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INTERNATIONAL JOURNAL OF
COMPUTATIONAL MATHEMATICAL IDEAS
ISSN: 0974-8652 Volume 2 / Number 1 / January 2010
--------------------------------------------------------------------------------------------------------------------
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
INTERNATIONAL JOURNAL OF
COMPUTATIONAL MATHEMATICAL IDEAS
ISSN: 0974-8652 Volume 2 / Number 2 / August 2010
--------------------------------------------------------------------------------------------------------------------
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
INTERNATIONAL JOURNAL OF
COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI)
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The INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI) is an international research journal, which publishes top-level work on computational aspects of
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We are pleased to announce the launch Fourth & Fifth issue of the INTERNATIONAL JOURNAL OF
COMPUTATIONAL MATHEMATICAL IDEAS (IJCMI).The IJCMI is a refereed Mathematics &Computer
science and Engineering journal devoted to publication of original research papers, research notes, and review
articles, with emphasis on unsolved problems and open questions in mathematics &Computer science and
Engineering. All areas listed on the cover of Mathematical Reviews, such as pure and applied mathematics &
Computer science and Engineering.
The IJCMI is an international research journal, which publishes top-level work on computational aspects of mathematics interface between applied mathematics, numerical computation, and applications of systems - oriented ideas to the physical, biological, social, and behavioral sciences. It includes but not limited to areas of mathematics such as algebra (especially group theory), combinatorics (especially graph theory), geometry, number theory and numerical analysis; computational complexity; cryptology; symbolic and algebraic computation; optimization; the mathematical aspects of: models of computation; automata theory; categories and logic in computer science; proof theory; type theory; semantics of programming languages; process algebra and concurrent systems; specification and verification; databases; rewriting; neural nets and genetic algorithms; computational learning theory; theorem proving, Applied Physics, Solid State Physics, Nuclear Physics Theoretical Physics and more... IJCMI Journal publishes research articles and reviews within the whole field of Mathematical Sciences& Computer Science and Engineering, and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The IJCMI will be published three issues in a year. The journal will be reviewed by two independent reviewers. In addition, the Journal may occasionally publish special issues on various topics in the areas of Mathematics &Computer science and Engineering, book reviews, conference reports, letters to the honorary editors, conference announcements, etc. Finally, The Editor-in-Chief and Honorary Editors wish to congratulate the authors of the published papers in IJCMI. Satyans Publications to starting Three more International Journals such as International Journal of Applied Sciences & Engineering ideas (IJASE), International Journal of Artificial Intelligence Ideas (IJAII), International Journal of Entrepreneurship Ideas (IJEI) for the inspiration of IJCMI. Thanks are due to the members of the Editorial Board for their precious feedback and advice. We hope that the new International Journal of Computational Mathematical Ideas will serve Mathematics & Computer science engineering research community as well as and this journal will be main media of presenting ideas and research work in their area. Suggestions to improve our efforts in order to deliver a better journal to the authors, readers and subscribers of this journal will always be appreciated.
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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:
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
partitions of a positive integer n having r parts
in which each part is even number.
1.7 ( ) :;rp S n ( );rp S n is the number of
partitions of a positive integer n having r parts
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+
+
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Note On r - partitions Of n In Which The Least Part Is k
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− − + .
Fixing ( )1 2a bλ = + , the remaining value
( ){ }2n a b− + of n can be expressed as the sum
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
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Note On r - partitions Of n In Which The Least Part Is k
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−= − − +
Fixing ( )1 3a bλ = + , the remaining value
( ){ }3n a b− + of n can be expressed as the sum
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
−
= =
− − −=∑∑
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Note On r - partitions Of n In Which The Least Part Is k
9
Proof: Put 1, 0a b= = in [ ]6.2
Corollary 6.2: Prove that
( )( )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
Corollary 6.3: Prove that
( )( )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
+ +
−
+
= +
= + + +
+ + + + +
+ + + + +
+ + + +
+ +
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Note On r - partitions Of n In Which The Least Part Is k
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⇒ + ≤ + < +
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Note On r - partitions Of n In Which The Least Part Is k
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+
+ + + + + + +
= + + =
Corollary 11.1: Prove that
( )
[ ]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
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Note On r - partitions Of n In Which The Least Part Is k
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.
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
13
INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 13-21 (2010)
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
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Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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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.
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Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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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
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Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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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,
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Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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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.
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INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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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
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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.
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Mixed Convection In A Porous Medium With Magnetic Field, Variable Viscosity And Varying Wall Temperature
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[3] Barletta.A, Lazzari.S, Magyare.E and Pop.I,
Mixed convection with heating effects in a
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[5]Gorsan.T, Revnic.C, Pop.I and Ingham.D.B,
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On Completely Prime and Completely Semi-prime Ideals in ΓΓΓΓ-near-rings 22
INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN: 0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 22-27 (2010)
ON COMPLETELY PRIME AND COMPLETELY SEMI-PRIME
IDEALS IN 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.
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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).
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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.
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(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
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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.
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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
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A Unified frame work for searching Digital libraries Using Document Clustering
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 28-32 (2010)
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
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Unified frame work for searching Digital libraries Using Document Clustering
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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).
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
A Unified frame work for searching Digital libraries Using Document Clustering
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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
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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).
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Reducibility For The Fiorini-Wilson-Fisk Conjecture
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 33-42 (2010)
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
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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.
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The following definitions are taken from Robertson et. al. [98] A match is a an unordered pair {e,f} of distinct edges of E(R). A matching is a nonempty set of matches {{e1, f1},{e2,f2},…,{ek, fk}} such that for any i ≠ j, the edges ej and fj are in the same component of R – {ei,fi}. Finally, a signed matching is a collection of ordered pairs {({e1,f1}, µ 1), ({e2, f2},
µ 2),…, ({ek, fk}, µ k)}, where the collection
{{e1, f1},{e2, f2},…,{ek, f k}} is a matching, and
where µ i ∈ {-1, 1} for 1 ≤ i ≤ k.The sign of a
match is used to differentiate whether both ends of a kempe chain have the same color or distinct colors.
If θ ∈ {-1, 0, 1} and k is a coloring of R we say
that k θ -fits a signed matching
M = {({e1, f1}, µ 1), ({e2, f2}, µ 2), … , ({ek, fk},
µ k)} if
(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
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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
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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
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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,
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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.
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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,
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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-
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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.
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Perceiving Plagiarism Using Weighted Window Approach- Performance Analysis
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 43-47 (2010)
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.
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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.
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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
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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)
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[8]. Decoo, W. Crisis on Campus: Confronting Academic Misconduct. Cambridge, MA., MIT Press, 2002. (Hesburgh LB 2344 .D43 2002). [9]. Harris, R. A. The Plagiarism Handbook.Los Angeles, Pyrczak Publishing, 2001 www.AntiPlagiarism.com [10]. Lathrop A. Student Cheating and Plagiarism in the Internet Era.Englewood, CO., Libraries Unlimited, 2000. (Hesburgh LB 3609 .L28 2000) [11]. C. Humes, J. Stiffler and M. Malsed. Examining Anti-Plagiarism Software: Choosing the Right Tool. Claremont-McKenna College technical report. 2003. [12]. Brian Martin. Plagiarism: policy against cheating or policy for learning? Nexus: Newsletter of the Australian Sociological Association, 16:2, pp. 1-12, 2004. [13]. L. Renard. Cut and paste 101: Plagiarism and the Net. Educational Leadership, 57:4, pp. 38-42, 2000 [14]. Arwin, C. and S. M. M. Tahaghogh (2006). Plagiarism Detection Across Programming Languages. ACM International Conference Proceeding Series, vol. 171. Proceedings of the 29th Australasian Computer Science Conference, Hobart, Australia, vol. 48 ,pp. 277 – 286. [15]. Niezgoda, S. and T. Way. (2006) SNITCH: A Software Tool for detecting Cut and Paste
Plagiarism. Proceedings of the 37th
Special Interest Group on Computer Science Education. Pp. 51 – 55. New York: Association for Computing Machinery.
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System Representation For Software Architecture Recovery
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 48-54 (2010)
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.
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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
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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).
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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.
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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
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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
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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
We would like to express our thanks to referees for valuable comments that improved the paper.
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.
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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
INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN: 0974-8652 / www.ijcmi.webs.com / VOL 2-NO1-PP 55-59 (2010)
EXPERIMENTAL AND THEORETICAL EVALUATION OF ULTRASONIC VELOCITIES IN
BINARY LIQUID MIXTURE CYCLOHEXANE + O-XYLENE AT 303.15, 308.15, 313.15 AND
318.15K.
Narendra K@, Narayanamurthy P# & Srinivasu Ch$
@Department of Physics, V.R.Siddhartha Engg.College, Vijayawada, Andhrapradesh, 520007, Email:- [email protected]
#Department of Physics, Acharya Nagarjuna University, Nagarjuna nagar, Guntur, Andhrapradesh $Department of Physics, Andhra Loyala College, Vijayawada, Andhrapradesh, 520008
ABSTRACT
A comparison of ultrasonic velocity evaluated from Nomoto’s relation, Vandael ideal mixing relation, impedence relation, Rao’s specific velocity relation and Junjie’s theory has been made in the binary mixture cyclohexnae with o-xylene at 303.15, 308.15, 313.15 and 318.15 K. Ultrasonic velocity and density of these mixtures have also been measured as a function of temperature and the experimental values are compared with the theoretical values. A good agreement is found between experimental and Vandael ideal mixing relation ultrasonic velocities. 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.
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Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15
and 318.15k.
56
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
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Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15
and 318.15k.
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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
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Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15
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
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
Experimental and theoretical evaluation of ultrasonic velocities in binary liquid mixture cyclohexane + o-xylene at 303.15, 308.15, 313.15
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.
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
Over View To Implementation Of Robotics With Voice Recognition
60
INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO 2-PP 60-64 (2010)
OVER VIEW TO IMPLEMENTATION OF ROBOTICS WITH VOICE RECOGNITION
Ande Stanly Kumar@, Dr.K.Mallikarjuna Rao#, Dr.A.Bala Krishna$, B.Venkatesh^
@Asst.Professr, Sri Venkateswara Institute of Science Information Technology,Tadepalligudem. [email protected], # Professor, JNTU College of Engineering, Kakinada. $ Professor,SRKR Engineering College,Bhimavaram. ^Asst.Professr, Sri Venkateswara Institute of Science Information Technology,Tadepalligudem. [email protected]
ABSTRACT
Automatic speech recognition by machine has been a goal of a research for a long time. Speech recognition is the process of
converting an acoustic signal, captured by a microphone or a telephone, to a set of words. The recognized words can be the final
results, as for applications such as commands & control, data entry, and document preparation. They can also serve as the input to
further linguistic processing in order to achieve speech understanding. There are many works carried out in this area. The speech
recognition system has also been implemented on some particular devices. Some of them are personal computer (PC), digital signal
processor, and another kind of single chip integrated circuit. In this paper we propose voice recognition to control robot.
Key words: Euclidean square distance, LPC, Voice recognition, Finger print.
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.
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Speech is a natural source of interface for human–
machine communication, as well as being one of the most
natural interfaces for human–human communication [8].
However, environmental robustness is still one of the main
barriers to the wide use of speech recognition. Speech
recognition performance degrades significantly under varying
environmental conditions for many application areas.
In 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
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formula above requires squaring the difference between the
two points, but since we are using fixed point arithmetic, we
found that squaring the difference produced too large of a
number causing our variables to overflow. Thus we
implemented a "pseudo Euclidean distance calculation" by
moving the sum out of the square root reducing the equation to
D =
PWM (Pulse Width Modulation) duty cycle calculation:
The motors in the robot were measured to have a 50 Hz PWM
frequency and movement was controlled by varying the duty
cycle from 5% to 10%. To generate the PWM signals we used
timer/counter1 in phase correct mode. The top value of
timer/counter 1 was set to be 20000 and using a /8 pre scalar
the PWM signal was set to have a frequency of 50Hz =
16MHz/(8*2*20000). To calculate the duty cycle the
following equation was used OCR1x = (20000 - 40000*duty
cycle). Where OCR1x is the value in the output compare
register 1 A or B.
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.
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
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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
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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.
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Novelty Of Extreme Programming
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO 2-PP 65-72 (2010)
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:
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� XP promises that they will be able to work on things that really matter, everyday. They won’t have to face scary situations alone. They will be able to do every thing in their power to make their system successful. They will make decisions that they can make best and they won’t make decisions that they can make best and they won’t make decisions they are not best qualified to make.
� 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
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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.
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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
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• 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.
[7]. William.A.Wood., William.L.K., Exploring XP for Scientific Research, IEEE Software, Vol.20, No. 3, pp30-36, 2003.
[8]. Martin Lippert., Stefan Roock., Adopting XP to Complex Application Domains, ACM, 2001.
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[9]. Glenn Vanderburg., A Simple Model of Agile Software Process –or- Extreme Programming Annealed, ACM, 2005.
[10]. Kuppuswami, S., Vivekanandan K., and Paul Rodrigues
(2003): A Sys-tem Dynamics Simulation Model to Find the
Effects of XP on Cost of Change Curve. In proceedings of
Fourth International Conference on Extreme Pro-
gramming and Agile process in Software Engineering,
(XP2003), May 25 – 29, 2003, Genova, Italy.
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO 2-PP 73-81 (2010)
RADIATION EFFECTS ON MHD FREE CONVECTION FLOW PAST A SEMI-INFINITE MOVING VERTICAL
POROUS PLATE WITH SORET AND DUFOUR EFFECT
G.Venkata Ramana Reddy@
and Dr. A.Rami Reddy#
@ Assistant Professor, LBR College of Engineering, Mylavaram, Krishna,A.P. Email: [email protected] # Associate Professor, LBR College of Engineering, Mylavaram, Krishna,A.P
ABSTRACT
In this paper, we deal with the interaction of Soret and Dufour effects on steady MHD free convection flow in a porous medium with dissipative fluid has received little attention. Hence, the object of the present chapter is to analyze the Soret and Dufour effects on steady MHD free convection flow past a semi-infinite moving vertical plate in a porous medium with viscous dissipation. The governing equations are transformed by using similarity transformation and the resultant dimensionless equations are solved numerically using the Runge-Kutta method with Shooting technique. The effects of various governing physical parameters on the fluid velocity, temperature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are shown in figures and tables and analyzed in detail.
Key words: MHD free convection, porous medium, vertical plate, and Nusselt number
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
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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
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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.
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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
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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
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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
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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
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(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
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(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
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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
We would like to express our thanks to referees for valuable comments that improved the paper.
REFERENCES
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[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.
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[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.
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[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.
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[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.
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INTERNATIONAL JOURNAL OF COMPUTATIONAL
MATHEMATICAL IDEAS ISSN:0974-8652 / www.ijcmi.webs.com / VOL 2-NO 2-PP 82-92 (2010)
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)
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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)
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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
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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
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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
=
⇒ − − − + − − + − =
− + − − − −
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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.
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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
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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
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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
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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
INTERNATIONAL JOURNAL OF COMPUTATIONAL MATHEMATICAL IDEAS ISSN:0974-8652
Pareto Distribution - Some Methods Of Estimation
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ACKNOWLEDGEMENT:
We would like to express our thanks to referees for valuable comments that improved the paper.
REFERENCES:
[1]. Balakrishnan, N. (1990). “Approximate maximum likelihood estimation for a generalized logistic distribution”, Journal of Statist. Plann. & inf., 26, 221-236. [2]. Bhattacharya.G..K.(1985). “The Asymptotics of maximum likelihood and related estimators based on Type II censored data”. Journal of American Statistical Association. 80, 398-404. [3]. Kantam R .R .L. & Sriram.B. (2003), “Maximum Likelihood Estimation from censored samples –some modifications in length biased version of exponential Model”, Statistical methods, Vol. 5, 63-78. [4]. Lloyd, E. H. (1952). “Least-squares estimation of location and scale parameters using order statistics”, Biometrika, 39, 88-95. [5]. Tiku. M.L., Tan. W .Y. and Balakrishnan. N. (1986). “Roboust Inference”, Marcel Decker, I. N.C. New York. [6]. Srinivasa rao. G. and Kantam. R. R .L. (2002) “A note on point estimation of system reliability exemplified for the Log-Logistic distribution”, Economic quality control, 19(2), 197-204,
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