IJCMI-VOL2-NO1-NO2-2010 (1)

INTERNATIONAL JOURNAL OF

COMPUTATIONAL MATHEMATICAL IDEAS

ISSN: 0974-8652 Volume 2 / Number 1 / January 2010

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CONTENTS

Research Papers Page No

A Note on r - partitions of n in which The Least Part is k 6-12

K.Hanuma Reddy

Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And 13-21

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

On Completely Prime And Completely Semi-Prime Ideals In ΓΓΓΓ-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 55-59

cyclohexane + o-xylene at 303.15, 308.15, 313.15 and 318.15k. Narendra K, Narayanamurthy P & Srinivasu Ch



ISSN: 0974-8652 Volume 2 / Number 2 / August 2010

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CONTENTS

Research Papers Page No

Over view to Implementation of robotics with Voice recognition 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 73-81

Porous Plate With Soret And Dufour Effect G.Venkata Ramana Reddy and Dr. A.Rami Reddy

Pareto Distribution - Some Methods Of Estimation 82-92 R. Subba Rao, R.R.L. Kantam, G.Srinivasa Rao


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ISSN: 0974-8652

INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652

A Note On r - partitions Of n In Which The Least Part Is k

6

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− , ( )r

p n , ( );rp e n , ( );rp o n and ( );rp S n

Subject classification: 11P81 Elementary theory of partitions. 1. Introduction: 1.1 Partition: A partition of a positive integer n is a finite sequence of non-increasing positive

integers 1 2, , ... , rλ λ λ such that 1

r

ii

nλ=

=∑

The number iλ is called the thi part of the

partition. The partition is also denoted as

1 2 r( , , ... , )n λ λ λ= .

1.2 Partition function: The partition function

( )p n is the number of partitions of n.

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

1.4 ( ) :r

p n ( )r

p n is the number of

r partiions− of a positive integer n .

Note: 1 2( ) ( ) ( ) ... ( )np n p n p n p n= + + +

1.5 ( ) :;rp o n ( );rp o n is the number of

partitions of a positive integer n having r parts

in which each part is odd number.

1.6 ( ) :;rp e n ( );rp e n is the number of


in which each part is even number.

1.7 ( ) :;rp S n ( );rp S n is the number of


in which each part is the element of the set S.

2. Theorem: Let ( ),r n N r n∈ ≤ and

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

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

( );r r

n brp S n p

a

−=

other wise

( ); 0rp S n = .

[ ]2.1

Proof: All parts in r partiions− of n multiplied

by a and added by b to get the partitions of n

whose parts are elements of S .

3. Theorem: Let ,r n N∈ and

{ }| , 1,2, ...,S am b a N b Z and m 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

na b

ar b+

+



7

[ ]3.1

Proof: Let 1 2, , ... , rλ λ λ be the first,

second,…, th

r parts of the r-partition of ‘n’ respectively.

So 1 2 r( , , ... , )n λ λ λ=

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.

If possible, let 1 1n

a bar b

λ = + ++

Since all the parts in each r partiions− are

monotonically increasing order, the least

possible value of each i λ for i = 2 to r is

1n

a bar b

+ ++

.

Then the sum of all parts in partition is

1n

r a bar b

+ ++

.

But 1n

r a bar b

+ ++

> n

This is contradiction.

Hence 1

na b

ar bλ = +

+

is the highest

integer.

4. Theorem: Let ,r n N∈ and

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

be the set of positive integers. Then prove that

( ){ }( )[ ]

1( ; ) ; 1

( ; ) 4.1

r r

r

p S n - p S n a b

p S n ar

− − +

= −

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

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

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

( ){ }1n a b− + 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 ( )1a b+ is equal to the number

of r partiions− of n ar− whose parts are

elements of S .

( ){ }( )1( ; ) ; 1

( ; )

r r

r

p S n - p S n a b

p S n ar

−∴ − +

= −

5. Theorem: Let , ,r n k N∈ and

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

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

elements of S with least part k is

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

where 1n

kar b

≤ ≤+

[ ]5.1

Proof: Let 1 2, , ... , rλ λ λ be the first,

second,…, thr parts of the r partiions− of n respectively.

So

1 2 r( , , ... , )n λ λ λ=

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 ( )1 1a bλ = + , the remaining value

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

of the remaining 1r − parts 2 3 , , ... , rλ λ λ in

( ){ }( )1 ; 1rp S n a b− − + ways.

i,e The number of r partiions− in which the

least part of the partition is ( )1 1a bλ = + is

{ }( )1 ;rp S n a b− − + .



of the remaining 1r − parts 2 3 r , , ... , λ λ λ in

( ){ }( )1 ; 2rp S n a b− − + ways.

Since all the parts in each r-partition are non

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

r partiions− with ( )1 2a bλ = + , ( )2 1a bλ = +

are to be eliminated from

( ){ }( )1 ; 2rp S n a b− − + r partiions− .Then,

the number of the r partiions− in which the



8

least part of the partition ( )1 2a bλ = + is

( ){ }( )1 ; 2rp S n a b− − +

-

( ) ( ){ }( )2 ; 3 2rp S n a b− − +

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

{ }( )1 ;rp S n ar a b−= − − +



of the remaining 1r − parts 2 3 , , ... , rλ λ λ in

( ){ }( )1 ; 3rp S n a b− − + ways.

( ) ( ){ }( )2 ; 4 2rp S n a b− − + r partiions− with

( )1 3a bλ = + , ( )2 1a bλ = + and

( ) ( ){ }( )2 ; 5 2rp S n a b− − +

-

( ) ( ){ }( )3 ; 6 3rp S n a b− − +

r partiions− with ( )1 3a bλ = + , ( )2 2a bλ = +

are to be eliminated

from ( ){ }( )1 ; 3rp S n a b− − + r partiions− .

Then, the number of the r partiions− in

which the least part of the partition

( )1 3a bλ = + is

( ){ }( )1 ; 3rp S n a b− − +

- ( ) ( ){ }( )2 ; 4 2rp S n a b− − +

-

( ) ( ){ }( )( ) ( ){ }( )

2

3

; 5 2

; 6 3

r

r

p S n a b

p S n a b

−

−

− +

− − +

( ) ( ){ }( )

( ) ( ){ }( )1

2

; 1 3

; 2 5

r

r

p S n a r a b

p S n a r a b

−

−

= − − − +

− − − − +

( ){ }( )

( ){ }( )1

2

; 2

; 3

r

r

p S n ar a b

p S n ar a b

−

−

= − − +

− − − +

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

{ }( )1 ; 2rp S n ar a b−= − − +

By induction we observe that the number of

r partiions− of n having the parts are elements

of S with least part k is

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

where 1n

kar b

≤ ≤ +

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

of r partiions− of n with least part k is

( )1 1 1 where 0r

np n k r k

r−

− − − ≤ ≤

Proof: Put 1, 0a b= = in [ ]5.1

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

( ); 2 1 2 where 01 2

np e n k r kr r

− − − ≤ ≤ −

Proof: Put 2, 0a b= = in [ ]5.1

Corollary 5.3: , ,Let n r k N∈ . Then the

number of r partiions− of n having the parts

are odd numbers with least part k is

( )1 ; 2 1 1

where 02 1

r o n k r

nk

r

p − − − −

≤ ≤ −

Proof: Put 2, 1a b= = − in [ ]5.1

6. Reduction theorem for ( ; )rp S n :

Let , ,r n k N∈ and

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

be the set of positive integers. then,

( ) { }( )1

1

(S; ) ; 1

n

ar b

r r

k

p n p S n k ar a b

+

−

=

= − − − +∑

[6.1] and

( ) { }( )1

1 1

(S; ) ; 1

n

ar bn

r

r k

p n p S n k ar a b

+

−

= =

= − − − +∑ ∑

[6.2]

Proof: From [ ]5.1 we can observe it

Corollary 6.1: Prove that

( )( )1

1 1

( ) 1 1

n

rn

r

r k

p n p n k r

−

= =

− − −=∑∑



9

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


( )( )2

1

1 1

(e; ) ; 2 1 2

n

rn

r

r k

p n p e n k r

−

= =

− − −=∑∑

Proof: Put 2, 0a b= = in [ ]6.2


( )( )2 1

1

1 1

(o; ) ; 2 1 1

n

rn

r

r k

p n p o n k r

−

−

= =

− − −=∑ ∑

Proof: Put 2, 1a b= = − in [ ]6.2

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

( ) ( )rp n p n r= − for 2n

r≤

Proof:

Case:1 Let 2n

r=

2n r⇒ =

1 2( ) ( ) ( ) ... ( )r rp n p n r p n r p n r= − + − + + −

1 2( ) ( ) ... ( )rp r p r p r= + + +

( )p r=

( )p n r= −

Case:2 Let 2n

r<

Since r n< and 2n

r<

2r n r⇒ < <

0 n r n⇒ < − <

1 2

1

( ) ( ) ( ) ...

( ) ( ) ... ( )

r

n r n r r

p n p n r p n r

p n r p n r p n r− − +

= − + − +

+ − + − + + −

1 2( ) ( ) ...

( ) 0 ... 0n r

p n r p n r

p n r−

= − + − +

+ − + + +

( )p n r= −

Hence (n) ( )rp p n r= − for 2n

r≤

Theorem 8: Let , ,n i j N∈ , then

2

( ) 1 ( )n

i

i j

p n p j+ =

= + ∑

Proof: Case 1: Let 2n m for m N= ∈

( ) (2 )p n p m=

1 2 3

1 1

2 2 2 1 2

(2 ) (2 ) (2 ) ...

(2 ) (2 ) (2 ) ...

(2 ) (2 ) (2 )

m m m

m m m

p m p m p m

p m p m p m

p m p m p m

− +

− −

= + + +

+ + + +

+ + +

{ }

{ }

{ }

{ }

1

1 2

1 2 3

1 2 1

(2 1)

(2 2) (2 2)

(2 3) (2 3) (2 3) ...

( 1) ( 1) ... ( 1)

( ) ( 1) ... (2) (1) 1

m

p m

p m p m

p m p m p m

p m p m p m

p m p m p p

−

= −

+ − + −

+ − + − + − +

+ + + + + + +

+ + − + + + +

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

1

1 2

1 2 3

1 2 1

1 2 1

1 2 1

1 2 3

1 2 1

(2 1)

(2 2) (2 2)

(2 3) (2 3) (2 3) ...

( 1) ( 1) ... ( 1)

( ) ( ) ... ( ) ( )

( 1) ( 1) ... ( 1) ...

(3) (3) (3)

(2) (2) (1) 1

m

m m

m

p m

p m p m

p m p m p m

p m p m p m

p m p m p m p m

p m p m p m

p p p

p p p

−

−

−

= −

+ − + −

+ − + − + − +

+ + + + + + +

+ + + + +

+ − + − + + − +

+ + +

+ + + +

=1+2

( )i

i j m

p j+ =

∑ +2 1

( )i

i j m

p j+ = −

∑ +…

+3

( )i

i j

p j+ =

∑ +2

( )i

i j

p j+ =

∑

2

2

1 ( )m

i

i j

p j+ =

= + ∑

2

1 ( )n

i

i j

p j+ =

= + ∑

Case 2: Let 2 1n m for m N= + ∈

1 2

3

1 2

2 1 2

2 1

( ) (2 1)

(2 1) (2 1)

(2 1) ... (2 1)

(2 1) (2 1) ...

(2 1) (2 1)

(2 1)

m

m m

m m

m

p n p m

p m p m

p m p m

p m p m

p m p m

p m

+ +

−

+

= +

= + + +

+ + + + +

+ + + + +

+ + + +

+ +



10

{ }

{ }

{ }

{ }

1

1 2

1 2 3

1 2

(2 )

(2 1) (2 1)

(2 2) (2 2) (2 2) ...

( 1) ( 1) ... ( 1)

( ) ( 1) ... (2) (1) 1

m

p m

p m p m

p m p m p m

p m p m p m

p m p m p p

=

+ − + −

+ − + − + − +

+ + + + + + + +

+ − + + + +

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

1

1 2

1 2 3

1 2

1 2

1 2 1

1 2 3

1 2 1

(2 )

(2 1) (2 1)

(2 2) (2 2) (2 2) ...

( 1) ( 1) ... ( 1)

( ) ( ) ... ( )

( 1) ( 1) ... ( 1) ...

(3) (3) (3)

(2) (2) (1) 1

m

m

m

p m

p m p m

p m p m p m

p m p m p m

p m p m p m

p m p m p m

p p p

p p p

−

=

+ − + −

+ − + − + − +

+ + + + + + +

+ + + +

+ − + − + + − +

+ + +

+ + + +

=1+2 1

( )i

i j m

p j+ = +

∑ +2

( )i

i j m

p j+ =

∑ +…+

2 1

2

1 ( )m

i

i j

p j+

+ =

= + ∑

2

1 ( )n

i

i j

p j+ =

= + ∑

Hence 2

( ) 1 ( )n

i

i j

p n p j+ =

= + ∑

Corollary 8.1: Let n be a natural number, then

( ) ( 1) ( )i

i j n

p n p n p j+ =

− − = ∑

Proof: Since 2

( ) 1 ( )n

i

i j

p n p j+ =

= + ∑

∴1

2

( 1) 1 ( )n

i

i j

p n p j−

+ =

− = + ∑

Hence ( ) ( 1) ( )i

i j n

p n p n p j+ =

− − = ∑

Theorem 9: If n N∈ then

Proof: If n is even

Let 2n m for m N= ∈

( )p n = 1( )p n + 2 ( )p n +…+ 1( )mp n− + ( )mp n

+ 1( )mp n+ +…+ 2 -2 ( )mp n + 2 1( )mp n− + 2 ( )mp n

= 1(2 )p m + 2 (2 )p m +…+ 1(2 )mp m− + (2 )mp m

+ 1(2 )mp m+ +…+ 2 -2 (2 )mp m + 2 1(2 )mp m−

+ 2 (2 )mp m

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

+ (1)p +1

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

= 1+ 1(3 -1)mp m− +

1

( )m

i

p i

=∑

= 1+1

2

3-1

2n

np

−

+2

1

( )

n

i

p i

=

∑

If n is odd

Let 2 1n m for m N= + ∈

( )p n = 1( )p n + 2 ( )p n +…+ 1( )mp n−

+ ( )mp n + 1( )mp n+ + 2 ( )mp n+ +…

+ 2 1( )mp n− + 2 ( )mp n + 2 1( )mp n+

= 1(2 +1)p m + 2 (2 +1)p m +…+ (2 +1)mp m

+ m+1(2 +1)p m + m+2 (2 +1)p m +…

+ 2m-1(2 +1)p m + 2 (2 +1)mp m + 2m+1(2 +1)p m

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

+ (2)p + (1)p +1

= 1+ (3 1)mp m + +i 1

( )m

p i=∑

= 1+ -1

2

3 -1

2n

np

+

1

2

1

( )

n

i

p i

−

=

∑

Hence

Theorem 10:

If 1m

r

=

,then

11

1

m

r

+ = +

for ,m r Z∈

Proof: Since 1m

r

=

2r m r⇒ ≤ < 1 1 2 1r m r⇒ + ≤ + < +



11

Since 1 1 2 1 2 1 2 2r m r and r r+ ≤ + < + + ≤ +

1 1 2 2r m r∴ + ≤ + < +

1

1 21

m

r

+⇒ ≤ <

+

1

11

m

r

+ ⇒ = +

Hence 1

1 11

m m

r r

+ = ⇒ = +

for ,m r Z∈

Theorem 11: If 1m

r

=

, then

1

( ) ( 1) ... ( )

( )

r r r

r

p m p m r p m tr t

p m tr t+

+ + + + + + +

= + +

Proof: Let 1t =

Since 1m

r

=

There fore

[ ]1 1 1 1

1 1 21 1 1

m r m r

r r r

+ + + + + = + = + = + + +

Since

[ ]

1( 1 1) ( 1)

( ) ( 1)

r r

r r

p m r p m r

p m r p m r

+ + + + = + +

+ + − + −

( 1) ( )r rp m r p m r r= + + + + −

( 1) ( )r rp m r p m= + + +

We assume that it is true for t s=

There fore

1

( ) ( 1) ... ( )

( 1)

r r r

r

p m p m r p m sr s

p m sr s+

+ + + + + + +

= + + +

Adding both sides by ( 1)rp m sr s r+ + + +

( ) ( 1) ...

( ) ( 1)

r r

r r

p m p m r

p m sr s p m sr s r

+ + + +

+ + + + + + + +

1( 1) ( 1)r rp m sr s p m sr s r+= + + + + + + + +

( ) ( )( )1 1 1 1 1r rp m r s p m r s+ = + + + + + + +

( ) ( )1 1 1 1 1r rp m r s p m r s r+ = + + + + + + + +

( )( )1 1 1 1rp m r s+ = + + + +

It is true for 1t s= +

There fore our statement is true for all t N∈

Hence

1

( ) ( 1) ... ( )

( ) 1

r r r

r

p m p m r p m tr t

mp m tr t when

r+

+ + + + + + +

= + + =


( )

[ ]1

( ) (2 1) ... 1

( 1)

r r r

r

p r p r p nr n

p n r+

+ + + + + −

= +

Proof: From theorem 11

1

( ) ( 1) ... ( )

( ) 1

r r r

r

p m p m r p m tr t

mp m tr t when

r+

+ + + + + + +

= + + =

Put m r= and

( )

[ ]1

( ) (2 1) ... 1

( 1)

r r r

r

p r p r p nr n

p n r+

+ + + + + −

= +

ACKNOWLEDGEMENT We would like to express our thanks to referees for valuable comments that improved the paper. REFERENCES [1]. George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. [2]. G. E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167–171. [3]. G. E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math., to appear. [4]. G. E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167–171. [5]. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, NewYork, 1976. [6]. A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, Ramanujan J. 7 (2003),343–366. [7]. K. Bringmann, F. G. Garvan and K. Mahlburg Partition statistics and quasiweak Maass forms, Internat. Math.Res. Notices, to appear. [8]. K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, Ann. of Math., to appear. [9]. K. S. Chua, Explicit congruences for the partition function modulo every prime, Arch. Math. (Basel) 81 (2003). [10]. F. J. Dyson, “Selected papers of Freeman Dyson with commentary,” Amer. Math. Soc., Providence, RI, 1996. [11]. F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. [12]. F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), 1–17. [13]. G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its



12

applications v.35, Cambridge, 1990. [14]. K.Hanuma Reddy: Note on convex polygons, proc of Joint Conference on Information Sciences 2007, USA, 1691-1697. [15]. G. H. Hardy and EM. Wright “An Introduction to the Theory of Numbers,” Oxford Univ. Press, London, 1979. [16]. S. Ramanujan, “The lost notebook and other unpublished papers,” Springer-Verlag, Berlin, 1988.


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

13



MIXED CONVECTION IN A POROUS MEDIUM WITH MAGNETIC FIELD,

VARIABLE VISCOSITY AND VARYING WALL TEMPERATURE

C.N.B.Rao$,V. Lakshmi Prasannam# , T.Raja rani* $S.R.K.R. Engineering College, Bhimavaram, Andhra Pradesh, India. #P. B Siddhardha P.G.College of arts and science, Vijayawada, *Sri Vishnu Engineering College for Women, Bhimavaram. E-Mail: [email protected]

ABSTRACT

Effects of magnetic field and variable viscosity on similarity solutions of mixed convection adjacent to a vertical flat plate in a porous medium are studied numerically. Excess of plate temperature over the ambient temperature and the free stream are assumed to vary as power functions of x , where x is the

distance measured along the plate. The flow and heat transfer quantities of the similarity solutions are

found to be functions of C, λ , µγ , RP where C is magnetic interaction parameter, λ is power of

index of the plate temperature, µγ is viscosity variation coefficient and RP, mixed convection

parameter is ratio of the Rayleigh number to the cleteP ′ number. The cases of assisting flow and

opposing flow are discussed. Dual solutions are found for negative values of RP, and ranges of values of RP are found for which either a unique solution, no solution or dual solutions exist. Skin friction and heat transfer coefficients are observed to diminish as the intensity of the magnetic field increases (or C takes diminishing values). The range of negative values of RP over which solutions exist is observed to

increase with decreasing values of C as well as with increasing values of λ and µγ .

Key Words: Mixed Convection; Variable viscosity; Magnetic field; varying wall temperature

Mathematics Subject Classification Codes: 76S05; 76R10; 76R05; 76DXX

NOMENCLATURE

=0B Magnetic flux

C - Magnetic interaction parameter

f - Dimensionless stream function

g - Acceleration due to gravity

K – Permeability

*K - Porous parameter, K

L2

2M – Hartmann number,

f

LB

µ

σ22

0

p – Pressure

xPe – cleteP ′ number,

m

xU

α∞

xRa – Rayleigh number,

mf

o xTTgK

αµ

βρ )( ∞∞ −

RP - Mixed convection parameter,

x

x

Pe

Ra

0T - Temperature of plate

∞T - Ambient temperature

fT - Reference temperature

vu, - Velocity components in x - and y -

directions

∞U - Free stream velocity

x , y – Cartesian coordinates



14

Greek Symbols

mα - Effective thermal diffusivity of the

porous medium

β - Coefficient of thermal expansion

µγ - Viscosity variation coefficient

η - Similarity variable

θ - Dimensionless temperature

µ - Dynamic viscosity

ρ - Fluid density

ψ - Stream function

−λ Power of index of plate temperature

Subscripts

0 – condition at the plate ∞ - condition at infinity f - condition at reference temperature u - upper solution l - lower solution I.INTRODUCTION Heat transfer studies in porous media find applications in several Engineering and technological systems (ref. [2]). In mixed convection flows, when the temperature of the plate varies as a power function of distance, similarity exists only if the free stream velocity also varies according to the same power function of distance as that of the plate temperature. In mixed convection flows, there arise four cases, of which two correspond to assisting flow and two to opposing flow (refer [10]), depending on the ambient and plate temperatures and the direction of the free stream. They are (i) hot plate assisting flow (ii) hot plate opposing flow (iii) cold plate assisting flow (iv) cold plate opposing flow. Of these four cases only two (i), (iv) are taken into consideration in this study. Reference [6] discussed the effect of variable viscosity on convective heat transfer in three different cases of natural convection, mixed convection and forced convection, taking fluid viscosity to vary inversely with temperature. However, the authors have confined their attention to the assisting flow case only. Reference [2] studied mixed convection boundary layer flow on a vertical surface in a porous medium, when both the temperature of the plate and the free stream velocity vary as the same power function of distance along the plate. Similarity solutions were found to be functions of two parameters

λ and ε where λ is the power of index of

the plate temperature and ε , the mixed

convection parameter is the ratio of the Rayleigh number to the cleteP ′ number. Both

assisting flow and opposing flow were discussed. Ranges of the values of ε for

different values of λ were presented for which

either a unique solution, dual solutions or no solution exist. The effects of λ and ε on the

flow and heat transfer characteristics were discussed. Reference [4] discussed mixed convection boundary layer flow over a vertical surface for the Darcy model when viscosity varies inversely as a linear function of temperature. Results of both assisting flow and opposing flow were presented which were discussed as functions of the mixed convection parameter ε and variable viscosity

parametercθ . In the opposing flow case, the

existence of dual solutions and boundary layer separation were noticed. There has been increasing attention to the study of magnetic field on convection flows in porous media as pointed out in [8]. Reference [7] studied free convection at a vertical plate in a porous medium in the presence of magnetic field, variable physical properties and varying plate temperature. Magnetic field effects on the free convection and mass transfer flow through a porous medium with constant suction and constant heat flux has been discussed in [1]. The effect of magnetic field and varying plate temperature on convective heat transfer past a vertical plate in porous medium has been discussed in [9]. Magneto hydrodynamic mixed convection flow has been analyzed in an annular region filled with a fluid saturated porous medium in [3]. A transverse magnetic field which acts radially is created by a stationary electric current that flows through a cylindrical shaped electrical cable present in the annular region. The effect of non uniform magnetic field on the flow and heat transfer of the Darcy model is discussed. Magneto hydrodynamic free convection in a horizontal cavity filled with a fluid saturated porous medium with internal heat generation has been studied in [5]. Assuming that the magnetic field is inclined at angle γ with the horizontal plane,

the flow and heat transfer are discussed as functions of inclination angle γ , Hartmann

number Ha, Rayleigh number Ra and aspect ratio a.



15

In the present paper the effects of magnetic field, variable viscosity and varying plate temperature on mixed convection at a vertical plate in a porous medium are studied. The plate temperature and the free stream velocity are assumed to vary as power functions of distance ( x ) along the plate, viscosity is assumed to

vary as a linear function of temperature and a magnetic field is assumed to act normal to the plate. Similarity solutions are obtained for the problem and both assisting flow and opposing flow are discussed. In the opposing flow case, dual solutions (referred to as upper and lower solutions) are obtained for certain values of the mixed convection parameter RP and ranges of values of RP are also obtained for which either a unique solution, dual solutions or no solution exist. Significant differences are noticed between the flow and heat transfer quantities related to the upper and lower solutions. II. FORMULATION AND SOLUTION Let a flat plate be embedded vertically in a porous medium saturated with a viscous incompressible homogeneous fluid. The porous medium is assumed to be homogeneous and is in thermal equilibrium with the surrounding fluid. Let a magnetic field of uniform strength be applied in a direction normal to the plate. Let X-axis be taken along the plate and Y-axis perpendicular to it. The temperature of the plate

( 0T ) is assumed to vary as a power function of

distance along the plate, as λxATT += ∞0

where ∞T is temperature of the ambient fluid,

A is a constant and λ is a real number. Fluid

viscosity is assumed to be a function of

temperature as )(tsf µµµ = , where fµ is

viscosity evaluated at the film temperature,

( ).1)( f

f

TTdt

dts −

+=

µµ

and

+= ∞

2

0 TTT f

is the film temperature.

A viscosity variation coefficient µγ is

introduced as

( ).10 ∞−

= TT

dT

d

ff

µ

µγ µ

Density of the fluid is assumed to be a function of temperature only in the body force term. The ambient fluid flows with a velocity

∞U parallel to the vertical plate, the flow

being vertically upwards. The physical model and coordinate system are presented in figure1.The governing equations of the present analysis and the boundary conditions are well known and are not presented here.

Taking the free stream velocity as λxbU =∞

where b is a constant, introducing

cleteP ′ number (xPe ) and nondimensional

functions ,f θ together with a similarity

variable η through the relations

=

−

−=

=

=

+

∞

∞

+

∞

2

11

0

2

1

1

2

)(

)2(

)(

m

m

m

x

x

x

y

TT

TT

x

f

xUPe

αη

ηθ

α

ψη

α

λ

λ

(1) the governing equations in the mixed convection case are obtained as

θθγθγ µµ ′=′′+′′

−+ )(

2

11 RPCfCfC

(2)

0)1(2 =′++′−′′ θλθλθ ff

(3) where

22

2

MK

KC

+= ,

,22

02

f

LBM

µ

σ=

( ),0

αµ

βρ

f

x

xTTgKRa ∞∞ −

= and



16

x

x

Pe

RaRP = .

The boundary conditions in terms of f and θ

are

→′→∞→

===

1,0,

,0,1,0

fas

fat

θη

θη

(4) Equation (2) can be integrated once using the

condition on f ′ at infinity to get

−+

+

−

=′

2

11

)(2

1

θγ

θγ

µ

µ

C

RPCC

f

(5)

Evaluating at 0=η , we get the slip velocity

)0(f ′ as

( )

µ

µ

γ

γ

C

CRPCf

+

−+=′

2

)(12)0( .

If 0=µγ then )(1)0( RPCf +=′

III.a PARAMETERS OF THE PROBLEM

AND THEIR EFFECT ON THE FLOW

AND HEAT TRANSFER: The flow and heat transfer depend on

the parameters µγ , λ , C and RP where RP is

the ratio of the Rayleigh number to the

cleteP ′ number.

The constant A appearing in the expression for the temperature of the plate can take positive as well as negative values and, as a result, the temperature of the plate can be higher or lower than the ambient temperature. In the present work, these correspond to assisting flow and opposing flow respectively. For liquids (except

for water near C04 ) the parameter µγ takes

negative values when ∞> TT0 and takes

positive values when ∞< TT0 , while for gases

it is vice versa. Irrespective of the values of 0T

and ∞T , zero value of µγ corresponds to

constant viscosity case. In this paper, solutions

are found for the values -1, 0, and 1 of µγ .

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

values for opposing flow. When RP is zero, the results correspond to the forced convection case. Calculations are done for a wide range of positive and negative values of RP. Enhanced flow can correspond to an increase in the value of RP, as an increase in the value of the parameter can be due to an increase in the

temperature difference ( ∞−TT0 ).

To determine certain important values for λ ,

the total heat convected in the flow, )(xQ at

any down stream location x is considered.

( ) dyuTTCxQ p∫∞

∞−=0

)( βρ .

This can be seen to be proportional to 2

13 +λ

x ,

like in the free convection case(ref. [7]). For uniform heat flux surface, )( xQ should vary

linearly with x and so3

1=λ . For an adiabatic

surface, )( xQ should be independent of x and

so3

1−=λ . Zero value of λ corresponds to

the isothermal case. In this study solutions are found for the values

15.0,3.0,0,2.0,3.0 and−− of λ .

When A is positive, an increase in the value of

λ can correspond to an increase in the

temperature of the plate, and, in a broader sense, it can result in enhanced flow. When there is no magnetic field, the parameter C takes the value unity and for increasing intensity of the magnetic field, the parameter takes values smaller than unity. In the present study, solutions are found for the values 0.1, 0.5 and 1 of C. Reduced flow can be expected for smaller values of C or for increased intensity of the magnetic field as the magnetic field lines obstruct the flow. The effect of simultaneous variation of the values of the parameters on the flow and heat transfer are presented in the discussion. III.b. NUMERICAL SOLUTION: The equations for f and θ , i.e., equations 3,5 are

integrated numerically subject to appropriate boundary conditions by Runge-Kutta-Gill method (Ref. [10]), together with a shooting technique. The accuracy of the method is tested by comparing appropriate results of the present analysis with available results. Our results for C

=1 and µγ =0 (i.e., no magnetic field, constant

viscosity) are in very good agreement with those of in ref. [2]. Also our results for C =1,

µγ =0 and λ =0 (i.e., no magnetic field,



17

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

)0(')0(),0( θ ′−′′′ andff ’ for positive

values of RP are presented in table I. Skin friction )0(f ′′ can be observed to be negative

for positive values of RP for all values of the other parameters under consideration. Absolute

value of )0(f ′′ decreases with increasing

values of RP and µγ while it increases with

increasing values of C..

Heat transfer coefficient ‘ )0(θ ′− ’ takes

increasing values with increasing values of RP and C while it takes decreasing values with

increasing values of µγ . 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 µγ in the isothermal

case ( λ = 0 ) while it increases with µγ 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

)0(f ′′ is referred to as the upper solution

and the one corresponding to a smaller value of

)0(f ′′ 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 µγ . 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, )0(f ′′ is observed to be

positive. For 0,0,5.0 === µγλC single

solution exists for 01.2 ≤≤− RP , dual

solutions exist for 0.27.2 −≤≤− RP and

no solution for 7.2−≤RP . Like the skin

friction )0(f ′′ , the heat transfer coefficient also

takes positive values when 0=λ (see figures

2(a) and 3(a)). Except for magnitude, behaviour of skin friction and heat transfer coefficient

when 1=µγ are similar to the corresponding

ones when µγ = 0.



18



19

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

example )05.0=λ , )0(f ′′ 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 µγ =0 the range over

which solutions exist is 1.07.2 −≤≤− RP . Like

the skin friction, heat transfer coefficient also

takes both positive and negative values with

changing values of RP, when 05.0=λ .

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,

)0(f ′′ can be seen to



20

diminish as µγ changes from 0 to -1, and

takes smaller values in the absence of the

magnetic field. From figures 10,11 heat transfer

coefficient ( )0(θ ′− ) and slip velocity

( )0(f ′ ) can be seen to increase as µγ

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 µγ =0 which coincide with

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

results of the two works for C = 1, λ =0 and

µγ =0.

VI.CONCLUSIONS:

Assisting flow (RP Positive)

1. For fixed values of µγλ ,,C , as RP increases

there is an increase in the magnitudes of

)0(f ′′ and ‘ )0(θ ′− ’. 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 λγ µ &, RP ,

)0(f ′′ as well as ‘ )0(θ ′− ’ decrease as ‘C’

decreases ( i.e., as the intensity of the magnetic

field increases).

Opposing flow (RP Negative) 1.

‘ )0(θ ′− ’ 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. )0(f ′′ takes positive as well as negative

values for certain values of the parameters.

Positive values of )0(f ′′ imply that the fluid

exerts a dragging force on the surface and

negative values imply the opposite.

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.

ACKNOWLEDGEMENTS

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.



21

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.1-

18,2000.

[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

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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

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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,

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[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.

[10] White,F.M, Viscous Fluid Flow, McGraw-

Hill Inc., New York, 1974.

[11] Vafai,K., Hand Book of Porous Media,2nd Ed, Taylor & Francis, New York.pp.379-381,2005.


On Completely Prime and Completely Semi-prime Ideals in ΓΓΓΓ-near-rings 22


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 ΓΓΓΓ-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. 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 Γ-near-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]. Now, we collect some existing fundamental definitions and results which are to be used in later sections.

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


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

23

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. (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. 1.4. 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). (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]. 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.

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.

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. 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

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.

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.

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.

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

(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-near-ring.

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.

(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*.

1.9. Definition: Let (M, +) be a group. A Γ-

near-ring M is said to be a maximal Γ-

near-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).



24

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,

then

j

j J

Γ∈

U ⊆ 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).

If property B holds, then

j

j J

Γ∈

U = 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 near-

ring (M, +, *γ). 1. 12. Observations: (i) Let (N, +, *) be a near-ring 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.

(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*-near-ring M, but the converse need not be true. 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).

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.

1.15. 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.

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.

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. 1.18. 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. 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.

2. γγγγ-Completely Prime and γγγγ-Completely

Semi-prime γγγγ-Ideals. Throughout this section we consider only zero-

symmetric right near-rings, and M denotes a Γ-near-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.



25

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

⇒ a ∈ I.

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.

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

(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).

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 γ-

ideal 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.]

2.4 Corollary: Let M be a Γ-near-ring, γ ∈ Γ

and A be a γ-ideal of M. Then A is γ-

completely semi-prime γ-ideal if and only if A

is the intersection of γ-completely prime γ-ideals of M containing A.

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 γ-

completely prime γ-ideals of M containing A. The proof is complete.

2.5 Definition: Let A be a proper ideal of M.

The intersection of all γ-completely prime γ-ideals 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). 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 semi-

prime ⇔ A = C-rad(A).

(ii) A γ-ideal A of a Γ-near-ring M is γ-

completely semi-prime ⇔ A = C-γ-rad(A).

2.7 Definitions: (i). A γ-ideal P of a Γ-near-

ring M is said to be a γ-prime γ-ideal of M (with

respect to γ ∈ Γ) if AγB ⊆ P for any two γ-

ideals A, B of M implies A ⊆ P or B ⊆ P.

(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.

2.8 Note: Let P be an γ-ideal of a Γ-near-ring M

and γ ∈ Γ. Then we have the following:

(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).

(ii). P is a γ-semi-prime γ-ideal of the Γ-near-

ring 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 γ-

ideal of M ⇔ S is the intersection of all γ - prime ideals P of M containing S. 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.

2.10 Definitions: Let I be a γ-ideal of a Γ-near-

ring 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. The following corollary follows from Theorem 1.18.

2.11 Corollary: Let P be a γ-ideal of a Γ-near-

ring M for γ ∈ Γ. Every minimal γ-

prime γ-ideal P of a γ-completely semi-prime γ-

ideal 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.

2.12 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.

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 γ-

completely semi-prime γ-ideal, then P is a



26

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.

2. 14 Corollary: Let γ ∈ Γ. If I is a γ-

completely semi-prime γ-ideal of M, then I is

the intersection of all γ-completely prime γ-

ideals 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)).

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 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. (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-near-ring, 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 γ-

ideal (0) of M is γ-completely semi-prime γ-

ideal, 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 γ-

completely 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.

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). 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.

Proof: Write S = P-γ-rad(A).

(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, P-

γ-rad(A)) is a γ-semi-prime γ-ideal. (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.

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) 471-475. [3]Booth G. L. & Groenewald N. J. “On Radicals of �-near-rings”, Math. Japan. 35 (1990) 417-425. [4] Nobusawa “On a Generalization of the Ring theory”, Osaka J. Math. 1 (1964) 81-89. [5] Pilz .G “Near-rings”, North Holland, 1983. [6] Ramakotaiah Davuluri “Theory of Near-rings”, 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. [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 g- prime Radical in Gamma rings”, Quaestiones Mathematicae, 12 (4) (1989) 415-423.

[11] Satyanarayana Bhavanari. “A Note on Γ-near-rings”, Indian J. Mathematics (B.N. Prasad Birth Centenary commemoration volume) 41(1999) 427-433.



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[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 Near- rings”, International Journal of Computational Mathematical Ideas, Vol.1, No.3 (2009) 107-112. [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 Near-ring 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


A Unified frame work for searching Digital libraries Using Document Clustering

28



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.

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

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.

Figure 1. Digital library system



29

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. 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.

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). Query-biased 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). 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

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 .

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: 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.

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).



30

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.

Generating TRD:

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

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.

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

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. 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.

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. 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

Data Source Pd Pg P Pc Balance

NG17-19 Overlapping groups

2998 15810 3 999 0.998

Classic CACM/CISI 7094 41681 4 1774 0.323

Obacal OHSUM 11162 11465 10 1116 0.437

Klb Wabace 2340 21839 6 390 0.013

La1 LA times 3204 31472 6 534 0.290

La2 Latimes 3075 31472 6 1047 0.282



31

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

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.

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 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.

ACKNOWLEDGEMENT

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 , Jan-2003.

[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 Conference, Melbourne, Australia (1998) 46–54.



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[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).


Reducibility For The Fiorini-Wilson-Fisk Conjecture

33



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

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.

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. 1.1.1.1 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



34

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 Fiorini-Wilson-Fisk graph and has at most one vertex-4-coloring. 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 PU {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-4-coloring 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}.

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-4-coloring 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. 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-4-colorable, 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.



35

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

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

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 ’.

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.

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 outer-facial 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)

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

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

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 kθ -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,

. . . , 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 non-

infinite face r, then ρ and ρ ’ must also share

both of the unique (because each of the finite



36

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 ej ≠ e0. 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,

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 }, pµ ) where

pµ = -1 if k(e ρ ) ≠ k (f ρ ) and pµ = 1

otherwise. Now we will show the set of ribs { 1ρ ,… pρ } defines a signed matching. First of

all, the set { 1ρ ,… pρ } defines a set of signed

matches M = M(k, θ ) = {({e 1p , f 1p }, 1pµ ),…,

({e pρ , f pρ }, pρµ )} 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 iρ ,f iρ } I {e jρ , f jρ } =θ , and 2) the

removal of {e iρ ,f iρ } from the ring could not

separate e jρ from f jρ . 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 1ρ ,…, e pρ , f 1ρ ,…,

f pρ } equals {f ∈ E(R) .k(f) ∈ {-1, 0, 1} –

{θ }}. The definition of iρµ also shows that for

every integer i (1 ≤ i ≤ p), k(e iρ ) = k (f iρ ) if

and only if iρµ = 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



37

is k and let { 1ρ ,… pρ } be the set of θ - ribs

induced by c which define M. If ({ei, fi}, µ i) is

the signed match associated with the θ - rib iρ ,

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 iρ =

g0,r1,g1,r2,…,rt,gt, we have k (g0) = k (g2) = k (g4)

= … = k (g 22 t ) = α and k (g1) = k (g3) = k

(g1) = … = k ( 1212 +−tg ) = 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 γ (e) = k (e)

if e is not in iρ , γ (e0) = γ (e2) = γ (e4) =… =

γ (

22 te ) = β and γ (g1) = γ (g3) = γ (g1) =

… = γ ( 1212 +−tg ) = 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 iρ 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 anyθ , 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.

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 ⊂ C1U 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 C1U 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 |AI 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 vertex-

set φ (V (G(K))). Then

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



38

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),

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

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,…gp-

1 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),

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

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.

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

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

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

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

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,



39

since G is an induced sub drawing of S, g ∈

E(G) and so φ (g) = g. Suppose without loss of

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

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)

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

MCS(u)I CS(X) = φ , then we say that u is C-

removable. 1. A configuration K is U-reducible if 1) every u ∈ U, is either D-removable or C-removable. 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 CHI CS ⊄ η ({u})

for some u ∈ U, then it can be shown that T has at least two vertex-4-colorings.

2) If CHI 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.



40

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 non-equivalent 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

CHI (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

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 near-

triangulation, 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

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),

cX(g)}. If X I {e, f, g} ≠ φ , say e ∈ X, then

φ (e) ∈ φ (X), so cX(f) = c(φ (f)) = c(φ (g)) =

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.

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,



41

x4} then either C is a short circuit or some pair of diagonally opposite vertices of C, say x1 and x3 are adjacent to each other and {x1, x2, x3} and {x3, x4, x1} form triangular faces in T. Since Y is sparse, Y has at most one edge in common which

each of these two faces and so |E(C)I Y | ≥ 2

and the inequality follows. If |E(C)| ≥ 6, then |X|

≥ 4 implies |E(C) - φ (X)| ≤ 2. So we may

assume |E(C)| = 1 and thus that |X| = 4. Let C = x1, x2, x3, x4, x1. We may assume |E(C)-Y | = 1 and so all 4 edges of Y are in E(C). Let int(C) denote the sub drawing of T induced by the vertices in one of the arc-wise connected

components of Σ - C and let ext(C) denote the the sub drawing of T induced by the vertices in

the other arc-wise connected component of Σ -

C.We may assume that either |V (int(C))| ≤ 1 or

|V (ext(C))| ≤ 1, or else C is a short circuit. If |V (int(C))| = 0 or |V (ext(C))| = 0, then there are only edges in one of the two disjoint regions of the sphere defined by C, but this will create triangular faces containing two edges of Y , a violation of the sparseness of Y . Thus we must have |V (int(C))| = 1 or |V (ext(C))| = 1. By symmetry, we may assume the former and we will let y denote the vertex for which V (int(C)) = {y}. Note that y has degree 1 and faces all the edges of Y and so cannot be a triad for Y . Since there is a triad v for Y , v ∈ V (ext(C)), v is incident to at least three vertices xi1 , xi2 , xi3 ⊂ {x1, x2, x3, x4, x1} which are, in turn endpoints of edges in Y . By relabeling, we may assume xi1 = x1 and xi2 = x2 and that i3 ∈ {3, 4}. If i3 = 4 then {v, x4, y, x1} form a short circuit. So assume that i3 = 3, and deduce that {v, x3, x4, x1, x1} is either a short circuit, or there is a degree 1 vertex w that 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 later holds and v has neighbors {x1, x2, x3,w} which form a short circuit. This completes the proof of the theorem. We will call any A - contract X with |X| = 4 and for which X has a triad a safe contract. Theorem 1.2.2 Every configuration in Appendix 1 is U-reducible. Moreover, for every u ∈ U that is C-removable, there is a safe u-contract X. Proof: Let K be a configuration in Appendix 1. The computer verifies that for every u 2 U, u is either D-removable or C-removable. When the color is C-removable, the computer finds a u-contract X and verifies that X is safe. After showing that every u ∈ U is either D-removable or C-removable, the computer verifies that the configuration K is U-reducible. If at least one of the u 2 U was C-removable, then U-reducibility for K is immediately established. Otherwise, the computer verifies that K is either D-reducible or C(4)-reducible

Theorem 1.2.3 Let T be a minimum counterexample. Then no configuration isomorphic to one in Appendix 1appears in T. Proof: Let T be a minimum counterexample, and suppose that K is a configuration in Appendix 1 which appears in T. By Theorem 3.3.1, we know that T is internally 6 - connected. Let H and S be as at the beginning of Section

1.2.2. We first notice that if CH I CS includes

two non-equivalent colorings, or if k ∈ CH I

CS for some k ∈ CS -U, then Lemma 1.2.2 implies that T would have at least two non-equivalent vertex-4- colorings. Therefore, we

may assume that CH I CS ⊂ (C* - CS) U

η ({u}), for some u ∈ U.

Assume first that no u ∈ U is C-removable. Therefore, every u ∈ U is D-removable, which

implies that u ∉ MCS(u) for every u ∈ U. Since

K appears in Appendix 1, Theorem 1.2.2 implies that K is either D-reducible or C-reducible. We first consider the case that K is D-reducible, and

therefore that MCS (C* - C) = φ . Since T is a

minimum counterexample, H has a vertex-4-

coloring and thus CH ≠ φ ,. If CH I CS =φ ,

then CH ⊂ (C* - CS) which implies that CH ⊂ MCS(C* - CS) since Lemma 1.1.3 implies the latter is consistent. This however, is a

contradiction. Assume then that CH I CS = η

({u}) for some u ∈ U. This also gives rise to a

contradiction, because then CH ⊂ (C* - CS) U

η ({u}) which implies CH ⊂ MCS(u) since by

Lemma 1.1.3, MCS(u) equals the maximal

critical subset of (C* - CS) U η ({u}) and CH is

critical by Theorem 1.1.3. Thus u ∈CH ⊂ MCS(u) which contradicts that u is D-removable. Now consider the case that K is C(4)-reducible,

and let X be a sparse subset of S such that φ (X)

is contractible in T and that CS(X) I MCS(C* -

CS) =φ . By Theorem 1.1.2, there is a tricoloring

of T modulo X which we denote by c. Let cH and cX be the colorings that are guaranteed to exist by

Lemma 1.2.4, let cH(R) be the lift of cH by φ

and let cX(R) be the restriction to R of the coloring cX. Lemma 1.2.4 says that cH(R) = cX(R)

and that cX(R) ∈ CH I CS. Also, we know that

cX(R) ∈ CS(X). Therefore cX(R) ∈ CH I CS I

CS(X). From this and our assumption that CH ⊂

(C* - CS) U η ({u}) for some u 2 ∈ , it follows

that cX(R) ∈ U and that CH ⊂ MCS(cX(R)) since by Theorem 1.1.3, CH is consistent and by Lemma 1.1.3, MCS(cX(R)) equals the maximal

consistent subset of (C* -CS) U {cX(R)}. Since

we are assuming that every u ∈ U is D-



42

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

⊂ C* - CS ⊂ (C* - CS) U η ({u}). It follows

then that u’ ∈ CH ⊂ MCS(u’), so u’ is not D-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.

ACKNOWLEDGEMENTS:

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.

[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 5-212. [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. [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 Fkorini-Wilson-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.

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


Perceiving Plagiarism Using Weighted Window Approach- Performance Analysis

43



PERCEIVING PLAGIARISM USING WEIGHTED WINDOW APPROACH-

PERFORMANCE 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.



44

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. 3.1 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. The problem with this manual approach is that it is labour intensive, requiring detailed, on-screen 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 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. 3.2.1 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. 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 his work without understanding the original content.



45

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

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

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.

5.3.1 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

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

PET INSTANCES PET INSTANCES

0 0 0 0

15 5 40 2

40 10 50 5

75 19 63 12



46

FIG 2 : MANUAL STATS

FIG 3: SNITCH STATS

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

TABLE 2: ANALYSIS TIME (RESULTS) The graphical representation is as follows

average analysis time

0

1

2

3

4

5

6

7

8

1 2 3 4 5

Series1

Series2

FIG 4 : ANALYSIS TIME COMPARISION RESULTS

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. 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: [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. [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 [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. [6]. Carroll, J. A Handbook for deterring Plagiarism in Higher Education. Oxford, Oxford Centre for Staff and Learning Development, 2002. [7]. Whitley, B. Academic Dishonesty: an Educator’s Guide. Mahwah, N.J., Erlbaum, 2002. (Hesburgh LB 3609.W45 2002)



47

[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

Plagiarism. Proceedings of the 37th

Special Interest Group on Computer Science Education. Pp. 51 – 55. New York: Association for Computing Machinery.


System Representation For Software Architecture Recovery

48



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

includes file, function, statement, expression, variable

2. 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.



49

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 (simpEnt-abs).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



50

iv)contains-resource

Entity-abs

attribute Example Description

Name “des” Name of the entity in the source code

File# 4 File number of the source code file where the entity is defined

line# 45 Line number of the entity in the source code file

Implement-id

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 type-abstraction, 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 file-abstraction L and either a function- abstraction 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 file-abstraction L and either a function- abstraction 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: N

s: {n1,n2,……..,nn} is the set of attributed nodes, obtained from the abstract domain model. R

s: {r1,r2,…..rm} is the set of attributed edges, obtained from the abstract domain model. A

s: alphabet for node attribute values such as node labels, node types, and their values. E

s: alphabet for edge attribute values

such as edge labels, edge types, and their values. µ

s: 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).



51

Ni(Gs) = ((type, function

abs),(des,”/a/..mgr”),(id,13),(line45),(file4))indicating that node I of the source graph G is an entity of type Function-abs 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, use-F),(#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:

sh-ents(F)={e | ∀ f ∈ F; ∃ rel :X • 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 }

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

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 source-graph 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}=sh- funcs({items}) {items}=sh-ents({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.



52

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]

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” “

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.

N jsr = { ni | ni ∈Ns ∧ ∃ nj ∈ Ns

• entAssoc (nj , ni) > 0 } ∪ { nj }

R jsr = { ns ; nt | ns , nt ∈ N j

sr ∧ ∃ rk

• rk = (ns , nt) ∧ rk ∈ 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



53

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

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

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

3.3.2(c) Source region Gsreg(1) after applying

Apriori

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



54

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

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

k that exist in the domain D n 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))

Where Gs = ( 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


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 (VLDB-94), pp. 487-499 santiago, Chile.

[6] Ahonen-Myka, H., Heinonen, O., Klemettinen, M., & verkamo, A.I. (1999). Finding co-occurring 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.


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


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. U2

exp/U2

imx 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

2. I. INTRODUCTION

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 physico-chemical 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 U2

exp/ U2 imx from unity based on the earlier studies16,17.

3. 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.



and 318.15k.

56

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

III. THEORY

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

density (ρ) by the following formula

βad = 2

1

Uρ --- (1)

The molar volumes of the binary mixtures were calculated using the equation

V = (X1M1+X2M2)/ρ --- (2) Nomoto Theory:

UNomoto =

3

1 1 2 2

1 1 2 2

x R x R

x V x V

+

+ --- (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 Free Length Theory (FLT):

UFLT = 1/ 2( )f

k

L x density --- (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 Collision Factor Theory (CFT):

γγγγGopal rao =

2

32

31 1 1

16mV RT MU

N MU RT

γ

γ

− + − Π

--- (5)

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

γγγγShaff’s =

2

32

31 1 1

16 3mV RT MU

N MU RT

γ

γ

− + − Π

--- (6)

γavg = (γGopal rao + γShaff’s) / 2

Bi = 4

3x Π x (γavg i)

3 x N

Space filling factor, Rf i = Bi / Vmi

Collision Factor, Si = Uexp / (1600 x Rf i)

Collision Factor Theory, UCFT = Uα x Smix x Rf mix --- (7) Vandael Theory:

U Vandael =

( )1/2

1 21 1 2 2 2 2

1 1 2 2

1

x xxM xM

MU MU

+ +

-- (8)

Where, x1, x2 – Mole fractions M1, M2 – Molecular weights U1, U2 – Ultrasonic velocities Junjie Theory:

UJunjie =

( )

1 1 2 2

1/2

2 21 11 1 2 2 2 2

1 2

m m

mm

xV x V

xVxVx M x M

U U

+

+ +

-- (9)

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

TABLE I - VALUES OF DENSITY (ρ), ADIABATIC COMPRESSIBILITY (β) AND

MOLAR VOLUME (Vm) FOR DIFFERENT MOLEFRACTIONS OF O-XYLENE WITH

CYCLOHEXANE AT DIFFERENT TEMPERATURES



and 318.15k.

57

X1 ρ x 103 (kg/m3) β x 1012 (m2 N-1) Vm (cm3 mol-1)

at 303.15 K

0.0000 0.7678 85.3058 109.6119

0.0908 0.7768 84.0621 110.9140

0.1835 0.7865 82.2063 112.1388

0.2781 0.7975 80.1706 113.2023

0.3747 0.8096 77.9252 114.1358

0.4734 0.8222 75.7054 115.0268

0.5742 0.8304 74.1076 116.5612

0.6772 0.8368 72.3779 118.3771

0.7824 0.8486 69.6749 119.4594

0.8900 0.8614 67.0067 120.4319

1.0000 0.8707 64.0815 121.9249

at 308.15 K

0.0000 0.7625 90.1520 110.3738

0.0908 0.7731 88.3109 111.4448

0.1835 0.7831 85.9713 112.6257

0.2781 0.7946 83.5235 113.6155

0.3747 0.8070 80.9327 114.5035

0.4734 0.8194 78.7345 115.4199

0.5742 0.8280 76.7718 116.8990

0.6772 0.8349 75.1008 118.6465

0.7824 0.8466 72.2496 119.7416

0.8900 0.8600 69.3633 120.6280

1.0000 0.8694 66.5164 122.1072

at 313.15 K

0.0000 0.7587 93.2715 110.9266

0.0908 0.7711 91.1762 111.7339

0.1835 0.7811 88.6288 112.9141

0.2781 0.7930 86.2194 113.8447

0.3747 0.8054 83.4881 114.7310

0.4734 0.8181 80.9494 115.5891

0.5742 0.8270 78.9435 117.0404

0.6772 0.8337 77.3219 118.8173

0.7824 0.8453 74.2945 119.8258

0.8900 0.8585 71.2342 120.8387

1.0000 0.8677 68.4567 122.3464

at 318.15 K

0.0000 0.7531 97.3125 111.7514

0.0908 0.7668 94.8148 112.3605

0.1835 0.7787 92.0530 113.2621

0.2781 0.7907 89.4967 114.1759

0.3747 0.8034 86.7270 115.0166

0.4734 0.8158 84.0702 115.9292

0.5742 0.8254 81.9252 117.2672

0.6772 0.8329 80.0900 118.9314

0.7824 0.8440 77.0418 120.1105

0.8900 0.8567 73.5534 121.0926

1.0000 0.8659 70.6254 122.6008



and 318.15k.

58

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 UFLT UCFT Uvandael UJunjie

303.15K

0.0000 1235.63 1235.62 948.71 1235.63 1235.63 1410.15

0.0908 1237.50 1245.69 961.29 1242.93 1239.63 1411.28

0.1835 1243.65 1255.81 978.13 1251.56 1244.67 1412.70

0.2781 1250.63 1265.98 997.37 1262.46 1250.83 1414.41

0.3747 1259.00 1276.21 1019.29 1275.29 1258.24 1416.41

0.4734 1267.50 1286.50 1042.14 1289.08 1267.04 1418.69

0.5742 1274.75 1296.84 1058.55 1296.19 1277.38 1421.28

0.6772 1284.95 1307.23 1075.25 1300.76 1289.46 1424.17

0.7824 1300.50 1317.68 1103.60 1313.96 1303.52 1427.37

0.8900 1316.25 1328.19 1133.82 1328.94 1319.83 1430.88

1.0000 1338.75 1338.75 1165.65 1338.75 1338.75 1434.71

308.15K

0.0000 1206.13 1206.12 919.67 1206.13 1206.13 1381.26

0.0908 1210.25 1216.63 935.64 1215.08 1210.49 1382.68

0.1835 1218.75 1227.22 654.40 1223.26 1215.90 1384.41

0.2781 1227.50 1237.89 975.37 1233.93 1222.46 1386.46

0.3747 1237.38 1248.64 998.56 1246.16 1230.30 1388.83

0.4734 1245.00 1259.48 1020.15 1258.53 1239.57 1391.54

0.5742 1254.25 1270.41 1038.52 1265.26 1250.45 1394.58

0.6772 1261.88 1281.43 1054.37 1269.63 1263.14 1397.97

0.7824 1278.63 1292.53 1082.48 1281.52 1277.91 1401.71

0.8900 1294.75 1303.72 1113.49 1296.17 1295.06 1405.82

1.0000 1315.00 1315.00 1143.26 1305.00 1315.00 1410.31

313.15K

0.0000 1188.75 1188.75 901.90 1188.75 1188.75 1364.76

0.0908 1192.63 1199.21 919.63 1205.33 1193.13 1366.05

0.1835 1201.88 1209.76 938.78 1217.48 1198.55 1367.67

0.2781 1209.38 1220.40 959.03 1231.88 1205.12 1369.61

0.3747 1219.50 1231.13 982.19 1246.33 1212.96 1371.88

0.4734 1228.75 1241.96 1005.36 1260.61 1222.22 1374.48

0.5742 1237.63 1252.88 1023.52 1267.95 1233.08 1377.43

0.6772 1245.50 1263.89 1038.37 1271.31 1245.75 1380.74

0.7824 1261.88 1275.00 1066.66 1281.33 1260.48 1384.42

0.8900 1278.75 1286.20 1097.81 1292.88 1277.60 1388.47

1.0000 1297.50 1297.50 1125.84 1297.50 1297.50 1392.91

318.15K

0.0000 1168.13 1168.13 879.71 1168.13 1168.13 1346.06

0.0908 1171.88 1178.70 900.00 1188.45 1172.65 1347.27

0.1835 1181.13 1189.39 919.74 1201.52 1178.21 1348.80

0.2781 1188.75 1200.18 939.94 1215.84 1184.92 1350.68

0.3747 1198.00 1211.07 962.47 1230.42 1192.91 1352.91

0.4734 1207.50 1222.07 985.08 1243.71 1202.34 1355.50

0.5742 1216.07 1233.19 1003.75 1251.88 1213.37 1358.46

0.6772 1224.38 1244.41 1019.78 1256.02 1226.23 1361.79

0.7824 1240.13 1255.74 1046.67 1264.70 1241.19 1365.52

0.8900 1259.75 1267.19 1079.23 1274.82 1258.56 1369.65

1.0000 1278.15 1278.75 1107.27 1278.75 1278.75 1374.21



and 318.15k.

59

TABLE III - U2exp/ U

2 imx VALUES WITH MOLEFRACTION FOR CYCLOHEXANE

WITH O-XYLENE AT DIFFERENT TEMPERATURES

X1 U2exp/ U

2 imx

303.15 K 308.15 K 313.15 K 318.15 K

0.0000 1.0000 1.0002 1.0000 1.0000

0.0908 0.9966 1.0008 0.9992 0.9987

0.1835 0.9984 1.0069 1.0056 1.0050

0.2781 0.9998 1.0116 1.0071 1.0065

0.3747 1.0014 1.0161 1.0108 1.0085

0.4734 1.0009 1.0147 1.0107 1.0086

0.5742 0.9961 1.0135 1.0074 1.0045

0.6772 0.9932 1.0070 0.9996 0.9970

0.7824 0.9956 1.0120 1.0022 0.9983

0.8900 0.9947 1.0125 1.0018 1.0019

1.0000 1.0000 1.0154 1.0000 1.0000

The ratio U2

exp/ 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 U2

exp/U2

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

[1] Prausnitz J M, Lichenthalar & Azevedo, Molecular Thermodynamics of fluid phase equilibria, second edition, (Prentice-Hall Inc, Englewood Cliffs, New Jersey) (1986)

[2] Acree W E (Jr), thermodynamics properties of non electrolyte solutions (Academic Press, New York)(1984)

[3] Rodriguez S, Lafuente C, Artigas H, Royo F M & Urieta J S, J Chem Thermodynamics, 31 (1999) 139

[4] Naidu P S & Ravindra Prasad K, J Pure Appl Ultrason, 24 (2002) 18. [5] Rao T S, Veeraiah N & Rambabu C, Indian J Pure & Appl Phys,

40(2002) 850

[6] Shipra Baluja & Swathi Oza, J Pure & Appl Ultrason, 24 (2002) 850 [7] Ali A, Yasmin A & Nain A K, Indian J Pure & Appl Phys. 40 (2002)

315 [8] Amalendu Pal, Gurcharan Dass & Harsh Kumar, J Pure & Appl

Ultrason, 23 (2001) 10 [9] Rastogi et al, Indian J Pure & Appl Phys, 40 (2002) 256 [10] Anwar Ali A, Anil Kumar Nain & Soghra Hyder, J Pure & Appl

Ultrason, 23 (2001) 73 [11] Nomoto O, J Phys Soc Japan, 4 (1949) 280 and 13 (1958) 1528 [12] Van Dael W & Vangeel E, Proc int conf on calorimetry and

thermodynamics, Warasa (1955) 555 [13] Shipra Baluja & Parsania P H, Asian J Chem, 7 (1995) 417 [14] Gokhale V D & Bhagavat N N, J Pure & Appl Ultrason, 11 (1989) 21 [15] Junjie Z, J China Univ Sci Techn, 14 (1984) 298 [16] Prakash A, prakash S & Prakash Q, Proc Nat Acd Sci, 55(A) 11 (1985)

114. [17] Sabeson R, Natarajan & Varadha Rajan R, Indian J Pure & Appl Phys,

25 (1987) 489 [18] Vogel A I, A text book of practical organic chemistry, 5th Edn (John

Wiley, New York)1989 [19] Riddick J A, Bunger W B & SokanoT K, techniques in chemistry, Vol

2, organic solvents, 4th Edition (John Wiley, New York) 1986 [20] Satyanarayana N, Satyanarayana B & Savitha jyostna T, J Chem Eng

Data, 52 (2007) 405. [21] Satyanarayana B, Savitha Jyostna T & Satyanarayana N, Indian J Pure

& Appl Phys, 44 (2006) 587. [22] Viswanatha Sarma A & Viswanatha Sastry J, J Acous Soc Ind, Vol

XXVII (1999) 309 [23] 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.


Over View To Implementation Of Robotics With Voice Recognition

60


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.

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.

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.

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.

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.



61

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 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.

Design:

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.

Mathematics for speech analysis: Speech Analysis:

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.

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.

Euclidean distance formula is:

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



62

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.

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.

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.

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.



63

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.

Implementation of Filter:

Fig.3. finger print implementation

Filter Implementation:

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

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 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.

PWM signal to move ROBOT:

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



64

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.

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.

[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.

[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.


Novelty Of Extreme Programming

65



NOVELTY OF EXTREME PROGRAMMING Ch.V.Phani Krishna@, S.Satyanarayana#, K.Rajasekhara Rao$ @ Sana Engg. College, Kodad, [email protected] # Dept of Mathematics, Sri Venkateswara Institute of Science & Information Technology, Tadepalligudem, [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.

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.

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.

Fundamentals of XP:

� Distinguishing between the decisions to be made by business interests and those to be made by project stake holders.

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

� Integrating and testing the whole system several times.

� Producing all software in pairs (pair programming)

� Simple design that constantly evolves to add needed flexibility and remove unneeded complexity.

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

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.

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

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

XP makes two sets of promises:

To Programmers:



66

� 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.

� To Customers and Managers:

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.

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.

XP is distinguished from other methodologies by:

� Its early, concrete, continuing feed back from short cycles

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

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

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

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

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

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

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

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.

XP addressing the risks encountered in software

development:

The basic problem of software development is risk.

(1) Schedule slips: - short releases

Within an iteration, XP plans with 1-3 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

(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.

(3) System goes sour: - Testing.

Repeated testing in XP ensures a quality base line.

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

Customer (program feature – by – Program feature)

(5) Business Misunderstood: - On – site customer

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

(6) Business changes: - Short Releases:

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.

(7) False feature Rich:



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XP insists that only the highest priority tasks are addressed.

(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.

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

XP development cycle:

� Pair of programmers program together.

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

� 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.

Four Variables:

There are four control variables in software development model

� Cost

� Time

� Quality

� 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.

� 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.

� 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.

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

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:

� Communication

� Simplicity

� Feedback

� Courage

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.

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.

Simplicity:

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.

Simplicity and communication have a wonderful mutually supporting relationship.

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),



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

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.

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.

Basic principles:

The fundamental principles are

� Rapid feed back

� Assume simplicity

� Incremental change

� Embracing change

� Quality work

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.

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

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

Incremental change:

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.

Some less central principles:

� Teach learning

� Small initial investment

� Play to win

� Concrete experiments

� Open, honest communication

� Work with people’s Instincts, not against them

� Accepted Responsibility

� Local Adaptation

� Travel light

� Honest Measurement

Teach learning:

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.

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



69

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

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.

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.

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.

Works with people’s Instincts, not against

them:

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.

Accepted Responsibility:

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

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

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.

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

� 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.

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.

Practices of XP:

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.



70

Small releases:

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

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.

Simple Design:

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

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.

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.

Pair Programming:

Writing code with two programmers at one machine .

Collective Code Ownership:

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

Continuous Integration:

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

40-Hour week:

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.

On-site Customer:

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

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.

Jeffries developed them further and has 13 practices.

Whole Team:

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.

� Caves and commons

� Fixed iterations and engineering tasks

� Write it on a card (RDP Technique)

� Spike Solutions

� All tests all the time

� Promiscuous Pairing

� Yesterday’s weather

� Track velocity and track progress

� Regression test

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

� No written design documentation

• Truth: no prescribed standards for how much

or what kinds of docs are needed.

� No design

• Truth: minimal explicit, upfront design:

design is an explicit part of every activity

through every day.

� XP is easy

• Truth: although XP does try to work with the

natural tendencies of developers, it requires

great discipline and consistency.

� XP is just legitimized hacking

• Truth: XP has extremely high quality

standards throughout the process.

� XP is the one, true way to build software



71

• Truth: It seems to be a sweet spot for certain

kinds of projects.

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.

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.

� 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.

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.

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

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

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.

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

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.

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[9]. Glenn Vanderburg., A Simple Model of Agile Software Process –or- Extreme Programming Annealed, ACM, 2005.

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(XP2003), May 25 – 29, 2003, Genova, Italy.


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

73



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

1. INTRODUCTION Combined heat and mass transfer (or double-diffusion) 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

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

light molecular weight ( )2 , eH H and of medium molecular

weight ( )2 ,N 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



74

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.

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

maintained at a constant temperature wT , which is higher than

the constant temperature T∞ of the surrounding fluid and the

concentration wC is greater than the constant

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

induced axial magnetic field with the flow is considered to be

negligible compared to the interaction of the applied field 0B ,

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

0u v

x y

∂ ∂+ =

∂ ∂

(2.1) Momentum equation

( ) ( )22

* 0

2

Bu u uu v g T T g C C u

x y y

σν β β

ρ∞ ∞

∂ ∂ ∂+ = + − + − −

∂ ∂ ∂

(2.2) Energy equation

2 2

2 2

1 m Tr

p s p

D kqT T T Cu v

x y y c y c c yα

ρ

∂∂ ∂ ∂ ∂+ = − +

∂ ∂ ∂ ∂ ∂

(2.3) Species equation

2 2

2 2

m Tm

m

D kC C C Tu v D

x y y T y

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂

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

( )0 0, , , 0w wu U v v x T T C C at y= = = = =

0, 0, ,u v T T C C as y∞ ∞→ → → → → ∞

where 0U is the uniform velocity and ( )0v 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 -

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 - the electrical conductivity of the fluid, 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

far away from the plate, 0B - the magnetic induction, k - the

thermal conductivity, pc - the specific heat at constant

pressure, Tk - the thermal diffusion ratio, sc - the

concentration susceptibility, mT - the mean fluid temperature,

mD - the mass diffusivity.

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

rq (transverse to the vertical surface). By using the Rosseland

approximation (Brewster [29]), the radiative heat flux rq is

given by



75

y

T

kq

e

sr

∂

′∂−=

4

3

4σ

(6)

where sσ is the Stefan-Boltzmann constant and ek - 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

by expanding 4T ′ into the Taylor series about ∞

′T , which

after neglecting higher order terms takes the form

434 34 ∞∞′−′′≅′ TTTT

(7)

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

32 2 2

2 2 2

16

3s m T

e p s p

T D kT T T T Cu v

x y y k c y c c y

σα

ρ∞′′ ′ ′ ′∂ ∂ ∂ ∂ ∂

+ = + +∂ ∂ ∂ ∂ ∂

(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 non- dimensionalization 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 to a system of coupled, non-

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 non-dimensional quantities are introduced.

( ) ( ) ( )00, , , ,

2 w w

U T T C Cy xU f

x T T C Cη ψ ν η θ η φ η

ν∞ ∞

∞ ∞

− −= = = =

− −

( ) ( )*

2 2

0 0

2

0

0

2 2, , , ,

2, , , Pr ,

w w

p

m

g T T x g C C xu v Gr Gm

y x U U

cBxM R Sc

U D k

β βψ ψ

νρσ ν

ρ

∞ ∞− −∂ ∂= =− = =

∂ ∂

= = = =

(2.

6)( )( )

( )( )

,m T w m T w

s p w m w

D k C C D k T TDu Sr

c c T T T C Cν∞ ∞

∞ ∞

− −= =

− −

where ψ is the stream function, θ - the non-dimensional

temperature function, φ - the non-dimensional concentration,

Gr - the thermal Grashof number, Gm - the solutal Grashof

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

uy

ψ∂=

∂ and v

x

ψ∂= −

∂.

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

0f ff Gr Gm Mfθ φ′′′ ′′ ′+ + + − =

(2.7)

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

(2.8)

0Sc f Sc Srφ φ θ′′ ′ ′′+ + =

(2.9) The corresponding boundary conditions are

, 1, 1, 1 0wf f f atθ φ η′= = = = =

0, 0, 0f asθ φ η′ → → → → ∞

where 0

0

2w

xf v

Uν= − is the dimensionless suction velocity

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

( ) ( )1

22 Re 0fC f−

′′=

The Nusselt number in non-dimensional form is

( )1

2(Re) 0Nu θ ′= −

The Sherwood number in non-dimensional form is

( ) ( )1

2Re 0Sh φ ′= −

where 0ReU 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 Runge-Kutta 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.



76

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 wf .

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

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 1Du ≤ , 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 wf on the velocity, temperature and

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 fC , 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



77

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 wf increases, the local skin-friction

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.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Gr = 1.0, 2.0, 3.0, 4.0

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

Fig.1. Velocity profiles for different values of Gr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Gm = 1.0, 2.0, 3.0, 4.0

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

Fig.2. Velocity profiles for different values of Gm

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

M = 0.0, 0.5, 1.0, 2.0

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

Fig.3. Velocity profiles for different values of M

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

K = 0.5, 1.0, 1.5, 2.0

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

Fig.4. Velocity profiles for different values of K

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Pr = 0.71, 1.0, 1.25, 1.5

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

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



78

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

θ

Pr = 0.71, 1.0, 1.25, 1.5

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Ec = 0.0, 0.01, 0.02, 0.03

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

θ Ec = 0.0, 0.01, 0.02, 0.03

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Du = 0.0, 0.2, 0.6, 1.0

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

θ

Du = 0.0, 0.2, 0.6, 1.0

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Sc = 0.3, 0.6, 0.78, 0.94

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

Fig.8(a). Velocity profiles for different values of Sc

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

φ

Sc = 0.3, 0.6, 0.78, 0.94

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

Sr = 1.0, 1.5, 2.0, 2.5

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



79

Fig.9(a). Velocity profiles for different values of Sr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

φ

Sr = 0.0, 1.0, 1.5, 2.0

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f '

f = 0.5, 1.0, 1.5, 2.0w

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

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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

θ

f = 0.5, 1.0, 1.5, 2.0w


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

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

η

φ

f = 0.5, 1.0, 1.5, 2.0w


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

Table 1 Numerical values of the skin-friction coefficient fC ,

Nusselt number Nu and

Sherwood number Sh for

Pr 0.71= , 0.01Ec = , 0.2Du = , 0.6Sc = , 1.0Sr =

Gr

Gm

M

K

wf

fC

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

0.82302

1.68650

1.88533

0.49068

0.48781

0.51154

0.86186

0.90193

0.91883

0.84005

0.83956

1.09368

0.43622

0.46479

0.47943

0.42136

0.41984

0.47301



80

Table 2 Numerical values of the skin-friction coefficient fC ,

Nusselt number Nu and

Sherwood number Sh for

2.0Gr = , 2.0Gm = , 0.5M = , 0.5K = , wf 0.5=

Pr

Ec

Du

Sc

Sr

fC

Nu

Sh

0.71

1.0 0.71

0.71

0.71

0.71

0.01

0.01

0.02

0.01

0.01

0.01

0.2 0.2 0.2 0.4 0.2 0.2

0.6 0.6 0.6 0.6 0.78

0.6

1.0 1.0 1.0 1.0 1.0 2.0

0.82302

0.75371

0.82406

0.84053

0.77315

1.00844

0.86186

1.10872

0.85906

0.82924

0.84694

0.93308

0.43622

0.29084

0.43788

0.45734

0.49949

0.29313

ACKNOWLEDGEMENT


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 Electro-Magneto 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 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.217-230. 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.

[12]. 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.

[13]. 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

[14]. medium, J. heat and Mass Transfer. Vol.43, pp.1265-1274.

[15]. 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.

[16]. 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.

[17]. Alam M.S., Ferdows M. and Maleque M.A. (2006), Dufour and Soret effects on steady free convection and mass transfer flow past a semi-infinite vertical plate in a porous medium, Int. J. of Applied Mechanics& Eng., Vol. 11, No.3, pp.535-545.

[18]. Gebharat B.(1962), Effects of viscous dissipation in natural convection , J. Fluid Mech., Vol. 14, pp.225-232.

[19]. Gebharat B. and Mollendorf J.(1969), Viscous dissipation in external natural

[20]. convection flows, J. Fluid. Mech.,Vol.38, pp.97-107.

[21]. 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.



81

[22]. Israel-Cookey C., Ogulu A. and Omubo-Pepple 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.

[23]. 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.432-448.

[24]. 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.


Pareto Distribution - Some Methods Of Estimation

82



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

0,1,,1),;()1(

fff σαµσ

µ

σ

αασ

α

xx

xf

−+=

+−

(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.

2. Estimation from Complete Sample

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

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 considered for estimation, is

0,1),;()1(

>

+=

+−

Xx

xfα

σσ

αασ

(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

+

+

+−=

∂

∂

σσσααn

xxxnL 1....... 1. 1log

log21

(2.2)

( ) ∑ −+

+=∂

∂

=

n

ii

i nz

zL

1 11

logα

σ

(2.3)



83

where σ

i

i

xz =

For M.L.Es of α and σ ,

0log

,0log

=∂

∂=

∂

∂

σα

LL

After simplification these equations become

∑

+

=

=

∧

n

i

ix

n

1

1logσ

α

(2.4)

( ) ∑ =−+

+=

n

ii

i nz

z

1

01

1 α

(2.5)

where σ

i

i

xz =

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

( )i

i

iz

zzh

+=

1

(2.6) of the log likelihood equation (2.5) for estimating σ by a linear expression say

iiiizzh δγ +≅)(

(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

( )

( ) ∑+−

∑+=

=

=∧

n

ii

n

iii

n

x

1

1

1

1

γα

δασ

(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)

Method I

,,3,2,1,1

nin

i

ipLet −−−−−=

+=

Let Zi, Zi’ be the solutions of the following

equations ''''''

)()(iiii

pzFandpzF ==

where

n

qppp

n

qppp

ii

ii

ii

ii+=−=

''',

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

( )

( ) 1

1

1

1

1

1

''''

''

−−

−=

−−

−=

α

α

ii

ii

pzand

pz

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

( )'''

''')(

ii

ii

i

zz

zhzh

−

−=δ

(2.9)

and ( )iiii

zzh δγ −=

(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)

Method II

Consider the Taylor’s expansion of

( )1

ii

i

zh z

z=

+ in the neighbourhood of ith

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

( )1

1 1i iz p α−

= − − ,

1+=

n

ip

i

( )iiii

zzh δγ −=

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)



84

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). 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

expressions of the log likelihood equation σ∂

∂ Llog

differ by little values. Also exact and approximate

values of 2

2log

σ∂

∂ L would differ little. Hence the

exact MLE, Modified MLEs by the two methods for the parameter σ shall have the same asymptotic bias and asymptotic variance (Bhattacharya, 1985). 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). A comparison of the sampling characteristics namely the bias, variance, and MSE

of ∧∧

21, σσ - 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.

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

distribution with unknown scale parameter σ and a

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

( )( 1)

1

1 1n r r

in r

i

xL x

ααα

σ σ

− +−−

−=

∝ + + ∏

+−

∑

++−

+=

−

−

=

σα

σα

σ

α

rn

rn

i

i

xr

xtConsL

1log.

1log)1(logtanlog1

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

( )0

1

.

1

1

0log

122

=∑

+

−

+

+−

−⇒

=∂

∂

−

=−

−rn

irn

rn

i

i

x

xr

x

xrn

L

σ

σ

α

σ

σ

α

σ

σ

( )0

1.

1

1

1

=∑+

−+

+−

−⇒

−

=−

−rn

irn

rn

i

i

z

zr

z

zrn

σ

α

σ

α

σ

( ) ( ) 01

.1

11

=∑+

−+

+−−⇒−

=−

−rn

irn

rn

i

i

z

zr

z

zrn αα

(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

approximate the expression i

i

z

z

+1 of the



85

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

( )iiii

zzh δγ +≅

(3.2) 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

0loglog

=∂

∂≅

∂

∂

σσ

LL

( ) ( ) ( ) 0)(11

=∑ +−++−−⇒−

=−−−

rn

irnrnrniii

zrzrn δγαδγα

( ) ( )

0

11 1

=−

∑ −∑−+−−⇒

−

−

−

=−

−

=

σδα

γασ

δγα

rn

rn

rn

irn

rn

i

i

ii

xr

rx

rn

( ) ( )

( ) 01

1

1

1

=∑ −+−

∑ −+−−⇒

−

=−−

−

=−

rn

irnrnii

rn

irni

xrx

rrn

δαδα

γαγασ

( )

( ) ( ) ∑ −+−−

∑ ++=⇒

−

=−

−

=−−∧

rn

irni

rn

irnrnii

rrn

xrx

1

1

1

1

γαγα

δαδασ

(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

principle is that the expression i

i

z

z

+1 is

approximated by a linear function in some neighbourhood of the population quantile. It can be seen that the construction of the neighbourhood over

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. 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.

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.

4 Estimation Based On Sample Quantiles

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’.

Accordingly we can get

k

c

n

BLUES for σ each with

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



86

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. 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

[ ] )1(11

11)()(

=+−=

+−=

−

−

σ

σ

α

α

ifx

xxFie

[ ] 11

1

−−=⇒−

αξ p

If σ

x is a standard Pareto variate, considering

σ

x

as ξ, we get

[ ]

[ ]

−−=⇒

−−==

−

−

11

11

1

1

α

α

σξ

σξ

p

px

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

[ ] 11

1

−−

=−

α

ξσ

p

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

[ ] 11

ˆ1

−−

=−

α

σ

P

xp

where x p is the pth 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

such a way that the variance of the σ̂ is the

minimum with respect to p. But the exact variance of

σ̂ is not analytically available. However the

asymptotic variance of σ̂ can be obtained as

fallows.

Asymptotic variance of

[ ]2

1

11

)(ˆ

−−

=−

α

σ

p

xRVASAp

From the asymptotic theory of order statistics, we know that the asymptotic variance of the Pth quantile in the sample is

[ ])1(2

1

21

)1()(

+−−

−

−=∴

α

αα p

ppxRAVSA

p

(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

(i. e.) 0)]([ =

∧

σRVASAdp

d

[ ]

[ ]

0

1

11)1(

)1(21

2

21

=

−

−−−

⇒+−

−

−

α

α

α

α P

PPP

dp

d

0

)1(

)1(22

)1(

)1(1

24

1

24

21

=

−

−−

+

−

−−

⇒++

α

α

α

α

αp

pp

p

p

2

2

1 1 1

2 2

1 1

2 2

[ ( )] 0

(1 ) 6 6 2 (1 ) 2 2 (1 )

2 (1 ) 4 (1 ) 12 16 2

p

dASVAR x

dp

p p p p p

p p p p

α α α

α α

α α α α

α α α α

=

⇒ − − − + − − + − =

− + − − − −



87

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

the asymptotic variance of σ̂

We have found that Newton Raphson method when applied to solve the above

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.



88

TABLE 1

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

n I α =2 α =3 α =4

5 1 0.00000 0.81650 0.00000 0.87358 0.00000 0.90360 5 2 0.02055 0.63246 0.00982 0.73681 0.00573 0.79527 5 3 0.07083 0.44721 0.03564 0.58480 0.02136 0.66874 5 4 0.17051 0.25820 0.09241 0.40548 0.05763 0.50813 5 5 0.64998 0.00000 0.50334 0.00000 0.40837 0.00000 10 1 0.00000 0.90453 0.00000 0.93530 0.00000 0.95107 10 2 0.00503 0.80904 0.00232 0.86825 0.00133 0.89947 10 3 0.01599 0.71351 0.00751 0.79848 0.00434 0.84469 10 4 0.03407 0.61791 0.01635 0.72547 0.00956 0.78608 10 5 0.06094 0.52223 0.03000 0.64850 0.01778 0.72266 10 6 0.09904 0.42640 0.05024 0.56652 0.03023 0.65299 10 7 0.15222 0.33029 0.08005 0.47782 0.04910 0.57471 10 8 0.22729 0.23355 0.12518 0.37924 0.07871 0.48327 10 9 0.33903 0.13484 0.19955 0.26295 0.13018 0.36721 10 10 0.75536 0.00000 0.60884 0.00000 0.50538 0.00000 15 1 0.00000 0.93541 0.00000 0.95647 0.00000 0.96717 15 2 0.00223 0.87082 0.00101 0.91191 0.00058 0.93318 15 3 0.00693 0.80623 0.00319 0.86624 0.00183 0.89790 15 4 0.01441 0.74162 0.00672 0.81932 0.00388 0.86117 15 5 0.02503 0.67700 0.01185 0.77101 0.00688 0.82280 15 6 0.03924 0.61237 0.01886 0.72112 0.01104 0.78254 15 7 0.05761 0.54772 0.02817 0.66943 0.01663 0.74008 15 8 0.08087 0.48305 0.04030 0.61564 0.02403 0.69502 15 9 0.11003 0.41833 0.05599 0.55934 0.03375 0.64678 15 10 0.14645 0.35355 0.07634 0.50000 0.04660 0.59460 15 11 0.19216 0.28868 0.10301 0.43679 0.06382 0.53728 15 12 0.25036 0.22361 0.13883 0.36840 0.08759 0.47287 15 13 0.32671 0.15811 0.18917 0.29240 0.12222 0.39764 15 14 0.43358 0.09129 0.26710 0.20274 0.17876 0.30214 15 15 0.80098 0.00000 0.65912 0.00000 0.55388 0.00000

20 1 0.00000 0.95119 0.00000 0.96719 0.00000 0.97529 20 2 0.00125 0.90238 0.00057 0.93381 0.00032 0.94994 20 3 0.00386 0.85356 0.00176 0.89982 0.00100 0.92389 20 4 0.00793 0.80475 0.00365 0.86518 0.00209 0.89708 20 5 0.01361 0.75593 0.00633 0.82983 0.00364 0.86944 20 6 0.02105 0.70711 0.00990 0.79370 0.00573 0.84090 20 7 0.03043 0.65828 0.01447 0.75673 0.00842 0.81134 20 8 0.04197 0.60945 0.02019 0.71883 0.01182 0.78067 20 9 0.05593 0.56061 0.02725 0.67989 0.01606 0.74874 20 10 0.07262 0.51177 0.03588 0.63981 0.02130 0.71538 20 11 0.09244 0.46291 0.04636 0.59841 0.02773 0.68037 20 12 0.11588 0.41404 0.05907 0.55551 0.03564 0.64346 20 13 0.14359 0.36515 0.07453 0.51087 0.04539 0.60428 20 14 0.17644 0.31623 0.09345 0.46416 0.05752 0.56234 20 15 0.21562 0.26726 0.11685 0.41491 0.07282 0.51697 20 16 0.26290 0.21822 0.14634 0.36246 0.09251 0.46714 20 17 0.32108 0.16903 0.18458 0.30571 0.11877 0.41113 20 18 0.39512 0.11952 0.23669 0.24264 0.15583 0.34572 20 19 0.49587 0.06901 0.31503 0.16824 0.21459 0.26269 20 20 0.82795 0.00000 0.69066 0.00000 0.58522 0.00000

γ δ γ δ γ δ



89

TABLE 2

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

n I α =2 α =3 α =4

5 1 0.00759 0.83333 0.00348 0.88555 0.00199 0.91287 5 2 0.03367 0.66667 0.01598 0.76314 0.00929 0.81650 5 3 0.08579 0.50000 0.04256 0.62996 0.02531 0.70711 5 4 0.17863 0.33333 0.09403 0.48075 0.05768 0.57735 5 5 0.35017 0.16667 0.20221 0.30285 0.13036 0.40825

10 1 0.00217 0.90909 0.00098 0.93844 0.00055 0.95346 10 2 0.00911 0.81818 0.00419 0.87478 0.00239 0.90453 10 3 0.02167 0.72727 0.01014 0.80872 0.00586 0.85280 10 4 0.04092 0.63636 0.01956 0.73984 0.01142 0.79772 10 5 0.06836 0.54545 0.03347 0.66758 0.01977 0.73855 10 6 0.10615 0.45455 0.05342 0.59118 0.03201 0.67420 10 7 0.15759 0.36364 0.08193 0.50946 0.04993 0.60302 10 8 0.22826 0.27273 0.12355 0.42055 0.07692 0.52223 10 9 0.32902 0.18182 0.18791 0.32094 0.12041 0.42640 10 10 0.48789 0.09091 0.30289 0.20218 0.20331 0.30151 15 1 0.00101 0.93750 0.00045 0.95789 0.00026 0.96825 15 2 0.00417 0.87500 0.00190 0.91483 0.00108 0.93541 15 3 0.00972 0.81250 0.00447 0.87073 0.00256 0.90139 15 4 0.01795 0.75000 0.00836 0.82548 0.00482 0.86603 15 5 0.02919 0.68750 0.01379 0.77896 0.00800 0.82916 15 6 0.04386 0.62500 0.02103 0.73100 0.01229 0.79057 15 7 0.06250 0.56250 0.03046 0.68142 0.01795 0.75000 15 8 0.08579 0.50000 0.04256 0.62996 0.02531 0.70711 15 9 0.11462 0.43750 0.05801 0.57630 0.03486 0.66144 15 10 0.15026 0.37500 0.07777 0.52002 0.04729 0.61237 15 11 0.19447 0.31250 0.10330 0.46050 0.06367 0.55902 15 12 0.25000 0.25000 0.13693 0.39685 0.08579 0.50000 15 13 0.32147 0.18750 0.18288 0.32759 0.11694 0.43301 15 14 0.41789 0.12500 0.25000 0.25000 0.16435 0.35355 15 15 0.56250 0.06250 0.36379 0.15749 0.25000 0.25000

20 1 0.00058 0.95238 0.00026 0.96800 0.00015 0.97590 20 2 0.00238 0.90476 0.00108 0.93545 0.00061 0.95119 20 3 0.00550 0.85714 0.00251 0.90234 0.00143 0.92582 20 4 0.01005 0.80952 0.00463 0.86860 0.00265 0.89974 20 5 0.01616 0.76190 0.00751 0.83419 0.00432 0.87287 20 6 0.02398 0.71429 0.01126 0.79906 0.00651 0.84515 20 7 0.03367 0.66667 0.01598 0.76314 0.00929 0.81650 20 8 0.04546 0.61905 0.02183 0.72636 0.01277 0.78680 20 9 0.05957 0.57143 0.02896 0.68861 0.01705 0.75593 20 10 0.07632 0.52381 0.03760 0.64980 0.02228 0.72375 20 11 0.09606 0.47619 0.04801 0.60980 0.02866 0.69007 20 12 0.11926 0.42857 0.06054 0.56844 0.03644 0.65465 20 13 0.14653 0.38095 0.07567 0.52551 0.04595 0.61721 20 14 0.17863 0.33333 0.09403 0.48075 0.05768 0.57735 20 15 0.21667 0.28571 0.11653 0.43380 0.07230 0.53452 20 16 0.26220 0.23810 0.14455 0.38415 0.09088 0.48795 20 17 0.31760 0.19048 0.18031 0.33105 0.11517 0.43644 20 18 0.38693 0.14286 0.22776 0.27328 0.14839 0.37796 20 19 0.47802 0.09524 0.29521 0.20855 0.19756 0.30861 20 20 0.61118 0.04762 0.40646 0.13138 0.28394 0.21822

γ δ γ δ γ δ



90

TABLE 3

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

α n BIAS VARIANCE MSE

MLE MMLE-I MMLE-II MLE MMLE-I MMLE-II MLE MMLE-I MMLE-II

2 5 0.17289 0.00784 0.35621 0.68399 0.52377 1.28796 0.71388 0.52383 1.41484

2 10 0.08545 0.01004 0.20054 0.26088 0.23200 0.40863 0.26818 0.23210 0.44884

2 15 0.05653 0.00637 0.13965 0.15975 0.14723 0.21541 0.16294 0.14728 0.23492

2 20 0.04262 0.00526 0.10723 0.11277 0.10649 0.14184 0.11458 0.10652 0.15333

3 5 0.11358 -0.00765 0.20724 0.46483 0.39623 0.60761 0.47773 0.39629 0.65055

3 10 0.05785 0.00617 0.12425 0.19810 0.18459 0.24237 0.20145 0.18463 0.25781

3 15 0.03881 0.00486 0.08977 0.12507 0.11938 0.14570 0.12657 0.11940 0.15376

3 20 0.02946 0.00440 0.07054 0.08965 0.08701 0.10095 0.09051 0.08703 0.10592

4 5 0.08450 -0.01788 0.14625 0.37894 0.34572 0.44431 0.38608 0.34604 0.46570

4 10 0.04394 0.00316 0.09012 0.17001 0.16379 0.19258 0.17194 0.16380 0.20070

4 15 0.02978 0.00342 0.06621 0.10915 0.10657 0.12061 0.11004 0.10658 0.12499

4 20 0.02271 0.00350 0.05266 0.07878 0.07784 0.08528 0.07930 0.07785 0.08805



91

TABLE 4

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

n r BIAS VARIANCE MSE

MLE MMLE-I

MMLE-

II MLE MMLE-I

MMLE-

II MLE MMLE-I MMLE-II

5 1 0.14677 -0.02913 0.20894 0.75861 0.58640 0.94666 0.78015 0.58725 0.99031

5 2 0.14934 -0.00262 0.17056 0.74603 0.56047 0.77531 0.76833 0.56047 0.80440

5 3 0.12808 -0.00075 0.13638 0.93202 0.73245 0.94696 0.94843 0.73245 0.96556

10 1 0.07112 -0.02030 0.12721 0.25736 0.22158 0.30009 0.26242 0.22200 0.31628

10 2 0.09819 0.01465 0.13395 0.30383 0.26172 0.32726 0.31347 0.26194 0.34521

10 3 0.08458 0.00637 0.10723 0.28847 0.24735 0.30036 0.29562 0.24739 0.31186

10 4 0.04250 -0.02704 0.05754 0.24044 0.20976 0.24795 0.24224 0.21049 0.25126

10 5 0.07066 0.00321 0.08056 0.33098 0.29022 0.33661 0.33597 0.29023 0.34310

10 6 0.05555 -0.00744 0.06133 0.35651 0.31527 0.36059 0.35959 0.31533 0.36435

10 7 0.03839 -0.02002 0.04159 0.45516 0.40512 0.45764 0.45663 0.40552 0.45937

10 8 0.03765 -0.01732 0.03917 0.65484 0.58754 0.65694 0.65626 0.58784 0.65848

15 1 0.04947 -0.01035 0.09825 0.16036 0.14506 0.18140 0.16280 0.14516 0.19105

15 2 0.06085 0.00433 0.09446 0.16020 0.14461 0.17209 0.16390 0.14463 0.18101

15 3 0.03111 -0.02191 0.05429 0.14769 0.13290 0.15483 0.14866 0.13338 0.15778

15 4 0.05043 -0.00044 0.06930 0.16553 0.15224 0.17473 0.16807 0.15224 0.17953

15 5 0.04336 -0.00582 0.05719 0.17373 0.15847 0.17941 0.17561 0.15850 0.18268

15 6 0.07767 0.02943 0.08910 0.19782 0.18005 0.20154 0.20385 0.18092 0.20948

15 7 0.04635 0.00088 0.05442 0.21385 0.19526 0.21685 0.21600 0.19526 0.21981

15 8 0.04085 -0.00244 0.04714 0.20875 0.19195 0.21156 0.21042 0.19196 0.21378

15 9 0.02159 -0.01924 0.02626 0.22239 0.20487 0.22437 0.22285 0.20524 0.22506

15 10 0.05605 0.01526 0.05940 0.28886 0.26696 0.29067 0.29200 0.26719 0.29420

15 11 0.02927 -0.00894 0.03156 0.31104 0.28833 0.31238 0.31190 0.28841 0.31337

15 12 0.04223 0.00482 0.04355 0.40826 0.37951 0.40933 0.41005 0.37953 0.41122

15 13 0.05583 0.01927 0.05644 0.63646 0.59306 0.63711 0.63958 0.59344 0.64030

20 1 0.04036 -0.00480 0.08016 0.11234 0.10486 0.12537 0.11397 0.10488 0.13179

20 2 0.03295 -0.00873 0.06336 0.10709 0.10014 0.11601 0.10818 0.10022 0.12003

20 3 0.03236 -0.00904 0.05447 0.10853 0.10025 0.11393 0.10958 0.10033 0.11690

20 4 0.04639 0.00733 0.06630 0.12636 0.11797 0.13240 0.12851 0.11802 0.13679

20 5 0.05041 0.01167 0.06595 0.12318 0.11505 0.12794 0.12572 0.11519 0.13229

20 6 0.04053 0.00350 0.05348 0.12842 0.11983 0.13225 0.13006 0.11984 0.13511

20 7 0.03492 -0.00041 0.04612 0.13426 0.12597 0.13818 0.13548 0.12597 0.14030

20 8 0.03444 -0.00094 0.04259 0.12860 0.11976 0.13043 0.12978 0.11976 0.13224

20 9 0.03900 0.00514 0.04632 0.14323 0.13466 0.14595 0.14475 0.13468 0.14810

20 10 0.04341 0.00962 0.04871 0.13905 0.13024 0.14055 0.14094 0.13033 0.14292

20 11 0.02592 -0.00598 0.03054 0.16622 0.15619 0.16788 0.16689 0.15623 0.16881

20 12 0.04569 0.01420 0.04957 0.18193 0.17112 0.18326 0.18402 0.17132 0.18572

20 13 0.02297 -0.00712 0.02585 0.17536 0.16514 0.17627 0.17589 0.16519 0.17694

20 14 0.03449 0.00490 0.03674 0.21107 0.19921 0.21204 0.21226 0.19923 0.21339

20 15 0.05060 0.02137 0.05232 0.26901 0.25418 0.26982 0.27157 0.25463 0.27255

20 16 0.01968 -0.00791 0.02086 0.29481 0.27904 0.29545 0.29519 0.27910 0.29589

20 17 0.03786 0.01050 0.03860 0.39870 0.37800 0.39931 0.40013 0.37811 0.40080

20 18 0.00903 -0.01691 0.00934 0.51541 0.48920 0.51567 0.51549 0.48948 0.51576



92

ACKNOWLEDGEMENT:


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,

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