proceedings of the national seminar on present trends in mathematics and its applications
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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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
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 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
drawn”. The paradoxical aspects are due to the fact that there are several different ways of
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 .
This can be written as a linear operator (and is known as the indefinite integral
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
Mathematical Society.
Proceedings of the National Seminar on Present Trends in Mathematics &
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
c. of the National Seminar on Present Trends in Mathematics and its Applications
1
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
. 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
drawn”. The paradoxical aspects are due to the fact that there are several different ways of
integer powers, leading to inequivalent results.
This can be written as a linear operator (and is known as the indefinite integral
is sufficiently continuous and
Lecture by
f. P. V. Arunachalam,
Chancellor,
Dravidian University,
Kuppam (Andhra Pradesh),
President, Indian
Mathematical Society.
its Applications,
12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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:
,
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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.
We can use mathematical induction to prove that this formula is true for all positive
=1 case: we see it gives:
,
and so the formula holds for n=1.
c. of the National Seminar on Present Trends in Mathematics and its Applications
3
last, and then performing the u1 and
We can use mathematical induction to prove that this formula is true for all positive
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
, by definition, we see that if we take the
derivative
respect to x,
.
Now, we can take the derivative of our formula with respect to x, using the variable
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
can be performed for n>0 even if n is not an integer. Recall, however, that for positive
, where is the gamma function. Thus, if we replace the
factorial in our formula with the corresponding gamma function, we obtain 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
c. of the National Seminar on Present Trends in Mathematics and its Applications
4
, by definition, we see that if we take the
Now, we can take the derivative of our formula with respect to x, using the variable
. Thus our formula holds for
>0 by induction.
can be performed for n>0 even if n is not an integer. Recall, however, that for positive
. Thus, if we replace the
factorial in our formula with the corresponding gamma function, we obtain 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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
semigroups are prevalent in mathematics, and have an interesting theory. Notice here
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.
As far as the existence of such a theory is concerned, the foundations of the subject
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
semigroups are prevalent in mathematics, and have an interesting theory. Notice here
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
As far as the existence of such a theory is concerned, the foundations of the subject
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
,
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.
Simply using the Gamma function to remove the discrete nature of the factorial
, or equivalently
gives us a natural candidate for fractional applications of the integral operator.
defined operator.
operator satisfies
c. of the National Seminar on Present Trends in Mathematics and its Applications
6
can be extended to all real values of n.
, which extends
, form the definite integral from
,
Simply using the Gamma function to remove the discrete nature of the factorial
)
f the integral operator.
,
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
7
differintegral operators.
is significantly more
in general.
(blue curve) together with the
, leads us to
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
.
c. of the National Seminar on Present Trends in Mathematics and its Applications
8
fferential operator need not be constrained only to real
th derivative yields the
yields integrals.
. Noting that
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
mass equation is needed when the control volume is not large enough compared to the
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
mass equation is needed when the control volume is not large enough compared to the
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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
This equation has been shown useful for modeling contaminant flow in heterogenous
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
Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John
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)
Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974).
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
c. of the National Seminar on Present Trends in Mathematics and its Applications
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:
This equation has been shown useful for modeling contaminant flow in heterogenous
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
Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John
The Fractional Calculus; Theory and Applications of Differentiation and Integration
to Arbitrary Order (Mathematics in Science and Engineering, V), by Keith B.
Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974).
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;
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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).
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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: lrkwmr@gmail.com
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and 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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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−=
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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−=
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
32
Fig. 3 θ with α at y = 2C/3 Fig. 4 θ with N at y = 2C/3
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
33
Fig. 5 θ with R at y = C/3 Fig. 6 θ with R at y = C/3
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
34
Fig. 7 θ with α at y = C/3 Fig. 8 θ with N at y = C/3
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
35
Fig. 9 θ with R at x = 1/3 Fig. 10 θ with EC at x = 1/3
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
36
Fig. 11 θ with α at x = 1/3 Fig. 12 θ with N at x = 1/3
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
37
Fig. 13 θ with R at x = 2/3 level Fig. 14. θ with EC at x = 2/3 level
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
38
Fig. 15 θ with α at x = 2/3 level Fig. 16 θ with N at x = 2/3 level
I II III IV I II III IV
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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).
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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,
Department of Mathematics,
Acharya Nagarjuna University,
Nagarjuna Nagar-522510,
Andhra Pradesh, India
Email:
bhavanari2002@yahoo.co.in
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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;
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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).
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
+ − + +
−
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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,
Department of Mathematics,
Manipal University, Manipal-
576 104, India,
Email:
kunchamsyamprasad@gmail.com
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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,
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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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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,
Nagarjuna University, 2000.
12. Zadeh L. A. “Fuzzy sets” Inform. & Control 8(1965) 338-353.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
62
Semisimple Hopf Algebras
and their Orbits
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Invited Lecture by
Prof. M. Sumanth Datt,
Department of Mathematics
and Statistics, University of
Hyderabad, Hyderabad-500
046, Andhra Pradesh
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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Proc. of the National Seminar on Present Trends in Mathematics and its Applications
66
Rough Sets
Invited Lecture by
Dr Kedukodi Babushri Srinivas,
Department of Mathematics,
Manipal University, Manipal-
576 104, India.
Email:
babushrisrinivas@yahoo.co.in
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Invited Lecture by
Dr Dasari Nagaraju,
Department of Mathematics,
HITS, Hindustan University,
Chennai.
Email:
dasari.nagaraju@gmail.com
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
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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.
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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.
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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.
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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.
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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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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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:
ihnrao@yahoo.com
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
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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,
Email: victorsugnanam@yahoo.co.in
Proceedings of the National Seminar on Present Trends in Mathematics & its Applications,
SGS College, Jaggaiahpet, A.P., India, November 11-12, 2010. (Editors: Dr Eswaraiah Setty
Sreeramula, Dr Satyanarayana Bhavanari and Dr Syam Prasad Kuncham)
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
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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 .
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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.
Email: bhavanari2002@yahoo.co.in
!!"!##$!$%& '(&) *
+( **,(*-+.
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 trans- formations 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 ashamidg@rediffmail.com ,
tanweerjalal@rediffmail.com, neyaznit@yahoo.co.in
!!"!##$!$ %& '(&) *
+( **,(*-+.
: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 se- quences.
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: pradeeptv5@gmail.com,
bhavanari2002@yahoo.co.in#
kunchamsyamprasad@gmail.com
!!"!##$!$ %& '(&) *
+( **,(*-+.
!"
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
Nagarjuna University, 1984.
[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,
Emailsasganie0@rediffmail.com,
neyaznit@yahoo.co.in,
sameergupkari@rediffmail.com
!"
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 trans- formations 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 gene- ralized 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: kunchamsyamprasad@gmail.com
bhavanari2002@yahoo.co.in
!!"!##$!$ %& '(&) *
+( **,(*-+.
!!
!!
!!
!!
!!
!!
!!
!!
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).
Email: simambi@gmail.com
!!"!##$!$ %& '(&) *
+( **,(*-+.
!!"!##$!$ %& '(&) *
+( **,(*-+.
!"
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: bhavanari2002@yahoo.co.in,
mohiddin_shaw26@yahoo.co.in
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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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(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
!!"!##$!$ %& '(&) *
+( **,(*-+.
!
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.
Example 4.4 Let X = a, b, c, Y = 1, 2, 3 and
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)>.
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) = 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)>.
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/.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.
Example 4.8 Let X = a, b, c, Y = 1, 2, 3 and
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)>.
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) = 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.
Email: bhavanari2002@yahoo.co.in
**: 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.
As F Calculated value is less than the F table value at 5% level of significance with (17,34)
degrees of freedom, so we conclude that there is no significant difference among treatments.
Yellow mosaic virus:
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.
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.
!!"!##$!$ %& '(&) *
+( **,(*-+.
!
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:swajo@vsnl.net
!
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
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1 2
3 %3
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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
!"
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
Acharya Nagarjuna University,
Andhra Pradesh, India
!!"!##$!$ %& '(&) *
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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
Department of Mathematics,
Akkineni Nageswara Rao College,
Gudivada-521301, A.P.
Email: asnmat1969@yahoo.in
.
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,
Department of Mathematics,
Andhra Loyola College
(Autonomous), Vijayawada- 520
008, Andhra Pradesh
Email:
venugopalparuchuri@gmail.com
V. Manjula,
B E D Dept., MIC College of
Technology, Kanchikacherla,
Email:
manju_adiraju@yahoo.co.in,
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:
drbsn@yahoo.co.in
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,
Email: nvn220463@yahoo.co.in
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,
Acharya Nagarjuna University,
Andhra Pradesh, India
N V Nagendram 1, Sri Viveka
Institute of Technology, Vijayawada,
Email: nvn220463@yahoo.co.in
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,
Acharya Nagarjuna University,
Andhra Pradesh, India
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:
swamy.pasham@rediffmail.com,
thotasrinivas.srinivas@gmail.com
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,
Department of Mathematics,
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:
raman93in@gmail.com
!!"!##$!$%& '(&) *
+( **,(*-+.
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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:
kunchamsyamprasad@gmail.com, syamprasad.k@manipal.edu
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: kbsrinivas77@yahoo.com
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: ram_jupudi@rediffmail.com
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.
Proc. of the National Seminar on Present Trends in Mathematics and its Applications
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: bhavanari2002@yahoo.co.in, bhavanari2005@rediffmail.com
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.
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