Proceedings of the National Seminar on Present Trends in Mathematics and Its Applications

Includes

Invited Lectures

Research Paper

Presentations

Survey Articles

Abstracts

Editors:

Dr Eswaraiah Setty Sreeramula

Dr Satyanarayana Bhavanari

Dr Syam Prasad Kuncham

Sponsored by UGC

Held at Smt. G.S. College, Jaggaiahpet,

Krishna Dist, Andhra Pradesh, India

November 11-12, 2010

Proceedings of the

National Seminar on

Present Trends in

Mathematics and its

Applications

Copy Right: Dr S. Eswaraiah Setty, Organizing Secretary, SGS College, Jaggaiahpet, A.P., India.

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any

means, without permission. Any person who does any unauthorized act in relation to this publication

may be liable to criminal prosecution and civil claims for damages.

First Published, 2010

Printed at

This book is meant for educational and learning purposes. The author(s) of the papers in this book

has/have taken all reasonable care to ensure that the contents of the book do not violate any existing

copyright or other intellectual property rights of any person in any manner whatsoever. In the event the

author(s) has/have been unable to track any source and if any copyright has been inadvertently

infringed, please notify the publisher in writing for corrective action.

The Organizing Committee, Editors and the publisher of the proceedings of the “National Seminar on

Present Trends in Mathematics and its Applications” are not responsible for the statements made or

opinion expressed by the authors in the proceedings of this Conference. The Organizing committee and

the Editors do not hold any responsibility for any omissions or typographical errors or violation of any

existing copyright or other intellectual property rights of any person in any manner whatsoever.

Proc. of the National Seminar on Present Trends in Mathematics and its Applications

Page i Contents

Invited Talks

S.NO. Invited Speaker Title of the Lecture Page

Nos.

1.

Prof. P. V. Arunachalam, Former Vice-

Chancellor, Dravidian University, Kuppam

(Andhra Pradesh), Former President, Indian

Mathematical Society.

Fractional calculus 1-11

2. Prof. L. Radha Krishna, UGC Centre for

Advanced Studies, Bangalore University,

Bangalore.

Metamathematics in Teaching and

Research 12-17

3.

Prof. D.R.V. Prasad Rao, Department of

Mathematics, Sri Krishnadevaraya University,

Anantapur, Andhra Pradesh.

Computational Fluid Dynamics-A

Study of Convection in

Rectangular Cavity by finite

Element Technique.

18-40

4. Prof. Bhavanari Satyanarayana, Department of

Mathematics, Acharya Nagarjuna University,

Nagarjuna University (Andhra Pradesh)

Dimension in Vector Spaces and

Modules 41-49

5. Prof. S. Sreenadh, Department of Mathematics,

Sri Venkateswara University, Tirupathi (Andhra

Pradesh)

Effect of Yield Stress, Elasticity

and Peristalsis on the Transport

Biofluids 50-57

6. Dr Syam Prasad Kuncham, Department of

Mathematics, Manipal University, Manipal-576

104, Karnataka.

Fuzzy Ideals of Gamma Nearrings 58-61

7. Dr. M. S. Dutt, Department of Mathematics and

Statistics, University of Hyderabad, Andhra

Pradesh

Semisimple Hopf Algebras and

their Orbits 62-65

8. Dr Babushri Srinivas Kedukodi, Department of

Mathematics, Manipal University, Manipal-576

104, Karnataka.

Rough Sets 66-68

9. Dr Nagaraju Dasari, Department of

Mathematics, HITS, Hisdustan University,

Padur, Chennai.

Finite Dimension in Associative

Rings 69-76

10. Prof. I. H. Nagaraja Rao, Sr. Professor,

Department of Mathematics, GVP College for P

G Courses, Visakapatnam, Andhra Pradesh.

On Some Results on Semi-

Complete Graphs 77-83

11. Dr Re. Victor Babu, Department of Statistics,

Acharya Nagarjuna University, Nagarjuana

Nagar-522 510, Andhra Pradesh.

Modified Second-Order Slope-

Rotatable Designs with Equi-

Spaced Levels-A Review, Survey

Article (Author: B. Re. Victor

Babu)

84-91


Page ii

Contributed Research Papers (Refereed)

S. No. Title / Author (s) Page No.

1. A Note on Semi-Prime Near-rings, (Author: Bhavanari Satyanarayana) 92-95

2. Fuzzy Numbers and Matrix Transformation (Authors: Abdul Hamid,

Neyaz Ahmad and Sameer Ahmad Gupkari)

96-100

3. Some Results on Completely Semi-Prime Ideals in Gamma Near-Rings

(Authors: Pradeep Kumar T.V., Satyanarayana Bhavanari, Syam Prasad

Kuncham and Mohiddin Shaw Sk.)

101-105

4. Almost Convergence and Some Matrix Transformations (Authors: Abdul

Hamid, Neyaz Ahmad and Tanweer jalal)

106-110

5. Generalized Fuzzy Ideals of Gamma Nearrings, (Authors: Syam Prasad

Kuncham, Satyanarayana Bhavanari and Subba Rao G.V.)

111-118

6. Global Relevant Weighing (GRW) - A Novel Term weighing Model for

Improved Document Clustering (Survey Article) (Authors: S. Sagar

Imambi, T. Sudha, and J.J.L.R. Bharathi Devi)

119-123

7. Prime Graph of an Integral Domain (Authors: Satyanarayana Bhavanari,

Mohiddin Shaw Sk., and Venkata Vijaya Kumari Arava)

124-134

8. On Fuzzy Continuous Functions in Intuitionistic Fuzzy Topological

Spaces, (Authors: Mamata Singh and Yashveer Singh)

135-140

9. Gamma Rings and m-systems, (Authors: Satyanarayana Bhavanari and

Shakira Sk.).

141-147

10. Reaction of Urdbean Genotypes on Growth in Rainfed Vertisols of Andhra

Pradesh – A Case Study, (Authors: B. Re. Victor Babu*, K. Rajya

Lakshmi* and G. Raghavaiah).

148-153

11. English Vocabulary development- An Experiment through Mathematics,

(Survey Article) (Authors: Suryanarayana Murty T.S.V.S., and Sastry

D.S.N.)

154-161

12. Cryptography and Security Visualization, (Survey Article) (Authors:

Swati Joglekar)

162-166

13. Advanced Predictive Data Mining and Text mining Models (Survey

Article) (Authors: S. Sagar Imambi and L. Padmavathi, )

167-171

14. Number and Infinity Concepts in Vedas (Survey Article)

(Authors: Satyanarayana Bhavanari and Satyanarayana K.)

172-173


Page iii

Abstracts of the Presentations

S. No. Abstracts of Presentations Pgs.

1. Heat and Mass Transfer in a Viscous Heat Generating Fluid Through a Porous

medium in a Triangular Duct (Authors: S. Eswaraiah Setty, S. Sivaiah,

D.R.V. Prasada Rao) 174

1. Ideals and Modules in Rings (Authors: Suryakumar U. and Satyanarayana)

2. Normalization of s-Ideals of Seminearrings (Author: P. Venugopala Rao) 175

3. Basics of Graph Theory and Applications (Author: V. Manjula)

4. Graphs and their Applications (Author: Pokkuluri Surya Prakash)

176 5. Fuzzy Ideals in BF-Algebras (Authors: B. Satyanarayana, D. Rames,

V.Vijaya Kumar, R.Durga Prasad, and M. Arokiasamy)

6. On Noetherian Regular δ- Near-rings and their Extensions (Authors:

Nagendram N V, Venkateswara Reddy Y., and Pradeep Kumar T.V.) 177

7. On P-Regular δ- Regular Near-rings and their Extensions (Authors:

Nagendram N V, Venkateswara Reddy Y., and Pradeep Kumar T.V.)

8. A Note on Goldie Near rings (Authors: P. Narasimha Swamy and T. Srinivas)

178 9. Certain Transformation Formulae for the General Triple Hyper Geometric

Series F3(X, Y, Z), (Authors: Pankaj Srivastava and R V G K Mohan)

10. On Different types of Semi-Complete Graphs 179

Students Presentation

S. No. Title Page

No.

1. Computer Representation of Sets (Author: V. Suvarchala)

180 2. Fuzzy Submodules and Fuzzy dimension in Modules over Associative

Rings (Author: Kavitha Nellore)

Contributed Research Papers (Without Refereed)

S. No. Title of the Presentation Page No.

3. Divisible Fuzzy Subgroups (Authors: N V Ramana Murty and Mariadas) 181-183


Page iv Page iv Preface

This Proceedings contains the substance of invited lectures and contributed research

presentations delivered at the UGC Sponsored Two day National Seminar on Present Trends in

Mathematics and its Application, which was held in the Smt. G. S. College, Jaggaiahpet

(in association with the Department of Mathematics, Acharya Nagarjuna University), November

11-12, 2010.

The main object of the seminar is to bring together eminent researchers of various fields of

Mathematics like: Fluid Dynamics / Graph Theory / Fuzzy sets theory / Near-rings / Gamma

Near-rings / Rings & Modules, for exchange of ideas.

Most of the Research papers in this volume have been refereed. The editors express their

gratitude to all the Management and staff of Smt. S G College, colleagues; research Scholars,

students, who helped in many ways for its success.

The event certainly provides an opportunity for young researchers to get strengthen their

collaborative works of common interest.

The hospitality provided by the Organizers of the Seminar, will be greatly appreciated.

Finally, the editors would like to thank the Press and Electronic media, for their extensive

coverage of news.

Editors

Dr Eswaraiah Setty Sreeramula

Dr Satyanarayana Bhavanari

Dr Syam Prasad Kuncham


Fractional C

Historical Introduction

Differentiation and integration are usually regarded as discrete operations, in the sense that we

differentiate or integrate a function once, twice, or any whole number of times. However, in

some circumstances it’s useful to evaluate a

1695, Leibniz raised the possibility of generalizing the operation of differentiation to non

integer orders, and L’Hospital asked what would be the result of half

Leibniz replied “It leads to a paradox, fr

drawn”. The paradoxical aspects are due to the fact that there are several different ways of

generalizing the differentiation operator to non

Pre-requisites

1. Repeated Integrals

Consider an antiderivative of the function

This can be written as a linear operator (and is known as the indefinite integral

or Volterra operator), which we will write here as

(Note that for the rest of this post, we assume

integrable on the regions in question).

Proceedings of the National Seminar on Present Trends in Mathematics &

SGS College, Jaggaiahpet, A.P., India, November 11

Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)

c. of the National Seminar on Present Trends in Mathematics and its Applications

Fractional Calculus

Historical Introduction



some circumstances it’s useful to evaluate a fractional derivative. In a letter to L’Hospital in

1695, Leibniz raised the possibility of generalizing the operation of differentiation to non

integer orders, and L’Hospital asked what would be the result of half-differentiating x.

Leibniz replied “It leads to a paradox, from which one day useful consequences will be


generalizing the differentiation operator to non-integer powers, leading to inequivalent results.

Integrals - Cauchy Formula

Consider an antiderivative of the function f(x), such as .


operator,

), which we will write here as J: so that.

(Note that for the rest of this post, we assume f(x) is sufficiently continuous and

integrable on the regions in question).

Invited Lecture

Prof. P. V. Arunachalam,

Ex. Vice–Chancellor,

Dravidian University,

Kuppam (Andhra Pradesh),

Former President, Indian


Proceedings of the National Seminar on Present Trends in Mathematics &

SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty



1



. In a letter to L’Hospital in

1695, Leibniz raised the possibility of generalizing the operation of differentiation to non-

differentiating x.

om which one day useful consequences will be


integer powers, leading to inequivalent results.


is sufficiently continuous and

Lecture by

f. P. V. Arunachalam,

Chancellor,

Dravidian University,

Kuppam (Andhra Pradesh),

President, Indian


its Applications,

12, 2010. (Editors: Dr Eswaraiah Setty



Now, let us consider repeated applications of the operator. For example,

We write the operator of J applied twice as

, and so on for

the n-th

antiderivative

The answer is yes. We can reduce it to a single integral. Observe in the case of

Treating this as a double integral, and considering the region of the

which it is integrated, we can reverse the order of integ

As f(u) is a constant with respect to

, and we have:

Similarly, we have


us consider repeated applications of the operator. For example,

.

We write the operator of J applied twice as . Similarly, we have

, and so on for , n any positive integer. Is there a way to find

without needing to perform n integrations?

The answer is yes. We can reduce it to a single integral. Observe in the case of

.

Treating this as a double integral, and considering the region of the tu plane over

which it is integrated, we can reverse the order of integration:

.

is a constant with respect to t, we find that the inner integral is simply

, and we have:

,


2

us consider repeated applications of the operator. For example,

given by

any positive integer. Is there a way to find

The answer is yes. We can reduce it to a single integral. Observe in the case of :

plane over

, we find that the inner integral is simply


which with reordering to make integration by

u2 integrals, we obtain (renaming

If we were to continue, we would find a pattern:

and so on, giving us the formula:

which is known as the Cauchy formula for repeated integration.

We can use mathematical induction to prove that this formula is true for all positive

integer n. First, the n=1 case: we see it gives:

and so the formula holds for


which with reordering to make integration by u3 last, and then performing the

integrals, we obtain (renaming u3 as u):

.

If we were to continue, we would find a pattern:

and so on, giving us the formula:

,

which is known as the Cauchy formula for repeated integration.


=1 case: we see it gives:

,

and so the formula holds for n=1.


3

last, and then performing the u1 and



Since

of both sides with respect to x,

Now, we can take the derivative of our formula with respect to x, using the variable

limit form of the Leibniz integral rule

which is our formula’s value for

given

Now, we see that the integral

can be performed for n>0 even if n is not an integer. Recall, however, that for positive

integer n, , where

factorial in our formula with the corresponding gamma function, we obtain the

formula

which can be computed for

Liouville differintegral, the most often used differintegral (operator combining

differentiation and integration) in

Fractional calculus is a branch of

of taking real number powers or

and the integration operator

other I-like glyphs and identities

In this context the term powers

same sense that f 2(x) = f(f(x)).

For example, one may ask the question of meaningfully inte


, by definition, we see that if we take the

derivative

respect to x,

.


Leibniz integral rule:

which is our formula’s value for . Thus our formula holds for

, and so it holds for all integer n>0 by induction.

Now, we see that the integral


, where is the gamma function. Thus, if we replace the


computed for n a positive real number. This is the basis of the

, the most often used differintegral (operator combining

ation) in fractional calculus.

is a branch of mathematical analysis that studies

powers or complex number powers of the differential operator

and the integration operator J. (Usually J is used instead of I to avoid confusion with

identities).

powers refers to iterative application or composition, in the

(x) = f(f(x)).

For example, one may ask the question of meaningfully interpreting


4

, by definition, we see that if we take the


. Thus our formula holds for

>0 by induction.


. Thus, if we replace the


,

a positive real number. This is the basis of the Riemann-

, the most often used differintegral (operator combining

that studies the possibility

differential operator

to avoid confusion with

refers to iterative application or composition, in the


as a square root of the differentiation

expression for some operator that when applied

effect as differentiation. More generally, one

for real-number values of

usual power of n-fold differentiation is recovered for

when n < 0.

There are various reasons for looking at this question. One is that in this way the

semigroup of powers Dn

semigroup (one hopes) with parameter

semigroups are prevalent in mathematics, and have an interesting theory. Notice here

that fraction is then a misnomer for the exponent, since it need not be

term fractional calculus has become traditional.

Fractional differential equations

through the application of fractional calculus.

Nature of the fractional derivative

An important point is that the fractional derivative at a point

when a is an integer; in non

at x of a function f depends only on the graph of

power derivatives certainly do. Therefore it is expected that the theory involves som

sort of boundary conditions

a metaphor, the fractional derivative requires some

As far as the existence of such a theory is concerned, the foundations of the subject

were laid by Liouville in a paper from 1832. The fractional derivative of a function to

order a is often now defined by means of the

Heuristics

A fairly natural question to ask is whether there exists an operator

derivative, such that

It turns out that there is such an operator, and indeed for any

operator P such that


of the differentiation operator (an operator half iterate

expression for some operator that when applied twice to a function will have the same

More generally, one can look at the question of defining

number values of a in such a way that when a takes an integer

fold differentiation is recovered for n > 0, and the −

There are various reasons for looking at this question. One is that in this way the n in the discrete variable n is seen inside

semigroup (one hopes) with parameter a which is a real number. Continuous


is then a misnomer for the exponent, since it need not be rational

has become traditional.

Fractional differential equations are a generalization of differential equations

through the application of fractional calculus.

Nature of the fractional derivative

An important point is that the fractional derivative at a point x is a local property

is an integer; in non-integer cases we cannot say that the fractional derivative

depends only on the graph of f very near x, in the way that integer

power derivatives certainly do. Therefore it is expected that the theory involves som

boundary conditions, involving information on the function further out. To use

a metaphor, the fractional derivative requires some peripheral vision.


in a paper from 1832. The fractional derivative of a function to

is often now defined by means of the Fourier or Mellin integral transforms.

A fairly natural question to ask is whether there exists an operator

.

It turns out that there is such an operator, and indeed for any a > 0, there exists an


5

half iterate), i.e., an

to a function will have the same

can look at the question of defining

integer value n, the

−nth

power of J

There are various reasons for looking at this question. One is that in this way the

is seen inside a continuous

which is a real number. Continuous


rational, but the

differential equations

local property only

integer cases we cannot say that the fractional derivative

, in the way that integer-

power derivatives certainly do. Therefore it is expected that the theory involves some

, involving information on the function further out. To use


in a paper from 1832. The fractional derivative of a function to

integral transforms.[1]

A fairly natural question to ask is whether there exists an operator H, or half-

, there exists an


or to put it another way, the definition of

To delve into a little detail, start with the

factorials to non-integer values. This is defined such that

Assuming a function f(x)

0 to x. Call this

Repeating this process gives

and this can be extended arbitrarily.

The Cauchy formula for repeated integration

leads to a straightforward way to a generalization for real

Simply using the Gamma function to remove the discrete nature of the factorial

function (recalling that

gives us a natural candidate for fractional applications o

This is in fact a well-defined operator.

It can be shown that the J


,

or to put it another way, the definition of can be extended to all real values of

To delve into a little detail, start with the Gamma function , which extends

integer values. This is defined such that

.

that is defined where x > 0, form the definite integral from

.

Repeating this process gives

and this can be extended arbitrarily.

Cauchy formula for repeated integration, namely

leads to a straightforward way to a generalization for real n.


, or equivalently

gives us a natural candidate for fractional applications of the integral operator.

defined operator.

operator satisfies


6

can be extended to all real values of n.

, which extends

, form the definite integral from

,


)

f the integral operator.

,


this relationship is called the semigroup property of fractional

Unfortunately the comparable process for the derivative operator

complex, but it can be shown that

Fractional derivative of a simple function

The half derivative (purple curve) of the function

first derivative (red curve).

Let us assume that f(x) is a monomial of the form

The first derivative is as usual

Repeating this gives the more general result that

Which, after replacing the


this relationship is called the semigroup property of fractional differintegral

Unfortunately the comparable process for the derivative operator D is significantly more

complex, but it can be shown that D is neither commutative, nor additive in general.

Fractional derivative of a simple function

The half derivative (purple curve) of the function f(x) = x (blue curve) together with the

is a monomial of the form

The first derivative is as usual

Repeating this gives the more general result that

Which, after replacing the factorials with the Gamma function, leads us to


7

differintegral operators.

is significantly more

in general.

(blue curve) together with the

, leads us to


So, for example, the half-derivative of

Repeating this process yields

which is indeed the expected result of

This extension of the above di

powers. For example, the

2nd derivative. Also notice that setting negative values for

Laplace transform

We can also come at the question via the

and

etc., we assert

For example


derivative of x is

Repeating this process yields

which is indeed the expected result of

This extension of the above differential operator need not be constrained only to real

powers. For example, the (1 + i)th derivative of the (1 − i)th derivative yields the

2nd derivative. Also notice that setting negative values for a yields integrals.

Laplace transform

e at the question via the Laplace transform. Noting that

.


8

fferential operator need not be constrained only to real

th derivative yields the

yields integrals.

. Noting that


as expected. Indeed, given the

(and short handing p(x) =

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they

solving fractional differential equations.

Riemann–Liouville integral

The classical form of fractional calculus is given by the

essentially what has been described above. The theory for

therefore including the 'boundary condition' of repeating after a period, is the

differintegral. It is defined on

coefficient to vanish (so, applies to functions on the

By contrast the Grünwald–

integral.

Functional Calculus

In the context of functional analysis

studied in the functional calculus

operators also allows one to consider powers of

of singular integral operators

dimensions is called the theory of

contemporary theories available, within which

See also Erdélyi-Kober operator

Applications

Fractional Conservation of Mass

As described by Wheatcraft and Meerschaert (2008

mass equation is needed when the control volume is not large enough compared to the


as expected. Indeed, given the convolution rule

) = xα − 1

for clarity) we find that

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they are oft

solving fractional differential equations.

Liouville integral

The classical form of fractional calculus is given by the Riemann–Liouville integral

essentially what has been described above. The theory for periodic functions

therefore including the 'boundary condition' of repeating after a period, is the

. It is defined on Fourier series, and requires the constant Fourier

coefficient to vanish (so, applies to functions on the unit circle integrating to 0).

–Letnikov derivative starts with the derivative instead of the

alculus

functional analysis, functions f(D) more general than powers are

functional calculus of spectral theory. The theory of pseudo

o allows one to consider powers of D. The operators arising are examples

singular integral operators; and the generalisation of the classical theory to

dimensions is called the theory of Riesz potentials. So there are a number of

contemporary theories available, within which fractional calculus can be discussed.

Kober operator, important in special function theory.

Fractional Conservation of Mass

As described by Wheatcraft and Meerschaert (2008)[2]

, a fractional conservation of



9

often useful for

Liouville integral,

periodic functions,

therefore including the 'boundary condition' of repeating after a period, is the Weyl

, and requires the constant Fourier

integrating to 0).

starts with the derivative instead of the

more general than powers are

pseudo-differential

. The operators arising are examples

; and the generalisation of the classical theory to higher

. So there are a number of

can be discussed.

a fractional conservation of



scale of heterogeneity and when the flux within the control volume is non

the referenced paper, the fractional conservation of mass equation for fluid flow is:

Fractional Advection Dispersion Equation

This equation has been shown useful for modeling contaminant flow in heterogenous

porous media [3][4][5]

WKB approximation

for the semiclassical approximation in one dimensional spatial system (x,t) the inverse

of the potential V − 1

(x) inside the Hamiltonian

integral of the density of states

.

Further Reading: Differintegral

References

• Fractional Integrals and Derivatives: Theory and Applications

A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor &

ISBN 2-88124-864-0

• Theory and Applications of Fractional Differential Equations

Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Els

2006.ISBN0-444-51832

(http://www.elsevier.com/wps/find/b

escription)

• An Introduction to the Fractional Calculus and Fractional Differential Equations

Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John

Wiley & Sons; 1 edition (May 19, 1

• The Fractional Calculus; Theory and Applications of Differentiation and Integration

to Arbitrary Order (Mathematics in Science and Engineering, V)

Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974).

ISBN 0-12-525550-0

• Fractional Differential Equations. An Introduction to Fractional Derivatives,

Fractional Differential Equations, Some Methods of Their Solution and Some of

Their Applications., (Mathematics in Science and Engineering, vol. 198), by Igor

Podlubny. Hardcover. Publisher: Academic Press; (October 1998)

558840-2

• Fractals and quantum mechanics

(http://link.aip.org/link/?CHAOEH/10/780/1

• Fractals and Fractional Calculus in Continuum Mechanics

F. Mainardi (Editor). Paperback: 348 pages. Publisher: Springer

(January 1998). ISBN 3


scale of heterogeneity and when the flux within the control volume is non

per, the fractional conservation of mass equation for fluid flow is:

Fractional Advection Dispersion Equation


or the semiclassical approximation in one dimensional spatial system (x,t) the inverse

inside the Hamiltonian H = p2 + V(x) is given by the half

integral of the density of states taken in units where

Differintegral, Fractional dynamics, Fractional Fourier

Fractional Integrals and Derivatives: Theory and Applications, by Samko, S.; Kilbas,

A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor &

0

Theory and Applications of Fractional Differential Equations, by Kilbas, A. A.;

Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Els

51832-0

http://www.elsevier.com/wps/find/bookdescription.cws_home/707212/description#d

An Introduction to the Fractional Calculus and Fractional Differential Equations


Wiley & Sons; 1 edition (May 19, 1993). ISBN 0-471-58884-9

The Fractional Calculus; Theory and Applications of Differentiation and Integration

to Arbitrary Order (Mathematics in Science and Engineering, V)


0

Fractional Differential Equations. An Introduction to Fractional Derivatives,

ractional Differential Equations, Some Methods of Their Solution and Some of

, (Mathematics in Science and Engineering, vol. 198), by Igor

Podlubny. Hardcover. Publisher: Academic Press; (October 1998)

Fractals and quantum mechanics, by N. Laskin. Chaos Vol.10, pp.780

http://link.aip.org/link/?CHAOEH/10/780/1)

Fractional Calculus in Continuum Mechanics, by A. Carpinteri (Editor),

F. Mainardi (Editor). Paperback: 348 pages. Publisher: Springer

ISBN 3-211-82913-X


10

scale of heterogeneity and when the flux within the control volume is non-linear. In

per, the fractional conservation of mass equation for fluid flow is:


or the semiclassical approximation in one dimensional spatial system (x,t) the inverse

is given by the half-

taken in units where

Fractional Fourier Transform.

, by Samko, S.; Kilbas,

A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor & Francis Books.

, by Kilbas, A. A.;

Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Elsevier, February

ookdescription.cws_home/707212/description#d

An Introduction to the Fractional Calculus and Fractional Differential Equations, by


The Fractional Calculus; Theory and Applications of Differentiation and Integration

to Arbitrary Order (Mathematics in Science and Engineering, V), by Keith B.


Fractional Differential Equations. An Introduction to Fractional Derivatives,

ractional Differential Equations, Some Methods of Their Solution and Some of

, (Mathematics in Science and Engineering, vol. 198), by Igor

Podlubny. Hardcover. Publisher: Academic Press; (October 1998) ISBN 0-12-

, by N. Laskin. Chaos Vol.10, pp.780-790 (2000).

, by A. Carpinteri (Editor),

F. Mainardi (Editor). Paperback: 348 pages. Publisher: Springer-Verlag Telos;


11

• Physics of Fractal Operators, by Bruce J. West, Mauro Bologna, Paolo Grigolini.

Hardcover: 368 pages. Publisher: Springer Verlag; (January 14, 2003). ISBN 0-387-

95554-2

• Fractional Calculus and the Taylor-Riemann Series, Rose-Hulman Undergrad. J.

Math. Vol.6(1) (2005).

• Operator of fractional derivative in the complex plane, by Petr Zavada,

Commun.Math.Phys.192, pp. 261-285,1998. doi:10.1007/s002200050299 (available

online or as the arXiv preprint)

• Relativistic wave equations with fractional derivatives and pseudodifferential

operators, by Petr Zavada, Journal of Applied Mathematics, vol. 2, no. 4, pp. 163-

197, 2002. doi:10.1155/S1110757X02110102 (available online or as the arXiv

preprint)

• Fractional differentiation by neocortical pyramidal neurons, by Brian N Lundstrom,

Matthew H Higgs, William J Spain & Adrienne L Fairhall, Nature Neuroscience, vol.

11 (11), pp. 1335 - 1342, 2008. doi:10.1038/nn.2212 (abstract).


12

Metamathematics in

Teaching and Research

Abstract: Corresponding to the three meanings of ‘meta’, three types of Metamathematics

(MM) are designed, herein captioned as MM-I, MM-II, MM-III. The topics for illustration

are chosen from prescribed syllabi, but the MM of the topics is outside the syllabus. It is

argued for the teaching to be effective, the discussion on the patterns of MM-II is desirable

since there are only four patterns of a mathematical discourse: Order out of order, Order out

of chaos, Chaos out of order and Chaos out of chaos. They are referred respectively as

Aesthetic, Significant, Exciting, and Ignorable mathematics. How aesthetic mathematics is

responsible for the invention of famous algebraic structures is demonstrated. The peer

recognition of a research paper depends on the MM-II aspects (viz Introduction and

Conclusion) of the successful calculations of the author. It is shown that MM-I does not exist

and MM-III runs the risk of producing unreadable master pieces.

1. Introduction

The three meanings of the prefix ‘meta’are the prepositions: ‘beyond’, ‘after’, and

‘behind’. Accordingly MM-I is devoted to the question of whether there is any

branch of knowledge, which is beyond mathematics. Main emphasis is on the role

of MM-II in teaching and research. Logical symbolism is the essence of MM-III.

2. Metamathematics – I

We consider the analogous topics metaphysics and metaengineering with examples,

where ‘meta’ refers to ‘beyond’. We explain the etymology of the word

‘engineering’ as ‘profiteering using geniuses. From the clause (in engineering) “A

rocket moves at the rate of 6000 miles per hour”, if we drop the words ‘A rocket

moves at the rate of’ the resulting phrase is “6000 miles per hour” which is a common

expression in physics. Consequently we observe that

metaengineering is physics. (1)

Invited Lecture by

Prof. L. Radha Krishna,

U.G.C. Centre for Advanced

Studies, Department of

Mathematics, Bangalore

University, Bangalore-

560001.

Email: [email protected]

Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,




13

This supports the view that ‘engineering is the realization of physics’. From the

phrase “6000 miles per hour”, if we drop the words ‘miles per hour’ (from physics)

we are left with the number “6000”. We generalize this situation as

metaphysics is mathematics. (2)

From the sentences (1), (2) we infer that mathematics is meta-metaengineering. If

we delete “6000”, then there exists nothing. We then claim that Metamathematics-I

does not exist! This supports the view that mathematics is the ultimate subject in

abstraction; it explains why UNESCO chose the caption WORLD MATHEMATICS

YEAR 2000, to celebrate the new millennium (2001-3000).

3. Metamathematics – II

a. MM-II for teaching

We develop the theme ‘after mathematics’ in the sense how to appreciate and enjoy

mathematical results ‘after completing the calculations’ or ‘after proving a theorem’

in a class room. Usually teachers are in a hurry to complete the syllabus and ignore

the introspection of the relevance and charm of their class room performance. It is

here that metamathematics helps the teacher to grow into an inspiring professional.

The four patterns of a mathematical discourse [Davis and Hersh:

In the context of mathematics, ‘Order’ means ‘predictability’ and ‘arrangement’; and

‘chaos’ means ‘confusion’, ‘mixedupness’, and ‘randomness’ (apparent).

i. AESTHETIC MATHEMATICS or ORDER out of

ORDER PATTERN (Recall teaching)

A simple example: From the equation of a circle

x2 + y

2 = 1 (ORDER)

we deduce the equation of the tangent to the circle at (x1, y1) as

xx1 + yy1 = 1 (ORDER OUT OF

ORDER).

If A, B are square matrices, then

(AB)-1

= B-1

A-1

(Oder [r.h.s.] out of order

[l.h.s]).

Algebra abounds with ‘Order out of order’ circumstances. The axioms of group are

an instance of order (associativity, existence of unit element, existence of inverse). If

we combine them with another order (the axioms of a field) —in a certain way— we


14

get the axioms of yet another order — the famous algebraic structure vector

space. We present this situation as a MM-II:

GROUP (FIELD) = VECTOR SPACE.

Thus ‘order out of order’ opens the flood gates of research in algebra. It follows that

MM-II is an effective introspection for finding the secrets of the following structures

by analogy:

GROUP (RING) = MODULE

RING (RING) = ALGEBRA

TOPOLOGICAL VECTOR SPACE (TOPOLOGICAL VECTOR SPACE) = FIBER

BUNDLE.

ii. EXCITING MATHEMATICS or CHAOS out of ORDER PATTERN (Recall politics)

We consider the following results:

√2 = 1. 41421356237…

e = 2.7182818828…

π = 3.141592653… .

On the left hand side of the three equalities there is order, simple to

present. However on the right hand sides there is apparent chaos. This pattern is

‘chaos out of order’. Such results are rare and exciting.

iii. IGNORABLE MATHEMATICS or CHAOS out of CHAOS PATTERN(Recall bull in

garbage)

Guinness book of world records in mathematics (1980) mentions the following

instances:

[1] Shakuntala Devi and computer were given to multiply two 14 digit numbers and

Devi gave the product correctly well before the computer could announce the answer:

7686369774870 x 2465099745779 = 18947668177995426462773730.

Here the left hand side and right handed side are both chaotic. No one will try to

remember the result. It is just ignored as it does not improve our knowledge.


15

[2] In 1984 Rajan Mahadevan recited π to 31,940 decimal places on All India

Radio. It is an example of tremendous memory power of the brain. This feat is

ignored since it has no utility beyond 5 digits.

iv. SIGNIFICANT MATHEMATICS or ORDER out ot CHAOS PATTERN (Recall

research)

All theorems that are taught in the classes belong to this variety. We treasure them

and repeatedly teach them for generations. We mention three theorems and how they

bring in order out of obviously chaotic situations.

[1] Pythogoras theorem (4 century B.C.): In ANY triangle ABC, we have

c2 = a

2 + b

2 – 2ab cos C. (3.1)

The word ‘any’ suggests the chaos —the uncountable infinity of choices of a, b, c.

The order is represented by the one formula (3.1). The teachers to be effective

should be able to describe the chaos before the discovery/ invention of the theorem

and the one pleasant order due to the theorem, after completing the proof of the

theorem. This famous theorem has now 370 proofs when C = 90.

[2] Cantor’s theorem (1874): If ℵ0 and ℵ1 denote the countable and uncountable

infinities respectively then = the cardinality (countable infinity) of the set of natural

numbers/ integers.

[3] Gelfond’s theorem (1934): If α is any algebraic number and is any irrational

number, then = where is a transcendental number.

Only two transcendental numbers e, π are well-known, but this formula gives the trick

for manufacturing an uncountable infinity of transcendental numbers!

b. MM-II in Reporting of Research

In a research paper hard core successful calculations are reported in the main body of

the paper. But metamathematical (after calculations) aspects of the paper have to be

inserted in the sections titled

Introduction and Conclusion,

which will be important for reviewers, critics and other research workers. Motivation

for the work, history and philosophy of the topic of research (the ‘Ph’ factor in

‘Ph.D.’ degree) special notation (if any), strategy for solution, type of abstraction,

type of generalization are included in ‘Introduction’. An appreciation of the


16

successful calculations (vide the four types of mathematics in Sec. 3), plausible future

applications, possible modifications of the premise and the ism (perspective) of the

paper are included in ‘Conclusion’. For a new research scholar the drafting of

Introduction and conclusion are difficult. The research guide comes to his

rescue. However a concerted effort in this direction will help in the increase of

Science Citation Index of the paper.

4. METAMATHEMATICS III

Consider the logical statements:

i] Complex analysis

ii] Real analysis x y x ≥ 0 y2 = x.

Since no one thinks in terms of the logical symbolism becomes a cumbersome

code and creates unpleasantness to the mind through thought

blockades. Replacement of the symbols of logic by words promotes

communication of ideas. Then i] means “Every complex number is the

product of a non-negative real number and a complex number of modulus

one”, and

ii] means “Every non-negative real number has a square root”. The book

Principia Mathematica by Russell and Whitehead is considered as an

“outstanding example of unreadable masterpiece’

because it is filled with logical symbols. After 362 pages of abstract notation

and no words, they prove 1 + 1 = 2! Thus MM-III runs the risk of

incomprehensibility and jargon.

5. Conclusion

A popular quotation is “Beauty is not in the holder, but in the beholder”.

The most beautiful equation in mathematics is

+ 1 = 0 (5.1)

and the beauty lies in the beholding mathematician’s knowledge of the

transcendental numbers


17

If we identify the symbols with nationalities of their inventors in (5.1), we have the

funny (nonmathematical) enjoyable relation

Swiss Italians Greeks + Arabs = Hindus!

The most powerful equation in mathematics is

E = mc2

since Einstein has revised our universe with just that equation, inaugurating the

mathematics of light, the tool of self orthogonal vector fields, the release of atomic

energy and the mathematical study of Microcosmos and Macrocosmos, where high

speeds comparable to the velocity of light exist. It is no wonder that the

mathematician Einstein has been elected as the person of the second millennium

(1001-2000) by TIME.

We have thus seen that “Charm is not in Mathematics (calculations) per se (in itself)

but in Metamathematics” as evidenced by MM-I, MM-II and MM- III for

inspired teaching as well as enlightened reporting of research, as described in

Sections 2, 3, 4.

4. References

Davis, P.J. and Hersh, R., The Mathematical Experience, Cambridge:

Birkhauser, 1981

Radhakrishna, L., Write Mathematics Right: Principles of Professional

Presentation, Narosa, Delhi (to appear in 2011).

Acknowledgment

The author wishes to thank Prof. B.Satyanaraayana and Dr Eswaraiah Setty for

inviting him to deliver this talk in the UGC sponsored seminar on Present Trends in

Mathematics and its Applications on 11 and 12 November 2010 at Smt. G.S. College,

Jaggaaiahpet.


18

Computational Fluid

Dynamics-A Study of

Convection in Rectangular

Cavity by Finite Element

Technique

1. Introduction

One of the powerful methods to analyse and understand any real phenomenon is to

formulate the best suited mathematical model based on certain hypothesis like

continuum hypothesis etc., which can be solved making use of either exact methods

or possible approximate methods. The mathematical may consists of an ordinary or

partial differential equation or an integral equation or an integro- differential equation

depending on the nature of the phenomenon and the physical conditions associated

with it. The equation together with the prescribed conditions refers to a boundary

value problem.

Such a boundary value problem may or may not be exactly solvable using the

available methods. The solvability of the boundary value problem depends on the

nature of the equation as well as the shape of the boundaries involved. In few cases

the governing equation is a linear differential equation and the boundaries involved

are smooth known shapes and hence can be solved exactly by standard methods.

However, many real phenomena are governed by non-linear differential equations

which are not amenable for exact solutions. Even if the equations are linear the

boundaries are complicated and hence unsolvable exactly. Under these circumstances,

it is desirable to evolve techniques which yield atleast reasonable Approximate

solutions for the boundary value problem. This lead to adoption of numerical

techniques which have gained vital importance in the recent times as a core subject in

all applied sciences.

The Finite element method overcomes the defect of the traditional variatonal method

in which we approximate the solution keeping the entire domain in to consideration.

The approximate solution obtained in such a manner may lead to small errors in

Invited Lecture by

Prof. D.R.V. Prasada Rao,

Department of Mathematics,

Sri Krishnadevaraya

University, Anantapur-515

003, Andhra Pradesh, India.





19

certain partitions of the domain while gives rise to large error in the remain partitions.

In order to minimize the error and obtain approximation solution valid in each

element of the whole domain it is desirable to sub-divide the domain into finite

number of elements and develop the approximation solution to each of such elements

and assemble these solution using the inter element continuity and equilibrium

requirements as well as the boundary conditions imposed in the problem.

The finite element method was initially developed as an adhoc engineering procedure

for constructing matrix solutions to stress and displacement calculations in structural

analysis. Very few fluid dynamic problems can be expressed in a variatonal form.

Consequently most of the finite applications in fluid dynamics have used the Galerkin

finite element formulation. A traditional engineering interpretation of finite element

method is given by Zienkiewicz and its applications to fluid mechanics are treated by

Thomasset and Baker. A mathematical perspective of this method is provided by

Strang and Fix, Oden and Reddy, Mitchell and Wait. In the recent times this method

and its applications to problems of mechanics have been quite popularized by Reddy,

Jain et al and Fletcher. The Galerkin finite element method has two important

features. Firstly, the approximate solution is written directly in terms of the nodal

unknowns.Secondly the approximating function or the shape functions are chosen

exclusively from low order piecewise polynomials restricted to contiguous elements.

Steps involved in the finite element Analysis

1. Discretization of the given domain into a collection pre-selected sub-domains (i.e.

finite elements under discretization the following steps are involved.

(a) Construct the finite element mesh of pre-selected elements.

(b) Numbering the nodes and elements

(c) Generate the geometric properties needed for the problem

2. Derivation of element equations for all typical element in the mesh we have

(a) Construct the variational formulation of the given b.v.p.

(b) Select the variational method of approximation using proper

approximation function and

(c) Obtain the element equation in the matrix viz stiffness matrix involving

arbitrary co-efficients involved in the approximate solution

3. Assembling of element equations to obtain the global matrix equation.

This assembling involves


20

(a) Identify the equilibrium conditions among the secondary variables.

(b) Identify the interlement continuity conditions among the primary variables

by relating element to global nodes.

(c) Assembling the element equations using the above steps in terms of the

global nodal values.

4. Imposition of the boundary conditions of the problem

5. Inverting the global matrix equation.

6. Post processing the solution and discuss the error analysis

During the last few decades the MHD heat transfer has been developed in few of its

extensive applications in Geophysics and Astrophysics. The problem of Steady flow

of mercury in pipes across a magnetic field was first investigated both theoretically

and experimentally by Hartmann and Lazarus. Further investigations

In this paper we discuss the convective heat transfer steady flow of a conducting fluid

through a rectangular vertical duct under a transverse magnetic field. The equations

for the velocity and induced magnetic field are suitably coupled. The walls of the duct

normal to the direction of the applied magnetic field are thermally insulated and those

parallel to the field are maintained at constant temperature.

Heat generation in a porous media due to the presence of temperature dependent heat

sources has number of applications related to the development of energy resources. It

is also important in engineering processes pertaining to flows in which a fluid

supports an exothermic chemical or nuclear reaction. Proposal of disposing the

radioactive waste material by burying in the ground or in deep ocean sediment is

another problem where heat generation in porous medium occurs.

The investigation of heat transfer in enclosures containing porous media began with

the experimental work of Verschooor and Greebler (27). Verschooor and Greebler

(27) were followed by several other investigators interested in porous media heat

transfer in rectangular enclosures. In particular Bankwall(2) has published a great deal

of practical work concerning heat transfer by natural convection in rectangular

enclosures completely filled with porous media. Burns, Cheng (9) described a porous

medium heat transfer flow in a rectangular geometry.

Recently Badruddin et al(1) have investigated the radiation effect and viscous

dissipation on convective heat transfer in porous cavity.

In this paper we investigate the radiation effect on the free convective flow and heat

transfer in as viscous dissipative fluid in a saturated porous medium with temperature


21

gradient dependent heat sources, enclosed in a rectangular duct. The Darcy model is

used for the momentum transport. Making use of the incompressibility non-

dimensional momentum and energy equations are derived in terms of the stream

function and temperature. The Galerkin finite element method with triangular

elements is employed to obtain iterative solution of the said coupled non-linear

equations. The temperature field at different horizontal and vertical levels is obtained

and their behavior is investigated for variations in the governing parameters. The local

rate of heat transfer along the side wall is obtained and its variations for different

parameters are discussed.

2. Formulation:

We consider the mixed convection flow of a viscous incompressible fluid in a

saturated porous medium confined in the rectangular duct (Fig. 1) whose base length

is a and height b. The heat flux on the base and top walls is maintained constant. The

Cartesian coordinate system θ(x,y) is chosen with origin on the central axis of the

duct and its base parallel to x-axis.

We assume that

i) The convective fluid and the porous medium are everywhere in local

thermodynamic equilibrium.

ii) There is no phase change of the fluid in the medium.

iii) The properties of the fluid and of the porous medium are homogeneous

and isotrophic.

iv) The porous medium is assumed to be closely packed so that Darcy’s

momentum law is adequate in the porous medium.

v) The Boussinesq approximation is applicable.

Under these assumption the governing equations are given by


22

0=′∂′∂

+′∂′∂

y

v

x

u (2.1)

′∂′∂

−=′x

p

µ

ku (2.2)

′+′∂′∂

−=′ gρy

p

µ

kv (2.3)

x

qvu

Ky

TQ

y

T

x

TK

y

Tv

x

Tuc r

p ∂∂

−+

+∂∂

+

′∂′∂

+′∂′∂

=

′∂′∂′+

′∂′∂′ )( 22

2

2

2

2

1

22 µρσ (2.4)

2

; )(1 000ch TT

TTT+

=−′−=′ βρρ (2.5)

where u′ and v′ are Darcy velocities along θ(x, y) direction. T′, p′ and g′ are the

temperature, pressure and acceleration due to gravity, Tc and Th are the temperature

on the cold and warm side walls respectively. ρ′, µ, ν, and β are the density,

coefficients of viscosity, kinematic viscosity and thermal expansion of he fluid, k is

the permeability of the porous medium, K1 is the thermal conductivity, Cp is the

specific heat at constant pressure and Q is the strength of the heat source. The

boundary conditions are

u′ = v′ = 0 on the boundary of the duct

T′ = Tc on the side wall to the right

T′ = Th on the side wall to the left (2.6)

0=∂

′∂y

T on the top ( y = 0) and bottom

0== vu walls(y = 0)which are insulated.

We now introduce the following non-dimensional variables

x′ = ax; ; y′ = by ; c = b/a

u′ = (ν/a) u ; v′ = (ν/a)v ; p′ = (ν2ρ/a2)p

T′ = T0 + θ (Th – Tc) (2.7).

Invoking Rosseland approximation for radiation


23

qr = x

T

R ∂∂

4'*

3

4

βσ

Expanding T4 in Taylor’s series about Te and neglecting higher order terms

T4 ≈ 4T 3

eT - 3T 4eT

The governing equations in the non-dimensional form are

x

p

a

Ku

∂∂

−=2

(2.8)

222

)(

v

TTkag

v

kag

y

p

a

kv

ch θβ −+−

∂∂

−= (2.9)

( )22

2

2

2

2

)(3

41 vuE

yyx

N

yv

xuP C ++

∂∂

∂∂

+∂∂

+=

∂∂

+∂∂ θαθθθθ

(2.10)

In view of the equation of continuity we introduce the stream function ψ as

xv

yu

∂∂

−=∂∂

=ψψ

; (2.11)

Eliminating p from the equation (2.8) and (2.9) and making use of (2.11) the

equations in terms of ψ and θ are

xRa

∂∂

−=∇θψ2 (2.12)

∂∂

+

∂∂

+∂∂

+∂∂

+∂∂

+

+=

∂∂

∂∂

−∂∂

∂∂ 22

2

2

2

2

)(3

41

3

41

xyE

yyx

NN

yxxyP C

ψψθαθθθψθψ (2.11)

where

2

3)(

v

aTTgG ch −

=β

(Grashof number)

P = µ cp / K1 (Prandtl number)

α = Qaz/K1 (Heat source parameter)

( )

2ν

β KaTTgRa

cg −= (Rayleigh Number)

32

1

:4

3

e

R

T

KN

σβ

= (Radiation parameter)


24

3. Finite Element Analysis and Solution of the Problem:

The region is divided into a finite number of three node triangular elements, in each of

which the element equation is derived using Galerkin weighted residual method. In

each element fi the approximate solution for an unknown f in the variational

formulation is expressed as a linear combination of shape function. ( ) ,3,2,1=kNi

k

which are linear polynomials in x and y. This approximate solution of the unknown f

coincides with actual values at each node of the element. The variational formulation

results in a 3 x 3 matrix equation (stiffness matrix) for the unknown local nodal values

of the given element. These stiffness matrices are assembled in terms of global nodal

values using inter element continuity and boundary conditions resulting in global

matrix equation.

In each case there are r distinct global nodes in the finite element domain and fp (p =

1,2,……r) is the global nodal values of any unknown f defined over the domain then

∑∑==

Φ=r

p

p

i

ff1

i

p

3

1

,

Fig (ii)

where the first summation denotes summation over s elements and the second one

represents summation over the independent global nodes and


25

,i

N

i

p N=Φ if p is one of the local nodes say k of the element ei

= 0, otherwise.

fp’ s are determined from the global matrix equation. Based on these lines we now

make a finite element analysis of the given problem governed by (2.12) and (2.13)

subjected to the conditions (2.14) – (2.16).

Let ψi and θi

be the approximate values of ψ and θ in an element θi.

iiiiiiiNNN 332211 ψψψψ ++= (3.1)

iiiiiiiNNN 332211 θθθθ ++= (3.2)

Substituting the approximate value ψi and θi

for ψ and θ respectively in (2.13), the

error

∂∂

+

∂∂

+∂∂

+

∂∂

∂∂

−∂∂

∂∂

−∂∂

+∂∂

+=22

2

2

2

2

1 3

41

xyE

yyxxyp

yx

NE C

iiiiiii ψψθαθψθψθθ

Under Galerkin method this error is made orthogonal over the domain of ei to the

respective shape functions (weight functions) where

0 1 =Ω i

i

k

i

i dNEe ς

Ω

∂∂

+

∂∂

++

∂∂

∂∂

−∂

∂∂

∂−

∂∂

+∂∂

+∫ dxy

Eyxxy

pyx

NN C

iiiiizizik

s

ie

22

22

3

41

ψψαθθψθψθθ

(3.3)

Using Green’s theorem we reduce the surface integral (3.3) without affecting ψ terms

and obtain

Ω

∂∂

+

∂∂

+−

∂∂

∂∂

−∂

∂∂

∂−

∂

∂

∂

∂+

∂

∂

∂

∂

+∫ dxy

Eyxxy

Npyy

N

xx

NNN C

iiii

ki

iik

iiki

ks

ie

22

3

41

ψψαθθψθψθθ

iy

i

x

iik

s

idn

yn

xN Γ

∂

∂+

∂

∂∫Γ

θθ

(3.4)

where ΓI is the boundary of ei.

Substituting L.H.S. of (3.1) and (3.2) for ψi and θi

in (3.4) we get

Ω

∂∂

∂∂

−∂∂

∂∂

−∂∂

∂∂

+∂∂

∂∂

+∫ ∑∑ dy

N

x

N

x

N

y

NsP

y

N

y

N

x

N

x

NNe

i

L

i

m

i

L

i

mi

m

i

k

i

L

i

L

i

ks

e

i

Li

3

411 ψ


26

∂∂

+

∂∂

+Ω− ∫∑22

xy

EdNsN Cilk

i ψψθα

i

kiy

i

x

i

s

e

i

k

s

i Qdny

nx

Ni

=Γ

∂∂

+∂∂

∫Γ=θθ

(l, m, k = 1,2,3) (3.5)

where

i

k

i

k

i

k

i

k

i

k QQQQQ ,321 ++= ’s being the values of i

kQ on the sides s = (1,2,3) of the

element ei. The sign of i

kQ ’s depends on the direction of the outward normal w.r.t the

element.

Choosing different i

kN ’s as weight functions and following the same procedure we

obtain matrix equations for three unknowns (i

pQ ) viz.,

)())((i

k

i

p

i

p Qa =θ (3.6)

where )(i

pka is a 3 x 3 matrix, )(),(i

k

i

p Qθ are column matrices.

Repeating the above process with each of s elements, we obtain sets of such matrix

equations. Introducing the global coordinates and global values for i

pθ and making use

of inter element continuity and boundary conditions relevant to the problem the above

stiffness matrices are assembled to obtain a global matrix equation. This global matrix

is r x r square matrix if there are r distinct global nodes in the domain of flow

considered.

Similarly substituting ψi and θi

in (2.12) and defining the error

xE i θψ Ra2

2 −=∇= (3.7)

and following the Galerkin method we obtain

0Ra 22

=Ω

∂∂

+∂∂

+∂∂

∫ dxyx

Niiziz

ik

s

ie

θθθ

(3.8)

Using Green’s theorem (3.8) reduces to

Ω

∂∂

+∂∂

∂∂

+∂∂

∂∂

∫ dx

N

yy

N

xx

N i

ki

ii

k

ii

ks

ei

Ra θψψ

i

i

x

i

k

s

iy

i

x

i

i

k

s

i dnNdny

nx

Ni

Γ+Γ

∂

∂+

∂

∂Γ= ∫Γ θ

ψψ (3.9)


27

In obtaining (3.9) the Green’s theorem is applied w.r.t derivatives of ψ without

affecting θ terms.

Using (3.1) and (3.2) in (3.9) we have

Ω∂∂

+Ω

∂∂

∂∂

+∂∂

∂∂

∫∑∫∑ i

i

Ls

e

i

L

L

i

i

k

i

m

i

m

i

ks

e

i

m

m

dx

Nd

y

N

y

N

x

N

x

N

ii

i

kNRa θψ

i

ki

ii

k

s

iy

i

x

i

i

k

s

i dNdny

nx

Ni

Γ=Ω+Γ

∂

∂+

∂

∂Γ= ∫Γ θ

ψψ (3.10)

In the problem under consideration, for computational purpose, we choose uniform

mesh of 10 triangular element (Fig. ii). The domain has vertices whose global

coordinates are (0,0), (1,0) and (1,c) in the non-dimensional form. Let e1,

e2…..e10 be the ten elements and let θ1, θ2, …..θ10 be the global values of θ and ψ1,

ψ2,……ψ10 be the global values of ψ at the ten global nodes of the domain (Fig. ii).

4. Shapes Hape Functions and Stiffness Matrices

Range functions in ji

n,

; i = element, j = node.

xn 311,1

−= C

yxn

33

2,1−=

C

yn

31

1,2−=

C

yn

31

2,2+−=

C

yxn

331

3,2+−= xn 32

1,3−=

C

yxn

331

2,3−+−=

C

yn

3

3,3=

C

yn

31

1,4−= xn 32

2,4+−=

C

yxn

332

3,4+−= xn 32

1,5−=

C

yxn

331

2,5−+−=

C

yn

3

3,5=

xn 321,6

−= C

yxn

33

2,6−=


28

C

yn

31

3,6+=

C

yn

32

1,7−=

xn 322,7

+−= C

yxn

331

3,7+−=

xn 331,8

−= C

yxn

331

2,8−+−=

C

yn

31

3,9+−=

TEMPERATURE LEFT MATRIX:

=

10 10108

999896

888786

78777672

6867666362

58565553

484443

38363332

2827262322

18171211

00000000

0000000

0000000

000000

00000

000000

0000000

000000

00000

000000

aa

aaa

aaa

aaaa

aaaaa

aaaa

aaa

aaaa

aaaaa

aaaa

A

TEMPERATURE RIGHT MATRIX :

=

10 1

19

18

17

16

15

14

13

12

11

b

b

b

b

b

b

b

b

b

b

B

LEFT MATRIX OF MOMENTUM EQUATION:

−−+−

−−++−−

−−

−−

=

−

−

0190000000

0091100000

00181100000

009010000

0030001000

009000100

009000010

0003000001

2

2

1

12

1

1

c

c

cc

c

cc

c

c

C

yxn

33

2,9−=


29

RIGHT MATRIX OF MOMENTUM EQUATION:

=

10 1

19

18

17

16

15

14

13

12

11

d

d

d

d

d

d

d

d

d

d

D

5. RESULTS:

The temperature distribution is evaluated for different variation of the governing

parameter Ra, N, α, ε. The rectangular duet is of narrow or wide gap according as the

aspect ratio C is less or higher than 0.5. The finite element technique is applied by

using linear triangular element and expressions in the unknown are bi-linear functions

of x & y. These linear expressions involving the global nodal value of the respective

unknowns are determined through the global matrix equations.

The profiles for the temperature distribution (θ) for different values of Ra, N, α & ε

are shown in figures (1 - 10) at different horizontal levels within the depth for Prandtl

number P = 0.71. The actual temperature is greater or lesser than the mean boundary

temperature according as the non-linear temperature is positive or negative. Figs 1-4

depicts the behaviour of temperature at (different levels) (x) different horizontal

levels y = c/3 & y = 2c/3 and vertical levels x = 1/3 & x = 2/3.

Fig 1 represents the variation of the temperature at the horizontal level y = 2c/3. It is

found that the temperature is negative for all values of Rayleigh number Ra. The

temperature which attains the maximum at x = 1/3 decays gradually to attain the

prescribed value at x = 1. At y = c/3 the temperature is also positive these by

indicating that the actual temperature is greater than the ambient temperature. Also

the actual temperature experiences an enhancement with increase in Rayleigh number

Ra. (Fig. 5).

The variation of θ with radiation parameter N is shown in figs 4 & 8. We find that the

actual temperature experiences a enhancement with increase in the radiation

parameter at both the horizontal levels. The variation of θ with heat generating

sources α is shown in figs 3 & 7 at the levels y = 2c/3 & y = c/3. An increase in α

results in a depreciation with α. Figs 2 & 6 represents the variation of θ with viscous

dissipation parameter ε. It is found that an increase ε leads to an depreciation at y =

2c/3 and enhances at y = c/3. It is clearly evident that the values of the actual

temperature at y = c/3 level is much greater than the horizontal y = 2c/3 level. The

variation of temperature at different vertical levels x = 1/3 & x = 2/3 are presented in

figs (9-16) for different values of Ra, N, α & ε. Figs 9 & 10 represents the variation of

θ with Rayleigh number Ra at vertical levels x = 1/3 & x = 2/3. It is found that the

temperature at the vertical level x = 1/3 experiences an enhancement with increase in


30

Ra, while at higher vertical level x = 2/3 the temperature enhances in the region 0 ≤ y

≤ 0.198 for Ra ≥ 100 and depreciates with higher Ra ≥ 110. In the region (0.264,

0.528) the temperature enhances for Ra ≤ 90 and depreciates with Ra ≥ 100, while in

the remaining region adjacent to y = 2c/3 it depreciates with all values of Ra (Fig 13).

The variation of temperature with radiation parameter N at vertical levels is shown in

figs (12&16).It is found that at x = 1/3 level, the temperature enhances with increase

in the radiation parameter N. From fig 16, we notice that at higher vertical level x =

2/3 the temperature.

The variation of temperature with α is shown in figs (11 & 15) at x = 1/3 and x = 2/3.

It is found that the temperature reduces with increase in the strength of heat

generating source at x = 1/3 & 2/3 levels.

Figs 10 & 14, we observe that the temperature at both the vertical levels experiences

an enhancement with increase in the viscous dissipation parameter ε. In general we

notice that the temperature at x = 2/3 is much higher than that at the vertical level x =

1/3.

The rate of heat transfer at three different we segments viz., Nu1, Nu2 & Nu3 are

evaluated for different values of Ra, α, ε & N. The variation of the rate of heat

transfer with respect to the Rayleigh number shows that the rate of heat transfer in the

first & mid level experiences a depreciation with increase in Ra while the rate of heat

transfer in the upper level enhances with increase in Ra. With respect to the variation

of Nu the heat source parameter α we notice that the rate of heat transfer in the lower

segment depreciates with increase in the strength of heat source while in the middle &

upper levels the rate of heat transfer enhances with α.

The variation of Nu with viscous dissipation parameter ε shows that the rate of heat

transfer in the lower level enhances with increase in ε while at the middle & upper

level Nu depreciates with ε ≤ 0.05 & enhances with higher ε ≥ 0. 08.

The variation of Nu with respect to the radiation parameter N exhibits that the rate of

heat transfer at all the three levels experiences a depreciation with increase in the

radiation parameter N. The inclusion of the radiation effect is to depreciate the rate of

heat transfer at all the three levels. We notice that the rate of heat transfer depreciates

as we move along the y = c in the vertical direction.


31

Fig. 1 Temperature θ with y = 2C/3 Fig. 2 θ with EC at y = 2C/3

I II III IV I II III IV

Ra 180 200 300 500 EC 0.001 0.003 0.005 0.008

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.67 0.737 0.804 0.871 0.938

x

θθθθ

I

II

III

IV

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.67 0.737 0.804 0.871 0.938

x

q

I

II

III

IV


32

Fig. 3 θ with α at y = 2C/3 Fig. 4 θ with N at y = 2C/3


α 0 2 4 6 N 0.5 1.5 5 10

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.67 0.737 0.804 0.871 0.938

x

θθθθ

I

II

III

IV

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.67 0.737 0.804 0.871 0.938

x

θθθθ

I

II

III

IV


33

Fig. 5 θ with R at y = C/3 Fig. 6 θ with R at y = C/3


Ra 100 200 300 500 EC 0.001 0.003 0.005 0.008

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.333 0.555 0.777 0.999

x

θθθθ

I

II

III

IV

0.3

0.35

0.4

0.45

0.3

3

0.3

9

0.4

4

0.5

0

0.5

5

0.6

1

0.6

6

0.7

2

0.7

7

0.8

3

0.8

8

0.9

4

0.9

9

x

θθθθ

I

II

III

IV


34

Fig. 7 θ with α at y = C/3 Fig. 8 θ with N at y = C/3


α 0 2 4 6 N 0.5 1.5 5 10

0.25

0.27

0.29

0.31

0.33

0.35

0.37

0.39

0.41

0.430

.3

0.4

0.4

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1.0

x

θθθθ

I

II

III

IV

0.25

0.27

0.29

0.31

0.33

0.35

0.37

0.39

0.3 0.5 0.7 0.9

x

θθθθ

I

II

III

IV


35

Fig. 9 θ with R at x = 1/3 Fig. 10 θ with EC at x = 1/3


Ra 100 200 300 500 EC 0.001 0.003 0.005 0.008

0

0.01

0.02

0.03

0.0 0.1 0.2 0.3

y

θθθθ

i

ii

iii

iv

0.01

0.02

0.03

0 0.1 0.2 0.3

y

θθθθi

ii

iii

iv


36

Fig. 11 θ with α at x = 1/3 Fig. 12 θ with N at x = 1/3


α 0 2 4 6 N 0.5 1.5 5 10

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.0 0.1 0.2 0.3 0.4

y

θθθθ

i

ii

iii

iv

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.0 0.1 0.2 0.3

y

θθθθ

I

II

III

IV


37

Fig. 13 θ with R at x = 2/3 level Fig. 14. θ with EC at x = 2/3 level


Ra 102 2x10

2 3x10

2 5x10

2 EC 0.01 0.03 0.05 0.08

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.133 0.266 0.399 0.532 0.665

y

θθθθ

I

II

III

IV

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.133 0.266 0.399 0.532 0.665

y

θθθθ

I

II

III

IV


38

Fig. 15 θ with α at x = 2/3 level Fig. 16 θ with N at x = 2/3 level


α 0 2 4 6 N 0.5 1.5 5 10

Table.1, Nusselt Number(Nu) at x=1 at different levels, P=0.71

Nu1 1.976226 1.953228 1.9447176 1.98261584

Nu2 1.934462 1.927718 1.9334484 1.99213184

Nu3 1.8926984 1.902208 1.9221792 2.00164788

R 102 2x10

2 3x10

2 4x10

2

Table.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.133 0.266 0.399 0.532 0.665

y

θθθθ

i

ii

iii

iv

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.133 0.266 0.399 0.532 0.665

y

θθθθ

i

ii

iii

iv


39

Nusselt Number(Nu) at x=1 at different levels

P=0.71

Nu1 1.9447176 1.93806 1.9317016 1.9227224

Nu2 1.9334484 1.926156 1.9190424 1.9087096

Nu3 1.9221792 1.9142516 1.9063836 1.8946968

Ec 0.001 0.003 0.005 0.008

Table.3

Nusselt Number(Nu) at x=1, P=0.71

Nu1 1.9325692 1.9447176 1.956812 1.9688452

Nu2 1.8681528 1.9334484 1.998646 2.0638759

Nu3 1.8037363 1.9221792 2.040577 2.1589048

α 0 2 4 6

Table.4

Nusselt Number(Nu) at x=1

P=0.71

Nu1 1.948655 1.944724 1.9449448 1.9493868

Nu2 1.954436 1.933448 1.9125445 1.8961772

Nu3 1.960219 1.922179 1.8801428 1.8429684

N 0.5 1.5 5 10

References:

1. Badruddin,I.A ,Zainal,Z.A, Aswatha Narayana, Seetharamu,K.N : “ Heat transfer in porous

cavity under the influence of radiation and viscous dissipation”,Int.Comm in Heat & Mass

Transfer 33(2006), pp.491-499.

2. Bankwall, C.G. : “Heat transfer in fibrous materials”, Eval.3, pp. 235-243 (1973).

3. Bankwall, C.G. : “Guarded Lot plate apparatus for the investigation of thermal insulations,

Document D5, National British Building Research.


40

4. Brinkman, H.C. : “Calculation of the viscous force exerted by a concentrating fluid on a dense

swan of particles”, Applied Science Research Al, pp.27-34 (1947).

5. Burns P.J., Chow, L.C. and Tien, C.L. : “Convection in a vertical channel filled with porous

insulation”, Int J. Heat and Mass Transfer, Vol. 14, pp.1-105 (1979).

6. Cheng, P : Heat Transfer in Geo-thermal system, “Advances in Heat Transfer”, v.14, pp. 1-

105 (1979).

7. Combarnous, M : “National Convection in porousmedia and Geo=thermal systems”, 6th

Int.

Heat transfer conf; Totonto, p.45-59 (1978).

8. Darcy, P : Less Entaines publiques dele Ville de Dijon, Paris (1956).

9. Desai, C.S. and Avel, J.F. : “Introduction to the finite element method”, A Numerical Method

for Engineering Analysis, Van & Reinhold, New York (1972).

10. Desai, C.S. and Christian, J.T. (eds) : “Seepage in porous media”, Numerical Methods in

Geotechnical Engineering. New York, (forth coming).

11. Desai, C.S. : Overview, trends and projections : Theory and applications of the proceedings of

symposium on application of FEM in Geotechnical Engineering, Desai, C.S. (eds), USA

Engineer Waterways Experiment Station, Vicksburg (1972).

12. Desai, C.S. : Finite element procedures for Seepage analysis using an iso-parametric element,

proceedings of symposium on application of FEM in Geotechnical Engineering, Desai C.S.

(eds), USA Engineer waterways Experimental Station, Vicksburg.

13. Hin-Sun law Masliyah, J.H. and Nandakumar, K : “Effects of Non-uniform Heating on

laminar Mixed convection in Duets”, Heat Transfer, v-109, pp.131-137 (1987).

14. Padmalatha : Ph.D. Thesis on Finite element analysis of laminar convection through a porous

medium in duets, S.K.University, Anantapur, AP, (1997).

15. Rajesh Rajamani, Srinivas, G and Seetharamu, K.N. : International Journal for Numerical

Methods in Fluids, Vol.11, pp.331-339 (1990).

16. Verschoor, J.D. and Greebler, P : Heat Transfer by Gas conduction and radiation in fibrous

insulation, Trans. Am. Soc. Mech. Engrs, pp. 961-968 (1952).

17. Zienkiewicz, D.C., Mayer, P and Cheng, Y.K. : “Solution of an in metric seepage by

finite elements”, Journal of Engineering Mechanics Division, ASCE, 92, No. EMI

(1966).


41

Dimension in Vector

Spaces and Modules

(Dedicated in memory of Prof. Dr A. W. Goldie)

Introduction: It is well known that the dimension of a vector space is defined as the

number of elements in the basis. One can define a basis of a vector space as a maximal

set of linearly independent vectors or a minimal set of vectors, which span the space. The

former, when generalized to modules over rings, becomes the concept of Goldie

Dimension. We discuss some results and examples related to the dimension in Vector

Spaces as well as Modules over Rings.

Section-1: Elementary Concepts in Vector Spaces

1.1 Definition: An Abelian group (V, +) is said to be a vector space over a field F if

there exists a mapping from F × V to V (the image of (α, v) is denoted by αv)

satisfying the following conditions: (i) α(v + w) = αv + αw; (ii) (α + β) v = αv +

βv; (iii) α(βv) = (αβ)v; and (iv) 1.v = v for all α, β ∈ F and v, w ∈ V (here 1 is the

multiplicative identity in F).

1.2 Note: We use F for field. The elements of F are called scalars and the elements of

V are called vectors.

1.3 Remark: Let (v, +) be a vector space over F. Let α ∈ F. Define f : V → V by

f(v) = αv for all v ∈ V. Then (i) f is a group homomorphism (or group

endomorphism). (ii) If α ≠ 0 then f is an isomorphism.

Invited Lecture by

Prof Bhavanari Satyanarayana,

A.P. Scientist Awardee; Fellow,

A.P. Academy of Sciences,


Acharya Nagarjuna University,

Nagarjuna Nagar-522510,

Andhra Pradesh, India

Email:

[email protected]





42

1.4 Examples: (i) Let K be a field and F be a subfield of K. Then K is a vector space

over F.

(ii) Let F be a field. Write V = Fn = (x1, x2,..., xn) / xi ∈ F, 1 ≤ i ≤ n. Define α(x1,

x2,..., xn) = (αx1, αx2, ..., αxn) for α ∈ F and (x1, x2, ..., xn) ∈ Fn. Then F

n is a vector

space. If we take F = R, the field of real numbers, then we conclude that the n-

dimensional Euclidean space Rk is a vector space over R.

(iii) Let F be a field. Consider F[x], the ring of polynomials over F. Write Vn = f(x) /

f(x) ∈ F[x] and deg.(f(x)) ≤ n. Then (Vn, +) is an Abelian group where “+” is the

addition of polynomials. Now for any α ∈ F and f(x) = a0 + a1x + ... + anxn ∈ Vn ,

define α(f(x)) = αa0 + αa1x + ... + αanxn. Then Vn is a vector space over F.

1.5 Definition: Let V be a vector space over F and W ⊆ V. Then W is called a

subspace of V if W is a vector space over F under the same operation. (Equivalently,

W is a subspace if it satisfies the condition: v, w ∈ W, α, β ∈ F ⇒ αv + βw ∈

W).

1.6 To construct a quotient space of V by W: Let V be a vector space and W be a

subspace of V. Define ~ on V as a ~ b iff a – b ∈ W. Clearly this ~ is an

equivalence relation and a + W is the equivalence class containing a ∈ V. Write

V/W = a + W / a ∈ V. Define + on V/W as (a + W) + (b + W) = (a + b) + W.

Since V is an Abelian group we have that (V/W, +) is also an Abelian group. Now to

get vector space structure, let us define the scalar product between α ∈ F and a + W ∈

V/W as α(a + W) = αa + W. Now V/W becomes a vector space over F and it is called

the quotient space of V by W.

Linear Independence and Bases

1.7 Definition: Suppose V is a vector space over F. (i) If vi ∈ V and αi ∈ F for 1

≤ i ≤ n, then α1v1 + α2v2 + ... + αnvn is called a linear combination of v1, v2, ..., vn.

(ii) For S ⊆ V, we write L(S) = α1v1 + α2v2 + ... + αnvn / n ∈ N, vi ∈ S and αi ∈ F

for 1 ≤ i ≤ n = the set of all linear combinations of finite number of elements of

S. This L(S) is called the linear span of S.

1.8 Note: (i) S ⊆ L(S); (ii) L(S) is a subspace of V; (iii) S ⊆ T ⇒ L(S) ⊆ L(T);

(iv) L(S ∪ T) = L(S) + L(T); (v) L(L(S)) = L(S); (vi) L(S) is the smallest subspace

containing S.

1.9 Definitions: (i) The vector space V is said to be finite-dimensional (over F) if

there is a finite subset S in V such that L(S) = V.


43

(ii) If V is a vector space and vi ∈ V for 1 ≤ i ≤ n, then we say that vi, 1 ≤ i ≤ n

are linearly dependent over F if there exists elements ai ∈ F, 1 ≤ i ≤ n, not all of them

equal to zero, such that a1v1 + a2v2 + ... + anvn = 0.

(iii) If the vectors vi, 1 ≤ i ≤ n are not linearly dependent over F, then they are said

to be linearly independent over F.

1.10 Lemma: Let V be a vector space over F. If v1, v2, ..., vn ∈ V are linearly

independent, then every element v in their linear span has a unique representation as v =

λ1v1 + λ2v2 + ... + λnvn with λi ∈ F, 1 ≤ i ≤ n.

1.11 Corollary: Let vi ∈ V, 1 ≤ i ≤ n and W = L(vi / 1 ≤ i ≤ n). If v1, v2, ... , vk are

linearly independent, then we can find a subset of vi / 1 ≤ i ≤ n, of the form S = v1,

v2, ... , vk, vi1, vi2, ... vir such that (i) S is linearly independent and (ii) L(S) = W.

1.12 Definition: (i) A subset S of a vector space V is called a basis of V if the

elements of S are linearly independent, and V = L(S); and

(ii) Let S be a basis for a vector space V. If S contains finite number of elements,

then V is a finite dimensional vector space. If S contains infinite number of elements

then V is called an infinite dimensional vector space;

(iii) If V is a finite dimensional vector space, and S is a basis for V, n = |S|, then the

integer n is called the dimension of V over F, and we write n = dim V.

1.13 Lemma: If V is finite dimensional and if W is a sub space of V, then (i) W is

finite dimensional, (ii) dim W ≤ dim V, and (iii) dim (V/W) = dim V – dim W.

1.14 Corollary: If A and B are finite dimensional sub spaces of a vector space V.

Then (i) A + B is finite dimensional; and (ii) dim (A + B) = dim A + dim B –

dim (A ∩ B).

Section-2: Elementary concepts in Modules

2.1 Definition: Let R be an associative ring. An Abelian group (M, +) is said to be a

module over R if there exists a mapping f : R × M → M (the image of (r, m) is

denoted by rm) satisfying the following three conditions:

(i) r(a+b) = ra + rb; (ii) (r+s)a = ra + sa; and (iii) r(sa) = (rs)a for all a, b ∈ M and

r, s ∈ R. Moreover if R is ring with identity 1, and if 1m = m for all m ∈ M,

then M is called a unital R–Module.

2.2 Example: (i) Every ring R is a module over it self;


44

(ii) Every group is a module over Z;

(iii) Every vector space over a field F, is a module over the ring F;

(iv) Let (G, +) be an Abelian group. Write R = f: G → G / f is a group

homomorphism.

Define (f + g)(x) = f(x) + g(x) for all x ∈ G and f, g ∈ R. Then (R, +) becomes an

additive Abelian group. The zero function is the additive identity and (-f) is the

additive inverse of f ∈ R where –f is defined by (-f)(x) = -(f(x)) for all x ∈ G.

Define (f.g)(x) = f(g(x)) for all f, g ∈ R and x ∈ G. Then (R, .) is a semigroup. The

distributive laws f(g + h) = fg + gh and (f + g)h = fh + gh hold good. So (R, +, .)

becomes a ring with identity (Here identity function on G acts as identity element in R).

For any f ∈ R and a ∈ G, the element fa (the image of a under f) is in G. Now G

is a module over R.

(v) Let R be a ring and L a left ideal of R. Define a ~ b ⇔ a – b ∈ L for any a, b

∈ R. Then ~ is an equivalence relation and the equivalence class containing a is [a] =

a + L.

Write M = a + L / a ∈ R. If we define (a + L) + (b + L) = (a + b) + L on M, then

(M, +) is an Abelian group. Here 0 + L is the additive identity and (- a) + L is the

inverse of (a + L) in M. For any r ∈ R, a + L ∈ M, if we define r(a + L) = ra + L,

then M is an R-module. It is called quotient module of R by L.

2.3 Definitions: (i) Let M be an R-Module. A subgroup (A, +) of (M, +) is said to be

a submodule of M if r ∈ R, a ∈ A then ra ∈ A.

(ii) If M is an R-module and M1, M2, …, Ms are submodules of M, then M is said to

be the direct sum of Mi, 1 ≤ i ≤ s if every element m ∈ M can be written in a

unique manner as

m = m1 + m2 + … + ms where mi ∈ Mi, 1 ≤ i ≤ s.

(iii) An R-Module M is said to be cyclic if there exists an element a ∈ M such that M

= ra / r ∈ R.

(iv) An R-module is said to be finitely generated if there exist elements aj ∈ M, 1 ≤

j ≤ n such that M = r1a1 + … + rnan / rj ∈ R, for 1 ≤ j ≤ n.

2.4 Definition: (i) If K, A are submodules of M, and K is a maximal submodule of

M such that K ∩ A = (0), then K is said to be a complement of A (or a complement

submodule in M).

(ii) A non-zero submodule K of M is called essential (or large) in M (or M is an

essential extension of K) if A is a submodule of M and K ∩ A = (0), imply A =

(0).


45

2.5 Remark: (i) If V is a vector space and W is a subspace of V, then W has no

proper essential extensions.

[Verification: Let W1

be a proper essential extension of W. Let v ∈ W1

\W.

Clearly v ≠ 0. Now we verify that Fv ∩ W = (0). In a contrary way, take

0 ≠ x ∈ Fv∩W. Now x = av, 0 ≠ a ∈ F and av = x ∈ W ⇒ v = a-1(av) ∈ W, a

contradiction.

Hence Fv∩W = (0). Since Fv ⊆ W1 and W is essential in W

1 , we have that Fv =

o which implies v = o, a contradiction. Thus W has no proper essential extensions.]

(ii). If W, W1 are two subspaces of V such that W is essential in W1 , then W =

W1

(iii). Every subspace W is a complement.

[Verification: Since W has no proper essential extensions, by the Theorem 3 of [5],

we have that W is a complement.]

Section -3: Finite Goldie Dimension in Modules

Hence forth, R denotes a fixed (not necessarily commutative) ring with 1.

3.1 Definition: (i) M has finite Goldie dimension (abbr. FGD) if M does not contain a

direct sum of infinite number of non-zero submodules. [Equivalently, M has FGD if for

any strictly increasing sequence H0 ⊆ H1 ⊆ … of submodules of M, there exists an

integer i such that Hk is an essential submodule in Hk+1 for every k ≥ i].

(ii) A non-zero submodule K of M is said to be an uniform submodule if every non-

zero submodule of K is essential in K.

With the concepts defined above, Goldie proved the following Theorem.

3.2 Theorem: (Goldie): If M is a module with finite Goldie dimension, then there exist

uniform submodules U1, U2, …, Un whose sum is direct and essential in M. The

number ‘n’ is independent of the uniform sumodules. The number ‘n’ of the above

theorem is called the Goldie dimension of M, and is denoted by dim M.

3.3 Remark: (i) Let W be a subspace of V. Then W is uniform ⇔ dim W = 1.

[Verification: Suppose W is uniform. Let o ≠ w ∈ W. Now Fw ⊆ W and dim

(Fw) = 1. Suppose dim W ≥ 2. Then there exist linearly independent elements w1 , w2

∈ W. Now Fw1 ∩ Fw2 = (o) and Fw1 ⊆ W, Fw2 ⊆ W, W is uniform ⇒ w1 = o or

w2 = o, a contradiction. Hence dim W < 2. This shows that dim W = 1.


46

Converse: Suppose dim W = 1. If W is not uniform, then there exists non-zero

subspaces W1 and W2 contained in W such that W1 ∩ W2 = (0). Let 0 ≠ v1

∈ W1 and 0 ≠ v2 ∈ W2. α1v1 + α2v2 = 0 ⇒ α1v1 = -α2v2 ∈ W1 ∩ W2 = (0)

⇒ α1v1 = 0 and α2v2 = 0 ⇒ α1 = α2 = 0. This shows that v1, v2 are linearly

independent vectors in W. This shows that dim W ≥ 2, a contradiction to the fact

that dim W = 1. Thus W is uniform.

(ii) For any subspace W, we have that dim W = 1 ⇔ W is indecomposable.

Verification: Now suppose W is uniform. If W is not indecomposable, then there

exists non-zero subspaces W1 and W2 of W such that W = W1 ⊕ W2. Now W1,

W2 are subspaces of W such that W1 ∩ W2 = (0). Since W is uniform either W1

= 0 or W2 = 0, a contradiction. This shows that W is indecomposable.

Converse: Suppose W is indecomposable. Suppose W is not uniform. Then W

contains two non-zero subspaces W1 and W2 such that W1 ∩ W2 = (0). If W1 +

W2 = W, then W = W1 ⊕ W2, a contradiction. Now suppose W1 ⊕ W2 ⊊ W. Let

v ∈ W \ (W1 ⊕ W2). It can be verified that (Fv) ∩ (W1 ⊕ W2) = (0). Hence there

exists a subspace such that Fv ∩ (W1 ⊕ W2) = (0). Let B be a subspace maximal with

respect to the property (W1 ⊕ W2) ∩ B = (0). Now W1 ⊕ W2 ⊕ B is essential in W

and so by Remark 2.5(ii), W1 ⊕ W2 ⊕ B = W, a contradiction to the fact that W is

indecomposable. Hence W is uniform.

3.4 Note (i): As in vector space theory, for any submodules K, H of M such that

K ∩ M = (0), the condition dim (K + H) = dim K + dim H holds.

(ii) If K and H are isomorphic, then dim K = dim H.

(iii) When we observe the following example, we will learn that the condition

dim (M/K) = dim M – dim K does not hold for a general submodule K of M.

3.5 Example: Consider Z, the ring of integers. Since Z is uniform Z-module, we have

that dim Z = 1. Suppose p1, p2, …, pk are distinct primes and consider K, the

submodule generated by the product of these primes. Now Z/K is isomorphic to the

external direct sum of the modules Z/(pi) where (pi) denotes the submodule of Z

generated by pi (for 1 ≤ i ≤ k) and so dim Z/K = k. For k ≥ 2, dim Z – dim K = 1-1 =

0 ≠ k = dim (Z/K). Hence, there arise a type of submodules K which satisfy the

condition dim (M/K) = dim M – dim K. In this connection, Goldie obtained the

following Theorem.

3.6 Theorem: (Goldie [1]): If M has finite Goldie dimension and K is a complement

submodule, then dim (M/K) = dim M – dim K.


47

On the way of getting the converse for Theorem 3.6, the concept ‘E-irreducible

submodule of M’ was introduced in Satyanarayana [1].

3.7 Definition: A submodule H of M is said to be E-irreducible if H = K ∩ J

where K and J are submodules of M, and H is essential in K, imply H = K or

H = J.

3.8 Note: Every complement submodule is an E-irreducible submodule, but the

converse is not true.

3.9 Example: Consider Z, the ring of integers and Z12 the ring of integers module 12.

The principle submodule K of the Z-module Z12 generated by 2, is E-irreducible

submodule, but it is not a complement submodule.

It is proved in Reddy & Satyanarayana [1] that:

3.10 Theorem: (Reddy – Satyanarayana): If K is a submodule of an R-module M and

f: M→M/K is the canonical epimorphism, then the conditions given below are

equivalent:

(i) K = M or K is not essential, but E-irreducible;

(ii) K has no proper essential extensions;

(iii) K is a complement;

(iv) For any submodule K1 of M containing K, we have that K1 is a complement in M

⇔ f(K1) is complement in M/K; and

(v) f(S) is essential in M/K for any essential submodule S of M.

Moreover, if M has FGD, then each of the above conditions (i) to (v) are equivalent

to

(vi) M/K has FGD and dim (M/K) = dim M – dim K.

3.11 Note: The converse of the Theorem 3.6, is a part of the Theorem 3.10.

As consequence of Theorem 3.10, we have the following Theorem 3.12.

3.12 Theorem: (Reddy – Satyanarayana [1]): If M is an R-module, then the following

conditions are equivalent:

(i) M is a completely reducible module;

(ii) Every submodule of M is a complement submodule


48

(iii) Every proper submodule of M is not an essential submodule, but it is an E-

irreducible

sumodule;

(iv) Every proper submodule of M has no proper essential extensions;

(v) For any submodule K of M with the canonical epimorphism f : M → M/K, we have

that: K1 is a complement submodule in M ⇔ f(K

1) is a complement submodule in M/K;

and

(vi) For any submodule K of M with the canonical epimorphism f : M → M/K, we have

that: S is an essential submodule in M imply f(S) is an essential submodule in M/K.

Moreover, if M has finite Goldie dimension, then the above conditions are equivalent to

each of the following:

(vii) M has the descending chain condition on its submodules and M is completely

reducible; and

(viii) For any submodule K of M, we have that M/K has finite Goldie dimension and

dim (M/K) = dim M – dim K.

E-direct systems:

3.13 Definition: A family Mii∈I of submodules of M is said to be an E-direct

system if, for any finite number of elements i1, i2, …, ik of I there is an element

i0 ∈ I such that 0iM ⊇

1iM + … +

kiM and 0iM is non-essential submodule of

M.

3.14 Theorem: (Satyanarayana [1]): For an R-module M the following two conditions

are equivalent: (i) M has FGD; and

(ii) Every E-direct system of non-zero submodules of M is bounded above by a non-

essential submodule of M.

References

Chatters A.W & Hajarnivas C.R

[1] "Rings with Chain Conditions", Research Notes in Mathematics, Pitman

Advanced

publishing program, Boston-London-Melbourne, 1980.

Goldie A.W

[1] "The Structure of Noetherian Rings", Lectures on Rings and Modules,

Springer – Verlag,

New York, Lecture Notes, 246 (1974) 213-31.

Lambek J


49

[1] "Lectures on Rings and Modules", Blaisdell Publishing Co., 1966.

Pilz G

[1] Near-rings, North-Holland pub., 1983.

Reddy Y.V and Satyanarayana Bh

[1] "A Note on Modules", Proc. Japan Acad., 63-A (1987) 208-211.

Satyanarayana Bh

[1] “A Note on E-direct and S-inverse Systems”, Proc. Japan Academy

64A(1988)

292 – 295.

Satyanarayan Bhavanari & Mohiddin Shaw Sk.

[1] “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Muller,

Germany, 2010, (ISBN 978-3-639-23197-7)

Satyanarayana Bhavanari, Mohiddin Shah Sk, Eswaraiah Setty S, and Babu

Prasad M.

[1] “A generalization of Dimension of Vector Space to Modules over Associative

Rings”, International Journal of Computational Mathematical Ideas, Vol. 1.,

No. 2 (2009) 39 – 46 (India). (ISSN : 0974 – 8652)

Satyanarayana Bh and Syam Prasad K

[1] “A Result on E-direct systems in N-groups ”, Indian J. Pure & Appl. Math. 29

(1998)

285-287.

[2] "On Direct & Inverse Systems in N-groups", Indian J. Math. (BN Prasad Birth

Commemoration Volume) 42 (2000) 183 - 192.

[3] Discrete Mathematics with Graph Theory (for B.Tech / B.Sc/ M.Sc.,(Math.))

Prentice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).

Sharpe D.W and Vamaos P

[1] "Injective Modules", Cambridge University Press, 1972.

Varada Rajan K

[1] "Dual Goldie Dimension", Communications in Algebra, 7(1979) 565-610.


50

Effect of Yield Stress,

Elasticity and Peristalsis

on the Transport of

Bio-fluids

Introduction:

Viscous flows of biofluids through elastic tubes are investigated extensively because of

their important applications in biology, engineering and medicine. Among many

discoveries in the area of fluid dynamics, Poiseuille law considered to be very important

as it describes the relation between the flux and the pressure gradient. According to

Poiseuille’s law, the flux of a viscous incompressible fluid through a rigid tube is a linear

function of the pressure difference between the ends of the tube. However in the vascular

beds of mammals, the pressure flow relation is always non-linear. This non-linearity has

been ascribed to the elastic nature of the blood vessels. It is reported that the transport of

blood takes place in small blood vessles due to the mechanism of peristalsis. Some

electrochemical reactions are sepeculated to be responsible for this phenomenon. This

mechanism also occurs in swallowing of food through oesophagus and stomach,the flow

of urine in the ureter, etc. Ramachandra Rao[1], Vajravelu et.al.[ 7, 8,9,10], Shapiro

et.al. [ 4], Usha et.al.[6] and Subba Reddy et.al.[5] and many others investigated on

several peristaltic flows in tubes and channels. But the wall properties of peristaltic flow

are not studied in detail.

The aim of this talk is to develop mathematical models which explaining the influence of

yield stress, elasticity, peristalsis etc. in the biofluid flow through a tube.

FLOW OF HERSCHEL-BULKLEY FLUID IN AN ELASTIC TUBE

Formulation of the Problem and Solution:

Invited Lecture by

Prof. S. Sreenadh, Department of

Mathematics, Sri Venkateswara

University, Tirupati, Andhra

Pradesh.





51

Consider the Poiseuille flow of a Herschel-Bulkley fluid in an elastic tube of radius a(z). The

flow is axisymmetric. The axisymmetric geometry facilitates the choice of the cylindrical

coordinate system (R,Θ,Z) to study the problem. Here we concentrate on the difference of the

external pressure to the inlet pressure and the outlet pressure, the flow is inherently unsteady in

the laboratory flame (R,Θ,Z) and becomes steady in the wave frame (r, θ, z). The transformation

between these two frames is given by

r = R, θ = Θ , z = Z-ct, p(z) = P(Z,t)

The basic equations and boundary conditions in non-dimensional form are: (dropping the bars)

(1)

Where (2) is the yield stress.

and the non-dimensional boundary conditions are: is finite at r=0 (3)

u=0 at r=a (4)

Where the non-dimensional quantities are , , , , ,

!"# , $ %&'# ,( )&'# , * , + , (5)

Solving eq(1) and (2) subject to the conditions (3) and (4) we obtain the velocity field as -. /0 12.- * 3/0 2.- 3/04 (6)

Where 5 and 6 . Using the condition 0 at r = r0

The upper limit of the plug flow region is obtained as -.

Also by using the condition 8 at r=a we obtain -

Hence , 0 < τ < 1 (7)

Using relation (7) and taking r=r0 in Eq.(6), we get the plug flow velocity as


52

5 2.-3/ :!"/0 1 /0

for 0 ≤ r ≤ r0 (8)

The volume flux Q through cross-section in the wave frame is given by ( < 5 = < = >*/0?/ (9)

Where > @:!"-:!"/0 11 -@0/0-/0-/0? 4 and 5 (10)

For a Herschel-Bulkley fluid, we assume that the flux Q is related to the pressure gradient A5A by

the relation ( BC C A5A/ (11)

Now, from (9) and (11) , we observe that BC C >*/0? (12)

Integrating (11) with respect to z from z=0 and using the inlet condition p(0)=p1, we obtain (": < BCD": =CD 5"@55@5 (13)

where p’=p(z)-p0 This equation determines p(z) implicitly in terms of Q and z . To find Q, we set

z=1 and p(1)=p2 in (13) to obtain

(": < BCD": =CD 5"@55@5 (14)

Now, using (12) in (14), we have ( > < *C C? 0 =CD 5"@55'@5 (15)

Where n= 1\k.

Eq. (19) can be solved if we know the form of the function a*(p’). If the stress or tension T(a) in

the tube wall is known as a function of ‘a’ , then a(p’) can be found using the equilibrium

condition , C C (16)

From the experimental data for human iliac artery, the expansion of T(a) is obtained as (Rubinow

and Keller [1]) +* E* 1 E-* 1F (17)

Where t1=13 and t2=300, When we substitute (20) in (19) we have C C 1E* 1 E-* 1F4


53

=CD GH"' E- 24*? 15*- 20* 10 '3L =* (18)

Using (21) in (18), we get

( > M *? 0 N E*- E- O4*? 15*- 20* 10 1*-PQ =* 5"@5

5'@5

Integrating this we have ( >R* R*- (19)

Here a1 = a(p1-p0) and a2 = a(p2-p0) are determined by solving (21) with p=p1 and p=p2,

respectively. The function g(a) and the constant F is defined by

> 1 02 1S 1 T1 21 2 1S

22 1S3 3 1S V R* E *? 3S E- W4*? 0F3S 5 15*? 0X3S 4 20*? 0?3S 3 10*? 0-3S 2 *? 3S Y

PERISTALTIC FLOW OF A VISCOUS FLUID IN AN ELASTIC TUBE

Mathematical Formulation and Solution:

Consider the peristaltic transport of a Bingham fluid in a elastic tube of radius ‘a’. The

flow is axisymmetric. Cylindrical polar coordinate system (R, θ,Z) is used. The wall deformation

due to the infinite train of peristaltic waves is represented by

R= 2

( , ) sin ( )H Z t a b Z ctπλ

= + − ------------------(1)

Where b is the amplitude , λ is the wavelength and c is the wave speed.

The transformation between the laboratory frame (R,θ,Z) to the wave frame (r,θ,z) is

r = R, θ = Θ , z = Z-ct, p(z) = P(Z,t) ------------------(2)

Introducing the non-dimensional quantities defined by


54

*

av

ww

u=

*

0

rr

a=

*

2

av

pp

uρ=

*

0

zz

a=

*

0

hh

a=

*

0

aa

a=

*

2

0 av

TT

a uρ=

* ctt

λ=

b

aφ =

*

0

qq

a c= ------------------(3)

Where Uav is the average velocity and a0 is the radius of the tube in the absence of elasticity the

governing equations (dropping the asterisks) are:

0p

r

∂=

∂ ------------------(4)

2

2

1R

w w p

r r r z

∂ ∂ ∂+ =

∂ ∂ ∂ ------------------(5)

The dimensionless boundary conditions are :

0w

r

∂=

∂ at r = 0 ------------------(6)

w= -1 at r = h ------------------(7)

Solving eqs (4) and (5) subjected to the boundary conditions (6) and (7), we obtain the velocity as

Z [ .\\'X 1 '

\\' 1, where p

Pz

∂= −

∂ ------------------(8)

The volume flux q through each cross-section in the wave frame is given by:

$ 2 < Z= [ .\\]^ *DD-_ ----------------(9)

The above relationship can be rewritten for elastic tube as : ( vide Rubinow)

$ B 5 *DD- ------------------(10)

Taking elastic property and peristaltic movement of the tube wall into consideration, we can take

' " 4( )

8

a aRσ +

= ------------------(11)

Where '

a is the change in the radius of the tube due to elasticity and ''

a is the change in radius

of the tube due to peristalsis. As the flow is of Poiseuille type, at each cross section, the radius '

a

is a function of pressure 0p p− ,'

0( )a p p− and the wall deformation due to the infinite train of

peristaltic waves is represented by '' ( ) 1 sin 2a z zφ π= + which is a function of z.

Using (13) in (12), we have


55

2 21 2 sin 2 sin 2

dpq z z

dzφ π φ π σ+ + + = − ---------------- (12)

Integrating from z=0 at the inlet pressure p (0) =p1 we have

0

1 0

2 2

0

cos 2 sin 4( ) '

8 2

p p

p p

z z zqz z p p dp

φ π π φ φ φ σπ π π

−

−

+ − − + + = − −∫ ---------------- (13)

Where 0'( ) ( )p z p z p= −

For one wave length we have z=1 then p (1) =p2, (12) becomes

1 0

2 0

2' '( ) 1

2

p p

p p

q p dpφσ

−

−

= + −∫ ---------------- (14)

Here σ is a function of p-p0 and z . So Eq(14) can be solved if we know the form of the function

a(p’). If the stress or tension T (a) in the tube wall is known as a function of ‘a’, then a (p’) is

found using the equilibrium condition.

0'( ) ( )'

Tp z p z p

a= − = ------------------ (15)

Roach & Burton [2] determined the static pressure-volume relation of a 4 cm long piece of the

human external iliac artery, and converted it into a tension versus length curve. Using least

squares method Rubinow and Keller [3] given the following equation:

' 5

1 2( ) ( ' 1) ( ' 1)T a t a t a = − + − ------------------ (16)

Where t1=13 and t2=300. When we substitute (16) in (15) , the latter becomes

5

0 1 2'

1[ ( ' 1) ( ' 1) ]p p t a t a

a− = − + − ------------------ (17)

Now (16) yields the results

2

1 2( ) ( ) 1

2q g a g a

φ= − + − ------------------ (18)

'3 '' 4 '8 ' 7 '6'' ' '' 2 ' ''3 ' '' '' 2 ''

1 2'

'5 ' 4''3 '' '' 4 ''3 '' 2 ''

'3 ' 2'' 4 ''3 '' 2 '' 4 ''3 ''

( ) [ 2 6 4 log ] [ (16 15) (24 60 20)3 2 7 6

(16 90 10) (4 60 120 40 )5 4

( 15 80 60 1) (20 40 43 2

a a a a ag a t a a a a a a t a a a

a

a aa a a a a a

a aa a a a a a

= + + + − + + − + − +

+ − − + − + − +

− + − + + − + '' 4 '' 2

'' 4''3 '

'

) ( 10 6 )

4 log

a a a

aa a

a

+ − + +

−


56

2.2 2.4 2.6 2.8 3

20000

1–Trapezoidalwave

2-Multisinusoidalwave(n=6)

3-Triangularwave

4-Squarewave

5- Sinusoidal

1

2

34

5

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

Radius

Flux

HB

Power law

Bingham

Newtonian

80000

60000

40000

g versus a1 for different types of waves

Radius a’

Graphs are drawn to find the effects of yield stress, elasticity and pressure difference on the

pumping phenomenon of Newtonian and Herschel-Bulkley fluids with and without

Peristalsis.

Conclusions:

• On comparing the flux for various fluids like Newtonian (N), Bingham (B), Power law

(P), and Herschel-Bulkley (H) we find that, H>P>B>N.

• Flux increases as the yield stress and the elastic nature of the tube increases.

• If the flowing fluid in the elastic tube is a Herschel-Bulkley fluid, then the flux increases

as the p2-p0 increases. If the flowing fluid is a Newtonian fluid and if it is pumped by a

peristaltic wave then the flux decreases as p2-p0 increases.

• As the amplitude of the peristaltic wave increases then the flux increases, if the fluid

pumped is a Newtonian fluid.

• If we define, 1–Trapezoidal wave, 2-Multisinusoidal wave, 3-Triangular wave, 4-Square

wave, 5- Sinusoidal wave, then the relation in flux is found to be 1<2<3<4<5.

References:

1) RAMACHADRA RAO, A., & PADMAVATHI,K., 1997 Mathematical models foe

Catheter movement in blood vessels, J. Math. Math. Biosci.., 1, 57-78.

2) ROACH, M. R. & BURTON, A. C. 1957, Can. J. Biochem. Physiol. 35,

3) RUBINOW, S. I. AND KELLER, B. JOSEPH. 1972 Flow of a Viscous Fluid Through ad

Elastic Tube with Applications to Blood Flow, J.theor.Biol.35, 299-313.

4) SHAPIRO, A.H., JAFFRIN,M.Y. & WEINBERG, S.L. 1969 Peristaltic pumping with

long wavelengths at low Reynolds number J .Fluid Mech. 37, 799 – 825.

120000

100000

Flux for different waves


57

5) SUBBA REDDY, M. V., SREENADH, S., & RAMACHANDRA RAO, A., 2007

Peristaltic motion of a Power-law fluid in an asymmetric channel, Int. J. Nonlinear Mech.

42 , 1153-1161.

6) USHA, & RAMACHANDRA RAO, A. 1997 Peristaltic transport of two-layered power-

law fluids ASME J. Biomech. Engg. 104, 182 – 186.

7) VAJRAVELU, K., HEMADRI REDDY, R. SREENADH, S. & MURUGESAN, K. 2009

Peristaltic Transport of a Casson fluid in contact with a Newtonian Fluid in a Circular

Tube with permeable wall, Int. J.Fluid Mech. Research, 36, 244-254.

8) VAJRAVELU, K., SREENADH,S. & RAMESH BABU, V. 2005 Peristaltic transport of

a Herschel-Bulkley fluid in a channel Appl. Math and Comput 169, 726 – 735.

9) VAJRAVELU, K., SREENADH,S. & RAMESH BABU, V. 2006 Peristaltic transport of

a Herschel-Bulkley fluid in contact with a Newtonian Fluid, Quarterly. Appl. Math, L

XIV,No.4, 593-604.

10) VAJRAVELU,K., SREENADH.S. & RAMESH BABU, V. 2005 Peristaltic transport of

a Herschel- Bulkley fluid in an inclined tube Int. J. Nonlinear Mech. 40, 83 – 90.


58

Fuzzy Ideals of

Gamma Near-rings

1. Introduction

The concept of fuzzy subset was introduced by Zadeh [6]. Later several authors like

[1, 3] were studied the concept: fuzziness in different algebraic systems, particularly in

the theory of rings and near-rings. For preliminary definitions and results, refer [2, 3, 4,

6]. A non-empty set N with two binary operations + and . is called a near-ring if N is an

additive group (not necessarily abelian), multiplicative semigroup satisfying one

distributive law (we consider right distributive law).

The concept of Gamma nearring was introduced by Satyanarayana [4] and further studied

in Satyanarayana [5, 6] . The definition given as follows.

Let (M, +) is a group (not necessarily Abelian) and Γ is 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; (ii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α , β ∈ Γ.

Throughout this talk, M stands for a Γ-near-ring. A normal subgroup (I, +) of (M, +) is

called (i) a left ideal if aα(b + i) – aαb ∈ A for all a, b ∈ M, α ∈ Γ , i ∈ I; (ii) a right

ideal if iαa ∈ A for all a ∈ M, α ∈ Γ , i ∈ I; and (iii) an ideal if it is both a left and a

right ideal. M is said to be zero-symmetric if aα0 = 0 for all a ∈ M and α ∈ Γ, where 0 is

the additive identity in M.

Invited Lecture by

Dr Kuncham Syam Prasad,


Manipal University, Manipal-

576 104, India,

Email:

[email protected]





59

We now review some fuzzy logic concepts. A fuzzy set in a set M is a function

µ: M→ [0, 1]. We shall use the notation µt, called a level subset of µ which is defined as

µt = x ∈ M µ(x) ≥ t where t ∈ [0, 1]. Let X and Y are two non empty sets and f a

function of X into Y. Let µ and σ be fuzzy subsets of X and Y respectively. Then f(µ),

the image of µ under f is a fuzzy subset of Y defined by

(f(µ))(y) = yf(x)

sup=

µ(x) if f-1(y) ≠ φ

= 0 if f-1(y) = φ.

And f -1

(σ), the preimage of σ under f is a fuzzy subset of X defined by (f -1

(σ))(x) =

σ(f(x)) for all x ∈ X.

1.1 Definition: A fuzzy set µ in a Γ-near-ring M is called a fuzzy left (resp. right) ideal

of M if (i) µ is a fuzzy normal subgroup with respect to addition, (ii) µ(xα(y+z)–xαy) ≥

µ(z) (resp. µ(xαy) ≥ µ(x) ) for all x, u, v ∈ M and α ∈ Γ.

1.2 Results: Let µ be a fuzzy ideal of M. Then (i) µ(0) ≥ µ(x); (ii) µ(x + y) = µ(y + x);

(iii) µ(x−y) = µ(0) implies µ(x) = µ(y), for all x, y ∈ M.

1.3 Theorem [3.5 of [1]]: Let µ be a fuzzy set in M. Then µ is a fuzzy left (resp. right)

ideal of M if and only if each level subset µt, t ∈ im (µ ), of µ is a left (resp. right) ideal

of M.

1.4 Theorem [3.2 of [1]]: Let µ be a fuzzy left (resp. right) ideal of M. Then the set

Mµ = x ∈ Mµ(x) = µ(0) is a left (resp. right) ideal of M.

1.5 Theorem [3.3 of [1]]: Let A be non-empty subset of M and µA be a fuzzy set in M

defined by µA (x) = ∈

otherwise t,

A xif s, , for all x ∈ M and s, t [0, 1] with s > t. Then µA is a

fuzzy left (resp. right) ideal of M if and only if A is a left (resp. right) ideal of M.

Moreover MµA = A.

1.6 Definition [5]: Let M and N are Γ-near-rings. A map θ: M → N is called a Γ-near-

ring homomorphism if θ(x + y) = θ(x) + θ(y) and θ(xαy) = θ(x)αθ(y) for all x, y ∈ M

and α ∈ Γ.

1.7 Theorem: If µ is a fuzzy ideal of M and a ∈ M then µ(x) ≥ µ(a) for all x ∈ <a>.

Hint: For a ∈ M, <a> = U∞

=0i

iA , where Ak+1 = Ak*∪Ak

+∪Ak0 ∪Ak

++ and A0 = A,


60

where Here Ak* = n+x–n n ∈ N, x ∈ Ak ; Ak

+ = n1α(n2+a)–n1αn2 n1, n2 ∈ M,

a ∈ Ak, α ∈ M; Ak0 = x – y x, y ∈ Ak; Ak

++ = xαmx ∈ Ak, α ∈ Γ and m ∈ M.

2. Fuzzy Cosets

2.1 Definition: Let µ be a fuzzy ideal of M and m ∈ M. Then a fuzzy subset

m+µ defined by (m+µ)(m1) = µ(m

1−m) for all m1 ∈ M is called a fuzzy coset of the fuzzy

ideal µ.

2.2 Proposition: Let µ be a fuzzy ideal of M. Then (i) x + µ = y + µ if and only if

µ(x−y) = µ (0), (ii) If x + µ = y + µ, then µ(x) = µ(y), (iii) Every fuzzy coset of a fuzzy

ideal µ of M is constant on every coset of ordinary ideal Mµ, (iv) If z ∈ Mµ, then (x +

µ)(z) = µ(x).

2.3 Theorem: Let µ be a fuzzy ideal of M. Then the set of fuzzy cosets M/µ of µ is a

Γ-near-ring with respect to the operations defined by

(x+µ)+(y+µ) = (x+y)+µ; and (x+µ)α(y+µ) = xαy+µ for all x, y ∈ M and α ∈ Γ.

2.4 Proposition: Let µ be a fuzzy ideal; the fuzzy subset θµ of M/µ, is defined by

θµ(x+µ) = µ(x) for all x ∈ M, is a fuzzy ideal of M/µ .

2.5 Theorem: If µ is a fuzzy ideal of M then the map θ: M → M/µ, defined by

θ(x) = x+µ, x ∈ M, is a Γ-near-ring homomorphism with kernel Mµ = x ∈ M µ(x) =

µ(0).

2.6 Theorem: The Γ-near-ring M/µ is isomorphic to the Γ-near-ring M/Mµ. The

isomorphic correspondence is given by x+µ→ x+Mµ.

2.7 Lemma: Let µ and σ be two fuzzy ideals of M such that σ ⊇ µ and σ(0) = µ(0). Then

the fuzzy subset θµ of M/µ defined by θσ(x+µ) = σ(x) for all x ∈ M is a fuzzy ideal of

M/µ such that θσ ⊇ θµ.

2.8 Notation: The fuzzy ideal θσ of M/µ is denoted by σ/µ.

2.9 Lemma: Let µ be a fuzzy ideal of M and θ be a fuzzy ideal of M/µ such that θ ⊇ θµ.

Then the fuzzy subset σθ of M defined by σθ(x) = θ(x+µ) for all x ∈ M is a fuzzy ideal of

M such that σθ ⊇ µ.

2.10 Correspondence Theorem: Let µ be a fuzzy ideal of M. There exists an order

preserving bijective mapping between the set P of all fuzzy ideals σ of M such that σ ⊇ µ

and σ(0) = µ(0) and the set Q of all fuzzy ideals θ of M/µ such that θ ⊇ θµ.

Proof: (Hint) Write P = σ σ is a fuzzy ideal of M, σ ⊇ µ, σ(0) = µ(0) and


61

Q = θ θ is a fuzzy ideal of M/µ, θ ⊇ θµ.

Define η: P → Q by η(σ) = θσ. One can verify that η is an order preserving bijective

correspondence.

2.11 Proposition: Let h: M → M1 be an epimorphism and σ is a fuzz ideal of M

1 such

that µ = h-1(σ). Then the map ψ: M/µ → M1/σ defined by ψ(x+µ) = h(x) + σ is a Γ-

near- ring isomorphism.

Acknowledgements

The author wishes to thank Dr Eswaraiah Setty (Organizing Secretary) Prof. Bhavanari Satyanaraayana

(Academic Secretary) for inviting him to deliver this talk in the UGC sponsored seminar on Present Trends

in Mathematics and its Applications on 11 and 12 November 2010 at Smt. G.S. College, Jaggaaiahpet, AP.

References

1. Jun Y. B., Sapanci M., and Ozturk M. K. “Fuzzy Ideals in Gamma Near-rings”,

Tr. J. of Mathematics, 22(1998), 449-459.

2. Pliz G. “Near-rings”, North Holland, 1983.

3. Salah Abou-Zaid “On Fuzzy Subnear-rings and Ideals”, Fuzzy sets and Systems,

44(1991) 139-146.

4. Satyanarayana Bh., “Contributions to Near-ring Theory”, Doctoral Thesis,

Nagarjuna University, 1984.

5. Satyanarayana Bh. “The f-Prime Radical in Γ-Near-rings, South East. Bull.

Math., (1999) 23: 507-511.

6. Satyanarayana Bh. “A Note on Γ- Near-rings”, Indian J. Mathematics, 41(3),

1999, 427-433. 7. Satyanarayana Bh., and Syam Prasad Kuncham ‘Fuzzy Cosets of Gamma Nearrings’,

Turkish Journal of Mathematics, (29) 11-22, 2005.

8. Syam Prasad K., and Satyanarayana Bh. ‘A Note on IFP N-Groups’ Proc. of 6th

Ramanujam Symposium on Algebra & Applications, 62-65, 1999.

9. Syam Prasad K., and Satyanarayana Bh. “On Fuzzy Prime ideal of a Gamma Nearring”,

Soochow Jr. Mathematics, 31(1) 121-129, 2005.

10. Satyanarayana Bh., Syam Prasad and Kumar TVP “On IFP N-Groups and Fuzzy IFP

Ideals”, Indian J. Mathematics, 46 (1) 11-19, 2004.

11. Syam Prasad K. “Contributions to Near-ring Theory II”, Doctoral Thesis,


12. Zadeh L. A. “Fuzzy sets” Inform. & Control 8(1965) 338-353.


62

Semisimple Hopf Algebras

and their Orbits




Invited Lecture by

Prof. M. Sumanth Datt,

Department of Mathematics

and Statistics, University of

Hyderabad, Hyderabad-500

046, Andhra Pradesh


63


64


65


66

Rough Sets

Invited Lecture by

Dr Kedukodi Babushri Srinivas,


Manipal University, Manipal-

576 104, India.

Email:

[email protected]





67


68


69

Finite Dimension in

Associative Rings

Introduction

It is well known that the dimension of a Vector Space is defined as the number of

elements in the basis. A. W. Goldie (University of Leeds) [2] generalized the dimension

concept to modules over rings. A Module M is said to have finite Goldie dimension

(FGD, in short) if M does not contain a direct sum of infinite number of non-zero

submodules. Goldie proved a structure theorem for modules which states that “a module

with FGD contains uniform submodules U1, U2, …, Un whose sum is direct and essential

in M”. The number n obtained here is independent of the choice of U1, U2, …, Un and it

is called as Goldie dimension of M. The concept Goldie dimension in Modules was

studied by several authors like Satyanarayana, Syam Prasad, Nagaraju (refer [3, 10]).

If we consider ring as a module over itself, then the existing literature tells about

dimension theory for ideals (i.e., two sided ideals) in case of commutative rings; and left

(or right) ideals in case of associative (but not commutative) rings. So at present we can

understand the structure theorem for associative rings in terms of one sided ideals only

(that is, if R has FGD with respect to left (right) ideals, then there exist n uniform left (or

right) ideals of R whose sum is direct and essential in R). This result cannot say about

the structure theorem for associative rings in terms of two sided ideals. To fill this gap,

Satyanarayana, Nagaraju, Balamurugan & Godloza [5] started studying the concepts:

complement, essential, uniform, finite dimension with respect to two sided ideals of R.

1. Essential Ideals

1.1 Definition (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): (i) Let I, J be

two ideals of R such that I ⊆ J. We say that I is essential (or ideal essential) in J if it




Invited Lecture by

Dr Dasari Nagaraju,


HITS, Hindustan University,

Chennai.

Email:

[email protected]


70

satisfies the following condition: K ⊴ R, K ⊆ J, I ∩ K = (0) imply K = (0). If I is

essential in J, then we write I ≤e J. Here K ⊴ R represents K is an ideal of ring R.

(ii). If K ⊴ R, A ⊴ R and K is a maximal element among the ideals I of R with

respect to the property I ∩ A = (0), then we say that K is a complement of A (or a

complement in R).

(iii) If A is essential ideal of R and A ≠ R, then we say that R is a proper essential

extension of A.

1.2 Note: Let I and J be ideals of R. Then I ≤e J ⇔ I ∩ K = (0), K ⊴ R ⇒ K ∩ J = (0).

1.3 Result (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]):

(i) The intersection of finite number of essential ideals is essential;

(ii) If I, J, K are ideals of R such that I ≤e J, and J ≤e K, then I ≤e K;

(iii) I ≤e J ⇒ I ∩ K ≤e J ∩ K;

(iv) If I ⊆ J ⊆ K, then I ≤e K if and only if I ≤e J, and J ≤e K; and

(v) Suppose R1, R2 are two rings and f: R1 → R2 is a ring isomorphism. If A is an ideal

of R1, then A ≤e R1 ⇔ f(A) ≤e R2.

1.4 Notation: (i) For any subset A of R, we write

A+ = ra / a ∈ A, r ∈ R, A

0 = a- b / a, b ∈ A , A

* = ar / a ∈ A, r ∈ R.

(ii) Let φ ≠ X ⊆ R. We write X1 = X, X2 = *1X ∪ 0

1X ∪ +1X ∪ X1. For any i ≥ 3, define

Xi = *1iX − ∪ +

−1iX ∪ 01iX − . Let a ∈ X = X1. Now 0 = a - a ∈ 0

1X ⊆ X2. For any x ∈ X2,

x = x - 0 ∈ 02X ⊆ X3 and so on X2 ⊆ X3. In this way, we get that X1 ⊆ X2 ⊆ X3…..

1.5 Note: If φ ≠ X ⊆ R, then the ideal generated by X, <X> = U∞

=1i

iX .

1.6 Lemma (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): Let K ⊴ R, L ⊴ R

such that K ∩ L = (0). Let a ∈ K, b ∈ L. Then for any a1 ∈ <a> there exist b1 ∈ <b>

such that a1 + b1 ∈ <a + b>.

Hint: By above note and the Principle of Mathematical induction we can get the result.

1.7 Lemma (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): (i) L1, L2, K1, K2

are ideals of R such that Li ⊆ Ki for i = 1, 2 and K1 ∩ K2 = (0). Then L1 ≤e K1 and

L2 ≤e K2 ⇔ L1 + L2 ≤e K1 + K2; and (ii) Let K1, K2, … Kt, L1, L2, … Lt are ideals of R


71

such that the sum K1 + K2 + … + Kt is direct and Li ⊆ Ki for 1 ≤ i ≤ t. Then

L1 + L2 + … + Lt ≤e K1 + K2 + … + Kt ⇔ Li ≤e Ki for 1 ≤ i ≤ t.

Hint: (i) Assume that L1 ≤e K1 and L2 ≤e K2. Write A1 = L1 + K2 and A2 = K1 + L2. We

show that A1 ≤e K1 + K2 and A2 ≤e K1 + K2. It follows that L1 + L2 = A1 ∩ A2 ≤e K1 +

K2. Converse is clear as it follows from the definition. And (ii) follows by using (i) and

Mathematical induction on t.

2. Uniform Ideals 2.1 Definition (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): A non-zero ideal

I of R is said to be uniform if (0) ≠ J ⊴ R, and J ⊆ I ⇒ J ≤e I.

2.2 Theorem (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): (i) I is an uniform

ideal ⇔ L ⊴ R, K ⊴ R, L ⊆ I, K ⊆ I, L ∩ K = (0) ⇒ L = (0) or K = (0).

(ii) Let R1 and R2 be two rings and f: R1 → R2 be ring isomorphism. If U is ideal of

R1, then U is uniform in R1 ⇔ f(U) is uniform in R2.

(iii) Let H and K be two ideals of R such that H ∩ K = (0). For an ideal U of R

contained in H, we have that U is uniform ⇔ (U + K)/K is uniform in R/K.

(iv) If U and K are two ideals of R such that U ∩ K = (0), then U is uniform in R ⇔

(U + K)/K is uniform in R/K.

2.3 Remark (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): Let K be an

uniform ideal of R and L ⊴ R such that L ⊆ K. Then either L = (0) or L is uniform.

3. Finite Dimension with respect to two sided Ideals

3.1 Definition (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): (i) We say that

R has Finite Dimension on Ideals (FDI, in short) if R does not contain a direct sum of

infinite number of non-zero (two sided) ideals of R. (ii) Let (0) ≠ K ⊴ R. We say that K

has Finite Dimension on Ideals of R (FDIR, in short) if K does not contain a direct sum

of infinite number of non-zero ideals of R. It is clear that if R has FDI, then every

ideal K of R has FDIR.

3.2 Theorem (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): K has FDIR ⇔

for any strictly increasing sequence H1, H2, … of ideals of R contained in K, there is

an integer i such that Hk ≤e Hk+1 for every k ≥ i.


72

3.3 Lemma (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): Suppose R has FDI

and (0) ≠ K ⊴ R. Then K contains a uniform ideal.

3.4 Theorem (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): Let 0 ≠ H ⊴ R.

Suppose R has FDI.

(i) (Existence) There exist uniform ideals U1, U2, … Un whose sum is direct and

essential in H;

(ii) If Vi, 1 ≤ i ≤ k are uniform ideals of R, such that vi ⊆ H and the sum of vi’s is

direct, then k ≤ n.

(iii) (Uniqueness) if Vi, 1 ≤ i ≤ k are uniform ideals of R whose sum is direct and

essential in H, then k = n.

3.5 Definition (Satyanarayana, Nagaraju, Balamurugan & Godloza [5]): The number n of

the above Theorem is independent of the choice of the uniform ideals. This number n is

called the dimension of R, and is denoted by dim R.

3.6 Theorem (Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): Suppose R has FDI.

(i). If H ⊴ R, K ⊴ R and H ⊆ K, then dim H ≤ dim K;

(ii) If (0) ≠ Ai is an ideal of R for all i, 1 ≤ i ≤ t whose sum is direct, and Ai ⊆ H,

1 ≤ i ≤ t, then dim H ≥ t;

(iii) H is uniform ⇔ dim H = 1;

(iv) If H is a non-zero ideal of R, then dim H ≥ 1;

(v) If Ii, 1 ≤ i ≤ k are uniform ideals of R whose sum is direct, then k ≤ dim R.

Moreover dim H = maxk / there exist uniform ideals Ii, 1 ≤ i ≤ k of R whose sum is

direct, Ii ⊆ H, 1 ≤ i ≤ k;

(vi). If n = dim R, then the number of summands in any decomposition of a given ideal I

of R as a direct sum of non-zero ideals of R is at most n.; and

(vii) If f: R → S is an isomorphism and R has FDI, then S has FDI and dim R = dim S.

3.7 Result (Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): If H and K are ideals of

R with H ∩ K = (0), then dim (K + H) = dim K + dim H.


73

Using the above Result and the principle of mathematical induction, we get the following

Corollary.

3.8 Corollary (Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): (i). If I1, I2, …, In

are ideals of R whose sum is direct, then dim(I1 ⊕ I2 ⊕ … ⊕ In) = dim I1 + dim I2 + … +

dim In.

(ii). Suppose Ri, 1 ≤ i ≤ n are rings and R = i

n

iR

1=⊕ is the direct sum of the rings

Ri, 1 ≤ i ≤ n. Then each Ri has FDI if and only if R has FDI. If R has FDI, then

dim R = ∑=

n

i

iR1

dim .

3.9 Theorem (Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): If R has FDI with

dim R = n and H ⊴ R, then the following conditions are equivalent:

(i). H ≤e R; (ii). dim H = dim R; and (iii). H contains a direct sum of n uniform ideals.

3.10 Proposition (Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): If R has FDI and

if an ideal H of R has no proper essential extensions, then R/H has FDI and dim(R/H)

≤ dim R.

3.11 Proposition (Satyanarayana, Nagaraju, Godloza & Sreenadh [6]): Suppose R has

FDI and K ⊴ R.

(i) K is a complement ideal ⇔ K has no proper essential extensions; and

(ii) If K is a complement, then R/K has FDI, and dim(R/K) ≤ dim R.

3.12 Theorem (Satyanarayana, Nagaraju & Mohiddin Shaw [7]): Let K be an ideal of R

and π: R → R/K be the canonical epimorphism. Then the following three conditions are

equivalent:

(i) K is a complement;

(ii) For any ideal K1 of R containing K, we have that K

1 is a complement in R ⇔

π(K1) is complement in R/K; and

(iii) For any essential ideal S of R, π(S) is essential in R/K.


74

4. Dimension of the Quotient Ring R/K

4.1 Lemma (Satyanarayana, Nagaraju & Mohiddin Shaw [7]): Let R be a Ring with FDI.

If A is an ideal of R such that dim(R/A) = 1 and A is not essential in R, then dim(R/A)

= dim R - dim A.

It is well known that if V is a finite dimensional vector space and W is a subspace of V,

then dim(V/W) = dim V - dim W. This dimension condition may not hold for a general

ideal W of a Ring V where “dim” denotes the “finite dimension”. For this, observe the

following examples.

4.2 Examples: Write R = ℤ, the ring of integers. Since every ideal of ℤ is essential in ℤ,

it follows that ℤ is uniform and so dim R = 1.

(i) Write K = 6ℤ. Now K is an uniform ideal of R. So dim K = 1 and dim R - dim K =

1 - 1 = 0. Now R/K = ℤ/6ℤ ≅ ℤ6 ≅ ℤ2 + ℤ3 and so dim(R/K) = 2. Thus dim(R/K) = 2 ≠

0 = dim R - dim K.

(ii) Let p, q be distinct primes and consider H, the ideal of ℤ generated by the product of

these primes (that is, H = pqℤ). Now H is uniform ideal and so dim H = 1. It is known

that ℤ/H = ℤpq ≅ ℤp ⊕ ℤq, and ℤp, ℤq are uniform ideals. So dim(ℤ/H) = 2. Thus

dim (ℤ/H) = 2 ≠ 0 = 1 - 1 = dim ℤ - dim H.

Hence, there arise a type of ideals K which satisfy the condition dim(R/K) = dimR–dimK.

4.3 Theorem (Satyanarayana, Nagaraju & Mohiddin Shaw [7]): If R has FDI and K is a

complement ideal, then dim(R/K) = dim R – dim K.

5. E-Irreducible Ideals

5.1 Definition (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): An

ideal I of R is said to be E-irreducible if I = J ∩ K, where J and K are ideals of R, and I is

essential in K, imply I = K or I = J.

5.2 Note: Every complement ideal is an E-irreducible ideal but the converse is not true.


75

5.3 Example: Consider ℤ, the ring of integers and ℤ12, the ring of integers modulo 12.

The principle ideal I of ℤ-module ℤ12 generated by 2 is an irreducible ideal, but not a

complement ideal.

6. Some Equivalent Conditions

6.1 Theorem (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): Let

I be an ideal of R. Then the following conditions are equivalent:

(i). I = R or I is not an essential ideal but it is an E-irreducible ideal;

(ii). I has no proper essential extensions; and (iii). I is a complement.

As a consequence we have the following corollary.

6.2 Corollary (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): Let

I be an E-irreducible ideal of R. Then

(i). I is an essential ideal or I has no proper essential extensions; and

(ii). I is an essential ideal or I is a complement.

6.3 Theorem (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): Let R

be a ring with FDI, and I is an ideal of R such that dim R = dim I + dim(R/I). Then I has

no proper essential extensions of R.

6.4 Lemma (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): If R has

FDI and I is an E-irreducible ideal of R, then R/I has FDI and dim(R/I) ≤ dim R.

6.5 Lemma (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): If I is

essential in R, then I is E-irreducible ⇔ R/I is uniform.

6.6 Theorem (Satyanarayana, Nagaraju, Mohiddin Shaw & Eswaraiah Setty [8]): If I is

an ideal of a ring R and f: R → R/I is the canonical epimorphism, then the conditions

given below are equivalent:

(i). I = R or I is not essential but E-irreducible.

(ii). I has no proper essential extensions.

(iii). I is a complement.

(iv). For any ideal I1 of R containing I, I

1 is a complement in R ⇔ f(I

1) is complement

in R/I.

(v). f(S) is essential in R/I for any essential ideal S of R.


76

References

[1]. Faechini Alberto (1998) “Module Theory” (Progress in Mathematics, Vol.167),

Birkhauser Verlag, Switzerland.

[2]. A. W. Goldie (1972) "The Structure of Noetherian Rings", Lectures on Rings and

Modules, Springer-Verlag, New York.

[3]. Bh. Satyanarayana "A note on E-direct and S-inverse Systems", Proc. of the Japan

Academy, 64-A (1988) 292-295.

[4]. Satyanarayana Bhavanari and Mohiddin Shaw Shaik "Fuzzy Dimension of Modules

over Rings (Monograph)", VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-

639-23197-7).

[5]. Satyanarayana Bh., Nagaraju D., Balamurugan K. S., & Godloza L. "Finite

Dimension in Associative Rings", Kyungpook Mathematical Journal, 48 (2008) 37-

43. (SOUTH KOREA).

[6]. Satyanarayana Bhavanari, Nagaraju Dasari, Godloza Lungisile, & S. Sreenadh

“Some Dimension Conditions in Rings with Finite Dimension”, The PMU Journal of

Humanities and Sciences 1 (2010) 69-75 (INDIA).

[7]. Satyanarayana Bhavanari, Nagaraju Dasari & Mohiddin Shaw Shaik. “On the

Dimension of the Quotient Ring R/K where K is a complement”, Communicated.

[8]. Satyanarayana Bhavanari, Nagaraju Dasari, Mohiddin Shaw Sk., & Eswaraiah Setty S

“E-Irreducible Ideals and Some Equivalent Conditions” Proceedings of the

International Conference on Challenges and Applications of Mathematics in Science

and Technology (CAMIST 2010), [NIT Rourkela, 11-01-2010 to 13-01-2010],

McMillan Advanced Research Series, India, pp. 681-687 (INDIA).

[9]. Bh. Satyanarayana and K. Syam Prasad (2009) “Discrete Mathematics and Graph

Theory”, Printice Hall of India, New Delhi (ISBN: 978-81-203-3842-5).

[10].Bh. Satyanarayana, K. Syam Prasad and D. Nagaraju (2006) "A Theorem on

Modules with Finite Goldie Dimension", Soochow J. Maths 32(2), pp 311-315.


77

On Some Results on Semi-

Complete Graphs

Invited Lecture by

Prof. I H Nagaraja Rao,

Sr. Professor, Department of

Mathematics, G V P College for

P G Courses, Rishikonda,

Visakhapatnam, India, Email:

[email protected]





78


79


80


81


82


83


84

Modified Second-Order

Slope-Rotatable Designs

with Equi- Spaced

Levels-A Review

Abstract

This article presents a review on modified second-order slope-rotatable designs (SOSRD)

with equi-spaced levels. It presents different methods of construction of modified

SOSRDs with equi-spaced levels, using central composite designs, balanced incomplete

block designs (BIBD), symmetrical unequal block arrangements (SUBA) with two

unequal block sizes, pairwise balanced designs (PBD) etc. Comparisons of different

methods of constructions of modified SOSRD with equi-spaced levels for 153 ≤≤ v are

given.

Introduction

Response surface methodology is a statistical technique very useful in design and

analysis of scientific experiments. In many experimental situations the experimenter is

concern with explaining certain aspects of a functional relationship

exxxfY v += ),...,,( 21 , where Y is the response, vxxx ,...,, 21 are v factors and e is the

random error. The function f(.) is called response surface or response function. Designs,

which are used, for the study of response surface methods, are called response surface

designs. Response surface methods are useful where several independent variables

influence a dependent variable. The independent variables are assumed to be continuous

and controlled by the experimenter. The response is assumed to be as random variable.

For example, if a chemical engineer wishes to find the temperature ( 1x ) and pressure ( 2x )

that maximizes the yield (response) of his process, the observed response Y may be

written as a function of the factors temperature ( 1x ) and pressure ( 2x ) as exxfY += ),( 21

.

In many applications of Response Surface Methodology, good estimation of the

derivatives of the response function may be as important or perhaps more important than

estimation of mean response. Certainly, the computation of a stationary point in a

Invited Lecture by

B. Re. VICTOR BABU

Department of Statistics, Acharya

Nagarjuna University,






85

second-order analysis or the use of gradient techniques for example, steepest ascent or

ridge analysis depends heavily on the partial derivatives of the estimated response

function with respect to the design variables. Since designs that attain certain properties

in Y (estimated response) do not enjoy the same properties for the estimated derivatives

(slopes), it is important for the user to consider experimental designs that are constructed

with the derivatives in mind.

The concept of rotatability, which is very important in response surface second-order

designs, was proposed by Box and Hunter (1957). A design is said to be rotatable if the

variance of the response estimate is a function only of the distance of the point from the

design center. The study of rotatable designs is mainly emphasized on the estimation of

differences of yields and its precision. Estimation of differences in responses at two

different points in the factor space will often be of great importance. If differences in

responses at two points close together is of interest then estimation of local slope (rate of

change) of the response is required. Estimation of slopes occurs frequently in practical

situations. For instance, there are cases in which we want to estimate rate of reaction in

chemical experiment, rate of change in the yield of a crop to various fertilizer doses, rate

of disintegration of radioactive material in an animal etc. Hader and Park (1978)

introduced slope-rotatable central composite designs (SRCCD). Different methods of

constructions of SOSRDs were suggested by various authors, including Victorbabu

(2002a, b, c, 2003, 2007), Victorbabu and Narasimham (1991, 1993a, b,1994, 2000-01),

and so on. Further, Victorbabu (2005a) studied modified slope-rotatable central

composite designs. Different methods of constructions of modified SOSRDs were

suggested by Victorbabu (2005b, 2006a, b, 2008a, 2009b) and so on. Specifically,

Victorbabu (2009a) introduced modified SOSRD with equi-space levels using central

composite designs and balanced incomplete block designs. Different methods of

constructions of modified SOSRD with equi-spaced levels were suggested by Victorbabu

(2008 b, c) and so on.

1. Second-order Slope-rotatable Designs

A second-order response surface model is D = ))x(( iu for fitting,

∑ ∑<

++∑=

+∑=

+=i j

ue

jux

iux

ijb

v

1i

2iu

xii

bv

1iiu

xi

b0

bu

Y (2.1)

where iux denotes the level of the thi factor ( v)1,2,..,i = in the thu run N)1,2,..,(u = of

the experiment, s'eu are uncorrelated random errors with mean zero and variance

Second-order slope-rotatable design: A second-order response surface design D is said

to be a SOSRD, if the variance of the estimate of first order partial derivative )x/Y( iu ∂∂with respect to each of independent variables )x( i is only a function of the distance


86

)(22 ∑=

v

i

ixd of the point )...,,,( 21 vxxx from the origin (centre) of the design. Such a

spherical variance function for estimation of slopes in the second-order response surface

is achieved if the design points satisfy the following conditions (cf. Hader and Park

(1978, Victorbabu and Narasimham , 1991).

∑=

∏=

∑ ≤=N

1u

v

1i

4iαforodd,is

iαanyif0iα

iux (2.2)

2Nλconstant

N

1u

2iu

x(i) ==∑=

,

4cNλconstant

N

1u

4iu

x(ii) ==∑=

, for all i (2.3)

jifor,4

NλconstantN

1u

2ju

x2iu

x ≠==∑=

(2.4)

( ) 22vλ

4λ1vc >−+ (2.5)

0]4)5c(v[])3c()c5(v[ 22

24 =+−λ+−−−λ (2.6)

where c, 42 andλλ are constants and the summation is over the design points.

1. Modified SOSRD

The usual method of construction of SOSRD is to take combinations with unknown

constants, associate a v2 factorial combinations or a suitable fraction of it with factors

each at 1± levels to make the level codes equidistant. All such combinations form a

design. Generally SOSRDs need at least five levels (suitably coded) at 0, 1± , a± for all

factors (0,0, …,0 – chosen center of the design, unknown level ‘a’ to be chosen suitably

to satisfy slope-rotatability). Generation of design points this way ensures satisfaction of

all the conditions even though the design points contain unknown levels.

Alternatively by putting some conditions indicating some relation among ∑ 2

iux , ∑ 4

iux

and ∑ 2

ju

2

iu xx some equations involving the unknown levels are obtained and their

solution gives the unknown levels. In SOSRD the conditions used are )(4)( iiij bVbV =

and ∑∑=

2

ju

2

iu

4

iu

xx

xc . Other conditions are also possible though, it seems, not yet exploited.

We shall investigate the condition (∑ 2

iux )2

= N∑ 2

ju

2

iu xx i.e., (N 2λ )2 = N(N 4λ ) i.e.,

4

2

2 λλ = to get another series of symmetrical response surface designs which provide more


87

precise estimates of response at specific points of interest than what is available from the

corresponding existing designs. By applying this new condition 4

2

2 λλ = in equation

(2.6), we get c=1 or c=5. The non-singularity condition (2.5) leads to c=5. It may be

noted 4

2

2 λλ = and c=5 are equivalent conditions. Further,

( )N4

4v)b(V

2

0

σ+= ,

4

2

i

N)b(V

λσ

= , 4

2

ijN

)b(Vλ

σ= ,

4

2

iiN4

)b(Vλ

σ= ,

4

2

ii0N4

)b ,b( Covλ

σ−= .

It is seen that if 4

2

2 λλ = , then 0)b,bCov( ijii = and other covariances are zero.

These modifications of the variances and covariances affect the variance of estimated

response at specific points considerably.

=

∂∂

i x

YV

2

4

2

4

N

dσ

λλ

+.

1. Construction of modified SOSRD with equi-spaced levels using central

composite designs The most widely used design for fitting a second order model is the central composite

design. Central composite designs are constructed by adding suitable factorial

combinations to those obtained from v

p2x

2

1 fractional factorial design (here )v(t2 =

v

p2x

2

1 denotes a suitable fractional replicate of v2 , in which no interaction with less

than five factors is confounded). In coded form the points of )2(2 )v(tv factorial have

coordinates (±a, ±a, … , ±a) and 2v axial points have coordinates of the form (±b, 0,

…,0), (0, ±b, …,0), …, (0,0, …, ±b) etc., The corresponding equi-spaced levels design of

the composite type is obtained by changing the axial points from )0,...,0,0,b(± etc., to

)0,...,0,0,a2(± . The axial points may be replicated an times and central points to be

replicated 0n times.

Theorem 4.1 A central composite design with design points

0

1

a

t(v) n22a,0,...0)(na)2a,...,a,( UU ±±±± will be a v-dimensional modified SOSRD with


88

equi-spaced levels in ( )

t(v)

2

a

t(v)

2

8n2N

+= design points, if 3t(v)

a 2n −= and

( )a

t(v)

t(v)

2

a

t(v)

0 2vn22

8n2n −−

+= .

2. Construction of modified SOSRD with equi-spaced levels

using BIBD Let ( )λk,r,b,v, be a BIBD, t(k)2 denote a fractional replicate of 2

k in 1± levels in

which no interaction with less than five factors is confounded. Let [ ])λk,r,b,(v,-a

denote the design points generated from the transpose of the incidence matrix of BIBD,

and [ ] t(k)2)λk,r,b,(v,-a are the b2t(k) design points generated from BIBD by

“multiplication” (see Raghavarao, 1971, pp.298-300). Choose the additional unknown

combinations ,0)2a,0,0,...(± by permuting over the different factors and multiply with 12

associate combinations to obtain the additional design points. Repeat this set of additional

design points say ‘an ’ times, where

an is determined so as to satisfy the conditions of

modified slope rotatability and U denotes the union of the design points generated from

different sets of points and )(n0 be the number of central points in the design.

Theorem 5.1 The design points, [ ]0

n1,0)22a,0,0,...(a

nt(k)

2λ)k,r,b,(v,a UU ±− will

give a v-dimensional modified SOSRD with equi-spaced levels in t(k)

2

a

t(k)

λ2

)8n(r2N

+=

design points if, 5t(k)

a r)2(5λn −−= and v2nb2λ2

)8n(r2n a

t(k)

t(k)

2

a

t(k)

0 −−+

= .

3. Modified SOSRD with equi-spaced levels using SUBA with two

unequal block sizes SUBA with two unequal block sizes (cf.

Raghavarao, 1971): The arrangement of v-treatments in b blocks where 1b

blocks of size 1k and 2b blocks of size 2k b)b(b 21 =+ , is said to be a

symmetrical unequal block arrangements with two unequal block sizes if

(i) every treatment occurs v

ik

ib

blocks of size ik ( i = 1,2), and

(ii) every pair of first associate treatments occurs together in u blocks of size

1k and in (λ-u) blocks of size 2k while every pair of second associate

treatments occurs together in λ blocks of size 2k .


89

From (i) each treatment occurs in rv

kb

v

kb 2211 =+

blocks in all.

),b,b,k,k,r,b,v(2121

λ are known as the parameters of the SUBA with two unequal

block sizes.

Let there be a SUBA with two unequal block sizes with parameters

),b,b,k,k,r,b,v(2121

λ and )k,sup.(kk 21= . Let us write the design in the form of a

vb× matrix, the elements of which are zero and ‘ a ’. If in any block a particular

treatment occurs the element in that block corresponding to that treatment will be ‘ a ’,

otherwise, zero. We denote these design points generated from the transpose of the

incidence matrix of a SUBA with two unequal block sizes by ( )[ ]λ,b,b,k,kr,b,v,a 2121− .

Let t(k)2 denotes the number of design points of a fractional factorial design of k2 in ± 1

levels, such that no interaction with less than five factors is confounded, and

( )[ ] t(k)

2121 2λ,b,b,k,kr,b,v,a − is the b2t(k)

design points generated from the SUBA with

two unequal block sizes by ‘multiplication’. Choose the additional unknown

combinations ,0)2a,0,0,...(± by permuting over the different factors and multiply with 12

associate combinations to obtain the additional design points. Repeat this set of additional

design points say ‘ an ’ times, where an is determined so as to satisfy the conditions of

modified slope rotatability.

Theorem 6.1 The design points,

[ ]0

n1,0)22a,0,0,...(a

nt(k)

2λ),2

b,1

b,2

k,1

kr,b,(v,a UU ±− will give a v-dimensional

modified SOSRD with equi-spaced levels using SUBA with two unequal block sizes in

t(k)

2

a

t(k)

λ2

)8n(r2N

+= design points if, 5t(k)

a r)2(5λn −−= and

v2nb2λ2

)8n(r2n a

t(k)

t(k)

2

a

t(k)

0 −−+

= .

4. Construction of modified SOSRD with equi-spaced levels

using PBD Let there be an equi-replicated PBD with parameters λ),k,...,k,kr,b,(v, p21 and

)k,...,k,sup.(kk p21= . Let us write the design in the form of a vb× matrix, the elements

of which are zero and ‘ a ’. If in any block a particular treatment occurs the element in

that block corresponding to that treatment will be ‘ a ’, otherwise, zero. We denote these

design points generated from the transpose of the incidence matrix of a PBD by

( )[ ]λ,k...,,k,kr,b,v,a p21− . Let t(k)2 denotes the number of design points of a fractional

factorial design of k2 in ± 1 levels, such that no interaction with less than five factors is

confounded. ( )[ ] t(k)

p21 2λ,k...,,k,kr,b,v,a − is the b2t(k)

design points generated from the

PBD by ‘multiplication’. Choose the additional unknown combinations ,0)2a,0,0,...(± by

permuting over the different factors and multiply with 12 associate combinations to obtain


90

the additional design points. Repeat this set of additional design points say ‘ an ’ times,

where an is determined so as to satisfy the conditions of modified slope rotatability.

Theorem 7.1 The design points,

0n1,0)22a,0,0,...(

an

t(k)2λ),

pk,...,

2k,

1kr,b,(v,a UU ±−

will give a v-dimensional

modified SOSRD with equi-spaced levels using PBD in t(k)

2

a

t(k)

λ2

)8n(r2N

+= design points

if, 5t(k)

a r)2(5λn −−= and v2nb2λ2

)8n(r2n a

t(k)

t(k)

2

a

t(k)

0 −−+

= .

Comparison of different methods of modified SOSRD with equi-spaced levels

Number of

factors ‘v’

CCD

(2009)

BIBD

(2009)

SUBA with

two unequal

PBD

(2008c) 3 32 -- -- --

4 64 -- -- --

5 64 -- -- --

6 128 600

(6,15,10,4,6

484

(6,15,8,2,4,6

--

7 256 242

(7,7,4,4,2)

-- --

8 256 432

(8,14,7,4,3)

588

(8,24,9,2,4,1

392

(8,15,6,4,3,29 512 -- 392

(9,15,6,3,4,6

392

(9,15,6,4,3,210 512 392

(10,15,6,4,2

578

(10,25,8,4,3,

--

11 512 726

(11,11,6,6,3

-- --

12 512 768

(12,33,11,4,

864

(12,15,7,4,6,

784

(12,16,6,6,5,13 -- 980

(13,39,15,5,

-- 784

(13,16,6,6,5,14 -- -- -- 1728

(14,15,7,7,6,15 -- 1728

(15,15,7,7,3

784

(15,16,6,5,6,

784

(15,16,6,6,5,

References 1. Box, G.E.P. and Hunter, J.S. (1957), Multifactor experimental designs for exploring

response surfaces, Annals of Mathematical Statistics, 28, 195-241.

2. Hader, R.J. and Park, S.H. (1978). Slope-rotatable central composite designs,

Technometrics, 20, 413-417.

3. Raghavarao, D. (1971). CONSTRUCTIONS AND COMBINATORIAL PROBLEMS

IN DESIGN OF EXPERIMENTS, JOHN Wiley, New York.

4. Victorbabu, B. Re. (2002a). A note on the construction of four and six level second-order

slope-rotatable designs, Statistical Methods, 4, 11-20.

5. Victorbabu, B. Re. (2002b). Construction of second-order slope-rotatable designs using

symmetrical unequal block arrangements with two unequal block sizes, Journal of the

Korean Statistical Society, 31, 153-161.

6. Victorbabu, B. Re. (2002c). Second-order slope-rotatable designs with equi-spaced levels,

Proceedings of Andhra Pradesh Akademi of Sciences, 6, 211-214.


91

7. Victorbabu, B. Re. (2003). On second-order slope-rotatable designs using incomplete block

designs, Journal of the Kerala Statistical Association, 14, 19-25.

8. Victorbabu, B. Re., (2005a). Modified slope-rotatable central composite designs, Journal

of the Korean Statistical Society, Vol. 34, 153-160.

9. Victorbabu, B. Re., (2005b). Modified second-order slope-rotatable designs using

pairwise balanced designs, Proceedings of Andhra Pradesh Akademi of Sciences, Vol. 9

(1), 19-23.

10. Victorbabu, B. Re., (2006a). Modified second-order slope-rotatable designs using BIBD,

Journal of the Korean Statistical Society, 35 (2), 179-192.

11. Victorbabu, B. Re., (2006b). Construction of modified second-order rotatable designs and

second-order slope-rotatable designs using a pair of balanced incomplete block designs,

Sri Lankan Journal of Applied Statistics, 7, 39-53.

12. Victorbabu, B. Re., (2007). On second-order slope-rotatable designs –A Review, Journal

of the Korean Statistical Society, 33 (3), 373-386.

13. Victorbabu, B. Re., (2008a). On modified second-order slope-rotatable designs using

incomplete block designs with unequal block sizes, Advances and Applications in

Statistics, 8(1), 131-151.

14. Victorbabu, B. Re., (2008b). Modified second-order slope-rotatable designs with equi-

spaced levels using symmetrical unequal block arrangements with two unequal block

sizes, Journal of the Kerala Statistical Association, 19, 31-39.

15. Victorbabu, B. Re., (2008c). Modified second-order slope-rotatable designs with equi-

spaced levels using pairwise balanced designs, Ultra Scientist of Physical Sciences,

20(2) M, 257-262.

16. Victorbabu, B. Re., (2009a). Modified second-order slope-rotatable designs with equi-

spaced levels, Journal of the Korean Statistical Society, 39, 59-63.

17. Victorbabu, B. Re. (2009b). On modified second-order slope- rotatable designs, ProbStat

Forum, 2, 115-131.

18. Victorbabu, B. Re. and Narasimham, V.L. (1991). Construction of second-order slope-

rotatable designs through balanced incomplete block designs, Communications in

Statistics -Theory and Methods, 20, 2467-2478.

19. Victorbabu, B. Re. and Narasimham, V.L. (1993a). Construction of second-order slope-

rotatable designs using pairwise balanced designs, Journal of the Indian Society of

Agricultural Statistics, 45, 200-205.

20. Victorbabu, B. Re. and Narasimham, V.L. (1993b). Classification and parameter bounds

of second-order slope-rotatable designs, Reports of Statistical Application Research,

Union of Japanese Scientists and Engineers, 40, 12-19.

21. Victorbabu, B.Re. and Narasimham, V.L. (1994). A new type of slope-rotatable central

composite design, Journal of the Indian Society of Agricultural Statistics, 46, 315-317.

22. Victorbabu, B. Re. and Narasimham, V.L. (2000-01). A new method of construction of

second-order slope-rotatable designs, Journal of Indian Society for Probability and

Statistics, 5, 75-79.

A Note on Semi-Prime

Near-Rings

Abstract

It is well known that the algebraic system Near-ring is a system satisfying all the axioms of a Ring except

possibly one of the distributive law and commutativity of addition. The concepts Semi-prime ideal, and

Essential ideal play an important role in the theory of Near-rings. The aim of the present short paper is to

prove the following result: If I is a proper ideal of a semi-prime near-ring N, then there exists an essential

ideal J of N such that I = J ∩ (P - rad(I)).

A.M.S. Subject Classification: 16 D 25, 16 Y 30, and 16 Y 99.

Key Words: Near-ring, Semi-prime Ideal, Semi-prime Near-ring, Complement Ideal, Essential

Ideal.

1. Introduction

In this section, we present some fundamental definitions and results from the literature.

An algebraic system (N, +,.) is said to be a near-ring if it satisfies the following three conditions:

(i) (N, +) is a group; (ii) (N, .) is a semi-group; (iii) (x + y)z = xz + yz for all x, y, z ∈ N.

Further if N satisfies the condition that “x0 = 0 for all x ∈ N”, where ‘0’ is the identity element

of (N, +), then N is called a zero-symmetric near-ring. We abbreviate (N, +, .) by N. Throughout

N stands for a zero-symmetric right near-ring. If A1, A2, …, An are subsets of N, then A1A2…An

denotes the set a1a2…an / ai ∈ Ai for 1 ≤ i ≤ n. In the special case when all Ai’s are the same

set A, we denote A1A2…An by An.

An ideal P of N is called a prime ideal if for any two ideals J and I of N, IJ ⊆ P, implies either

I ⊆ P or J ⊆ P. An ideal S of N is said to be semi-prime if for any ideal I of N, I2 ⊆ S, implies

I ⊆ S.

Authors: Satyanarayana Bhavanari,

Department of Mathematics, Acharya

Nagarjuana University, Nagarjuna Nagar-522

510, Andhra Pradesh, India.


!!"!##$!$%& '(&) *

+( **,(*-+.

A near-ring N is said to be (i) prime near-ring if (0) is a prime ideal; and (ii) Semi-prime near-

ring if (0) is a semi-prime ideal. The intersection of all prime ideals of N is called the prime

radical of N and it is denoted by P-rad(N). For any proper ideal I of N, the intersection of all

prime ideals of N containing I, is called the prime radical of I and is denoted by P-rad(I).

1.1 Remark: Let I and J be two ideals of N such that I ⊆ J. Now P-rad (I) = the intersection

of prime ideals containing I ⊆ the intersection of prime ideals containing J = P-rad (J). Thus I

⊆ J implies P-rad(I) ⊆ P-rad(J).

1.2 Result (Sambasiva Rao [3]): An ideal I of a Near-ring N is a semi prime ideal if and only if

I = P-rad(I).

2. Essential Ideals

We start this section the definitions ‘essential ideal’ and ‘complement ideal’. We also list two

necessary results from the literature.

2.1 Definition: Let I, J be two ideals of N such that I ⊆ J. We say that I is essential (or ideal

essential) in J if it satisfies the following condition:

K N, K ⊆ J, I ∩ K = (0) imply K = (0). We write as I ≤e J.

2.2 Definitions: If K N, A N and K is a maximal element among the ideals I of N

with respect to the property I ∩ A = (0), then we say that K is a complement of A (or a

complement in N).

Equivalently, the ideal K is said to be a complement of an ideal I of N if (i) K I = (0); and (ii)

K1 is an ideal of N containing K properly, imply K

1 I (0).

2.3 Remark (Remark 1.3 of Satyanarayana, Godloza and Vijaya Kumari [9]): Let I & J be

ideals of N.

(i) I ≤e J if and only if I ∩ K = (0), K N J ∩ K = (0).

(ii) B is a complement in N ⇔ there exists an ideal A of N such that B ∩ A = (0) and

K1 ∩ A ≠ (0) for any ideal K1 of N with B K1. In this case B + A ≤e N.

(iii) If A ∩ B = (0), and C is an ideal of N which is maximal with respect to the property

C A, C ∩ B = (0), then C ⊕ B is essential in N, where ⊕ denote the direct sum. Moreover,

C is a complement of B containing A.

2.4 Result (Result 1.4 of Satyanarayana, Godloza and Vijaya Kumari [9]): (i) The intersection of

finite number of essential ideals is essential;

(ii) If I, J, K are ideals of N such that I ≤e J, and J ≤e K, then I ≤e K;

(iii) I ≤e J I ∩ K ≤e J ∩ K.

(iv) If I ⊆ J ⊆ K, then I ≤e K if and only if I ≤e J, and J ≤e K; and

(v) Suppose N1, N2 are two near-rings and f : N1 → N2 is a near-ring isomorphism. If A is an

ideal of N1, then A ≤e N1 ⇔ f(A) ≤e N2.

Let us fix an ideal I of N. By Zorn’s Lemma, the set of all ideals H of N satisfying H I = (0)

contains a maximal element, say K. Again by using Zorn’s Lemma the set of all ideals X of N

satisfying X I and X K = (0) contains a maximal element, say K*. Then we have the

following lemma.

2.5 Lemma (See Lemma 1.4 of Satyanarayana [7]) : (i). K is a complement of I; (ii) K* is a

complement of K; (iii) K + I and K + K* are essential ideals; and (iv) I is essential in K

*.

3. Semi-prime Near-rings

In this section we prove the main theorem of this paper. Before proving our main theorem, we

prove the following two lemmas.

3.1. Lemma: If N is a semiprime near-ring, then every complement ideal is semiprime.

Proof: Let K be a complement of a non-zero ideal. Suppose K = (0). Since N is a semiprime

near-ring, by definition, K = (0) is a semiprime ideal of N. Now suppose that K is a non zero

ideal of N. Let K be a complement of a non-zero ideal B of N. This means that K is a maximal

among all the ideal J with respect to the property that J ∩ B = (0). In a contrary way, suppose

that K is not a semiprime ideal. Then there exists an ideal A of N such that A2 ⊆ K and A K.

Since A K, and K is a complement ideal, we have that (K + A) ∩ B ≠ (0). Let 0 ≠ a ∈ (K + A)

∩ B. Now <a>2 ⊆ (K + A)

2 ⊆ (A2 + K) ⊆ K. Since a ∈ B, we also have that <a>

2 ⊆ B.

Therefore <a>2 ⊆ (K ∩ B) = (0). Since N is a semiprime near-ring, we get that a = 0, a

contradiction to the selection of the element a in N. This proves that K is a semiprime ideal.

Hence every complement ideal is a semiprime ideal.

3.2 Lemma: If N is a semiprime near-ring, then every proper ideal I of N is essential in its prime

radical P-rad(I).

Proof: By Lemma 2.5, there exists two ideals K and K* such that K is a complement of I; K

* is

a complement of K containing I; and I ⊕ K, K* ⊕ K both are essential ideals of N. Also we

have that I is essential in K*. By above Lemma 3.1, K

* is a semiprime ideal. Since I ⊆ K*, by

Remark 1.1, we have that P-rad(I) ⊆ (P-rad(K*)). Since K* is a semiprime ideal, it follows that

P-rad(K*) = K*. Now it is clear that I ⊆ P-rad(I) ⊆ P-rad(K*) = K*. Since I is essential in K*,

and I ⊆ P-rad(I) ⊆ K*, by result 2.4(iv), it follows that I is essential in P-rad(I).

Now we are ready prove the main result of this short paper.

3.3 Theorem: Let I be a proper ideal of a semiprime near-ring N. Then there exists an essential

ideal J of N such that I = J ∩ (P - rad(I)).

Proof: If I essential in N, then I = J ∩ (P-rad(I)), where J = I.

Suppose I is not an essential ideal. Let K be a complement of I. By Remark 2.3(iii), we have

that J = I ⊕ K is essential in N. By modular law.

(P-rad(I)) ∩ J = (P-rad(I)) ∩ (I ⊕ K) = I + (K ∩ (P-rad(I))).

Now K ∩ (P-rad(I)) is an ideal of N contained in P-rad(I) and I ∩ (K ∩ (P-rad(I)) = (0). By

Lemma 3.2, we have that I is essential in P-rad(I). By the definition of essential ideal, we have

that K ∩ (P-rad(I)) = (0). It follows that I = J ∩ (P-rad(I)), and I is essential in J.

Acknowledgements

The author acknowledges the financial assistance from the UGC, New Delhi under the grant F.

No.34-136/2008 (SR) dated 30th

December 2008. The author thanks the referee for valuable

comments that improved the paper.

References

1. Pilz G. “Near-rings”, North Holland, New York, 1983.

2. Reddy Y. V. & Satyanarayana Bhavanari “On finiThe f-prime Radical in Near-rings”, Indian J.

Pure & Appl. Math. 17 (1986) 327-330.

3. Sambasiva Rao V. “A Characterization of Semi-prime Ideals in Near-rings”, Journal of

Australian Mathematical Society 32 (1982) 212-214.

4. Sambasiva Rao V. & Satyanarayana Bhavanari "The Prime Radical in Near-rings", Indian

J. Pure & Appl. Math. 15 (1984) 361- 364.

5. Satyanarayana Bhavanari "Tertiary Decomposition in Noetherian N-groups", Communications in

Algebra, 10 (1982) 1951 – 1963.

6. Satyanarayana Bhavanari "Primary Decomposition in Near rings", Indian J. Pure & Appl. Math.

15 (1984) 127-130.

7. Satyanarayana Bhavanari "On Finite Spanning dimension in N-groups", Indian J. Pure &

Appl. Maths. 22 (8) 633-636, August 1991. (Zbl 0748.16024).

8. Satyanarayana Bhavanari “Contributions to Near-ring Theory”, VDM Verlag Dr Muller,

Germany, 2010 (ISBN: 978-3-639-22417-7).

9. Satyanarayana Bh., Godloza L and Vijaya Kumari A. V., “Finite Dimension in Near-rings”,

Journal of AP Society for Mathematical Sciences, 1 (2) (2008) 62-80.

10. Satyanarayana Bhavanari & Richard Weigandt "On the f-prime Radical of Nearrings",

in the book ‘Nearrings and Nearfields (Editors: Hubert Kiechle, Alexander Kreuzer and

Momme Johs Thomsen) (Proc. 18th International Conference on Nearrings and

Nearfields, Universitat Bundeswar, Hamburg, Germany, July 27-Aug 03, 2003), Springer

Verlag, Netherlands, 2005, pp 293-299.

11. Satyanarayana Bhavanari & Syam Prasad Kuncham “Discrete Mathematics and Graph Theory”,

Printice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).

Fuzzy Numbers and

Matrix

Transformation

Abstract

In this paper we characterize the classes where , or ( )l R∞ and denotes

the set of fuzzy analytic sequences.

Mathematical Subject Classification: 46A45

Keywords: Sequence spaces, Fuzzy numbers, analytic sequences, Matrix transformations.

1. Preliminaries, Background and Notation:

A sequence space is defined to be a linear space with elements in another space. Throughout the

paper , and denotes the set of non-negative integers, the set of real numbers and the set of

complex numbers respectively. Let ω denote the space of all sequences (real or complex ) ; l∞

and c respectively denotes the space of all bounded sequences , the space of convergent

sequences .

Let X and Y be two nonempty subsets of ω . Let ( ), ( , )nk

A a n k= ∈ be an infinite matrix of

real or complex numbers. We write ( ) ( ) .n n nk k

k

Ax A x a x= = Then Ax = ( )n

A x is called the A -

transform of x , whenever ( )n nk k

k

A x a x= < ∞ for all n. We with limn

Ax = lim ( )n

nA x . If x X∈

implies Ax Y∈ , we say that A defines a matrix transformations from X into Y , denoted by

Authors: Abdul Hamid, Tanweer Jalal and Neyaz

Ahmad, Department of Mathematics, National

Institute of Technology, Hazatbal, Jammu and Sri

Nagar (India)-190006,

Emails [email protected] ,

[email protected], [email protected]

!!"!##$!$ %& '(&) *

+( **,(*-+.

:A X Y→ . By ( ):X Y , we mean the class of all matrices A such that :A X Y→ .The matrix domain

AX of an infinite matrix A in a sequence space X is defined as

( ) :

A kX x x Ax Xω= = ∈ ∈

The concept of Fuzzy theory was introduced by Zadeh [5] and later by several authors (see, [1],

[3], [4] ).A fuzzy number is a function from to [0,1] so that

a) is normal, i.e., there exists an such that ,

b) is a fuzzy convex, i.e., for any and ,

c) d) is upper semi continuous, i.e., for every , is open in the

usual topology on .

e) The closure of is compact.

These property imply that for each ! , the level set " " is

nonempty compact convex subset of with the compact support. Let # denote the set of all

closed boun $ $ $% on , and also & ' &()$ *) )$ *)+ . Let , denote the

set of all fuzzy numbers. The linear structure of , induces addition - and scalar

multiplication interms of -level sets by -% % -% and % %for each .

Define a map . , / , 0 by . - 123456 - and hence (, .+ is a

complete metric space.For - , define - if and only if - for %. A sequence is called a fuzzy sequence if the terms are fuzzy numbers.

A sequence of fuzzy numbers is said to be convergent to the fuzzy number zero, if for

every there exists a positive integer 7 such that

. 8 for 9 and let the set of all fuzzy sequences.

A sequence of fuzzy numbers is said to be convergent to the fuzzy number , if for

every there exists a positive integer 7 such that

. 8 for 9 and let the set of all fuzzy sequences.

A sequence of fuzzy numbers is said to be bounded if the set : 7 of fuzzy

numbers is bounded and let : denote the set of fuzzy bounded sequences.

A sequence of fuzzy numbers is said to be entire if . 0 as 0 and let

the set of all fuzzy entire sequences.

A sequence of fuzzy numbers is said to be analytic if . %;< is bounded and let

= the set of all fuzzy entire sequences.

Main Result: Throughout the paper we write,

- > ?@@@ and AA .?@ , where $ ?@ is an infinite matrix.

Theorem 2: - = if and only if B C> .?@ @ D;< is bounded. (1)

Proof: Sufficient Condition: Since )E E :, it is enough to prove the sufficiency

for : .Since : there is a constant F such that

.?@ F GH We take G .Now, we have

A-A;< I.> ?@@ @ J;< IA> ?@@ A.@ J

;<

F;<> A?@A@ ;< F;

<(> .?@ @ +;< K9, by (1),

which proves sufficiency.

Necessary Condition: It is enough to prove the necessity for . Suppose that (1) is not

satisfied, then by selecting a subsequence and suppose that B 0 monotonically.

(2)

The matrix L ?MN to applicable to each member of) , the series > .?@ @ , O P,

must all be convergent. (3)

Also putting QN and QR for all S T U so that QN VW.We have XM ?MN.

In this case , if XM W,we must have C> .?@ @ D;< KN for fixed U andO (4)

We shall construct a sequence QN VW with the condition .Q@ U (5)

and show that the corresponding XM Y W .This will prove the necessity of (1).

We choose O by (2) such that

BMZ C> .(?MZ@ +@ D;<Z [ (6)

and Uby (3) such that

C> .(?Z@ +N\NZ] D;<Z (7)

*2^ then we have

C> .(?Z@ +NZN\ D;<Z BM; C> .(?Z@ +N\N;] D

;<Z [ (8)

Now,

)XMZ);_Z I`(> ?Z@@ NZN\ +J

;_Z I`(> ?Z@@ N\NZa; +J

;_Z

I)> ?Z@NZN\ )`@ J;_Z I)> ?Z@N\NZ] )`@ J

;_Z

I> )?Z@)NZN\ J;_Z I> )?Z@)N\NZ] J ;_Z

C> .(?Z@ +NZN\ D;<Z C> .(?Z@ +N\NZ] D

;<Z

, by (7) & (8)

When U is already fixed we have by (4)

C> .?@ NZN\ D;< K Kb cKNZ dNZ O (10)

ChoosingO O, by (2) such that

BM; C> .(?M;@ +@ D;<; edNZ f (11)

Then, U U by (3) such that

C> .(?M;@ +N\NZ] D;_; (12)

But we have

g`h> ?;@@ N;N\NZ] ij;_;

BM; C> .(?;@ +NZN\ D;<; C> .(?;@ +N\ D

;<;

edNZ f dNZ dNZ e (13)

Now, we have

)XM;);_; g`h> ?M;@@ N;N\NZ] ij

;_; I`(> ?;@@ NZN\ +J

;_;

I`(> ?M;@@ N\N;] +J;_;

gk> ?M;@N;N\NZ] k`@ j;_; I)> ?M;@NZN\ )`@ J

;_;

!""

I)> ?M;@N\N;] )`@ J;_;

g> )?M;@)N;N\NZ] j;_; I> )?Z@)NZN\ J

;_; I> )?M;@)N\N;] J ;_;

l> .(?M;@ +N;N\NZ] m;_; C> .(?M;@ +NZN\ D

;_;

C> .(?M;@ +N\N;] D;_;

dNZ e dNZ , by (12) & (13).

* proceeding in this way we shall get )XMn);_n o .Hence, AXMA;_ 0 as o 0 , through the

subsequence Op . Hence, XM > ?MNN QN Y qW.This is contradiction and hence there by

proving the necessity of (1) and the proof is complete.

References:

[1] Basarir, M. and Mursaleen, “ Some Sequence Spaces of Fuzzy Numbers Genera-

ted by Infinite Matrices ” , The Journal of Fuzzy Mathematics , 11(3),(2003)757-

764.

[2] Maddox, I.J., “ Elements of Functional Analysis” , Cambridge University Press ,

1970.

[3] Matloka, M., “Sequences of Fuzzy Numbers ”, BUSEFAL, 28,(1986) 23-27.

[4] Nanda, S., “ On Sequences of Fuzzy Numbers ” , Fuzzy sets and Systems, 33,

(1989) 123-126.

[5] Zadeh, L. A ., “ Fuzzy Sets ” , Inform and Control 8 , (1965) 338-353.

!"!

Some Results on

Completely Semi-Prime

Ideals in Gamma Near-

Rings

Abstract

In this paper we considered the algebraic system Γ-near-ring that was introduced by Satyanarayana.

Γ-near-ring is a more generalized system than both near-ring and Γ-ring. The aim of this short paper is to

study some important results related to the concepts: Prime and Semi- prime Ideals in Γ-near-rings.

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], Satyanarayana, Pradeep Kumar & Srinivasa Rao [12 ] also

contributed to the theory of Γ-rings. A generalization of both the concepts near-ring and Γ-ring, namely

Γ-near-ring was introduced by Satyanarayana [ 9 ] and later studied by several authors like: Booth [ 2 ],

Booth & Godloza [3], Syam Prasad [15], Satyanarayana, Pradeep kumar, Sreenadh, and Eswaraiah Setty

[13].

Authors: Pradeep Kumar T.V.

*, Satyanarayana

Bhavanari@

, Syam Prasad Kuncham# and Mohiddin

Shaw Sk@

.

*: Department of Mathematics, ANU College of

Engineering and Technology, Acharya Nagarjuna

University, Nagarjuna Nagar-522 510, AP., @

: Department of Mathematics, Acharya Nagarjuna

University, Nagarjuna Nagar-522 510, AP. #: Department of Mathematics, Manipal University,

Manipal-576 104.

Emails: [email protected],

[email protected]#

[email protected]

!!"!##$!$ %& '(&) *

+( **,(*-+.

!"

This short paper is divided into three sections. The section – 3 contains new results related to the concept:

completely semi-Prime ideal. In the sections 1 and 2, we collect some existing definitions and examples

which are to be used in the later section – 3.

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 semi-group; 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) – nn

1 ∈ I for all I ∈ I and n, n1 ∈ N (Equivalently,

n(I + n1) – nn

1 ∈ 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 this fact by I N.

1.3. Definitions: (i) An ideal P of N (with P ≠ N) is said to be a prime ideal of N if it

satisfies the condition: I, J are 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.

For other fundamental definitions and results in near-rings, we refer Pilz [5], Satyanarayana

& Syam Prasad [14].

1. 4. Definition: (Satyanarayna [9]): 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:

1.5. Example (Satyanarayana [10]): 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, we have that (M, +) is non-abelian. Hence forth, M stands for a zero symmetric

Γ-Near-ring.

!"

2. Some relations between Semi-prime, Completely semi-prime

Ideals

2.1. Definitions (Satyanarayana [9, 11]): An ideal A of M is said to be (i) prime if B and C are

ideals of M such that BΓC ⊆ A implies B ⊆ A or C ⊆ A; (ii) completely prime if aΓ b ⊆ A, a, b ∈ M,

implies either a ∈ A or b∈ A; (iii) semi-prime if BΓB ⊆ A implies B ⊆ A; (iv) completely semi-prime

ideal if it satisfies condition: aΓa ⊆ A a∈ A.

2.2. Definitions ( Satyanarayana [9, 11]): (i) An element a in M is said to be a nilpotent element, if there

exists a positive integer n such that (aΓ)na = aΓaΓa...Γa = 0.

(ii) An ideal A of M is said to be a nilpotent ideal, if there exists a positive integer n such that (AΓ)nA =

AΓAΓA...ΓA = 0. We denote the sum of all nilpotent ideals of M by SN(M).

2.3. Lemma: (i) If J ⊆ M and J2 ⊆ I , I is completely semi-prime ideal, then J ⊆ I (in particular every

completely semi-prime ideal is a semi-prime ideal).

(ii) If I is completely semi-prime ideal of M, then aΓb ⊆ I bΓa ⊆ I.

Proof: (i) Let a ∈ J. Now aΓa ⊆ JΓJ = J2 ⊆ I a ∈ I (Since I is completely semi-prime ideal of I).

Therefore J ⊆ I.

(ii) Suppose I is completely semiprime ideal of M and a, b ∈ M such that aΓb ⊆ I.

Now aΓb ⊆ I (aΓb)Γa ⊆ I (Since I is a right ideal) MΓ(aΓb)Γa ⊆ MΓI ⊆ I

(since M is zero symmetric Γ-near-ring and I is a left ideal) bΓ(aΓb)Γa ⊆ I

(bΓa)Γ(bΓa) ⊆ I (bΓa)2 ⊆ I (bΓa) ⊆ I (by (i)). The proof is complete.

2.4 Lemma: (i) If a ∈ M and I is an ideal of M, then

(I : a) = x ∈ M / xΓa ⊆ I is a left ideal of M.

(ii). If I is a completely semi-prime ideal of M, then (I : a) is an ideal of M.

Proof: (i). Take x , y ∈ (I : a). Then xΓa ⊆ I and yΓa ⊆ I. We have to verify that (x - y)Γa ⊆ I. Let γ ∈Γ. Consider (x - y)γa = xγa - yγa ∈ xΓa - yΓa ⊆ I - I = I. This shows that x - y ∈ (I : a). Let m∈ M.

Now (m + x - m)γa = mγa + xγa - mγa ∈ I (since xγa ∈ I, and I is a normal subgroup of (M, +)). This is

true for all m ∈ M and γ ∈ Γ.

Hence m + x - m ∈ (I : a). Therefore (I : a) is a normal subgroup of (M , +). Let m, m1 ∈ M, x ∈ (I : a)

and γ ∈ Γ. Now we verify that mγ(m1 + x) - mγm

1 ∈ (I : a). Consider (mγ(m1 + x) - mγm

1)γa = mγ(m

1 +

x)γ a - mγm1γa = mγ(m

1γa + xγa) - mγm1γa ∈ I (since xγa ∈ I, m

1γa ∈ M and I is left ideal of M)

Therefore (I : a) is a left ideal of M.

!"

(ii) Suppose that I is completely semi-prime ideal of M. Now we verify that (I : a) is a right ideal. Let x

∈ (I : a). We have to show that xΓM ⊆ (I : a). For this take m ∈ M, γ ∈ Γ. To show xγm ∈ (I : a), we

have to verify that (xγm)Γa ⊆ I. Let β ∈ Γ. Since x ∈ (I : a), we have xΓa ⊆ I. By Lemma 2.3 (ii), aΓx

⊆ I. Since I is left ideal MΓ(aΓx) ⊆ I.

Now mβaΓx ⊆ I. By Lemma 2.3 (ii), xΓmβa ⊆ I xγmβa ∈ I . This is true for all β ∈ Γ. Therefore

xγmΓa ⊆ I. This shows that (I : a) is a right ideal. The proof is complete.

2.5 Theorem: If S is a semi-prime ideal of M, then the following are equivalent:

(i) If xΓx ⊆ S, then <x>Γ<x> ⊆ S.

(ii) S is completely semi-prime ideal of M.

(iii) If xΓy ⊆ S, then <x>Γ<y> ⊆ S.

Proof: (i) (ii): Suppose (i). That is if xΓx ⊆ S, then <x>Γ<x> ⊆ S. Let x ∈ M and xΓx ⊆ S. <x>Γ<x> ⊆ S (by (i)) <x> ⊆ S (since S is a semi-prime ideal) x ∈ S. Hence S is a completely

semi-prime ideal of M. This proves (i) (ii).

(ii) (iii): Suppose S is completely semi-prime. Let x, y ∈ M such that xΓy ⊆ S.

Now x ∈ (S : y). By Lemma 3.2(ii), (S : y) is an ideal of M. Now x ∈ (S : y), and (S : y) is an ideal <x> ⊆ (S : y) <x>Γy ⊆ S. By Lemma 3.1(ii), we have that yΓ<x> ⊆ S. Now by Lemma 3.2(ii), we

have that <y>Γ<x> ⊆ S. By Lemma 3.1(ii), it follows that <x>Γ<y> ⊆ S.

(iii) (i): By taking y = x in (iii), we get (i). The proof is complete.

ACKNOWLEDGEMENTS

The second author and fourth author acknowledge the financial assistance from the UGC, New Delhi

under the grant F.No: 34-136/2008(SR), dt: 30-12-2008. The authors thank the referee for valuable

comments that improved the paper.

References

[1] Barnes W.E. “On the Gamma-rings of Nobusawa”, Pacific J. Math 18 (1966) 411- 422.

[2] Booth G.L.“A note on Γ-Near-rings”,Stud.Sci.Math.Hunger 23 (1988) 471-475.

[3] Booth G.L. and Godloza L. “ On Primeness and Special Radicals of Gamma rings”, Rings

and Radicals, Pitman Research notes in Math series (contains selected lectures

presented at the international conference on Rings and Radicals, held at Hebei, Teachers

University, Shijazhuang, Chaina, August 1994) pp 123–130.

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

!"

[5] Pilz .G “Near-rings”, North Holland, 1983.

[6] Pradeep Kumar T.V “Contributions to Near-ring Theory - III”, Doctoral Dissertation,

Acharya Nagarjuna University, 2006

[7] Ramakotaiah Davuluri “Theory of Near-rings”, Ph.D. Diss., Andhra university, 1968.

[8] Satyanarayana Bh. "A Note on Γ-rings", Proceedings of the Japan Academy 59-A(1983)

382-83.

[9] Satyanarayana Bh. “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya


[10] Satyanarayana Bh. “A Note on Γ-near-rings”, Indian J. Mathematics (B.N. Prasad

Birth Centenary commemoration volume) 41(1999) 427-433.

[11] Satyanarayana Bh. “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany,

2010 (ISBN: 978-3-639-22417-7).

[12] Satyanarayana Bh., Pradeep Kumar T.V. and Srinivasa Rao M. “On Prime left ideals in Γ-

rings”, Indian J. Pure & Appl. Mathematics 31 (2000) 687-693.

[13] Satyanarayana Bh., Pradeep kumar T.V., Sreenadh S., and Eswaraiah Setty S. “On

Completely Prime and Completely Semi-Prime Ideals in Γ-Near-Rings”, International

Journal of Computational Mathematical Ideas Vol. 2, No 1(2010) 22 – 27.

[14] Satyanarayana Bhavanari & Syam Prasad Kuncham “Discrete Mathematics & Graph

Theory”, Prentice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).

[15] Syam Prasad K. “Contributions to Near-ring Theory II”, Doctoral Dissertation

Acharya Nagarjuna University, 2000.

!"

Almost Convergence and

Some Matrix

Transformations

Abstract

The sequence space ( )bv p have been defined and the classes ( ( ): )bv p l∞ , ( ( ): )bv p c and 0

( ( ): )bv p c of

infinite matrices have been characterized by Abdullah and Malkowsky (see , [1] ). The main purposes of

the present paper is to characterize the classes ( ( ): )bv p f∞ , ( ( ): )bv p f and0

( ( ): )bv p f where f∞ , f and

0f denote respectively the spaces of almost bounded sequences, almost convergent sequences and almost

convergent null sequences with real or complex terms.

2000 AMS Mathematical Subject Classification: 46A45; 46B20; 40C05.

Keywords and phrases: Sequence space of non-absolute type, almost convergent sequences,

and Matrix mappings.

1. Introduction and Preliminaries: A sequence space is defined to be a linear

space with elements in another space. Throughout the paper , and denotes the set of non-

negative integers, the set of real numbers and the set of complex numbers respectively. Let ω

denote the space of all sequences ( real or complex ) ; l∞ and c respectively denotes the space

of all bounded sequences , the space of convergent sequences . A linear Topological space X

over the field of real numbers is said to be a paranormed space if there is a subadditive

function :h X → such that ( ) 0,h θ = ( )h x = ( )h x− and scalar multiplication is continuous, that

is, 0n

α α− → and ( ) 0n

h x x− → imply ( ) 0n n

h x xα α− → for all α in and 'x s in X , where

is a zero vector in the linear space X .

!!"!##$!$ %& '(&) *

+( **,(*-+.

Authors: Abdul Hamid Ganie, Ahmad

Sheikh and Sameer Gupkari, Department

of Mathematics, National Institute of

Technology, Hazatbal, Jammu and Sri

Nagar (India)-190006,

[email protected],

[email protected],

[email protected]

!"

Let X and Y be two non-empty subsets of ω .Let ( ), ( , )nk

A a n k= ∈ be an infinite matrix of

real or complex numbers. We write ( ) ( ) .n n nk k

k

Ax A x a x= = Then Ax = ( )n

A x is called the A

-transform of x , whenever ( )n nk k

k

A x a x= < ∞ for all n. We write lim lim ( )n

n nAx A x= .If x X∈

implies Ax Y∈ , we say that A defines a matrix transformations from X into Y , denoted by

:A X Y→ . By ( ):X Y , we mean the class of all matrices A such that :A X Y→ .The matrix

domain A

X of an infinite matrix A in a sequence space X is defined as

( ) :A k

X x x Ax Xω= = ∈ ∈

Let :S l l∞ ∞→ be the shift operator defined by 1

( )n n

Sx x += for all 0,1,2,...n∈ = . A Banach

limit L is defined on l∞ as a non negative linear functional such that ( ) ( )L sx L x= and ( ) 1L e = ,

( )(1,1,1,...)e = [2]. A sequence space is said to be almost convergent to the generalized limit α

if all Banach limits of x are α [4].We denote the set of almost convergent sequences by f i.e.,

: lim ( )mnm

f x l t x α∞= ∈ = , uniformly in n

where , 1,

0

1( ) , 0

1

m

mn j n n

j

t x x tm

+ −=

= =+ and limf xα = − .

Nanda [6] has defined a new set of sequences f∞as follows

:sup ( )mn

mn

f x l t x∞ ∞

= ∈ < ∞

We call f∞the set of all almost bounded sequences.

Abdullah and Malkowsky[1] have defined the sequence space ( )bv p and characterize the matrix

classes ( ( ): )bv p l∞ , ( ( ): )bv p c and 0

( ( ): )bv p c .In this paper we characterize the matrix classes

( ( ): )bv p f∞ , ( ( ): )bv p f .and

0( ( ): )bv p f . The space ( )bv p is defined (see,[ 1]) as

( ) ( ) :kp

k k

k

bv p x x xω

= = ∈ ∆ < ∞

.

Main Results: For brevity in notation, we write

0

1( ) ( ) ( , , )

1

m

mn n i k

j k

t Ax A x a n k m xm

+=

= =+ where,

,

0

1( , , ) ; ( , , )

1

m

n j k

j

a n k m a n k mm

+=

= ∈+

we give the following lemma which will be needed in proving the main results.

Lemma 2.1 [6]: f f∞⊂ .

!"

Theorem 2.1: Let 1k

p M< ≤ < ∞ for every k ∈ .Then ( ( ): )A bv p f∞∈ if and only if

,

sup ( , , )kk qq

n m k

a n k m B−

∈

< ∞

(2.1)

Proof: Sufficiency: Suppose the conditions (2.1) holds and ( ).x bv p∈ .Using the inequality

which holds for any 0C > and any two complex numbers ,a b

1q p

ab C aC b−≤ + , where 1p > and 1 1 1p q− −+ = (see, [4]), we have

( ) ( , , )mn k

k

t Ax a n k m x=

( , , )k kk

q pq

k

k

B a n k m B x− ≤ + , where, 1 1 1

k kp q− −+ = .

Taking ,

supm n

on both sides and using (2.1) we get Ax f∞∈ for every ( )x bv p∈ , i.e., ( )( ) :A bv p f∞∈ .

Necessity: Suppose that ( )( ) :A bv p f∞∈ and ( ) sup ( )n mn

m

q x t Ax= .It is easy to see that for 0,n

n q≥

is a continuous seminorm on ( ) ( )n

bv p and q is pointwise bounded on ( )bv p . Suppose that (2.1) is

not true. Then there exists ( )x bv p∈ with supn

n

q = ∞ .By Principle of condensation of singularities

[7], the set ( ) : sup n

n

x bv p q∈ = ∞ is of second category in ( )bv p and hence nonempty, that is,

there is ( )x bv p∈ with supn

n

q = ∞ .But this contradicts the fact that ( )n

q is point wise bounded on

( )bv p .Now by the Banach-Steinhauss theorem, there is a constant M such that

( ) ( )n

q x Mg x≤ (2.2)

We define a sequence x and y as follows

1 1sgn ( , , ) ( , , ) ( 1) ( 1) 1 ( )k

k k

q

q p

k jy a n k m a n k m j M for n j k n j

−− −= + − + ≤ ≤ and

0

k

k k

j

x y=

= for 0,1,2,...k =

Then one can see that ( )x bv p∈ .Applying this sequence to (2.2) we get the necessity of (2.1) and

completes the proof of the result.

Theorem 2.2: 1k

p M< ≤ < ∞ for every k ∈ .Then ( ( ): )A bv p f∈ if and only if

(i) the condition (2.1) of Theorem 2.1 holds ;

(ii) there is a sequence ( )kβ of scalars such that

lim ( , , )k

ma n k m β= , uniformly in n .

Proof: Sufficiency: Suppose that the conditions (i)-(ii) hold and ( )x bv p∈ . We observe for 1j ≥ ,

!"

1

( , , ) ( , , )kk k k

jqq q q

k k

a n k m B a n j m B− −

=

≤ < ∞ , for every n .

Therefore, 1

lim lim ( , , )k kk k

jq qq q

kj k

k k

B a n k m Bβ − −

=

≤

( , , )k k

q q

k

a n k m B−≤ < ∞ ,( 1 1 1

k kq p− −+ = ).

Consequently reasoning as in the proof of the sufficiency of Theorem 2.1, the series

( , , )k

k

a n k m x and k k

k

xβ converges for every ,n m and for every ( )x bv p∈ .Now, for given 0ε >

and ( )x bv p∈ ,choose a fixed 0

k ∈ such that

0

1

1

k

Hp

k

k k

x ε∞

= +

<

, where sup

kk

H p= . (2.3)

Then, there is some 0

m ∈ , by condition (ii) such that

[ ]0

1

( , , )k

k

k

a n k m β ε=

− < , for every 0

m m≥ and uniformly in n . (2.4)

Now, since ( , , )k

k

a n k m x and k k

k

xβ converges (absolutely) uniformly in ,n m and for

( )x bv p∈ , we have that

[ ]0 1

( , , )k k

k k

a n k m xβ∞

= +

−

converges uniformly in ,n m and ( )x bv p∈ .

Hence by conditions (i) and (ii) we have

[ ]0 1

( , , )2

k

k k

a n k mε

β∞

= +

− < ( )0for all m m≥ , uniformly in n .

Therefore, [ ]0 1

( , , ) 0 ( )k

k k

a n k m mβ∞

= +

− → → ∞ uniformly in n i.e.,

lim ( , , )k k k

mk k

a n k m x xβ= (2.5)

uniformly in n .Hence, Ax f∈ , which proves sufficiency.

Necessity: Suppose that ( ( ): )A bv p f∈ . Then , since f f∞⊂ ( by Lemma 2.1 ), the necessities of

condition (i) is immediately obtained from Theorem 2.1 . To prove the necessity of (ii), consider

the sequence (0,0,...,1 ,0,0,...) ( )kth place

ke bv p−= ∈ , condition (ii) follows immediately by(2.5) and

the proof is complete.

!!"

Note that if we replace f by 0

f , then Theorem 2.2 is reduced to the following corollary:

Corollary:0

( ( ): )A bv p f∈ if and only if condition (i) and (ii) of above Theorem holds along with

0k

β = for each k ∈ .

Reference

[1] Abdullah M. Jarrah, and Malkowsky, E., the Space ( )bv p , its β − dual and matrix

transformations, Collect. Math .,(55)(2004),151-162.

[2] Banach, S., Theries des operations linaries, Warszawa, 1932.

[3] Lascarides, C.G., Maddox, I.J., Matrix transformations between some classes of

sequences, Proc. Camb. Phil. Soc.,(68) (1970) , 99-104.

[4] Lorentz,G.G., A contribution to the theory of divergent series, Acta

Math.,(80)(1948),167-190.

[5] Mursaleen, Infinite matrices and almost convergent sequences, Southeast Asian Bulletin

of Math. 19(1)(1995),45-48.

[6] Nanda, S., Matrix transformations and almost boundedness, Glasnik Mat.,

14(34)(1979),99-107.

[7] Yasida, K., Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1966.

!!!

Generalized Fuzzy

Ideals of Gamma Near-

rings

Authors: Syam Prasad Kuncham

*,

Satyanarayana Bhavanari**, and Venkata

Subba Rao Gunda**

*: Department of Mathematics, Manipal

University, Manipal-576 104, India.

**: Department of Mathematics, Acharya

Nagarjuna University, Nagarjuna Nagar-522

510, Andhra Pradesh.

Emails: [email protected]

[email protected]

!!"!##$!$ %& '(&) *

+( **,(*-+.

!!

!!

!!

!!

!!

!!

!!

!!

Global Relevant Weighing

(GRW) -A Novel Term

weighing Model for

Improved Document

Clustering

Abstract:

This paper describes new model for estimating terms weights in a Medline and Pubmed

documents and shows how the classification accuracy is improved with this method. The method

uses global relevant weight as term weighing schema .Experiments performed with different

weighing schemas shows that the new global weighing method outperforms the tradition term

weighing approaches.

1. Introduction:

Text Mining is the process to extract meaningful data from the text, and, thus, make the

information contained in the text accessible to the various data mining algorithms. Medline and

Pub med repositories are rich in medical literature. Automatic extraction of useful information

from these online sources remains a challenge because these documents are unstructured and

expressed in a natural language form i.e. in text format.

It is virtually impossible for researchers to obtain all the information that is important and

available for their work. There are several text mining approaches for handling the vast amount

of textual domain-specific information available, some of them are Document clustering , Text

classification .

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 between

members. Members from different clusters have a low degree of association [8]. We can use

Authors: Sagar Imambi

*, Sudha T

**., and

Bharathi Devi*, *: Department of Computer

Science T J P S College, Guntur (Andhra

Pradesh), **: Department of Computer Science,

Vikram Simhapuri University, Nellore (Andhra

Pradesh).


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+( **,(*-+.

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+( **,(*-+.

!"

document clustering in many applications. In our previous work [5], we showed how document

clustering is used for information access in digital libraries.

The goal of Text classification is to build a set of models that can correctly predict the class of

the different text documents. The input to these methods is a set of documents (i.e., training

data), the classes which these documents belong to, and a set of terms describing different

characteristics of the documents. There are several classification systems, which will analyze

structured data from biomedical databases and unstructured data from open access abstracts and

full text documents and provide the voluble knowledge to doctors.[4].

Text classification process involves following steps

• Document representation

• Term selection

• Term weighing

• Classifier learning.

Term weighing plays an important role in both classification and clustering. We propose new

Term weighing approach which improves accuracy of analysis (classification or clustering).

2 Document representations

Vector space model is used to represent Documents in n-dimensional space. Vector space model

uses term-document matrix. The representation of documents in this model is as follows

D=[d1,d2,d3,d4……dm]

di=(ti1,ti2,ti3,ti4,….tin)

where D indicates Total document set with m elements and di is set of n terms. Each term ti in jth

document identifies features of the documents. In the vector space model a document is located

as point in an n- dimensional vector space. Dimension is equal to the number of features (terms)

in the document set. The occurrence of term represents its proportional significance in

representing the document.

3. Term weighing

Term weighting is an important factor in the performance of information retrieval systems. Many

weighting methods have been developed within text search, and their variety is astounding..

Term weighing is the process of computing weight of every term in the document set. A

weighting scheme is composed of three different types of term weighting: local, global, and

normalization. Local weights are functions of how many times each term appears in a document,

!!

global weights are functions of how many times each term appears in the entire collection, and

the normalization factor compensates for discrepancies in the lengths of the documents.

There are many different local weight schemas available. Some of them listed in Table 1.

Formula NAME

1 if fij >0, 0 if if fij =0 Binary

1+log fij if fij >0, 0 if if fij =0 LOG

fij Document frequency

Table 1: Local weights

Global weighting tries to give a “discrimination value” to each term. Many schemes are based on

the idea that the less frequently a term appears in the whole collection, the more discriminating it

is [6].

4. Proposed Global weighing method:

Following steps are applied to derive the global weight associated with the terms in the each

documents

Phase 1: The document is tokenized for punctuation, special symbols and word abbreviations.

Common words are also removed.

Phase 2: Calculate term frequency using any traditional methods.

The tf–idf weight (term frequency–inverse document frequency) is a weight often used text

mining. This weight is a statistical measure used to evaluate how important a word is to a

document collection The importance increases proportionally to the number of times a word

appears in the document but is offset by the frequency of the word in the corpus.

dtDd

Dtf dt

∈∈× log,

!

where tft,d is term frequency of term t in document d , | D | is the total number of documents in

the document set; dtDd ∈∈ is the number of documents containing the term t.

Phase 3 Calculate global weight using the formula proposed by us.

Global weight of term tij = local weight of term tij * max (Pi), where Pi is the probability of the

term tij belongs to class Ci.

5. Experimental Result:

In our experiment we used 1000 documents, collected from the Pubmed. We partitioned the

dataset into a training set of 600 and a test set of 400 documents. Sample document is showed in

fig1. Pmid is the identification number from the Pubmed. The documents are labeled by 4

categories, which represent the complications of Type 2 diabetes.

Fig 1 Sample document

We used both local global weights to represent the documents in vector space. After applying the

classification algorithm we measured the effectiveness in terms of precision. Precision is the

fraction of the documents retrieved that are relevant to the user's information need. The results

are tabulated in table 2.

!

Schema Weighing method Precision

1 Local –Binary 0.9363

2. Local-Log 0.9452

3. Local -DF 0.91

4 Global Relevant 0.9623

The local weight combined with the global weight makes the difference. Our global weight

works well in document classification as its precision rate is high compared to other local

weighting schemas.

6. Conclusions:

The Term weighing schema plays an important role in document classification and in Text mining

applications. We proposed the global weight schema based on the probability of term relevance. Our

results show that the accuracy and precision are high when global relevant weight schema is used. We

experimented on Collection of diabetic literature from PUBMED.

References:

1. Dhillon I., Mallela S., Kumar R.,A Divisive Information-Theoretic Feature Clustering

Algorithm for Text Classification, Journal of Machine Learning Research 3, (2003) 1265-1287.

2. S.Sagar Imambi, T.Sudha - A Unified frame work for searching Digital libraries Using Document

Clustering –International Journal of Computational Mathematical ideas Vol 2-No1-(2010) 28-32

3. Nordiannah et.al-Term weighting Schemes Experiment Based on SVD for Malay Text retrieval-

International journal of Computer science and Network security , Vol 8.No.10, (2008).

4. Srinivasa K.G et.al –Feature Extraction using Fuzzy C-Means Clustering for Data mining systems

- International journal of Computer science and Network security Vol 6 No 3A (2006).

5. S.Sagar Imambi, T.Sudha-Clinical Decision Support System for Heart Patients-International

Journal of Computer Science, System Engineering and Information Technology, Vol 2-No2.

(2009) 165-169

6. W. B. Croft and D. J. Harper. Using probabilistic models of document retrieval without

relevance information. , J. Documentation, 35(4)( 1979) 285-295

7. S.Sagar Imambi, T.Sudha -.Building Classification System to Predict Risk factors of Diabetic

Retinopathy Using Text mining - International Journal on Computer Science and Engineering,

Vol. 02, No. 07 ( 2010 ,to print)

8. Christian Borgelt and Andreas Nurnberger-Experiments in Document clustering using Cluster

Specific Term weighing, Citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.88.4757& rep.

!

Prime Graph of an

Integral Domain

Abstract

Satyanarayana, Syam Prasad and Nagaraju [ 10 ] introduced the concept ‘Prime Graph of R’

(denoted by PG(R)), where R is a given associative ring. Later it was studied by Satyanarayana,

Mohiddin Shaw, Mallikarjun and Pradeed Kumar [6] and constructed Prime Graphs related to

the ring of integers modulo n. The aim of this present paper is to study the concept of prime

graph of an integral domain and also to study the prime graph of the product ring R × 2 where R

is an integral domain. We present necessary examples.

Key Words: Associative ring, Integral Domain, Graph, Prime Graph.

1. INTORDUCTION

The concept “prime graph” of an associative ring, which is defined by Satyanarayana, Syam

Prasad and Nagarajua [10] is a new bridge between “graph theory” and the algebraic concept

“ring theory”. This paper is a study the prime graph of integral domain. In the present paper we

continue the study in [6, 10]. This paper is divided into three sections. In the present Section-1,

we collect necessary definitions, and results from the literature that are used in the next sections.

Section-2 and 3 contains new theorems.

Authors: Satyanarayana Bhavanari

* , Mohiddin Shaw

Sk.* and Vijaya kumara Arava** .

*: Department of Mathematics, Acharya Nagarjuna

University, Nagarjuna Nagar, Andhra Pradesh, India.

**: Department of Mathematics, J M J College, Tenali

(Guntur Dist), Andhra Pradesh, India

E-mail: [email protected],

[email protected]

!!"!##$!$ %& '(&) *

+( **,(*-+.

!

1.1 Definition: A non empty set R is said to be a ring (or an associative ring) if there exists two

binary operations + and . on R satisfying the three conditions: (i) (R, +) is an Abelian group; (ii)

(R, .) is a semi-group; and (iii) a.(b + c) = a.b + a.c, and (a + b).c = a.c + b.c for any a, b, c ∈ R.

More over, if a ring R satisfies the condition a.b = b.a for all a, b ∈ R, then we say that R is a

commutative ring. If R contains the multiplicative identity, then we say that R is a ring with

identity.

1.2 Definition: Let R be a ring, and φ ≠ I ⊆ R. Then (i) I is said to be a left ideal of R if I is a

subgroup of (R, +) and ra ∈ I for every r ∈ R, a ∈ I; (ii) I is said to be a right ideal of R if I is a

subgroup of (R, +) and ar ∈I for every r ∈ R, a ∈ I; and (iii) I is said to be an ideal (or two sided

ideal) of R if I is both left and right ideal.

1.3 Definition: An ideal P of R is said to be prime if A, B are two ideals of R, and AB ⊆ P A

⊆ P or B ⊆ P (equivalently, a, b ∈ R and aRb ⊆ P a ∈ P or b ∈ P).

1.4 Definition: An integral domain is a commutative ring with identity such that for any two

elements a and b of the ring, ab = 0 implies either a = 0 or b = 0.

1.5 Definitions: A linear graph (or simply a graph) G = (V, E) consists of a set of objects V

= v1, v2,.. called vertices, and another set E = e1, e2, … whose elements are called edges

such that each edge ek is identified with an unordered pair (vi, vj) of vertices. vi and vj are called

the end points of ek. If V and E are finite sets then the graph G is said to be a finite graph. A

graph G is said to be a null graph if it contains no edges, that is E = φ. An edge connecting two

vertices u and v is denoted by vu or uv . A graph G is said to be simple if it contains no loops

and multiple edges. The number of edges incident to a vertex v is called the degree of the vertex

v, and it is denoted by d(v). Each maximal connected subgraph of a graph G is called a

component of the graph G. The distance between the two vertices x and y is denoted by d(x, y).

In this paper we consider only simple graphs.

1.6 Definitions: Let G = (V, E) be a graph and φ ≠ X ⊆ V. Write E1 = xy ∈ E / x, y ∈ X.

Then G1 = (X, E1) is a subgraph of G and it is called as the subgraph generated by X (or the

maximal subgraph with vertex set X ). If v1, v2, v3 are vertices, and the maximal subgraph with

vertex set v1, v2, v3 forms a triangle, then we say that the set v1, v2, v3 is a triangle (or forms

a triangle). A complete graph is a simple graph in which each pair of distinct vertices is joined

by an edge. The complete graph on n vertices is denoted by Kn. It is clear that a complete graph

is a regular graph of degree (n - 1), where n is the number of vertices.

A graph G = (V, E) is said to be a star graph if there exists a fixed vertex v such that

E = vu / u ∈ V and u ≠ v. A star graph is said to be n-star graph if the number of vertices in

the graph is n. A connected graph without circuits is called a tree. It is clear that every star

graph is a tree. (Th. 13.8, Page 347 [9]) A given connected graph G is an Euler graph if and only

if all the vertices of G are of even degree.

!

For further concepts related to Ring Theory and Graph Theory, we refer Herstein [1], Lambek

[2], Narsing Deo [3] and Satyanarayana & Syam Prasad [9].

1.7 Definition (Satyanarayana, Syam Prasad & Nagaraju [10]): Let R be a ring. A graph

G(V, E) is said to be a prime graph of R (denoted by PG(R)) if V = R and E = xy / xRy = 0

or yRx = 0, x ≠ y.

1.8 Examples: Consider n, the ring of integers modulo n.

(i) Note that PG(R), when R = n, 1 ≤ n ≤ 5 contains no triangles; and this graph is a star graph.

(ii) The graph PG(6) is given in Figure 1.8 (i).

(ii) The graph PG(8) is given in Figure 1.8 (ii).

1.9 Observations: Let R be a ring and PG(R) be its prime graph.

(i) There are neither self loops nor multiple edges in PG(R), and so PG(R) is a simple

graph.

"

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!

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!

(ii) Since 0Rx = 0 for all 0 ≠ x ∈ R, there is an edge from ‘0’ to x for all x ∈ V = R. So

degree(0) = | R \ 0| = |R| - 1. Also note that there exists a subgraph which is a n-star graph.

For any two non-zero elements x, y in R, there are edges one from x to 0, and another from 0 to

y. This shows that the graph PG(R) is a connected graph. Moreover, d(0, x) = 1 and d(x, y) ≤ 2

for any two non-zero elements x, y ∈ R.

(iii) xRy ≠ 0 if and only if d(x, y) = 2.

(iv) The set 0 is a dominating set for PG(R). Hence the domination number of PG(R) is

equal to 1.

Section-2: Integral Domain R and PG(R)

We start this section with following Lemma

2.1 Lemma: If R is an integral domain then PG(R) is a star graph with number of vertices

R .

Proof: If R is an integral domain ⇔ (for a, b R, ab = 0 implies a = 0 or b = 0) and

(a0 = 0 for all a R) ⇔ (ab 0 for all 0 a and 0 b in R) and (a0 = 0 for all a R)

⇔ (There is no edge between any two non-zero vertices in PG(R)) and (there is an edge

between 0 and a for all a R) ⇔ PG(R) is star graph with centre ‘0’.

AN APPLICATION TO p:

2.2 Theorem: Let p be a prime number. Then p is a field and hence an integral domain.

PG(p) is a star graph with number of vertices p and centre ‘0’. Conversely any star graph with

p vertices is isomorphic to the graph PG(p).

Verification: From the Lemma 2.1, it follows that PG(p) is a star graph with number of

vertices p and centre 0.

Conversely, suppose G = (V,E) is a star graph with V = a, 1v , 2v , …, 1pv − . Consider p =

0, 1, 2, …, p-1. Now PG(P) = ( V*, E) where V

* = 0, 1, 2, …, p-1 and E = oi

/ 1ip-1. Define f: V*→ V by f(0) = a and f(n) = 1nv + for 1np-1. It is clear that f

produces an isomorphism between PG(p) and G. This completes the proof.

!

Section -3: Prime Graph of R ×××× 2 where R is an integral domain.

Let R be an integral domain and 2 is the ring of integers modulo 2.

For (a, b), (c, d) R × 2 we define (a, b) + (c, d) = (a + c, b + d) and

(a, b)⋅(c, d) = (a⋅c, b⋅d). Then R × 2 becomes the product ring, and the zero element of R × 2 is

(0, 0). Since (0, 0) (1, 0) and (0, 0) (0, 1) are two elements in R × 2 with (1, 0)⋅(0, 1) =

(0,0). Therefore R × 2 is not an integral domain.

Now we study PG(R × 2) and prove some results.

3.1 Note: Suppose R = 2. Then PG(R × 2) is

(i) It is clear from the diagram that PG(2 × 2) = (4-star graph with centre (0, 0)) ∪ one

edge.

We verify this fact algebraically.

Consider the subgraphs H and K of PG(2 × 2). H is a 4-star graph .

So PG(2 × 2) = H ∪ K = (4-star graph) ∪ one edge.

(ii). Any graph of the form (4-star graph) ∪ one edge is isomorphic to PG(2 × 2).

Verification: Consider a graph G = (4-star graph) ∪ one edge.

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Let the 4-star graph in G be H*. Then H

* = ( V

*, E

*) where V

*= a, b, c, d and

E* = ab ac , ad (with centre 0). If the additional edge is aa then it is a loop, a contradiction

(as we consider simple graphs only). If the additional edge is either ab , ac , or ad then we get

multiple edges, a contradiction. Hence the additional edge must be either bc or cd or db . We

verify the result for the case: the additional edge is bc . A similar argument valid for other

cases to conclude. Now the graph G considered is G = (V, E) where V = V* = a, b, c, d and E

= E* ∪ bc = ab , ac , ad , bc

[Now we verify that PG(2 × 2) and G are isomorphic.

Define f: V(PG(2 × 2)) → V by f((0,0)) = a, f((1,0)) = b, f((0,1)) = c and f((1,1)) = d.

This mapping f produces an isomorphism. This completes the proof.

3.2 Theorem: Let R be an integral domain with R = n. Then PG(R × 2) contains two

particular elements (0, 0) = a, (say), (0, 1) = b (say) such that V(PG(R × 2)) = 2n and

PG(R × 2) = [ the 2n-star graph with R × 2 as vertex set and centre a] ∪ [the n-star graph with

vertex set (x, 0) / 0 x R with centre b].

Proof: Since R × 2 = R × 2 = (n). (2) = 2n we have that V(PG(R × 2)) = 2n.

Writ a = (0, 0) and b = (0, 1). For any (0, 0) (x, y) R × 2 we have that (0, 0) (x, y) = (0.x,

0.y) = (0, 0). Thus there exists an edge between (0, 0) and (x, y), for all (0, 0) (x, y) R

× Z2. Now H = (V1, E1) where V1 = R × 2 and E1 = (0,0)( , )x y / (0, 0) (x, y) V.

H is a 2n-star graph which is a subgraph of PG(R × 2). Also note that product of (0, 1) and (x,

0) is equal to (0, 0) and so there is an edge between (0, 1) and (x, 0). Now write K = (V2, E2)

where V2 = (0, 1) ∪ (x, 0) / 0 x R and E2 = (0,1)( ,0)x / 0 x R , is a n-star

subgraph of PG(R × 2).

Now H ∪ K PG(R × 2).

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To show the other part, take an edge ( , )( , )c d x y in PG(R × Z2). Now ( , )( , )c d x y is an edge

implies (c, d)(x, y) = (0, 0) cx = 0 and dy = 0.

(c = 0 or x = 0 (since R is an integral domain) and d = 0 or y = 0 (since d, y 2))

Case (i): Suppose c = 0 and d = 0.

Then a = (c, d) = (0, 0). In this case ( , )( , )c d x y = (0,0)( , )x y E(H) = E1.

Case (ii): Suppose c = 0 and y = 0. If d = 0 then ( , )( , )c d x y = (0,0)( , )x y E(H) = E1.

If d = 1 then ( , )( , )c d x y = (0,1)( , )x y E(K) = E2.

Case (iii): Suppose x = 0 and d = 0. If y = 0 then ( , )( , )c d x y = ( ,0)(0,0)c E(H) = E1.

If y = 1 then ( , )( , )c d x y = ( , 0)(0,1)c E(K) = E2.

Case (iv) Suppose x = 0 and y = 0. ( , )( , )c d x y = ( , )(0,0)c d E(H) = E1.

This shows that E(PG(R × 2)) = E(H) ∪ E(K).

Hence PG(R × 2) = H ∪ K where H and K are subgraphs, H is 2n-star graph, K is a n-star

graph.

3.3 Note: In the proof of above theorem we arrived at two subgraphs H and K of

PG(R × 2). We can state that E(H) ∩ E(K) = φ and a∉V(K).

3.4 Remark: The graph PG(R × 2) where R an integral domain, satisfy the following

properties:

(i) V(G) = 2n where n = R

(ii) It contains two particular vertices a, b V(G) with a b.

(iii) There exists a subgraph H of G such that H is a 2n-star graph (with centre a).

(iv) There exists a subgrph K of G such that K is a n-star graph (with centre b).

(v) G = H ∪ K.

!!

Observe the converse statement of the above theorem. There arises the question “which type of

graphs are isomorphic to PG(R × 2) where R is an integral domain. This will be an open

question.

The following theorem provides a partial answer.

3.5 Theorem: Suppose G is a graph satisfying the following conditions

(i) V(G) = 2p, where p is a prime number.

(ii) G contains two particular vertices a*, b

* with a

* b

*.

(iii) H* is a 2p-star graph (with centre a

*) which is a subgraph of G.

(iv) K* is a p-star graph of G (with center b*) and a* ∉ V(K).

(v) G = H* ∪ K

*.

Then G is isomorphic to PG(P × 2).

Proof: Given that V(G) = 2p, where p is a prime number. Suppose that V(K*) = b

*,

1x , 2x , …,

1px − . Since V(G) = 2p and a* V(G) \ V(K

*) there exists

px , 1px + , … ,

2 2px −

such that V(G) = a*, b

* , 1x , 2x , …,

1px − , px ,

1px + , … , 2 2px − . Given that H

* is a 2p-star

graph with centre a*. So E(H

*) = * *a b ∪ *

ia x / 1i(2p-2). Given that K

* is a p-star

graph with centre b*. So E(K*) = *

ib x / 1i(p-1).

Now E(G) = E(H*) ∪ E(K

*). We have to show that G PG (P × 2). Write R = p. Since p is

prime, we have that R is an integral domain.

We follow the notation used in the proof of the Theorem 3.2

H = (V1, E1) where V1 = R × 2 = P × 2 and E1 = (0,0)( , )x y / (0, 0) (x, y) V1

= (0, 0)( ,0)i / 1i(p-1) ∪ (0,0)( ,1)i / 0i(p-1). H is a 2n-star graph which is a subgraph

of PG(P × 2). K = ((V2, E2) where V2 = (0, 1) ∪ (i, 0) / 1i(p-1) and E2 =

(0,1)( ,0)i / 1i(p-1). K is a n-star graph which is a subgraph of PG(P × 2). Also

PG(P × 2) = H ∪ K. Now define f : V(PG(P × 2) → V(G) by f((0, 0)) = a*, f((0, 1) = b

*,

f( (i, 0)) = ix for 1i(p-1), f((i, 1)) = 1p ix + − for 1i(p-1).

This f produces an isomorphism

(i) between H and H* ; (ii) between K and K* ; and (iii) between PG(P × 2) = H ∪ K and

H* ∪ K

* = G. This completes the proof.

!

3.6 Example: Let p = 3 and consider R = 3 and R × 2 = 3 × 2. The diagram for

PG(3 × 2) is given below.

Note that H is a 2p-star graph (that is 6-star graph), K is a 3-star graph, E(H)∩E(K) = φ and

PG(P × 2) = H ∪ K.

3.7 Example: Consider the graph G given in Fig-3.7(i). .

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Now V(G) = 2p, where p = 5, a prime number. H* is 10-star graph (2p-star graph with p = 5)

and a* is the centre. K

* is 5-star graph with centre b

*. G = H

* ∪ K*. Thus G satisfies the

hypothesis of the Theorem 3.5, with p = 5, a prime number. Let us observe through this example

that G PG(5 × 2). The graph for PG(5 × 2) is given in Fig-3.7(iv)

Observe that f: V(PG(P × 2) → V(G) defined (as in the proof of Theorem 3.5 ) as follows:

f((0, 0)) = a*, f((0,1)) = b

*, f((i, 0) = xi for 1i4, f((1, 1)) = x5, f((2, 1)) = x6, f((3,1)) = x7, f((4,

1)) = x8. This f produces an isomorphism between the given graph G and PG(5 × 2).

Conclusion: If R is an integral domain then it is proved that the related graph PG(R) is a star

graph with number of vertices R. This concept applied and observed for p (as p is an

integral domain). The prime graph of the product ring R × 2 was studied and proved that it is

equal to the disjoint union of two star graphs. A partial converse to this result was obtained.

Necessary examples were presented.

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Acknowledgements

The third author thanks the University Grants Commission (UGC), SERO, Hyderabad, for

providing financial assistance under the grant No. F.MRP-3129/09(MRP/UGC-SERO), Sept,

2009.

References

[1]. Herstein I. N, “Topics in Algebra”, Vikas Publishing House, (1983).

[2]. Lambek. J, “Lectures on Rings and Modules”, Blaisdel. Publ. Co., (1966).

[3]. Narsing Deo, “Graph Theory with Applications to Engineering and Computer

Science”, Prentice-Hall of India Pvt. Ltd., (1997).

[4]. Satyanarayana Bhavanari, Godloza Lungisile and Nagaraju Dasari (2008)“Ideals and

Direct Product of Zero Square Rings”, East Asian Mathematical Journal., 24, 377-387.

[5]. Satyanarayana Bhavanari, Mohiddin Shaw Sk., Mallikarjun Bhavanari, and Venkata

Pradeep Kumar Tumurukota “A Graph Related to the Ring of Integers Modulo n”, ACTA

CINCIA INDICA, Accepted.

[6]. Satyanarayana Bhavanari, Mohiddin Shaw Sk., Mallikarjun Bhavanari, and Venkata

Pradeep Kumar Tumurukota “On a Graph Related to the Ring of Integers Modulo n”,

Proceedings of International Conference on Challenges and Applications of Mathematics in

Science and Technology(CAMIST), January 11-13,(2010). (Publisher: Macmilan Research

Series, 2010) 688-697.

[7]. Satyanarayana Bhavanari and Nagaraju Dasari, Balamurugan K. S., & Godloza

Lungisile, "Finite Dimension in Associative Rings", Kyungpook Mathematical Journal, 48,

(2008), 37-43.

[8]. Satyanarayana Bhavanari & Syam Prasad Kuncham (2003) "An Isomorphism

theorem on Directed Hypercubes of Dimension n", Indian J. Pure & Appl. Mathematics, 34, PP

1453-1457.

[9]. Satyanarayana Bhavanari and Syam Prasad Kuncham (2009) “Discrete Mathematics

and Graph Theory”, Printice Hall of India, New Delhi.(ISBN: 978-81-203-3842-5).

[10]. Satyanarayana Bhavanari, Syam Prasad Kuncham and Nagaraju Dasari “Prime

Graph of a Ring”, Journal of Combinatories, Informations & System Sciences, 35 (2010).

Accepted.

!

On Fuzzy Continuous

Functions in Intuitionistic

Fuzzy Topological Spaces

Abstract:

The purpose of this paper is to introduce several types of fuzzy continuity between intuitionistic

fuzzy topological spaces; namely fuzzy somewhat continuity, fuzzy almost-somewhat continuity,

fuzzy weakly-somewhat continuity.

Keywords and phrases: Intuitionistic fuzzy set, intuitionistic fuzzy topological space,

intuitionistic fuzzy regular open set, fuzzy somewhat continuity, fuzzy almost-somewhat

continuity, fuzzy weakly-somewhat continuity.

AMS subject classification: 54A40

1. Introduction

The concept of fuzzy sets was introduced by Zadeh [7]. In [1, 2, 3, 4,] Atanassov introduced the

fundamental concept intuitionistic fuzzy sets. Later, this concept was generalized to intuitionistic

L-fuzzy sets by Atanassov-Stoeva [2, 3]. On the other hand Coker [3] introduced the notion of

intuitionistic fuzzy topological spaces, fuzzy continuity and some other related concepts. In this

paper, we introduced fuzzy somewhat continuity, fuzzy almost-somewhat continuity, fuzzy

weakly-somewhat continuity. Then we give definitions of several types of somewhat continuity

and counter-example between intuitionistic fuzzy topological spaces.

Let X be a set and I = [0,1]. IX denote the set of all mappings λ: X → Y. A member of IX is

called a fuzzy subset of X. Unions and intersections of a fuzzy set denoted by ∨ and ∧respectively are defined by

∨ λi = sup λi (x): i∈J and x∈X,

∧ λi = inf λi (x): i∈J and x∈X.

Authors: Mamta Singh

* and Yashveer Singh

**,

*: Department of Mathematical Sciences and

Computer Applications, Bundelkhand

University, Jhansi (UP), India

**: Institute of Engineering and Technology,

Bundelkhand University, Jhansi (UP), India

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+( **,(*-+.

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2. Preliminaries

Definition 2.1 [4] Let X be a nonempty fixed set. An intuitionistic fuzzy set (IFS for short) A is

an object having the form

A = <x, µA(x), A(x)> : x ∈ X

where the functions µA : X→ I and A : X→ I denote the degree of membership (namely µA(x))

and the degree of nonmebership (namely A(x)) of each element x ∈ X to the set A,

respectively , and 0 µA(x) + A(x) 1 for each x ∈ X. For the sake of simplicity, we shall use

the symbol A = <x, µA,A> for the IFS A =<x, µA(x), A(x)> : x ∈ X.

Definition 2.2 [4] Let X be a nonempty set and the IFS’s A and B be in the form

A = <x, µA(x), A(x)> : x ∈ X, B = <x, µB(x), B(x)> : x ∈ X

and letAi : i ∈ j be an arbitrary family of IFS’s in X .Then

(a) A ⊆ B iff ∀ x∈X [µA(x) µB(x) and A(x) B(x)];

(b) A = B iff A ⊆ B and B ⊆ A;

(c) = <x, A(x), µA(x)> : x ∈ X;

(d) ∩ Ai = <x, ∧µAi(x), ∨Ai(x)> : x ∈ X;

(e) ∪ Ai = <x, ∨µAi(x), ∧Ai(x)> : x ∈ X;

(f) 0~ = <x, 0, 1> : x ∈ X and 1~= <x, 1, 0> : x ∈ X.

Now we shall define the image and preimage of IFT’s. Let X, Y be two nonempty sets and

f : X Y be a function.

Definition 2.3 [5] (a) If B = <y, µB(y), B(y)> : y ∈ Y is an IFS in Y, then the preimage of B

under f denoted by f-1

(B), is the IFS in X defined by

f -1

(B) = <x, f -1

(µB)(x), f -1

(B)(x)> : x ∈ X.

(b) If A = <x, A(x), A(x)> : x ∈ X, is an IFS in X, then the image of A under f denoted by

f(A) is the IFS in Y defined by

f(A) = <y, f(A )(y), f - (A)(y)> : y ∈ Y, where f - (A) = 1 – f(1 - A).

Definition 2.4 [5] An intuitionistic fuzzy topology (IFT for short) on a nonempty set X is a

family τ of IFS’s in X satisfying the following axioms :

(T1) 0~, 1 ∈ τ,

!

(T2) G1 ∩ G2 ∈ τ for any G1, G2 ∈ τ,

(T3) ∪ Gi ∈ τ for any arbitrary family Gi : i ∈ j ⊆ τ.

In this case the pair (X, τ) is called an intuitionistic fuzzy topological space (IFTS for short) and

each IFS inτ is known as an intuitionistic fuzzy open set (IFOS for short) in X .

Definition 2.5 [5] The complement of an IFOS A in an IFTS (X, τ) is called an intuitionistic

fuzzy closed set (IFCS for short) in X.

Definition 2.6 [5] Let (X, τ) be an IFTS and A = <x, µA(x), A(x)> be an IFS in X. Then the

fuzzy interior and fuzzy closure of A are defined by

cl(A) = ∩ K : K is an IFCS in X and A ⊆ K, int(A) = ∪ G : G is an IFOS in X and G ⊆ A.

It can be also shown that cl(A) is an IFCS and int(A) is an IFOS in X, and

(a) A is an IFCS in X r cl(A) = A,

(b) A is an IFOS in X r int(A) = A.

Definition 2.7 [6] An IFS A in an IFTS X is called

(a) an intuitionistic fuzzy regular open set of X if int(cl(A)) = A,

(b) an intuitionistic fuzzy regular closed set of X if cl(int(A)) =A.

Every intuitionistic fuzzy regular open (closed) set is an intuitionistic fuzzy regular open (closed)

set.

Theorem 2.8 [6] (a) The interior of an IFCS is an intuitionistic fuzzy regular open set.

(b) The closure of an IFOS is an intuitionistic fuzzy regular closed set.

3. Some type of fuzzy continuity in IFTS’s

Throughout this section (X, τ), (Y, ) will denote IFTS’s and f : X Y will denote a function.

Definition 3.1 [5] f is said to be fuzzy continuous if the preimage of each IFS in is an IFS in τ.

Definition 3.2 [6] A function f is called a fuzzy almost continuous function, if for each

intuitionistic fuzzy regular open set A of Y, f -1

(A) ∈ τ.

Definition 3.3 [6] The following are equivalent:

(a) f is a fuzzy almost continuous function,

(b) f -1

(B) is an IFCS, for each intuitionistic fuzzy regular closed set B of Y,

(c) f -1(B) s int(f -1(int(cl(B)))), for each IFOS B of Y,

(d) cl(f -1

(cl(int(B)))) s f -1

(B), for each IFCS B of Y.

!

Definition 3.4 [6] A function f is called a fuzzy weakly continuous function if for each IFOS B

of Y, f -1

(B) s int(f -1

(cl(B))).

4. Some types of fuzzy somewhat continuity in IFTS’s

Throughout this section (X, τ), (Y, ) will denote IFTS’s and f : X Y will denote a function.

Definition 4.1 A function f is said to be fuzzy somewhat continuous if for any IFOS A in Y for

which f -1

(A) 0~ we have int(f -1

(A)) 0~.

A fuzzy continuous function is always fuzzy somewhat continuous. But the converse is not true.

Example 4.2 Let X = a, b, c, Y = 1, 2, 3 and

G1 = <x,(a/.4,b/.4,c/.5),(a/.4,b/.4,c/.4)>, G2 = <x,(a/.2,b/.3,c/.4),(a/.5,b/.5,c/.5)>,

U1 = <y,(1/.5,2/.4,3/.5),(1/.4,2/.4,3/.3)>, U2 = <y,(1/.4,2/.2,3/.4),(1/.5,2/.4,3/.5)>.

Then the family τ = 1~, 0~, G1, G2 of IFS’s in X is an IFT on X and the family = 1~, 0~, U1,

U2 of IFS’s in Y is in IFT on Y. If we define the function f: X Y by f(a) = 2, f(b) = 3, f(c) = 1,

then

f -1

(U1) = <x, (a/.4, b/.5, c/.5), (a/.4, b/.3, c/.4)> 0~

int(f -1

(U1)) = G1 0~.

f -1(U2) = <x, (a/.2, b/.4, c/.4), (a/.4, b/.5, c/.5)> 0~

int(f -1

(U2)) = G2 0~.

Thus f is fuzzy somewhat continuous, but not fuzzy continuous since

f -1(U2) = <x, (a/.2, b/.4, c/.4), (a/.4, b/.5, c/.5)> Y . Definition 4.3 A function f is said to be fuzzy almost somewhat continuous if for any IFOS A in

Y for which f -1

(A) 0~ we have int(f-1

(int(cl(A))) 0~.

A fuzzy somewhat continuous function is always fuzzy almost somewhat continuous. But the

converse is not true in general.


G1 = <x,(a/.5,b/.5,c/.4),(a/.4,b/.3,c/.4)>, G2 = <x,(a/.5,b/.35,c/.4),(a/.4,b/.5,c/.5)>,

U1 = <y,(1/.5,2/.6,3/.5),(1/.4,2/.4,3/.3)>, U2 = <y,(1/.4,2/.5,3/.4),(1/.5,2/.4,3/.5)>.


U2 of IFS’s in Y is in IFT on Y. If we define the function f : X Y by f(a) = 1, f(b) = 3, f(c) =

2, then

f -1

(U1) = <x, (a/.5, b/.5, c/.6), (a/.4, b/.3, c/.4)> 0~

!

int(f -1

(int(cl(U1)) )) = 1~ 0~.

f -1

(U2) = <x, (a/.4, b/.4, c/.5), (a/.5, b/.5, c/.4)> 0~

int(f -1

(int(cl (U2)))) = 1~ 0~.

Thus f is fuzzy almost somewhat continuous, but not fuzzy somewhat continuous, since

int(f -1

(U1) ) = G1 0~, int(f -1

(U2) ) = 0~.

Corollary 4.5 Every fuzzy almost continuous function is also fuzzy almost somewhat

continuous.

Proof: Let A be an IFOS of Y such that f -1(A) 0~. Since f is fuzzy almost fuzzy continuous by

theorem 3.3 f -1(A)s int(f -1(int(cl(A)))). On the other hand, we obtain int(f -1(int(cl(A)))) 0~

from f -1

(A) 0~. This show that f is fuzzy almost somewhat continuous.

It is shown in the following example that the converse of the above corollary is not true, in

general.

Example 4.6 Let X = a, b, c, Y = 1, 2, 3 and G1 = <x,(a/.4,b/.4,c/.5),(a/.4,b/.4,c/.4)>,

G2 = <x,(a/.2,b/.4,c/.3),(a/.5,b/.5,c/.5)>,

U1 = <y,(1/.3,2/.2,3/.4),(1/.3,2/.35,3/.4)>, U2 = <y,(1/.3,2/.2,3/.5),(1/.2,2/.2, 3/.4)>.


U2 of IFS’s in Y is in IFT on Y. If we define the function f : X Y by f(a) =2, f(b) = 3, f(c) = 1,

then

f -1(U1) = <x, (a/.2, b/.4, c/.3), (a/.35, b/.4, c/.3)> 0~

int(f -1

(int(cl(U1)) )) = G2 0~.

f -1

(U2) = <x, (a/.2, b/.5, c/.3), (a/.2, b/.4, c/.2)> 0~

int(f -1(int(cl(U2)))) = 1~ 0~.

Thus f is fuzzy almost somewhat continuous, but not fuzzy almost continuous, since for U1 ⊂ Y

IFROS

f -1

(U1) = <x, (a/.2, b/.4, c/.3), (a/.35, b/.4, c/.3)> Y . Definition 4.7 A function f is said to be fuzzy weakly somewhat continuous if for any IFOS A in

Y for which f -1

(A) 0~ we have int(f -1

(cl(A))) 0~.

A fuzzy almost somewhat continuous function is always fuzzy weakly somewhat continuous.

But the converse is not true.


G1 = <x,(a/.4,b/.5,c/.5),(a/.3,b/.4,c/.4)>, G2 = <x, (a/.5,b/.5,c/.5),(a/.2,b/.3,c/.1)>,

!"

U1 = <y,(1/.5,2/.4,3/.5),(1/.4,2/.4,3/.3)>, U2 = <y,(1/.4,2/.2,3/.4),(1/.5,2/.4,3/.5)>.


U2 of IFS’s in Y is in IFT on Y. If we define the function f: X Y by f(a) = 1, f(b) = 3, f(c) = 2,

then

f -1(U1) = <x, (a/.5, b/.5, c/.4), (a/.4, b/.3, c/.4)> 0~

int(f -1

(cl(U1)) )) = 1~ 0~.

f -1

(U2) = <x, (a/.4, b/.4, c/.2), (a/.5, b/.5, c/.4)> 0~

int(f -1(cl (U2)))) = G2 0~.

Thus f is fuzzyweakly somewhat continuous, but not fuzzy almost somewhat continuous, since

int(f -1

(int(cl(U1) ))) = 1~ 0~

int(f -1

(int(cl(U2) ))) = 0~.

Corollary 4.9 Every fuzzy weakly continuous function is also fuzzy weakly somewhat

continuous.

Proof. Let A be an IFOS of Y such that f -1

(A) 0~. Since f is fuzzy weakly continuous by

definition 2.4. we have f -1

(A) s int (f -1

(cl(A))). On the other hand, we obtain int(f -1

(cl(A))) 0~ from f

-1(A) 0~ This show that f is fuzzy weakly somewhat continuous.

It is shown in the following example that the converse is not true, in general.

Example 4.10 Refer to example 4.8. Then f is fuzzy weakly somewhat continuous, but not fuzzy

weakly continuous, since f -1

(U2) tint(f -1

(cl(U2))).

References

[1] K. Atanassov, Intuitionistic Fuzzy Sets, VIIITKR’s Session, Sofia, 1989, (Bulgaria).

[2] K.Atanassov, S.Stoeva, Intuitionistic Fuzzy Sets, Polish Symposium on Interval and Fuzzy

Mathematics,Poznan, 1983,23-26.

[3] K.Atanassov, S.Stoeva,Intuitionistic L-FuzzySets,Cybernetics and System

Research(1984),539-540.

[4] K. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Set and System 20(1986) 87-96.

[5] D.Coker, An Introduction to Intuitionistic Fuzzy Topological Spaces, to appear in Fuzzy Sets

and Systems.

[6] H. Gurcay, A.H. Es, and D.Coker, On Fuzzy Continuity Intuitionistic Fuzzy Topological

Spaces, to appear in The Journal of Fuzzy Mathematics.

[7] L.A. Zadeh, Fuzzy Sets, Information and Control, 18 (1965) 338-353.

!!

Gamma Rings and

m-Systems

Abstract

In this paper, we collect some results related to the concepts: Γ-ring, ideal of a Γ-ring,

Γ-homomorphisms, m-system, prime ideal, g-system, prime radical. Finally, in the last section,

we present the proof of a new theorem: If M, M1 are two Γ-rings, f: M → M

1 a Γ-epimorphism

and S ⊆ M, then S is an m-system in M ⇔ f(S) is an m-system in f(M).

1. Introduction

Historically, the first step towards Γ-rings was taken by Nobusawa 1964 and the next step was

taken by Barnes 1966. Γ-rings of Barnes were much studied. Many authors studied the system

Γ-ring in different aspects. The radical theory in Γ-rings was studied by several authors like

Booth, Godloza, Satyanarayana, Pradeep Kumar and Srinivasa Rao.

This paper is divided into three sections. In Sections 1 and 2, we provide a collection of well

known related definitions, examples and some results related to these concepts in Gamma Ring

Theory. In Section-3, we study the concept m-systems, and proved a new theorem related to m-

systems in Gamma rings.

In this section, first, we present the definition of Γ-ring and related examples. The definition of

Γ-ring in the sense of Nobusawa [1964] is as follows:

1.1 Definition : Let M be an additive group whose elements are denoted by a, b, c, ... and Γ

another additive group whose elements are α, β, γ, ... . Suppose that aαb is defined to be an

!!"!##$!$ %& '(&) *

+( **,(*-+.

Authors: Satyanarayana Bhavanari

* and

Shakira Sk.**,

*: Department of Mathematics, Acharya

Nagarjuna University, Nagarjuna Nagar-522

510, A.P., India.


**: Department of Mathematics, Sanketika

Institute of Technology & Management,

Behind Cricket Stadium , P M Palem,

Visakhapatnam, A.P.

!

element of M and that αaβ is defined to be an element of Γ for every a, b, α and β. If the

products satisfy the following three conditions for every a, b, c ∈ M, α, β ∈ Γ: (i) (a + b)αc

= aαc + bαc; a(α + β)b = aαb + aβb; aα(b + c) = aαb + aαc;

(ii) (aαb)βc = aα(bβc) = a(αbβ)c; and (iii) If aαb = 0 for all a and b in M, then α = 0,

then M is called a ΓΓΓΓ-ring.

The definition of Γ-ring in the sense of Barnes [ 1966] is as follows:

1.2 Definition: Let M and Γ be additive Abelian groups. M is said to be a ΓΓΓΓ-ring if there exists

a mapping M x Γ x M → M (the image of (a, α, b) is denoted by aαb) satisfying the following

conditions (i) and (ii):

(i) (a + b)αc = aαc + bαc; a(α + β)b = aαb + aβb; aα(b + c) = aαb + aαc; and

(ii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ.

The concept Γ-ring in the sense of Barnes is a generalization of the concept Γ-ring in the sense of

Nobusawa. Several authors preferred to study the Γ-ring in the sense of Barnes.

Henceforth, all Γ-rings considered (in this paper) are Γ-rings in the sense of Barnes. A natural

example of a Γ-ring can be constructed in the following way:

1.3 Example (Barnes [ 1 ]): Let (X, +), (Y, +) be two Abelian groups. Write M = Hom(X, Y),

Γ = Hom(Y, X). M and Γ are additive Abelian groups with respect to the usual addition of

mappings. (That is (f + g)(x) = f(x) + g(x) for all x ∈ X).

Let a, b ∈ M and α ∈ Γ. Then b: X → Y, α: Y → X and a: X → Y. Suppose aαb is the usual

composition of mappings. Since a, α, b are homomorphisms, we have that aαb: X →

Y is a homomorphism. Therefore aαb ∈ M. With these operations, it is easy to verify that M is

a Γ-ring.

1.4 Example: Let R be a ring. Write M = R, Γ = R. Take a, b, c ∈ M, α, β, γ ∈ Γ. aαb is the

product of a, α, b in R. So aαb ∈ R = M. Then M is a Γ-ring.

1.5 Example: Let M be any additive non Abelian group. Take Γ = 0. Define aαb = a0b =

0M and αbβ = 0b0 = 0Γ for all a, b ∈ M, α, β ∈ Γ. Then it is easy to verify that M is a Γ-ring.

1.6 Example: Suppose M be a right R-module and suppose there exists m ∈ M such that (0:m)

= 0. Take Γ= HomR(M, R). Define aγb = a.γ(b) for all a, b ∈ M, γ ∈Γ. Then it is easy to verify

that M is a Γ-ring.

!

1.7 Example: Let U, V be vector spaces over the same field F. Write M = Hom(U, V), Γ =

Hom(V, U). Then M is a Γ-ring with respect to point wise addition and composition of

mappings.

1.8 Notation: Let M be a Γ-ring. For A ⊆ M, B ⊆ M, ∆ ⊆ Γ we denote the set aαb / a ∈ A, α

∈ ∆, b ∈ B by A∆B. The set AΓB will be denoted by AB.

1.9 Definitions: (i) A subset A of a Γ-ring M is said to be a right ideal of M if A is an additive

subgroup of M and AΓM ⊆ A. (ii) A subset A of a Γ-ring M is said to be a left ideal of M if A

is an additive subgroup of M and MΓA ⊆ A. (iii) If A is both left and right ideal of M then A is

said to be an ideal of M. The smallest left ideal containing a ∈ M is denoted by al . This is the

intersection of all left ideals of the Γ-ring M containing the element a. We may also call this left

ideal as the left ideal generated by the element a ∈ M. The smallest left ideal containing a

subset X of M is denoted by Xl. We may also call this left ideal as the left ideal generated by

the subset X of M. The smallest ideal containing a subset X of M is denoted by X. We may

also call this ideal as the ideal generated by the subset X of M. The ideal a is denoted by

a.

1.10 Definition: Let M be a Γ-ring and I an ideal of M. Consider M/I = x + I / x ∈ M, the

quotient group of M with respect to the addition subgroup I. Define (x + I)γ(y + I) = xγy + I for

all x, y ∈ M and γ ∈ Γ. Then M/I becomes a Γ-ring. It is called as quotient ΓΓΓΓ- ring of M with

respect to the ideal I.

1.11 Definition (Barnes [ 1 ]): Let M, M1 be two Γ-rings. A mapping h: M → M1 is said to be a

Γ-ring homomorphism (or Γ-homomorphism) if it satisfies the following two conditions: (i) h(a

+ b) = h(a) + h(b) for all a, b ∈ M; and (ii) h(aγb) = h(a)γh(b) for all a, b ∈ M and γ ∈ Γ.

1.12 Theorem (Barnes [1]): (i) If M, M1 are Γ-rings and f: M → M

1 is a Γ-homomorphism, then

ker f = x ∈ M / f(x) = 0 is an ideal of M.

(ii) If f is onto Γ-homomorphism, then f(A) is an ideal of M1 if A is an ideal of M; and B = x /

f(x) ∈ B1 is an ideal of M for all ideals B

1 of M.

1.13 Definition: A subset S of a Γ-ring M is said to be an m-system if S = φ or if a, b ∈ S

implies a b ∩ S ≠ φ.

1.14 Examples: Consider M = Γ = , the ring of integers. Then M is a Γ-ring.

!

(i) If n > 0 and n is a prime number then n is a prime ideal of the Γ-ring M.

(ii) Let n ∈ such that n > 0. Write S = n, n2, n

3, .... Now we verify that S is an m-system.

Let x, y ∈ S x = nk, y = n

s for some k, s. Now n

k+s+1 = n

knn

s = xny ∈ x y and so n

k+s+1

∈ x y ∩ S. Therefore x y ∩ S ≠ φ. Hence S is an m-system.

1.15 Definition: An ideal P of M is said to be a prime ideal if for any two ideals A, B of M,

AB ⊆ P A ⊆ P or B ⊆ P.

1.16 Theorem (Barnes [1]): An ideal P of a Γ-ring M is prime if and only if C(P) = M\P is an m-

system.

1.17 Theorem (Barnes [1]): If I and P are ideals of a Γ-ring M, I ⊆ P and P is prime, then P/A is

prime in M/A. Conversely, if P1 is a prime ideal of M/A and f: M→ M/A is the canonical Γ-

epimorphism, then P = f –1

(P1) is a prime ideal of M.

1.18 Definition: Let A be an ideal of M. The prime radical of A (denoted by r(A)) is defined as

the set of all elements x of M such that every m-system containing x contains an element of A.

The prime radical of M is defined as the prime radical of the zero ideal.

1.19 Result (Barnes [1]): If A is any ideal of the Γ-ring M, then the prime radical r(A) is equal to

the intersection of all prime ideals containing A.

2. g-Systems

In this section we present the definitions: ideal mapping g, g-system, g-prime ideal and g-prime

radical. We also present some results related to these concepts.

2.1 Definition (Hsu [ 5 ]): Let M be a Γ-ring. We define g as a function of M into the family of

all ideals of M satisfying the following two conditions: (i) a ∈ g(a); and (ii) x ∈ g(a)+A

g(x) ⊆ g(a)+A for any element a ∈ M and for any ideal A of M. We fix such an ideal mapping g

on M.

2.2 Lemma (Satyanarayana [ 9 ]): g(x) = g(0) + (x) for all x ∈ M.

Proof: Let x ∈ M. x ∈ g(0) + (x) g(x) ⊆ g(0) + (x). Since 0, x ∈ g(x) + (0) we have g(0)

⊆ g(x) + (0) = g(x) and (x) ⊆ g(x) + (0). So g(0) + (x) ⊆ g(x) + (0) = g(x).

Therefore g(x) = g(0) + (x) for all x ∈ M.

!

2.3 Definitions: A subset S of the Γ-ring M is said to be a g-system if either S = φ or S contains

an m-system S1

such that g(a) ∩ S1 ≠ φ for every element a of S where S

1 is called a kernel of S.

Any m-system is a g-system with kernel itself. (Hsu [ 5 ]) An ideal P of the Γ-ring M is said to

be a g-prime if either M\P is a g-system in M or P = M.

(Satyanarayana [ 9 ]) A Γ-ring M is said to be prime ΓΓΓΓ-ring if (0) is a prime ideal of M. A Γ-

ring M is said to be a g-prime ΓΓΓΓ-ring if the ideal (0) is a g-prime ideal.

2.4 Note: (i) Let P be a prime ideal M\P is an m-system (by Theorem 1.16) M\P is a g-

system (by Definitions 2.3) P is a g-prime ideal;

(ii) The converse of (i) is not true. (For this observe the following Example 2.5).

The following example provides an example of a g-prime ideal which is not prime.

2.5 Example: Let M = , the set of all integers and Γ = 0, ± 2, ± 4, ... . Clearly M is a Γ-

ring. Let P = 32. We define g(a) = a, 2n for all a ∈ M where n is a fixed positive integer.

Let S1 = 2, 2

2, 2

3, ... . Now as in Example 2.3.2 (ii), we can verify that S

1 is an m-system.

Let x ∈ M\P. Then 2 n ∈ g(x) ∩ S1. Hence M\P is a g-system with kernel S1. Therefore P is a

g-prime ideal. Clearly 3 3 ⊆ 32 = P. But 3 ⊆ P. Therefore P is not a prime ideal. Thus

we conclude that P is a g-prime ideal but not a prime ideal.

2.6 Lemma (Hsu [5]): For any g-prime ideal P, a, b ∈ M we have that g(a)g(b) ⊆ P a ∈ P or

b ∈ P.

2.7 Definition: The set x ∈ M / every g-system containing x contains 0 is called the g-prime

radical of M. It is denoted by rg(M).

2.8 Theorem (Hsu [5]): rg(M) = the intersection of all g-prime ideals of M.

2.9 Lemma (Satyanarayana [9]): If A is an ideal of M containing g(0), then the following two

conditions are equivalent: (i) A is prime; and (ii) A is g-prime.

2.10 Theorem (Satyanarayana [9]): If A is an ideal of M, then either rg(A) = A or rg(A) = r(A).

Moreover, rg(M) = (0) or rg(M) = r(M).

2.11 Theorem (Satyanarayana [9]): If r(M) ≠ 0, then the following two conditions are

equivalent: (i) rg(M) = r(M); and (ii) every g-prime ideal is a prime ideal.

!

1. Some Results on m-systems

In this section, we obtain some new results on m-systems. For proving our main theorem, we

prove three lemmas. We start this section with the following lemma.

3.1 Lemma: Suppose f: M → M1 is a Γ-epimorphism from the Γ-ring M onto the Γ-ring M

1. If

a ∈ M, then f(a) = f(a) .

Proof: Let a ∈ M. Then f(a) ∈ M1. Since f(a) is an ideal of M

1, by Theorem 1.12, we have

that f –1(f(a)) is also an ideal of M. Now f(a) ∈ f(a) a ∈ f –1(f(a)) a ⊆ f –1(f(a))

(since f –1

(f(a)) is an ideal) f(a) ⊆ f(a). Now we prove the other part. Since a is an

ideal of M, by Theorem 1.12, we have that f(a) is an ideal of M1. Now f(a) ∈ f(a) and f(a) is

an ideal f(a) ⊆ f(a). Hence f(a) = f(a).

3.2 Lemma: Let M, M1 be two Γ-rings and f: M → M

1 is a Γ-ring epimorphism. If S is an m-

system in M, then f(S) is an m-system in f(M) = M1.

Proof: Let S be an m-system in M. We have to show that f(S) is an m-system in M1. To verify

that f(S) is an m-system, take a1, b

1 ∈ f(S). Since f is onto there exist a, b ∈ S such that f(a) = a1,

f(b) = b1. Since S is an m-system and a, b ∈ S, there exist a

* ∈ a, γ ∈ Γ, b* ∈ b such that

a*γb

* ∈ a b ∩ S. Now a* ∈ a f(a

*) ∈ f(a) ⊆ f(a) (by Lemma 3.1) = a1 Similarly

f(b*) ∈ f(b) = b1 . Now f(a*)γf(b*) ∈ a1 b1. Also f(a*)γf(b*) = f(a*γb*) (since f is Γ-ring

homomorphism)∈ f(S) (since a*γb

* ∈ S). Therefore f(a*)γf(b

*) ∈ a1 b1 ∩ f(S). Now we

proved that a1, b

1 ∈ f(S) implies that a1 b1 ∩ f(S) ≠ φ. This shows that f(S) is an m-system in

M1. The proof is complete.

3.3 Lemma: Let M, M1be two Γ-rings. If S* is an m-system in f(M) = M1, then f -1(S*) is an m-

system in M.

Proof: Let S* be an m-system in f(M). Now we have to verify that f

–1(S

*) is an m-system in M.

To verify this, let a, b ∈ f -1

(S*). Now f(a), f(b) ∈ S

*. Since S

* is an m-system, it follows that

f(a) f(b) ∩ S* ≠ φ f(a) f(b) ∩ S

* ≠ φ (by the above Lemma 3.1) f(a1)γf(b1) ∈ S* for

some a1 ∈ a, b1 ∈ b, γ ∈ Γ f(a1γb1) ∈ S* (since f is Γ-homomorphism) a1γb1 ∈ f

-1(S

*)

a b ∩ f -1

(S*) ≠ φ (since a1 ∈ a and b1 ∈ b). Therefore f

-1(S

*) is an m-system. The

proof is complete.

3.4 Theorem: Let M, M1 be two Γ-rings, f: M → M1 a Γ-epimorphism and S ⊆ M. Then S is m-

system in M ⇔ f(S) is an m-system in f(M).

Proof: Combination of Lemmas 3.2 and 3.3.

!

Conclusion: In the first part of the paper, we collected some results related to the concepts: Γ-

ring, ideal of a Γ-ring, Γ-homomorphisms, m-system, prime ideal, g-system, prime radical.

Finally, in the last section, we presented the proof of a new theorem (3.4): If M, M1 are two Γ-

rings, f: M → M1 a Γ-epimorphism and S ⊆ M, then S is an m-system in M ⇔ f(S) is an m-

system in f(M). We presented necessary examples.

Acknowledgements: The first author acknowledges the financial assistance from the UGC, New

Delhi under the grant F. No.34-136/2008 (SR) dated 30th

December 2008.

References:

[1] BARNES W.E "On the Γ-rings of Nobusawa" Pacific J. Math 18 (1966) 411-422.

[2] BOOTH G.L "A Contribution to the Radical Theory of Gamma Rings". Ph. D, Thesis, University of

Stellenbosch, South Africa,1985.

[3] BOOTH G.L. & GODLOZA L "On Primeness and Special Radicals of Γ-rings", Rings and radicals,

Pitman Research notes in Math series (contains selected lectures presented at the international

conference on Rings and Radicals, held at Hebei, Teachers University, Shijazhuang, Chaina, August

1994) pp 123-130.

[4] FACCHINI ALBERTO "Module Theory", Progress in Mathematics, Vol.167, Birkhäuser Verlag,

Switzerland, 1998.

[5] HUS D.F "On Prime Ideals and Primary Decompositions in Γ-rings", Math. Japonicae, 2 (1976) 455-

460.

[6] NOBUSAWA "On a Generalization of the Ring theory" Osaka J. Math. 1(1964) 81-89.

[7] PRADEEP KUMAR T.V. "On g1-γ-Prime Left Ideals and related Prime Radical in Γ-rings", M.

Phil., dissertation, Acharya Nagarjuna University, 1998.

[8] SATYANARAYANA BHAVANARI "A Note on Γ-rings" Proc. Japan Acad. 59-A (1983) 382-33.

[9] SATYANARAYANA BHAVANARI "A Note on g-prime Radical in Gamma rings", Quaestiones

Mathematicae, 12(4) (1989) 415-423.

[10] SATYANARAYANA BH., PRADEEP KUMAR T.V. & SRINIVASA RAO M. "On Prime Left

Ideals in Gamma rings", Indian J. Pure and Appl. Math. 31(6) (2000) 687-693.

[11] SATYANARAYANA BHAVANARI, “Contributions to Near-ring Theory”,

VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7)

[12] WIEGANDT RICHARD "Radical Theory of Rings", The Mathematics Student 51 (1983)145-185.

!

Reaction of Urdbean

Genotypes on Growth in

Rainfed Vertisols of

Andhra Pradesh – A Case

Study

Abstract

A Field experiment was conducted during Kharif (rainy) season in rain fed vertisols of Regional

Agricultural Station (RARS), LAM, Guntur (A.P.) to study the “Reaction of different genotypes

of Urdbean to leaf curl and Yellow mosaic virus” to elicit the information on relative

resistance/tolerance to the biotic agents. Eighteen genotypes of Urdbean were tested by using

Randomized Block Design with three replications. The data on Leaf curl, and Yellow mosaic

virus were tested (or) subjected to analysis of variance using ‘34’ error degrees of freedom. Leaf

curl on eighteen genotypes of Urdbean did not differ significantly. The growth of the genotypes

in terms of Yellow mosaic virus differs significantly from each other. Among the eighteen

genotypes tested, OBG33 is least susceptible to Yellow mosaic virus followed by NDU3-5 and

KUG50 and COBG672.

Introduction

India is the largest producer and consumer of Pulses in the world accounting for 25% of Global

production, 27% of Global consumption. It is also the importer with 11% share of world imports

during 1995-2001. The percapita availability of pulses has declined from 69 grams/day, in 1961

to 36 grams a day in the recent times, in a country.

Andhra Pradesh is one of the major pulses producing state in the country. Pulse crops are subject

to attack by more than 150 species of insects both in field and storage, 25 species cause serious

damage. Among them thrips are the major groups of insect pests causing damage both by direct

feeding and transmitting leaf curl disease caused by Peanut bud necrosis virus.

Identification of genotypes resistant to thrips and viral diseases is considered a better eco-

friendly option for their management. Keeping in view this important component of integrated

pest/disease managements, effects were made to screen genotypes of Urdbean (18) to identify

Authors: B. Re. Victor Babu

*, K. Rajya

Lakshmi* and G. Raghavaiah

**

*:Department of Statistics, Acharya Nagarjuna

University, Nagarjuna Nagar-522 510, Andhra

Pradesh.

**: Regional Agricultural Research Station,

Acharya N.G. Ranga Agricultural University,

Lam, Guntur-522 034 (Andhra Pradesh)

!!"!##$!$ %& '(&) *

+( **,(*-+.

!

promising resistant/or tolerant genotypes under natural infestation. Therefore, a field trial was

conducted at regional research station, Lam with the following objectives.

1. To study the Reaction of Urdbean genotypes to Leaf curl and Yellow mosaic virus.

2. It was carried the information on relative resistance/tolerance to the biotic agents during

Kharif.

Materials and Methods

A Field experiment entitled “Reaction of Urdbean genotypes on Leaf Curl, Yellow Mosiac Virus

using Randomized Block Design in rain fed vertisols of Andhra Pradesh” was conducted at

RARS, Lam, Guntur during Kharif season. The research station is situated at 16o18’ Northern

latitude, 80o29’Eastern longitude and at an altitude of 31.5 meters above mean sea level. The

experimental site was fairly uniform in topography and well drained. Normally south west

monsoon rains start in the second week June and end by September last week. The weekly mean

meteorological Data recorded during the crop period (July-September) gap period class B

METEROLOGICAL observatory of RARS lam are presented. The soil of the experimental site

is clayey in texture and slightly alkaline in reaction. The experimental site was low in available

Nitrogen, medium in available Phosphorous and high in available Potassium. Since, a single

factor needs to be evaluated with the limited number of treatments Randomized Block Design

was selected, which was replicated three times. The various treatments existed, for testing the

reactions are as follows.

TREATMENTS:

1. NDU-3-4 2. COBG-662 3. COBG-653 4.LBG-20

5. TU-6 6. TU-17-14 7. OBG-33 8. LBG-752

9. VALLABH 10. KUG-50 11.TU-17-19 12.WBG-26

13. OBG-32 14. PANT-2-3 15.NDU-3-5 16.LBG-623

17. COBG-672 18. COBG-671

The biometric data thus collected on Urdbean genotypes were subjected to analysis of variance

as per Randomized Block Design as given by Gomez and Gomez (1784).

ANOVA TABLE:-

Sources of

Variation

Degrees of

freedom Sum of Squares Mean Sum of

Squares F calculated F tabulated

Replications (r-1) R R1 = R/(r-1) R1/E1 F((r-1),*)

Treatments (t-1) T T1 = T/(t-1) T1/E1 F((t-1),*)

Error * ** E1 =uuu

Total (rt-1) T. S. S.

!"

Where R = Replicate Sum of Squares >vwxp yH zH

T = Treatment Sum of Squares >|x yH zH

Total Sum of squares >>~Rb – c.f, r = number of replications,

t= number of treatments

Calculated F value was compared with tabulated F value at 5% level of significance and if the

data were found significant, the critical difference was calculated by the following formula,

Standard Error = K Where EMSS=Error mean sum of squares

Mean standard Error (S.E.M+) =Standard Error Difference (S.E.D) = [ (S.E. M+)

Critical difference at 5% = (S.E. D) X t (error d.f. at 5% level of significance)

Coefficient of variation (C.V.) = (KQ Grand Mean

If the Coefficient of variation is above 25% then the experiment is wrong.

RESULTS AND DISCUSSIONS:

The Reaction of the Urdbean entries on viral diseases determined in terms of Leaf curl and

Yellow mosaic virus measured with various methods.

Leaf curl:

Total number of plants in each row and number of leaf curl infected plants in each row were

counted and the percentage of leaf curl infected plants per each genotype was computed.

Yellow Mosaic Virus (YMV):

Total number of plants in each row (genotype) and number of Yellow Mosaic virus infected

plants were counted and the percentage of YMV infected plants is calculated.

The Urdbean genotypes did not different significantly in leaf curl. But they differ significantly in

Yellow mosaic virus. The genotypes exhibited difference in growth with respect to Yellow

mosaic virus OBG33 is least susceptible followed by NDU3-5 and KUG50 and COBG672.

Conclusions:

2. Leaf curl did not affect the growth of the yield.

3. Yellow mosaic virus affects the growth of the yield.

!!

References:1. All India Co-ordinate Pulses Improvement Project, Annual Progress report for

1981-1982, Punjab Agricultural University, Ludhiana.

2. Amin P.W. (1985). Apparent resistance of ground cultivar ROBUT 33-1 to

Bud necrosis disease.

3. Chhabra K.S. and Kooner B.S. (1998). Insect Pest Management (IPM) in Mungbean

and Black gram and strategies. In IPM system in agriculture, Pulses limited, New

Delhi.

4. Cochran W.G. and Cox G.M. (1957). Experimental Designs, John Wiley & sons,

INC, New York.

5. Das, M.N. and Giri N.C. (1986). Design and Analysis of Experiments, Wiley Eastern

Limited, India.

6. Kalpana G, Vijaya Lakshmi K, Retna Sudhakar T, and Ramesh T (2002). Evaluation

of rice Agriculture against rice thrips, stenches to thrips biform is bagnall. Indian

Journal of Plant Protection, Vol.30, No.1, Page No.81-83.

7. Ranga Swamy R. (1995). A Text book of Agricultural Statistics, Wiley Eastern

Private limited, India.

TABLE-1: URDBEAN ENTRIES

Leaf curl Yellow mosaic virus

TREATMENTS R1 R2 R3 R1 R2 R3

NDU3-4 16.32 19.46 32.33 41.32 37.94 53.73

COBG662 30.00 22.22 29.93 45.00 57.67 52.24

COBG653 25.77 17.56 21.97 42.30 39.11 37.17

LBG20 43.74 21.13 23.73 33.46 34.82 36.33

TU-6 13.81 18.34 22.46 43.34 53.73 18.24

TU-17-14 20.70 22.22 27.90 20.70 28.59 54.09

OBG33 16.11 18.72 17.85 18.72 13.05 14.54

LBG752 25.62 33.71 19.91 25.70 36.03 38.94

VALLABH 22.55 24.80 23.73 59.02 61.00 56.79

KUG50 19.09 22.55 26.99 19.09 10.14 30.98

TU-17-19 24.50 25.03 24.50 33.83 42.82 29.40

COBG-26 34.02 27.69 20.27 39.58 31.31 16.43

OBG32 35.24 19.46 28.04 32.96 28.11 33.09

PANT-2-3 28.59 16.74 22.22 42.53 64.38 22.22

NDU3-5 24.12 29.00 21.39 16.74 23.81 18.44

LBG623 45.00 38.65 34.33 30.00 19.28 37.11

COBG672 26.13 24.43 31.88 47.75 49.08 36.21

COBG671 17.76 17.76 23.73 01.62 08.53 56.98

Original values of Leaf curl and Yellow mosaic virus are transformed by using Arc sin values.

!

In Table-1 we present the converted values, The original data is presented (Table-2) as follows

and it is presented in Parenthesis.

Null Hypothesis:

H01: There is no significant difference between replications.

H02: There is no significant difference between treatments.

Analysis of Variance Table:

CONCLUSION:

Leaf Curl:

As F Calculated value is less than the F table value at 5% level of significance with (2,34)

degrees of freedom, so we conclude that there is no significant difference among replications.


degrees of freedom, so we conclude that there is no significant difference among treatments.

Yellow mosaic virus:


degrees of freedom, so we conclude that there is no significant difference among replications.

As F Calculated value is greater than the F table value at 5% level of significance with (17, 34)

degrees of freedom, so we conclude that there is a significant differenc between treatments. i.e.,

So

urce

s o

f

va

ria

tio

n

Deg

rees

o

f

free

do

m

Su

m o

f

Sq

ua

res

Mea

n S

um

of

Sq

ua

res

F

Ca

lcu

late

d

va

lue

Leaf

Curl

Yellow

mosaic

virus

Leaf

Curl

Yellow

mosaic virus

Leaf Curl Yellow

mosaic

virus

Leaf Curl Yellow

mosaic

virus

Replic

ates

2 2 0093.56 0083.94 46.7801 41.9643 1.4797 0.2770 3.5546

Treatm

ents

17 17 1150.57 6626.73 67.6805 389.8078 2.1408 2.5726 2.4563

Error 34 34 1074.91 5151.73 31.6149 151.5215

Total 53 53 2319.04 11862.39

Leaf Curl Yellow mosaic virus

Grand Mean Not Significant 34.7406

S E M Not Significant 7.1068

S E D Not Significant 10.0506

C.D. Not Significant 16.9947

C.V. Not Significant 35.4324

!

OBG 33 is least susceptible to Yellow mosaic virus followed by NDU3-5 and KUG50 and

COBG672.

TABLE: 2

Leaf Curl Yellow mosaic virus

TREATMENTS R1 R2 R3 R1 R2 R3

NDU3-4 (7.9) (11.1) (28.6) (43.6) (37.8) (65)

COBG662 (25) (14.3) (24.9) (50) (71.4) (62.5)

COBG653 (18.9) (9.1) (14) (45.3) (39.8) (36.5)

LBG20 (47.8) (13) (16.2) (30.4) (32.6) (35.1)

TU-6 (5.7) (9.9) (14.6) (47.1) (65) (9.8)

TU-17-14 (12.5) (14.3) (21.9) (12.5) (22.9) (65.6)

OBG33 (7.7) (10.3) (9.4) (10.3) (5.1) (6.3)

LBG752 (18.7) (30.8) (11.6) (18.8) (34.6) (39.5)

VALLABH (14.7) (17.6) (16.2) (73.5) (76.5) (70)

KUG50 (10.7) (14.7) (20.6) (10.7) (3.1) (26.5)

TU-17-19 (17.2) (17.9) (17.2) (31) (46.2) (24.1)

COBG-26 (31.3) (21.6) (12) (40.6) (27) (8.0)

OBG32 (33.3) (11.1) (23.1) (29.6) (22.2) (30.8)

PANT-2-3 (22.9) (8.3) (14.3) (45.7) (81.3) (14.3)

NDU3-5 (16.7) (23.5) (13.3) (8.3) (16.3) (10)

LBG623 (50) (39) (31.8) (25) (10.9) (36.4)

COBG672 (19.4) (17.1) (27.9) (54.8) (57.1) (34.9)

COBG671 (9.3) (9.3) (16.2) (0.8) (2.2) (70.3)

!

English Vocabulary

Development-An

Experiment through

Mathematics

I. INTRODUCTION:

Enriching knowledge of vocabulary and the ability to spell those words are significant tasks of

teaching English as a foreign language in our schools. Besides routine teaching of those

vocabulary skills in the class, integration of English vocabulary with teaching of other core

subjects whenever there is chance and if possible, through play way, reinforces children’s

learning of English vocabulary. School mathematics is one subject which can provide such

opportunity. Such integrated teaching can serve two purposes, firstly it takes away the

monotony of learning mathematics; secondly it reinforces what has been learnt in English

language periods or can give a prior feel of some of the vocabulary items before they are

introduced in regular English periods.

II. AIM:

The purpose of this paper is to share with teachers and teacher educators a few ideas regarding

the scope of integrating English vocabulary development with the teaching of a few topics in

school mathematics which was born out of experience while demonstrating the teaching of

‘sets’, a topic for 8th

standard to B.Ed. trainees.

This is not strictly an empirical study. This is rather a quasi experimental one indicating the

possibility of integrated teaching of English with other subjects, and its advantages and

establishing feasibility of sound experimental study by those who are interested in this area.

III. EXPANSION OF VOCABULARY:

Vocabulary means the words that we use in day to day life for expressing our feelings and

thoughts. Learning a language is not merely learning words, but yet we cannot speak a language

Authors: T. S. V. S. Suryanarayana Murthy,

Ganita Avadhani & S.A. Maths, Mukteswaram-

533 211, Ainavilli Mandal, E.G. Dist., Andhra

Pradesh.

Dr. D.S.N. Sastry, Retd. Principal, A J College

of Education, Machilipatnam, Krishna Dist,

Andhra Pradesh.

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!

unless we know its words. H.W. Beecher says, “ words are pegs to hang ideas on.” Pupils

should do not restrict themselves to a limited vocabulary but on the other hand, should increase

their vocabulary. Word building is an ability that all the pupils should acquire and be as efficient

and skilled as to making their own acquisition an asset to them. The English language teacher

can employ a number of exercises to help pupils enlarge their vocabulary. Language games

play a significant role in the expansion of vocabulary of any language. So is the case with

English language also. Voluntary involvement, healthy competitive spirit, thrill and pleasure are

the principles of a game. Any language learning activity, which possesses these characteristics,

can be called a language game. The following is an example of English Language games.

Eg: Find the word: In this game, the teacher writes a word ‘postpone’ on the board. Then he

will ask the students to find out the words hidden in this word. He may initially work out an

example of how they can write a new word using some of the letters in the given word as for

instance, ‘stop’. Similarly from the word ‘postpone’ , words like post, stop, one, top, pot, pose,

tone, etc. can be formed. The student who writes the largest number of words will be the winner.

In the above paragraphs, part of the work of an English teacher for the development of

vocabulary is described. Even a mathematics teacher can play a vital role in the development of

vocabulary , while applying mathematical principles. He can combine mathematics exercises

with language games by substituting letters for numbers etc. Exercises in “sets” serves as good

example.

IV. SETS:

The main principles of ‘sets’ are given hereunder.

Set : A set is well defined collection of objects.

Element : The objects in the set are called elements or members of the set.

Describing a set : It is customary to name a set by capital letters such as A,B,X, etc.

In order to define a set all the elements of the set are written in a row,

separated by commas and then enclosed in braces. . The elements of

a set are represented with small letters such as a,b,c, etc.in the case of

letters. The elements are represented by numbers also.

While writing the set consisting of elements a,a,b,b,c,c,d,d, we do not repeat

the elements which occur more than one time. The set of letters given

above is indicated by a,b,c,d.

!

Equality of two sets: Two sets ‘A’ and ‘B’ are equal if and only if every element in ‘A’ belongs

to ‘B’ and every element in ‘B’ belongs to ‘A’.

V. EXERCISES IN SETS INTEGRATED WITH VOCUBALARY GAMES:

The items discussed in the preceding paragraph (set, element, describing a set, and equal sets) are

the ‘key terms’ in the present context. Brief explanation of the above will project the role to be

played by a mathematics teacher for the development of vocabulary of English language.

The letters in the word mathematics. The collection of letters in the above word consists simply

eight distinct letters m, a, t, h, e, i, c, s. Here in the word ‘mathematics’ the letters m, a, t occur

more than one time and hence are dropped.

If a set N = o, n, w is given, a student can write – now, own, won. Mathematically these

words are equal sets as each word consists of all the elements of the given set.

When a set is given, it is possible to prepare a list of meaningful words using the elements

(letters) one at a time or more than once. The mathematics teacher’s role in the development of

English vocabulary starts at this point. Encouraging the students to write as many meaningful

words as possible is the primary duty of the mathematics teachers, which in turn develops the

vocabulary of his students.

V. APPLICATION OF THE INSIGHT IN THE CLASSROOM:

The idea came into the mind during a demonstration lesson in mathematics. The demonstration

lesson was being given to the B. Ed. trainees of the 2001-2002 batch. The pupils were 8th

standard students of the practicing school of the College of Education where the author is

working. The topic was “SETS”. The pupils being taught the lesson were a group of 16 boys and

24 girls.

Observing the steps of mathematics lesson, as usually key concepts of the topic, ‘sets’ were

introduced. A little practice was also given to the students using numbers and letters as

elements of sets. At this stage, the students were found somewhat restless. Then the author got

the idea of releasing the students from the monotony of learning sets, strictly as a mathematics

lesson. Then the students were told that they would play a game. Two exercise items of sets in

letters, each consisting of four sub-items, were displayed to the class, which are given below:

VII. ITEMS:

I. Write meaningful words from the following sets using all the elements of each set

once only.

(a) a, r, t (b) a, e, t (c) a, e, m, t (d) a, d, e, r

II. Write as many meaningful words as possible from the following sets using the

elements one at a time or more than once, of each set.

(a) a, e, r (b) a, e, s, t (c) o, p, s, t (d) a, e, l, p

!

Then the students were given instructions for building meaningful words from each exercise sub-

item. The instructions were somewhat in the following lines:

VI. INSTRUCTIONS:

All the sub-items of the first test item test the ability of the pupils to write meaningful words

using all the letters given in the test item. Mathematically, this represents writing of equal sets. In

other words, the pupil has to use all the letters (elements) given in a particular test item and to

write meaningful words. The probable words one can write in each sub-test item are presented

hereunder:

I. (1) art; rat, tar.

(2) eat, ate, tea.

(3) team, tame, meat, mate.

(4) read, dear, dare.

In the second test item, the pupils have to use all the letters (elements) given in the test item once

or more than once to write meaningful words. If any word is given, to write it in the form set,

the usual procedure is to eliminate repeated letters. For example, the word apple is given to

write in the form of a set, the set consisting of the letters (elements) a, p, l, ewill be written.

The main intention of the investigator is to test the ability of the pupils to write meaningful

words by using all the letters once or more than once given in the test item. The probable words

that can be written by following the procedure discussed above, sub-item wise are presented

hereunder:

II. (1) ear, era, are, area, rare, rear, arrear.

(2) east, eats, sate, seat, state, tease, attest, estate.

(3) opts, post, pots, spot, stop, tops, spots, stops.

(4) leap, pale, plea, apple, appeal, peal.

Keeping in view the time limit for demonstration lesson, the students were asked to work out

only first item consisting of the four sub-items in class. The class became lively. Thee was a

little humdrum for consultation among themselves. After six minutes, the author asked some

students to give out the words they wrote. Finally all the possible words that could be written

were displayed. The author could find expressions for surprise, disappointment, pride and

happiness, depending on whether they left the word, misspelt word, wrote all the words correctly

without misspelling. Later the children were asked to work out the second item at home on the

same sheet of paper on which the first item was done.

The next day with the permission of the Head Master and the mathematics class teacher, the

mathematics period was taken. The answer sheets were self-evaluated by the students in a few

minutes. It was found that a few words like attest, arrear were not attempted by the students.

Giving cues and by goading, the children were encouraged to complete those words also.

The number of boys and girls who wrote each word of the sub-item was arrived at by scanning

the answer sheets of the 40 students. The data with percentages has been tabulated ad presented

in Table nos. 1 to 8.

!

Table-1

art rat tar

boys 14 12 4

% 87.5 75 25

girls 13 23 7

% 54.17 95.83 29.19

total 27 35 11

% 67.5 87.5 27.5

87.5% of boys wrote this word ‘art’ where as the girls % is 54.17

75% of boys wrote the word ‘rat’ where as the girls % is 95.83

25% of boys wrote the word ‘tar’ where as the girls % is 29.19

Table-2

eat ate tea

boys 16 13 16

% 100 81.25 100

girls 21 16 19

% 87.57 66.72 79.23

total 37 29 35

% 92.5 72.5 87.5

100% of boys wrote this word ‘eat’ where as the girls % is 87.57

81.25% of boys wrote the word ‘ate’ where as the girls % is 66.72

100% of boys wrote the word ‘tea’ where as the girls % is 79.23

Table-3

team tame meat mate

boys 11 1 14 1

% 62.5 6.25 87.5 6.25

girls 12 0 21 1

% 50 0 87.57 4.17

total 23 1 35 2

% 57.5 2.5 87.5 5

62.5% of boys wrote this word ‘team’ where as the girls % is 50

6.25% of boys wrote the word ‘tame’ where as the girls % is 0

87.5% of boys wrote the word ‘meat’ where as the girls % is 87.57

6.25% of boys wrote the word ‘mate’ where as the girls % is 4.17

!

Table-4

read dare dearboys 14 1 13% 87.5 6.25 81.25girls 15 9 17% 62.55 37.53 70.89total 29 10 30

% 72.5 25 75

87.5% of boys wrote the word ‘read’ where as the girls % is 62.55

6.25% of boys wrote the word ‘dare’ where as the girls % is 37.53

81.25% of boys wrote the word ‘dear’ where as the girls % is 70.89

Table-5

ear are era area rare rear arrear

boys 10 12 6 0 1 0 0

% 62.5 75 37.5 0 6.25 0 0

girls 21 17 0 0 2 4 0

% 87.57 70.89 0 0 8.34 16.68 0

total 31 29 6 0 3 4 0

% 77.5 72.5 15 0 7.5 10 0

62.5% of boys wrote this word ‘ear’ where as the girls % is 87.57

75% of boys wrote the word ‘are’ where as the girls % is 70.89

37.5% of boys wrote the word ‘era’ where as the girls % is 0

0% of boys wrote the word ‘area’ where as the girls % is also 0

6.25% of boys wrote this word ‘rare’ where as the girls % is 8.34

0% of boys wrote the word ‘rear’ where as the girls % is 16.68

0% of boys wrote the word ‘arrear’ where as the girls % is also 0

Table-6

east eats sate seat state taste tease attest estate

boys 5 1 0 0 1 1 0 0 0

% 31.25 6.25 0 0 6.25 6.25 0 0 0

girls 18 0 0 1 3 0 0 0 0

% 75.06 0 0 4.17 12.5 0 0 0 0

total 23 1 0 1 4 1 0 0 0

% 57.5 2.5 0 2.5 10 2.5 0 0 0

!"

31.25% of boys wrote this word ‘east’ where as the girls % is 75.06

6.25% of boys wrote the word ‘eats’ where as the girls % is 0

0% of boys wrote the word ‘sate’ where as the girls % is also 0

0% of boys wrote the word ‘seat’ where as the girls % is 4.17

6.25% of boys wrote this word ‘state’ where as the girls % is 12.5

6.25% of boys wrote the word ‘taste’ where as the girls % is 0

0% of boys wrote the word ‘tease’ where as the girls % is also 0

0% of boys wrote the word ‘at least’ where as the girls % is also 0

0% of boys wrote the word ‘estate’ where as the girls % is also 0

Table-7

opts post pots spot stop tops spots stopsboys 1 16 4 11 16 12 0 0% 6.25 100 25 68.75 100 75 0 0girls 0 22 0 6 21 16 0 0% 0 91.66 0 25.02 87.57 66.72 0 0total 1 38 4 17 37 28 0 0

% 2.5 95 10 42.5 92.5 70 0 06.25% of boys wrote this word ‘opts’ where as the girls % is 0

100% of boys wrote the word ‘post’ where as the girls % is 91.66

25% of boys wrote the word ‘pots’ where as the girls % is 0

68.75% of boys wrote the word ‘spot’ where as the girls % is 25.02

100% of boys wrote this word ‘stop’ where as the girls % is 87.57

75% of boys wrote the word ‘tops’ where as the girls % is 66.72

0% of boys wrote the word ‘spots’ where as the girls % is also 0

0% of boys wrote the word ‘stops’ where as the girls % is also 0

Table-8

leap pale peal plea apple appeal palea

boys 14 2 0 8 1 0 0

% 87.5 12.5 0 50 6.25 0 0girls 22 2 0 0 14 0 0

% 91.66 8.34 0 0 58.38 0 0

total 36 4 0 8 15 0 0

% 90 10 0 20 37.5 0 0

!!

87.5% of boys wrote this word ‘leap’ where as the girls % is 91.66

12.5% of boys wrote the word ‘pale’ where as the girls % is 8.34

0% of boys wrote the word ‘peal’ where as the girls % is also 0

50% of boys wrote this word ‘plea’ where as the girls % is 0

6.25% of boys wrote the word ‘apple’ where as the girls % is 58.38

0% of boys wrote the word ‘appeal’ where as the girls % is also 0

0% of boys wrote the word ‘palea’ where as the girls % is also 0

Note: Only % comparisons are made statistical significance between these two persons is not

established.

To find out the effect of this technique of integrated teaching on the students, the next day, the

same class was taken again and the students were asked to work out the same exercise items.

The answer sheets were collected and scanned. Surprisingly it was found that all most all the

students wrote all possible words for all the eight sets, including the zero words in the tables.

Hence this information is not presented in tables for comparison with the first time exercise

results given in tables 1 to 8. It is presumed that because of the new situation, because of the

play way, and because of peer learning possibility, the students could acquire new words and

retain them in their memory. By this, the author felt that the initial idea is practicable and useful;

and it is feasible to conduct a thorough experimental study to confirm it.

At this stage the following advantages can be stated which are due to the integrated teaching

technique.

1. It gives variety to the teaching –learning of mathematics.

2. Mathematical exercises can be made pleasant by integrating them with language games

like word building.

3. Vocabulary items picked up by the students during the practice are retained well by them.

There is scope for team teaching. For better results, Mathematics and English teachers

should plan the exercises together.

!

Cryptography and Security

Visualization

Introduction:

Security administration can be costly and prone to error because administrators usually specify

access control lists for each user on the system individually. With RBAC security is managed at

a level that corresponds closely to organisations structure. Each user is assigned one or more

roles and each role is assigned one or more privileges that are permitted to users in that role.

Security administration with RBAC consists of determining the operations that must be executed

by persons in particular jobs and assigning employees to the proper roles.

Complexities introduced via mutually exclusive roles or role hierarchies are handled by the

RBAC software making security administration easier.

Field of security is a challenging field as technology changes everyday. There is a need to secure

computers and networks from the hackers by developing new algorithms. Cryptography is an art

of achieving security by encoding messages to make them not readable. Plaintext refers to

message (data) to be sent. When the plaintext message is codified using any suitable scheme, the

resulting message is called ciphertext. The process of encoding plaintext messages into

ciphertext messages is called encryption. Decryption is exactly opposite of encryption.

Decryption transforms a ciphertext message back into plaintext. Every encryption and decryption

process has two aspects, the algorithm and key used for encryption and decryption.

Cryptographic mechanisms are of two types , Symmetric and Asymmetric key cryptosystems

(Public key Cryptosystem). Cryptography plays major role in many information technology

applications like electronic mail, electronic banking, and electronic commerce.

Cryptography achieves security using following goals.

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Authors: Swati Joglekar

Fergusson College, Pune

Correspondence address:[email protected]

!

1. : Information cannot be observed by an unauthorised party. This is accomplished via

public-key and symmetric key encryption.

2. Data Integrity: Transmitted data within a given communication session cannot be altered

in transit due to error or unauthorised party. This is accomplished by Message

Authentication Codes.

3. Message Authentication: Parties within a given communication session must provide

certifiable proof validating the authenticity of a message. This is accomplished via the

use of Digital Signature.

4. Non-repudiation: Neither a sender nor the receiver of a message deny transmission. This

is accomplished via Digital Signature.

5. Entity Authentication: Establishing the identity of an entity, such as a person or device.

6. Access Control: Controlling access to data and resources, Access is determined based on

the privilege assigned to the data and resources as well as the privilege of the entity

attempting to access the data and resources.

The locks and keys technique combines features of access control lists and capabilities. A piece

of information (the lock) is associated with the object and a second piece of information (the

key) is associated with those subjects authorized to access the object. When a subject tries to

access an object, the subject’s set of keys is checked. If the subject has a key corresponding to

any of the object’s locks, access of the appropriate type is granted. Locks and keys may change

in response to system constraints, general instructions about how entries are to be added and any

factor other than a manual change. Cryptographic implementation of locks and keys is

suggested. The object o is enciphered with a cryptographic key. The subject has deciphering

key. To access the object, the subject deciphers it. This provides a simple way to allow n

subjects to access the data (or-access). Encipher n copies of data using n different keys, one per

subject. The object o is then represented as o’ , where o’ = ( E1(o),E2(o),……,En(o)). The

system can easily access except on the request of n subjects (and –access).Iterate the cipher

using n different keys, one per subject: o’ = E1(E2(……(En(o))……).

Role based access control mechanism:

File

System Mgt.

Network

Mgt.

Backup

Recovery

Kerberos

Mgt.TCPIP

Mgt.

System Management by root

!

Users are said to be the owners of the objects under their control. For many organisations end

users do not own the information for which they are allowed access.

Access priorities are controlled by the organisation and are often based on employee functions

rather than data ownership. In many organisations the processes are unclassified but have

sensitive information. In these environments security objectives often support higher level

organisational policies which are derived from existing laws, ethics, regulations or generally

accepted practices. Such environments usually require the ability to access information

according to how that information is labelled based on its sensitivity.

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With role based access control, access decisions are based on the roles that individual users have

as part of an organization. The process of defining roles should be based on a through analysis

of how an organization operates and should include input from a wide spectrum of users in an

organization

Using RBAC users are granted membership into roles based on their competencies and

responsibilities in the organisation. The operations that a user is permitted to perform are based

on the user’s role. Roles can have overlapping responsibilities and privileges; user’s belonging

to different roles may need to perform common operations. Role hierarchies can be established

to provide for the natural structure of an organisation. A role hierarchy defines roles that have

unique attributes and that may contain other roles (one role may implicitly include the

operations that are associated with another role)

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RBAC allows carrying out a broad range of authorized operations and providing great flexibility

and breadth of application.

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Advanced Predictive Data

Mining and Text mining Models

1. Introduction:

Use of mathematical models increase the performance of any process. We can adopt the

mathematical models in the fields like: data mining, text mining, and Business Intelligence. This

will increase awareness of domain knowledge to researchers. Data mining is the automated

discovery of strategic hidden patterns in large amounts of data using intelligent data analysis

methods [1]. The data mining revolution started with large volume data storage became cheaper

and analysis technology became more advanced. The two primary goals of data mining tend to

be prediction and description. Prediction involves some variables or fields in the data set to

predict unknown or future values of other variables of interest. Description focuses on finding

patterns describing the data that can be interpreted by humans. In this paper, we are presenting

some predictive data mining and text mining models.

2. Predictive models:

Predictive analysis is data mining technology that uses our data to build a predictive model

specialized for our domain. This process learns from the collective domain information. The

Knowledge gained is encoded as the predictive model itself. This knowledge and information

can be represented in various formats like number format for example mean, median, mode,

count or it could be in a graphical format such as histogram, line, pie chart, trend lines or moving

averages.

Authors: S. Sagar Imambi and

L. Padmavathi, T J P S College, Guntur,

Andhra Pradesh.

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Fig1. Predictive analysis model

Predictive Data Mining combines database analysis with multivariate statistics and artificial

intelligence. In recent years, predictive data mining has become an essential tool for strategic

decision making among several fields of applications. In business applications predicting future

customer behavior, classifying customer segments and forecasting events are some of

applications. There are many techniques like regression and classification for predictive data

analysis. Mapping the problem into a mathematical model requires some intellectual talent. We

have to carefully choose right model for right problem.

3. Classification:

In a classification, we are given historical data with class labels and unlabeled data. Each labeled

example consists of multiple independent attributes and one target attribute as dependent

attribute. The value of the target attribute is a class label. The unlabeled examples consist of the

independent attributes only. The goal of classification is to construct a model using the historical

data that accurately predicts the class of the unlabeled examples. The historical data with class

labels is considered as train data and data without the class labels is called as test data. Different

classification algorithms use different techniques for finding relations between the predictor

attributes' values and the target attribute's values in the build data. A classification model can

also be used on build data with known target values, to compare the predictions to the known

answers; such data is also known as test data or evaluation data. This technique is called testing a

model, which measures the model's predictive accuracy. Some of Classification techniques are

Logistic Regression, Bayesian Methods, Discriminate Analysis, Neural Net, kNN, CART

Many automated prediction methods exist for extracting patterns from sample cases in text

mining, specifically text categorization. The documents are encoded in terms of features in some

numerical form, requiring a transformation from text to numbers. For each case, a uniform set of

measurements on the features are taken by compiling a dictionary from the collection of training

documents. Prediction methods look at samples of documents with known topics, and attempt to

find patterns for generalized rules that can be applied to new unclassified documents. Once the

0

1 2

3 %3

0

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data is in a standard encoding for classification, any standard data mining method, such as

decision trees or nearest neighbors, can be applied.[2].

3.2 Neural net

Algorithms based on neural networks have lot of applications. In Data mining generally the

following neural networks architectures are used. 1) Multi layered Feed forward and 2) Kohonen

self organizing maps. Feed-forward networks regards the perception back-propagation model

and the function network as representatives, and mainly used in the areas such as prediction and

pattern recognition. Self-organization networks uses adaptive resonance theory (ART) model and

Kohonen model as representatives, and mainly used for cluster analysis.

Neural networks are a proven technology for solving complex classification problems. Credit

companies often deploy neural networks to spot fraudulent credit card activity and identity

theft. Other companies deploy neural networks to identify defecting customers in order to

maximize their customer retention.

Neural networks are used to discover marketing opportunities, segment customers and to

discover more complex relationships in your data. With this technology, we can develop more

accurate and effective predictive models for better decision-making.

3.3 Discriminant Analysis

Discriminant Analyis (DA), a multivariate statistical technique is commonly used to build a

predictive or descriptive model of group discrimination based on observed predictor variables

and to classify each observation into one of the groups. In DA multiple quantitative attributes are

used to discriminate single classification variable.

In computerized face recognition, each face is represented by a large number of pixel values.

Linear discriminant analysis is primarily used here to reduce the number of features to a more

manageable number before classification. Each of the new dimensions is a linear combination of

pixel values, which form a template.

In marketing, discriminant analysis was once often used to determine the factors which

distinguish different types of customers and/or products on the basis of surveys or other forms of

collected data

4. Regression

Regression creates predictive models. The difference between regression and classification is

that regression deals with numerical/continuous target attributes, whereas classification deals

with discrete or categorical target attributes. If the target attribute contains continuous values, a

regression technique is required. If the target attribute contains categorical values, a classification

technique is called for. Some of regression models are Linear Regression, kNN, CART, Neural

Net. The most common form of regression is linear regression, in which a line that best fits the

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data is calculated, that is, the line that minimizes the average distance of all the points from the

line. This line becomes a predictive model when the value of the dependent variable is not

known; its value is predicted by the point on the line that corresponds to the values of the

independent variables for that record. The support vector machine has the baseline form of a

linear discriminator.

K Nearest Neighbor is one of those algorithms that are very simple to understand .The kNN

algorithm predicts the outcome y for an example x by finding the k labeled examples (xi, yi)

closest to x and returning: the average outcome of y. kNN can be used for both Classification and

regression.

CART builds classification and regression trees for predicting continuous dependent variable

(regression) and categorical predictor variables (classification). The classic CART algorithm was

popularized by Breiman et al. (Breiman, Friedman, Olshen, & Stone, 1984)[1]. CART is

powerful because it can deal with incomplete data, multiple types of features like input features

and predicted features. The trees it produces often contain rules which are humanly readable. The

basic CART building algorithm is a greedy algorithm in that it chooses the locally best

discriminatory feature at each stage in the process. This is suboptimal but a full search for a fully

optimized set of question would be computationally very expensive.

5. Data mining and Text Mining Applications.

• Retrieving Documents from digital libraries

Text mining can be used to improve the comprehensiveness and relevance of information

retrieved from databases.

• Identify Infrastructure

Text mining can be used to identify the elements of the infrastructure of a technical

discipline. These infrastructure elements are the authors, journals, organizations and other

group or facilities that contribute to the advancement and maintenance of the discipline.

• Geospatial,

Data mining techniques are used to handle spatiotemporal data, robust geographic

concept hierarchies and granularities, and sophisticated geographic relationships,

including non-Euclidean distances, direction, connectivity, attributed geographic space

and constrained interaction structures [6]

• Identify Technical Themes / Relationships with literature

Text mining can be used to identify technical themes, their inter-relationships, their

relationships with the infrastructure and technical taxonomies through computational

linguistics. By categorizing phrases and counting frequencies.

• Technology Forecasting In the process of retrieving and relating useful text data, text mining can also provide the

time series for trend extrapolation. It can be used to identify state-of-the-art Research &

Development (R&D) emphases.

!!

• Telecommunications:

Data mining techniques are used to billing, fraud detection and consumer marketing. of

data

• Prediction Relationships: Prediction relationships include the application of data

mining techniques to help researchers predict storm tracts and changes in intensity as

storms approach land, to identify conditions that will result in droughts, floods or fire

potential.

Conclusion: We studied the various predictive data mining models and presented them in

this paper. These new concepts are beginning to guide to build models for various problems in

the fields of banking, Business, Telecommunication, Geospatial and literature mining e.t.c.

References:

[1] Jiawei Han and Micheline Kamber, Data Mining: Concepts and Techniques, Kaufmann

Publishers ( 2006)

[2] A Roadmap to Text Mining and Web Mining. http://www.cs.utexas.edu/users/pebronia/text-

mining/

[3] Jeffrey W. Seifert, Data Mining an Over View , CRS Report for Congress(2004)

[4] Brigitte Mathiak and Silke EcksteinFive Steps to Text Mining in Biomedical Literature

Proceedings of the Second European Workshop on Data Mining and Text Mining in

Bioinformatics(2003)

[5] Shannon R. Anderson, The Dangers of Using Data Mining Technology to Prevent

Terrorism(2002)

[6] Y. W. Huang, F. Yu, C. Hang, C.-H. Tsai, D.-T. Lee, and S.-Y. Kuo. Securing,web

application code by static analysis and runtime protection. In Proceedings, of the 13th conference

on World Wide Web, (2004.),pp 40-52.

!

Number and Infinity

Concepts in Vedas

Introduction: The Vedas are the earliest records of human wisdom. Vedas consist of the output

related to the results of continuous observations and research of the earlier mankind during the

period approximately 6000 BC – 19000BC. Thus Vedas consists of prehistoric bulk of records

those carry, then believed scientific statements pertaining to several branches of sciences

including mathematics. In particular, Vedic Mathematics is the mathematical knowledge of

ancient Hindus passed down through generations (initially verbally) in the form of hymn/slokas

(verses) in Sanskrit.

Four Parts of the Veda: The Veda was divided into four parts by Sage Veda Vyasa in around

3000BC. The parts are called 1. Rig Veda. 2. Yajur Veda, 3. Sama Veda and 4. Atharvan

Veda. We treat the whole Vedic literature as single unit.

Development of Number system in Vedas: The Rig-Veda contains some statements in the form

of hymn that are related to the fractions.

Jyestha aha camasa dva (1/2) karoti kaniyan trin (1/3) kunavametyaha

kanistha aha caturas karoti (1/4) tvasta ubha vastat panayad vaco vah

(Rig-Veda/mandala-4/Sukta 33/Mantra 5)

The Yajur Veda contains some concepts related to Arithmetic and Geometric Progressions. The

concept of digits to the extent of several millions, can be clearly seen in all the Vedas.

Zero and Infinity: Zero as a member of the numerical system and zero as a symbol of

emptiness, both can be inferred in the philosophical context of the Vedic literature.

The concept of infinitely big (Ananta) and infinitesimal (Paramanu) entities are very much

available in the Vedas.

Authors: Satyanarayana Bhavanari * and

Satyanarayana K.@

*: Department of Mathematics,

@: Department of Sanskrit



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In the Upanishad entitled “Taittiriya Upanishad”, there presents a hymn ‘anoran i yam mahato

mahiyan’ which means that the Almighty is as big as a ‘Ananta’ (infinity) and as small as

Paramanu (atom). The same concepts were later adopted in mathematics as infinity and zero.

The Yajur Veda contains the following hymn.

“um purnamadah purnamida purnat purna mudacyate purnasya purnamadaya purnameva

avasisyate (yajurveda / adhyaya 4 / kandika 3)

This hymn means: Infinity comes out of infinity and when infinity is subtracted from infinity, the

result remains also infinity. Cleary this is similar to the modern concept of infinity. The

detailed meaning of hymn is: the knowledge in this universe/nature is infinite. The human

beings can manifest the knowledge from the existing knowledge of the universe. Even though

the knowledge manifested by human kind is infinite, the infinite knowledge remains

unobserved/unstudied in nature.

In this context, we may recollect the words of Swami Vivekananda “At any point of time the

knowledge manifested by Human beings is finite”.

We can understand this hymn/sloka with the present day mathematical concepts.

Let N be the set of all natural numbers. The set = 1, 2, 3, 4, …. is infinity.

Let us consider the set of all even natural numbers. This set is

2 = 2, 4, 6, 8, …., again infinity. Suppose we take all the even natural numbers from the set

of all natural numbers. Then the remaining set is

- 2 = 1, 3, 5, 7, …., and it is also infinity.

Note that it appears that 2 is half of , but the modern set theory explains that the number of

elements in N is same as the number of elements in 2. The same meaning was available in this

hymn.

Conclusion: The authors made an attempt to explain few concepts related to the number system

that available in the earlier records: Vedas.

Heat and Mass Transfer in a Viscous

Heat Generating Fluid Through a

Porous Medium in a Triangular

Duct

Abstract: In this paper we investigate a free

convective heat and mass transfer flow of an

incompressible viscous fluid through a heat

generated saturated porous medium enclosed in a

Triangular duct using Brinkman model. The lower

half portion of the duct is insulated while the upper

half is maintained at constant temperature and

concentration. The Galerkin finite element method with six nodded triangular elements is

adopted to obtain the iterative solutions for the stream function, temperature and

concentration by the coupled non linear equations. The flow being symmetric, the

expressions for velocity, temperature and concentration are obtained in typical quadrant

of triangular duct. Their behavior is discussed for variation in governing parameters Ra,

D-1

, N, Sc and . The local rate of heat transfer and mass transfer are also investigated

computationally.

Key words: Vertical duct, Porous medium, Brinkman model, Galerkin FEM.

___________________________________________________

Ideals and Modules in Rings

Abstract: In this paper we study one of the most

important aspects of rings namely ideals, discussion

of prime and maximal ideals and various properties

of these are investigated. We also describe the

characterization for Jacobson radical ring. Also we

shall formulate and study modules over rings and

quotient modules. We prove that if M be a

submodule of A-module M then there is an order

preserving one to one correspondence between the

submodules of M containing M and submodule of

M/M. We also explain the basic properties of Artinian and Noetherian rings.

S. Eswaraiah Setty*, S. Sivaiah**,

D.R.V. Prasada Rao@

,

*: Reader, Department of

Mathematics, Smt. G.S. College,

Jaggaiah pet, Krishna Dist, (A.P.)

**: Principal & Professor, Malla

Reddy P.G. College, Secunderabad,

Andhra Pradesh.

@: Professor, Department of

Mathematics, S.K. University,

Anantapur, Andhra Pradesh.

U.Suryakumar and

A. Satyanarayana


Akkineni Nageswara Rao College,

Gudivada-521301, A.P.


.

Normalization of fuzzy s-ideals of

Seminearrings

Abstract: In this Paper, the algebraic system

Seminearring has been considered, which is a

generalization of both a semiring and a nearring.

The normal fuzzy s-ideal of a seminearring with

absorbing zero has been introduced which is

analogue as that defined for rings, semirings. A

necessary and sufficient condition for a fuzzy s-

ideal of a seminearring to be normal is obtained

and some related results are proved.

_____________________________________________________________________

Basics of Graph Theory and Applications

Abstract: This paper concerns the importance of Graph Theory in teaching. . Concepts

and notations from discrete mathematics are useful in studying and describing objects and

problems in branches of computer science, such as computer algorithms, programming

languages, cryptography. Graph theory plays an important role in several areas of

computer science such as switching theory and logic design, artificial intelligence,

computer graphics, operating systems etc

In writing this paper I was guided by my

experience and interest in teaching Discrete

mathematics and and Graph theory the

motivation is I have been teaching this subject

to computer science and MCA students

Important findings: 1) In mathematics and

computer science, graph theory is the study of

graphs: mathematical structures used to model

pairwise relations between objects from a

certain collection. 2) Graphs especially trees, binary trees are used widely in the

representation of data structures 3)The term graph is used to denote the diagram of

a real valued function=f(x) but in Graph theory we use the term graph as an

object Method of derivation is procedure given in the text books with necessary

formulae and their application.

P.Venu Gopala Rao,


Andhra Loyola College

(Autonomous), Vijayawada- 520

008, Andhra Pradesh

Email:

[email protected]

V. Manjula,

B E D Dept., MIC College of

Technology, Kanchikacherla,

Email:

[email protected],

ph no: 9948233772

Graphs and their Applications

What are graphs? Graphs are diagrams that represent systems of connections or inter-

relations among two or more things by a number of distinctive dots, bars or lines. Also,

graphs denote a series of points discrete or continuous, as in forming a curve or surface

that represents a value of a given function. By the help of graphs, a certain data can be

effectively represented. Thereby, it is possible to easily grasp the true significance of a

set of figures. Further, by means of graphs, dry and uninteresting statistical facts can be

presented attractively. Furthermore, graphical representation of data is appealing and

evokes interest in people. Above all, graphs

enable people to easily understand difficult

theories in various subjects including

economics. Graph theory is applied in different

areas like linguistics, social sciences, computer

science, physics, chemistry and biology. Thus,

graphs with their different applications are very

useful to people in different walks of life. The

need of the hour is (1) to encourage the

study of graphs, in particular & mathematics, in general, (2) to enable people to

understand different related subjects & their applications in a better light and (3) to make

the best use of them in the present modern world.

____________________________________________________________________

On Fuzzy Ideals in BF-Algebras

Abstract: In this paper, we introduce the

concept of p-ideal, implicative ideal and

positive implicative ideals in BF-algebras and

obtain some results.

Mathematics subject classification: 03B52,

03F35, 03G25.

Keywords: BF-algebra, BF-subalgebras,

ideals.

Pokkuluri Suryaprakash,

Former Lecturer, S.C.I.M.

Government Degree College,

Tanuku, West Godavari District,

Andhra Pradesh.

B. Satyanarayana a,* , D. Rames

a,

M. V. Vijaya Kumar a , R. Durga

Prasad a, M. Arokiasamy

b

a Department of Applied

Mathematics, Acharya Nagarjuna

University Campus, Nuzvid-521201,

Andhra Pradesh, INDIA. Email:

[email protected]

b Department of Mathematics, Andhra

Loyola College, Vijayawada, Andhra

Pradesh, INDIA.

On Noetherian Regular -Near rings and their Extensions

Abstract: Recall that a commutative ring N is said

to be a Noetherian Regular -Near Ring if every

prime ideal of N is strongly prime. We say that a

commutative Noetherian Near Ring N is

Noetherian Regular Near Ring is a Noetherian

Regular -Near Ring if assasinator of every right

ideal (i.e., a right N-module) is strongly Prime

Ideal. Let us recall that a prime ideal P of a ring N

is said to be divided if it is comparable under set

inclusion to every ideal of N. A -Near Ring N is

called a “Regular -Near Ring” if a sub-direct

product of subdirectly irreducible -Near Ring

Ni is isomorphic to a -Near Ring N. Since,

each Ni is isomorphic image of N -Near Ring

and N has the IFP follows then N is a Regular -Near Ring. Let N be a semi prime

commutative Noetherian Q-Algebra, be an automorphim of N such that N is a (x) –

ring and is -Near – Ring then (i) if for any U ∈ S.spec(N) with (U) = U and (U)

subset of U implies o(U) ∈ S.spec(N), then N is a semi-Noetherian Regular -Near –

Ring implies N(x; , ) is a semi-Noetherian Regular -Near – Ring. (ii) if N is a semi-

Noetherian Regular Near – Ring then o(N) is a Noetherian Regular -Near – Ring.

__________________________________________________________________

On P-Regular -Near Rings and their

Extensions

Abstract: In this paper, we studied the concepts of

P-Regular Near-Rings, P-Regular -Near Rings and

their extensions and we obtain some fundamental

results of P-Regular -Near Rings.

Key Words : Near Ring, Regular Near-Ring, P-

Ring, P-Regular-Near Ring, P-Regular -Near Ring.

N V Nagendram 1, Sri Viveka

Institute of Technology, Vijayawada,


Y. Venkateswara Reddy, ANU

College of Engineering and

Technology, Acharya Nagarjuna

University, Andhra Pradesh, India

T. V. Pradeep Kumar, ANU College

of Engineering and Technology,



N V Nagendram 1, Sri Viveka

Institute of Technology, Vijayawada,


Y. Venkateswara Reddy, ANU

College of Engineering and

Technology, Acharya Nagarjuna

University, Andhra Pradesh, India

T. V. Pradeep Kumar, ANU College

of Engineering and Technology,



A Note of Goldie Near-rings

Abstract: If M is a −K module with d.c.c. on

−K subgroups and satisfying the property ,)(P

then it is shown that M has a submodule which is

uniform. Further, if M satisfies the Goldie

condition, then it is shown that there exists

minimal elements nxxx ,, 21 in M such that

⊕><⊕>< 21 xx ><⊕ nx is direct and M is

an essential extension of .21 ><⊕⊕><⊕>< nxxx

Keywords: Essential extension, Uniform, Direct sum of submodules, Minimal elements.

AMS Subject classification (2000): 16Y30.

_________________________________________________________

Certain Transformations

Formulae for the General Triple

Hyper Geometric Series F3(X, Y,

Z)

Abstract: In the present paper we have

established Transformation formulae for the Triple Hyper Geometric series of three

variables and several special cases have been discussed.

Keywords: Lauricella Function, Hypergeometric Series

Mathematics Subject Classification: Primary 33C05, Secondary 11F111

__________________________________________________________________

P. Narasimha Swamy and T. Srinivas

Department of Mathematics, Kakatiya

University, Warangal-506009, Andhra

Pradesh, India.

Email:

[email protected],

[email protected]

Pankaj Srivastava and R V G K

Mohan,

Department of Mathematics, Motila

Nehru National Institute of

Technology, Allahabad, India

On Different Types of Semi-

Complete Graphs

Abstract: In this presentation, semi-complete

graphs are classified into Weak semi-

complete,

Strong semi-complete, Super Strong semi-

complete graphs.

Main Results:

1. Any Wheel Wn (n 4) is Strong semi-complete but not Super Strong semi-complete.

2. For n 4 , Kn is Super Strong semi-complete.

3. For any n 5 , Kn –e,where ‘e’ is any edge of Kn, is Super Strong semi-complete.

4. Characterization for a semi-complete graph to be a) Weak semi-complete, b) Strong

semi-complete, c) Super Strong semi-complete, are obtained.

I. H. Naga Raja Rao, Senior Professor

&Director, G.V. P. College for P.G.

Courses, Rushikonda, Visakhapatnam

S. V. Siva Rama Raju, Assistant

Professor, S&H Department, Visakha

Institute of Technology & Science,

Sontyam , Visakhapatnam

Proc. of the National Seminar on Present Trends in Mathematics and its Applications 180

Computer Representation of Sets

A set can be represented in a computer using

characteristic function which is now defined.

Characteristic function: Let A be any subset of the universal set U. Then the

function is called the characteristic function of A.

Computer representation of sets in arrays: A sequence is a list of objects arranged

in other, such as first element, second element, third element and so on. Let U= x1,

x2, x3, . . . ., x n be the universal set and A be a subset of U. List the elements of

A in some order (the order we choose is of no importance). Then the characteristic

function fA assigns 1 to xi if xi ∈A and assigns 0 otherwise. Thus fA can be

represented by a sequence of 0’s and 1’s of length n. Universal set U is represented

in a computer as an array A of length n. Assignment of 0 or 1 to each location A[k]

of the array specifies a unique subset of U.

For example, let U=1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10 and S=1 ,3 , 5 ,7 ,9 .Then

Here U and S are represented by the arrays of length 10 as in Fig.1.

U=

S=

Fig.1. Computer representation of sets U and S.

_______________________________________________

Fuzzy Submodules and Fuzzy

Dimension in Modules over

Associative Rings

Abstract: We discuss the existing concepts

and results related to fuzzy Submodules and fuzzy dimension in Modules over

Associative Rings.

1 1 1 1 1 1 1 1 1 1

1 0 1 0 1 0 1 0 1 0

Student Presentation

V. Suvarchala, M. Sc.

First Semester, Department

of Mathematics, ANU

Author: Kavitha Nellore,


Acharya Nagarjuna University

Divisible Fuzzy

Subgroups

Abstract

This paper makes an attempt to study Divisible Fuzzy Subgroups. It is clear about the concept of divisible

groups in the crisp abelian group theory. In group theory, Divisibility is the most important and basic

concept which led to the development of Purity and Pure-injectivity. Therefore, it is necessary to study

this concept in case of Fuzzy Algebra also. Basic concepts and some of the interesting properties of

divisible fuzzy subgroups have been studied. Throughout this paper, all groups are to be considered as

additive abelian groups, and X denotes a non-empty set. All the notations and basics followed from [1].

1. Basic concepts

Definition 1.1: A function from a set X into the closed interval [0,1] is known as Fuzzy subset of

X and the set [0,1]X is referred to as the Fuzzy power set of X.

Definition 1.2: Suppose Y X⊆ and [0,1].a ∈ Then the set Ya ∈ [0,1]X is defined as

( )0 \

Y

a for x Ya x

for x X Y

∈=

∈ .

In particular, if Y is singleton, say y,

then ya is known as a Fuzzy point or Fuzzy-singleton and it is denoted by .ay

Definition 1.3: Suppose [0,1]Xµ ∈ and [0,1].a ∈ Then, the set ( ) | ,a x x X x aµ µ= ∈ ≥ is

called the a - cut or a − level set of .µ

Definition 1.4: Suppose [0,1]Xµ ∈ . Then, the set ( ) | , 0x x X xµ∈ > is called the support of µ

and is denoted by *.µ

Definition 1.5: Let G be an abelian group. Then, a Fuzzy subset µ of G is called a Fuzzy

subgroup of G if ( ) ( ) ( ) , ,x y x y x y Gµ µ µ+ ≥ ∧ ∀ ∈ and ( ) ( ) .x x x Gµ µ− ≥ ∀ ∈

The set of all

Fuzzy subgroups of G are denoted by [0,1] .G

Authors: Ramana Murty N.V., and Mariadas M.,

Department of Mathematics, Andhra Loyola

College, Vijayawada-520008. Email:

[email protected]

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+( **,(*-+.

Definition 1.6: A Fuzzy subgroup µ is called divisible if for all ax µ⊆ with [ ]0 0,1 ,a< ∈ and

for all ,n ∈ there exists ay such that ay µ⊆ and ( ) .a an y x=

2. Main Results

Lemma 2.1: A Fuzzy subgroup µ of a group G is divisible if and only if aµ is divisible for all

( ) | 0 0 .a a a µ∈ < ≤

Proof: Suppose µ is divisible. Therefore, for all ax µ⊆ with [ ]0 0,1 ,a< ∈ and for all ,n ∈

there exists ay such that ay µ⊆ and ( ) .a an y x= Since ( )0 0 ,a µ< ≤ the equation ny x= is

solvable in aµ for all n ∈ . Hence aµ is divisible. Conversely suppose that aµ is divisible. So,

the equation ny x= is solvable in aµ for all n ∈ . This implies the equation ( )a an y x= is

solvable in µ for all ,n ∈ since ( )0 0 .a µ< ≤ Hence µ is divisible.

Lemma 2.2: If a Fuzzy subgroup µ of a group G is divisible, then its support *µ is divisible.

Proof: Suppose µ is divisible. Therefore, for all ax µ⊆ with [ ]0 0,1 ,a< ∈ and for all ,n ∈

there exists ay such that ay µ⊆ and ( ) .a an y x= Therefore *x µ∈ and so there exists *y µ∈

such that .ny x= Hence *µ is divisible.

Lemma 2.3: If µ is a Fuzzy subgroup of a group G, its support *µ is divisible and µ is constant

on *µ \0, then µ is divisible.

Proof: Suppose *µ is divisible and µ is constant on *µ \0. Let ax µ⊆ with 0a > and .n∈

Since *µ is divisible, for *x µ∈ there exists *y µ∈ such that .x ny= So, if 0,y = then 0.x =

Therefore, the result is true. Let 0.y ≠ Since µ is constant on *µ \0 and 0a > , we have

( ) ( ) ,y x aµ µ= ≥ by definition of *µ . Thus, ay µ⊆ and ( ).a ax n y= Hence µ is divisible.

Lemma 2.4: If µ is any Fuzzy subgroup of a group G, then for all ,x y G∈ and n∈ , ny x=

implies that ( ) ( )x yµ µ= for all divisible Fuzzy subgroups µ of G if and only if G is torsion-

free.

Proof: Suppose the given condition on µ holds. Now, we show that G is torsion-free. To show

this, Let T be the Torsion part of G and x be any element in T. Therefore, there exists n∈

such that 0.nx = Since 01 is divisible,

( ) ( )0 01 1 0 1.x = = Hence 0.x = Thus, 0T =

implies that G is torsion-free. Conversely, Let G be torsion-free and µ any divisible fuzzy

subgroup of G. Suppose that .nx y= Let ( ).a xµ= Then ( ) .a an y x= Now, there exists y′ in G

such that a any x′ = and .y µ′ ⊆ Thus .ny x′ = Since G is torsion-free, y y′= and so .ay µ⊆

Thus ( ) ( ) ( ) .a x ny y aµ µ µ= = ≥ ≥ Hence ( ) ( ).x yµ µ=

Similarly, in case of torsion divisible groups, we have

Lemma 2.5: For all fuzzy divisible subgroups µ of ( )Z p∞ (a Quasi-cyclic group), for all

( ) , \ 0 ,x y Z p∞∈ for all ,n∈ ny x= implies ( ) ( ).x yµ µ=

Theorem 2.6: A Fuzzy subgroup µ is divisible if and only if it is constant on the additive group

of rational numbers .

Proof: Suppose µ is divisible. Let .x ∈ Then, there exists n∈ such that .nx m= ∈ By the

Lemma 2.4, ( ) ( ).x mµ µ= Now, 1m m⋅ = and so ( ) ( )1mµ µ= by Lemma 2.4. Thus

( ) ( )1 .xµ µ= Hence µ is constant on . Conversely, suppose that µ is constant on . Let

ax µ⊆ and .n∈ Now, there exists y in such that ny x= and so ( ) .a an y x= since µ is

constant on , ( ) ( ).y xµ µ= Thus .ay µ⊆ Hence µ is divisible.

Similarly, in case of a torsion divisible subgroup ( )Z p∞ we have

Theorem 2.7: A Fuzzy subgroup µ is divisible if and only if it is constant on ( )Z p∞ \0,

where ( )Z p∞ is a Quasi-cyclic group.

Example 2.8: Let ( ) ( ).G Z p Z p∞ ∞= ⊕ Define a fuzzy subset µ of G by

( )

( )

( ) ( )

1 0,0

1 0 \ 0,02

0

if x

x if x Z p

otherwise

µ ∞

=

= ∈ ⊕

. Then, we see that µ is a fuzzy subgroup of G. So,

( )* 0 Z pµ ∞= ⊕ and it is a fuzzy subgroup of G. Hence *µ is divisible and µ is constant on

( ) * \ 0, 0 .µ Therefore, by Lemma 2.3 µ is a divisible fuzzy subgroup of G.

References:

1. Sidky, F. I. and Mishref, M. A. A., Divisible and Pure and fuzzy subgroups, Fuzzy Sets and Systems, Vol.

34(1990), 377-382.

2. Sidky, F. I. and Mishref, M. A. A., Fuzzy cosets and cyclic and abelian fuzzy subgroups, Fuzzy Sets and systems,

Vol. 43 (1991), 243-250.

3. Malik, D. S. and Mordeson, J. N., Fuzzy subgroups of abelian groups, Chinese J. of Math. Taipei, Vol.19(1992),

129-145.

4. Mordeson, J. N., Invariants of fuzzy subgroups, Fuzzy Sets and Systems, Vol. 63(1994), 81-85.


With best Complements from Dr Bhavanari Research Scholars Association 184

Dr Kuncham Syam Prasad: He awarded Gold Medal for his first rank in M. Sc., Mathematics in

1994. He is a recipient of CSIR-Senior Research Fellowship. He got awarded M. Phil., (Graph

Theory) in 1998 and Ph.D., (Algebra - Nearrings) in 2000 under the guidance of Dr Bhavanari

Satyanarayana (AP SCIENTIST Awardee). He published Nineteen research papers in reputed

journals and presented research papers in thirteen National Conferences and six International

Conferences in which five of them were outside India: U.S.A (1999), Germany (2003), Taiwan (2005),

Ukraine (2006), Austria (2007), Bankok (2008), Indonesia (2009). He also visited the Hungarian Academy

of Sciences, Hungary for joint research work with Dr Bhavanari Satyanarayana (2003) and the National

University of Singapore (2005) for Scientific Discussions. He authored nine books (UG/PG level). He is also

a recipient of Best Research Paper Prize for the year 2000 by the Indian Mathematical Society for his research

in Algebra. He received INSA Visiting Fellowship Award (2004) for the collaborative Research Work.

Presently working as Associate Professor of Mathematics, Manipal University, Karnataka, India. E-mail:

[email protected], [email protected]

Dr. Tumurukota Venkata Pradeep Kumar: He got awarded M.Phil., (ΓΓΓΓ-ring theory) and Ph.

D., (Near-ring Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST

Awardee). He published five research papers in Indian and Abroad International Journals. He

attended three National Conferences and one International Conference. At present he is working as

Assistant Professor in ANU College of Engineering & Technology.

Dr. Dasari Nagaraju: He completed his Ph. D., (Ring Theory) He is a Project Associate in UGC-Major

Research Project (2004-2007) under the Principal Investigatorship of Dr Bhavanari Satyanarayana (AP

SCIENTIST Awardee). He published eight research papers. He Worked in Rajiv Gandhi University

(AP), Periyar Maniammai University (Tanjavur). Presently working in Hindusthan University, Chennai.

Dr. Kedukodi Babushri Srinivas: He is a Associate Professor in Mathematics, Manipal University, Karnataka.

His educational qualifications are DOEACC ‘O’ LEVEL from DOEACC Society, Department of

Electronics, Govt. India, M. Sc., and P.G.D.C.A. from Goa University. He qualified in the Joint

CSIR-UGC JRF(JRF-NET), Maharashra State Eligibility Test (SET) for Lectuership (accredited by

UGC) and GATE in Mathematics. He got Ph.D., (Fuzzy and Graph Theoritic aspects of Near-

rings, 2009) under the guidance of Dr Kuncham Syam Prasad and Dr Bhavanari Satyanarayana

. He attended a number of workshops/Seminars in Mathematics and pubished

four research papers in international Journals like: Soft Computing, Communications in Algebra. He presented

papers/delivered Lectures in International Conferences held at Ukraine (2006), Austria (2007), Bankok (2008),

Indonesia (2009). He was awarded first prize for the poster presentation during the MU Scientific meet for the

Ph.D., students. E-mail: [email protected]

Mr. M. B. V. Lokeswara Rao: He completed his M. Sc., (Mathematics) from ANU with third rank. He got

awarded M. Phil., (Matrix Near-rings) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST

Awardee) with A grade. He is an elected General Secretary of “Association for Improvement of

Maths Education (AIMEd., Vijayawada)”. He published one research paper in Matrix Near-rings.

Mr. Sk. Mohiddin Shaw: He completed his M. Phil., (Module Theory) under the guidance of Dr

Bhavanari Satyanarayana (AP SCIENTIST Awardee). He visited Institute of Mathematical

Sciences (Chennai), IIT (Chennai), ISI (Calcutta), IIT (Guwahati) and Burdwan University (West Bengal) for his

research purpose. He attended eight Conferences/Seminars/ Workshops. He worked as a faculty in

the ANU P.G Centre at Ongole. He published four research papers in Ring Theory.

Mr. J. L. Ramprasad: Awarded with Kavuru Gold Medal for College first in B. Sc., Course and

with JCC Gold Medal for Town first. Qualified in GATE-2001 Examination with Percentile score

of 85.73. Awarded with M. Phil., (Module Theory) in May 2005 under the guidance of Sri. Dr

Bhavanari Satyanarayana . He authored two books at PG level. He published a research

paper in USA. Presently working as a Lecturer in P.G. Department of P. B. Siddhartha College, Vijayawada.

E-mail: [email protected]

Mr. K. S. Balamurugan: He got First Rank in B. Sc., and Second Rank in M. Sc., course. He awarded with

M. Phil., (Ring Theory) under the guidance of Dr Bhavanari Satyanarayana in 2006.

He is working as Sr. Lecturer in RVR & JC College of Engineering. He published one research paper in Ring

Theory. Mrs. T. Madhavi Latha: Her educational are B. Sc., B. Ed., M. Sc., Ed., M. Phil., PGDCA and

IELTS: 7.5. She was a NCERT scholarship holder during 1992-94. She got Visista Acarya Puraskar

Award in 1997 by Amalapuram Educational Society. She was the author of 3 books. She attended

various National and International seminars both on Education and Mathematics. She worked as a

resource person for various academic programmes. Presently she is working as a PGT in APSWR JC.


With best Complements from Dr Bhavanari Research Scholars Association 185

Mrs. Sk. Shakeera: She got M. Phil., degree (2007) in ΓΓΓΓ-ring theory under the guidance of Dr

Bhavanari Satyanarayana

Mr. D. Srinivasulu: He got his M. Phil., degree (Graph Theory) under the guidance of Dr

Bhavanari Satyanarayana .

Brief Biodata of Prof. Dr SATYANARAYANA BHAVANARI, ANU

• Got 2nd

Rank securing 75% of marks in M.Sc., Maths (1977-79), ANU.

• Got 1st Rank in Certificate Course in Statistics, ANU.

• Undergone Certificate Courses in Electronic Computers (i). Indian Statistical

Institute, Calcutta (1986); and (ii) Annamalai University.

• Awarded CSIR-JRF (1980-82), CSIR-SRF (1982-85), UGC-Research Associateship (1985), CSIR-

POOL OFFICER (1988), INSA Visiting Fellowship Award 2005, and ANU – Best Research Paper

Award-2006, AP State Scientist-2009 Award (by DST New Delhi & APCOST Hyderabad), • Fellow, AP Akademi of Sciences.

• Awarded Five Ph.D., degrees and Ten M.Phil., degrees under his supervision.

• One Research Student (Dr. Kuncham Syam Prasad, working in Manipal Academy of Higher Education,

Deemed University) got the National Award: IMS Award - 2000) for best research paper in Algebra.

• Life member of Eight Mathematics Associations.

• Elected President (2005-2007, 2007-2009) of the Association for Improvement of Maths Education

(AIMEd.,), Vijayawada.

• Director of the National Seminar on Algebra and its Applications, organized by the Department of

Maths, ANU, Jan 05-06, 2006.

• Published 27 General Articles in periodicals.

• Authored / Edited 33 books (for B.Com. / M.A. (Eco.) / B.C.A / M.Sc.(Maths) (including a book on

Discrete Mathematics & GT, published by Prentice Hall of India, New Delhi)), Three books published by

VDM VERLAG DR MULLER, GERMANY.

• Honorary Editor for the two Mathematical Periodicals (in Telugu Language): “Ganitha Chandrica” &

“Ganitha Vahini” Published from Andhra Pradesh.

• Member Secretary and Managing Editor of “Acharya Nagarjuna International Journal of

Mathematics & Information Technology”, Acharya Nagarjuna University.

• Got Paul Erdos No. 3. Collaborative Distance with Einstein = 5

• Attended 13 International Conferences (INCLUDING ICM-2010) and 24 National Conferences.

• Principal Investigator of 3 Major Research Projects (Sponsored by U G C, New Delhi).

• Published 57 research papers (in Algebra / Fuzzy Algebra / Graph Theory) in National and

International Journals.

• Introduced the algebraic system “Gamma near-ring” in 1984.

• Visiting Fellow at Tata Institute of Fundamental Research, Bombay, May 1989.

• Visiting Professor at Walter Sisulu University (WSU), Umtata, South Africa, March 26 – April 10, 2007.

• Visited Austria (1988), Hongkong (1990), South Africa (1997), Germany (2003) Hungary (2003),

Taiwan (2005), Singapore (2005), Hungary (2005), Ukraine (2006), and South Africa (2007) on official

works (to deliver lectures / Collaborative research work).

• Selected Scientist (By Hungarian Academy of Sciences, Budapest; and University Grants Commission,

New Delhi, 2003) to work with Prof. Richard Wiegandt at A.Renyi Institute of Mathematics (Hungarian

Academy of Sciences) during June 05- Sept. 05, 2003. A research paper on Radical theory of Near-rings

was published with the co-authorship of Prof. Wiegandt (in the Book: Nearrings and Near-fields,

Springer, Netherlands, 2005, pp.293-299).

• Selected Sr. Scientist (By Hungarian Academy of Sciences, Budapest; and Indian National Science

Academy, New Delhi), Aug. 16 – Sept. 05, 2005.

Contact: Ph: 0863-2232138 (R), 0863-2346456 (Office); Cell: 98480 59722.

E-mail: [email protected], [email protected]

Prof. Dr Bhavanari Satyanarayana has 27 yrs Teaching experience in

Acharya Nagarjuna Univ. Authored 33 books (including a book by

Prentice Hall of India, New Delhi, and three books by VDM Verlag

Dr Muller, Germany). He has Published 57 Research papers (Algebra/

Fuzzy Algebra/Graph Theory) in International Journals. Member of

several Editorial Boards, Mathematical Journals. AP SCIENTIST–2009

Awardee. Fellow, AP Akademi of Sciences. Scientist UGC-HAS (Hungarian

Academy of Science), 2003. Sr Scientist INSA–HAS-2005. Principal

Investigator of 3 MAJOR Research Projects (UGC). Introduced the

concept “Gamma near-ring”. He has visited Austria (1988), Hongkong

(1990), South Africa (1997), Germany (2003) Hungary (2003), Taiwan

(2005), Singapore (2005), Hungary (2005), Ukraine (2006), and South

Africa (2007) on official works (to deliver lectures / Collaborative

research work). He has guided for five Ph.D.,s and ten M.Phil.,s.

Bio Data of Editors

Dr Kuncham Syam Prasad has 10 years of Teaching

experience in Manipal University, Manipal, (Karnataka). He

has published 20 Research Papers in International Journals

besides his contribution to 10 books (one with Prentice Hall

India Ltd.). He was a recipient of Best Paper Prize by the

Indian Mathematical Society (year 2000). He has guided one

Ph.D., under Manipal University. He has visited USA,

Germany, Hungary, Taiwan, Ukraine, Austria, and Indonesia

etc., for various academic conferences and interactions.

Dr Sreeramula Eswaraiah Setty, has obtained Ph.D., from

Srikrishnadevaraya University, and presently working as a Reader in

the Department of Mathematics, S. G. S. College, Jaggaiahpet. He

has more than 20 years of teaching/research experience in Under

Graduate Teaching. He was a member/convener of various academic

committees. He has attended about ten academic conferences/

workshop/refresher courses. His keen interest includes

extracurricular activities and social activities.