer Science and Engineering, and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The IJCMI will be published three issues in a year. The journal will be reviewed by two independent reviewers. In addition, the Journal may occasionally publish special issues on various topics in the areas of Mathematics &Computer science and Engineering, book reviews, conference reports, letters to the honorary editors, conference announcements, etc. Finally, The Editor-in-Chief and Honorary Editors wish to congratulate the authors of the published papers in IJCMI. Satyans Publications to starting Three more International Journals such as International Journal of Applied Sciences & Engineering ideas (IJASE), International Journal of Artificial Intelligence Ideas (IJAII), International Journal of Entrepreneurship Ideas (IJEI) for the inspiration of IJCMI. Thanks are due to the members of the Editorial Board for their precious feedback and advice. We hope that the new International Journal of Computational Mathematical Ideas will serve Mathematics & Computer science engineering research community as well as and this journal will be main media of presenting ideas and research work in their area. Suggestions to improve our efforts in order to deliver a better journal to the authors, readers and subscribers of this journal will always be appreciated. Dr.S.Satyanarayana Editor-In-Chief –IJCMI E-Mail:[email protected] ISSN: 0974-8652

INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652

INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 6-12 (2010)

A NOTE ON

r − partitions OF n IN WHICH THE LEAST PART IS k

K.Hanuma Reddy@ Lecturer in mathematics, Hindu College, Guntur, A.P-522002, India, Mail: [email protected] ABSTRACT Partitions play an important role in Number Theory. It has wide applications in various fields. An attempt is made to develop a theorem on the number of r − partiions of positive integer n in which the least part is k , a reduction theorem on r − partiions and some more results on r − partiions are derived. Key words: p ( n ) , r − partiions , p r ( n ) , pr ( e ; n ) , pr ( o ; n ) and pr ( S ; n ) Subject classification: 11P81 Elementary theory of partitions.

pr ( S ; n ) :

pr ( S ; n ) is the number of

1. Introduction:

1.7

1.1 Partition: A partition of a positive integer n is a finite sequence of non-increasing positive

partitions of a positive integer n having r parts in which each part is the element of the set S.

r

integers λ1 , λ2 , ... , λr such that ∑ λi = n

2.

i =1

The number

λi

is called the

i th

 n − br    a 

pr ( S ; n ) = pr 

1.2 Partition function: The partition function p ( n ) is the number of partitions of n.

pr ( n) :

pr ( n )

and

e the set of positive integers. If a | n − br , then

partition. The partition is also denoted as n = (λ1 , λ2 , ... , λr ) .

1.4

r, n ∈ N ( r ≤ n)

S = {am + b | a ∈ N , b ∈ Z and m = 1,2,..., n} b

part of the

1.3 r-partition: A partition containing called r − partiions .

Let

Theorem:

other

wise

pr ( S ; n ) = 0 .

[ 2.1]

r parts is

Proof: All parts in r − partiions of n multiplied

is the number of

r − partiions of a positive integer n .

by a and added by b to get the partitions of n whose parts are elements of S .

Note: p ( n ) = p1 ( n) + p2 ( n ) + ... + pn ( n )

3. Theorem: Let r , n ∈ N and

1.5

pr ( o ; n ) :

pr ( o ; n ) is the number of

S = {am + b | a ∈ N , b ∈ Z and m = 1,2,..., n}

partitions of a positive integer n having r parts in which each part is odd number. 1.6

pr ( e ; n ) :

be the set of positive integers. Then, the highest least part of r − partiions of n in which the parts are the elements of the set S is

pr ( e ; n ) is the number of

a

partitions of a positive integer n having r parts in which each part is even number.

A Note On r - partitions Of

 n   ar + b  + b

n In Which The Least Part Is k 6


(

})

{

∴ pr ( S ; n ) - pr −1 S ; n − a (1) + b

[3.1] Proof:

= pr ( S ; n − ar )

λ1 , λ2 , ... , λr be the first,

Let

second,…,

5. Theorem: Let r , n , k ∈ N and

th

S = {am + b | a ∈ N , b ∈ Z and m = 1,2,..., n}

r parts of the r-partition of ‘n’ respectively. So n = (λ1 , λ2 , ... , λr )

be the set of positive integers. then, the number of r − partiions of n having the parts are

All the distinct r − partiions of n are arranged in such a way that all the parts and corresponding parts in each r − partiions are monotonically increasing.

elements of S with least part k is

(

pr −1 S ; n − ( k − 1) ar − {a + b}

  n   + b + 1   ar + b  

where 1 ≤ k ≤

If possible, let λ1 = a

)

 n   ar + b 

[5.1]

Since all the parts in each r − partiions are monotonically increasing order, the least possible value of each λ i for i = 2 to r is

Proof: Let λ1 , λ2 , ... , λr be the first, second,…,

r th parts of the

  n   + b + 1 . a    ar + b  

r − partiions of n respectively.

So

n = (λ1 , λ2 , ... , λr )

Then the sum of all parts in partition is

  n      + b  + 1 .   ar + b      n    But r a   + b  + 1 > n   ar + b   

All the distinct r − partiions of n are arranged in such a way that all the parts and corresponding parts in each r − partiions are monotonically increasing. Fixing λ1 = a (1) + b , the remaining value

This is contradiction.

n − {a (1) + b} of n can be expressed as the sum

r a

Hence

 n 

λ1 = a  +b  ar + b 

of the remaining r − 1 parts λ2 , λ3 , ... , λr in

is the highest

(

)

pr −1 S ; n − {a (1) + b} ways.

integer.

i,e The number of r − partiions in which the least part of the partition is λ1 = a (1) + b is

4. Theorem: Let r , n ∈ N and

S = {am + b | a ∈ N , b ∈ Z and m = 1,2,..., n}

pr −1 ( S ; n − {a + b} ) .

be the set of positive integers. Then prove that

(

})

{

pr ( S ; n) - pr −1 S ; n − a (1) + b

= pr ( S ; n − ar )

Fixing λ1 = a ( 2 ) + b , the remaining value n − {a ( 2 ) + b} of n can be expressed as the sum

[4.1]

of the remaining r − 1 parts

Proof: The number of r − partiions of n whose parts

(

Since all the parts in each r-partition are non

(

decreasing, pr − 2 S ; n − {a ( 3) + b ( 2 )}

equal to the number of ( r − 1) − partitions of

{

}

is

equal

to

the

of r − partiions of elements of S .

n − ar

whose parts are

(

{

})

pr −1 S ; n − a ( 2 ) + b

r − partiions .Then,

the number of the r − partiions in which the

number


)

r − partiions with λ1 = a ( 2 ) + b , λ2 = a (1) + b are to be eliminated from

whose parts are elements of

S and the number of r − partiions of n whose parts are elements of S with least part is

not a (1) + b

)

pr −1 S ; n − {a ( 2 ) + b} ways.

are elements of S with least part a (1) + b is

n − a (1) + b

λ2 , λ3 , ... , λr in



least part of the partition λ1 = a ( 2 ) + b is

(

 n  where 1 ≤ k ≤    ar + b 

})

{

pr −1 S ; n − a ( 2 ) + b -

(

Corollary 5.1: Let n, r , k ∈ N . Then the number

}) = pr −1 ( S ; n − a ( r − 1) − {a ( 2 ) + b}) = pr −1 ( S ; n − ar − {a + b}) {

pr − 2 S ; n − a ( 3 ) + b ( 2 )

of r − partiions of n with least part k is

n pr −1  n − ( k − 1) r − 1 where 0 ≤ k ≤   r  Proof: Put a = 1, b = 0 in [5.1]

Fixing λ1 = a ( 3) + b , the remaining value

{

}

n − a ( 3) + b of n can be expressed as the sum

λ2 , λ3 , ... , λr in

of the remaining r − 1 parts

( ) pr − 2 ( S ; n − {a ( 4 ) + b ( 2 )})

Corollary 5.2: Let n, r , k ∈ N . Then, the number of r − partiions of n having the parts are even numbers with least part k is

pr −1 S ; n − {a ( 3) + b} ways.

n pr −1  e ; n − 2 ( k − 1) r − 2  where 0 ≤ k ≤    2r 

r − partiions with

λ1 = a ( 3) + b , λ2 = a (1) + b and

(

pr − 2 S ; n − {a ( 5 ) + b ( 2 )}

Proof: Put

)

(

)

 n  where 0 ≤ k ≤    2r − 1  Proof: Put a = 2, b = −1 in [5.1]

)

from pr −1 S ; n − {a ( 3) + b} r − partiions . Then, the number of the r − partiions in which the least part of the partition λ1 = a ( 3) + b is

(

pr −1 S ; n − {a ( 3) + b}

6. Reduction theorem for pr ( S ; n) : Let r , n, k ∈ N and

)

(

- pr − 2 S ; n − {a ( 4 ) + b ( 2 )}

S = {am + b | a ∈ N , b ∈ Z and m = 1, 2,..., n} be the set of positive integers. then,

)

 n   ar +b   

-

(

[5.1]

are odd numbers with least part k is pr −1 o ; n − 2 ( k − 1) r − 1

r − partiions with λ1 = a ( 3) + b , λ2 = a ( 2 ) + b are to be eliminated

(

in

Corollary 5.3: Let n, r , k ∈ N . Then the number of r − partiions of n having the parts

-

pr −3 S ; n − {a ( 6 ) + b ( 3)}

a = 2, b = 0

)

 pr − 2 S ; n − {a ( 5 ) + b ( 2 )}   − pr −3 S ; n − {a ( 6 ) + b ( 3)} 

(

   

pr (S; n) =

) = pr −1 ( S ; n − a ( r − 1) − {a ( 3) + b}) − pr − 2 ( S ; n − a ( r − 2 ) − {a ( 5 ) + b}) = pr −1 ( S ; n − ar − {a ( 2 ) + b}) − pr − 2 ( S ; n − ar − {a ( 3 ) + b}) = pr −1 ( S ; n − a ( r − 1) − ar − {a ( 2 ) + b})

pr −1 ( S ; n − ( k − 1) ar − {a + b} )

k =1

[6.1] and  n  n  ar +b 

p(S; n) =

∑∑ r =1

pr −1 ( S ; n − ( k −1) ar − {a + b} )

k =1

[6.2] Proof: From [5.1] we can observe it

= pr −1 ( S ; n − 2 ar − {a + b} )

By induction we observe that the number of r − partiions of n having the parts are elements

Corollary 6.1: Prove that n n  r 

of S with least part k is pr −1 ( S ; n − ( k − 1) ar − {a + b} )


∑

p (n) =

∑∑ p ( n − ( k − 1) r − 1) r −1

r =1 k =1



Proof: Put a = 1, b = 0 in [ 6.2]

Proof: Case 1: Let n = 2m for m ∈ N

p(n) = p(2m) = p1 (2m) + p2 (2m) + p3 (2m) + ...

Corollary 6.2: Prove that

+ pm −1 (2m) + pm (2m) + pm +1 (2m) + ... + p2 m − 2 (2m) + p2 m −1 (2m) + p2 m (2m)

n n  2 r 

∑ ∑ p ( e ; n − 2 ( k − 1) r − 2)

p(e; n) =

= { p1 (2m −1)}

r −1

r =1 k =1

+{ p1 (2m − 2) + p2 (2m − 2)} +{ p1 (2m − 3) + p2 (2m − 3) + p3 (2m − 3)} + ...

Proof: Put a = 2, b = 0 in [ 6.2] Corollary 6.3: Prove that  n  n  2 r −1 

p(o; n) =

∑∑ r =1

+{ p1 (m +1) + p2 (m +1) + ... + pm−1 (m +1)} + p(m) + p(m −1) + ... + p(2) + p(1) +1

pr −1 ( o ; n − 2 ( k − 1) r − 1)

= { p1 (2m −1)}

k =1

+ { p1 (2m − 2) + p2 (2m − 2)}

Proof: Put a = 2, b = −1 in [ 6.2]

+ { p1 (2m − 3) + p2 (2m − 3) + p3 (2m − 3)} + ... + { p1 (m + 1) + p2 (m + 1) + ... + pm−1 (m +1)}

Theorem 7: If r , n ∈ N and r < n , then

+ { p1 (m) + p2 (m) + ... + pm−1 (m) + pm (m)}

n ≤2 r

pr (n) = p(n − r ) for

+ { p1 (m −1) + p2 (m −1) + ... + pm−1 (m −1)} + ... + { p1 (3) + p2 (3) + p3 (3)} + { p1 (2) + p2 (2)} + p1 (1) +1

Proof: n =2 r ⇒ n = 2r pr (n) = p1 (n − r ) + p2 (n − r ) + ... + pr (n − r )

Case:1

∑

+

∑

pi ( j ) +

i+ j =2m

∑

pi ( j ) +

pi ( j )

i+ j =2

2m

= 1+

= p(r )

∑

pi ( j )

i+ j=2

= p( n − r )

n

= 1+

n Let < 2 r

∑

pi ( j )

i+ j=2

Case 2: Let n = 2m + 1 for m ∈ N p (n) = p (2m + 1)

n Since r < n and < 2 r ⇒ r < n < 2r ⇒ 0< n−r < n pr (n) = p1 (n − r ) + p2 (n − r ) + ...

= p1 (2m + 1) + p2 (2m + 1) + p3 (2m + 1) + ... + pm (2m + 1) + pm +1 (2m + 1) + pm + 2 (2m + 1) + ... + p2 m −1 (2m + 1) + p2 m (2m + 1)

+ pn −r (n − r ) + pn− r +1 (n − r ) + ... + pr (n − r )

+ p2 m +1 (2m + 1)

= p1 (n − r ) + p2 (n − r ) + ... + pn −r (n − r ) + 0 + ... + 0 = p( n − r ) Hence

pi ( j ) +…

i + j = 2 m −1

i + j =3

= p1 (r ) + p2 (r ) + ... + pr (r )

Case:2

∑

=1+

Let

pr (n) = p(n − r ) for

n ≤2 r

Theorem 8: Let n, i , j ∈ N , then n

p ( n) = 1 +

∑

pi ( j )

i+ j=2




= { p1 (2m)}

+ pm +1 (n) +…+ p2 m -2 (n) + p2 m −1 (n) + p2 m (n)

+ { p1 (2m − 1) + p2 (2m − 1)} + { p1 (2m − 2) + p2 (2m − 2) + p3 (2m − 2)} + ...

= p1 (2m) + p2 (2m) +…+ pm −1 (2m) + pm (2m)

+ { p1 (m + 1) + p2 (m + 1) + ... + pm (m + 1)} +

+ pm +1 (2m) +…+ p2 m -2 (2m) + p2 m −1 (2m)

p(m) + p(m − 1) + ... + p(2) + p(1) + 1

+ p2 m (2m)

= { p1 (2m)}

= pm −1 (2m + m − 1) + p(m) + p(m -1) +…+ p(2)

+ { p1 (2m − 1) + p2 (2m − 1)}

+ p(1) +1

+ { p1 (2m − 2) + p2 (2m − 2) + p3 (2m − 2)} + ... + { p1 (m + 1) + p2 (m + 1) + ... + pm ( m + 1)}

=1+ pm −1 (2m + m -1) + { p(1) + p(2) + ... + p(m)}

+ { p1 (m) + p2 (m) + ... + pm ( m)}

m

+ { p1 (m − 1) + p2 ( m − 1) + ... + pm −1 (m − 1)} + ... +{

= 1+ pm −1 (3m -1) +

p1 (3) + p2 (3) + p3 (3)}

i =1

+ { p1 (2) + p2 (2)} + p1 (1) + 1

∑

=1+

∑

pi ( j ) +

i + j = 2 m +1

 3n  = 1+ p n  -1 + −1 2  2 

pi ( j ) +…+

i + j =2m

∑

pi ( j )

i+ j=2

∑

pi ( j )

i+ j=2

p ( n) = 1 +

∑

pi ( j )

= p1 (2m +1) + p2 (2m +1) +…+ pm (2m +1)

i+ j=2

Corollary 8.1: Let n be a natural number, then

p(n) − p(n − 1) =

i =1

+ p2 m −1 (n) + p2 m (n) + p2 m +1 (n)

n

Hence

∑ p(i)

p ( n) = p1 (n) + p2 (n) +…+ pm −1 (n) + pm (n) + pm +1 (n) + pm + 2 (n) +…

n

= 1+

n 2

If n is odd Let n = 2m + 1 for m ∈ N

2 m +1

= 1+

∑ p(i)

∑

+ pm+1 (2m +1) + pm+2 (2m +1) +…

pi ( j )

+ p2m-1 (2m +1) + p2 m (2m +1) + p2m+1 (2m +1)

i+ j=n

= pm (2m + m +1) + p(m) + p(m -1) +… + p(2) + p(1) +1

n

∑

Proof: Since p(n) = 1 +

pi ( j )

m

= 1+ pm (3m + 1) + ∑ p (i )

i+ j=2

i =1

n −1

∑

∴ p(n − 1) = 1 +

pi ( j )

n −1

 3n -1  2 = 1+ p n -1  + ∑ p (i ) 2  i =1 2 

i + j =2

Hence

p(n) − p(n − 1) =

∑

pi ( j )

i+ j=n

Hence

Theorem 9: If n ∈ N then

Theorem 10: m  m + 1 If   = 1 ,then   = 1 for m, r ∈ Z r  r +1 

Proof: If n is even Let n = 2m for m ∈ N

m Proof: Since   = 1 r ⇒ r ≤ m < 2r ⇒ r + 1 ≤ m + 1 < 2r + 1

p ( n) = p1 (n) + p2 (n) +…+ pm −1 (n) + pm (n) A Note On r - partitions Of



Proof: From theorem 11 pr (m) + pr (m + r + 1) + ... + pr (m + tr + t )

Since r + 1 ≤ m + 1 < 2r + 1 and 2r + 1 ≤ 2r + 2 ∴ r + 1 ≤ m + 1 < 2r + 2 m +1 ⇒1≤ T∞ and takes

T0 < T∞ , while for gases it is vice versa. Irrespective of the values of T0 positive values when

and T∞ , zero value of γ µ corresponds to constant viscosity case. In this paper, solutions are found for the values -1, 0, and 1 of γ µ .

viscosity) are in very good agreement with those of in ref. [2]. Also our results for C =1, γ µ =0 and λ =0 (i.e., no magnetic field,

The mixed convection parameter RP takes positive values for assisting flow and negative


16


case ( λ = 0 ) while it increases with

constant viscosity and isothermal plate) agree very well with those of ref. [4 ]. IV. DISCUSSION OF THE RESULTS: Qualitatively interesting results related to the shear stress, heat transfer coefficient, velocity and temperature are presented, some of them in the form of tables I,II and others in the form of figures 2 to 11. Quantities such as the Nusselt number and drag coefficient can be readily obtained from the heat transfer coefficient and skin friction. Variations in f ′ ( 0 ), f ′′ ( 0 ) and '−θ ′( 0 ) ’ for positive values of RP are presented in table I. Skin friction f ′′ ( 0 ) can be observed to be negative for positive values of RP for all values of the other parameters under consideration. Absolute value of f ′′ ( 0 ) decreases with increasing values of RP and

γ µ when

λ

takes positive values. In the following, more attention is paid to the discussion of the dual solutions of the opposing flow case. For a given value of RP, the solution corresponding to a relatively larger value of f ′′ ( 0 ) is referred to as the upper solution and the one corresponding to a smaller value of f ′′ ( 0 ) as the lower solution. The changes in skin friction with negative values of the mixed convection parameter RP are shown in figures 2(a),2(b) for different values of the parameters C, λ and γ µ . The corresponding changes in heat transfer coefficient are shown in figures 3(a),3(b) respectively. One curve each corresponding to ref. [2] are presented in figures 2(a),2(b) and one curve corresponding to ref.[4] in fig.3(a). From the figures the range of values of RP over which solutions exist can be seen to be more when fluid viscosity is taken to be temperature dependent than when it is constant. Similarly the range is more in the presence of magnetic field than in its absence. In the isothermal case, when viscosity is a constant as well as variable and in the presence as well as absence of magnetic field, f ′′ ( 0 ) is observed to be positive. For C = 0 . 5 , λ = 0 , γ µ = 0 single

γ µ while it increases with

increasing values of C..

solution

exists

for − 2.1 ≤ RP ≤ 0 ,

dual

− 2.7 ≤ RP ≤ − 2.0 and no solution for RP ≤ − 2.7 . Like the skin solutions exist for

friction f ′′ ( 0 ) , the heat transfer coefficient also takes positive values when λ = 0 (see figures 2(a) and 3(a)). Except for magnitude, behaviour of skin friction and heat transfer coefficient when γ µ = 1 are similar to the corresponding ones when γ µ = 0.

Heat transfer coefficient ‘ − θ ′(0) ’ takes increasing values with increasing values of RP and C while it takes decreasing values with increasing values of γ µ . In table II are presented the ranges of values of RP for which either no solution, a single solution or dual solutions exist. The range of values can be seen to be more for gasses than for liquids. The range can also be seen to increase with increasing values of λ . The range decreases with increasing values of γ µ in the isothermal Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature

17



18


example λ = 0.05) , f ′′ ( 0 ) is observed to take both positive and negative values with changing negative values of RP, and dual solutions exist for a wide range of values of RP. For C =0.5, λ =0.05 and γ µ =0 the range over which solutions exist is − 2 . 7 ≤ RP ≤ − 0 . 1 . Like the skin friction, heat transfer coefficient also takes both positive and negative values with changing values of RP, when λ = 0.05 . Plots of shear stress for the upper and lower solutions for different values of the parameters are shown in the figures 4(a),4(b),5(a) and 5(b). Considerable differences can be noticed in the behaviour of the shear stress for the upper and lower solutions (see figures 4(a) and 4(b)). Curves of figure 4(a) correspond to those for liquids while those of figure 4(b) correspond to gases. From figures 5(a) and 5(b) and also from numerical results, it can notice that, for positive values of RP, the shear stress at the plate becomes negative thereby indicating separation of the boundary layer. Fluid velocity profiles for the two solutions of the opposing flow case are presented in figures 6(a), 6(b) (for liquids) and in figures 7(a) and 7(b) (for gases). It can be observed that the hydrodynamic boundary layer thickness of the lower solution is much larger than that of the upper solution. Qualitative differences between the two solutions can also be observed in the vicinity of the plate. Fluid temperature profiles corresponding to the upper and lower solutions are presented in figure 8 for certain negative values of RP. It can be noticed that thermal boundary layer thickness of the lower solution is much larger than that of the upper solution. Variations in the lower solutions with changing values of the parameters are significant than those in the other solution. From figure 9, f ′′ ( 0 ) can be seen to

Unlike in the isothermal case, when the plate temperature is variable (for


19


results of the two works for C = 1, γ µ =0.

λ =0

and

VI.CONCLUSIONS: Assisting flow (RP Positive) 1. For fixed values of λ , C , γ µ , as RP increases there is an increase in the magnitudes of f ′′ ( 0 ) and ‘ − θ ′(0) ’. Increase in ‘RP’ can mean increase in the buoyancy force and and this can cause an increase in fluid velocity and hence an increase in the skin friction and heat transfer coefficient. 2. For fixed values of

γ µ , RP & λ ,

f ′′ ( 0 ) as well as ‘ − θ ′(0) ’ decrease as ‘C’ decreases ( i.e., as the intensity of the magnetic field increases). Opposing flow (RP Negative) 1. ‘ − θ ′(0) ’ decreases with diminishing values of RP. This may be due to the buoyancy force that works against the flow and hence the retardation in the heat transfer process. 2. f ′′ ( 0 ) takes positive as well as negative values for certain values of the parameters. Positive values of f ′′ ( 0 ) imply that the fluid exerts a dragging force on the surface and negative values imply the opposite.

diminish as

3. Dual solutions exist for certain values of RP. Significant differences are observed between upper and lower solutions. Ranges of values of RP for which unique solution or dual solutions exist is observed to change considerably with changing values of the parameters.

γ µ changes from 0 to -1, and

takes smaller values in the absence of the magnetic field. From figures 10,11 heat transfer coefficient ( − θ ′(0) ) and slip velocity (

f ′(0) ) can be seen to increase as

γµ

ACKNOWLEDGEMENTS

changes from 0 to -1, and both the quantities assume larger values in the absence of magnetic field. V.COMPARISION WITH AVAILABLE RESULTS: Results of the present analysis agree well with appropriate results of references[2] and [4]. In figures 2 and 3 of our analysis are shown curves for C = 1, λ =0 and γ µ =0 which coincide with

T.Raja Rani wishes to thank the authorities of Sri Vishnu Engineering College for Women and also authorities of S.R.K.R Engineering College for their encouragement and also for providing the facilities for research. T.Raja Rani also conveys her thanks to Mr.K.S.Sreenivasa babu of S.R.K.R.Engineering College for his help in numerical computations.

those presented in references [2] and [4]. Comparison of numerical results of our analysis with those presented in table 2 of reference [4] has revealed excellent agreement between the


20


[10] White,F.M, Viscous Fluid Flow, McGrawHill Inc., New York, 1974.

REFERENCES [1] Acharya.M, Dash.G.C and Singh.L.P., Magnetic field effects on the free convection and mass transfer flow through porous medium with constant suction and constant heat flux, Indian J pure appl. Math., vol.31(1),pp.118,2000.

[11] Vafai,K., Hand Book of Porous Media,2nd Ed, Taylor & Francis, New York.pp.379381,2005.

[2] Aly,E.H, Elliot,L, Ingham,D.B, Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium, European Journal of Mechanics B/Fluids., vol.22, pp. 529-543, 2003. [3] Barletta.A, Lazzari.S, Magyare.E and Pop.I, Mixed convection with heating effects in a vertical porous annulus with a radially varying Magnetic field, Int. J. Heat. Transfer., vol.51, pp.5777-5784, 2008. [4] Chin,K,E, Nazar,R, Arifin,N.M, Pop,I, Effect of Variable Viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium, International Communications in Heat and Mass Transfer., vol.34,pp. 464-473, 2007. [5]Gorsan.T, Revnic.C, Pop.I and Ingham.D.B, Magnetic field and internal heat generation effects on the Free convection in a rectangular cavity filled with a porous medium, International Communications in Heat and Mass Transfer, vol 52, pp.1525-1533,2009. [6] Lai,F.C, and Kulacki,F.A,(1990), The Effect of Variable Viscosity on Convective Heat Transfer along a Vertical Surface in a Saturated Porous medium, Int.J.Heat Mass Transfer. vol.33, pp.1028- 1031,1990. [7] Lakshmi Prasannam.V, Raja Rani,T and C.N.B.Rao, Free convection in a porous medium with magnetic field, variable physical properties and varying wall temperature, accepted for publication in International Journal of Computational Mathematical ideas. [8] Nield.D.A, Bejan.A, Convection in Porous Media, Third ed., Springer, New York,2006. [9] Sobha.V.V and Ramakrishna.K, Convective Heat Transfer Past a Vertical Plate Embedded in a Porous Medium with an Applied Magnetic Field, IE(I) Journal-MC, 84, pp.133-134, 2003. Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature

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INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN: 0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 22-27 (2010)

ON COMPLETELY PRIME AND COMPLETELY SEMI-PRIME IDEALS IN Γ-NEAR-RINGS Satyanarayana Bhavanari@, Pradeep Kumar T.V.#, Sreenadh Sridharamalle$, Eswaraiah Setty Sriramula^ @

Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510,A.P. India. e-mail: [email protected] # Department of Mathematics, ANU College of Engineering, Acharya Nagarjuna University $ Department of Mathematics, S.V. University, Tirupathi, A.P. India. ^ Department of Mathematics, SGS College, Jaggaiahpet, Krishna DT., A.P., India. ABSTRACT In this paper we considered the algebraic system Γ-near-rings that was introduced by Satyanarayana. “Γ-near-ring” is a more generalized system than both near-ring and gamma ring. The aim of this short paper is to study and generalize some important results related to the concepts: completely prime and completely semi-prime ideals, in Γ-near-rings. We included examples when ever necessary. AMS Subject Classification: 16 D 25, 16 Y 30, and 16 Y 99 Key Words: Gamma near-ring, γ-ideals, γ-semi prime ideals 1.1 Definition: An algebraic system (N, +, .) is called a near-ring (or a right near-ring) if it satisfies the following three conditions: (i) (N, +) is a group (not necessarily Abelian); (ii) (N, .) is a semigroup; and (iii) (n1 + n2)n3 = n1n3 + n2n3 (right distributive law) for all n1, n2, n3 ∈ N.

1. Introduction In recent decades interest has arisen in algebraic systems with binary operations addition and multiplication satisfying all the ring axioms except possibly one of the distributive laws and commutativity of addition. Such systems are called “Near-rings”. A natural example of a near-ring is given by the set M(G) of all mappings of an additive group G (not necessarily abelian) into itself with addition and multiplication defined by (f + g)(a) = f(a) + g(a); and (fg)(a) = f(g(a)) for all f, g ∈ M(G) and a ∈ G. The concept Γ-ring, a generalization of ‘ring’ was introduced by Nobusawa [ 4 ] and generalized by Barnes [1]. Later, Satyanarayana [8, 9], Satyanarayana, Pradeep Kumar & Srinivasa Rao [14] also contributed to the theory of Γ-rings. A generalization of both the concepts near-ring and the Γ-ring, namely Γnear-ring was introduced and studied by Satyanarayana [ 9, 11, 12 ], and later studied by several authors like: Booth [2 ], Booth & Groenewald [ 3], Syam Prasad [16].

In general n.0 need not be equal to 0 for all n in N. If a near-ring N satisfies the property n.0 = 0 for all n in N, then we say that N is a zerosymmetric near-ring. 1.2. Definitions: A normal subgroup I of (N, +) is said to be (i) a left ideal of N if n(n1 + i) – nn1 ∈ I for all i ∈ I and n, n1 ∈ N (Equivalently, n(i + n1) – nn1 ∈ I for all i ∈ I and n, n1 ∈ N); (ii) a right ideal of N if IN ⊆ I; and (iii) an ideal if I is a left ideal and also a right ideal.

Now, we collect some existing fundamental definitions and results which are to be used in later sections.

If I is an ideal of N then we denote it by I ⊴ N.

On Completely Prime and Completely Semi-prime Ideals in Γ-near-rings

22


1.3. Definitions: (i) An ideal (left ideal) P of N (with P ≠ N) is said to be a prime (prime left) ideal of N if it satisfies the condition: I, J are ideals (left ideals) of N, IJ ⊆ P, implies I ⊆ P or J ⊆ P.

f1g1f2 + f1g1f3. To see this, fix 0 ≠ z ∈ G and u ∈ X. Define Gu: G → X by gu(x) = u for all x ∈ G and fz:X → G by fz(x) = z for all x ∈ X. Now for any two elements f2, f3 ∈ M, consider fzgu(f2+ f3) and fzguf2 + fzguf3. For all x ∈ X, [fzgu(f2+ f3)] (x) = fz[gu(f2(x) + f3(x))] = fz(u) = z and [fzguf2 + fzguf3](x) = fzguf2(x) + fzguf3(x) = fz(u) + fz(u) = z + z. Since z ≠ 0, we have z ≠ z + z and hence fzgu(f2+ f3) ≠ fzguf2 + fzguf3. Thus we have that M is a Γ-near-ring which is not a Γ- ring.

(ii) An ideal P of N is said to be completely prime if for any a, b ∈ N, ab ∈ P ⇒ a ∈ P or b ∈P (iii) An ideal S of N is said to be semi-prime if for any ideal I of N, I2 ⊆ S implies I ⊆ S. (iv) An ideal S of N is said to be completely semi-prime ideal if for any element a ∈ N, a2∈ S implies either a ∈ S.

1.7. Definition: Let M be a Γ-near-ring. Then a normal subgroup I of (M, +) is called (i) a left ideal if aα(b + i) - aαb ∈ I for all a, b ∈ M, α ∈ Γ and i ∈ I; (ii) a right ideal if iαa ∈ I for all a ∈ M, α ∈ Γ, i ∈ I; and (iii) an ideal if it is both a left and a right ideal.

1.4. Definitions: (i) For any proper ideal I of N, the intersection of all prime(Completely Prime, respectively) ideals of N containing I, is called the prime(Completely Prime, respectively) radical of I and is denoted by P-rad(I) (C-rad(I) , respectively).

Let M be a Γ-Near-ring and α ∈ Γ. Satyanarayana [ 11 ] defined a binary operation “*α” on M by a *α b = aαb for all a, b ∈ M. Then (M, +, *α) is a near-ring. So we may consider every element α ∈ Γ as a binary operation on M such that (M, +, *α) is a near-ring. Also for any α, β ∈ Γ, we have (a *α b) *β c = a *α (b *β c) for all a, b, c ∈ M.

(ii) The Prime (Completely Prime, respectively) radical P-rad(0)(C-rad(0) , respectively) is also called as Prime (Completely Prime, respectively) radical of N and we denote this by P-rad(N) (C-rad(N), respectively). For some other fundamental definitions and results, we refer Pilz [5], Satyanarayana [9, 13], Satyanarayana and Syam Prasad [15].

Conversely, if (M, +) is a group and Γ is a set of binary operations on M satisfying

1.5. Definition: (Satyanarayna [9, 11, 12, 15]): Let (M, +) be a group (not necessarily Abelian) and Γ be a non-empty set. Then M is said to be a Γ-near-ring if there exists a mapping M × Γ × M → M (the image of (a, α, b) is denoted by aαb), satisfying the following conditions: (i) (a + b)αc = aαc + bαc; and (ii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ.

(i) (M, +, * ) is a near-ring for all * ∈ Γ; and (ii) (a *1 b) *2 c = a *1 (b *2 c) for all a, b, c ∈M and for all *1, *2 ∈ Γ, then (M, +) is a Γ-nearring. 1.8. Remark: (i) If *α, *β are operations on M with a *α b = a *β b for all a, b ∈ M, then the functions *α, *β are one and the same. So in this case, we have *α = *β.

M is said to be a zero-symmetric Γ-near-ring if aα0 = 0 for all a ∈ M and α ∈ Γ, where 0 is the additive identity in M.

(ii) Suppose that (M, +) is a Γ-near-ring and also (M, +) is a Γ*-near-ring with the following property: α ∈ Γ implies there exists β ∈ Γ* such that a *α b = a *β b for all a, b ∈ M. Then we may consider this case as α = β and so Γ ⊆ Γ*.

A natural example of Γ-near-ring is given below: 1.6 Example (Satyanarayana [11]): Let (G, +) be a non - abelian group and X be a non-empty set. Let M = {f / f: X → G}. Then M is a group under point wise addition.

1.9. Definition: Let (M, +) be a group. A Γnear-ring M is said to be a maximal Γnear-ring if M cannot be a Γ*-near-ring for any Γ ⊂ Γ* (Here it is assumed that the restriction of the mapping M × Γ* × M → M to M × Γ × M is the mapping M × Γ × M → M).

Since G is non-abelian, then (M, +) is non abelian. Let Γ be the set of all mappings of G into X. If f1, f2 ∈ M and g ∈ Γ, then, obviously, f1gf2 ∈ M. But f1g1(f2 +f3) need not be equal to


23


1.14. Definition: Let I be an ideal of N. Then a prime (completely prime, respectively) ideal of N containing I is called a minimal prime (minimal completely prime, respectively) ideal of I if P is minimal in the set of all prime (completely prime, respectively) ideals containing I.

1. 10. Theorem (Th. 1.3 of Satyanarayana [ 11 ]): Let (M, +) be a group and P = {* / * is a binary operation on M such that (M, +, *) is a near-ring and M * M = M}. Then there exists a partition {Γi / i ∈ I} of P such that (M, +) is a maximal Γi-near-ring for all i ∈ I. Conversely, if {Γj}j∈J be a disjoint collection of sets such that (M, +) is a maximal Γj-near-ring for each j ∈ J with M * M = M for all * ∈ Γj and for all j ∈ J,

UΓ

1.15. Theorem ( Th. 1.4 of [ 13 ]): Let I be an ideal of a near-ring N. Then I is a semi-prime ideal of a N ⇔ I is the intersection of all minimal prime ideals of N ⇔ I is the intersection of all prime ideals containing I.

j

j∈J

then ⊆ P. Moreover (Say property B: If Γ is a nonempty set such that (M, +) is a maximal Γ-near-ring implies Γ = Γj for some j ∈ J).

UΓ If property B holds, then

j∈J

1.16. Theorem (Cor. 5.1.10 of Satyanarayana [ 9 ]) : Let N be a near-ring and A an ideal of N. Then A is completely semi-prime ideal if and only if A is the intersection of completely prime ideals of N containing A.

j

= P.

1.11. Definition: Let M be a Γ-near-ring and γ ∈ Γ. A subset A of M is said to be a γ-ideal of the Γ-near-ring M if A is an ideal of the nearring (M, +, *γ).

1.17. Theorem (Theorem 2.2(b) of Satyanarayana [ 13]): An ideal P of N is prime and completely semi-prime ⇔ it is completely prime.

1. 12. Observations: (i) Let (N, +, *) be a nearring which is not zero symmetric. Then there exists a ∈ N such that a * 0 ≠ 0. Write Γ = {*}. Then N is a Γ-near-ring with aα0 ≠ 0 for some a ∈ N, α ∈ Γ. Therefore, in this case, N cannot be a zero symmetric Γ-near-ring. (ii) Let M be a Γ-near-ring and (I, +) a normal subgroup of (M, +). It is clear that I is an ideal of the Γ-near-ring M if and only if I is an ideal of the near-ring (M, +, *α) for all α ∈ Γ. In other words, I is an ideal of the Γ-near-ring M if and only if I is a γ-ideal of M for all γ ∈ Γ.

1.18. Theorem (Lemma 2.7 of Satyanarayana [ 13 ]): Every minimal prime ideal P of a completely semi-prime ideal I is completely prime. Moreover, P is minimal completely prime ideal of I. 1.19.Theorem (Theorem 2.8 of Satyanarayana [ 13 ]): Let I be a completely semi-prime ideal of N. Then I is the intersection of all minimal completely prime ideals of I. 1.20. Theorem (Theorem 2.9 of Satyanarayana [ 13 ]): If P is a prime ideal and I is a completely semi-prime ideal, then P is minimal prime ideal of I if and only if P is minimal completely prime ideal of I.

(iii) Let M be a Γ-near-ring. For any Γ* ⊆ Γ we have that M is a Γ*-near-ring. Every ideal I of the Γ-near-ring M is also an ideal of Γ*-nearring M, but the converse need not be true.

1.21. Corollary: (Corollary 2.10 Satyanarayana [ 13 ] ): If I is a completely semi-prime ideal of N, then I is the intersection of all completely prime ideals of N containing I.

To see this, we observe the following example. 1. 13. Example: Consider G = {0, 1, …, 7} the group of integers modulo 8 and a set X = {a, b}. Write M = {f / f: X → G such that f(a) = 0} = {fi / 0 ≤ i ≤ 7} where fi: X → G is defined by fi(b) = i, fi(a) = 0 for 0 ≤ i ≤ 7. Consider two mappings g0, g1 from G to X defined by g0(i) = a for all i ∈ G, and gi(i) = a if i ∉ {0, 3}, g1(3) = g1(7) = b. Write Γ = {g0, g1} and Γ* = {g0}. Now M is a Γ-near-ring and also Γ*-near-ring. Now Y = {f0, f2, f4, f6} is an ideal of the Γ*near-ring M but not an ideal of the Γ-near-ring M (since f2 ∈ Y and f3g1(f1 + f2) f3g1f1 = f3 ∉ Y).

2. γ-Completely Prime and γ-Completely Semi-prime γ-Ideals. Throughout this section we consider only zerosymmetric right near-rings, and M denotes a Γnear-ring. 2.1 Definition: Let γ ∈ Γ. A γ-ideal I of M is said to be (i) γ-completely prime if a, b ∈ M, aγb ∈ I ⇒ a ∈ I or b ∈ I.


24


(ii) γ-completely semi-prime if a ∈ M, aγa ∈ I ⇒ a ∈ I.

respect to γ ∈ Γ) if AγB ⊆ P for any two γideals A, B of M implies A ⊆ P or B ⊆ P.

2.2 Note: Let M be a Γ-near-ring and γ ∈ Γ. Write N = M. Now (N, +, *γ) is a near-ring. Let I be a γ-ideal of M.

(ii). A γ-ideal S of a Γ-near-ring M is said to be a γ-semi-prime γ-ideal of M (with respect to γ ∈ Γ) if AγA ⊆ S for any γ-ideal A of M implies A ⊆ S.

(i) I is a γ-completely prime γ-ideal of M if and only if I is a completely prime ideal of the nearring (N, +, *γ).

2.8 Note: Let P be an γ-ideal of a Γ-near-ring M and γ ∈ Γ. Then we have the following:

(ii) I is a γ-completely semi-prime γ-ideal of M if and only if I is a completely semi-prime ideal of the near-ring (N, +, *γ).

(i). P is a γ-prime γ-ideal of the Γ-near-ring M ⇔ P is a prime ideal of the near-ring (M, +, *γ).

2.3 Remark: Every γ-completely prime γ-ideal of M is a γ-completely semi-prime γ-ideal of M. [Verification: Let I be a γ-completely prime γideal of M. Let a ∈ M. Suppose aγa ∈ I. Since I is γ-completely prime, we have that a ∈ I. Thus I is a γ-completely semi prime γ-ideal of M.]

(ii). P is a γ-semi-prime γ-ideal of the Γ-nearring M ⇔ P is semi-prime ideal of the near-ring (M, +, *γ). (iii).Suppose that S is a γ-ideal of M. Then (by Theorem1.15) we have that S is γ-semi-prime γideal of M ⇔ S is the intersection of all γ prime ideals P of M containing S.

2.4 Corollary: Let M be a Γ-near-ring, γ ∈ Γ and A be a γ-ideal of M. Then A is γcompletely semi-prime γ-ideal if and only if A is the intersection of γ-completely prime γideals of M containing A.

The following corollary follows from Theorem 1.17. 2.9 Corollary: A γ-ideal P of a Γ-near-ring M is γ-prime and γ-completely semi-prime ⇔ it is γ-completely prime.

Proof: A is γ-completely semi-prime γ-ideal ⇔ A is completely semi-prime ideal of the near-ring (M, +, *γ) (by Remark 2.3 ) ⇔ A is the intersection of all completely prime ideals of the near-ring (M, +, *γ) containing A (by Theorem 1.16) ⇔ A is the intersection of all γcompletely prime γ-ideals of M containing A. The proof is complete.

2.10 Definitions: Let I be a γ-ideal of a Γ-nearring M for γ ∈ Γ. I is called a minimal γ-prime (γ-Completely Prime, respectively) γ-ideal of M if it is minimal in the set of all γ-prime (γ-Completely Prime, respectively) γ-ideals containing I.

2.5 Definition: Let A be a proper ideal of M. The intersection of all γ-completely prime γideals of M containing A of M, is called as the γ-completely prime radical of A and it is denoted by C-γ-rad(A). The γ-completely prime radical of M is defined as the γ-completely prime radical of the zero ideal, and it is denoted by C-γ-rad(M).

The following corollary follows from Theorem 1.18. 2.11 Corollary: Let P be a γ-ideal of a Γ-nearring M for γ ∈ Γ. Every minimal γprime γ-ideal P of a γ-completely semi-prime γideal I is a γ-completely prime γ-ideal. More over P is a minimal γ-completely prime γ-ideal of I. The following corollary follows from Theorem 1.19.

2.6 Note: From Theorem 1.16, and Theorem 2.4 we conclude the following: (i) An ideal A of a near-ring is completely semiprime ⇔ A = C-rad(A).

2.12 Corollary: Let γ ∈ Γ. If I is γ-completely semi-prime γ-ideal of M, then I is the intersection of all minimal γ-completely prime γ-ideals of I.

(ii) A γ-ideal A of a Γ-near-ring M is γcompletely semi-prime ⇔ A = C-γ-rad(A).

2.13 Corollary: Let γ ∈ Γ and P be a γ-ideal of M. If P is a γ-prime γ-ideal and I is a γcompletely semi-prime γ-ideal, then P is a

2.7 Definitions: (i). A γ-ideal P of a Γ-nearring M is said to be a γ-prime γ-ideal of M (with


25


minimal γ-prime γ-ideal of I if and only if P is a minimal γ-completely prime γ-ideal of I. Let γ ∈ Γ. By applying the Corollary 1.21 to the near-ring (M, +, *γ) we get the following.

(i). Since S = P-γ-rad(A) is equal to the intersection of all γ-prime γ-ideals of M containing S, by Note 2.8(iii), it follows that S is a γ-semi-prime γ-ideal. Thus we conclude that the γ-prime radical of a γ-ideal A (that is, Pγ-rad(A)) is a γ-semi-prime γ-ideal.

2. 14 Corollary: Let γ ∈ Γ. If I is a γcompletely semi-prime γ-ideal of M, then I is the intersection of all γ-completely prime γideals of M containing I (that is, I = ∩ {P / P is a γ-completely prime γ-ideal of M such that I ⊆ M} = C-γ-rad(I)).

(ii). Follows from (i), by taking A = (0).

Acknowledgements The first author acknowledges the financial assistance from the UGC, New Delhi under the grant F.No. 34-136/2008(SR), dt 30-12-2008. The authors thank the referee for valuable comments that improved the paper.

2.15 Example: Let us consider the Example 2.11 of Satyanarayana [13]. In this example, (G, +) is the Klein four group where G = {0, a, b, c}. We define multiplication on G as follows: 0 a b C . 0 0 0 0 0 a a a a A b 0 a b C c a 0 c B

References [1] Barnes W.E. “On the -rings of Nobusawa”, Pacific J. Math 18 (1966) 411- 422. [2] Booth G.L. “A note on Γ -Near- rings”, Stud. Sci. Math. Hunger 23 (1988) 471475.

This (G, +, .) is a near-ring which is not zero symmetric. The ideal {0, a} is only the nontrivial ideal and also it is completely prime.

[3]Booth G. L. & Groenewald N. J. “On Radicals of -near-rings”, Math. Japan. 35 (1990) 417-425.

(i) Write M = G, the Klein four group and G = {0, a, b, c}. Define multiplication on G as above. If we write Γ = {.}, then M is a Γ-nearring, which is not a zero symmetric Γ-near-ring (because aγ0 = a.0 ≠ 0). It is clear that for γ ∈ Γ, the γ-ideal {0, a} of M is only the nontrivial γ-completely prime γ-ideal. The γideal (0) of M is γ-completely semi-prime γideal, but not γ-completely prime γ-ideal (because cγa = c.a = 0 and a ≠ 0 ≠ c). Hence the γ-completely semi-prime γ-ideal (0) can not be written as the intersection of its minimal γcompletely prime γ-ideals. From this example 2.15, we can conclude that if M is not a zero symmetric Γ-near-ring, then the corollary 2.14 need not be true.

[4] Nobusawa “On a Generalization of the Ring theory”, Osaka J. Math. 1 (1964) 81-89. [5] Pilz .G 1983.

“Near-rings”,

North Holland,

[6] Ramakotaiah Davuluri “Theory of Nearrings”, Ph.D. Diss., Andhra univ.,1968. [7] Sambasivarao.V and Satyanarayana.Bh. “The Prime radical in near-rings”, Indian J. Pure and Appl. Math. 15(4) (1984) 361-364. [8] Satyanarayana Bh. "A Note on Γ-rings", Proceedings of the Japan Academy 59-A (1983) 382-83.

2.16 Notation: Let A be a γ-ideal of M. The intersection of all γ-prime ideals containing A is called the γ-prime radical of A and it is denoted by P-γ-rad(A). The γ-prime radical of M is defined as the γ-prime radical of the zero ideal (0). So P-γ-rad(M) = P-γ-rad(0).

[9] Satyanarayana Bhavanari. “Contributions to Near-ring Theory”, VDM Verlag Dr Mullar, Germany, 2010 (ISBN: 978-3-639-22417-7). [10] Satyanarayana Bhavanari “A Note on gprime Radical in Gamma rings”, Quaestiones Mathematicae, 12 (4) (1989) 415-423.

2.17 Theorem: Let A be an ideal of M. Then (i). P-γ-rad(A) is a γ-semi-prime γ-ideal. (ii). The γ-prime radical of M is a γ-semi-prime γ-ideal.

[11] Satyanarayana Bhavanari. “A Note on Γnear-rings”, Indian J. Mathematics

Proof: Write S = P-γ-rad(A).


26


(B.N. Prasad Birth Centenary commemoration volume) 41(1999) 427-433. [12] Satyanarayana Bhavanari "The f-prime radical in Γ-near-rings", South-East Asian Bulletin of Mathematics 23 (1999) 507-511. [13] Satyanarayana Bhavanari “A Note on Completely Semi-prime Ideals in Nearrings”, International Journal of Computational Mathematical Ideas, Vol.1, No.3 (2009) 107112. [14] Satyanarayana Bhavanari, Pradeep Kumar T.V. and Srinivasa Rao M. “On Prime left ideals in Γ-rings”, Indian J. Pure & Appl. Mathematics 31 (2000) 687-693. [15] Satyanarayana Bhavanari & Syam Prasad Kuncham “Discrete Mathematics and Graph Theory”, Printice Hall of Inida, New Delhi, 2009. [16] Syam Prasad K. “Contributions to Nearring Theory II”, Doctoral Dissertation Acharya Nagarjuna University, 2000. [17] Venkata Pradeep Kumar T. “Contributions to Near-ring Theory III”, Doctoral Dissertation, Acharya Nagarjuna University, 2008


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A UNIFIED FRAME WORK FOR SEARCHING DIGITAL LIBRARIES USING DOCUMENT CLUSTERING Shaik Sagar Imambi@, Thatimakula Sudha# @ #

Dept. of Computer Science, TJPS College, Guntur, A.P., India Sri Padmavathi Mahila Univesity, Tirupathi, A.P., India

Abstract The increasing interest in processing larger collections of documents from digital libraries has led to a new emphasis on document clustering problem. Document clustering is a technique for identifying clusters or groups of documents which share some common features or have overlapping content. These groupings of documents can be useful in document retrieval from digital libraries. We have developed the retrieval frame work for searching digital libraries, called UFDC. It is an advanced, an efficient and effective search facility for digital information. It combines conventional information retrieval and full-text searching techniques. Automatically linked Top Ranked Document(TRD) clusters are generated from the digital library information . By grouping together TRDs that share a common topic, UFDC provides an effective means of finding and tracking documents. Keywords: Data mining, Digital library, Document clustering, UDFC frame work. graph partitioning (Karypis, 2002), mixture of vMFs (Banerjee et al., 2003), information bottle-neck (IB) clustering (Slonim and Tishby, 2000), and co-clustering using bipartite spectral graph partitioning (Dhillon, 2001). One of the challenging research issues in Digital Libraries is the facilitation of efficient and effective access to large amounts of available information. Digital library system architecture is shown in fig 1. Document clustering [1] and automatic text summarisation [2] are two methods which have been used in the context of information access in digital libraries.

INTRODUCTION Digital Library systems are software systems which help in the management of metadata and data. They also provide end-user services for activities such as submission, discovery and retrieval of digital objects. In most cases these systems have developed out of the needs of libraries to manage digital equivalents of their holdings. The existing generation of such software tools – systems such as Greenstone (Witten and Bainbridge, 2002), DSpace (Tansley, et al., 2003) and Eprints (University of Southampton,2006) – have made it possible for non-programmers to easily set up and manage a digital archive. The increasing interest in processing larger collections of documents from digital libraries has led to a new emphasis on designing more efficient and effective techniques, leading to an explosion of diverse approaches to the document clustering problem. Some of the approaches are (multi-level) self-organizing map (Kohonen et al., 2000), spherical k-means (Dhillon and Modha, 2001), bisecting k-means (Steinbach et al., 2000), mixture of multinomials (Vaithyanathanand Dom, 2000; Meila and Heckerman, 2001), multi-level

Figure 1. Digital library system

A Unified frame work for searching Digital libraries Using Document Clustering

28


clustering algorithms. The unified framework for searching data from digital libraries is based on the clustering data and models. In short, the scope of this paper is to help users to evaluate the quality and feasibility of using cutting edge clustering methods implemented for digital libraries. In order to achieve this, we designed a unified frame work for document clustering. This frame work should be aimed at clustering experiments on medium to large scale digital library websites which are already indexed .

1. DOCUMENT CLUSTERING Document clustering is defined as the automatic discovery of document clusters/groups in a document collection, where the formed clusters have a high degree of association (with regard to a given similarity measure) between its members. Members from different clusters will have a low degree of association (ref 5). Document clustering generates groupings of potentially related documents. By taking into account inter document relationships; users have the possibility to discover documents that might have otherwise been left unseen in the digital libraries.(ref 3). Document clusters, effectively, reveal the structure of the document space.

Challenges Although commercial information retrieval systems utilizing existing clustering algorithms, document clustering is far from a trivial or solved problem. The clustering process is filled with challenges like:

This space however may not help users understand how their search terms relate to the retrieved documents. Therefore, the information space offered by document clusters to users is essentially not representative of their queries. Clusters are represented as probabilistic models in a model space that is conceptually separate from the data space. For partitioned clustering, the view is conceptually similar to the Expectation Maximization (EM) algorithm. For hierarchical clustering, the graph-based view helps to visualize critical/important distinctions between similarity-based approaches and model-based approaches.

Selecting appropriate features of the documents that should be used for clustering. Selecting an appropriate similarity measure between documents. Selecting an appropriate clustering method, utilising the above similarity measure. Implementing the clustering algorithm in an efficient way that makes it feasible in terms of required memory and CPU resources. Finding ways of assessing the quality of the performed clustering. Finding feasible ways of updating the clustering if new documents are added to the collection. Finding ways for applying the clustering to improve the information retrieval task at hand.

2. TEXT SUMMARIZATION Text summarization, in the context of information access, offers short previews of the contents of documents, so that users can make a more informed assessment of the usefulness of the information without having to refer to the full text of documents (ref 2 ,4). Particular classes of summarisation approaches, query-oriented or query-biased approaches, have proven effective in providing users with relevance clues (ref 4). Querybiased summaries present to users textual parts of documents (usually sentences) which highly match the user’s search terms. The effectiveness of such summaries in the context of interactive retrieval on the World Wide Web has been verified by (ref 4).

3. UDFC FRAMEWORK The proposed framework includes query generation view, generating top ranked documents, view of Clustering Top Ranked Document (TRD) and a view of hardware that leads to several useful extensions. The four main views of Unified frame work of Document clustering (UFDC).

Goal: In this paper we present Unified Frame work for Clustering Documents (UFDC) by comparing its effectiveness at providing access to useful information. The framework also suggests several useful variations of existing A Unified frame work for searching Digital libraries Using Document Clustering

29


or query of each of the required documents. Individual TRD are linked to the original documents (or to representations of the original documents, such as titles, summaries, etc.) in which they occur, so that users can access the original information. User interaction with TRS clusters, individual documents and other document representations can be monitored. The information collected can be used to recommend new documents to users, and to select candidate terms to be added to the query from the documents and clusters viewed. We use a standard iterative clustering technique to compute N clusters of documents. The N seeds for the initial cluster centers are obtained by a full hierarchical clustering of the best-ranked 100 documents resulting from the query, in TRS clusters. This type of implicit feedback has been used by (ref 6) in order to utilise information from the interaction of users with query based document summaries, and is effective in enabling users to access useful information.

Figure 2. UFDC frame work Generating query The overall objective of our approach is to utilise information resulting from the interaction of users with in the personalized information space. Users can access documents, or other shorter representations of documents such as titles and query-biased summaries, by selecting individual sentences.

D. Hardware view: The hardware was designed in a modular framework. This allows for more complicated operations to be created from smaller and simpler operations. For instance, the cosine distance module is created by linking together a controller, a dot product, and a normalization circuit.

Generating TRD: Sentences within document will discuss query terms in the context of the same (or similar) topics. This can assist users in better Data

Source

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41681

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1774

0.323

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11162

11465

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1116

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understanding the structure and the contents of the information space which corresponds to the top-retrieved documents. This may be especially useful in cases where users have a vague, not well-defined information need.

4. EXPERIMENTAL RESULTS We used the 20-digital libraries data and a number of datasets from the CLUTO toolkit2. These datasets provide a good representation of different characteristics: number of documents ranges from 204 to 19949, number of words from 5832 to 43586, number of classes from 3 to 20. A summary of all the datasets used in this paper is shown in Table 1.

Clustering TRD view: The main function of TRD clusters is to provide effective access to retrieved documents by acting as an abstraction of the query information space. Essentially, TRD clusters form a second level of abstraction, where the first level corresponds to summaries

Table 1 : Summary of text datasets(for each data set d is the total number of documents , g the total number of words , K the number of


30


classes, C the average number of documents per class and Balance the state ratio of the smallest classes to the large class

of accuracy, efficiency, and speed. Feature work includes the development of suitable scheduling strategies for document cluster architectures. A dynamic, temporal or permanent retrieval of the images between the nodes balances the workload. Moreover, suitable operators for dynamic feature extraction and similarity metrics are necessary.

For each dataset, we assess the results by locating the result clusters that are affiliated with the user’s area of interest. We then calculate precision for each located cluster defined as the number of relevant documents compared to the total number of documents in the cluster. We thereafter calculate the combined recall of the clusters defined as the number of relevant documents found in these clusters compared with the total number of relevant documents in the search result. For each case, we have gone through the entire search result and identified the top ranked documents that were relevant to the area of interest based on the way the search word was used in the document. The result of this is a list of relevant and irrelevant pages with duplicates removed. The times taken for searching same data set with the same given query, in normal search and in experimental search are tabled in table2.

ACKONOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper. REFERENCES [1]. T. Heskes, ” Self-organizing maps, vector quantization, and mixture modeling.” IEEE Trans. Neural Networks, 12(6):1299–1305, November 2001. [2]. Young wang et al , “Document clustering with semantic Analysis” ,Proceeding of 39th Hawai International conference on System Science,2006 [3]. T. S. Jaakkola and D. Haussler, “Exploiting generative models in discriminative classifiers”, Advances in Neural Information Processing Systems, volume 11, pages 487–493. MIT Press, 1999. [4]. David M Blei Andrew Yng and Michael I JD , “ Latent Dirichlet allocation”, Journal of Machine Learning Research 3, 993-1022 , Jan2003. [5]. K. Jain, M. N. Murty, and P. J. Flynn,”Data clustering: A review”, ACM Computing Surveys, 31 (3): 1999: 264–323, [6]. Tapher H Haveliwala, “Topic Sensitive page Rank – A context sensitive ranking algorithms for web search” , IEE transactions on Knowledge & Data Engineering 15(4)-784-796 ,2003. [7]. White, R.W., Ruthven, I., Jose, J.M, “ A task-oriented study on the influencing effects of query-biased summarisation in web searching”. Journal of Information Procsessing & Management , 2003:350-362. [8]. Zamir, O., Etzioni, O, “ Web document clustering: A feasibility demonstration”, In: Proceedings of the 21st Annual ACM SIGIR

Table 2: Normal and Experimental Search component Times in Seconds

Conclusion: Document clustering has been studied intensively because of its wide applicability in areas such as web mining, search engines, information retrieval, and topological analysis. Most traditional clustering methods do not satisfy the special requirements for document clustering. We presented an unified frame work for document clustering technique which can be implemented in searching text from digital libraries. The novelty of this approach is that it exploits top ranked documents from digital libraries; organize the cluster hierarchy, and reducing the dimensionality of document sets. The experimental results show that our approach outperforms its competitors in terms


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Conference, Melbourne, Australia (1998) 46–54. [9]. White, R.W., Ruthven, I., Jose, J.M, “ Finding relevant documents using top ranking sentences: an evaluation of two alternative schemes.”, In: Proceedings of the 24th Annual ACM SIGIR Conference, Tampere, Finland (2002) 57–64 [10]. Radev, D.R., Jing, H., Budzikowska, M, ” Centroid-based summarization of multiple documents: sentence extraction, utility-based evaluation, and user studies”. In: Proceedings of the ANLP/NAACL Workshop on Summarization, Seattle, U.S.A. (2000). [11]. O. Kao, “Towards Cluster Based Image Retrieval”, In Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA), CSREA Press ,2000, pp 1307-1315, [12]. Reuter, “Methods for parallel execution of complex database queries”, Journal of Parallel Computing, Volume 25, 1999:pp 2177-2188, [13]. Zha, H.,”Generic summarization and key phrase extraction using mutual reinforcement principle and sentence clustering” , In: Proceedings of the 25th Annual ACM SIGIR Conference, Tampere, Finland (2002) 113–120. [14]. Kural, Y., Robertson, S.E., Jones, S.,” Deciphering cluster representations.” Information Procsessing & Management, Volume 37 (2001) 593–601. [15]. O. Kao, I. la Tendresse, “CLIMS - A system for image retrieval by using colour and wavelet features” , Proceedings of the First Biennial International Conference on Advances in Information System (2004).


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REDUCIBILITY FOR THE FIORINI-WILSON-FISK CONJECTURE S.Satyanarayana@, J.Venkateswara Rao#, V.Amarendra Babu$ @

Department of Mathematics, Sri Venkateswara Institute of Science & Information Technology, Tadepalligudem-534101, E-mail:[email protected] # Professor of Mathematics, Mekelle University Main Campus, P.O.Box No.231, Mekelle, Ethiopia, Email: [email protected] $ Assistant Professor of Mathematics, Acharya Nagarjuna University, Email:[email protected]

ABSTRACT In this paper we define what we mean by reducibility for the Fiorini-Wilson-Fisk Conjecture and outline the logic used to establish the reducibility of every configuration in U. Of course, a computer actually verifies the reducibility of each configuration, as it would be too difficult using the present techniques to do so by hand. Essentially, reducibility for the Fiorini-Wilson-Fisk Conjecture is just a strengthening of reducibility for the Four Color Theorem, and in fact many of the configurations that were reducible for the Four Color Theorem are also reducible for the Fiorini-Wilson-Fisk Conjecture. Keywords:- Tricolorings, Contracts, Colorings of Rings, Reducibility. Mathematics Subject Classification:- 05C15, 05C35, 05C90 Theorem 1.1.1 (Tait) Let G is a triangulation or near-triangulation. The following statements are equivalent, (i) The vertices of G can be 4-colored. (ii) The drawing G has a tri coloring. (iii) The dual of G can be edge-3-colored.

1.1.1 Tri colorings and Notation Recall that two functions c and c| with identical domain and range={1, 2,…,K} are equivalent if {c-1({1}), c-1({2}),…, c-1({k})} = {c|-1({1}), c|1 ({2}),…, c|-1({k})}. We will use this frequently when the functions represent colorings. If A is a set of functions with domain D and range R = {1,…, k}, then η (A) will denote the set of all functions with domain D and range R that are equivalent to some coloring in A. Let T be a triangulation or near-triangulation, and let F(T) denote the set of all faces of T that are bounded by exactly 3 edges. A tri coloring of T is a function c :F(T) → {-1, 0, 1} such that for every f ∈ F(T), and for any two distinct edges I and j incident to f, c(i) ≠ c(j). The next theorem establishes a connection between Tri colorings and vertex colorings of a graph and edge colorings of the dual of the graph.

1.1.1.1 Tri colorings and Contracts The reducibility part of the recent Robertson et al. proof of the Four Color Theorem essentially proceeds by induction. Without going into details, the contraction of edges is critical in their argument to produce smaller graphs. To avoid notational difficulties, they introduced the idea of a tri coloring of T modulo X, where T is a triangulation or near-triangulation and X is a set of edges in T. As the definition will reveal, the set X represents the set of edges to be contracted. Following the definitions of Robertson et al. [98], a set X ⊂ E(T) is said to be sparse if no two edges of X are incident to a common finite face


33


k (e) = 1 if {c(Vi), c(Vj)} = {1, 4} or {c(Vi), c(Vj)} = {2, 3}. Claim. k is a tri coloring of T modulo X. To prove this, let r be a triangular face in F(T) incident to the vertices {x, y, z} and edges e = {x, y}, f = {x, z} and g = {y, z}. If {e, f, g} I X = φ then x, y and z are in three distinct vertices of H, say x ∈ Vi, y ∈ Vj and z ∈ Vk. In the vertex-4coloring of H which defines k, Vi, Vjand Vk receive different colors and thus k can be seen to assign difierent colors to e,f,and g. If one of the edges incident with r is in X, say g ∈ X, then the vertices y andz are in the same vertex, say Vi of H. If x were in the same component of T(X) as yor z, then since e, f ∉ X, there would be a circuit C in T such that |E(C) - X| = 1.Since X is sparse, x is in a distinct vertex. Hence, k (e) and k (f) are well defined andequal to each other. This completes the proof of the claim that k is a tri coloring ofT modulo X and hence completes the proof of the theorem. If X ⊂ E(T) is sparse and |E(C) - X| ≥ 2 for all circuits C in T then we say that X is contractible in T.

of T, and if T is a near-triangulation, then no edge of X is incident to the infinite face of T. If X is sparse, then a tri coloring of T modulo X is a coloring k:E(T)- X → {-1, 0, 1} such that for every finite face r ∈ F(T) 1.) If r does not have any edges in common with X then k assigns distinct colors to the three edges of r. 2.) If r has exactly one edge in common with X, then k(e) = k(f) for the other two edges e and f of r. Recall that a counterexample is defined to be a planar graph which is not a vertex FioriniWilson-Fisk graph and has at most one vertex-4coloring. A minimum counterexample is a counterexample with a minimum number of vertices. The following theorem, adapted from [48], captures the idea that if one can contract edges in a minimum counterexample so that no loops are created, then the resulting graph has a tri coloring. Theorem 1.1.2 Let T be a minimum counterexample, and let X ⊂ E(T) be a non-empty, sparse set such that there is no circuit C of T for which |E(C) – X| = 1. Then there is a tri coloring of T modulo X. Proof. Let T(X) be the sub graph of T consisting of the vertices of T and the edges of X. Let V1,V2,…, Vp be the vertex sets of the components of T(X). Let H be the graph obtained by deleting multiple edges in the graph with vertex set {V1,…, Vp} and with Vi adjacent to Vj if and only if there is an edge in E(T) − X which joins two vertices vi and vj with vi ∈ Vi and vj ∈ Vj . Claim. H is loopless. If there was a loop f joining the vertex Vi to itself then there would be an edge f’ ∈ E(T) - X which joins two distinct vertices x, y of T that are both in Vi. Since Vi is a vertex set of a component of the graph T(X), there is a path in T joining x and y and consisting entirely of vertices in T. The circuit P U {f’} violates the condition that |E(C) – X| ≠ 1 for every circuit C in T. Thus H is loopless. Since X is nonempty, p < |V (T)| and since T is a minimum counterexample and H is loopless, H is either a vertex Fiorini-Wilson-Fisk graph or H has at least two vertex-4-colorings that are not permutations of one another. Either way, H has a vertex-4coloring c. Use the standard Tait coloring to define a coloring k . E(T) - X → {-1,0,1}, that is for an edge e of E(T) - X with endpoints u ∈ Vi and v ∈ Vj, define. k (e) = -1 if {c(Vi), c(Vj)} = {1, 2} or {c(Vi), c(Vj)} = {3, 4}. k (e) = 0 if {c(Vi), c(Vj)} = {1, 3} or {c(Vi), c(Vj)} = {2, 4}.

1.1.2 Colorings of a Ring Let S be a free completion of a configuration K with ring R. If c is a tri coloring of S, the restriction of c to the ring defines a coloring of that ring. A basic part of the theory of reducibility is the consideration of these ring colorings. Let the vertices of R be 1,2,…,r and the edges of R be e1,e2,…,er, where ei has endpoints i and i + 1 for i = 1,…, r - 1 and er has endpoints r and 1. A coloring of R is a function k : E(R) → {-1, 0, 1}. Let C*(R) denote the set of colorings of R. We will sometimes abbreviate C*(R) by C*. By a restriction to R of a tri coloring c of S, we mean the function c|R with domain E(R) and range {-1,0,1} that agrees with c on the edges of R. Also, if k . E(R) → {-1,0,1} is a coloring of R, then we define an extension of k into a tri coloring of S, to be a tri coloring of S which agrees with fi on E(R). We let C(S) denote the set of restrictions to R of Tri colorings of S. A restriction to R of a tri coloring of S has either one or more extensions into Tri colorings of S. Let the set of restrictions to R of Tri colorings of S which have exactly one extension into a tri coloring of S be denoted by U(S) or just U if the free completion S is understood from the context. If T is a triangulation that is uniquely vertex-4colorable, and if the free completion S appears in T then it follows that the restriction to R of the corresponding tri coloring of T must be an element of U.


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The following definitions are taken from Robertson et. al. [98] A match is a an unordered pair {e,f} of distinct edges of E(R). A matching is a nonempty set of matches {{e1, f1},{e2,f2},…,{ek, fk}} such that for any i ≠ j, the edges ej and fj are in the same component of R – {ei,fi}. Finally, a signed matching is a collection of ordered pairs {({e1,f1}, µ 1), ({e2, f2}, µ 2),…, ({ek, fk}, µ k)}, where the collection {{e1, f1},{e2, f2},…,{ek, f k}} is a matching, and where µ i ∈ {-1, 1} for 1 ≤ i ≤ k.The sign of a match is used to differentiate whether both ends of a kempe chain have the same color or distinct colors. If θ ∈ {-1, 0, 1} and k is a coloring of R we say that k θ -fits a signed matching M = {({e1, f1}, µ 1), ({e2, f2}, µ 2), … , ({ek, fk}, µ k)} if

Therefore α ’ does not γ ’- fit k, because if it did, condition ii) of consistency would imply that α ’ ∈ C. This establishes that C is A - critical and completes the proof of Lemma 1.1.1. Consistency and criticality are defined in terms of colorings of a circuit, but the near triangulations to which we want to apply the ideas of consistency may have their infinite face bounded by something other than a circuit. This does not turn out to be a serious obstacle to using consistency as we shall now see. Let R be a circuit with vertices {1, 2,…, r} and edges e1, e2,…, er where edge ei joins vertex i to vertex i + 1 for 1 ≤ i < r, and edge er joins vertex r to vertex 1. Let H be a near triangulation with outerfacial walk W = v1, f1, v2, f2, v3,. . . , vr, fr, v1, where v1, v2, . . . , vr are vertices, not necessarily distinct and where {f1, f2,…, fr} are edges such that fi joins vi and vi+1 for 1 ≤ i ≤ r - 1 and where fr joins the vertices vr and v1. Let φ . E(R)

U i ≤ i ≤ k {ei, fi} = {e ∈ E(R) . k (e) =

→ {f1, f2,…, fr} be defined by φ (ei) = fi. Also

θ } and (ii) For each ({ei, fi}, µ i) ∈ M, k (ei) = k (fi) if and only if µ i = 1. A set C of colorings of R is consistent if for every k ∈ C and every θ , θ ’ ∈ {-1, 0, 1} there is a signed matching M such that i) k θ -fits M. ii) C contains every coloring of R that θ ’-fits M. Let A ⊂ c * . A set of colorings C of R is said to be A-critical if for every k ∈ C and every θ , θ ’ ∈ {-1, 0, 1}, there is a signed matching M such that i) k θ -fits M., ii) C contains every edge that θ ’-fits M, and iii) there are not two colorings α , α ’ ∈ A and integers γ , γ ’ ∈ {-1, 0, 1} such that both α γ fits M and α ' γ ’-fits M and α is not equivalent to α ’.

suppose that k is a tri coloring of H and define a function λ on the edges of the circuit E(R) by λ (e) = k( φ (e)). Following [98], we say that

(i) E(R) -

φ wraps R around H and that the coloring λ of E(R) is a lift of k. The next theorem is an important result which uses ideas of both Kempe and Birkhoff. Theorem 1.1.3 Let H be a near triangulation with outer facial walk W as above, and let φ wrap the circuit R around H. The set C of all lifts of Tri colorings of H is consistent. Proof; Let k ∈ C and let θ ∈ {-1, 0, 1}. We will construct a signed matching M = M(k, θ ) such that k θ -fits M. Following Robertson et. al. [98], we define a θ - rib to be a sequence g0,r1,g1,r2,…,rt,gt such that (i) g0, g1, … , gt are distinct edges of H. (ii) r1,r2,. . . ,rt are distinct finite faces of H. (iii) If t > 0 then g0,gt are both incident with the infinite face of H, and if t = 0 then g0 is incident with no finite face of H. (iv) For 1 ≤ i ≤ t, ri is incident with gi-1 and with gi. (v) For 0 ≤ i ≤ t, k(gi) ≠ θ . Any two distinct θ -ribs ρ = g0, r1,g1,r2,g3,…,gt and ρ ’ = g’0,r’1,g’1,…,r’t’ ,g’t’ must have {g0, g1,

Lemma 1.1.1 If |A I C| ≤ 1, then C is A critical if and only if C is consistent. Proof; If C is A - critical, then it is clearly consistent. Conversely, let C be a consistent set. We must show that under the hypothesis, C is critical. Let k ∈ C and θ ∈ {-1, 0, 1}. Since C is consistent, there is a signed matching M such that k θ -fits M and C contains every coloring that θ ’- fits M. Now let α , α ’ ∈ A, and let, γ , γ ' ∈ {-1, 0, 1}. Since |A I C| ≤ 1, it follows that one of either α or α ’ is not in C. Without loss of generality, assume that α ’ ∉ C.

. . . , gt}

I {g’0, g’1,… , g’t’ }= θ , and {r0, r1,…,

rt} I {r’0, r’1, …, r’t’} = θ . To see this, note two things. First, if ρ and ρ ’ share a common noninfinite face r, then ρ and ρ ’ must also share both of the unique (because each of the finite


35


si+1} = {sj , sj+1}, it follows that the edge ej-1 is in one of the faces si or si+1. Therefore ej ∈ {ei-1, ei, ei+1}. Now the choice of j insures that ej-1 ∉{e0, e1, . . . , ej-2}. Thus, it must be that i + 1 = j-1, so j = i + 2 and si+1 = sj-1. Thus ei = ej is incident to si, si+1 = sj-1 and si+2 = sj and since each edge is incident to at most two faces, two of the three faces si,si+1 and si+2 must actually be the same face. By the construction, si ≠ si+1 which forces si = si+2. But then ej-1 = ei-1 which contradicts the choice of j. Thus, every edge of the sequence is distinct. Since the graph is finite, this means that the sequence must terminate on an edge ek other than e0 which is incident to the infinite face. The sequence e0,s1,e1,s2,. . .,sk,ek is a rib which contains e0, as desired. This shows that each θ - rib ρ defines a pair of edges {e ρ , f ρ } which are both incident to the infinite face and which both receive colors from {-1, 0, 1} - { θ }. We can thus use ρ to define a signed match, namely ({e ρ , f ρ }, µp ) where µp = -1 if k(e ρ ) ≠ k (f ρ ) and µp = 1 otherwise. Now we will show the set of ribs { ρ 1 ,… ρp } defines a signed matching. First of all, the set { ρ 1 ,… ρp } defines a set of signed

faces is a triangle) edges incident to r that are colored with the colors in {-1, 0, 1} – { θ }, because of (iv). Second, if ρ and ρ ’ share a common edge g then ρ and ρ ’ also share both of the faces that are incident to g, because of (iv) and (v). Using these two facts, we can show that if any two θ - ribs share either an edge or a finite face, then the two θ - ribs are identical. Because of (iii), unless a rib consists of a single edge, it contains at least two edges incident to the infinite face. Because of (ii) and (iv), a rib does not contain more than two edges which are incident to an infinite face. Thus, if a rib is not a single edge, then it has exactly two edges that are incident to the infinite face and colored with colors in {-1, 0, 1} - { θ }. Conversely, we claim every edge that is incident to the infinite face and is colored with a color in {-1, 0, 1} – { θ } is in some rib. To see this, let e0 be an edge incident to the infinite face which is colored α ∈ {-1, 0, 1}-{ θ } and let γ ∈ {-1, 0, 1} - { θ , α }. If e0 is not incident to a finite face, then e0 is itself a rib by (iii). So suppose that e0 is incident to a unique finite face s1 that is a triangle. Because the given coloring is a tri coloring, s1 has exactly one edge e1 ≠ e0, which receives the color . This edge is incident to a face s2 ≠ s1. If s2 is the infinite face, then e0,s1,e1 is a rib and we have proven that e0 is in a rib. If s1 is not an infinite face, then because the coloring is a tri coloring and because s1 is a triangle, there is an edge e2 ≠ e1 which receives the color θ and is incident to a face s3 ≠ s2. In this way we generate an alternating sequence of edges and faces e0,s1,e1,s2,. . .. If the sequence ever selects an infinite face sk+1 it terminates. We now show that the construction of this sequence guarantees that all of the edges e0, e1,…, are distinct. If not, there would be integers 0 ≤ i < j such that ei = ej . Of all such pairs (i, j) choose one with the smallest j and subject to that choose among those the one with the largest i. As a first case, assume i = 0. Thus ej is incident to the infinite face. It follows that s1 = sj and e1 = ej-1. By the choice of j, it cannot be that ej-1 ∈ {e0, e1, . . . , ej-2} and so j - 2 = 0. It follows that s1 = sj = s2. This however, contradicts the construction. Thus we have shown that i > 0, and in addition that if e0,…, ej-1 are distinct and ej is incident to the infinite face, then e j ≠ e 0. Of the two faces incident to ei, the face si-1 precedes the face si+1 is the sequence. Similarly, the face sj precedes the face sj+1 and both are incident to ei = ej . Clearly {si, si+1} = {sj, sj+1}. The face sj is incident to an edge ej-1 ≠ ej that receives a color in {-1, 0, 1} - { θ }. Since {si,

matches M = M(k, θ ) = {({e p1 , f p1 }, µp1 ),…, ({e ρp , f ρp }, µρp )} in the manner defined above. Because of the planarity of the graph, and the fact that two ribs are either disjoint or identical, it must be that for every i ≠ j, 1 ≤ i, j ≤ p, 1) {e ρi ,f ρi } I {e ρj , f ρj } = θ , and 2) the removal of {e ρi ,f ρi } from the ring could not separate e ρj from f ρj . Thus M is a signed matching. Now we show that k θ - fits M. First, because every edge incident to the infinite face and receiving a color in {-1, 0, 1} – { θ } must be in a rib, it follows that {e ρ 1 ,…, e ρp , f ρ 1 ,…, f ρp } equals {f ∈ E(R) .k(f) ∈ {-1, 0, 1} – { θ }}. The definition of every integer i (1

≤

i

≤

µρi also shows that for p), k(e ρi ) = k (f ρi ) if

and only if µρi = 1. This proves that k θ - fits M = M(k). We now finish the proof that C is consistent. First, for every k ∈ C and every θ ∈ {-1, 0, 1}, our construction using ribs has produced a signed matching M = M(k, θ ) such that k θ - fits M. So let θ ’ ∈ {-1, 0, 1} and let k’ be another coloring that θ ’- fits M(k, θ ) = M. Define the coloring k’’ as follows. k ‘’(e) = θ if k ‘(e) = θ ’, k ‘’(e) = θ ’ if k ‘(e) = θ and k ‘’(e) = k ‘(e) if k ‘(e) 2 {-1, 0, 1} – { θ , θ ’}. It follows that k ‘’ θ - fits M. Let c be the coloring of H whose lift Reducibility For The Fiorini-Wilson-Fisk Conjecture

36


is k and let { ρ 1 ,… ρp } be the set of θ - ribs induced by c which define M. If ({ei, fi}, µ i) is the signed match associated with the

Let k ∈ C1 U C2 and let θ ∈ {-1, 0, 1}. Without loss of generality, we may assume k ∈ C1. Therefore there is a matching M that k θ - fits and such that every other coloring k’ which θ ’fits M is in C1 ⊂ C1 U C2. Now suppose by way of contradiction that there are two non-equivalent colorings α 1, α 2 and two integers λ 1, λ 2 ∈ {-1, 0, 1} such that α 1 λ 1 fits M and α 2 λ 2- fits M. This however violates the A - criticality of either C1 or C2. This proves that C1 U C2 is A - critical. If |A| ≤ 1 the set C is A - critical if and only if it is consistent and so by choosing A = φ , we deduce that the union of two consistent sets is consistent. Now let B ⊂ C*(R). The union of all A - critical subsets of B is A - critical and certainly contains every A - critical subset of B. This completes the proof of Lemma 1.1.2. Lemma 1.1.3 If |A I C| ≤ 1, then the maximal consistent subset of C equals the maximal A critical subset of C. Proof: Let MCSA(C) denote the maximal A critical subset of C and let MCS φ (C) denote the maximal consistent subset of C. We know that MCSA(C) is consistent so MCSA(C) ⊂ MCS φ (C). Also, by Lemma 1.1.1 and the fact that |CA| ≤ 1, MCS φ ,(C) is A - critical. Hence MCS φ (C) ⊂ MCSA(C), and thus the theorem holds. 1.2 Proving Reducibility 1.2.1 Using a Corresponding Projection Let K be a configuration that appears in a triangulation T and has free completion S and ring R. In general S will not appear in T, but suppose for illustration that it does. The ring R will naturally split T up into two near triangulations, one of them S and the other which we denote by H. However, it may be the case that S does not appear in T. The next lemma is a technical result to show that we may still in a certain sense decompose T into the near triangulation H and the free completion S. Lemma 1.2.1 Let K be a configuration which appears in a triangulation T and has free completion S with ring R and let φ be the corresponding projection of S into T. Let H be the graph obtained from T be deleting the vertexset φ (V (G(K))). Then

θ - rib ρi ,

then the fact that k’’ θ ’- fits implies that 1a) Either k ‘’(ei) = k ‘’(fi) ≠ k (ei) or 1b) k ‘’(ei) = k ‘’(fi) = k (ei) or 1c) k ‘’(ei) ≠ k ‘’(fi), and k ‘’(ei) ≠ k (ei) or 1d) k ‘’(ei) ≠ k ‘’(fi), and k ‘’(ei) = k (ei). where either 1a) or 1b) hold if µ i = 1 and either 1c) or 1d) hold of µ i = -1. In the θ - rib ρi = g0,r1,g1,r2,…,rt,gt, we have k (g0) = k (g2) = k (g4) = … = k (g 2 2t ) = α and k (g1) = k (g3) = k



(g1) = … = k ( g 2 t −21  + 1 ) = β where { α , β } = {-1, 0, 1} – { θ }. There is another tri coloring γ of H that can be obtained by exchanging the colors α and β along H, namely if e is not in ρi , γ (e0) = γ (e2) =

γ (e) = k (e) γ (e4) =… =   γ ( e 2  2t  ) = β and γ (g1) = γ (g3) = γ (g1) = … = γ ( g 2 t −21  + 1 ) = α . Using this idea, we define a new tri coloring c’’ of H by exchanging the colors α , β ∈ {-1, 0, 1} – { θ } along each rib ρi for which either 1a) or 1c) holds. The lift of c’’ will be k’’ and thus k’’ ∈ C. Moreover, by defining a coloring c’ of H from the coloring c’’ by swapping the colors θ and θ ’, that is defining c’(x) = θ if c’’(x) = θ ’, c’(x) = θ ’ if c’’(x) = θ and c’(x) = c’’(x) otherwise, we see that c’ is also tri coloring of H whose lift equals k’. Thus k’ ∈ C as desired. This shows that C is consistent and completes the proof of Theorem 1.1.3. Lemma 1.1.2 Let R be a ring and let A ⊂ C*(R). The empty set is an A – critical set. Also, the union of two A - critical sets is an A - critical set and in particular, the union of two consistent sets is consistent. Finally, for any subset B of colorings of R, the maximally A - critical subset of B exists, that is, there is a subset of B which is A - critical, and such that every other A - critical subset of B is contained in it. Proof: Let A ⊂ C*(R). The statement that the empty set is A - critical is vacuously true. Let C1, C2 ⊂ C*(R) be two A - critical sets, and assume that for i = 1, 2, and any θ , θ ’, γ , γ ’ ∈ {-1, 0, 1} and any k ∈ Ci, there is a signed matching M such that i) k θ - fits M and ii) every k’ that θ ‘- fits M is in Ci and iii) no two non-equivalent colorings α i ∈ A γ I - fit M for both i = 1 and i = 2.

1) H is a near triangulation and φ wraps R around H. 2) If X ⊂ E(S) is sparse in S, then φ (X) is sparse in T. Proof; Since φ fixes G(K) and G(K) is connected, all of G(K) lies in the same face of the


37


3) The edges φ (gi), φ (hi) and φ (gi-1) are all

drawing T -V (G(K)). This and the fact that T is a triangulation implies that H = T - φ (V (G(K))) = T - V (G(K)) is a near triangulation. Let V (R) = {r1, r2,…, rq} and E(R) = {e1, e2,…, eq} where for i = 1, 2, . . . , q - 1, ei has endpoints ri and ri+1 and eq = {rq, r1}. Suppose that r1,r2,…,rq is the clockwise order of appearance of the vertices of V (R). Consider the alternating sequence W of vertices and edges in T . φ (r1),

incident to the finite face ri in T, for 1 ≤ i ≤ p. All of this implies that φ (g0), φ (g1),…, φ (gp) is a portion of any clockwise listing in T of edges incident to φ (v) in T. Moreover, from the claim above, φ (ri) is a face of T that is contained in the infinite face of H. Hence, the edges φ (g1),

φ (g2),…, φ (gp-1) are all edges of the infinite face of H, and it therefore follows that φ (gp) follows φ (g0) in any clockwise listing in H of edges incident to φ (v). We now show that the sequence φ (r1), φ (e1), φ (r2), φ (e2),…, φ (rq), φ (eq), φ (r1) is a

φ (e1), φ (r2), φ (e2),…, φ (rq), φ (eq), φ (r1). By property (iii) of projections, φ (ri) is incident to φ (ei) in T for each i ∈ 1,…, q - 1 because ri is incident to ei in S for each i ∈ 1,…, q - 1. For the same reason φ (eq) is incident to φ (r1). Thus W is a closed walk in H. We now prove some things that will help in establishing that W is a facial walk. We claim that every finite face r ∈ F(S) - F(G) has the property that φ (r) is contained in the infinite

facial walk in H that bounds the infinite face. From what we have just shown, φ (ei-1) follows

φ (ei) in the clockwise listing in T of edges incident to φ (ri) for 1 ≤ i ≤ q (and where when

face of H. It suffices to show that φ (r) is incident to some vertex of V (G), because the infinite face of H will contain that vertex in its interior. By property 2) of known Lemma r is incident to a vertex x ∈ V (G). By property (iii) of projections, φ (r) is incident to φ (x) = x in T. This proves that claim. Let u ∈ V (R). By property 6) of Lemma 4.2.1, a clockwise listing in S of edges incident to u is g0, g1,…, gp where g0, gp ∈ E(R), p ≤ 2 and g1,…gp1 2 E(G). We assume that the endpoints of gi in V (S) - v are xi for 0 ≤ i ≤ p. Also, for 1 ≤ i ≤ p, we label the unique finite face of S that is incident to gi-1 and gi as ri and the unique edge in E(S) that is incident to ri as hi. Thus, for 1 ≤ i ≤ p - 1, hi+1, gi, hi is a portion of a clockwise listing in S of edges incident to xi. From the fact that φ is the extension of the natural edge function, and the fact that the natural edge function preserves the embedding in S at xi, it follows that φ (hi+1),

i = 1, we interpret ei-1 as er). This completes the proof that W is a facial walk of the infinite face of G and thus shows that φ wraps R around H. Now let X ⊂ E(S) be a sparse set of edges. Each edge in X must have at least one endpoint in V (G). Therefore, every edge of X is in the domain of the natural edge function of Lemma 4.11. Since φ is an extension of the natural edge function, φ preserves that embedding at every x ∈ V (G), which implies that if e, f ∈ X share a common endpoint x ∈ V (G), but do not share a common face, then φ (e) and φ (f) have common endpoint x in T but are not in a common face of T. This implies that if φ (e) and φ (f) are in the same face r’ of T for some distinct edges e, f ∈ X, then φ (e) and φ (f) do not have a common endpoint in V (G). However, φ (e) and φ (f) both have endpoints in V (G) since e and f do. Let xe denote the endpoint of e in V (G) and xf the endpoint of f in V (G), and let z be the common endpoint of φ (e) and φ (f) in T. We digress briefly to show that there is an r ∈ F(S) such that φ (r) = r’. Now r’ is adjacent to a vertex in V (G), (in fact two, xe and xf ). Property 4) of the natural edge function guarantees that φ restricted to the edges of S which have at least one endpoint in V (G) is a one to one and onto function into the set of edges T with at least one endpoint in V (G). From the way φ was defined

φ (gi), φ (hi) is a portion of the clockwise listing in T of edges incident to φ (xi) = xi for 1 ≤ i ≤ p - 1. Also, the fact that ≤ is a projection, implies that φ (ri) is a face of T that is incident to the edges fi(gi-1), fi(hi), fi(gi) for 1 ≤ i ≤ p. We claim that for every 1 ≤ i ≤ p, φ (gi) follows φ (gi-1) in the clockwise listing in T of edges incident to φ (v). This however follows from three facts. 1) φ (hi) follows φ (gi) in any clockwise listing

for faces, this implies that φ (r) = r’. Let g be the edge in S which has endpoints xe and xf and which is incident to the face r. Because φ is a projection, φ (g) is incident to r’. Also,

in T of edges incident to xi for 1 ≤ i ≤ p. 2) φ (gi-1) follows φ (hi) in any clockwise listing in T of edges incident to xi-1 for 1

≤ i ≤ p.


38


since G is an induced sub drawing of S, g ∈ E(G) and so φ (g) = g. Suppose without loss of

such that |X| = k, φ (X) is contractible and no tri coloring of S modulo X is in the set MCS φ (C* - CS). 3. If u ∈ U and u ∉ MCS(u) then we say that u is D-removable. 4. If u ∈ U and there is a sparse set X ⊂ E(S) such that φ (X) is contractible and

generality that g follows φ (e) in every clockwise listing in T of edges incident to xe, and that φ (f) follows g in every clockwise listing in T of edges incident to xf . Now the edge e either precedes or follows the edge g in any clockwise listing in S of edges incident to xe. If e follows g in every clockwise listing in S of edges incident to xe then φ (e) would both precede and follow

MCS(u) I CS(X) = φ , then we say that u is Cremovable. 1. A configuration K is U-reducible if 1) every u ∈ U, is either D-removable or Cremovable. 2) At least one u ∈ U is C-removable or the configuration K is either D-reducible or C(4)-reducible. Notice that each of the above types of reducibility depends only on R and S and not on H. This has the very practical application that if /V (R)/ is relatively small, say 14 or 11, it is feasible computationally to calculate Maximum Critical Subsets like MCS(u). This coupled with the following observations which relate CH to MCS(u) are the key to reducibility because calculations on a small piece of the triangulation T yield information about the rest of the triangulation which could be immense. Let us recall that the operator fi was defined at the beginning of the chapter. 1) If it is not the case that CH I CS ⊄ η ({u}) for some u ∈ U, then it can be shown that T has at least two vertex-4-colorings. 2) If CH I CS ⊂ η ({u}) for some u ∈ U, then CH ⊂ MCS(u) by Lemma 1.1.3. Thus if T is a minimum counterexample, then CH fi MCS(u). This will turn out to be valuable because the induction hypothesis can be used to color H. Robertson et al. used D-reducibility and C(k)reducibility for 1 ≤ k ≤ 4 to prove the Four Color Theorem [98]. Notice that reducibility for the Four Color Theorem (Types 1. and 2.) is defined for entire configurations while reducibility for the Fiorini- Wilson-Fisk Conjecture must first be defined in terms of individual colors in U (types 3. and 4.) and only then defined for an entire configuration (type 1.) This means that proving reducibility for the Fiorini-Wilson-Fisk conjecture will tend to be more computationally intensive than proving it for the Four Color Theorem because in principle, each color in U needs to be considered. 1.2.3 Proving Reducibility As noted above, S will not, in general, appear in T, but suppose again for illustration that it does. The ring R will appear in T and will naturally split T up into two near triangulations, one of them S and the other which we denote by H.

φ (g) in every clockwise listing in T of edges incident to xe. This however is impossible because dT (xe) = γ K(xe) ≥ 1. Therefore, it must be the case that e precedes g in any clockwise listing in S of edges incident to xe. For similar reasons, f must follow g in any clockwise listing in S of edges incident to xf . Thus e and f share a triangular face with each other and with g, which contradicts that X is sparse. This completes the proof of the lemma. 1.2.2 Defining Various Types of Reducibility We now introduce the notation that will be used for the rest of this chapter. Let K be a configuration with free completion S and ring R. Suppose that K appears in the triangulation T, that φ is a corresponding projection of S into T, that H is the near triangulation T - V (K(G)), and that φ wraps R around the outer facial walk of H. Finally, let X be a sparse subset of E(S). We now define various sets of colorings of R. Let C* be the set of all colorings of the ring R, let CS be the set of restrictions to R of Tri colorings of S, and let U ⊂ C be the set of colorings of R which extend to a unique tri coloring of S. Note that C*(R) - CS is the set of colorings of R which do not extend into S. The set CS(X) will denote the set of restrictions to R of tri coloring of S modulo X. Also, let CH denote the set of lifts of Tri colorings of H. By Lemma 1.1.2, for any B ⊂ C, there is a maximal U - critical subset of B which we denote by MCSU (B) or just MCS(B) for short. The notation MCS φ (B) will denote the maximal consistent subset of B. Finally, for u ∈ U we denote the set MCSU ((C* - CS) U {u}) by MCS(u). With these definitions in place, we now define various types of reducibility, the first two of which appear in the literature and are suficient to prove the Four Color Theorem, and the third, fourth and fifth of which are introduced to prove the Fiorini-Wilson-Fisk Conjecture. 1. The configuration K is D-reducible if MCS φ (C* - CS) = φ . 2. The configuration K is C(k)-reducible if there exists a sparse set set X ⊂ E(S)


39


I {e, f, g} ≠ φ , say e ∈ X, then φ (e) ∈ φ (X), so cX(f) = c( φ (f)) = c( φ (g)) =

Denoting by CS the set of restrictions to R of tri coloring of S and CH the set of restrictions to R of Tri colorings of H, it is clear that T will have a tri coloring if and only if CS I CH ≠ φ . Many of the results of this section will use this simple principle in one way or another. The next theorem proves that this simple principle can be applied even when only a projection φ of S appears in T. Lemma 1.2.2 Let d be a coloring of R. Then d ∈ CS I CH if and only if T has a tri coloring whose restriction to φ (R) is d. The proof of this is straightforward and we omit it. The usefulness of our definitions of reducibility hinge on the following lemma, as was alluded to in Section 4.2.2. Lemma 1.2.3 Either T has at least two nonequivalent vertex-4-colorings or there is a u ∈ U such that CH ⊂ MCS(u). Proof. By Lemma 1.2.2, if |CH I CS| ≥ 2, or if CH I (CS - U) ≠ φ , then T has at least two distinct vertex-4-colorings. Hence, we may assume there is a u ∈ U such that CH ⊂ (C* - CS) S U ({u}). Now MCS(u) ⊂ (C* - CS) U η ({u}) and also MCS(u) is consistent by Lemma 1.1.3. Thus Theorem 1.1.3 implies that CH ⊂ MCS(u) since CH is consistent. Lemma 1.2.4 Let φ be a corresponding projection of S into T. If c is a tricoloring of T modulo φ (X), then there are functions cX and cH, such that cX is a tricoloring of S modulo X and cH is a tricoloring of H. In addition, cX(e) = c( φ (e)) for every e ∈ E(S), and cH(e) = c(e) for all e ∈ E(H). Finally, the restriction of c to φ (E(R)), the

cX(g)}. If X

cX(g). Thus cX is a tricoloring of S modulo X. From the definitions of cX and cH, cX(e) = c( φ (e)) = cH( φ (e)) for e ∈ E(R). Hence the restriction of cX to R (which equals the restriction of cH to φ (E(R)) is in the set CS I CH. This completes the proof of Lemma 1.2.4 Let A ⊂ C*. Generalizing Robertson et al. we say that a set X ⊂ E(S) - E(R) is an A - contract if it is a nonempty, sparse set and if no tricoloring modulo X of S is in the set MCS( (C* - C) U A ). If A = {a} we call an A-contract simply an a contract. If A = φ , then we say that X is a contract. The free completion S of a configuration does not necessarily appear in the triangulation T even if the configuration does. However, Theorem 4.3.1 shows that there is a projection of S into T. It is conceivable that a contract X in S might produce loops if the corresponding edges were contracted in T and Theorem 1.1.2 would not be applicable. The following method of Robertson et al. gives an easy to check sufficient condition for a contract X ∈ S not to produce loops after being projected into T. An edge e is said to face a vertex v if v is not an endpoint of e and both v and e are incident to a common face. A vertex v ∈ V (S) is a triad for X if (i) v ∈ V (G(K)) (ii) There are at least three vertices of S adjacent to v and incident to a member of X (iii) If γ K(v) = 1, then there is an edge of X that does not face v. Theorems 1.2.1 Let K be a configuration with free completion S and ring R and suppose that K appears in an internally 6 connected triangulation T. Let φ be a corresponding projection of S into T and let X ⊂ E(S) be a sparse subset with |X| = 4 such that there is a vertex of G(K) which is a triad for X. Then for every circuit C in T, |E(C) - φ (X)| ≥ 2 or there is a short circuit in T.

restriction of cX to E(R) and the lift of cH by φ are all the same ring coloring,and this ring coloring is in CH I CS. Proof: Note that by Lemma 1.2.1, H is a neartriangulation, φ wraps R around H and φ (X) is sparse in T. Since E(H) I φ (X) = φ , c restricted to H is a tricoloring of H, which we henceforth denote cH. The tricoloring c also defines a tricoloring cX of S modulo X as follows. cX(e) = c( φ (e)) for e ∈ E(S). By property (iii) of projections, if r ∈ F(S), and r is incident to the distinct edges e, f, g ∈ E(S) then φ (r) is a face

Proof: Let Y = φ (X). Lemma 1.2.1 guarantees that Y is sparse in T. Let C be a circuit in T. Since T is loopless, |E(C)| > 1. If |E(C)| = 2 then because all faces are triangles, C cannot bound a face and must therefore be a short circuit. If |E(C)| = 3 then C must bound a face, for otherwise it would be a short circuit. Thus, the sparseness of Y implies that Y has at most one edge in common with E(C) and so the desired inequality holds. If |E(C)| = 4 and C = {x1, x2, x3,

in F(T) which is incident to edges the edges φ (e), φ (f) and φ (g). If X T {e, f, g} = φ , then

φ (X) I { φ (e), φ (f), φ (g)} = , so {1, 0, 1} = {c( φ (e)), c( φ (f)), c( φ (g))} = {cX(e), cX(f),


40


x4} then either C is a short circuit or some pair of Theorem 1.2.3 Let T be a minimum diagonally opposite vertices of C, say x1 and x3 counterexample. Then no configuration are adjacent to each other and {x1, x2, x3} and isomorphic to one in Appendix 1appears in T. {x3, x4, x1} form triangular faces in T. Since Y is Proof: Let T be a minimum counterexample, and sparse, Y has at most one edge in common which suppose that K is a configuration in Appendix 1 which appears in T. By Theorem 3.3.1, we know each of these two faces and so |E(C) I Y | ≥ 2 that T is internally 6 - connected. and the inequality follows. If |E(C)| ≥ 6, then |X| Let H and S be as at the beginning of Section ≥ 4 implies |E(C) - φ (X)| ≤ 2. So we may 1.2.2. We first notice that if CH I CS includes assume |E(C)| = 1 and thus that |X| = 4. Let C = two non-equivalent colorings, or if k ∈ CH I x1, x2, x3, x4, x1. We may assume |E(C)-Y | = 1 C 1.2.2 S for some k ∈ CS -U, then Lemma and so all 4 edges of Y are in E(C). Let int(C) implies that T would have at least two nondenote the sub drawing of T induced by the equivalent vertex-4colorings. Therefore, we vertices in one of the arc-wise connected I C (C* C ) may assume that C ⊂ components of Σ - C and let ext(C) denote the H S S U the sub drawing of T induced by the vertices in η ({u}), for some u ∈ U. the other arc-wise connected component of Σ Assume first that no u ∈ U is C-removable. Therefore, every u ∈ U is D-removable, which C.We may assume that either |V (int(C))| ≤ 1 or implies that u ∉ MCS(u) for every u ∈ U. Since |V (ext(C))| ≤ 1, or else C is a short circuit. If |V K appears in Appendix 1, Theorem 1.2.2 implies (int(C))| = 0 or |V (ext(C))| = 0, then there are that K is either D-reducible or C-reducible. We only edges in one of the two disjoint regions of first consider the case that K is D-reducible, and the sphere defined by C, but this will create triangular faces containing two edges of Y , a therefore that MCS (C* - C) = φ . Since T is a violation of the sparseness of Y . Thus we must minimum counterexample, H has a vertex-4have |V (int(C))| = 1 or |V (ext(C))| = 1. By coloring and thus CH ≠ φ ,. If CH I CS = φ , symmetry, we may assume the former and we then CH ⊂ (C* - CS) which implies that CH ⊂ will let y denote the vertex for which V (int(C)) = MCS(C* - CS) since Lemma 1.1.3 implies the {y}. Note that y has degree 1 and faces all the latter is consistent. This however, is a edges of Y and so cannot be a triad for Y . Since contradiction. Assume then that CH I CS = η there is a triad v for Y , v ∈ V (ext(C)), v is incident to at least three vertices xi1 , xi2 , xi3 ⊂ ({u}) for some u ∈ U. This also gives rise to a {x1, x2, x3, x4, x1} which are, in turn endpoints of contradiction, because then CH ⊂ (C* - CS) U edges in Y . By relabeling, we may assume xi1 = η ({u}) which implies CH ⊂ MCS(u) since by x1 and xi2 = x2 and that i3 ∈ {3, 4}. If i3 = 4 then Lemma 1.1.3, MCS(u) equals the maximal {v, x4, y, x1} form a short circuit. So assume that critical subset of (C* - CS) U η ({u}) and CH is i3 = 3, and deduce that {v, x3, x4, x1, x1} is either a short circuit, or there is a degree 1 vertex w that critical by Theorem 1.1.3. Thus u ∈ CH ⊂ MCS(u) which contradicts that u is D-removable. is adjacent to {v, x3, x4, x1, x1}. We may assume {v, x3, x4, x1, x1} is not a short circuit in T, so the Now consider the case that K is C(4)-reducible, later holds and v has neighbors {x1, x2, x3,w} and let X be a sparse subset of S such that φ (X) which form a short circuit. This completes the is contractible in T and that CS(X) I MCS(C* proof of the theorem. CS) = φ . By Theorem 1.1.2, there is a tricoloring We will call any A - contract X with |X| = 4 and of T modulo X which we denote by c. Let cH and for which X has a triad a safe contract. cX be the colorings that are guaranteed to exist by Theorem 1.2.2 Every configuration in Appendix 1 is U-reducible. Moreover, for every u ∈ U that Lemma 1.2.4, let cH(R) be the lift of cH by φ is C-removable, there is a safe u-contract X. and let cX(R) be the restriction to R of the Proof: Let K be a configuration in Appendix 1. coloring cX. Lemma 1.2.4 says that cH(R) = cX(R) The computer verifies that for every u 2 U, u is and that cX(R) ∈ CH I CS. Also, we know that either D-removable or C-removable. When the c X(R) ∈ CS(X). Therefore cX(R) ∈ CH I CS I color is C-removable, the computer finds a uCS(X). From this and our assumption that CH ⊂ contract X and verifies that X is safe. After showing that every u ∈ U is either D-removable (C* - CS) U η ({u}) for some u 2 ∈ , it follows or C-removable, the computer verifies that the that cX(R) ∈ U and that CH ⊂ MCS(cX(R)) configuration K is U-reducible. If at least one of since by Theorem 1.1.3, CH is consistent and by the u 2 U was C-removable, then U-reducibility Lemma 1.1.3, MCS(cX(R)) equals the maximal for K is immediately established. Otherwise, the consistent subset of (C* -CS) U {cX(R)}. Since computer verifies that K is either D-reducible or we are assuming that every u ∈ U is DC(4)-reducible Reducibility For The Fiorini-Wilson-Fisk Conjecture

41


removable, it follows that cX(R) is D-removable, and hence that cX(R) ∉ MCS(cX(R)). This however is a contradiction because we know that cX(R) ∈ CH ⊂ MCS(cX(R)). This completes the proof of Theorem 1.2.3 in the case when no u ∈ U is C-removable. We may assume then that there is a u ∈ U which is not D-removable and hence is C-removable. By Theorem 1.2.2, we know that there is a safe u - contract X. We now show that we may assume CH ⊂ (C* - CS) U η ({u}). If not, then from our previous assumptions we know that there is a u’ ∈ U with u ≠ u’ such that CH ⊂ (C* - CS) U η ({u’}). Now CH ⊄ C* - CS, otherwise CH

[4] Akbari, S.: Behzad, M.; Haijiabolhassen, H.Mahmoodian Uniquely total colorable graphs. Graphs Combin 13, (1997) 305-314 . [5] S.Satyanarayana. “Complete Works on Four Colour theorem (Research Note Book)”, Satyan’s Publications in Progress. [6] S.Satyanarayana “Programming Approach on Discharge on Four Colour Theorem”, “International Journal of Computational Mathematical Ideas, Vol 1 No 1, (2009) PP 5212.

⊂ C* - CS ⊂ (C* - CS) U η ({u}). It follows then that u’ ∈ CH ⊂ MCS(u’), so u’ is not D-

[7] S.Satyanarayana “Programming Approach on Reduce on Four Colour Theorem”, “International Journal of Computational Mathematical Ideas, Vol 1 No 1, (2009) PP 213-236.

removable. Therefore u0 is C-removable, by Theorem 1.2.2. Thus, we could let u’ play the role of u. This proves that that we may assume u ∈ CH ⊂ MCS(u) ⊂ (C* - CS) U η ({u}). Since X is a safe contract, Theorem 1.1.2 guarantees that T has a tricoloring modulo X which we denote by c. Using Lemma 1.2.4 and its notation, we write cH(R) for the lift of cH by φ , and cX(R) for the restriction to R of the coloring c Lemma 1.2.4 guarantees that cX(R) = cH(R) and that cX(R) ∈ CH T CS(X). Since CH ⊂ MCS(u), it follows that cX(R) ∈ MCS(u) I CS(X). This is a contradiction however, because X is a u-contract implies that CS(X) I MCS(u) =φ . This completes the proof of Theorem 1.2.3.

[8] S.Satyanarayana, Dr.J.Venkateswara Rao, “On Planar Coloured Graphs”, “International Journal of Computational Mathematical Ideas, Vol 1 No 2, (2009) PP 6-8. [9] S.Satyanarayana, Dr.J.Venkateswara Rao, Dr.A.Rami Reddy, “Theorems on Structure of Minimum counter example to the FkoriniWilson-Fisk Conjecture”, “International Journal of Computational Mathematical Ideas, Vol 1 No 2, (2009) PP 20-25. [10] S.Satyanarayana, Dr.J.Venkateswara Rao, “Configurations, Projections and Free Completions on Uniquely Planar Coloured Graphs”, “International Journal of Computational Mathematical Ideas”, Vol 1 No 3, (2009) PP 80-86.

Acknowledgements:

[11] Xu, Shaoji, The size of Uniquely colorable graphs. J.Combin. Theory (B) 50, (1990) 319-320.

We would like to express our thanks to referees for valuable comments that improved the paper. The first Author wish to thank Principal & Management, Sri Venkateswara Institute of Science & Information Technology, Tadepalligudem for their encouragement and cooperation in the preparation of their research paper. References: [1] G.Chartrand, D.Geller. “On uniquely colorable planar graphs”, J.Combin. Theory 6 (1969) 271-278. [2] A.G.Thomason. “Hamiltonian cycles and uniquely edge colourable graphs”, Ann.Discrete Math. 3 (1978) 259-268.

[3] Akbari S. “Two conjectures on uniquely totally colorable graphs”. Discrete Math.1-3 (2003). 41-45. Reducibility For The Fiorini-Wilson-Fisk Conjecture

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PERCEIVING PLAGIARISM USING WEIGHTED WINDOW APPROACHPERFORMANCE ANALYSIS Bobba Veeramallu@, T. Pavan Kumar#, Prof.V.Srikanth$, Prof.K.Rajasekhara Rao^ @

Dept. of CSE, KLEF University, [email protected] Dept. of IST, KLEF University, [email protected] $ Dept. of IST, KLEF University, [email protected] ^ Dept. of CSE, KLEF University, [email protected] #

ABSTRACT Plagiarism of data from the internet is a rapidly growing problem in this competitive world. Most of the students get accustomed to sometimes get their work done with a “cut and paste” approach in assembling a paper in part. It perverts learning and assessment of subject. Detection of cut and paste plagiarism is a time consuming and cumbersome task when it is done manually, and can be greatly aided by automated software tools. This paper presents an approach on the implementation of a software tool called SNITCH that uses a fast and accurate plagiarism detection algorithm using the Google Web API. Several issues related to plagiarism detection software are discussed and in addition to it the performance and accuracy study are also dealt with. Keywords: Plagiarism, algorithm, design 1. INTRODUCTION Plagiarism is a pervasive form of academic dishonesty in collegiate settings. Since it distorts learning and assessment, deterring and detecting it are crucial to maintaining academic integrity. Plagiarism fundamentally warps two essential aspects of education, learning and assessment. Students who submit plagiarized work deprive themselves of the learning opportunities afforded by authentic academic productions and by assessments of those productions by educators [2, 3]. Large class sizes and an increase in writing assignments that result from writing across the curriculum combine to make detection of plagiarism burdensome. The rapid increase of written material on the Internet and its ease of appropriation contribute to the problem. Detection of cut and paste plagiarism is time consuming when done by hand, so plagiarism detection software has emerged in response[13]. This paper describes the design of an algorithm for automated

plagiarism detection and an associated software tool called SNITCH. 2. DEFINITION OF PLAGIARISM “Plagiarize” derives from a Latin root meaning “to kidnap”[4]. It is a form of dishonesty that misrepresents intellectual property and that deprives the creator of intellectual property due recognition. In academia, it is an unpardonable mistake. Examples of plagiarism include failure to give appropriate acknowledgement when using other’s words and presenting other’s line of thinking[5]. Plagiarism substitutes the physical labour of theft and misrepresentation for the labour and growth of learning. Moreover, plagiarism erodes the sense of community that is essential to free academic inquiry. Misrepresentation masks the true identity of members of the community.


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to try to find similarities in the files that could indicate plagiarism. This form of software continues to prove an invaluable tool both for the detection of cheating and for grading assistance in several courses.

3. DETECTING PLAGIARISM Detecting plagiarism can be a time-consuming task, all characteristics that make the problem ideally suited to a software solution. In this, a brief discussion of the general, non-software-based approach to plagiarism detection is provided, followed by an overview of existing software solutions and issues to be considered in the design of such software solutions.

3.2.1 Existing Softwares The Eve2 software performs adequately for shorter papers, but its report generation feature can be inaccurate. Although Eve2 is really designed to determine how closely a student paper matches a single online source, for simply detecting the simple presence of plagiarism it is acceptable. TurnItIn is a well-respected service, where student papers are submitted via the Internet for analysis. Reports are generated and returned to the instructor, normally within four to six hours of submission. This service is expensive. The MyDropBox service is a recent and able competitor to TurnItIn, with a similar pricing strategy and turn-around time. Reports are generated within 24 hours of submission. But the cost varies according to the plans that are chosen.

3.1 Manual Approach In a time with a seemingly limitless cache of data from which to “borrow” from the Internet, an approach commonly used by educators to detect the cut and paste approach to plagiarism is to highlight suspicious excerpts in a paper, and then enter them into an online search engine. If identical excerpts are found in an online source, it is likely that the excerpt was plagiarized [6]. Certainly, if multiple such instances of identical excerpts are discovered in a single paper, a strong decision can be made that intentional plagiarism is present.

4. TECHNICAL ISSUES Plagiarism detection software that uses the Internet for its corpus is subject to effective countermeasures. One is that Web sites associated with the sale of term papers are not openly connected to the World Wide Web. Materials acquired through these sites are likely to escape detection. In addition, Web sites can deploy software that repels Web crawlers such as those used by TurnItIn[8,9]. Another issue is that an instructor can recognize where some copied material has been slightly revised by replacing, adding or deleting one or more words to avoid detection. Software can duplicate this approach, although it is a difficult problem to solve. Because search engines allow for “wildcards” a simple approach of replacing any short or common words in the passage with a wildcard may be an effective technique to detect cut and paste plagiarism. Other technical issue is that of false positives and false negatives. Corpus-based programs, such as TurnItIt.com, do an excellent job of finding matches between student submissions and items in their database. These programs, however, do not distinguish between matches that are properly cited and those that do not, contain a high index of plagiarism. Software tools that are available for use by instructors could also be used by students, such that students may try to defeat automated plagiarism detection by using such a program while submitting their work [10]. Because of this the student submits

The problem with this manual approach is that it is labour intensive, requiring detailed, onscreen reading and re-reading of each paper, coupled with the repeated use a search engine including copying and pasting selected passages from each paper using a mouse and keyboard. Although this tedious approach is perhaps less exhausting than referring textbooks looking for potential matches, or being intimately familiar with enough such textbooks and other sources to recognize stolen excerpts. 3.2 Software Approach Software has been developed that reduces a lot of the labour intensive aspects of manual approach of detecting plagiarism. There are a number of commercially available software tools and services that perform automated checking. But the cost is relatively high, the turn-around time is sufficiently long, or both, reducing the availability to those educators with budget constraints. These available automated approaches often assume that large sections of a paper, or even entire papers, would be copied verbatim[7]. Yet, for technical oriented research papers, such as in computer science and engineering disciplines, a cut and paste approach where paragraphs, sentences or even phrases can be gathered into a report is easier to get away with. Software tools have been successfully used for detection of plagiarism in student programming assignments for many years. Two program files are compared after some compiler like pre-processing


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matches will be found if the window is larger that the plagiarized passage. Increasing or decreasing N will increase or decrease the thoroughness, and lengthen or shorten the time taken to analyze a paper, since the time to perform each search is determined by load on the Internet, and Google specifically, rather than the capabilities of the user’s own computer.

his work without understanding the original content. The issues and approaches raised here, and the features and techniques used in previous tools and manual methods, provide the motivation for the design and implementation of the SNITCH software. 5. DESIGN OF SNITCH

5.3 Evaluation Here in the design of SNITCH the design of the plagiarism analysis and detection algorithm used in SNITCH is discussed in detail.

An initial evaluation of the SNITCH software was performed to measure its effectiveness at detecting instances of plagiarism in custom-designed plagiarism benchmarks and a sampling of typical computer science student term papers. Results are compared with results for the same papers using the only other available practical and cost-effective software tool, Eve2. No formal comparison was done with online subscription services due to cost constraints.

5.1 Algorithm The algorithm developed for SNITCH uses a sliding window technique and average length per word metric to identify potential instances of plagiarism. In general, the algorithm uses the following steps: Open a document Analyze the document o Read a window containing the first/next W words o Measure the number of characters for each word o Calculate the Weight of the window, the average number of characters per word for the words in the window o Associate this Weight with this particular window for later use. o Repeat for all such windows in the document, shifting the window forward in the document by 1 word Search for plagiarized passages o Order windows in decreasing order, and eliminate overlapping windows. o Rank all windows in decreasing order by Weight. o Select the top N weighted windows, and search the Internet for each, gathering the top search result (if any) for each Generate a report – Create an HTML document containing statistics of search time, number of searches performed, percentage of document found to be plagiarized, and other Pertinent statistics. Include the original document with embedded HTML tags linking plagiarized passages to their sources on the Internet [11]. The algorithm is parameterized to allow variation of the size of the sliding window (W) and number of searches performed (N), to enable finetuning on a per-user basis. Decreasing W will lead to more potential candidates, but may increase false positive results because the fewer words there are in a search phrase, the more likely they could occur by chance. Increasing W can improve the confidence in individual search results, but if set too high, it may reduce that likelihood that any

5.3.1 Comparison of Performance Experiments using four synthetic benchmark term papers and a sampling of 10 actual student term papers were performed. The synthetic benchmarks consisted of carefully crafted documents containing known amounts and instances of cut and paste plagiarism representing hypothetical papers containing high, moderate, minimal, and no plagiarism[4,13]. Actual student papers were manually analyzed using careful online detective work, and were divided into similar groupings based on the prevalence of plagiarism that was found. These student papers were all rough drafts, and any plagiarism detected was later removed by the Students. Three experiments were performed to measure analysis speed and accuracy of SNITCH on the synthetic and real documents, and in comparison with the commercial plagiarism detection program Eve2. Manual stats

SNITCH stats

PET

INSTANCES

PET

INSTANCES

0

0

0

0

15

5

40

2

40

10

50

5

75

19

63

12

TABLE 1: ANALYSIS RESULTS The graphical representation that shows the performance of SNITCH is better than that of the manual approach.


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6. CONCLUSION In this paper we have noticed that SNITCH program provides an efficient and accurate alternative to commercial tools and services, producing good accuracy and faster analysis at affordable cost. The availability of SNITCH will increase the threat of detection and prevents individuals from using cut and paste plagiarism. FIG 2 : MANUAL STATS 7. FUTURE WORK This SNITCH tool can compare only twenty documents at a time. In future we enhance the performance of SNITCH tool so that it can be used to look into more files with a better accuracy and speed even for pdf files. ACKONOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper. REFERENCES:

FIG 3: SNITCH STATS

[1]. Freedman, M. (2004, March). A tale of Plagiarism and a new paradigm. Phi Delta Kappan 85 (7), 545-549. Retrieved May 6, 2004 from Academic Search Elite.

5.3.4 Comparison of performance of SNITCH and Eve2 Eve2

SNITCH

6:45

0.15

6:45

0.18

7:00

0.38

7:30

0.44

[2]. Kansas College Give First ‘XF’ Grade to Plagiarist. (2003, December 8).Community College Week. Retrieved May 6, 2004 from Academic Search Elite. [3]. Scanlan, P. (2003, Fall). Student online plagiarism: how do we respond. College Teaching, 54 (4) 161-164. Retrieved May 6, 2004 from Academic Search Elite [4]. Kellogg, A. (2002, February 15).Students plagiarize online less than many think, a new study finds.Chronicle of Higher Education, 48, A44. Retrieved May 5, 2004 from Academic Search Elite

TABLE 2: ANALYSIS TIME (RESULTS) The graphical representation is as follows average analysis time 8

[5]. Howard, R. (2002, January). Don’t police plagiarism. Just teach! Education Digest, 67 (5), 46-50. Retrieved May 5, 2004 from Academic Search Elite.

7 6 5 Series1 4

Series2

3

[6]. Carroll, J. A Handbook for deterring Plagiarism in Higher Education. Oxford, Oxford Centre for Staff and Learning Development, 2002.

2 1 0 1

2

3

4

5

[7]. Whitley, B. Academic Dishonesty: an Educator’s Guide. Mahwah, N.J., Erlbaum, 2002. (Hesburgh LB 3609.W45 2002)

FIG 4 : ANALYSIS TIME COMPARISION RESULTS


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[8]. Decoo, W. Crisis on Campus: Confronting Academic Misconduct. Cambridge, MA., MIT Press, 2002. (Hesburgh LB 2344 .D43 2002). [9]. Harris, R. A. The Plagiarism Handbook.Los Angeles, Pyrczak Publishing, 2001 www.AntiPlagiarism.com [10]. Lathrop A. Student Cheating and Plagiarism in the Internet Era.Englewood, CO., Libraries Unlimited, 2000. (Hesburgh LB 3609 .L28 2000) [11]. C. Humes, J. Stiffler and M. Malsed. Examining Anti-Plagiarism Software: Choosing the Right Tool. Claremont-McKenna College technical report. 2003. [12]. Brian Martin. Plagiarism: policy against cheating or policy for learning? Nexus: Newsletter of the Australian Sociological Association, 16:2, pp. 1-12, 2004. [13]. L. Renard. Cut and paste 101: Plagiarism and the Net. Educational Leadership, 57:4, pp. 38-42, 2000 [14]. Arwin, C. and S. M. M. Tahaghogh (2006). Plagiarism Detection Across Programming Languages. ACM International Conference Proceeding Series, vol. 171. Proceedings of the 29th Australasian Computer Science Conference, Hobart, Australia, vol. 48 ,pp. 277 – 286. [15]. Niezgoda, S. and T. Way. (2006) SNITCH: A Software Tool for detecting Cut and Paste th

Plagiarism. Proceedings of the 37 Special Interest Group on Computer Science Education. Pp. 51 – 55. New York: Association for Computing Machinery.


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SYSTEM REPRESENTATION FOR SOFTWARE ARCHITECTURE RECOVERY Shaheda Akthar@, Sk.MD.Rafi# @

Assoc.Professor, Sri Mittapalli College of engineering, Affiliated to JNTU, Kakinada, Email:[email protected] # Asst.Professor, Sri Mittapalli Institute of Technology for women, Affiliated to JNTU, Kakinada. Email:[email protected]

ABSTRACT The source code of the system needs to be analyzed in every step. For analyzing, system should be represented in the form of a tree using any of the models that suits for our system because the source code of the soft ware system will be highly detailed. So we can’t analyze the source code without using any model. So, for analyzing a model is used to describe the entities and relationships .so, to represent that model a graph is used. In this chapter we are considering a graph known as Attributed Relational graph [ARG]. Software systems uses graphical representation because it will be very easy for them to trace out the errors ,if any the source code of the system should be represented in the form of graph and that graph should be divided in to smaller sub graphs based on their properties. After representing the system as a graph, we have to find out the entities that were related, To partition the graph into smaller modules. Related entities can be found out using association properly. After finding out the similar entities [1], file level and function level measurements are defined. Keywords:ARG,XML,DTD,Domainmodel,datamining,association. 1.1. Graph representation of a software system A Graph is a collection of nodes (vertices) and links (edges). So, when a software system is to be designed as an attributed relational graph, there should be a specific domain model to represent the nodes and links of the graphs. 1.1.1. Specified Domain Model A Specific domain model [2] is used to represent the software system which consists of entities, in the form of graphs, charts etc. For any domain software system a domain model should be proposed. For example, for every programming language also domain model should be proposed. 1.1.2. Domain Model for Programming Language The basic theme is to obtain the ER graph of a system at the source code level. The domain model should be represented in terms of classes and associations. This can be done by 1. Considering source code constructs as the domain model classes which

2.

includes file, function, statement, expression, variable Finding Relationship between source code entities as an association between model classes. XML notation can also be used to define a domain model using Document Type Definition (DTD). The major advantage in using these type of representations is, XML can easily validate data. After representing a software system using a domain model it should be analyzed. A software system should be analyzed at two levels i) File level ii) Function level files and directories are to be used to make analysis at the file level. Global variables, aggregate data types and functions are to be considered to make analysis at the function level.


48


Entities and relations of the specified domain model will be analyzed in directly if both the file level and function level analysis is carried out in a proper way. In Abstract domain model the different types of entities are a subset of the types of entities in the software system’s source code, and each relation in the abstract domain model is an aggregation of one are

more relations in the software system’s source code. The advantage of this domain model is that it is simpler than the detailed source-code domain model. It is language independent for procedural programming paradigm; and yet it is adequate for architecture level Analysis.

Here the file entities with other type of entities are separated using separate entity-abs. this separation is because of these were of the same granularity. From the above figure, we can observe the following properties. i) Entity-abs in a class ii) Relation-abs inherits all the properties of that are identified by every entity in the abstract domain model. iii) File entity with other type of entities are separated using simple entity-abs (simpEntabs).this separation is because

of these were of different granularities. iv) Each relation contains the attributes from and to, to denote the source and destination entity for that relation. Two types of relations exists i)file level and ii)function level

File level relations File level relations are of 4 types. i) import-resource ii) export-resource iii) use-resource System Representation For Software Architecture Recovery

49


iv)contains-resource

Entity-abs attribute

Example

Description

Name

“des”

File#

4

Name of the entity in the source code File number of the source code file where the entity is defined

line#

45

Line number of the entity in the source code file

Implementid

13

Unique identifier of the object in the source-level domain model

Function level relations Relation use-F: this relation is defined between two function-abstractions F and F1, denoting that the source code function f calls the source code function of f1 Relation use-T: this is defined between a function abstraction F and a data type abstraction T, denoting that function f reads the value of a variable v and the variable v is of type t, and t is the implementation of data type abstraction T. Relation use-V: this is defined between a function-abstraction F and a variable abstraction V, denoting that the function f reads the value of global variable v. File level relationships: Relation cont-R(contain-resource): A file-abstraction is called a composite entity and a function-abstraction, a typeabstraction, or a variable-abstraction is called a simple-entity such that a composite entity contains a set of simple entities. this relation is defined between a fileabstraction L and either a functionabstraction F, or a data type-abstraction T, or a variable-abstraction V, denoting that the source-file l defines the implementation of function f. Relation use-R(use resource): this relation is defined between a fileabstraction L and either a functionabstraction F, or a data type-abstraction T, or a variable-abstraction V, denoting that source-file l defines a function f1 ,

and function f1 calls function f, and function f is the implementation of function abstraction F. Relation imp-R (import-resource): is defined between a file-abstraction L and entity-abstraction R, denoting that L uses R but does not contained in R1. Relation exp-R (export-resource): is defined between a file-abstraction L and entity-abstraction R, denoting that L contains R and another file-abstraction L1 uses R. 3.1.3. Source Graph To analyze the software system, a source graph should be modeled. The notation for Attributed Relational Graph (ARG) that is presented in is adopted to define all graphs. The attributed relational graph representation of the source-graph is a six-tuple Gs={Ns, Rs, Es, µ s, €s)2 that is defined as: Ns: {n1,n2,……..,nn} is the set of attributed nodes, obtained from the abstract domain model. Rs: {r1,r2,…..rm} is the set of attributed edges, obtained from the abstract domain model. As: alphabet for node attribute values such as node labels, node types, and their values. Es: alphabet for edge attribute values such as edge labels, edge types, and their values. µs: Ns->(As × As)p: a function for returning the “node attribute, node attribute value” pairs where p is a constant and denotes the number of node attributes. €s: Rs-> (Es × Es)q: a function for returning the “ edge attribute, edge attribute value” pairs where p is a constant and denotes the number of edge attributes.

Figure 1.1: an attributed relational graph representation of a source graph Gs={Ns, Rs, Es, µ s, €s).


50


Ni(Gs) = ((type, function abs),(des,”/a/..mgr”),(id,13),(line45),(fil e4))indicating that node I of the source graph G is an entity of type Functionabs with des “/a/../mgr” and idF10 and it has been defined in the line 45 of the source file 4; and E j (Gs)=((from,n2),(to,n8),(type, useF),(#line89),(file#4) indicating that edge j of the source graph Gs is an object of type use-F that relates the function n2 to the function n8 with a function call relation in line89 of file 4. 3.2. Computing maximal association Maximal association can be extracted by data mining and is considered as an interesting property for grouping the entities in to cohesive modules. Maximal association is defined in a group of entities in the form of a maximal set of entities that all share the same relation to every member of another maximal set of entities. For every set of functions, denoted as F, we can determine a set of shared entities, denoted as E, where every function f in F has a relation r to an entity e in E. for example, two functions f and g can share the data type t and variable v by the relations use-T and use-V, respectively. The operation sh-ents(F) returns the set of shared entities E for the set F as follows: F; ∃ rel :X sh-ents(F)={e | ∀ f ∈ • X ∈ {use-F,use-T,use-V} ∧ (f,e) ∈ rel } Similarly, for every set E of entities we can determine a set of functions F, where every function f in F has a relation r to an entity e in E. the operation sh-funcs(E) returns the set of sharing functions F for the set E as follows: sh-funcs (E)={f | ∀ e ∈ E; ∃ rel :X • X ∈ {use-F,use-T,use-V} ∧ (f,e) ∈ rel }

Data mining [3] refers to a collection of algorithms for discovering interesting relations among data in large databases [4]. Frequent items can be found out by applying association rules [5], which is an implication of the form x ≈> y, where x and y are disjunctive item sets which are subsets of set of items I={i1,i2,i3……iN},N ≥ 2.The association rules [6]are generated by frequent-itemsets and the frequent itemsets can be grouped by the Apriori algorithm. A k-itemset whose elements are contained in every basket of a group of baskets. The cardinality of this group of baskets must be greater than a user-defined threshold called minimum-support. In order to apply the Apriori algorithm on the source-graph Gs, we define B(Gs) as the basket representation of the source-graph Gs=(Ns,Rs): B (Gs) = {b: Function-abs; I: set (Entity-abs) Figure 3.2(a),(b),(c) Application of data mining in extracting frequent itemsets.(d) Representation of the frequent itemsets for system analysis. The above figure illustrates the process of generating frequent itemsets from the sourcegraph Gs. In figures (a),(b), the entities and relationships in source-graph Gs are represented as a data base of baskets and items B(Gs). in figure (c), two frequent-itemsets generated by the apriori algorithm on B(Gs) are shown. Each frequent-itemset is presented as a tuple ({baskets, {items}), where {baskets} is the set of functions and {items} is the set of “relation and entity” pairs, such that: {baskets}=shfuncs({items}) {items}=shents({baskets}) Hence the set of functions in {baskets} and the set of entities in {items} are related by maximal association. Finally figure (d) represents a small portion of frequent 5-itemsets extracted from a software system’s source-graph. The first line is interpreted as: all the functions F774, F800, F807 use functions F209, F811, F812, and use aggregate type T5 and global variable V259. Each frequent 5-itemset is equivalent to a concept with intended size 5. 3.1 Similarity measure between two entities A similarity measure[7] is defined so that two entities that are alike possess higher similarity value than two entities that are not alike. the clustering research literature provides a rich collection of techniques for extracting groups of related software entities using different similarity metrics namely association metrics, correlation metrics, and probabilistic metrics but the jaccard metric produces better clusters than the others.

A of functions F and a set of entities E are related by maximal association, iff: F = sh-funcs (E) ∧ E = sh-ents (F) 2.1. Data Mining System Representation For Software Architecture Recovery

51


Therefore, the association property is defined between either: sharing entities (clustering), or shared entities (data mining). To apply the association rules to a graph an associated graph of graph nodes is defined, when two more source nodes share one or more sink nodes. A source node is a node where an edge emanates from it. A sink node is a node where an edge points to it. In this sense, the whole group of source nodes and sink nodes are denoted as an associated group. By considering the source nodes as the “basketset” and the sink nodes as the “itemset” “

3.2Entity association similarity measure The entity association measure is an extension to the notion of association in the clustering and data mining domains that are briefly compared below: Clustering: The association similarity is defined between two entities are the proportion of the numbers of shared and total attribute-values, figure3.3 (a). [8] Data mining: The association rule is defined between two sets of items as the proportion of the numbers of the shared and total baskets, figure 3.3(b). [4]

Fig.,3.3 Association Rules 3.3.2. Source region A source region Gsreg=(Nsreg,Rsreg,Asreg,Esreg,Usreg,Esreg) of a source graph Gs=(Ns,Rs,As,Es,Us,Es) is a sub graph of Gs.In the source region Gsreg(j) each node ni!=nj satisfies the association property entAssoc(nj,ni)>0 with respect to node nj.We call ni the main seed of the source region Gsreg(j) and use it as the identity of this source region.

∧

= { ni | ni ∈ Ns ∃ nj ∈ Ns • entAssoc (nj , ni) > 0 } ∪ { nj } R jsr = { ns ; nt | ns , nt ∈ N jsr ∧ ∃ rk

N

sr j

• rk = (ns , nt)

∧r

k

∈ Rs }

For a given source graph Gs=(Ns,Rs) we generate | Ns | source regions.

A source graph Gs=(Ns,Rs) at functional level represented as an ARG


52


determine the maximum association value of each node with respect to the source regions main seed. Applying Apriori: In the source region Gsreg(1) with node 1 as main seed in fig 3.3.2(d) We consider two associated groups with nodes 1,7,10,2,13 with entAssoc of 4;and 1,6,10,7,2 with entAssoc of 3,5. The similarity value of node 10 to the main seed node 1 is 4 and is obtained from the first associated group

Fig 3.3.2(a) Source region with 1 as main seed

3.3.2(c) Source region Gsreg(1) after applying Apriori Fig 3.3.2(b) Source region with 6 as main seed In these figures fig 3.3.2(a) and fig 3.3.2(b) represent 2 source regions Gsreg(1) and Gsreg(6) of the source graph Gs. Each node of Gsreg (1) is a member of an associated group with respect to main seed n1. However it is not clear what is the highest association value of each node with regard to main seed n1 since each node can be a member of several associated groups, thus different association values in each group. The Apriori algorithm computes all the associated groups in a source region and allows to 3.3.2 (d) Source region Gsreg(6) after applying APriori At phase I of the incremental graph matching process the user may select a main seed nj that corresponds to the source region Gsreg(j) for the current matching phase i. 3.4 Similarity measure between two groups of entities Group association can be defined as a similarity measure between two groups of system entities gi, gj based on the similarity between two entities in a graph.

Three methods of similarity measures between two groups of entities are commonly used in clustering. They are 1. Single linkage 2. complete linkage 3. group average similarity In single linkage method, the maximum or minimum similarity between every pair of entities, one in each group, is considered as the similarity value between two groups. The single linkage computes higher similarity value for the groups that are non-compact and isolated, where as, complete


53


linkage computes higher similarity values for cohesive and compact groups. To avoid the extremes, the group average similarity method defines the similarity between two groups as the average of similarities between all pairs of entities that are made up of one entity from each group. In this thesis, the group average similarity method is adopted to compute the proposed group association metric group assoc, as follows: In this equation, the first summation iterates over every entity in group g i and the second summation iterates over every entity in group g k in order to add the similarity values

References [1] Douglas B. West. Introduction to Graph Theory. Prentice Hall, 1996. Page 19. [2] Allan Terry et al. An annotated repository schema, domain-specific software architecture. Technical report, TFS and ARDEC, October 1993. [3] Baeza-Yates, R., & Ribeiro-Neto, B. (1990). Modern Information retrieval, ACM Press, New York. [4] Bayardo Jr., R.j (1997). Brute-force mining of high-confidence classification rules. In Heckerman, D., Manila, H., & Pregibon, D. (Eds.), Proceedings of the Third International Conference on Knowledge discovery and Date Mining (KDD-97), pp. 123-126 Newport Beach, CA. AAAI Press. [5] Agrawal, R., & Srikant, R. (1994). Fast algorithms for mining association rules. In Proceedings of the 20th International conference on very Large Databases (VLDB94), pp. 487-499 santiago, Chile. [6] Ahonen-Myka, H., Heinonen, O., Klemettinen, M., & verkamo, A.I. (1999). Finding cooccurring text phrases by combining sequence and frequent set discovery. In Feldman, R. (Ed.), Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI-99) workshop on Text Mining: Foundations, Techniques and applications, pp.1-9 Stockholm, Sweden. [7] Arun Lakhotia. A unified framework for expressing software subsystem classification techniques. Journal of Systems and Software, 36(3):211–231, 1997. [8] Brian S. Everitt. Cluster Analysis. John Wiley, 1993. [9] Agrawal, R… Imielinsky, T., & Swami, A. (1993). Mining association rules between sets of items in large databases. In Proceedings of the 1993 ACM SIGMOD International Conference on Management of Data (SIGMOD-93), pp. 207-216. [10] Angell, R.C., Rreund, G. E., & Willet, P.(1983). Automatic spelling correction using a trigram similarity measure. Information Processing and Management. 19(4), 255-261. [11] Anil K. Jain. Algorithms for Clustering Data. Prentice Hall, Englewood Cliffs, N.J.,1988. [12] Architectural level. In Proceedings of the 17th ICSE, pages 186–195, 1995. [13] Bayardo Jr., R. J., & Agrawal, R.(1999). Mining the most interesting rules. In Proceedings of the Fifth International Conference on Knowledge Discovery and Data Mining (KDD-99), pp.145-154 San Diego, CA.

sim j, m between every pair of entities, one entity in each group. sim j, m refers to the similarity value between node n j in group g i and n m in group g k. for every entity n m € g k that does not exist in the domain of n j the similarity value sim j, m between n j and n m is zero. Therefore, only those entities in g n k that exist in the domain D j are considered for similarity calculation between two groups. The terms |g i| and |g k| denote to the cardinality of each group.

4. System representation A soft ware system can be represented at a higher-level of abstraction in the form of a source graph Gs along with the collection of domains, which is defined as two-tuple; System= (Gs, D(Ns)) s Where G = ( Ns, Rs) ∧ D(Ns) = [Dnj | j ∈ [1 ..| Ns| ] ] , D(Ns) is an ordered sequence of entity domains D n j by the average similarity of each domain, where each domain is a search space for a module recovery. In this model the matching process searches only with the appropriate domains not the whole source graph.

ACKNOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper.


54


INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN: 0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 55-59 (2010)

EXPERIMENTAL AND THEORETICAL EVALUATION OF ULTRASONIC VELOCITIES IN BINARY LIQUID MIXTURE CYCLOHEXANE + O-XYLENE AT 303.15, 308.15, 313.15 AND 318.15K. Narendra K@, Narayanamurthy P# & Srinivasu Ch$ @

Department of Physics, V.R.Siddhartha Engg.College, Vijayawada, Andhrapradesh, 520007, Email:[email protected] # Department of Physics, Acharya Nagarjuna University, Nagarjuna nagar, Guntur, Andhrapradesh $ Department of Physics, Andhra Loyala College, Vijayawada, Andhrapradesh, 520008 ABSTRACT

A comparison of ultrasonic velocity evaluated from Nomoto’s relation, Vandael ideal mixing relation, impedence relation, Rao’s specific velocity relation and Junjie’s theory has been made in the binary mixture cyclohexnae with o-xylene at 303.15, 308.15, 313.15 and 318.15 K. Ultrasonic velocity and density of these mixtures have also been measured as a function of temperature and the experimental values are compared with the theoretical values. A good agreement is found between experimental and Vandael ideal mixing relation ultrasonic velocities. U2exp/U2imx has also been calculated for non-ideality in the mixtures. The relative applicability of these theories to the present system discussed. The results are explained in terms of intermolecular interactions occurring in these binary liquid mixtures. Keywords - Ultrasonic velocity, Binary liquid mixtures, O-xylene, Theories of ultrasonic velocity. AMS_82D15 3.

2. I. INTRODUCTION

II. EXPERIMENTAL DETAILS

The chemicals were redistilled and purified by the standard methods described 18,19. Liquid mixtures of different known compositions were prepared by mixing measured amounts of the pure liquids in cleaned and dried flasks. Ultrasonic velocity was measured by a single crystal variable path interferometer (Mittal enterprises, Model F-80X) at a frequency of 3 MHz. The working principle used in the measurement of speed of sound through medium was based on the accurate determination of the wavelength of ultrasonic waves of known frequency produced by quartz crystal in the measuring cell20,21. The apparatus is standardized first with distilled water then with benzene at various temperatures, the results obtained are found to be in good agreement with reported values in the literature. An electronically digital operated constant temperature bath has been used to circulate water through the double walled measuring cell made up of steel containing the experimental solution at the desired temperature. The accuracy of the velocity measurements is ±5ms-1. The densities of pure liquids and liquid mixtures were measured by employing a 25ml specific gravity bottle at all the temperatures and weights were taken to an accuracy of ±0.1mg.

Ultrasonic study of liquid and liquid mixtures has gained much importance during the last two decades in assessing the nature of molecular interactions and investigating the physicochemical behaviour of such systems. A survey of literature1-5 indicates that excess values of ultrasonic velocity, adiabatic compressibility and molar volume in liquid mixtures are useful in understanding the interactions between the molecules. Several reseachers6-10 carried out ultrasonic investigations on liquid mixtures and correlated the experimental results of ultrasonic velocity with the theoretical relations11-15 and interpreted the results in terms of molecular interactions. Velocities in the binary liquid mixture cyclohexane with o-xylene using the above theoretical relations are compared with the experimental values of ultrasonic velocities at four temperatures 303.15, 308.15, 313.15 and 318.15K. An attempt has been made to study the molecular interaction from the deviation in the value of U2exp/ U2 imx from unity based on the earlier studies16,17.

Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15 and 318.15k.

55


Collision Factor Theory, UCFT = Uα x Smix x Rf mix ---

The measurements were made at all the temperatures with the help of thermostat with an accuracy of ±0.1K.

(7)

III. THEORY Vandael Theory:

The adiabatic compressibility has been determined by using the experimentally measured ultrasonic velocity (U) and density (ρ) by the following formula

1 ρU 2

βad =

---

U Vandael =

(1) Where,

The molar volumes of the binary mixtures were calculated using the equation V = (X1M1+X2M2)/ρ --(2)

--

1/2

 x1 x   + 2 2  ( xM 1 1 +xM 2 2)  2  1 1 MU 2 2  MU 

(8)

x1, x2 – Mole fractions M1, M2 – Molecular weights U1, U2 – Ultrasonic velocities

Junjie Theory:

Nomoto Theory:

 x R + x2 R2  UNomoto =  1 1   x1V1 + x2V2 

1

3

UJunjie = ---

(3)

Where R1 and R2 are the radiuses of 1st and 2nd liquid V1 and V2 are the molar volumes for pure liquids R1 = molar volume of 1st liquid x (velocity)1/3 R2 = molar volume of 2nd liquid x (velocity)1/3

x1Vm1 +x2 Vm2 1/2

 x1Vm1 x2Vm2  ( x1 M1 +x2 M2 )  2 + 2  U2    U1

--

(9)

Where,

x1, x2 – Mole fractions M1, M2 – Molecular weights Vm1, Vm2 – Molar volumes U1, U2 – Ultrasonic velocities

Free Length Theory (FLT): UFLT =

k L f x (density )1/ 2

---

IV. RESULTS AND DISCUSSION The values of density, viscosity, adiabatic compressibility and molar volume for different mole fractions of o-xylene with cyclohexane at different temperatures are given in Table 1.

(4)

Where k is a constant and its values are 631, 636.5, 642 and 647 at 300C, 350C, 400C and 450C respectively. Lf – Free length

TABLE I - VALUES OF DENSITY (ρ), ADIABATIC COMPRESSIBILITY (β) AND MOLAR VOLUME (Vm) FOR DIFFERENT MOLEFRACTIONS OF O-XYLENE WITH CYCLOHEXANE AT DIFFERENT TEMPERATURES

Collision Factor Theory (CFT):

γGopal rao =

3Vm  γ RT  1−  16ΠN  MU2   

 MU2   1+  −1  γ RT  

---

(5)

3Vm  γ RT   MU2   1− γShaff’s = 3  1+  −1 --16ΠN  MU2   3γ RT     

(6)

3

Where, N = Avagadro’s Number = 6.023 x 10 23 γ = Cp/Cv = Ratio of principle Specific heats R = gas constant = 8.3144 x 107 M = Molecular weight of the liquid U = Ultrasonic velocity T = Absolute temperature Vm= Molar volume

γavg = (γGopal rao + γShaff’s) / 2 Bi =

4 x Π x (γavg i)3 x N 3

Space filling factor, Rf i = Bi / Vmi Collision Factor, Si = Uexp / (1600 x Rf i) Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15 and 318.15k.

56


ρ x 103 (kg/m3)

X1

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

0.7678 0.7768 0.7865 0.7975 0.8096 0.8222 0.8304 0.8368 0.8486 0.8614 0.8707

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

0.7625 0.7731 0.7831 0.7946 0.8070 0.8194 0.8280 0.8349 0.8466 0.8600 0.8694

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

0.7587 0.7711 0.7811 0.7930 0.8054 0.8181 0.8270 0.8337 0.8453 0.8585 0.8677

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

0.7531 0.7668 0.7787 0.7907 0.8034 0.8158 0.8254 0.8329 0.8440 0.8567 0.8659

β x 1012 (m2 N-1) at 303.15 K 85.3058 84.0621 82.2063 80.1706 77.9252 75.7054 74.1076 72.3779 69.6749 67.0067 64.0815 at 308.15 K 90.1520 88.3109 85.9713 83.5235 80.9327 78.7345 76.7718 75.1008 72.2496 69.3633 66.5164 at 313.15 K 93.2715 91.1762 88.6288 86.2194 83.4881 80.9494 78.9435 77.3219 74.2945 71.2342 68.4567 at 318.15 K 97.3125 94.8148 92.0530 89.4967 86.7270 84.0702 81.9252 80.0900 77.0418 73.5534 70.6254

Vm

(cm3 mol-1)

109.6119 110.9140 112.1388 113.2023 114.1358 115.0268 116.5612 118.3771 119.4594 120.4319 121.9249 110.3738 111.4448 112.6257 113.6155 114.5035 115.4199 116.8990 118.6465 119.7416 120.6280 122.1072 110.9266 111.7339 112.9141 113.8447 114.7310 115.5891 117.0404 118.8173 119.8258 120.8387 122.3464 111.7514 112.3605 113.2621 114.1759 115.0166 115.9292 117.2672 118.9314 120.1105 121.0926 122.6008


57


The experimental values along with the values calculated theoretically using the relations of Nomoto’s, Free length theory, Collision factor theory, Vandael ideal mixing, Junjie relation for cyclohexane+ o-xylene at the temperatures 303.15, 308.15, 313.15 and 318.15 K are given in Table 2. TABLE II – EXPERIMENTAL AND THEORETICAL VALUES IN CYCLOHEXANE +O-XYLENE SYSTEM AT DIFFERENT TEMPERATURES

X1

Uexp

UNomoto

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

1235.63 1237.50 1243.65 1250.63 1259.00 1267.50 1274.75 1284.95 1300.50 1316.25 1338.75

1235.62 1245.69 1255.81 1265.98 1276.21 1286.50 1296.84 1307.23 1317.68 1328.19 1338.75

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

1206.13 1210.25 1218.75 1227.50 1237.38 1245.00 1254.25 1261.88 1278.63 1294.75 1315.00

1206.12 1216.63 1227.22 1237.89 1248.64 1259.48 1270.41 1281.43 1292.53 1303.72 1315.00

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

1188.75 1192.63 1201.88 1209.38 1219.50 1228.75 1237.63 1245.50 1261.88 1278.75 1297.50

1188.75 1199.21 1209.76 1220.40 1231.13 1241.96 1252.88 1263.89 1275.00 1286.20 1297.50

0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

1168.13 1171.88 1181.13 1188.75 1198.00 1207.50 1216.07 1224.38 1240.13 1259.75 1278.15

1168.13 1178.70 1189.39 1200.18 1211.07 1222.07 1233.19 1244.41 1255.74 1267.19 1278.75

UFLT 303.15K 948.71 961.29 978.13 997.37 1019.29 1042.14 1058.55 1075.25 1103.60 1133.82 1165.65 308.15K 919.67 935.64 654.40 975.37 998.56 1020.15 1038.52 1054.37 1082.48 1113.49 1143.26 313.15K 901.90 919.63 938.78 959.03 982.19 1005.36 1023.52 1038.37 1066.66 1097.81 1125.84 318.15K 879.71 900.00 919.74 939.94 962.47 985.08 1003.75 1019.78 1046.67 1079.23 1107.27

UCFT

Uvandael

UJunjie

1235.63 1242.93 1251.56 1262.46 1275.29 1289.08 1296.19 1300.76 1313.96 1328.94 1338.75

1235.63 1239.63 1244.67 1250.83 1258.24 1267.04 1277.38 1289.46 1303.52 1319.83 1338.75

1410.15 1411.28 1412.70 1414.41 1416.41 1418.69 1421.28 1424.17 1427.37 1430.88 1434.71

1206.13 1215.08 1223.26 1233.93 1246.16 1258.53 1265.26 1269.63 1281.52 1296.17 1305.00

1206.13 1210.49 1215.90 1222.46 1230.30 1239.57 1250.45 1263.14 1277.91 1295.06 1315.00

1381.26 1382.68 1384.41 1386.46 1388.83 1391.54 1394.58 1397.97 1401.71 1405.82 1410.31

1188.75 1205.33 1217.48 1231.88 1246.33 1260.61 1267.95 1271.31 1281.33 1292.88 1297.50

1188.75 1193.13 1198.55 1205.12 1212.96 1222.22 1233.08 1245.75 1260.48 1277.60 1297.50

1364.76 1366.05 1367.67 1369.61 1371.88 1374.48 1377.43 1380.74 1384.42 1388.47 1392.91

1168.13 1188.45 1201.52 1215.84 1230.42 1243.71 1251.88 1256.02 1264.70 1274.82 1278.75

1168.13 1172.65 1178.21 1184.92 1192.91 1202.34 1213.37 1226.23 1241.19 1258.56 1278.75

1346.06 1347.27 1348.80 1350.68 1352.91 1355.50 1358.46 1361.79 1365.52 1369.65 1374.21


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INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652 2

2

TABLE III - U exp/ U imx VALUES WITH MOLEFRACTION FOR WITH O-XYLENE AT DIFFERENT TEMPERATURES

X1 0.0000 0.0908 0.1835 0.2781 0.3747 0.4734 0.5742 0.6772 0.7824 0.8900 1.0000

303.15 K 1.0000 0.9966 0.9984 0.9998 1.0014 1.0009 0.9961 0.9932 0.9956 0.9947 1.0000

U2exp/ U2 imx 308.15 K 313.15 K 1.0002 1.0000 1.0008 0.9992 1.0069 1.0056 1.0116 1.0071 1.0161 1.0108 1.0147 1.0107 1.0135 1.0074 1.0070 0.9996 1.0120 1.0022 1.0125 1.0018 1.0154 1.0000

CYCLOHEXANE [6] [7] [8]

318.15 K 1.0000 0.9987 1.0050 1.0065 1.0085 1.0086 1.0045 0.9970 0.9983 1.0019 1.0000

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

The ratio U2exp/ U2 imx is used as an important tool to measure the non-ideality in the mixtures, especially in those cases where the properties other than sound velocity are not known22. A perusal of Table 2 indicate large deviations from ideality, which may be due to the existence of strong tendency for the formation of association in liquid mixtures through hydrogen bonding as reported by Shukla et al 23. The deviations between theoretical and experimental value of ultrasonic velocities decrease with increase of temperature due to breaking of hetero and homo molecular clusters at higher temperatures24. On increasing the temperature, the ultrasonic velocity values decreases in the binary liquid mixture. This is probably due to the fact that the thermal energy activates the molecule, which would increase the rate of association of unlike molecules. Hence the complex formation through hydrogen bonding will occur9. In the present work it is evident to say experimental values of ultrasonic velocities are nearer to ultrasonic velocities as calculated from Vandael ideal mixing relation followed by CFT theory and Nomoto’s relation. Further from Table.3 U2exp/U2imx indicates small deviations from ideality, that means no strong interactions in liquid mixtures through hydrogen bonding. This is probably due to weak interactions in the liquid mixture.

[20] [21] [22] [23]

Shipra Baluja & Swathi Oza, J Pure & Appl Ultrason, 24 (2002) 850 Ali A, Yasmin A & Nain A K, Indian J Pure & Appl Phys. 40 (2002) 315 Amalendu Pal, Gurcharan Dass & Harsh Kumar, J Pure & Appl Ultrason, 23 (2001) 10 Rastogi et al, Indian J Pure & Appl Phys, 40 (2002) 256 Anwar Ali A, Anil Kumar Nain & Soghra Hyder, J Pure & Appl Ultrason, 23 (2001) 73 Nomoto O, J Phys Soc Japan, 4 (1949) 280 and 13 (1958) 1528 Van Dael W & Vangeel E, Proc int conf on calorimetry and thermodynamics, Warasa (1955) 555 Shipra Baluja & Parsania P H, Asian J Chem, 7 (1995) 417 Gokhale V D & Bhagavat N N, J Pure & Appl Ultrason, 11 (1989) 21 Junjie Z, J China Univ Sci Techn, 14 (1984) 298 Prakash A, prakash S & Prakash Q, Proc Nat Acd Sci, 55(A) 11 (1985) 114. Sabeson R, Natarajan & Varadha Rajan R, Indian J Pure & Appl Phys, 25 (1987) 489 Vogel A I, A text book of practical organic chemistry, 5th Edn (John Wiley, New York)1989 Riddick J A, Bunger W B & SokanoT K, techniques in chemistry, Vol 2, organic solvents, 4th Edition (John Wiley, New York) 1986 Satyanarayana N, Satyanarayana B & Savitha jyostna T, J Chem Eng Data, 52 (2007) 405. Satyanarayana B, Savitha Jyostna T & Satyanarayana N, Indian J Pure & Appl Phys, 44 (2006) 587. Viswanatha Sarma A & Viswanatha Sastry J, J Acous Soc Ind, Vol XXVII (1999) 309 Shukla B D, Jha L K & Dubey G P, J Pure & Appl Ultrason, 13 (1991) 72 Nikkam P S, Jagdale B S, Sawant A B & Mehdi hasa, J Pure & Appl Ultrason, 22 (2000) 115.

V.ACKNOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper. VI. CONCLUSIONS An estimation of ultrasonic velocities of the binary mixtures at four temperatures reveals that it agrees well with Vandael ideal mixing relation followed by CFT and Nomoto’s relation. REFERENCES [1] [2] [3] [4] [5]

Prausnitz J M, Lichenthalar & Azevedo, Molecular Thermodynamics of fluid phase equilibria, second edition, (Prentice-Hall Inc, Englewood Cliffs, New Jersey) (1986) Acree W E (Jr), thermodynamics properties of non electrolyte solutions (Academic Press, New York)(1984) Rodriguez S, Lafuente C, Artigas H, Royo F M & Urieta J S, J Chem Thermodynamics, 31 (1999) 139 Naidu P S & Ravindra Prasad K, J Pure Appl Ultrason, 24 (2002) 18. Rao T S, Veeraiah N & Rambabu C, Indian J Pure & Appl Phys, 40(2002) 850


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INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO 2-PP 60-64 (2010)

OVER VIEW TO IMPLEMENTATION OF ROBOTICS WITH VOICE RECOGNITION Ande Stanly Kumar@, Dr.K.Mallikarjuna Rao#, Dr.A.Bala Krishna$, B.Venkatesh^ @

Asst.Professr, Sri Venkateswara Institute of Science Information Technology,Tadepalligudem. [email protected], Professor, JNTU College of Engineering, Kakinada. $ Professor,SRKR Engineering College,Bhimavaram. ^ Asst.Professr, Sri Venkateswara Institute of Science Information Technology,Tadepalligudem. [email protected] #

ABSTRACT Automatic speech recognition by machine has been a goal of a research for a long time. Speech recognition is the process of converting an acoustic signal, captured by a microphone or a telephone, to a set of words. The recognized words can be the final results, as for applications such as commands & control, data entry, and document preparation. They can also serve as the input to further linguistic processing in order to achieve speech understanding. There are many works carried out in this area. The speech recognition system has also been implemented on some particular devices. Some of them are personal computer (PC), digital signal processor, and another kind of single chip integrated circuit. In this paper we propose voice recognition to control robot. Key words: Euclidean square distance, LPC, Voice recognition, Finger print. They h introduced a novel method for isolating the rove of higher order polynomials in Linear predictive systems. Y.M. Lam et al. implemented [3] fixed point implementations for speech recognition, they achieved recognition rate of 81.33%. SoshiIba et al. proposed [4] the framework takes a three-step approach to the robot programming i.e multi-modal recognition, intention interpretation, and prioritized task execution.

Introduction: The term "voice recognition" is sometimes used to refer to speech recognition where the recognition system is trained to a particular speaker - as is the case for most desktop recognition software, hence there is an element of speaker recognition, which attempts to identify the person speaking, to better recognize what is being said. Speech recognition is a broad term which means it can recognize almost anybody's speech such as a call-centre system designed to recognize many voices. Voice recognition is a system trained to a particular user, where it recognizes their speech based on their unique vocal sound.

In previous works, speech recognition system was implemented [5] on ATMEL 89C51RC microcontroller to control the movement of Wheelchair. They used the LPC model for speech recognition and achieved recognition rate of 78.57%. Thiang implemented [6] the speech recognition for controlling movement of Mobile Robot ATmega162 Microcontroller. Used Techniques were Linear Predictive Coding (LPC) combined with Euclidean Squared Distance and Hidden Markov Mode (HMM). In this project, highest recognition rate achieved was 87%. Stanly & Ande implemented [7] Voice Recognition Robotic Car with filters and finger print conversion method. Coming to this project, it describes continuation work to the previous works.

The first speech recognizer appeared in 1952 and consisted of a device for the recognition. Literature:

Treeumnuk & Dusadee, implemented [1] the Speech Recognition on FPGA with segmentation technique. Sriharuksa & Janwit implemented [2] a complete design and layout of an ASIC Design of Real Time Speech Recognition. Over View To Implementation Of Robotics With Voice Recognition

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Speech is a natural source of interface for human– machine communication, as well as being one of the most natural interfaces for human–human communication [8]. However, environmental robustness is still one of the main barriers to the wide use of speech recognition. Speech recognition performance degrades significantly under varying environmental conditions for many application areas.

In order to analyze speech, we needed to look at the frequency content of the detected word. To do this we used several 4th order Chebyshev band pass filters. To create 4th order filters, we cascaded two second order filters using the following "Direct Form II Transposed" implementation of a difference equations.

In this paper, speech recognition system is implemented to recognize the word used as the command for controlling the movement of robot. The proposed novel method will increase the recognition rate. Especially monitor the need of Embedded Systems in Industrial applications to control the movement of either simple or Bulky devices. There are two approaches used to recognize the speech signal. The first approach is Linear Predictive Coding combined with Euclidean Squared Distance (ESD). In this approach LPC is used as the feature extraction method and Euclidean Squared Distance is used as the recognition method. The second approach is Hidden Markov Model, which is used to build reference model of the words and also used as the recognition method. Feature extraction method used in the second approach is a simple segmentation and centroid value. Both approaches work on time domain. Experiments have to do in several variations of observation symbol number and number of samples. The hoist & crane can move in accordance with the voice command. Maximum recognition rate will be expected here by introducing a novel method.

Where the coefficient a’s and b’s were obtained through Matlab using the following commands. [B,A] = cheby2(2,40,[Freq1, Freq2]); (Where 2 defines a 4th order filter, 40 defines the stop band ripple in decibels, and Freq1 and Freq2 are the normalized cutoff frequencies). [sos2, g2] = tf2sos (B2, A2,'up','inf'); Fingerprint Calculation: Due to the limited memory space on the Mega32, we needed a way to encode the relevant information of the spoken word. The relevant information for each word was encoded in a “fingerprint”. To compare fingerprints we used the Euclidean distance formula between sampled word fingerprint and the stored fingerprints to find correct word.

Design:

Euclidean distance formula is:

Fig. 1. Layout robotic system The design had been done in the field of robotics and there exists a line follower robots, sensor robots and used speech to control a robot. It would make a robot which obeys human speech commands and performs errands.

P=(

)

and Q = (

)

Where P is a dictionary fingerprint and Q is the sampled word fingerprint and p i and q i are the data points that make up the fingerprint. To see if two words are the same we compute the Euclidean distance between them and the words with the minimum distance are considered to be the same. The

Mathematics for speech analysis: Speech Analysis:

Over View To Implementation Of Robotics With Voice Recognition

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formula above requires squaring the difference between the two points, but since we are using fixed point arithmetic, we found that squaring the difference produced too large of a number causing our variables to overflow. Thus we implemented a "pseudo Euclidean distance calculation" by moving the sum out of the square root reducing the equation to

D= PWM (Pulse Width Modulation) duty cycle calculation: The motors in the robot were measured to have a 50 Hz PWM frequency and movement was controlled by varying the duty cycle from 5% to 10%. To generate the PWM signals we used timer/counter1 in phase correct mode. The top value of timer/counter 1 was set to be 20000 and using a /8 pre scalar the PWM signal was set to have a frequency of 50Hz = 16MHz/(8*2*20000). To calculate the duty cycle the following equation was used OCR1x = (20000 - 40000*duty cycle). Where OCR1x is the value in the output compare register 1 A or B.

Fig.2.flow chart for voice recognition The Basic algorithm of our code was to check the ADC input at a rate of 4 KHz. If the value of the ADC is greater than the threshold value it is interpreted as the beginning of a half a second long word. The sample word passes through 8 band pass filters and is converted into a fingerprint. The words to be matched are stored as fingerprints in a dictionary so that sampled word fingerprints can be compared against them later. Once a fingerprint is generated from a sample word it is compared against the dictionary fingerprints and using the modified Euclidean distance calculation finds the fingerprint in the dictionary that is the closest match. Based on the word that matched the best the program sends a PWM signal to the car to perform basic operations like left, right, go, stop, or reverse.

Hardware/Software tradeoffs: The signal coming out of the microphone needed to be amplified. We had two different versions of operational amplifier, LMC 711 and LM 358. The LMC711 has a slew rate of 0.015 V/µ s, on the other hand LM 358 has 0.3V/µ s. The LM358 has a better slew rate and it gave us better response to input signals so we used it when we designed our amplification circuit. The signal processing of speech requires lot of computations, which implies we need fast processor, but we had to operate at 16 M Hz. In order to minimize the number of cycles we used filtering the audio signal we had to write most of the code in assembly. We wrote all of 10 digital filters in assembly which made them very efficient and significantly improved our performance over a C code implementation.

Initial-Threshold Calculation: At start up as part of the initialization the program reads the ADC input using timercounter0 and accumulates its value 256 times. By interpreting the read in ADC value as a number between 1 to 1/256, in fixed point, and accumulating 256 times. The average value of ADC was calculated without doing a multiply or divide. Three average values are taken each with a 16.4msec delay between the samples. After receiving three average values, the threshold value is to be four times the value of the median number. The threshold value is useful to detect when a word has been spoken or not.


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first 2 KHz since this usually contains the first and second speech formants, (resonant frequencies). This also allowed us to sample at 4 KHz and gave us almost enough time to implement 10 filters. We thought we needed ten filters each with approximately a 200 Hz bandwidth so that we would have enough frequency resolution to properly identify words. Originally we had 5 filters that spanned from 0 – 4 KHz and were sampling at 8 KHz, but this scheme did not produce consistent word recognition.

Fingerprint Generation: The program considers a word detected if a sample value from the ADC is greater than the threshold value. Every sample of ADC is typecast to an int and stored in a dummy variable A in. The A in value passes through 8 4th order Chebyshev band pass filters with a 40 dB stop band for 2000 samples (half a second) once a word has been detected. When a filter is used its output is squared and that value is accumulated with the previous squares of the filter output. After 125 samples the accumulated value is stored as a data point in the fingerprint of that word. The accumulator is then cleared and the process is begun again. After 2000 samples 16 points have been generated from each filter, thus every sampled word is divided up into 16 parts. Our code is based around using 10 filters and since each one outputs 16 data points every fingerprint is made up of 160 data points.

Fingerprint Comparison: Once the fingerprints are created and stored in the dictionary when a word was spoken, it was compared against the dictionary fingerprints. In order to do the comparison, we called a lookup() function. The lookup() function did a pseudo Euclidean distance formula by calculating the sum of the absolute value of the difference between each sample finger print a finger print from the dictionary. The dictionary has multiple words in it and the lookup went through all of them and picked the word with the smallest calculated number. We had originally used the square of the correct Euclidean distance calculation, d = Σ(pi – qi) 2. The words we finally used in our dictionary were Let's Go, (sound of a finger snapping), daiya [right – in Hindi], rukh [stop – in Hindi], peiche [back – in Hindi]. We had originally used English words, go, left, right, stop, and back, but many of these words seemed to be very similar in frequency as far as our algorithm was concerned. We then went to vowels and had better success, but we still wanted to use words that were directions and so we went to Hindi The set of words that we used were mostly orthogonal, but in Hindi left is baiya, which sound very similar to daiya and so that could not be used. We had previously had success with snapping so we used that for left.

Implementation of Filter:

Fig.3. finger print implementation PWM signal to move ROBOT: Filter Implementation: Once a word is recognized, its time to perform an action based on the recognized word. To perform an action we generated a PWM signal using timercounter1. Control of the PWM signal generation is done by the car control() function. For our robot, we needed to generate two different PWM signals, one for moving the car front/back and another one to steer left or right. We also need to send a default PWM signal to pause a robot. We chose timercounter1 because it has two different compare registers, OCR1A and OCR1B and can output two unique PWM signals. We used Phase correct mode

We chose a 4th order Chebyshev filter with 40 dB stop band since it had very sharp transitions after the cutoff frequency. We designed 10 filters a low pass with a cutoff of 200 Hz, a high pass with a cutoff of 1.8 KHz, and eight band passes that each had a 200 Hz bandwidth and were evenly distributed from 200Hz to 1.8 KHz. Thus we had band pass filters that went from 200-400 Hz, 400-600, 600 – 800 and so on all the way to the filter that covered 1.6 Khz – 1.8 Khz. We designed our filters in this way because we felt that most of the important frequency content in words was within the


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[2]. Thiang, “Implementation of Speech Recognition on MCS51Microcontroller for Controlling Wheelchair” proceedings of International conference on Intelligent and advanced systems. Kuala Lumpur, Malaysia, 2007 [3]. Y.M. Lam, M.W. Mak, and P.H.W. Leong , “Fixed point implementations of Speech Recognition Systems”. Proceedings of the International Signal Processing conference.Dallas. 2003. [4]. Treeumnuk, Dusadee. (2001). Implementation of Speech Recognition on FPGA.Masters research study, Asian Institute of Technology, 2001).Bangkok: Asian Instituteof Technology. [5]. Soshi Iba, Christiaan J. J. Paredis, and Pradeep K. Khosla. “Interactive MultimodalRobot Programming”. The International Journal of Robotics Research (24), pp 83 –104, 2005. [6]. Sriharuksa, Janwit. (2002). An ASIC Design of Real Time Speech Recognition.(Masters research study, Asian Institute of Technology, 2002). Bangkok: AsianInstitute of Technology.

to generate PWM signals because it is is glitch free, which is better for the motor. To find out a frequency and a duty cycles at which car turns forward/backward and left/right, we attached an oscilloscope probe to a car’s receiver. We sent different signals to the receiver using the car’s remote control an d measured the frequency and duty cycle for different motions. From the measurements, we found that car PWM frequency was 50Hz (period of 20ms) and had the following properties. Conclusion: The Embedded systems design covers a very wide range of microprocessor designs Our task is to design a control module for a robot. The robot is a simple two wheel robot that uses two stepper motors for driving. The robot can be programmed to drive autonomously a certain path. A list of driving commands are first downloaded from a PC to the robot, after which the robot will drive automatically through the program and to provides a framework to specify a system. At the beginning of our project, we set a goal to recognize five words, at the end of project we got ive words to be recognized. However our five words needed to be orthogonal to each other because our filters were not giving a high enough resolution and inaccuracy in fingerprint calculations due to using fix point arithmetic made the lookup function to be error prone. As a result, we had to pick various different words that sound apart. If we had to do this again instead of trying to use the Euclidean distance formula to match words we would like to try do perform a correlation of the two fingerprints. A correlation is less sensitive to amplitude differences and is a better way of identifying patterns between two objects. If we had faster process chip, we could modified our algorithm to add more filters, perform Fourier transform, or floating point arithmetic in order to improve our results.

[7]. Lawrence Rabiner, and Biing Hwang Juang, Fundamentals of Speech Recognition.Prentice Hall, New Jersey, 1993.Speech recognition by machine. By William Anthony Ainsworth, Institution of Electrical Engineers. [8] Andre Harison and Chirag Shah Voice recognition by robot. [9]. www.speechrecognition.com -/ united states. [10]. Frank Vahid and Tony Givargis, Embedded System Design: A Unified Hardware/Software Approach.

Acknowledgement We would like to express our thanks to referees for valuable comments that improved the paper. References: [1]. Thiang, “Limited speech recognition for controlling movement of Mobile Robot Implemented on ATmega162 Microcontroller” proceedings on International conference on Computer and Automation Engineering.2009. Over View To Implementation Of Robotics With Voice Recognition

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NOVELTY OF EXTREME PROGRAMMING Ch.V.Phani Krishna@, S.Satyanarayana#, K.Rajasekhara Rao$ @

Sana Engg. College, Kodad, [email protected] Department of Mathematics, Sri Venkateswara Institute of Science & Information Technology, Tadepalligudem534101, E-mail:[email protected] $ K.L. College of Engineering, Vijayawada, [email protected] #

ABSTRACT Extreme Programming is one of the most discussed to topics in the software development community. In this paper, we discussed the fundamentals of Extreme Programming, how Extreme Programming distinguished from other methodologies, how Extreme Programming addresses the risks encountered in software development, the values and basic principles of Extreme Programming, advantages and disadvantages of Extreme Programming. Then we will see how Extreme Programming uses a set of practices to build an effective software development team that produces quality software in a predictable and repeatable manner. Simple design that constantly evolves to add needed flexibility and remove unneeded complexity.

Introduction: Extreme Programming was visualized and developed to address the specific needs of software development conducted by small teams in spite of vague rapidly changing requirements.

Putting a token (nominal) system into production quickly and growing it in whatever directions prove most valuable.

This new light weight methodology challenges many conventional principles & opinions, including the long held assumption that the cost of changing a piece of software necessarily rises dramatically over the courses of time. XP recognizes that projects have to work to achieve cost reduction and make use of savings once they have been earned.

Reasons why XP is controversial: Does not force team members to specialize because - every XP programmer participates in (all of these practices all the critical activities everyday. Do not conduct complete – up – front analysis and design because – XP project make analysis and design decisions through development.

Fundamentals of XP: Distinguishing between the decisions to be made by business interests and those to be made by project stake holders.

Delivering business value is the heart beat that drives XP projects.

Writing unit tests before programming and keeping them running at all times.

Do not write and maintain implementation documentation because – communication in XP projects occurs face to face or through efficient tests or carefully written coding.

Integrating and testing the whole system several times. Producing all programming)

software

in

pairs

(pair

XP makes two sets of promises: To Programmers:

Novelty Of Extreme Programming

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XP promises that they will be able to work on things that really matter, everyday. They won’t have to face scary situations alone. They will be able to do every thing in their power to make their system successful. They will make decisions that they can make best and they won’t make decisions that they can make best and they won’t make decisions they are not best qualified to make.

Novelty of XP: Putting all the practices under one umbrella Making sure they are practiced as thoroughly as possible Making sure the practices support each other to the greatest possible degree.

To Customers and Managers:

XP addressing the risks encountered in software development:

XP promises that they will get the most possible value out of every programming week. Every few weeks they will able to see concrete progress on goals they care about. They will be able to change the direction of project in the middle of development without incurring exorbitant costs.

The basic problem of software development is risk. (1) Schedule slips: - short releases Within an iteration, XP plans with 13 days tasks, so the team can solve problems even during iteration. XP calls for implementing the highest priority features first, so any features that slip past the release will be of lower value

XP promises to reduce project Risk, improve responsiveness to business changes, improve productivity throughout the entire life of a system & add fun to building software in teams – all at the same time.

(2) Project cancelled: - same as above XP asks the customer to choose the smallest release that makes the most business sense, so there is a less chance to go wrong & the value of the software is greatest.

XP is distinguished from other methodologies by: Its early, concrete, continuing feed back from short cycles

(3) System goes sour: - Testing.

Its incremental planning approach, which comes up with an overall plan that in expected to evolve through the life of the project

Repeated testing in XP ensures a quality base line.

Its ability to flexibly schedule the implementation of functionality, responding to changing business needs.

(4) Defect Rate: Testing (both programmer as well as customer perspectives) Programmer (Testing function – b y – function)

Its dependence, on automated tests written by programmers and customers to monitor the progress of development and to catch defects early.

Customer (program feature – by – Program feature)

Its dependence on oral communication, tests and source code to communicate system structure and goal

Specification of project is continuously refined during development, so learning by the customer & the team can be reflected in the software.

(5) Business Misunderstood: - On – site customer

Its dependence on an evolutionary design process that lasts as long as the system lasts

(6) Business changes: - Short Releases:

Its dependence on the close collaboration of programs with ordinary skills.

XP shortens release cycle, so there is less change during the development of a single release. During the release the customer is welcome to substitute a new functionality for functionality not yet completed.

Its dependence on practices that work with both the short-term instincts of programmers and the long term interests of the project.

(7) False feature Rich:


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XP insists that only the highest priority tasks are addressed.

Scope: - Less scope makes it possible to deliver better quality. It also let us delivers sooner or cheaper.

(8) Staff turnover: - Pair programming XP asks programmers to accept responsibility for estimating and completing their own work, gives them feedback about the actual time taken, so that their estimates can improve.

Four values: Project will be successful when the team follows a style that celebrates a consistent set of values that serve human and commercial needs:

Thus there is less chance of or a programmer to get frustrated by being asked to do the obviously impossible.

Communication Simplicity

XP development cycle:

Feedback

Pair of programmers program together.

Courage

Development is driven by tests. Until, all the tests run, the process of adding functionality is not succeeded. Then coding activity begins.

Communication: XP aims to keep the right communications flowing by employing many practices that can’t be done without communicating. They are practices that make short term sense, like unit, testing, pair programming and task estimator. The effect of testing, pairing and estimating is that programmers and customers and managers have to communicate.

Pairs don’t just make the test cases run. They also evolve the design of the system. Changes are not restricted to any particular area. Pairs add value to the analysis, design, implementation, testing of the system. They add that value wherever the system needs it. Integration immediately follows development, including integration testing.

This doesn’t mean that communications don’t sometimes get logged in an XP project. People get scared, make mistakes, get distracted XP employs a coach whose job is to notice when people are not communicating and reintroduce them.

Four Variables: There are four control variables in software development model Cost Time

Simplicity:

Quality

The second XP value is simplicity. XP is making a bet. It is betting that it is better to do a simple thing today and pay a little more tomorrow to change it if it needs it than to do a more a complicated thing today that may never be used any way.

Scope Interactions Between the variables: Cost: - Giving a project too little money and it won’t be able to solve the customer’s business problem, on the other band too much money too soon creates more problems than it solves.

Simplicity and communication have a wonderful mutually supporting relationship.

Time: - More time to deliver can improve quality and increase scope. Give a project too little time and quality suffers, with scope, time and cost not far behind.

Feed back: The third value in XP is feedback. Concrete feed back about the current state of the system is absolutely priceless. Optimism is an occupational hazard of programming. Feedback is the treatment. The programmers have minute – by – minute feed back about the state of their system. When customers write new ‘stories” (descriptions of feature),

Quality: - Quality is an important control variable we can make very short – term gains by deliberately sacrificing quality, but the cost – human, business, and technical is enormous.


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problem today, and trust your ability to add complexity in the future where you need it.

the programmers immediately estimate them, so the customers have concrete feedback about the quality of their stories.

Incremental change:

The person who tracks progress watches the completion of the tasks to give the whole team feedback about whether they are likely to finish everything they set out to do in a span of time. The customers and testers write functional tests for all the stories implemented by the system. They have concrete feed back about the current state of their system. The customers review the schedule every frequently to see if the terms over all velocity matches the plan and to adjust the plan. The system is put into production as soon as it makes sense, to the business can begin to “feel” what the system is like in action & discover how it can best be exploited. Concrete feedback works together with communication and simplicity.

Any problem is solved with a series of the smallest changes that make a difference. Hence the adoption of changes in XP must be taken in little steps. Embracing change: The best strategy is the one that preserves the most obtains while actually solving your most pressing problem.

Quality work: Of all the four Project variables – Quality is not really a free variable. The only possible values are “excellent” and insanely excellent depending on whether lives are at stake or not otherwise we don’t enjoy our work & the project goes down the drain.

Courage: When combined with communication, simplicity and concrete feedback, courage becomes extremely valuable. Communication supports courage because it opens the possibility for more high – risk, high – reward experiments. Simplicity supports courage. Concrete feedback supports courage because of feeling much safer trying radical surgery on the code.

Some less central principles: Teach learning Small initial investment Play to win Concrete experiments Open, honest communication

Basic principles:

Work with people’s Instincts, not against them

The fundamental principles are

Accepted Responsibility

Rapid feed back

Local Adaptation

Assume simplicity

Travel light

Incremental change

Honest Measurement

Embracing change

Teach learning:

Quality work

We will focus on teaching strategies for learning how much testing you should do. Also how much design refactoring and everything else you should do.

Rapid feed back: Learning psychology teaches that the time between an action and its feedback is critical to learning. So, our principle to get feed back, interpret it and put what is learned back into the system as quickly as possible.

Small investment (Initial): Too many resources too early in a project are a recipe for disaster. Tight budgets force programmers & customers to pare requirements and approaches. Resources can be too tight. If you don’t have the resources to solve even one interesting problem, then the system you create is guaranteed not to be

Assume simplicity: Treat every problem as if it can be solving with ridiculous simplicity. This is the hardest principle for programmers to swallow. XP says to do a good job of solving today’s


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going along. Along the way a person told what to do will find a thousand of expressing their frustration, most of them to the detriment of the team & many of them to the detriment of the person

interesting. If you have some one dictating scope, dates, quality & cost, there you are unlikely to be able to navigate to a successful conclusion.

The alternate is that responsibility be accepted, not given you are past of a team, and if the team comes to the conclusion that a certain task need doing, someone will choose to do it, no matter how odious.

Play to win: The difference between playing to win and playing not to lose, most software development I see is played not to lose. Lots of paper gets written. Lots of meetings are held. Everyone is trying to develop “by the book”, not because it makes any particular sense, but because they want to be able to say at the end’ that it wasn’t their fault, they were following the process. Software development played to win does every thing that helps the team to win and doesn’t do anything that doesn’t help to win.

Local adaptation: This is an application of accepted responsibility to your development process. Adopting XP means that you get to decide how to develop i.e. deciding on something today and being aware of whether it still works tomorrow. You have to change and adapt. Travel light: The artifacts to be maintained are

Concrete experiments: Every time you make a decision and you don’t test it, there is some probability that the decision is wrong, the more decisions you make the more these risks compound. The result of a discussion of requirements should also be a series of experiments. Every abstract decision should be tested.

Few

Simple

Valuable XP team becomes intellectual nomads, always prepared to quickly pack up the tents and follow the herd. XP team gets used to traveling light. They don’t carry much in the way of baggage except what they must have to keep producing value for the customer – tests and code.

Open, Honest Communication: Programmers have to be able to tell each other where there are problems in the code. They have to be free to express their fears, and get support. They have to be free to deliver bad news to customers and management to deliver it early, and not be punished.

Honest measurement: Our quest for control over software development has led us to measure, which is fine, but it has led us to measure at a level of detail that is not supported by our instruments.

Works with people’s Instincts, not against them:

Practices of XP:

People like winning. People like interacting with other people. People like learning. People like being past of a team. People like being in control – people like being trusted. XP celebrates what programmers seen to do when left to their own devices, with just enough to keep the whole process on track, XP matches observations of programmers in the wild.

The values of XP are implemented by employing 12 practices as elucidated by Kent Beck. Planning game: Determining the scope of project and releases by combining business priorities with the technical estimates according to the changing requirements.

Accepted Responsibility: Primate dominance displays work only so long in getting people to act like they are


69


The customer if located at the same site as a domain expert to help the programmer’s team in the production of system.

Small releases: Enforce a simple system into production quickly, and then release new versions on short cycles.

Coding standards: Programmers write all code in accordance with the standards agreed upon by the development team to ensure that communication is made through code.

System Metaphor: A shared story which helps the programmers as well as customers understand the basic elements on which the system works and their relationships.

Jeffries developed them further and has 13 practices.

Simple Design:

Whole Team:

Keeping the system design as simple as possible and remove excess complexity as soon as possible.

All people who take part in the project gather at one place to develop the system as a team. In addition to the above said practices there are certain implicit practices.

Testing: Continuously writing and running required tests, each time a new code is written are changing the existing code including unit testing and customer written functionality testing.

Caves and commons Fixed iterations and engineering tasks Write it on a card (RDP Technique) Spike Solutions

Refactoring: Improving the design of project without changing the functionality of there existing code by removing the duplication of the code and by improving communication, simplification and flexibility.

All tests all the time Promiscuous Pairing Yesterday’s weather Track velocity and track progress

Pair Programming:

Regression test

Writing code with two programmers at one machine .

There are certain misconceptions regarding XP. But the real truth is ….

Collective Code Ownership:

No written design documentation

All programmers accepting responsibility for all code therefore being able to make changes to any piece of code at any time when necessary.

• Truth: no prescribed standards for how much or what kinds of docs are needed. No design

Continuous Integration:

• Truth: minimal explicit, upfront design: design is an explicit part of every activity through every day.

Soon after completing a task, the system should be integrated and run several times continuously.

XP is easy

40-Hour week:

• Truth: although XP does try to work with the natural tendencies of developers, it requires great discipline and consistency.

To keep programmers active, creative and fresh, no programmer should work more than 40-hours per week. No programmer should do more than a week’s overtime in a row.

XP is just legitimized hacking • Truth: XP has extremely high standards throughout the process.

On-site Customer:

XP is the one, true way to build software


70

quality


Keep the customer in charge of what is developed and when.

• Truth: It seems to be a sweet spot for certain kinds of projects.

To quickly demonstrate the benefits of XP, implement it first on a new project with no legacy code.

Advantages:

XP is indeed flexible. Changes in the priorities can be done repeatedly with very little notice and customers will be served with what they requested.

Results and output can be forecasted to customers at every schedule.

XP team can inculcate satisfaction in customer by revealing short releases and adapting changes requested by the customer.

The unit tests written by the developer team and acceptance tests written by the customer increases the degree of confidence in the product.

Conclusion: Tacit knowledge and communication among all team members are highlighted in XP. XP practices such as Pair Programming and extensive testing further reinforce this insight, as well as minimizing documentation. XP puts a high premium on customer satisfaction. Taking the customer within the team and receiving feedback frequently are ways to accomplish it. This way customer’s suggestions can be taken into account throughout the development project. The customer also participates in testing. Thus, XP is developed to provide a favorable setting for programmers to be able to respond rapidly to changing customer requirements.

Disadvantages: Extreme flexibility exerts heavy responsibility on the customers to produce strategic plans. Responding to sudden changes is part and parcel of XP, but adopting changes for every iteration is confusing and may not be a sound business practice.

ACKONOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper.

The lack of emphasis on documentation within XP does not take into account end user needs for documentation, including user guides, integration kits, reference texts and fact sheets.

REFERENCES: [1]. Lindstrom, L., & Jeffries, R. (2004). Extreme programming and agile software Methodologies. Information Systems Management, 21(3), 41. [2]. Beck, Kent and Martin Fowler. Planning Extreme Programming, Addison-Wesley, Boston MA, October 2000. [3]. Armitage, J. (2004). Are agile methods good for design?`. Interactions, 11(1), n/a. Retrieved September 9, 2006, from Proquest database. [4]. Jeffries,R.What is extreme programming? http://www.xprogramming.com [5]. Osamu Kobayashi., Mitsuyoshi Kawabata., Makoto Sakai., Eddy Parkinson., Analysis of the Interaction between Practices for introducing XP effectively, ACM, 2006. [6]. Grenning.,J., Launching Extreme Programming at a Process Intensive Company, IEEE Software, Vol 18, No.6, pp. 27-33, 2001. [7]. William.A.Wood., William.L.K., Exploring XP for Scientific Research, IEEE Software, Vol.20, No. 3, pp30-36, 2003. [8]. Martin Lippert., Stefan Roock., Adopting XP to Complex Application Domains, ACM, 2001.

Problems encountered in the implementation of XP:

Overly engineering

Overly complex integration

Unrepresentative acceptance testing

Coding assistant

Hard to test software

Obtuse specification

Recommendations: Have a contingency plan to manage resistant participants who are not won over to XP. Make the necessary physical changes to the work place to foster too key tenets of XP: Pair Programming and constant customer developer communication.


71


[9]. Glenn Vanderburg., A Simple Model of Agile Software Process –or- Extreme Programming Annealed, ACM, 2005. [10]. Kuppuswami, S., Vivekanandan K., and Paul Rodrigues (2003): A Sys-tem Dynamics Simulation Model to Find the Effects of XP on Cost of Change Curve. In proceedings of Fourth International Conference on Extreme Programming and Agile process in Software Engineering, (XP2003), May 25 – 29, 2003, Genova, Italy.


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INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO 2-PP 73-81 (2010) RADIATION EFFECTS ON MHD FREE CONVECTION FLOW PAST A SEMI-INFINITE MOVING VERTICAL POROUS PLATE WITH SORET AND DUFOUR EFFECT

G.Venkata Ramana Reddy@ and Dr. A.Rami Reddy# @

#

Assistant Professor, LBR College of Engineering, Mylavaram, Krishna,A.P. Email: [email protected] Associate Professor, LBR College of Engineering, Mylavaram, Krishna,A.P ABSTRACT

In this paper, we deal with the interaction of Soret and Dufour effects on steady MHD free convection flow in a porous medium with dissipative fluid has received little attention. Hence, the object of the present chapter is to analyze the Soret and Dufour effects on steady MHD free convection flow past a semi-infinite moving vertical plate in a porous medium with viscous dissipation. The governing equations are transformed by using similarity transformation and the resultant dimensionless equations are solved numerically using the Runge-Kutta method with Shooting technique. The effects of various governing physical parameters on the fluid velocity, temperature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are shown in figures and tables and analyzed in detail. Key words: MHD free convection, porous medium, vertical plate, and Nusselt number two dimensional natural convective flow of an incompressible, electrically conducting fluid along an infinite vertical porous plate embedded in a porous medium. Helmy [7] studied MHD unsteady free convection flow past a vertical porous plate embedded in a porous medium. Elabashbeshy [8] studied heat and mass transfer along a vertical plate in the presence of magnetic field. Chamkha and Khaled [9] investigated the problem of coupled heat and mass transfer by magnetohydrodynamic free convection from an inclined plate in the presence of internal heat generation or absorption. In the above all studies, the level of concentration of foreign mass assumed very low, so that the Soret and Dufour effects can be neglected. However, expectations are observed therein. The Soret effect, for instance, has been utilized for isotropic separation, and in mixture between gases with very

1. INTRODUCTION Combined heat and mass transfer (or doublediffusion) in fluid-saturated porous media finds applications in a variety of engineering processes such as heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and processes, geothermal and geophysical engineering such moisters migration in a fibers insulation and nuclear waste disposal and others. Double diffusive flow is driven by buoyancy due to temperature and concentration gradients. Bejan and Khair [1] investigated the vertical free convection layer flow in a porous media owing to combained heat and mass transfer. Lai and Kulacki [2] used the series expansion method to investigate coupled heat and mass transfer in natural convection from a sphere in a porous medium. The suction and blowing effects on free convection coupled heat and mass transfer over a verrtical plate in a saturated porous medium was studied by Raptis et al. [3] and Lai and Kulacki [4], respectively. Magnetohydrodynamic flows have applications in meteorology, solar physics, cosmic fluid dynamics, astrophysics, geophysics and in the motion of earths core. In addition from the technological point of view, MHD free convection flows have significant applications in the field of stellar and planetary magnetospheres, aeronautical plasma flows, chemical engineering and electronics. An excellent summary of applications is to be found in Huges and Young [5]. Raptis [6] studied mathematically the case of time varying

light molecular weight weight

( H2 , He )

and of medium molecular

( N 2 , air ) . The Dufour effect was found to be of order

of considerable magnitude such that it cannot be ignored [10]. The Soret effect arises when the mass flux contains a term that depends on the temperature gradient. The analogous effect that arises from a concentration gradient dependent term in the heat flux is called the Dufour effect. Dursunkaya and Worek [11] studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface. In view of the importance of above mentioned effects, Kafoussias and Williams [12] studied the Soret and Dufour


73


effects on free convective and mass transfer boundary layer flow with temperature dependent viscosity. Anghel et al. [13] investigated the Dufour and Soret effects on free concentration boundary layer flow over a vertical surface embedded in a porous medium. Postelnicu [14] studied numerically the influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Recently, Alam and Rahman [15] investigated the Dufour and Soret effects on mixed convection flow past a vertical porous flat plate with variable suction. Alam et al. [16] studied Dufour and Soret effects on steady free convection and mass transfer flow past a semi-infinite vertical plate in a porous medium. In most of the studies mentioned above, viscous dissipation is neglected. Gebhart [17] has shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux at the plate. Gebhart and Mollendorf [18] considered the effects of viscous dissipation for external natural convection flow over a surface. Soundalgekar [19] analyzed viscous dissipative heat on the two-dimensional unsteady free convective flow past an infinite vertical porous plate when the temperature oscillates in time and there is constant suction at the plate. Israel Cookey et al. [20] investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction.

∂u ∂v + =0 ∂x ∂y (2.1) Momentum equation

u

(2.2) Energy equation

u

Species equation

u

Cw

u = U 0 , v = v0 ( x ) , T = Tw , C = Cw at y = 0 u → 0, v → 0, T → T∞ , C → C∞

greater

than

the

as

y→∞

U 0 is the uniform velocity and v0 ( x ) is the velocity of suction at the plate and u , v are the velocity components in x, y directions respectively, ρ - the fluid density, g-

where

the acceleration due to gravity, β and β* - the thermal and concentration expansion coefficients respectively, K ′ - the permeability of the porous medium, T - the temperature of the fluid in the boundary layer, ν - the kinematic viscosity,

σ-

T∞ - the temperature of the fluid far away from the plate, α - the thermal diffusivity, C - the species concentration in the boundary layer, C∞ - the species concentration in the fluid

T∞ of the surrounding fluid and the

is

∂C ∂C ∂ 2C D k ∂ 2T +v = Dm 2 + m T ∂x ∂y ∂y Tm ∂y 2

(2.4) The boundary conditions for the velocity, temperature and concentration fields are

maintained at a constant temperature Tw , which is higher than

concentration

∂T ∂T ∂ 2T 1 ∂ qr Dm kT ∂ 2C +v =α 2 − + ∂x ∂y ∂y ρ c p ∂ y cs c p ∂y 2

(2.3)

2. MATHEMATICAL ANALYSIS A steady two-dimensional hydromagnetic flow of a viscous incompressible, electrically conducting and viscous dissipating fluid past a semi-infinite moving vertical porous plate embedded in a porous medium is considered. The flow is assumed to be in the x - direction, which is taken along the semi-infinite plate and y - axis normal to it. Initially, it is assumed that the plate and the fluid are at the same temperature T and the concentration C . The surface is the constant temperature

σB2 ∂u ∂u ∂2u +v =ν 2 + gβ (T −T∞ ) + gβ* ( C −C∞ ) − 0 u ρ ∂x ∂y ∂y

the electrical conductivity of the fluid,

B0 - the magnetic induction, k - the thermal conductivity, c p - the specific heat at constant far away from the plate,

constant

concentration C∞ . It is assumed that the interaction of the

pressure,

induced axial magnetic field with the flow is considered to be negligible compared to the interaction of the applied field B0 ,

concentration susceptibility,

kT - the thermal diffusion ratio,

cs - the

Tm - the mean fluid temperature,

Dm - the mass diffusivity.

with the flow. It is also assumed that all the fluid properties are constant except that of the influence of the density variation with temperature and concentration in the body force term (Boussinesq’s approximation). Also, there is no chemical reaction between the diffusing species and the fluid. Then, under the boundary layer approximations, the governing equations are Continuity equation

Thermal radiation is assumed to be present in the form of a unidirectional flux in the y-direction i.e.

q r (transverse to the vertical surface). By using the Rosseland approximation (Brewster [29]), the radiative heat flux q r is given by


74


qr = −

number, M - the magnetic field parameter, , Pr - the Prandtl number, Du - the Dufour number, Sc - the Schmidt number , Sr - the Soret number. The mass conservation equation (2.1) is satisfied by the Cauchy-Riemann Equations

4σ s ∂ T ′ 4 3k e ∂ y (6)

where σ s is the Stefan-Boltzmann constant and k e - the mean absorption coefficient. It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If temperature differences within the flow are sufficiently small, then Equation (6) can be linearized

u=

In view of the Equation (2.6) , and following the analysis of Chamkha and Issa [21], the equations (2.2), (2.3) and (2.4) reduce to the following non-dimensional form

4

by expanding T ′ into the Taylor series about T∞′ , which after neglecting higher order terms takes the form 3

T ′4 ≅ 4T∞′ T ′ − 3T∞′

f ′′′ + ff ′′ + Gr θ + Gm φ − Mf ′ = 0 (2.7)

θ ′′ + Pr f θ ′ + radiationterm + Pr Du φ ′′ = 0

4

(2.8)

φ ′′ + Sc f φ ′ + Sc Sr θ ′′ = 0

(7)

(2.9) The corresponding boundary conditions are

In view of Equations (6) and (7), Equation (3) reduces to

f = f w , f ′ = 1, θ = 1, φ = 1 at η = 0 f ′ → 0, θ → 0, φ → 0 as η → ∞

∂T′ ∂T′ ∂ 2 T ′ 16σ s T∞′3 ∂ 2T ′ Dm kT ∂ 2C u +v =α + + ∂x ∂y ∂ y 2 3ke ρ c p ∂ y 2 cs c p ∂y 2 (8) The Equations (2.2) to (2.4) are strongly coupled, parabolic and nonlinear partial differential equations. An analytical solution cannot be obtained and therefore we seek numerical solutions. Numerical computations are greatly facilitated by nondimensionalization of the equations. Proceeding with the analysis, we introduce the following similarity transformations and dimensionless variables which will convert the partial differential equations from two independent variables

( x, y )

and primes denote partial differentiation with respect to the variable. The skin-friction coefficient, Nusselt number and Sherwood number are important physical parameters for this type of boundary layer flow. The skin-friction coefficient in non-dimensional form is

C f = 2 ( Re )

to a system of coupled, non-

νρcp

2σB x ν , R= , Sc= , Pr = , Dm k ρU0 Dm kT ( Cw − C∞ ) D k (T − T ) 6) Du = , Sr = m T w ∞ cs c p (Tw − T∞ ) ν Tm ( Cw − C∞ ) where

ψ

is the stream function,

θ

1 2

f ′′ ( 0 )

1

Nu = −(Re) 2 θ ′ ( 0 ) The Sherwood number in non-dimensional form is 1

Sh = − ( Re ) 2 φ ′ ( 0 ) where

*

2 0

−

The Nusselt number in non-dimensional form is

U T −T C−C∞ η = y 0 , ψ = νxU0 f (η) , θ(η) = ∞ , φ(η) = , 2νx Tw −T∞ Cw −C∞

gβ(Tw −T∞) 2x gβ ( Cw −C∞) 2x ∂ψ ∂ψ u= , v=− , Gr = , G m , = U02 U02 ∂y ∂x

2x is the dimensionless suction velocity ν U0

where f w = −v0

linear ordinary differential equations in a single variable ( η ) i.e. coordinate normal to the plate. In order to write the governing equations and the boundary conditions in dimensionless form, the following nondimensional quantities are introduced.

M=

∂ψ ∂ψ and v = − . ∂x ∂y

Re =

U0 x

ν

is the Reynolds number

3. NUMERICAL SOLUTION The set of coupled non-linear governing boundary layer Equations (2.7) - (2.9) together with the boundary conditions (2.10) are solved numerically by using RungeKutta fourth order technique along with Shooting method. First of all, higher order non-linear differential Equations (2.7) - (2.9) are converted into simultaneous linear differential equations of first order and they are further transformed into initial value problem by applying the Shooting technique (Jain et al. [22]). The resultant initial value problem is solved by employing Runge-Kutta fourth order technique.

(2.

- the non-dimensional

temperature function, φ - the non-dimensional concentration, Gr - the thermal Grashof number, Gm - the solutal Grashof


75


the heated surface more rapidly than for higher values of Pr. Hence in the case of smaller Prandtl numbers as the boundary layer is thicker and the rate of heat transfer is reduced. The effect of the viscous dissipation parameter i.e., the Eckert number Ec on the velocity and temperature are shown in Figs. 6(a) and 6(b) respectively. The Eckert number Ec expresses the relationship between the kinetic energy in the flow and the enthalpy. It embodies the conversion of kinetic energy into internal energy by work done against the viscous fluid stresses. The positive Eckert number implies cooling of the surface i.e., loss of heat from the plate to the fluid. Hence, greater viscous dissipative heat causes a rise in the temperature as well as the velocity, which is evident from Figs. 6 (a) and 6 (b). For different values of the Dufour number Du, the velocity and temperature profiles are plotted in Figs. 7(a) and 7(b) respectively. The Dufour number Du signifies the contribution of the concentration gradients to the thermal energy flux in the flow. It is found that an increase in the Dufour number causes a rise in the velocity and temperature throught the boundary layer. For Du ≤ 1 , the temperature profiles decay smoothly from the surface to the free stream value. The influence of Schmidt number Sc on the velocity and concentration profiles are plotted in Figs. 8(a) and 8(b) respectively. The Schmidt number embodies the ratio of the momentum to the mass diffusivity. The Schmidt number therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers. As the Schmidt number increases the concentration decreases. This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid velocity. The reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the velocity and concentration boundary layers. These behaviors are clear from Figs. 8(a) and 8(b). Figs. 9(a) and 9(b) depict the velocity and concentration profiles for different values of Soret number Sr. The Soret number Sr defines the effect of the temperature gradients inducing significant mass diffusion effects. It is obvious that an increase in the Soret number Sr results in an increase in the velocity and concentration with in the boundary layer. Figs.10(a), 10(b) and 10(c) illustrate the influence of suction parameter f w on the velocity, temperature and

4. RESULTS AND DISCUSSION As a result of the numerical calculations, the dimensionless velocity, temperature and concentration distributions for the flow under consideration are obtained and their behaviour have been discussed for variations in the governing parameters viz., the thermal Grashof number Gr, solutal Grashof number Gm, magnetic field parameter M, permeability parameter K, Prandtl number Pr, Eckert number Ec, Dufour number Du, Schmidt number Sc, Soret number Sr and the suction parameter f w . The influence of the thermal Grashof number Gr on the velocity is presented in Fig.1. The thermal Grashof number Gr signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer. As expected, it is observed that there is a rise in the velocity due to the enhancement of thermal buoyancy force. Here the positive values of Gr correspond to cooling of the surface. Also, as Gr increases, the peak values of the velocity increases rapidly near the wall of the porous plate and then decays smoothly to the free stream velocity. Fig.2 presents typical velocity profiles in the boundary layer for various values of the solutal Grashof number Gm, while all other parameters are kept at some fixed values. The solutal Grashof number Gm defines the ratio of the species buoyancy force to the viscous hydrodynamic force. The velocity distribution attains a distinctive maximum value in the vicinity of the surface and then decreases properly to approach the free stream value. As expected, the fluid velocity increases and the peak value is more distinctive due to increase in the species buoyancy force. For various values of the magnetic parameter M, the velocity profiles are plotted in Fig.3. It can be seen that as M increases, the velocity decreases. This result qualitatively agrees with the expectations, since the magnetic field exerts a retarding force on the free convection flow. The effect of the permeability parameter K on the velocity field is shown in Fig. 4. The parameter K as defined in equation (2.6) is inversely proportional to the actual permeability K ′ of the porous medium. An increase in K will therefore increase the resistance of the porous medium (as the permeability physically becomes less with increasing K ′ ) which will tend to decelerate the flow and reduce the velocity. Figs.5(a) and 5(b) illustrate the velocity and temperature profiles for different values of Prandtl number Pr. The Prandtl number defines the ratio of momentum diffusivity to thermal diffusivity. The numerical results show that the effect of increasing values of Prandtl number results in a decreasing velocity. From Fig.5 (b), it is observed that an increase in the Prandtl number results a decrease of the thermal boundary layer thickness and in general lower average temperature with in the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse away from

concentration respectively. It is observed that an increase in the suction parameter results in a decrease in the velocity, temperature and concentration. The effects of various governing parameters on the skin friction coefficient C f , Nusselt number Nu and the Sherwood number Sh are shown in Tables 1 and 2. From Table 1, it is observed that as Gr or Gm increases, there is a rise in the local skin-friction coefficient, Nusselt number and


76


the Sherwood number. It is seen that, as M or K increases, there is a fall in the skin-friction coefficient, the Nusselt number and the Sherwood number. Also, it is noticed that as the suction parameter f w increases, the local skin-friction

1.4 M = 0.0, 0.5, 1.0, 2.0

1.2 1

coefficient decreases, while the Nusselt number and Sherwood number increase. From Table 2, it is observed that an increase in Pr leads to a decrease in the skin-friction and Sherwood number and an increase in the Nusselt number. It is also noticed that an increase in Ec or Du leads to an increase in the skin-friction and Sherwood number and a decrease in the Nusselt number. It is observed that an increase in the Schmidt number Sc reduces the skin-friction coefficient and Nusselt number and increases the Sherwood number. It is also seen that an increase in Soret number Sr leads to an increase in the skin-friction and Nusselt number and a decrease in the Sherwood number.

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 Sc = 0.6 Sr = 1.0 fw = 0.5

f'

0.8 0.6 0.4 0.2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

M

Fig.3. Velocity profiles for different values of 1.4 1.2 K = 0.5, 1.0, 1.5, 2.0

1.4

1 Gr = 2.0 Gm = 2.0 M = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 Sc = 0.6 Sr = 1.0 fw = 0.5

Gr = 1.0, 2.0, 3.0, 4.0

1.2

f'

0.8

1 Gm = 2.0 M = 0.5 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 Sc = 0.6 Sr = 1.0 f w = 0.5

f'

0.8

0.6 0.4

0.6

0.2

0.4

0

0.2

0

0.5

1

1.5

2

2.5

3

3.5

4

η

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig.1. Velocity profiles for different values of

K


η

Gr 1.4

1.4 1.2

Gm = 1.0, 2.0, 3.0, 4.0

1.2

Pr = 0.71, 1.0, 1.25, 1.5

1

1 Gr = 2.0 M = 0.5 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 Sc = 0.6 Sr = 1.0 fw = 0.5

f'

Gr = 2.0 Gm = 2.0 K = 0.5 Sc = 0.6 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 f w = 0.5

0.8 f'

0.8

0.6

0.6 0.4

0.4 0.2

0.2 0

0

0

0

0.5

1

1.5

2

2.5

3

3.5

η


0.5

1

1.5

2

2.5

3

3.5

4

η

4

Fig.5(a). Velocity profiles for different values of

Gm


77

Pr


Fig.7(a). Velocity profiles for different values of Du

1

1 0.8

0.8 0.6

Pr = 0.71, 1.0, 1.25, 1.5

θ

Du = 0.0, 0.2, 0.6, 1.0 0.6

0.4

θ

Gr = 2.0 Gm = 2.0 K = 0.5 Sc = 0.6 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 fw = 0.5

0.4

0.2

0

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 M = 0.5 Sc = 0.6 Sr = 1.0 f w = 0.5

0.2 0

0.5

1

1.5

2

2.5

3

3.5

4

η

Fig.5(b). Temperature profiles for different values of

Pr

0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

1.4

Fig.7(b). Temperature profiles for different values of Du

Ec = 0.0, 0.01, 0.02, 0.03

1.2

1.4 1

Sc = 0.3, 0.6, 0.78, 0.94

1.2

0.8 f'

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Sc=0.6 Du = 0.2 M = 0.5 Sr = 1.0 fw = 0.5

0.6

1 Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 fw = 0.5

f'

0.8

0.4

0.6 0.2 0

0.4 0

0.5

1

1.5

2

2.5

3

3.5

4

0.2

η

Fig.6(a). Velocity profiles for different values of Ec

0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

1


Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Sc=0.6 Du = 0.2 M = 0.5 Sr = 1.0 fw = 0.5

0.8

1

0.8

0.6 θ

Sc

Sc = 0.3, 0.6, 0.78, 0.94

Ec = 0.0, 0.01, 0.02, 0.03 0.6

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 f w = 0.5

φ

0.4

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

0.2

4

η

Fig.6(b). Temperature profiles for different values of Ec

0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

1.4

Fig.8(b). Concentration profiles for different values of

1.2

1.4

Du = 0.0, 0.2, 0.6, 1.0

1

Sr = 1.0, 1.5, 2.0, 2.5

1.2

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 M = 0.5 Sc = 0.6 Sr = 1.0 fw = 0.5

f'

0.8

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 Sc = 0.6 M = 0.5 f w = 0.5

f'

0.6

1 0.8 0.6

0.4

0.4

0.2 0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

0

η

0

0.5

1

1.5

2

2.5

3

η


78

3.5

4

Sc



Sr

1

1

0.8

Sr = 0.0, 1.0, 1.5, 2.0

fw = 0.5, 1.0, 1.5, 2.0

0.8

0.6 Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 Sc= 0.6

φ

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 Sc = 0.6 M = 0.5 fw = 0.5

0.6 φ

0.4 0.4

0.2 0.2

0 0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig.10(c). Concentration profiles for different values of

η

Fig.9(b). Concentration profiles for different values of

Sr

fw

Table 1 Numerical values of the skin-friction coefficient C f ,

1.4 1.2

Nusselt number Nu and Sherwood Pr = 0.71 , Ec = 0.01 , Du

fw = 0.5, 1.0, 1.5, 2.0

1 0.8 f'

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 Sc= 0.6

0.6

number Sh = 0.2 , Sc = 0.6 , Sr

for

= 1.0

0.4 0.2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

Fig.10(a). Velocity profiles for different values of 1

0.8 fw = 0.5, 1.0, 1.5, 2.0

θ

0.6

0.4

Gr = 2.0 Gm = 2.0 K = 0.5 Pr = 0.71 Ec = 0.01 Du = 0.2 M = 0.5 Sr = 1.0 Sc= 0.6

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

Gr

Gm

M

K

fw

Cf

Nu

Sh

2.0 4.0 2.0 2.0 2.0 2.0

2.0 2.0 4.0 2.0 2.0 2.0

0.5 0.5 0.5 1.0 0.5 0.5

0.5 0.5 0.5 0.5 1.0 0.5

0.5 0.5 0.5 0.5 0.5 1.0

0.8230 2 1.6865 0 1.8853 3 0.4906 8 0.4878 1 0.5115 4

0.8618 6 0.9019 3 0.9188 3 0.8400 5 0.8395 6 1.0936 8

0.4362 2 0.4647 9 0.4794 3 0.4213 6 0.4198 4 0.4730 1

fw

4

η

Fig.10(b). Temperature profiles for different values of

fw


79


concentration in the presence of magnetic field, Int. J. Eng. Sci., Vol.34, pp. 515-522. [9]. Chamkha A.J. and Khaled A.R.A. (2001), Similarity solutions for hydrodynami simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption, Heat Mass Transfer, Vol.37, pp.117-123. [10]. Eckert E.R.G. and Drake R. M. (1972), Analysis of Heat and Mass Transfer, [11]. McGraw-Hill Book Co., New York. pp.217230. [12]. Dursunkaya Z. and Worek W. M. (1992), Diffusion-thermo and thermal diffusion effects in transient and steady natural convection from a vertical surface, Int. J. Heat Mass Transfer, Vol.35, pp.2060-2065. [13]. Kafoussias N.G. and Williams E.M. (1995), Thermal-diffusion and Diffusion-thermo effects on free convective and mass transfer boundary layer flow with temperature dependent viscosity, Int. J. Engg. Science, Vol. 33, pp.1369-1376. [14]. Anghel M., Takhar H.S. and Pop I. (2000), Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous [15]. medium, J. heat and Mass Transfer. Vol.43, pp.1265-1274. [16]. Postelnicu A. (2004), Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer, Vol.47, pp.1467-1472. [17]. Alam M.S. and Rahman M.M. (2006), Dufour and Soret effects on mixed convection flow past a vertical porous flat plate with variable suction, Nonlinear Analysis: Modelling and Control, Vol.11, pp.435-442. [18]. Alam M.S., Ferdows M. and Maleque M.A. (2006), Dufour and Soret effects on steady free convection and mass transfer flow past a semiinfinite vertical plate in a porous medium, Int. J. of Applied Mechanics& Eng., Vol. 11, No.3, pp.535545. [19]. Gebharat B.(1962), Effects of viscous dissipation in natural convection , J. Fluid Mech., Vol. 14, pp.225-232. [20]. Gebharat B. and Mollendorf J.(1969), Viscous dissipation in external natural [21]. convection flows, J. Fluid. Mech.,Vol.38, pp.97-107. [22]. Soundalgekar V.M.(1972), Viscous dissipation effects on unsteady free convective flow past an infinite, vertical porous plate with constant suction, Int. J. Heat Mass Transfer, Vol. 15, pp.1253-1261.

Table 2 Numerical values of the skin-friction coefficient C f , Nusselt number Nu and Sherwood

number Sh

for

Gr = 2.0 , Gm = 2.0 , M = 0.5 , K = 0.5 , f w = 0.5

Pr

Ec

Du

Sc

Sr

Cf

Nu

Sh

0.7 1 1.0 0.7 1 0.7 1 0.7 1 0.7 1

0.0 1 0.0 1 0.0 2 0.0 1 0.0 1 0.0 1

0.2 0.2 0.2 0.4 0.2 0.2

0.6 0.6 0.6 0.6 0.7 8 0.6

1.0 1.0 1.0 1.0 1.0 2.0

0.8230 2 0.7537 1 0.8240 6 0.8405 3 0.7731 5 1.0084 4

0.8618 6 1.1087 2 0.8590 6 0.8292 4 0.8469 4 0.9330 8

0.4362 2 0.2908 4 0.4378 8 0.4573 4 0.4994 9 0.2931 3

ACKNOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper. REFERENCES [1]. Bejan A. and Khair K.R. (1985), Heat and mass transfer by natural convection in a porous medium, Int. J. Heat Mass Transfer, Vol. 28, pp.909-918. [2]. Lai F.C. and Kulacki F.A. (1990), Coupled heat and mass transfer from a sphere buried in an infinite porous medium, Int. J. Heat Mass Transfer, Vol. 33, pp.209-215. [3]. Raptis A., Tzivanidis G. and Kafousias N. (1981), Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction, Lett. Heat Mass Transfer, Vol. 8, pp. 417-424. [4]. Lai F.C. and Kulacki F.A. (1991), Coupled heat and mass transfer by natural convection from vertical surfaces in a porous medium, Int. J. Heat Mass Transfer, Vol. 34, pp.1189-1194. [5]. Huges W.F. and Young F.J. (1966), The ElectroMagneto Dynamics of fluids, John Wiley and Sons, NewYork. [6]. Raptis A. (1986), Flow through a porous medium in the presence of magnetic field, Int. J. Energy Res., Vol.10, pp. 97-101. [7]. Helmy K.A. (1998), MHD unsteady free convection flow past a vertical porous plate, ZAMM, Vol. 78, pp. 255-270. [8]. Elabashbeshy E.M.A. (1997), Heat and mass transfer along a vertical plate with variable temoerature and


80


[23]. Israel-Cookey C., Ogulu A. and OmuboPepple V.B.(2003), Influence of viscous dissipation on unsteady MHD free-convection floe past an infinite heated vertical plate in porous medium with time-dependent suction, Int. J. Heat Mass transfer, Vol.46, pp.2305-2311. [24]. Chamkha A.J. and Camille I. (2000), Effects of heat generation/absorption and thermophoresis on hydromagnetic flow with heat and mass transfer over a flat surface, Int. J. Numerical Methods in Heat and Fluid Flow, Vol.10, pp.432448. [25]. Jain M.K., Iyengar S.R. K. and Jain R.K. (1985), Numerical Methods for Scientific and Engineering Computation, Wiley Eastern Ltd., New Delhi, India.


81



PARETO DISTRIBUTION - SOME METHODS OF ESTIMATION R. Subba Rao@, R.R.L. Kantam#, G.Srinivasa Rao$ @

Shri Vishnu Engineering College for Women, Bhimavaram-534 202, Andhra Pradesh, INDIA, E-mail: [email protected] # Department of Statistics, Acharya Nagarjuna University, Nagarjuna Nagar – 522 510. Andhra Pradesh, INDIA, E-mail: [email protected] $ Assistant professor, College of Agriculture, KEREN, ERITREA, E-mail: [email protected]

ABSTRACT Pareto distribution of type IV is considered with a known location and shape parameters. Estimation of its scale parameter by the well known maximum likelihood method is modified by two different approaches in order to yield linear estimators. Estimation based on a single optimum quantile is also presented. The proposed methods are compared with respect to simulated sampling characteristics. Key words: Order statistics, M.L Estimation, Quantiles, Asymptotic Variance. 1. Introduction The Probability distribution function (P.d.f.) of Pareto (IV) distribution is given by

f (x ;σ , α ) =

α σ

  x − µ  1 +    σ 

−( α + 1)

 , x f µ, α f 1, 

given random sample in order to satisfy the requirement that X ≥ µ. Any other estimator of µ different from the first order statistic may not be that efficient, because it contains the maximum information about µ. Here without loss of generality we assume that µ is zero. Accordingly the density σconsidered f0 for estimation, is

f ( x ;σ , α ) =

(1.1) We start with general M.L. estimation of ‘α ‘and ‘σ’ taking µ as zero. As the estimating equations are to be solved by numerical iterative techniques we suggest some modifications to M.L. method from complete as well as censored samples. Discussion of complete sample situation is given in section 2, whereas section 3 deals with the situation of censored samples. Quantile estimation based on optimally selected sample quantiles is presented in section 4. Whenever the results are based on numerical computations, all such results are presented in the form of numerical tables towards the end with appropriate identification labels.

α  x  1 +   σ  σ 

− ( α + 1)

  

, X >0

(2.1) Let X1 < X 2 < X3 < X4 < -------- < Xn be an ordered sample of size ‘n’ from a Pareto distribution (2.1). The log likelihood equations to estimate α, σ from the given complete sample are given by

 x   x  ∂ logL n  x  = − log1+ 1  . 1+ 2 . ......1+ n   ∂α α  σ   σ   σ  (2.2) n zi ∂ log L = (1 + α ) ∑ −n i =1 1 + z ∂σ i

2. Estimation from Complete Sample

(2.3) The parameter µ in the p d f given by equation (1.1) is the threshold parameter and is generally estimated by the first order statistic in a Pareto Distribution - Some Methods Of Estimation

82


zi =

where

xi

Method I

σ

i Let p = , i = 1, 2, 3, − − − − − n, i n +1

For M.L.Es of α and σ ,

∂ logL ∂ logL = 0, =0 ∂α ∂σ

Let Zi, Zi’ be the solutions of the following equations '

n

∧

α =

'

''

where

x   ∑ log  1 + i  i =1 σ   n

pi qi

'

pi = pi −

(2.4)

z (1 + α ) i∑= 1 i − n 1 + zi n

,

n

' '

pi

= pi +

pi qi n

The solutions of zi ' and zi ' ' in our Pareto distribution are

= 0

'

xi

σ It can be seen that equation, (2.5) can be solved only by iterative method for σ. The MLE of α is an analytical expression involving σ. In order to overcome the iterative techniques that may some times lead to convergence problems we approximate the expression zi h(z i ) = 1 + zi (2.6) of the log likelihood equation (2.5) for estimating σ by a linear expression say

1

)−

(

'

and

zi = 1 − pi

zi = 1 − pi

(2.5) where z i =

''

F(zi ) = pi and F(zi ) = pi

After simplification these equations become

α

1

(

''

− 1 ''

)− α

− 1

The intercept γ i and slope δ i of the linear approximation in the equation (2.7) are respectively given by

( )

''

δi =

'

h ( zi ) − h zi ''

'

zi − zi

(2.9)

( )

− δi zi and γ i = h zi (2.10) The values of γi and δi in this method for n = 5, 10, 15 and 20 and for α = 2, 3and 4 are given in table (1)

h( z i ) ≅ γ i + δ i z i (2.7) in certain admissible ranges of Zi. Such approximations are not feasible for the log likelihood equation of α. Hence we develop our approximate ML method for estimation of σ with a known α. As per the parametric specifications we take α = 2, 3 and 4. After using the linear approximation given by equation (2.7) in the equation (2.5) and solving it for σ we get

∧

σ =

(1

+ α

n

)∑δ i =1

Method II Consider

zi h ( zi ) = 1 + zi

z = (1 − p

n

n − (α + 1 ) ∑ γ i = 1

Taylor’s

expansion

of

in the neighbourhood of ith

quantile of our standard Pareto population. We get another linear approximation for h(z), with δi = h '(zi ), 1 − α , i i

xi

i

the

pi =

i

(2.8) as an approximate MLE of σ , which is a linear estimator. We suggest two methods of finding γi, δi of equation (2.7 ). Similar methods are given in Srinivasa Rao and Kantam (2002) and Kantam and Sri Ram (2003)

)

−1

i n +1

γ i = h (z i ) − δ i z i Substituting these approximations in the equation (2.8) we get another linear estimator of σ with different values of γ i, δ i. The values of γi and δ i in this method for n = 5, 10, 15, and 20 and α = 2, 3 and 4 are given in Table ( 2)

Pareto Distribution - Some Methods Of Estimation

83


These two methods are asymptotically as efficient as exact maximum likelihood estimators as can be seen from the following narration (Tiku et al , 1986; Balakrishnan, 1990).

3 Estimation From Right Censored Samples As described in section 2, we develop estimation of the parameter σ from a censored sample with a known α.

In the suggested two methods of approximation, the function h(z) is linearised in the neighbourhood of the population quantile in two different senses. Because sample quantiles are consistent estimators of the corresponding population quantiles and sample moments are consistent estimators of the population moments, for large values of ‘n’, the neighbourhood of the population quantile becomes narrower, thereby giving more linearity of h(z) in that neighbourhood. That is the closeness of h (z) to γ + δ z is stronger, the larger the sample size. Hence the approximate and exact ∂ log L expressions of the log likelihood equation ∂σ differ by little values. Also exact and approximate 2 ∂ log L values of would differ little. Hence the 2 ∂σ exact MLE, Modified MLEs by the two methods for the parameter σ shall have the same asymptotic bias and asymptotic variance (Bhattacharya, 1985).

Let X1 < X2 < X3 < X4 < -------- < Xn be an ordered sample of size ‘n’ from a Pareto

a

distribution with unknown scale parameter σ and known shape parameter α. Let the largest ‘r’ observations be deleted so that X1 < X2 < X3 < X4 < ------- < Xn –r is Type II right censored sample (also called failure censored sample) from a Pareto distribution (2.1). The log likelihood function to estimate σ from the given censored sample is given from

 L ∝  

 −α r   ( 1+ xn−r )    

where the constant is independent of the parameters to be estimated. The log likelihood equation for estimating σ is given by

∂logL =0 ∂σ n −r (α +1) n− r x αr x n − r ⇒ − 2 ∑ i − 2. xn − r σ σ i =1 1+xi σ 1+

σ

A comparison of the sampling characteristics namely the bias, variance, and MSE ∧

 α   xi  ∏   1+  σ i =1  σ  

n−r   x  α logL=Cons tant + ∑  log  −(α +1) log 1+ i   i =1  σ   σ  xn −r  −α. r log1+   σ

However the same can not be said in small samples. Since the exact MLE of σ is an iterative solution of equation (2.5), its sampling variance can not be mathematically tractable. Hence, we have resorted to Monte Carlo simulation to get the empirical sampling characteristics of the exact M L E. We have computed the simulated bias, variance and M S E of exact M L E solving equation (2.5) iteratively for σ in 10,000 samples of size 5, 10, 15 and 20 each generated from standard Pareto distribution with α = 2, 3 and 4. These are given in table (3). The simulated bias, MSE, and variance of MML Es of σ are also given in table (3).

⇒

∧

of σ 1 , σ 2

−(α + 1)

n −r

- the two MMLEs together with those of

the corresponding exact MLE, reveal that MMLE of Method I is preferable to that of Method II as well as exact MLE in small samples as Method I recorded minimum values for bias variance and MSE. Coming to the actual magnitudes of these sample characteristics it is MMLE of method I that is closer to exact M L method rather than MMLE of Method II.

n−r

σ

−

(α +1)

n− r

σ

i =1

∑

n− r

⇒(n − r) − (α + 1) ∑

i =1

σ

zn − r zi αr − . =0 1+ zi σ 1+ zn − r

zn − r zi − αr . =0 1 + zi 1 + zn − r

(3.1) It can be seen that equation (3.1) cannot be solved analytically for σ .The M L E of σ has to be obtained as an iterative solution of (3.1). We zi approximate the expression of the 1 + zi Pareto Distribution - Some Methods Of Estimation

84

= 0


likelihood equation (3.1) for estimating σ by a linear expression as

h (z i ) (3.2)

≅

γi

which certain function is linearised depends on the size of the sample also. The larger the size, the closer the approximation. That is, the exactness of the approximation becomes finer and finer for large values of n. Hence, the approximate log likelihood equation and the exact log likelihood equation differ by little quantities for large n. Therefore, the solutions of exact and approximate log likelihood equations tend to each other as n → ∞. Hence the exact and modified M L Es are asymptotically identical (Tiku et al 1986). However, the same cannot be said in small samples. At the same time the small sample variance of exact M L E is not mathematically tractable. We therefore compared these estimates in small samples through Monte Carlo simulation.

+ δ i zi

where γi, δi are to be suitably found, to get a modified MLE of σ. Proceeding on the same lines as mentioned in section 2.2 approximate likelihood equation for σ is given by

∂ log L ∂σ

∂ log L ∂σ

≅

= 0

n− r

⇒(n−r) − (α+1) ∑ (γi +δi zi ) − αr (γn −r + δn −r zn −r ) = 0 i =1

⇒ (n − r) − (α + 1)

n− r

n −r

xi

i =1

i =1

σ

∑ γ i − ∑ δi

− α r γn − r

− α r δn − r

xn − r

σ

=0

Conclusions: In most of the situations it is the MMLE of Method – I that is rated as the most preferable method; the second preference going to exact MLE. The same trend is observed for other values of α also. Thus whether complete or censored sample, one can go for MLE with iterative solution or MMLE – I with linear analytical estimator. In large samples, as mentioned earlier all the three methods are equally efficient.

n− r ⇒ σ (n−r) − (α+1) ∑ γi − α r γn − r  i =1   n− r

− (α+1) ∑ δi xi − α r δn − r xn − r = 0 i =1

∧

⇒σ =

(α + 1) (n − r)

n− r

4 Estimation Based On Sample Quantiles

∑ δi xi + α r δn − r xn − r

i =1

− (α + 1)

The bias, the variance, the MSE of the estimates by the two methods of modification and that of the exact M L E obtained through simulation for n = 5, 10, 15 and 20 and α = 2, 3 and 4 with all possible considerations of right censored samples are given in table 4for α = 2only.

The concept of failure censored sample and estimation therefrom as described in section 3 can be modified slightly, with the notion of estimating unknown scale parameter σ based on selected order statistics in an optimum way. That is if ‘n’ is the given sample size and k is a positive integer less than ‘n’ best linear unbiased estimation based on a subset of k order statistics in the sample can be thought of using the theory of Lloyd (1952) if we have the moments of order statistics in a sample of size ‘n’. n   Accordingly we can get  c  BLUES for σ each with k    its own variance given by the formulae of Lloyd (1952). Among them, the BLUE with smallest variance is called estimator based on k – optimally selected order statistics. A revision of this procedure

n− r

∑ γi − α r γ n − r

i =1

(3.3) and the resulting MMLEs of σ from the censored sample by the two methods are similar to those given in section 2. The relevant values of slope and intercept are calculated for α = 2, 3, 4; n = 5, 10, 15, 20, and for all possible combinations of r, which run into 10 pages. Owing to the problem of space we are not including those tables here. In the two methods referred above, the basic zi principle is that the expression is 1 + zi approximated by a linear function in some neighbourhood of the population quantile. It can be seen that the construction of the neighbourhood over


85


may be as fallows. Suppose a population contains a single unknown parameter that requires estimation from a given sample of size ‘n’. We therefore select one order statistic from the sample optimally, corresponding to a population quantile. The optimality here is specified as the minimum asymptotic variance of the resulting estimator. If such an estimator is obtained, it is called quantile estimator or optimum estimation based on a single quantile. In the case of multi parameter populations the optimally selected sample quantiles will be as many as the number of independent unknown parameters that are to be estimated. The optimality criterion would be minimization of asymptotic generalized variance, taken as the trace of the asymptotic dispersion matrix or the determinant of the asymptotic dispersion matrix.

σˆ =

x  F ( x) = 1 −  1 +  σ   =1 −

[1

−1

[1 − P]

α

−1 where x p is the p quantile in the sample; which can be obtained as an ordered statistic in the sample whose suffix is [n.p ] + 1. The above choice of p has to be made in th

such a way that the variance of the σˆ is the minimum with respect to p. But the exact variance of

σˆ

is not analytically available.

asymptotic variance of fallows.

σˆ

Asymptotic variance of

σˆ =

∴ A S V A R (x p ) =

]

can be obtained as

A S VA R ( x p ) 2

p (1− p ) −1   α  [1 − p] α   

− 2 (α + 1)

2

− α

+ x

However the

−1   α − 1 [1 − p ]   From the asymptotic theory of order statistics, we know that the asymptotic variance of the Pth quantile in the sample is

In our present investigation, we consider only a single unknown parameter σ with other parameters assumed to be known. Let P be a real number between 0 and 1. Let ξ be the Pth quantile of standard Pareto population with known value of α. (i.e)ξ satisfies the equation F(ξ) = P

( ie )

xp

(4.1) For an optimum choice of a sample quantile we have to minimize the asymptotic variance given by (4.1) with respect to P − α

(if σ = 1 ) (i. e.)

∧ d [ A S VA R (σ )] = 0 dp

− 1

⇒

If

ξ

x

[1

=

−

p

]

α

− 1

is a standard Pareto variate, considering

σ as ξ, we get −1 x ξ = = [1 − p ] α − 1

2 −1      P ( 1 − P )  [1 − P ] α − 1  d     ⇒   = 0 − 2 (α + 1) −1 dp     2 α  α  [1 − P ]      

x

σ

σ

⇒

−1   ⇒ ξ = σ  [1 − p ] α − 1  

4

( 1 − p)

α

2

2p

+

α

+ 2

1   α 2 − ( 1 − p)    4

(1 − p )

α

+ 2

d2 [ASVAR(xp )]=0 dp2

for a given sample of size n, if σ is to be estimated, on the basis of a single sample quantile the above equation can be used as

σ =

1   α 1 − (1 − p)   

1 1 1   ⇒ (1− p) α  −6 pα − 6 p + 2α (1− p) α − 2α2 + 2α2 (1− p) α  =  

ξ −1

[1 − p ]

1

α

−1 where ξ is the population pth quantile. The above equation suggests a possible estimator for σ as

1

2α2 (1− p) α + 4α (1− p) α − 12 pα −16p −2α2


86

= 0


This equation has to be solved iteratively for P to get its optimum value corresponding to a minimum of

equation at α = 2, 3 and 4 gives that P = 0.000151, 0.000251, 0.000332. This shows that the sample size should be above thousand to get an asymptotic optimum sample quantile what ever may be the specified α.

the asymptotic variance of σˆ We have found that Newton Raphson method when applied to solve the above


87


TABLE 1 Intercept and slope of the approximation h(Zi) = γi+ δi Zi Method-I n

I

5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

α =2

γ

δ

γ

α =3

δ

γ

0.00000 0.81650 0.00000 0.87358 0.00000 0.02055 0.63246 0.00982 0.73681 0.00573 0.07083 0.44721 0.03564 0.58480 0.02136 0.17051 0.25820 0.09241 0.40548 0.05763 0.64998 0.00000 0.50334 0.00000 0.40837 0.00000 0.90453 0.00000 0.93530 0.00000 0.00503 0.80904 0.00232 0.86825 0.00133 0.01599 0.71351 0.00751 0.79848 0.00434 0.03407 0.61791 0.01635 0.72547 0.00956 0.06094 0.52223 0.03000 0.64850 0.01778 0.09904 0.42640 0.05024 0.56652 0.03023 0.15222 0.33029 0.08005 0.47782 0.04910 0.22729 0.23355 0.12518 0.37924 0.07871 0.33903 0.13484 0.19955 0.26295 0.13018 0.75536 0.00000 0.60884 0.00000 0.50538 0.00000 0.93541 0.00000 0.95647 0.00000 0.00223 0.87082 0.00101 0.91191 0.00058 0.00693 0.80623 0.00319 0.86624 0.00183 0.01441 0.74162 0.00672 0.81932 0.00388 0.02503 0.67700 0.01185 0.77101 0.00688 0.03924 0.61237 0.01886 0.72112 0.01104 0.05761 0.54772 0.02817 0.66943 0.01663 0.08087 0.48305 0.04030 0.61564 0.02403 0.11003 0.41833 0.05599 0.55934 0.03375 0.14645 0.35355 0.07634 0.50000 0.04660 0.19216 0.28868 0.10301 0.43679 0.06382 0.25036 0.22361 0.13883 0.36840 0.08759 0.32671 0.15811 0.18917 0.29240 0.12222 0.43358 0.09129 0.26710 0.20274 0.17876 0.80098 0.00000 0.65912 0.00000 0.55388 0.00000 0.95119 0.00000 0.96719 0.00000 0.00125 0.90238 0.00057 0.93381 0.00032 0.00386 0.85356 0.00176 0.89982 0.00100 0.00793 0.80475 0.00365 0.86518 0.00209 0.01361 0.75593 0.00633 0.82983 0.00364 0.02105 0.70711 0.00990 0.79370 0.00573 0.03043 0.65828 0.01447 0.75673 0.00842 0.04197 0.60945 0.02019 0.71883 0.01182 0.05593 0.56061 0.02725 0.67989 0.01606 0.07262 0.51177 0.03588 0.63981 0.02130 0.09244 0.46291 0.04636 0.59841 0.02773 0.11588 0.41404 0.05907 0.55551 0.03564 0.14359 0.36515 0.07453 0.51087 0.04539 0.17644 0.31623 0.09345 0.46416 0.05752 0.21562 0.26726 0.11685 0.41491 0.07282 0.26290 0.21822 0.14634 0.36246 0.09251 0.32108 0.16903 0.18458 0.30571 0.11877 0.39512 0.11952 0.23669 0.24264 0.15583 0.49587 0.06901 0.31503 0.16824 0.21459 0.82795 0.00000 0.69066 0.00000 0.58522 Pareto Distribution - Some Methods Of Estimation

88

α =4

δ

0.90360 0.79527 0.66874 0.50813 0.00000 0.95107 0.89947 0.84469 0.78608 0.72266 0.65299 0.57471 0.48327 0.36721 0.00000 0.96717 0.93318 0.89790 0.86117 0.82280 0.78254 0.74008 0.69502 0.64678 0.59460 0.53728 0.47287 0.39764 0.30214 0.00000 0.97529 0.94994 0.92389 0.89708 0.86944 0.84090 0.81134 0.78067 0.74874 0.71538 0.68037 0.64346 0.60428 0.56234 0.51697 0.46714 0.41113 0.34572 0.26269 0.00000


TABLE 2 Intercept and slope of the approximation F(t i) = γ i + δ I Method-I I α =2

α =3

n

I

γ

δ

5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.00759 0.03367 0.08579 0.17863 0.35017 0.00217 0.00911 0.02167 0.04092 0.06836 0.10615 0.15759 0.22826 0.32902 0.48789 0.00101 0.00417 0.00972 0.01795 0.02919 0.04386 0.06250 0.08579 0.11462 0.15026 0.19447 0.25000 0.32147 0.41789 0.56250 0.00058 0.00238 0.00550 0.01005 0.01616 0.02398 0.03367 0.04546 0.05957 0.07632 0.09606 0.11926 0.14653 0.17863 0.21667 0.26220 0.31760 0.38693 0.47802 0.61118

0.83333 0.66667 0.50000 0.33333 0.16667 0.90909 0.81818 0.72727 0.63636 0.54545 0.45455 0.36364 0.27273 0.18182 0.09091 0.93750 0.87500 0.81250 0.75000 0.68750 0.62500 0.56250 0.50000 0.43750 0.37500 0.31250 0.25000 0.18750 0.12500 0.06250 0.95238 0.90476 0.85714 0.80952 0.76190 0.71429 0.66667 0.61905 0.57143 0.52381 0.47619 0.42857 0.38095 0.33333 0.28571 0.23810 0.19048 0.14286 0.09524 0.04762

γ 0.00348 0.01598 0.04256 0.09403 0.20221 0.00098 0.00419 0.01014 0.01956 0.03347 0.05342 0.08193 0.12355 0.18791 0.30289 0.00045 0.00190 0.00447 0.00836 0.01379 0.02103 0.03046 0.04256 0.05801 0.07777 0.10330 0.13693 0.18288 0.25000 0.36379 0.00026 0.00108 0.00251 0.00463 0.00751 0.01126 0.01598 0.02183 0.02896 0.03760 0.04801 0.06054 0.07567 0.09403 0.11653 0.14455 0.18031 0.22776 0.29521 0.40646

α =4

δ 0.88555 0.76314 0.62996 0.48075 0.30285 0.93844 0.87478 0.80872 0.73984 0.66758 0.59118 0.50946 0.42055 0.32094 0.20218 0.95789 0.91483 0.87073 0.82548 0.77896 0.73100 0.68142 0.62996 0.57630 0.52002 0.46050 0.39685 0.32759 0.25000 0.15749 0.96800 0.93545 0.90234 0.86860 0.83419 0.79906 0.76314 0.72636 0.68861 0.64980 0.60980 0.56844 0.52551 0.48075 0.43380 0.38415 0.33105 0.27328 0.20855 0.13138


89

γ 0.00199 0.00929 0.02531 0.05768 0.13036 0.00055 0.00239 0.00586 0.01142 0.01977 0.03201 0.04993 0.07692 0.12041 0.20331 0.00026 0.00108 0.00256 0.00482 0.00800 0.01229 0.01795 0.02531 0.03486 0.04729 0.06367 0.08579 0.11694 0.16435 0.25000 0.00015 0.00061 0.00143 0.00265 0.00432 0.00651 0.00929 0.01277 0.01705 0.02228 0.02866 0.03644 0.04595 0.05768 0.07230 0.09088 0.11517 0.14839 0.19756 0.28394

δ 0.91287 0.81650 0.70711 0.57735 0.40825 0.95346 0.90453 0.85280 0.79772 0.73855 0.67420 0.60302 0.52223 0.42640 0.30151 0.96825 0.93541 0.90139 0.86603 0.82916 0.79057 0.75000 0.70711 0.66144 0.61237 0.55902 0.50000 0.43301 0.35355 0.25000 0.97590 0.95119 0.92582 0.89974 0.87287 0.84515 0.81650 0.78680 0.75593 0.72375 0.69007 0.65465 0.61721 0.57735 0.53452 0.48795 0.43644 0.37796 0.30861 0.21822


TABLE 3 Sample Characteristics of MLE and MMLE of σ from complete Samples.

BIAS

VARIANCE

MSE

α

n

2 2 2 2

5 10 15 20

0.17289 0.08545 0.05653 0.04262

0.52377 0.23200 0.14723 0.10649

1.28796 0.40863 0.21541 0.14184

0.71388 0.26818 0.16294 0.11458

0.52383 0.23210 0.14728 0.10652

1.41484 0.44884 0.23492 0.15333

3 3 3 3

5 10 15 20

0.11358 -0.00765 0.20724 0.46483 0.39623 0.05785 0.00617 0.12425 0.19810 0.18459 0.03881 0.00486 0.08977 0.12507 0.11938 0.02946 0.00440 0.07054 0.08965 0.08701

0.60761 0.24237 0.14570 0.10095

0.47773 0.20145 0.12657 0.09051

0.39629 0.18463 0.11940 0.08703

0.65055 0.25781 0.15376 0.10592

4 4 4 4

5 10 15 20

0.08450 -0.01788 0.14625 0.37894 0.34572 0.04394 0.00316 0.09012 0.17001 0.16379 0.02978 0.00342 0.06621 0.10915 0.10657 0.02271 0.00350 0.05266 0.07878 0.07784

0.44431 0.19258 0.12061 0.08528

0.38608 0.17194 0.11004 0.07930

0.34604 0.16380 0.10658 0.07785

0.46570 0.20070 0.12499 0.08805

MLE

MMLE-I MMLE-II

0.00784 0.01004 0.00637 0.00526

0.35621 0.20054 0.13965 0.10723

MLE

0.68399 0.26088 0.15975 0.11277

MMLE-I MMLE-II

MLE


90

MMLE-I MMLE-II


TABLE 4 Sample Characteristics of MLE and MMLE of σ from right censored samples (for α = 2)

BIAS n

r

VARIANCE

MMLEMMLE-I II -0.02913 0.20894 -0.00262 0.17056 -0.00075 0.13638

MLE 0.75861 0.74603 0.93202

MMLEMMLE-I II 0.58640 0.94666 0.56047 0.77531 0.73245 0.94696

MSE

5 5 5

1 2 3

MLE 0.14677 0.14934 0.12808

10 10 10 10 10 10 10 10

1 2 3 4 5 6 7 8

0.07112 0.09819 0.08458 0.04250 0.07066 0.05555 0.03839 0.03765

-0.02030 0.01465 0.00637 -0.02704 0.00321 -0.00744 -0.02002 -0.01732

0.12721 0.13395 0.10723 0.05754 0.08056 0.06133 0.04159 0.03917

0.25736 0.30383 0.28847 0.24044 0.33098 0.35651 0.45516 0.65484

0.22158 0.26172 0.24735 0.20976 0.29022 0.31527 0.40512 0.58754

0.30009 0.32726 0.30036 0.24795 0.33661 0.36059 0.45764 0.65694

0.26242 0.31347 0.29562 0.24224 0.33597 0.35959 0.45663 0.65626

0.22200 0.26194 0.24739 0.21049 0.29023 0.31533 0.40552 0.58784

0.31628 0.34521 0.31186 0.25126 0.34310 0.36435 0.45937 0.65848

15 15 15 15 15 15 15 15 15 15 15 15 15

1 2 3 4 5 6 7 8 9 10 11 12 13

0.04947 0.06085 0.03111 0.05043 0.04336 0.07767 0.04635 0.04085 0.02159 0.05605 0.02927 0.04223 0.05583

-0.01035 0.00433 -0.02191 -0.00044 -0.00582 0.02943 0.00088 -0.00244 -0.01924 0.01526 -0.00894 0.00482 0.01927

0.09825 0.09446 0.05429 0.06930 0.05719 0.08910 0.05442 0.04714 0.02626 0.05940 0.03156 0.04355 0.05644

0.16036 0.16020 0.14769 0.16553 0.17373 0.19782 0.21385 0.20875 0.22239 0.28886 0.31104 0.40826 0.63646

0.14506 0.14461 0.13290 0.15224 0.15847 0.18005 0.19526 0.19195 0.20487 0.26696 0.28833 0.37951 0.59306

0.18140 0.17209 0.15483 0.17473 0.17941 0.20154 0.21685 0.21156 0.22437 0.29067 0.31238 0.40933 0.63711

0.16280 0.16390 0.14866 0.16807 0.17561 0.20385 0.21600 0.21042 0.22285 0.29200 0.31190 0.41005 0.63958

0.14516 0.14463 0.13338 0.15224 0.15850 0.18092 0.19526 0.19196 0.20524 0.26719 0.28841 0.37953 0.59344

0.19105 0.18101 0.15778 0.17953 0.18268 0.20948 0.21981 0.21378 0.22506 0.29420 0.31337 0.41122 0.64030

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.04036 0.03295 0.03236 0.04639 0.05041 0.04053 0.03492 0.03444 0.03900 0.04341 0.02592 0.04569 0.02297 0.03449 0.05060 0.01968 0.03786 0.00903

-0.00480 -0.00873 -0.00904 0.00733 0.01167 0.00350 -0.00041 -0.00094 0.00514 0.00962 -0.00598 0.01420 -0.00712 0.00490 0.02137 -0.00791 0.01050 -0.01691

0.08016 0.06336 0.05447 0.06630 0.06595 0.05348 0.04612 0.04259 0.04632 0.04871 0.03054 0.04957 0.02585 0.03674 0.05232 0.02086 0.03860 0.00934

0.11234 0.10709 0.10853 0.12636 0.12318 0.12842 0.13426 0.12860 0.14323 0.13905 0.16622 0.18193 0.17536 0.21107 0.26901 0.29481 0.39870 0.51541

0.10486 0.10014 0.10025 0.11797 0.11505 0.11983 0.12597 0.11976 0.13466 0.13024 0.15619 0.17112 0.16514 0.19921 0.25418 0.27904 0.37800 0.48920

0.12537 0.11601 0.11393 0.13240 0.12794 0.13225 0.13818 0.13043 0.14595 0.14055 0.16788 0.18326 0.17627 0.21204 0.26982 0.29545 0.39931 0.51567

0.11397 0.10818 0.10958 0.12851 0.12572 0.13006 0.13548 0.12978 0.14475 0.14094 0.16689 0.18402 0.17589 0.21226 0.27157 0.29519 0.40013 0.51549

0.10488 0.10022 0.10033 0.11802 0.11519 0.11984 0.12597 0.11976 0.13468 0.13033 0.15623 0.17132 0.16519 0.19923 0.25463 0.27910 0.37811 0.48948

0.13179 0.12003 0.11690 0.13679 0.13229 0.13511 0.14030 0.13224 0.14810 0.14292 0.16881 0.18572 0.17694 0.21339 0.27255 0.29589 0.40080 0.51576


91

MLE 0.78015 0.76833 0.94843

MMLE-I MMLE-II 0.58725 0.99031 0.56047 0.80440 0.73245 0.96556


Acknowledgement: We would like to express our thanks to referees for valuable comments that improved the paper. References: [1]. Balakrishnan, N. (1990). “Approximate maximum likelihood estimation for a generalized logistic distribution”, Journal of Statist. Plann. & inf., 26, 221-236. [2]. Bhattacharya.G..K.(1985). “The Asymptotics of maximum likelihood and related estimators based on Type II censored data”. Journal of American Statistical Association. 80, 398-404. [3]. Kantam R .R .L. & Sriram.B. (2003), “Maximum Likelihood Estimation from censored samples –some modifications in length biased version of exponential Model”, Statistical methods, Vol. 5, 63-78. [4]. Lloyd, E. H. (1952). “Least-squares estimation of location and scale parameters using order statistics”, Biometrika, 39, 88-95. [5]. Tiku. M.L., Tan. W .Y. and Balakrishnan. N. (1986). “Roboust Inference”, Marcel Decker, I. N.C. New York. [6]. Srinivasa rao. G. and Kantam. R. R .L. (2002) “A note on point estimation of system reliability exemplified for the Log-Logistic distribution”, Economic quality control, 19(2), 197-204,


92

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