153406929...

Editors:

Prof. Dr Satyanarayana BhavanariM.Tech., B.Ed., M.Sc., Ph.D.,

A.P. Scientist Awardee 2009, Fellow - A.P. Akademi of Sciences

Glory of India Award & International Achievers Award (Thailand 2011)

Top 100 Professionals - 2011 (IBC, England)

Mr. V.V.N. Suresh Kumar M.Sc.,

Mr. Mohiddin Shaw Shaik M.Sc., M.Phil.,

Proceedings of theOne Day National

Seminar on Algebra

(200th Birth Anniversary

Celebrations of Evariste GALOIS)

(In collaboration with the Department of Mathematics

Includes

Invited Lectures

Research Paper Presentations

Abstracts

EVARISTE GALOIS(25 Oct 1811 TO 1831)

SRINIVASA RAMANUJAN

(22 Dec. 1887 to 1920)

K.B.N. College VIJAYAWADA, AP, India

25th October, 2011 ALBERT EINSTEIN

(1879-1955)

Proceedings of the National Seminar on ALGEBRA (200th Birth Anniversary Celebrations of Evariste GALOIS), KBN College, Vijayawada, A.P., (October 25, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana, Mr. VVN Suresh Kumar and Mr Mohiddin Shaw Shaik.)

CONTENTS

S.No Speaker/ presenters Title of the Talk Pages01 Prof. Dr P.V. Arunachalam

Founder Vice-Chancellor,Dravidian University, A.P.

Romantic Mathematician and his Mathematics(Inaugural Address)

01-04

02 Prof. Dr Bhavanari Satyanarayana,

Acharya Nagarjuna University

Finite Dimension of VectorSpaces and Modules(Key Note Address)

05–15

03 Mr. Mohiddin Shaw ShaikAcharya Nagarjuna University

Prime Graph of ℤn and

Zero Square Ring S(ℤ3)

16-20

04 Dr. T.V. Pradeep KumarAcharya Nagarjuna UniversityCollege of Engineering, ANU

On Gamma Near-Rings 21-24

05 Dr. Dasari NagarajuAssistant Professor

Manipal University, Jaipur Rajastan.

Zero Square Rings and Zero Square Ideals

25-28

06 Dr. G. GanesanAdikavi Nannaya University,

Rough Fuzzy Groupinduced by epimorphisms

29-32

07 Dr. Atul GaurUniversity of Delhi

Multiplication Modules 33-36

08 Dr. Syam PrasadManipal University

An Introduction to New Algebraic Structure Gamma

Near Rings

37-38

09 Mr. Venkata Subba RaoLecturer in Mathematics

Amalapuram

Fuzzy Ideals of Gamma Near Rings

39-41

10 Dr. V. AmbethkarUniversity of Delhi

Some Numerical Methods for solving Navier-Stokes

Equations

42-46

11 Kumari Bhavanari SatyaSriZhejiang University, Hangzhou

Republic Chaina.

Golden Ratio and human body

47-49

12 Dr.K.Satyanarayana & Prof Dr Bhavanari Satyanarayana

Acharya Nagarjuna University

A Prospective Theme of Mathematics

50-53

13 Bhavanari Satyanarayana& D Srinivasulu

A Note On 2-Quasi-total Graphs

53-53

14 M. Babu PrasadVijauawada

Some results on Fuzzy Ideals of M Gamma Modules

54-54

15 Mrs. S. Latha & J. PraveenaSt.Mary’s Eng., College

Efficient using graphs in computer sciences

54-54

16 Mr. Y. Sankar Rao & Mrs. S. LathaSt.Mary’s Eng., College

Efficient using graphs in computer sciences

54-54


CONTENTS 17 G.C. Rao & N. Rafi

AndhraUiversityTopological Characterization

of Normal Almost Distributive Lattice

55-55

18 P.Venu Gopala Rao Loyola College, Vijauawada

On Seminearrings and their Ideals

56-56

19 G.C. Rao & Naveen KumarAndhraUiversity

Dual PS Almost Distributive Lattice

57-57

20 Dr. Ramana MurthyLoyola College, Vijauawada

Divisible Semigroups 58-58

21 N.V. NagendramL.B.R. College of Enge.,

A Note on Inversive Localization in Noetherian Regular Delta Near Rings

59-59

22 S. Venu Madava SarmaK.L.University

A note on Hamiltonian Path and Hamiltonian Circuit

59-59

23 Mr. Nanda KumarAcharya Nagarjuna University

A Note on Fuzzy Ideals 59-59

24 A. Sudhakaraiah, E. Gnana Deepika, A. Sreenivasulu and

V. Rama Latha

Minimal Dominating Set of an Interval Graphs

59-59

25 Bhavanari MallikarjunKolla Ramasubramanyam and

Wudayagiri harsha

Nanotechnology: An Emerging Era

60-61


1A ROMANTIC MATHEMATICIAN AND HIS MATHEMATICS

Prof. Dr P.V. ARUNACHALAMFounder Vice-Chancellor,

Dravidian University, Kuppam, AP.

A Paris newspaper, dated may 31st 1811, carried the following news: “A deplorable duel yesterday robbed science of a young man who inspired the brightest hopes, but whose prodigious fame is only of a political nature. Young Evariste Galois fought a duel with an old friend, a very young man like him – a member of the Societe des Amis people-{Society of the Friends of People}. At point blank range, each of them was given a pistol and fired. Only one of the pistols was loaded.”Historians of mathematics still have vague ideas about why he fought the duel, some seeing it as a matter of honour about a young lady, possibly as set up by agents of the police to get rid of a dangerous revolutionary, and some seeing it as a set up by Galois himself to go out in a blaze of glory. Although his fame as a revolutionary was transient, his mathematics was timeless.Galois Theory and Galois Groups are common currency in mathematics to day. As an young man of twenty he joined the ranks of the galaxy of immortals in mathematics. He was Evariste Galois.

Evariste Galois, pronounced as ay-vah ReesTGalwah, was born on 25thOctober, 1811 in a Frenchvillage called Bourg-la-Reine, near Paris. Nicolas-Gabriel Galois was his father and Adelaide Marie Demante , his mother. Evariste’s mother was his sole teacher and taught him all subjects well before he was sent to the fourth grade in a famous boarding school in Paris.

Inspite of lackluster career at the school, as he failed twice in the entrance examination to the prestigious educational institution, the Ecole Polytechnique, the young Galois devoured the advance mathematical texts of Legendre and Lagrange. When he was just 17 years old, he began making fundamental discoveries in the theory of polynomial equations.

By looking at the series of mishaps in the short, but eventful life of Galois, it is amazing that the mathematical document that he wrote under the shadow of death the following morning, on the fateful night of May 29th 1832, did not go to oblivion, but survived for the benefit of posterity.

Galois was a radical republican during the turbulent days, after the fall of Napoleon in France. He was something of a romantic figure in the history of mathematics. For his rebellious and revolutionary activities he was arrested and served jail terms. Soon after his release from the prison he was to stay in a place to avoid the risk of contacting the cholera epidemic. At that point of time he fell desperately in love with a girl which led to the tragic end of his life.

While in teens Galois was attracted towards solving the quantic equation in Algebra. To understand and appreciate the achievement of Galois, the romantic and tragic mathematician, we need to review the state of the theory of equations in Algebra, at his time. We know that linear


2and quadratic equations were solved long ago, may be four thousand years before. We find ample evidence from the history of mathematics of all the ancient civilizations, to confirm this fact. Babylonians and Egyptians started elementary work in the theory of equations when they dealt with simple linear and quadratic equations involving only positive roots. Then entered Euclid and Archimedes with their geometric approaches to the problems. Thereafter Diophantus and Alkhwarizmi contributed in a big way and enriched the field. Then followed the works of Indian mathematics, anmathematicians: Bakhali Manuscript, Aryabhata- I, Brahmagupta, Varahamihira and Sridhara and some mathematicians from China. At the dawn of second millennium Omar Khayyam, Fibonacci and a few others from Europe and Middle East, worked in this area. Li Zhi (also known as Li Ye) of China did solve higher degree equations by numerical methods.

But till the times of Renaissance, the cubic and the quartic, defied the attempts ofmathematicians in producing a universal solution in terms of radicals. When we can find solutions for a polynomial equation with rational coefficients using only rational numbers andthe operations of addition, subtraction, division, multiplication and extraction of roots, we say that equation is soluble by radicals. In the sixteenth century the Italian mathematicians, Ferro (1501-1576), Tartaglia (1500-1557), Cardano (1501-1576), and Ludovico Ferrari (1522-1565), succeeded in breaking the impasse by their ingenious methods. In the subsequent centuries,mathematicians broke their heads to extend the methods of the Renaissance scholars, but their attempts proved futile. Mathematicians of repute like: Bolognese Rafael Bombelli (1526-1572), Francois Viete (1540-1603), Albert Girard (1595-1632), Rene Descartes (1596-1650) ,Newton (1642-1727), Leibniz (1646-1716), Michel Rolle (1652-1719), Paolo Ruffini (1765-1822), Daniel Bernoulli (1700-1782), Euler (1707-1783), Alexandre –TheophileVandermonde (1735-1796), Joseph -Louis Lagrange (1736-1813), Edward Waring (1736-1798), Augustin –Louis Cauchy (1789-1859), Carl Friedrich Gauss (1777-1855), William George Horner (1786-1837) and NeilsHenrik Abel (1802-1829), and others tried and recorded their partial success. Thus numerous mathematicians tried their hand at this, producing valuable insights into the deep structure of equations and offering several new methods for solving equations of degrees three and four. Others produced practical numerical methods for solving equations of degrees higher than four, using successive approximations. They expanded and enriched the field of the theory of equations enormously. But the sole credit of hitting the nail on its head goes to the young Galois who gave the final verdict on the polynomial equation that could not be solved in the most general way. He had the advantage of the availability of the works of Lagrange and Abel to make ground breaking advances which made him immortal in the world of modern mathematics.

Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. In fact Group Theory is the discovery of Galois.It originated in the study of symmetric functions - the coefficients of a polynomial with sign attached are the elementary symmetric polynomials in the roots. For example, (x - a) (x - b) =


3x2– (a + b)x + a.b, where (a + b), a.b are the elementary polynomials of degrees 1 and 2 in two variables. From this view point the Discriminant of a quadratic polynomial is a symmetric function in the roots which reflects properties of the roots. While Ruffin and Abel established that the general quantic could not be solved by radicals , however certain particular quinticscan be solved , and the precise criterion by which a given quantic or higher polynomial could be determined to be solvable or not was given by Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms , its Galois group had a certain structure that is, in other words, whether or not it was a solvable group? This group is always solvable for polynomials of degree four or less. This explains why there is no general formulaic solution for the quantic polynomial and polynomials of higher degrees.

How can permutations of solutions tell us anything about solvability? The fact that permutations may provide at least some new and meaningful information had long been known in the nonmathematical world. “Anagrams–words or phrases formed by the letters of others in different order –do just that. Take the name GALOIS, for instance. Allowing for two word anagrams leads us to such combinations as OIL GAS,GOAL IS,GO SAIL, and so on.

When Galois was asked about the solvability of an equation, he did not ask simply whether an equation could be solved or not, but rather what kind of a solution it might have. Finally what has Galois done? He had not found a new solution in the manner of the Renaissance mathematicians of Italy. He discovered a fundamental truth about equations of different degreesin the way Lagrange did and proved this truth, as Abel has done. What he had found was a method to determine whether any particular equation was solvable by radicals and that method was next to useless from practical point of view. “Galois revels in the impracticality of his results, which keeps them pure and beautiful, free of the messy, grungy work of prolonged calculations. And perhaps it is no wonder that the young man who styled himself as a romantic hero, who mused about the proximity of death and sacrificed his life for love and liberty, would always value beauty over practicality. Galois saw himself not just as a victim of unjust persecution but also as a prophet of future mathematics”. The future of mathematics according to Galois, lay not in producing specific results and resolving specific problems. The mathematicians of future, Galois believed, will be concerned only with the beautiful inner relations that govern the mathematical world itself, paying no heed to the messy and untidy demands of lengthy and complicated computations. We quote Matthew Pordage: “One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work”. What a profound truth lies hidden in the words of Matthew Pordage! “The mathematics of the future will be an analysis of analysis, and his own beautiful but impractical work was but a harbinger of what it would be like”. Galois indeed inspired and paved the path toward the type of mathematics cherished by the Bourbakians, a century later: “highly abstract, focused only on inner mathematical relations, and unconcerned with specific numerical results”.


4Galois started his phenomenal work by showing that every equation has its own symmetry profile—a group of permutations (now called the Galois group) that represents the symmetry properties of the equation. Earlier to Galois, equations were always classified only by their degree: as linear, quadratic, cubic, quantic and so on. Galois discovered that symmetry was a more important characteristic. The Galois group of an equation is the largest group of permutations the solutions (or roots) that leaves the values of certain combinations of these solutions unchanged. The maximum number of possible permutations of n solutions of a polynomial equation of degree n is n!. The group containing all of these permutations is the group we usually represent as Sn. Galois was able to show that for any degree n, we can always find equations for which Galois group is the full Sn. There are fifth degree equations, for instance, for which the Galois group is S5. We say that such equations possess the maximum symmetry possible.

Next to this, the remarkable feature of Galois’ approach lies in the innovative concept of a subgroup and normal subgroup. (Though he did not use the word normal subgroup essentially the concept was his in his own style in its formative days). To be brief, he demonstrated that general polynomial equation of degree n could be solved by radicals if and only if every subgroup N of the group of permutations Sn is a normal subgroup. Then demonstrated that every subgroup of Sn is normal for all n≤ 4, but not for any n greater than 5. This demonstrated the impossibility of finding a solution in radicals for all general polynomial equations of degree greater than 4.

Another stage in Galois’ innovative scheme, which escaped the imagination of Abel was, this question. What does it take for an equation to be solvable by a formula? Galois showed that to be delighted with the luxury of having a formulaic solution, equations must have a Galois group of a very particular type. Galois was excited to call it a solvable group. When the Galois group of an equation is solvable, the process of the solution of the equation can be broken up into simpler steps, each involving only the solutions of equations of lower degree.

To sum up we list three criteria in the scheme of Galois. (1) Maximum Symmetry Principle; (2) So Normal Subgroup; (3)Solvable Galois Group.Thus a family of problems dating back for centuries, concerning the solution of polynomial equations, has been wrapped up once for all. All though Galois’ work marked the end of the equation story, it inaugurated the group story. Therefore Galois’ work is both a remarkable end and an equally remarkable and marvelous beginning.

ReferencesMario Livio

[1] “The Equation That Couldn’t Be Solved”, Simon &Schuster Paperbacks- New York (2005)

Amir Alexander [1] “Duel At Dawn”, Harvard University Press, US (2010).


5

Key Note Address – Title:

Finite Dimension in Vector Spaces and Modules

Prof. Dr Bhavanari SatyanarayanaAP Scientist Awardee

Fellow, AP Akademy of SciencesSiksha Rattan Puraskar (New Delhi, Jan.2011)

Glory of India Award (Tailand, 2011)International Achiever’s Award (Tahiland, 2011)

One of the TOP-100 PROFESSIONALS (Cambridge, England, 2011)Rajiv Gandhi Excellence Award (New Delhi, 2011)

Deputy Director General, IBC, England, 2011.Head, Department of Mathematics,

Acharya Nagarjuna University, A.P., India

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

Introduction: It is well known that the dimension of a vector space is defined as the number of elements in its 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 (, v) is denoted by v) satisfying the following conditions: (i) (v + w) = v + w; (ii) ( + ) v = v + v; (iii) (v) = ()v; and (iv) 1.v = v for all , 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 F. Define f : V V by f(v) = v for all v V. Then (i) f is a group homomorphism (or group endomorphism). (ii) If 0 then f is an isomorphism.

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

Proceedings of the National Seminar on ALGEBRA (200th Birth Anniversary Celebrations of Evariste GALOIS), KBN College, Vijayawada, A.P., (October 25, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana, Mr. VVN Suresh Kumar

and Mr Mohiddin Shaw Shaik.) 6

(ii) Let F be a field. Write V = Fn = {(x1, x2,..., xn) / xi F, 1 i n}. Define (x1, x2,..., xn) = (x1, x2, ..., xn) for F and (x1, x2, ..., xn) Fn. Then Fn 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 F and f(x) = a0 + a1x + ... + anx

n Vn , define (f(x)) = a0 + a1x + ... + anx

n. 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, , F v + w 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 F and a + W V/W as (a + W) = a + 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 i F for 1 i n, then 1v1 + 2v2 + ... + nvn is called a linear combination of v1, v2, ..., vn.

(ii) For S V, we write L(S) = {1v1 + 2v2 + ... + nvn / n N, vi S and i 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.

(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 = 1v1 + 2v2 + ... + nvn with i 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 moduleover 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;

(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 essentialextension of K) if A is a submodule of M and K A = (0), imply A = (0). 2.5 Remark: (i) If V is a vector space and W is a subspace of V, then 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.

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.

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

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.

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 : MM/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(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(K1) 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 {Mi}iI 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.

Section 4: FUZZY DIMENSION IN MODULES:

4.1 Definition: Let M be a unitary R-module and : M [0, 1] is a mapping. is said to be a fuzzy submodule if the following conditions hold: (i) (m + m1) min{(m), (m1)} for all m and m1 M; and (ii) (am) (m) for all m M, a R.

This concept (given in Definition 4.1) of "fuzzy submodule" is a generalization of the definition of "fuzzy submodule" studied by Fu-Zheng PAN [ 1, 2 ], Golan [ 1 ] and Negoeta & Ralescu [ 1 ].

4.2 Proposition: If M is a unitary R-module, : M [0, 1] is a fuzzy set with (am) (m) for all m M, a R then the following two conditions are true. (i). for all 0 a R, (am) = (m) if a is left invertible; and (ii). (-m) = (m).

4.3 Corollary: If : M [0, 1] is a fuzzy submodule and m, m1 M, then (m - m1) min{(m), (m1)}.

4.4 Proposition: If : M [0, 1] is a fuzzy submodule, m, m1 M and (m) > (m1), then (m + m1) = (m1).

4.5 Corollary: If : M [0, 1] is a mapping satisfies the condition (am) (m) for all m M and a R, then the following conditions are equivalent: (i) (m - m1) min{(m), (m1)}; and (ii) (m + m1) min{(m), (m1)}.

4.6 Corollary: If : M [0, 1] is a fuzzy submodule and m, m1 M with (m) (m1), then (m + m1) = min{(m), (m1)}.

4.7 Proposition: If : M [0, 1] is a fuzzy submodule, then



(i) (0) (m) for all m M ; and (ii) (0) = mSupMm

.

Level Submodules

Now, we discuss few results on level submodules.

4.8 Theorem: A fuzzy subset of a module M is a fuzzy submodule t is a submodule of M for all t [0, (0)].

4.9 Definition: Let be any fuzzy submodule. The submodules t, t [0, 1] where t = {x M / (x) t} are called level submodules of .

4.10 Result : Let M1 M. Define (x) = 1 if x M1, = 0 otherwise. Then the following conditions are equivalent: (i) is a fuzzy submodule; and (ii) M1 is a submodule of M.

4.11 Proposition: Let be a fuzzy submodule of M and t, s (with t < s) be two level submodules of . Then the following two conditions are equivalent: (i) t = s; and (ii) there is no x M such that t (x) < s.

Minimal Elements

Now we introduce the concept "minimal element".

4.12 Definition: An element x M is said to be a minimal element if the submodule generated by x is minimal in the set of all non-zero submodules of M.

4.13 Theorem: If M has DCC on its submodules, then every nonzero submodule of M contains a minimal element.

There are modules which do not satisfy DCC on its submodules, but contains a minimal element. For this we observe the following example.

4.14 Example: Write M = ℤ ℤ6. Now M is a module over the ring R = ℤ. Clearly M have no DCC on its submodules. Consider a = (0, 2) M. Now the submodule

generated by a, that is, ℤa = {(0, 0), (0, 2), (0, 4)} is a minimal element in the set of all non-zero submodules of M. Hence a is a minimal element.

Every minimal element is an u-element. The converse is not true. For this observe the example 4.15 (given below). If M is a vector space over a field R, then every non-zero element is a minimal element as well as an u-element.



4.15 Example: Write M = ℤ as a module over the ring R = ℤ..

Since ℤ is a uniform module, and 1 is a generator, we have that 1 is an u-element. But 2ℤ is a

proper submodule of 1.ℤ = ℤ = M. Hence 1 cannot be a minimal element. Thus 1 is an u-element but not a minimal element.

4.16 Theorem: Suppose is a fuzzy submodule of M. (i) If a M, then for any x Ra we have (x) (a); and (ii) If a is a minimal element, then for any 0 x Ra we have (x) = (a).

4.17 Lemma: If x is an u-element of a module M with DCC on submodules, then there exist minimal element y Rx such that Ry e Rx.

4.18 Theorem: If M has DCC on its submodules, then there exist linearly independent minimal elements x1, x2, ….., xn in M where n = dim M, and the sum <x1> + …….+ <xn> is direct and essential in M. Also B = {x1, x2,….,xn} forms a basis for M.

Fuzzy Linearly Independent Elements

4.19 Definition: Let M be a module and a fuzzy submodule of M. x1, x2, …, xn M are said to be fuzzy -linearly independent ( or fuzzy linearly independent with respect to ) if (i) x1, x2, …, xn are linearly independent; and (ii) (y1 + … + yn) = min{(y1), …, (yn)} for any yi Rxi, 1 i n.

4.20 Theorem: Let be a fuzzy submodule on M. If x1, x2, …, xn are minimal elements in M with distinct -values, then x1, x2, …, xn are (i). linearly independent; and (ii). fuzzy -linearly independent.

Fuzzy Dimension

4.21 Definition: (i). Let be a fuzzy submodule on M. A subset B of M is said to be a fuzzy pseudo basis for if B is a maximal subset of M such that x1, x2,….., xk are fuzzy linearly independent for any finite subset { x1, x2,….., xk } of B.

(ii). Consider the set ℬ = {k / there exist a fuzzy pseudo basis B for with |B| = k}. If ℬ has no upper bound then we say that the fuzzy dimension of is infinite.

We denote this fact by S-dim() = ∞. If ℬ has an upper bound, then the fuzzy dimension of is

sup ℬ. We denote this fact by S-dim() = sup ℬ .

If m = S-dim() = sup ℬ , then a fuzzy pseudo basis B for with |B| = m, is called as fuzzy basisfor the fuzzy submodule .

4.22 Result: Suppose M has FGD and is a fuzzy submodule on M. Then



(i). |B| dim M for any fuzzy pseudo basis B for ; and (ii). S-dim () dim M.

4.23 Definition: A module M is said to have a fuzzy basis if there exists an essential submodule A of M and a fuzzy submodule on A such that S-dim() = dim M. The fuzzy pseudo basis of is called as fuzzy basis for M.

4.24 Remark: If M has FGD, then every fuzzy basis for M is a basis for M.

4.25 Theorem: Let M be a module with DCC on submodules. Then M has a fuzzy basis (In other words, there exists an essential submodule A of M and a fuzzy submodule of A such that S-dim() = dim M).

Acknowledgements

The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), and Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics) for inviting me to present this Key Note Address at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois).

Reference

Chatters A.W & Hajarnivas C.R [1] "Rings with Chain Conditions", Research Notes in Mathematics, Pitman Advanced publishing program, Boston-London-Melbourne, 1980.

Fu-Zheng Pan [1] "Fuzzy finitely generated Modules", Fuzzy sets and Systems 21 (1987) 105-113. [2] "Fuzzy Quotient Modules", Fuzzy sets and Systems 28 (1988) 85-90.

Golan J. S. [1] "Making Modules Fuzzy", Fuzzy sets and Systems 32(1989) 91-94.

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

Negotia C.A. & Ralescu D. A. [1] "Applications of Fuzzy sets to system Analysis", Birkhauser, Basel, 1975.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.



[2] Lecture on "Modules with Finite Goldie dimension and Finite Spanning dimension", International Conference on General Algebra, Krems, Vienna, Austria, August, 21-27, 1988.[3] "The Injective Hull of a Module with FGD", Indian J. Pure & Appl. Math. 20 (1989) 874-883.[4] "On Modules with Finite Goldie Dimension" J. Ramanujan Math. Society. 5 (1990) 61-75.[5] Lecture on "Modules with Finite Spanning Dimension", Asian Mathematical Society Conference, University of Hong Kong, Hong Kong, August 14-18, 1990.[6] "On Essential E-irreducible submodules", Proc., 4th Ramanujan symposium on Algebra and its Applications, University of Madras, Feb 1-3 (1995), pp 127-129.

Satyanarayan Bhavanari & Mohiddin Shaw Sk[1] "On Fuzzy Dimension of a Module with DCC on Submodules”, Acharya Nagarjuna International Journal of Mathematics and Information Technology, 01 (2004), 13-32.[2] “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. Maths (BN Prasad Birth

Commemoration Volume) 42 (2000) 183 - 192. [3] Discrete Mathematics with Graph Theory (for B.Tech/B.Sc/M.Sc.,(Maths)) Printice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).Satyanarayana Bhavanari, Syam Prasad K & Nagaraju D.

[1] "A Theorem on Modules with Finite Goldie Dimension", Soochow Journal of Mathematics, 32, No.2 (2006) 311-315.

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.Zadeh L.A. [1] "Fuzzy Sets", Information and Control, 8(1965) 338-353

THE HIGHEST FORM OF PURE THOUGHT IS IN MATHEMATICS

…..PLATO---------------------------------------------------------------------------------------------------------------

LIFE IS GOOD FOR ONLY TWO THINGS. DISCOVERING MATHEMATICS, AND TEACHING

MATHEMATICS ….. SIMEON POISSON


16

Prime Graph of ℤn and

Zero Square Ring S(ℤ3)----------------------------------------------------------------------------------------

1. INTRODUCTIONSatyanarayana, Syam Prasad and Nagaraju [1] introduced the concept ‘Prime Graph of R’ (denoted by PG(R)), where R is a given associative ring. This concept ‘prime graph of a ring’ is a new bridge between the graph theory and ring theory. This concept provides a geometric presentation of rings via graph theory. This paper is divided in to three sections. In Section-1, we collect some definitions and examples from Satyanarayana,

Syam Prasad and Nagaraju [1]. In Section-2, we construct the prime graph of S(ℤ2), and

observed certain important properties of S(ℤ2). In Ssection-3, we construct the prime

graph of S(ℤ3). 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 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 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 = . 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).1.5 Definitions: Let G = (V, E) be a graph and 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). In a graph G, a subset S of V(G) is

Short Talk -----------------

Mr. Mohiddin Shaw


17said to be a dominating set for G (or in G) if every vertex not in S has a neighbor in S. The domination number, denoted by (G) is defined as min{|S| / S is a dominating set in G}. This domination number is an important parameter in graph theory. If d(v) = k for every vertex v of a given graph G, for some fixed positive integer k, then the graph G is called a k-regular graph (or a regular graph, or a regular graph of degree k). 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 Hamiltonian path in a graph G is a path (with out repetition of vertices) which contains every vertex of G. A Hamiltonian cycle in a graph G is a cycle which contains every vertex of G. A graph G is called Hamiltonian if it has a Hamiltonian cycle. (Note.13.13, Page 351 [ 2 ])Every complete graph on more than two vertices is a Hamiltonian graph.

For further concepts related to Ring Theory and Graph Theory, we refer Herstein [1], Lambek [1], Narsing Deo [1] and Satyanarayana & Syam Prasad [ 2 ].1.6 Definition: 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, and x y}

1.7 ( Examples: Consider ℤn, the ring of integers modulo n.

(i) Let us construct the graph PG(R), where R = ℤ4. We know that R = ℤ4 = {0, 1, 2, 3}. So V(PG(R)) = {0, 1, 2, 3}. Since 0R1 = 0, 0R2 = 0, 0R3 = 0, we have that 01, 02, 03 E(PG(R)). There are no other edges, as there are no two distinct non-zero elements x, y R such that xRy = 0. So E(PG(R)) = { 01, 02, 03 }. Now PG(R) is given in Figure 1.7(iii).

(ii) Let us construct the graph PG(R), where R = ℤ6. We know that R = ℤ6 = {0, 1, 2, 3, 4, 5}. So V(PG(R)) = {0, 1, 2, 3, 4, 5}. Since 0R1 = 0, 0R2 = 0, 0R3 = 0, 0R4 = 0, 0R5 = 0, 2R3 = 0, 3R4 = 0, we have that 01, 02, 03, 04, 05, 23, 34 E(PG(R)). So E(PG(R)) = { 01, 02, 03, 04, 05, 23, 34 }. Now PG(R) is given in Figure 1.7(vi).

1.8 Definition: A ring R is said to be a zero square ring of type-1 if x2 = 0 for all x R, and there exists two elements a, b R such that ab 0. If R is a non-zero Boolean ring, then x2 = x for all x R. Hence every non-zero Boolean ring can not be a zero square ring of type-1.

0

1 2

3

PG(R) = PG(ℤ4)Fig. 1.7(iii)

0

1

2

3

4

5Fig. 1.7 (vi) PG(R) = PG(ℤ6)


181.9 Definition: Let R be a non null ring (that is, R2 0). Now define a new ring (denoted by S(R) where S(R) = R R R, define addition on S(R) component wise, and multiplication by (x1, y1, z1).(x2, y2, z2) = (0, 0, x1y2 - x2y1).

Stanley [ 1 ] mentioned that (S(R))2 0 (that is S(R) is not a null ring, if R is not a null ring) and a2 = 0 for all a S(R). Hence S(R) is a zero square ring of type-1.

2. Prime graph of the Zero Square Ring S(ℤ2).Now we construct the prime graph of the S(ℤ2).

2.1. Construction: To construct the prime graph of S(ℤ2), consider the vertex set V = { x0, x1, x2, x3, x4, x5, x6, x7 }, where x0 = (0,0,0), x1 = (0,0,1), x2 = (0,1,0), x3 = (0,1,1), x4 = (1,0,0), x5 = (1,0,1), x6 = (1,1,0), x7 = (1,1,1). The edge set is E(S) ={ xy / xRy = 0 or yRx = 0 and x ≠ y } = { x0x1, x0x2, x0x3, x0x4, x0x5, x0x6, x0x7, x1x2, x1x3, x1x4, x1x5, x1x6, x1x7, x2x3, x4x5, x6x7}.2.2 Note: (i). It is known that a non-empty graph with at least two vertices is bipartite if

and only if it has no odd cycles. PG(ℤ2) is not a bipartite graph because it contains cycles of odd length.

(ii). Since PG(ℤ2) contains some vertices of odd degree, we conclude that PG(ℤ2) is not an Euler Graph

(iii) PG(ℤ2) is not a Hamiltonian graph, because there is no circuit running through all the vertices exactly once.(iv). The domination number for this graph is one (Note that the set {x1} forms a dominating set for this graph).

3. Prime graph of the Zero Square Ring S(ℤ3).In this Section, we construct the prime graph of S(ℤ3). We use the following

notation throughout this section.

3.1 Notation: Consider the ring S(ℤ3) = ℤ3 x ℤ3 x ℤ3, with component wise addition and the product defined by (a1, a2, a3).(b1, b2, b3) = (0, 0, a1b2 - a2b1). The vertex set

V(S(ℤ3)) = { xi / 0 ≤ i ≤ 26}, where x0 = (0,0,0), x1 = (0,0,1), x2 = (0,0,2), x3 = (0,1,0), x4 = (0,1,1), x5 = (0,1,2), x6 = (0,2,0), x7 = (0,2,1), x8 = (0,2,2), x9 = (1,0,0), x10 = (1,0,1), x11 = (1,0,2), x12 = (1,1,0), x13 = (1,1,1), x14 = (1,1,2),x15 = (1,2,0), x16 = (1,2,1), x17 = (1,2,2), x18 = (2,0,0), x19 = (2,0,1), x20 = (2,0,2), x21 = (2,1,0), x22 = (2,1,1), x23 = (2,1,2), x24 = (2,2,0), x25 = (2,2,1), x26 = (2,2,2).

and the edge set E(S(ℤ3)) = { xy / x, y S(ℤ3), xS(ℤ3)y =0, x ≠ y }.

IF PEOPLE DO NOT BELIEVE THAT MATHEMATICS IS SIMPLE, IT IS ONLY BECAUSE THEY DO NOT

REALISE HOW COMPLICATED THE LIFE IS


19

Figure -A

x13

x12

x11

x16

x15

x14

x19

x18

x17x23

x22

x21

x26

x25

x24

x20

x19

x18

x8

x13x4

x3 x6

x7

x5

Everest Galois first used the term GROUP.The Algebraic System GAMMA-NEAR-RING (that paved a new for mathematical researchers) was

introduced by Bhavanari Satyanarayana in 1984.


20

AcknowledgementsThe author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this survey article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois).

ReferencesHerstein I. N[1]. “Topics in Algebra”, Vikas Publishing House, 1983.Lambek. J [1]. “Lectures on Rings and Modules”, Blaisdel. Publ. Co. 1966.Satyanarayana[1] “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7).Satyanarayana Bhavanari Godloza Lungisile and Nagaraju Dasari[ 1 ] “Ideals and Direct Product of Zero Square Rings”, East Asian Mathematical Journal., 24(2008) 377-387. Satyanarayana Bhavanari & Mohiddin Shaw Sk, [1] “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-639-23197-7)

Satyanarayan Bhavanari & Nagarju Dasari

[1] “Dimension and Graph Theoretic Aspects of Rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3-639-30558-6)Satyanarayana Bhavanari , Nagaraju Dasari, Balamurugan K. S., & Godloza. L[1] "Finite Dimension in Associative Rings", Kyungpook Mathematical Journal, 48 (2008) 37-43.Satyanarayan Bhavanari & Rama Prasad J L [1] “Fuzzy Prime Submodules”, VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-639-24355-0)Satyanarayana Bhavanari & Syam Prasad Kuncham[1] "An Isomorphism theorem on Directed Hypercubes of Dimension n", Indian J. Pure & Appl. Mathematics, 34 (2003) 1453-1457.[ 2 ] “Discrete Mathematics and Graph Theory”, Printice Hall of India, New Delhi, 2009.Satyanarayana Bhavanari, Syam Prasad Kuncham and Nagaraju Dasari[ 1 ] “Prime Graph of a Ring”, “ Prime Graph of a Ring”, Journal of Combinatories,

Informations & Systems Sciences, 35 (2010).Stanley Richard P[1].. "Zero Square Rings", Pacific Journal of Mathematics, 30 (1969) 811-824.Syam Prasad Kuncham & Satyanarayana Bhavanari “Dimension of N-groups and Fuzzy ideals in Gamma Near-rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3- 639-30558-624851-7).West D. B[ 1 ] . "Introduction to Graph Theory – 2nd Edition", Prentice Hall of India, New Delhi,2002.

NOETHER IS THE MOTHER OF MODERN ALGEBRA


21

On Gamma 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 -ring, a generalization of ‘ring’ was introduced by Nobusawa [1] and generalized by Barnes [ 1 ]. Later Satyanarayana [1], Satyanarayana, Pradeep Kumar & Srinivasa Rao [1 ] also contributed to the theory of -rings. A generalization of both the concepts near-ring and the -ring, namely -near-ring was introduced by Satyanarayana [ 2 ] and later studied by several authors like: Booth [ 1 ], Booth & Godloza [1], Syam Prasad [1], Satyanarayana, Pradeep kumar, Sreenadh, and Eswaraiah Setty [1].

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) – nn1 I for all i I and n, n1 N (Equivalently, n(i + n1) – nn1 I for all i I and n, n1 N); (ii) a right ideal of N if IN I; and(iii) an ideal if I is a left ideal and also a right ideal.

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

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 [1], Satyanarayana & Syam Prasad [1].

Short Talk_____________________________________________________

Dr. T.V. Pradeep KumarANU College of Engineering, ANU


22

1. 4. Definition: (Satyanarayna [2]): Let (M, +) be a group (not necessarily Abelian) and be a non-empty set. Then M is said to be a -near-ring if there exists a mapping M M M (the image of (a, , b) is denoted by ab), satisfying the following conditions: (i) (a + b)c = ac + bc; and (ii) (ab)c = a(bc) for all a, b, c M and , . M is said to be a zero-symmetric -near-ring if a0 = 0 for all a M and , where 0 is the additive identity in M.A natural example of -near-ring is given below:

1.5. Example (Satyanarayana [3]): Let (G, +) be a non-abelian group and X be a non-empty set. Let M = {f / f: X G}. Then M is a group under point wise addition.Since G is non-abelian, then (M, +) is non-abelian. In this paper, the first two sections contains some definitions and results from the literature. The third section contains few new results related to the important concept: completely semi-prime ideal.

2. Some Results on Gamma-Near rings

In this section M stands for -Near-ring. In this section we collect some more definitions and results from the literature.2.1. Definitions (6.2.3 of Satyanarayana [2, 5]): An ideal A of M is said to be (i) prime if B and C are ideals of M such that BC A implies B A or C A;(ii) completely prime if a b A, a, b M, implies either a A or b A.

2.2 Definition: An ideal A of M is said to be semi-prime if B is an ideal of M such that BB A implies B A. 2.3. Definition: An ideal I of M is said to be completely semi-prime ideal of M if it satisfies the following condition: aa I a I .

2.4. Theorem (Pradeep Kumar, Satyanarayana, Syam Prasad and Mohiddin Shaw [ 1 ] ): If S is a semi-prime ideal of M, then the following are equivalent:(i) If xx S, then <x><x> S. (ii) S is completely semi-prime ideal of M.(iii) If xy S, then <x><y> S.

2.5 Theorem (Satyanarayana, Pradeep kumar, Sreenadh and Eswaraiah Setty [1]): Let I be a completely semi-prime ideal of N. Then I is the intersection of all minimal completely prime ideals of I.

2.6 Theorem (Satyanarayana, Pradeep kumar, Sreenadh and Eswaraiah Setty [1]):: If P is a prime ideal and I is a completely semi-prime ideal, then P is minimal prime ideal of I if and only if P is minimal completely prime ideal of I.


23

Acknowledgements

The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this survey article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois).

References

Barnes W.E[1] . “On the -rings of Nobusawa”, Pacific J. Math 18 (1966) 411- 422.Booth G.L[1] .“A note on -Near-rings”,Stud.Sci.Math.Hunger 23 (1988) 471-475.Booth G.L. and Godloza L.[1] “ On Primeness and Special Radicals of -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.

Nobusawa[1] “On a Generalization of the Ring theory”, Osaka J. Math. 1 (1964) 81-89Pilz .G[1] “Near-rings”, North Holland, 1983.Pradeep Kumar T.V[1] “Contributions to Near-ring Theory - III”, Doctoral Dissertation, Acharya

Nagarjuna University, 2006Pradeep Kumar T.V., Satyanarayana Bhavanari, Syam Prasad K and Mohiddin Shaw Sk[1] “Some results on Completely Semi Prime ideals in Gamma Near-Rings”, 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) Ramakotaiah Davuluri[1] “Theory of Near-rings”, Ph.D. Diss., Andhra univ., 1968.Satyanarayana Bhavanari.[1] "A Note on -rings", Proceedings of the Japan Academy 59-A(1983) 382-83.[2] “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna

University, 1984. [3] “A Note on -near-rings”, Indian J. Mathematics (B.N. Prasad Birth Centenary

commemoration volume) 41(1999) 427-433.[4] “A Note on Completely Semi-Prime ideals in Near- Rings”, International Journal of Computational Mathematical Ideas Vol. , No 3(2009) 107 – 112.


24

[5] “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7).

. Satyanarayan Bhavanari & Mohiddin Shaw Sk, [1] “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-639-23197-7)Satyanarayan Bhavanari & Nagarju Dasari

[1] “Dimension and Graph Theoretic Aspects of Rings”, VDM Verlag Dr Muller,

Germany, 2011, (ISBN 978-3-639-30558-6)Satyanarayana Bh., Pradeep Kumar T.V. and Srinivasa Rao M.[1] “On Prime left ideals in -rings”, Indian J. Pure & Appl. Mathematics 31 (2000)

687-693.Satyanarayana Bh., Pradeep kumar T.V., Sreenadh S., and Eswaraiah Setty S

[1] . “On Completely Prime and Completely Semi- Prime Ideals in -Near-Rings”, International Journal of Computational Mathematical Ideas Vol. 2, No 1(2010) 22 – 27.

Satyanarayan Bhavanari & Rama Prasad J L “Fuzzy Prime Submodules”, VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-

639-24355-0)Satyanarayana Bhavanari & Syam Prasad Kuncham

[1] “Discrete Mathematics & Graph Theory”, Prentice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).

Syam Prasad K.

[1] “Contributions to Near-ring Theory II”, Doctoral Dissertation Acharya Nagarjuna University, 2000.

Syam Prasad Kuncham & Satyanarayana Bhavanari “Dimension of N-groups and Fuzzy ideals in Gamma Near-rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3- 639-30558-624851-7).

What ever you think that you will be if you think your self weak, weak you will if you think your self strong, strong you will be

----------------------------------------------------------------------------------------------------------

GOD IS PRESENT IN EVERY JIVA, THERE IS NO OTHER GOD BESIDES THAT “WHO SERVES JIVA, SERVES GOD IN DEED”


25

Zero Square Rings andZero Square Ideals-------------------------------------------------------------------------------------------

1. Introduction

In 1969, Richard P. Stanley has introduced the concept of zero square rings and he proved several interesting results related to this concept in Stanley [1]. Zero square rings were also studied by Vasantha [1, 2]. Stanley [1] calls ring R a zero square if x2 = 0 for all x R.

1.1 Example (Stanley [1]): Let S be a non-zero commutative ring. Write R= S S S, the direct product of three copies of the additive group S. The addition in R is defined component wise, and the multiplication by (x1, y1, z1).(x2, y2, z2) = (0, 0, x1y2 - x2y1) for all (x1, y1, z1), (x2, y2, z2) R. Then R is a zero square ring.1.2 Remark (Stanley [1]): (i) Every zero square ring is anti commutative (that is, xy = - yx for all x, y) and (ii) A zero square ring R is commutative if and only if 2R2 = 0.

In 2008, Satyanarayana Bhavanari, Godloza & Nagaraju [1] were defined and studied the concepts zero square ring (and Ideal) of type-1/type-2. Zero square ring of type-2 is same as the zero square ring studied by the earlier authors.

2. Zero Square Rings

2.1 Definition (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): (i) A ring R is said to be a zero square ring of type-1 if x2 = 0 for all x R and there exists two elements a, b R such that ab 0. (ii) A ring R is said to be a zero square ring of type-2 if x2 = 0 for all x R.2.2 Examples: (i) Let (G, +) be a group (not necessarily Abelian). Define a multiplicative operation on G by a.b = 0 for all a, b G, where 0 is the additive identity. Then (G, +, .) is a null ring. So (G, +, .) is a zero square ring of type-2, but not of type-1. We can conclude that every group can be made into a zero square ring of type-2.(ii) Suppose that R is a non-zero Boolean ring. Then x2 = x for all x R. So R is a non-null ring and for any x 0, we have x2 0. Hence every non-zero Boolean ring can neither be a zero square ring of type-1 nor a zero square ring of type-2.

(iii) Let S = ℤ and write R = S S S. Define addition on R component wise, and multiplication on R by (x1, y1, z1).(x2, y2, z2) = (0, 0, x1y2 - x2y1). By Example 1.1, it follows that x2 = 0 for all x R. But (1, 2, 3), (2, 3, 4) R and (1, 2, 3)(2, 3, 4) = (0, 0, 3 - 4) = (0, 0, -1) (0, 0, 0). Hence R is a zero square ring of type-1.

Short Talk_____________________________________________________

Dr. Dasari NagarajuHead/Maths.,

Manipal University Jaipur


26

2.3 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): Suppose R is a zero square ring of type-2, and A is a module over R. Then (i) aR A for all 0 a A. (ii) If A is irreducible, then AR = 0.2.4 Corollary: A primitive ring cannot be a zero square ring of type-2.2.5 Corollary: If R is a zero square ring of type-2, then rR R for all 0 r R.2.6 Corollary: Let R be a zero square ring of type-2.(i) If I is a non-zero right ideal of R, then I cannot be a monogenic right ideal.(ii) If I is a non-zero left ideal of R, then I cannot be a monogenic left ideal.2.7 Corollary: If R is a non-zero zero-square ring of type-2, then

(i) Rr R for all r R. (ii) rR R for all r R.

3. Zero Square Ideals

3.1 Definition (Satyanarayana Bhavanari., Godloza L. & Nagaraju D [1]): A proper ideal I of R is said to be a zero square ideal of type-1 (respectively, type-2) if the quotient ring R/I is a zero square ring of type-1 (respectively of type-2).3.2 Remark: (i) If R is a zero square ring of type-2, then every ideal I of R is a zero square ideal of type-2. The converse of this statement is not true. For this observe the following Example 3.3.(ii) If R is a zero square ring of type-2, then every ideal of R is also a zero square ring of type-2.3.3 Remark: Let I, J be two ideals of a ring R. If I, J are two zero square ideals of type-2, then I J is also a zero square ideal of type-2.

3.4 Note: A class ℬ of rings is said to be homomorphically closed if every homomorphic

image of R is in ℬ for all R in ℬ.

3.5 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): The class ℬ of all zero square rings of type-2 is homomorphically closed.3.6 Remark: Suppose I is an ideal of R, I is a zero square ideal of type-2 and also a zero square ring of type-2. Then x4 = 0 for all x R. 3.7 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): Let R be a zero square ring of type-2 and I an ideal of R. Then the following two statements are equivalent:

(i) R2 ⊈ I. (ii) I is a zero square ideal of type-1.3.8 Corollary: (i) Let I and J be ideals of a zero square ring R of type-2 with I J. If J is a zero square ideal of type-1, then I is also a zero square ideal of type-1.(ii) The intersection of any collection of zero square ideals of type-1 is also a zero square ideal of type-1.3.9 Corollary: In a zero square ring R of type-2,

(i) Every semi-prime ideal S of R is a zero square ideal of type-1.(ii) Every prime ideal P of R is a zero square ideal of type-1.

LET US HAVE NOBLE THOUGHTS FROM EVERY CORNER


27

3.10 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): If there exists a

chain R = I0 ⊋ I1 ⊋ I2 ⊋ … ⊋ Ik = (0) of ideals of R such that Is+1 is a zero square ideal

of type-2 in the ring Is, for all 0 s < k, then R is a nil ring. In particular, 2k

x = 0 for all x R.4. Zero Square Rings and Direct Products4.1 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): (i) If Ri, 1 i

k, are zero square rings of type-1, then

k

1iiR is also a zero square ring of type-1.

(ii) Each Ri, 1 i k, is a zero square ring of type-2 if and only if

k

1iiR is a zero square

ring of type-2.

4.2 Remark: The converse of the Theorem 4.1 (i) is not true, in general.

4.3 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [1]): Let Ri, 1 i k

be rings. The direct product

k

1iiR is a zero square ring of type-1 if and only if there

exists a non-empty subset I {1, 2, …, k} such that Ri is a zero square ring of type-1 for all i I and Rj is a zero square ring of type-2 but not of type-1 for all j {1, 2, …, k} \ I.4.4 Corollary: For any positive integer k, we have that R is a zero square ring of type-2 (respectively, type-1) if and only if Rk is a zero square ring of type-2 (respectively, type-1).

5. Zero Square Fuzzy Ideals of R

5.1 Definition (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [2]): A fuzzy ideal of R is said to be a zero square fuzzy ideal if it satisfies the following two properties: (i). (x2) = (0) for all x R; and (ii). (xy) (0) for some x, y R.

5.2 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [2]): is a fuzzy ideal of a ring R. The following conditions are equivalent:(i). R = {x R / (x) = (0)} is a zero square ideal of R;(ii). (x2) = (0) for all x R and there exist x, y R such that (xy) (0) (that is, is a zero square fuzzy ideal of R).

5.3 Note: Let be a fuzzy ideal of a ring R. For any y R, the coset y + is defined from R to [0, 1] by (y + )(x) = (x - y) for all x R. (i) R/ = the set of all cosets of = {y + / y R} forms a ring (the proof is parallel to Theorem 2.4 of Satyanarayana & Syam Prasad [1]).

HE/SHE WHO DO NOT HOPE TO WIN HAS ALREADY LOST


28

(ii) If we define : R/ [0, 1] by (x + ) = (x) for all x R, then is a fuzzy ideal of R/ (the proof is parallel to Lemma 2.6 of Satyanarayana & Syam Prasad [1]).5.4 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [2]): If is a zero square fuzzy ideal of R, then is a zero square fuzzy ideal of R/.5.5 Theorem (Satyanarayana Bhavanari, Godloza L. & Nagaraju D [2]): Let be a zero square fuzzy ideal of R and y R . If the fuzzy coset y + is a fuzzy ideal of R, then y + is a zero square fuzzy ideal of R.

ReferencesSatyanarayana Bhavanari [1] “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7).Satyanarayana Bhavanari., Godloza L and Nagaraju D.[1]. Ideals and Direct Product of Zero Square Rings, East Asian Math. J. 24 (2008),

377-387.[2]. Fuzzy Ideals of Zero Square Rings, The Int. Journal of Fuzzy Mathematics,

(2011), Accepted.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 and Nagaraju D[1]. Dimension and Graph Theoretic aspects of Rings (Monograph), VDM Verlag

Dr. Müller e.K., Germany (2011), (ISBN 978-3-639-30558-6). Satyanarayan Bhavanari & Rama Prasad J L [1] “Fuzzy Prime Submodules”, VDM Verlag Dr Muller, Germany, 2010,

(ISBN 978-3-639-24355-0)Satyanarayana Bh., and Syam Prasad K[1]. On Fuzzy Cosets of Gamma Near-rings” Turkish J. Mathematics 29 (2005) 11-22.[2]. Discrete Mathematics and Graph Theory, PHI, New Delh (2010) (ISBN: 978-81-

203-3842-5).Syam Prasad Kuncham & Satyanarayana Bhavanari[1] “Dimension of N-groups and Fuzzy ideals in Gamma Near-rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3- 639-30558-624851-7).Stanley R. P[1]. Zero Square Rings, Pacific Journal of Mathematics 30 (1969), 811-824.Vasantha K.[1]. A Zero Square Group Ring, Bull. Cal. Math. Soc. 80 (1988), 105-106.[2]. Semi-group Rings which are Zero Square Rings, News Bull. C.M.S, 12 (1989),

08-10.Zadeh L. A[1]. Fuzzy Sets, Inform. and Control, 8 (1965), 338-353.

RELIGION IS ONLY TO AWAKEN DIVINITY IN MAN


29

Rough Fuzzy Group induced by epimorphisms_________________________________________________________

1. INTRODUCTION

In 1982, Pawlak introduced theory of rough sets. This theory involves in several technical aspects based on decision-making. This concept can be viewed either algebraically or topologically. As fuzzy concept has become inevitable in technology, several scientists have been involved in hybridizing these two concepts. In 1989, Dubois and Prade gave their contributions on rough fuzzy and fuzzy rough sets.

In 2005, the group structure in rough fuzzy sets was discussed in [3]. This approach deals with defining a fuzzy ordered pair for each element of the given finite quotient group. In this paper, an effort is made to bring the homomorphism approach on rough fuzzy groups for developing further algebraic approaches.

This paper comprises four sections. Section two deals with rough sets and rough fuzzy sets. It also gives the basic theorems on groups, which are essential for this paper. Section three deals with rough fuzzy group on group of congruence modulo n on Z and it is extended for any finite group using fundamental theorem of homomorphism on groups [4].

In the forthcoming section, the definitions given by Pawlak on rough sets and Dubois and Prade on rough fuzzy sets are dealt.

2. ROUGH SETS AND ROUGH FUZZY SETS

In 1982, Pawlak introduced Rough Sets [5]. For a given partition W={X1,X2,…,Xt} on a finite universe of discourse U={x1,x2,…,xn}, the lower and upper rough approximations of any

subset A of U are given by :i iA X W X A and :i iA X W X A respectively. In

general, the lower approximation gives certainty and the upper approximation gives possibility. The region between the upper and lower approximations is termed as boundary. This theory found very good usage in information systems. By sensing the importance of rough sets, Dubois and Prade have introduced fuzzy concepts in rough sets. In 1989, they introduced rough fuzzy sets [1,2], which describe the approximation of fuzzy subset of universe of discourse under the given partition. The rough fuzzy approximations are defined as follows:

Consider the finite universe of discourse U={x1,x2,…,xn}.Let W={X1,X2,…,Xt} be an partition of

U. For any fuzzy subset A of U, define ( ) inf ( ) :A i A j j iX x x X and

( ) sup ( ) :i A j j iAX x x X . The lower and upper approximations of A are given by

1 2( ), ( ), , ( )A A A tA X X X and 1 2( ), ( ), , ( )tA A AA X X X respectively.

Invited TalkDr. G. Ganesan

Adikavi Nannaya University, A.P, India


30

For each Xi, ( ), ( )A i iAX X is called the fuzzy ordered pair [3].

The next section introduces group concept on rough fuzzy sets. Before entering into it, it is necessary to note the following remarks in groups.

(a) If N is a normal subgroup of a group (G, * ), then G/N is defined as G/N={Na/aG}. Here, it can be seen that G/N is a group under the operation which is given by NaNb=Na*b. The group thus obtained is called a quotient group.

(b) (Fundamental Theorem of Homomorphism) If f:GG’ is an epimorphism with kernel K, then G/k is isomorphic to G’.

3. ROUGH FUZZY GROUPS

In [3], rough fuzzy groups on Z is introduced by G.Ganesan and C. Raghavendra Rao. For any

positive integer n, consider the congruence class = 0,1,........, 1n

. For any p and q

in ,

define addition (modulo n) of p and q

by np q

=Rem(p+q,n) where Rem(p+q,n) is the

remainder obtained by dividing p+q by n Let F be any fuzzy subset of Z. Then every element of is associated with a fuzzy ordered pair with respect to F. Then can be expressed in terms of

fuzzy ordered pairs 0 ,0 , 1 ,1 ,...., 1 , 1l ul ul u

n n

with respect to F.

The closed fuzzy ordered pairs can be obtained by using the following membership functions. If

the fuzzy ordered pair of k

is ,l uk k

under the fuzzy number F, define

( 1) ( ) ( )

( ) ( ) ( )

maxmin( , )

s s sl s s s l l

lnl l

k p qp q k

and

( 1) ( ) ( )

( ) ( ) ( )

maxmin( , )

s s su s s s u u

unu u

k p qp q k

where

0

l lp p

;0

u up p 0

l lq q

; 0

u uq q

; 0l lk k

and 0u uk k

Here, it can be observed that the above process can be made to be iterative. That is, if one tries to

find ( 1) ( 1),

s sl uk k

, for some appropriate s, it can be seen that it is equal to ( ) ( ),

s sl uk k

. The

saturated (with respect to iteration) fuzzy ordered pair is called closed fuzzy ordered pair. Hence, by finite iteration, closed fuzzy ordered pairs are obtained for each element of . It can be illustrated by the following example.

3.1 Example: Consider the group = 0,1,2

. Let F be any fuzzy subset of Z, which induce the

fuzzy ordered pairs (0.3,0.7),(0.4,0.6) and (0.2,0.6) for 0, 1 2and

respectively.

Then1

0l

=max{min(0.3,0.3),min(0.2,0.4)}=0.3;

11l

=max{min(0.4,0.3),min(0.2,0.2)}=0.3;

12l

= max {min(0.3,0.2), min(0.4,0.4)} =0.4;

10u

=max{min(0.7,0.7),min(0.6,0.6)} =0.7;

11u

=max

{min(0.7,0.6), min(0.6,0.6)} =0.6; 1

2u

= max{min (0.7,0.6), min(0.6,0.6)}=0.6;


31

Hence, set of fuzzy ordered pairs in first iteration is given by 0.3,0.7 , 0.3,0.6 , 0.4,0.6 .

Here, it is further observed that the second iteration also gives the set of fuzzy ordered pairs

0.3,0.7 , 0.3,0.6 , 0.4,0.6 .

Hence, from the above example, it can be seen that the set of closed fuzzy ordered pairs is closed with respect to the above operation. Also, it is known that is closed with respect to addition modulo n. This group of closed fuzzy ordered pairs is called rough fuzzy group on .

Now, in order to generalize this approach to a finite group, it is necessary to note the following theorem.

3.2 Theorem: Consider a finite group (G,*) with cardinality m. Let = 0,1,........, 1k

be the

congruence group of Z under addition modulo k (k<m) and f: G an epimorphism with kernel K. Then o(G/K)=k

Proof: By fundamental theorem of homomorphism, G/K is isomorphic to . Hence, there is a one-one correspondence between and G/K. Thus o(G/K)=k.

3.3 Algorithm

Using above theorem, the following algorithm describes the procedure of constructing rough fuzzy group for a given finite group G; a congruence group of Z under addition modulo k and an epimorphism f :G.

1. begin2. Input ((G,*),,f, F)3. Compute the quotient group L=G/K4. By the axiom of choice, denote each element of L by any of it member. For

example, if aKx for some xG, then denote Kx by [a].5. Compute closed rough fuzzy ordered pair for each element of using the fuzzy

subset F of Z. 6. Associate the closed rough fuzzy ordered pair with the preimage of each element

of in G/K.7. The group L associated with the fuzzy ordered pairs is called the rough fuzzy

group, denoted by (G,*,,f,F).8. return

This algorithm is illustrated by using the following example.3.4 Example: Consider a group G={1,,2,3,4,5} where is the sixth root of unity, with the binary operation*, which is defined as below:

* 1 2 3 4 5

1 1 2 3 4 5

2 3 4 5 12 2 3 4 5 1 3 3 4 5 1 2

4 4 5 1 2 3

5 5 1 2 3 4


32

Here, ‘1’ acts as the identity. Let f be any epimorphism defined from G onto = 0,1,2

with

kernel K. It is given that G/K={{1, 3},{2, 5}, {,4}}. By axiom of choice, denote the sets {1, 3},{2, 5} and {,4} by [1],[2] and [] respectively. Suppose that the isomorphism

:G/K is given as ([1])= 0

; ([])= 2

and ([2])= 1

. Let F be any fuzzy subset of Z, which give the approximations for F be (0.3,0.4,0.2) and (0.7,0.6,0.6) respectively. Then the set of closed fuzzy pairs of is given by ((0.3,0.7),(0.4,0.6),(0.2,0.6)).Now, by considering the preimage of , for each element of G/K, the closed ordered pair can be associated. Thus, the rough fuzzy group can be obtained.

4. CONCLUSION

In this paper, the work is initiated to bring out useful approaches on rough fuzzy sets algebraically,

Acknowledgements

The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Organizing Secretary) and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for inviting me to present a talk in the Seminar on 25th October 2011 (the 200 Birth day of Galois).

REFERENCES

[1] Biswas R. ‘On Rough Fuzzy Sets ’, Bulletin of the Polish Academy of Sciences, Mathematics; Vol 42(1994); No: 4, 352-355

[2] Dubois D, Prade H, ‘Rough fuzzy Sets and Fuzzy Rough Sets’, International Journal of General Systems, 17, pp 191-209, 1989

[3] G.Ganesan, C.Raghavendra Rao, ‘Rough Fuzzy Groups’, Indian Journal of Mathematics and Mathematical Sciences, Vol I, pp:1-8, 2005

[4] Michael Artin, “Algebra”, Prentice-Hall of India Pvt. Ltd., 1996[5] Z.Pawlak, ‘Rough Sets’, International Journal of Computer and Information Sciences, 11,

pp:341-356, 1982

MATHEMATICS IS AN INDEPENDENT WORLD CREATED OUT OF PURE

INTELLIGENCE


37

An Introduction to new AlgebraicStructure Gamma Near-ring

Dr. Kuncham Syam Prasad,

Manipal University, Manipal, Karnataka, India.

Near-rings are one of the generalized structures of rings. Substantial work on near-rings

related to group theory and ring theory was studied by Zassenhaus and Wielandt in 1930. World

War-II interrupted the study of near-rings, but in 1950s, the research of near-ring redeveloped by

Wielandt, Frohlich, and Blackett. Since then, work in this area has grown and was diversified

to include applications to projective geometry, groups with near-ring operators, automata theory,

combinatorial geometry, formal language theory, nonlinear interpolation theory, optimization

theory. A generalization of both the concepts Near-ring and the ring, namely gamma near-ring

was introduced by Satyanarayana Bhavanari (in the year 1984).

Let M be a additive group (not necessarily abelian) and a non-empty set (sometimes

called the set of operators). Then M is said to be a-Nearring if there exists a mapping M

MM (the image of (a, , b) is denoted by ab), satisfying the conditions:(i) (a + b)c =

ac + bc; and (ii) (ab)c = a(bc) for all a, b,c M and , . A natural example of

gamma near-ring is obtained in by taking an additive group (G, +) and a non-empty set X. Let

M be the set of all mappings from X into G. Then M is a group under point-wise addition.Let

be the set of all mappings of G into X. For all mappings f1, f2 M and g , we have f1gf2 M.

Further, it is clear that for all f1, f2, f3 M and g1, g2, i) (f1gf2)g2f3 = f1g1(f2g2f3); andii)

(f1+f2)g1f3 = f1g1f3 + f2g1f3.But f1g1(f2 +f3) need not be equal to f1g1f2 + f1g1f3. To see this, fix 0

z G and u X. Define Gu: G X by gu(x) = u for all x G and fz:X G by fz(x) = z for

all x X. Now for any two elements f2, f3 M, we have [fzgu(f2+ f3)](x) = fz[gu(f2(x) + f3(x))] =

fz(u) = z and[fzguf2 + fzguf3](x) = fzguf2(x) + fzguf3(x) = fz(u) + fz(u) = z + z. Since z 0, we have

z z + z and hence fzgu(f2+ f3) fzguf2 + fzguf3. Therefore the left distributive law fails here. It


38is obvious that the conditions: (f1gf2)g2f3 = f1g1(f2g2f3); and are true. Hence M is a right gamma

near-ring.

Gamma near-ring theory has been developedin many directions by many scholars and has

evoked great interest among mathematiciansworking in different fields of algebraic systems.

There have been wide-ranging applicationsof the theory of associate algebras, sequential

mechanics, and formal languages.

Some of the researchers working in the field of Gamma Nearrings are:

Bhavanari Satyanarayana, (Introduced the concept in the year 1984), India

Dr G L Booth (studied the radical theory), 1987, South Africa

Dr Groenewald Nico (Studied the equiprime ideals), 1991, South Africa

Dr Veldsman Stefan (Studied the special radicals), 1994, Oman

Dr Godloza (Studied Special radicals), 1996, South Africa

Dr Kuncham Syam Prasad (Studied the fuzzy aspects), 2000, India

Dr Pradeep Kumar Tumurukota (Studied prime ideals), 2004, India

Dr VijayaKumari A. (Studied the fuzzy ideals of gamma near-ring module), 2009, India

Dr Yong Bae Jun (Studied the fuzzy aspects), 2001, South Korea

Dr Cho Yong Uk (Studied gamma derivations), 2003, South Korea

Dr Kim Kyung Ho (Studied gamma derivations), 2003, South Korea

Dr Ozturk (Studied the fuzzy aspects of gamma near ring ideals), 2001, Turkey

Dr Tamizh Chelvam (Studied generalized gamma near-fields), 2002, India

Dr N. Meenakumari (Studied generalized gamma near-fields), 2002, India

Dr M Shabir (Studied idempotent gamma near-rings), 1999, India

Dr Datta T K (Studied semiprime and irreducible ideals), 2003, India

Dr Om Prakash (Studied Ideals of gamma near-rings), 2009, India

A new Algebraic Structure Gamma Near-ring was introduced by Bhavanari Satyanarayana in 1984.

Proceedings of the National Seminar on ALGEBRA (200th Birth Anniversary Celebrations of Evariste GALOIS), KBN College, Vijayawada, A.P., (October 25, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana,

Mr. VVN Suresh Kumar and Mr Mohiddin Shaw Shaik.) 39

FUZZY IDEALS OF GAMMA NEAR RINGS------------------------------------------------------------------------------------------------------------

INTORDUCTION

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical computer science, controlengineering, information sciences, coding theory, Net Work Security, etc. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader frame work of fuzzy setting.

A near-ring is an algebraic system that satisfy all the axioms of an associative ring, except posibly the commutativity of addition and one of the two distributive laws. A comprehensive reference of the theory of near-rings appears in Pilz [1983].

The concept of Gamma Near-ring was introduced by Satyanarayana [1984], which is a generalization of the concept near-ring and a gamma ring. Later this concept was attracted many mathematicians from different parts of the world, in doing their research.

The concep “fuzzy set” which was introduced by Zadeh [1967] is applied to many mathematical branches. Rosenfeld [1] inspired the fuzzification of algebraic structures and introduced the notion of fuzzy subgroups.

Abou-Zaid [1] introduced the notion of fuzzy sub-nearring and studied the theory of fuzzy ideals in near-rings. The authors Satyanarayana and Syam Prasad [1, 2] considered fuzzy notion of a gamma near-ring, and obtained fuzzy cosets, prime ideals, and some related results.

A new type of fuzzy subgroup (viz, (∈, ∈q)–subgroup) which was an important generalization of Rosenfeld’s fuzzy subgroup, was introduced by Bhakat and Das [1] by using the combined notions ‘belongingness’ and ‘quasicoincidence’ of fuzzy points and fuzzy sets.

Our aim of this paper is to introduce and study the new sort of fuzzy subnearring/

ideal of a gamma near-ring called (viz, ((∈,∈q) – sub nea-rring), (∈,∈q) – ideal) of a gamma near-ring.

We recall the following definition form Pilz [1]. A non-empty set N with two binary operations + and is called a near-ring if (1) (N, -) is a group (not necessarily abelian)(2) (N, ) is a semigroup.(3) (a+b)c = a b + b c for all a, b, c N.

Survey Article--------------------------------

Presenter: Venkata Subba Rao Gunda

Amalapuram, A.P



2. Gamma Near Rings

2.1 Definition (Satyanarayana (1984)): Let (M, +) be a group (not necessarily Abelian) and be a non empty set. Then M is said to a -near-ring if there exists a mapping M M M (the image of (a, , b) is denoted by ab) satisfying the following conditions:

(i) (a + b)c = ac + b c; (ii) (ab)c = a(bc) for all a, b, c M and , .

2.2 Example (Satyanarayana (1984)): For a natural example, let us consider a group (G, +) and a non-empty set X. Write M = {f: X G} and = {g: G X}. With respect to the pointwise addition, (M, +) becomes a group. At this point, it is easy to verify that, if (G, +) is non-Abelian then so is (M, +) . Also (M, +) becomes a -near-ring.

2.3 Definitions (Satyanarayana (1984)): Let M be a -near-ring. Then a normal subgroup I of (M, +) is called (i) a left ideal if a (b + i) - ab I for all a, b M, , iI ; (ii) a right ideal if ia I for all a M, , iI ; and (iii) an ideal if it is both left and right ideal. M is said to be a zero- symmetric -near-ring if a0 = 0 for all a M and , where 0 is the additive identity in M.

2.4 Notation: For any two subsets A, B of M the set { ab | aA, , bB} is denoted by either AB or AB. {xA| xB} is denoted by A/B. For any subset X of M, the smallest ideal containing X is denoted by <X>. If X = {a} then <X> is denoted by <a>.

3. ( , q) – Fuzzy Ideals of a Gamma Near rings

3.1 Definition: Let M be a gamma near ring and be a fuzzysubset of M. We say that a fuzzy sub near-ring of M if for all x, y M, (i) (x-y) (x) (y); (ii) (xy) (x) (y). is called a fuzzy ideal of M if is a fuzzy sub near-ring of M and (iii) (y + x - y) (x); (iv) (xy) (x) and (v) (x(y+ a) - xy) (a).

3.2 Definition: A fuzzy set of M of the form

xyif0

xyif)0(t)y( is said to be

a fuzzy point with support x and value t and it is denoted by xt. A fuzzy point xt belong to (respectively be quasi-coincident with ) a fuzzy set , written as xt (respectively xtq) if (x) t (respectively (x) + t 1).

3.3 Proposition (Syam Prasad, Satyanarayana, and Venkata Subba Rao (2010)):For all x, y, a M (where M is a gamma near-ring), we have (i) (x-y) (x) (y) 0.5 if and if and only if (xt, ys implies (x - y) min{t,s} q) (ii) (xy) (x) (y) 0.5 if and if and only if (xt, ys and implies (xy) min{t,s} q). (iii) (y + x - y) (x) 0.5 if and if and only if (xt implies (y + x - y)t q),(iv) ( xy) (x) 0.5 if and if and only if (xt implies (y + x - y)t q),



(v) (x(y+ a) - xy) (a) 0.5 if and if and only if (at implies x(y+ a) - xy)t q).

Acknowledgements

The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this survey article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois).

ReferencesBhakat S.K.., and Das P., [1] “(( , q) – Fuzzy subgroup”, Fuzzy sets and Systems, 80 (1996) 359 – 368.Booth G.L[1] .“A note on -Near-rings”,Stud.Sci.Math.Hunger 23 (1988) 471-475.Booth G.L., Groenewald N.J. [1] “Equiprime Gamma Nearring”, quaestiones Mathematicae, 14(1991) 411-417.Jun Y.B.., Spanci M., and Ozturk M.A.,[1] “Fuzzy Ideals of Gamma Nearrings”, Turkish Journal of Mathematics, 22 (1998) 449- 459.Pilz G., [1] Near-rings, North-Holland pub., 1983.Rosenfeld A., [1] “Fuzzy Groups” Journal of Math Anal Appl, 35 (1971) 512-517.Salah Abou-Zaid,[1] “On Fuzzy Subnearrings and Ideals”, Fuzzy Sets and Systems, 44 (1991) 139-146.Satyanarayana Bhavanari.[1] "A Note on -rings", Proceedings of the Japan Academy 59-A(1983) 382-83.[2] “Contributions to Near-ring Theory”, Doctoral Dissertation, Acharya Nagarjuna University, 1984. Published by VDM Verlag Dr Muller, Germany, 2010 (ISBN: 978-3-639-22417-7).[3] “The f-Prime Radical in Gamma Nearring”, South East. Bull.Math. (1999)23: 507-511.[4]“A Note on -near-rings”, Indian J. Mathematics (B.N. Prasad Birth Centenary commemoration volume) 41(1999) 427-433.[5] “A Note on Completely Semi-Prime ideals in Near- Rings”, International Journal of Computational Mathematical Ideas Vol. , No 3(2009) 107 – 112.Satyanarayana Bh and Syam Prasad K[1] “Fuzzy Cosets pf Gamma Nearrings”, Turkish Journal of Mathematics, 29 (2005)11-22.

[2] Discrete Mathematics with Graph Theory (for B.Tech/B.Sc/M.Sc.,(Maths)) Printice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).Syam Prasad Kuncham & Satyanarayana Bhavanari [1] “Dimension of N-groups and Fuzzy ideals in Gamma Near-rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3- 639-30558-624851-7).Zadeh L.A. [1] "Fuzzy Sets", Information and Control, 8(1965) 338-353.



SOME NUMERICAL METHODS FOR SOLVING NAVIER-STOKES EQUATIONS-----------------------------------------------------------------------------------------------------------------ABSTRACT

The main object of this talk is to explain Navier-Stokes equations and difficulties involved while seeking solution for these equations. It has been explained briefly how to overcome the difficulties associated with the solution of these equations. We are giving an outline of the important methods available for solving the Navier-Stokes equations. In particular emphasis has been given to finite difference; finite volume and finite element methods. Alsomethod of solution to obtain the unknown variables of the Navier-Stokes equations is given. Some applications have been given and explained briefly.

1. Navier-Stokes Equations Navier-Stokes Equations named after Claude-Louis and George Gabriel Stokes,

describe the motion of fluid substances. These equations arise from applying the Newton’s second law to fluid motion together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. TheNavier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics.The Navier-Stokes Equations along with the continuity equation in 3-Dims are

2

2

2

2

2

21

z

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

u

(1)

2

2

2

2

2

21

z

v

y

v

x

v

y

p

z

vw

y

vv

x

vu

t

v

(2)

2

2

2

2

2

21

z

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

w

(3)

0

z

w

y

v

x

u(4)

2. Difficulties involved in the Navier-Stokes equations

(i) Non-linearity: Equations (1)-(3) are nonlinear in nature due to the convective parts.

Invited Talk------------------------------------

Dr. V. AmbethkarDepartment of Mathematics

University of Delhi



(ii) Pressure gradient: The main hurdle to overcome in the calculation of velocity field is the unknown pressure field. As long as there is a correct pressure field, there is no difficulty in solving the momentum equation. So our challenging task is to find the correct pressure field.(iii)The pressure filed is indirectly linked with the continuity equation (4).When the correct pressure field is plugged into the momentum equations the resulting velocity field satisfies the continuity equation.

3. Remedy- Numerical Methods: Over view

. Finite difference method ;Finite volume or Control Volume method

. Finite element method;Stream function-vorticity method;Spectral methods

4. Finite Difference Method:• This is the oldest method of the first three. Techniques published as early as 1910 by L.

F. Richardson. The first ever numerical solution by using this was the flow past a circular cylinder by Thom (1933). Advantage is easy to implement. Disadvantage is restricted tosimple grids and does not conserve momentum, energy, and mass on coarse grids.

• Finite difference: basic methodology

•

• e

domain

is

• Discretized into a series of grid points.• A “structured” (ijk) mesh is required.• The governing equations (in differential form) are discretized (converted to algebraic

form).• First and second derivatives are approximated by truncated Taylor series expansions.• The resulting set of linear algebraic equations is solved either iteratively or

simultaneously.• Draw backs: This method best suits for uniform grids and hence a regular computational

domains. This method fails to apply for complex geometries.



5. Finite volume method (FVM): This method is a generalization of the finite difference method but use the integral form of governing equations of flow, rather than their differential form. This has an advantage of handling complex geometries.

• Finite volume: basic methodology:

.Divide the domain into control volumes. Using finite volume method, the solution domain

is subdivided into a finite number of small control volumes (cells) by a grid. Integrate the differential equation over the control volume and apply the divergence theorem.• To evaluate derivative terms, values at the control volume faces are needed: have to make

an assumption about how the value varies.• Result is a set of linear algebraic equations: one for each control volume.• Solve iteratively or simultaneously using TDMA, SOR etc.• Initial or Boundary Conditions: Initial condition involves knowing the state of pressure

(p) and initial velocity (u) and v at given points in the flow. • Boundary conditions such as walls, inlets and outlets largely specify what the solution

will be.• Algorithms Used while finding pressure and velocities:

• SIMPLE,SIMPLER,SIMPLEC,PISO.

6. Finite element method (FEM):

• Earliest use was by Courant (1943) for solving a torsion problem.• Clough (1960) gave the method its name.• Method was refined greatly in the 60’s and 70’s, mostly for analyzing structural

mechanics problem.• FEM analysis of fluid flow was developed in the mid- to late 70’s.• Advantages: highest accuracy on coarse grids. Excellent for diffusion dominated

problems (viscous flow) and viscous, free surface problems.• Disadvantages: slow for large problems and not well suited for turbulent flow.7. Stream function-Vorticity Method.In this method, pressure gradient is eliminated from equations (1) and (2) by differentiating w.r.t y and x and subtract. This method is applicable for two- dimensional flow only. Then using the definitions of stream function and vorticity equation. We finally get

2 and 2Dt

D(5)



Which can be solved by finite difference method with appropriate boundary conditions as explained above.

8. CONCLUSIONSWhatever may be the method used, we will be able to get finally the numerical computations for the flow variables in a given physical domain like velocity and pressure for incompressible case and density even for case of unsteady compressible case. Once these flow variables and its numerical values available, then we can discuss the several situations associated with the given physical problem and accordingly we can give the conclusions based on the given data.

APPLICATIONS:

1. Transient Free Convection from a Heated Vertical Plate: The governing equations for

semi-infinite heated vertical plate in infinite fluid that is initially at temperature T and at rest.

0

y

v

x

u( Continuity equation)

TTgy

u

y

uv

x

uu

t

u 2

2

(x-Momentum equation)

2

2

y

T

y

Tv

x

Tu

t

T (Energy equation)

Method of Solution: By using finite difference approach, we can finally get the values of u,v,T.2. Development of a fully coupled control-volume finite element method for the incompressible Navier–Stokes equations .Method of solution: By using the finite volume method, the given Navier-Stokes equation have been solved (Ref 5). Similar applications available for other methods used to solve Navier-stokes equations.

Acknowledgements

The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Organizing Secretary) and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for inviting me to present a talk in the Seminar on 25th October 2011 (the 200 Birth day of Galois).

REFERENCES

1. Computer Simulation of Flow and Heat Transfer, Tata McGra Hill,1998, New Delhi,India.

2. K.Muralidhar and T.Sundarajan, Narosa Pub.House, 2003,New delhi,India.



3. John C. Tannehill, Dale A.Anderson, Richard H.Pletcher, Computational Fluid Mechanics and Heat Transfer,Taylor and Francis,1997, USA.

4. John D. Anderson,Jr., Computational fluid dynamics :The basics and Applications, McGraw-Hill,1995,USA.

5. Idriss Ammara and Christian Masson,Development of a fully coupled control-volume finite elementmethod for the incompressible Navier–Stokes equations, Int. J. Numer. Meth. Fluids 2004; 44:621–644.

------------------------------------------------------------------------- Analysis of Two-Phase N-policy M/M/1 Queueing System with Secong Optional Service and Server Authors: K. Chandan, K.Satish Kumar and B. Basaveswara RaoDepartment of Statistics, Acharya Nagajuna University, Guntur Dist)Andhra Pradesh, India. kotagirichandan@gmail.com.------------------------------------------------------------------------------------------------------------ Abstract : This paper deals with an analysis of single server N-policy two phase M/M/1 queueing system with second optional service and server startup. The arrival occurs individually according to a Poisson process and the service times follow an exponential distribution. The customers receive individual service in first phase and proceed to the optional batch service in second phase. The server is turned off each time the system empties, as and when the queue length reaches or exceeds N (threshold), the server immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing first phase (individual) service. The distribution of the system size is derived through probability generating function and obtained other system characteristics. The sensitivity analysis has been carried out to examine the effect of values of the parameters in the system.

AcknowledgementsThe author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this survey article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois).

Mathematics – the unshaken Foundation of Sciences, and the plentiful Fountain of

Advantage to human affairs ---- Issac Barrow

Oral Prasentation-------------------------------



GOLDEN RATIOAND HUMAN BODY

Author: Satyasri Bhavanari Zhejiang University, Hangzhou, Republic of China

In mathematics and the arts, we say that two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

It is also known as the divine proportion.

The golden ratio is an irrational mathematical constant, approximately 1.6180339887. The

golden ratio is denoted by the Greek lowercase letter phi ( ) , while its reciprocal, or , is denoted by the uppercase variant Phi( ).

Section-1: Some Natural Examples:

1.1. GOLDEN RECTANGLE: Suppose the rectangle is divided into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle, in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us:

a/b = (a+b)/a

Survey Article----------------------------------------------------

Presenter: Satyasri BhavanariMBBS V Yr,

Zhejiang University, Hangzhou,

Republic of China

“Virtue is the knowledge of goodness” “Sin is the ignorance of goodness”.



1.2 Many buildings and works of art have the Golden Ratio in them,

such as the Parthenon in Greece.

1.2. FIBONACCI NUMBERS: There is a close relationship between the golden ratio and the fibonacci numbers. The Fibonacci Sequence is the series of numbers. Discovered by Leonardo Fibonacci. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...The characteristic of these numbers is each number is formed by the sum of preceding two numbers. The Rule is xn = xn-1 + xn-2.

If you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio. In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

1.3. Golden ratio is exhibited by Egypt pyramids, Leonardo DaVinci’s portrait of Monalisa, Sunflower, the snail. A book ‘Universal Principles of design’ also approximates the golden ratio when it is opened.

1.4. Honeybees: A honeybee colony consists of a queen, a few drones and lots of workers. The female bees (queens and workers) all have two parents, a drone and a queen. Drones, on the other hand, hatch from unfertilized eggs. This means they have only one parent. Therefore, Fibonacci numbers express a drone's family tree in that he has one parent, two grandparents, three great-grandparents and so forth. 1.5. The APPLE company’s IPod used the golden ratio.

Section-2: GOLDEN RATIO IN HUMAN BODYMost of our human body parts follow the numbers one, two, three and five. A human being

has one nose, two eyes, three segments to each limb and five fingers on each hand.

2.1. Our fingers have three sections. The proportion of the first two to the full length of the finger gives the golden ratio (with the exception of the thumbs).

2.2. We can also see that the proportion of the middle finger to the little finger is also a golden ratio.



2.3. We have two hands, and the fingers on them consist of three sections. There are fivefingers on each hand, and only eight of these are articulated according to the golden number: 2, 3, 5, and 8 fit the Fibonacci numbers.

2.4. The DNA molecule in which all the physical features of living beings are stored, consists of two intertwined perpendicular helices. The length of the curve in each of the helices is 34Amstrong and the Width is 21 ang. 1 angstrom= 100millionth of a centimeter. 21 and 34 are two consecutive fibronacci numbers.

Golden ratio applies to idealized human body which scientists and artists agree. The proportions and measurements of the human body can also be divided up in terms of the golden ratio.

2.5. The important example of the golden ratio in the average human body is that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618.

2.6. Some other golden proportions in the average human body are:(i). (The distance between the finger tip and the elbow) / (distance between the wrist and the elbow).(ii). (The distance between the shoulder line and the top of the head) / (head length).(iii). (The distance between the navel and the top of the head) / (the distance between the shoulder line and the top of the head).(iv). (The distance between the navel and knee) / (distance between the knee and the end of the foot).(v). (The total width of the two front teeth on the upper jaw)/(by their length).(vi). (The width of the first tooth and the second tooth of the upper jaw) /(the width of the first tooth of the upper jaw).(vii). (The length of face)/(the width of face).(viii). (The distance between the lips and where the eyebrows meet)/(the length of nose).

(ix). (The length of face)/(the distance between the jaw and where the eyebrows meet). (x). (The length of mouth)/(the width of nose).

(xi). (The distance between the eyes)/(the distance between eyebrows).

Not only the outer parts of the body, but some of the inner structures of the body also follow the golden ratio.

2.7. Example: In a study carried out between 1985 and 1987, the American physicist Bruce West and professor of medicine Aure Gold Burger revealed the existence of golden ratio in the lung. One feature of bronchus that constitute the lung, is that they are not of equal length. The windpipe divides into two unequal bronchi, one long on the left and the other short on the right. It was determined that the proportion of the long bronchus to the short bronchus was 1.618.Acknowledgements : The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this survey article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois). References: Some websites in the net through google search engine.


50

A PROSPECTIVE THEME OF MATHEMATICS

Prof. Dr K.Satyanarayana (Chairman, BOS. Dept. of Telugu &Sanskrit); and

Prof. Dr Bhavanari Satyanarayana (Head, Department of Mathematics), Acharya Nagarjuna University Nagarjuna Nagar-522510 (A.P.)

Sanskrit has contributed a lot in the field of Script, Language, Literature,Religion, Philosophy, Mythology, Cosmology, Astronomy, Ayurveda, Jurisprudence,Art, Architecture and Inscriptions etc. The Vedas are the store house of knowledge and a treasure of ancient and modern Mathematics. The famous book ‘Vedanga Jyothisha’ explains the importance of mathematics in the following beautiful verse:

“Yatha sikha Mayuranam Naganam manayo yathaTadvat vedanga sasthranam Ganitam murdhani sthitam”

Similar to a plumage on the head of a peacock, Gems on the head of a serpent, the subject: mathematics occupying a prominent place on the top of all vedangasastras.

The word Ganita derived from the Sanskrit root ‘Gan samkhyane’ which indicates Numeration. This is undoubtedly emphasizes a great contribution ofmathematics to every field. The familiar great scientist: Albert Einstein said “we owe to Indians who taught us how to count, without which no worthwhile Scientific discovery could have been made”. This sentence makes us understand the intellectual caliber of Indian Mathematicians of the past.

Now let us examine some of the great Sanskrit scholars who have contributed for the development of mathematics. Mahaveeracharya, a great scholar in sanskritexplains the importance of mathematics in the following beautiful stanzas in his familiar work ‘Ganitasara samgraha’

“Lowkike vaidike vapi tatha samayikepi yah Vyaparastatra sarvatra samkhyanamupayujyate Bahunirvipralapaih kimsthrilokye sacaracare Yatkincitvastu tatsarvam Ganitena vina nahi”

The above stanzas are explaining that all the applications in this world are much inter linked with mathematics. To acquire good knowledge in spiritual and secular things Mathematics is the main source and base. The ten numerals such as Sunya, Eka, Dwi, Thri, Catur, Pancha, shat, Sapta, Asta, Nava are well explained in Yajur Veda 17th chapter. These are used in making highest numbers such as Sata (102

= hundred)), Sahasra (103 = Thousand), Ayuta (104 = ten thousand), Niyuta (105 = hundred thousand), Paryuta (106 = million), Arbuda (107 = ten million), Nyarbuda(108 = hundred million), Samudra (109 = Billion), Madhya (1010 = ten Billion), Anta(1011 = hundred Billion) and Parartha (1012 = One Trillion). According to ‘Lalitha


51 Vistara’ the book of Buddihists, the biggest number Tallakshna ( that is1053). In the work of Jaina ‘Anuyogadwara’ the biggest number is 10140. Zero is well discussed in Vedas, and Puranas. The santhi Manthra of Esa-vasyopanishad explains the importance of zero in this verse ‘Om purnamadah purnamidam purnat purnamudacyate

Purnasya purnamadaya Purnamevaava sishyate’.

Thus the importance of zero is well explained in Sanskrit by quoting varioussecular usages. There are many expressions which depict the numbers very well. For example, to explain zero and one in ancient period, they used to mention Sky and Moon or the Earth respectively. To denote the number two they used eyes, hands,bubbles or ears which are two in number. In this way Sanskrit is bearing full of beautiful numerical expressions which denote the numbers in a clear way. Hence, we may proud to say that India is the base for the Numerology or Numerical Education.

Sanskrit explains the eight fold path which is used in mathematics asSamkalanam (Addition), vyavakalanam (Subtraction), Gunanam (Multiplication), Bhaagah (Division), Vargah (Root), Vargamulam (Square Root), Ghanam (Cube) and Ghanamulam (Cube Root). The salient features of Square root and cube root are well explained by the great Indian Sanskrit scholars like: Arya Bhatta and Brahma Gupta. In Taittariya samhita; Brahmanas, Samantara srenis and Gunottara srenis these number concepts are well narrated/presented.

The great Scholar Prof. Macdonnel said that “the invention of Algebra and Astronomy by Indians” occupied the firm place in this world. Algebra has its popularity not only in Arabian countries but also in European countries from this great India. The eminent scholars Arya Bhatta, Apastamba, Boadhayana, Katyayana, Bramha Gupta and Bhaskaracharya rendered a good service in popularizing Algebrain this Universe. Later course of time, the subject Algebra occupied a prominent place in china and Japan.

In Vedic-samhitas Arithmetics, Algebra and Geometry etc, are well explained with the help of easy formumulae. Many applications are found in ‘Bouddhayana Sulabha Sutras’. Thus we may say that India is far a-head in the field of Geometry, Algebra and Arithmetic from the ancient period. The “Vedic Mathematics” written by well versed scholars providing so many easy solutions for the problems in the field of Modern Mathematics. The progress of Mathematics isfrom B.C period first century in India. The familiar work “Aryabhateeyam’’ was notified as the first and familiar work in Mathematics and Astrology written by the well versed Indian scholar Aryabhatta who born in 476 A..D. This familiar work “Aryabhateeyam’’ pocessing four padas (namely Dasagitika pada, Ganita pada,Kalakriya pada and Golapada). In the “Ganita pada” the topics: Ankaganitam,Bijaganitam and Rekhaganitam were described with many formulas in an easy manner. This is also bearing the introduction of the numbers, root, square root, cube, cube root, etc.. The formulas for the construction of triangle, square and circles are narrated.. The famous book ‘Maha siddantham’ written by AryaBhatta describesfamous astrological Mathematics of those days. He mentions in a stanza ‘Bhugolah Sarvathovruthih’ (‘Arya Bhateeya’ in the sixth verse of Golapada) that the earthrotates round the Sun. He also mentioned that the rotation of the earth around the sun takes place 23 hours 56 minutes and 4.1 seconds. The value of ‘Pi’ (in chapter-2 of


52 Ganitha Pada in Arya Bhateeya) is well approximated as 3.1416. The area of triangle is said as “Tribhujasya phalasariram samatalakoti bhujartha samvargah”, which

means 1

2 b h.

The great scholar Varaha Mihira (in 505 A.D) one of the nine Gems of Vikramaditya’s court explained in his work ‘Pancha Siddhanthika that “The earth with five elements is appearing like a ball in a cage” and hence, this sentence states the earth is round.

Zero was not the brain child of the western world. But, as the product of Indian Mathematician (called Brahma Gupta (598-670 A.D) who was born in Bhinmal of Gujarath, India) there are two familiar books ‘Brahma sphuta siddntha(628 A.D)’ and ‘Kahandakhadyaka (665 A.D)’. The first one is having 24 chapters and 1008 verses. In this great work 12th chapter explains Anka ganitam,Kshetramiti. In 18th chapter algebra was discussed. The author mentioned that if we divide any number with zero then the remainder is infinity (in the present days, dividing any number with zero is undefined one). These two great treatises are translated in to Arabic by the scholars, in later course of time.

Also the well versed Mathematician “Sridharacharya: contributed a lot for mathematics by his familiar books “Trisati and Ganithsarah” He has written many formulas to know the ratio, interest, simple and compound interest, time and distance. He attained the name and fame in the field of Mathematics during his times.

In 1200 A.D., the book Lilavathi was written by Bhaskaracarya. Sripati (who was an expert in Astrology) wrote two famous books ‘Ganita Tilaka’ and ‘Bijaganita’, the first one contains 125 verses that narrate various questions. A famous commentary of Jaina Mathematician “Simha tilaka suri” is available to this great work. `

The famous mathematician: Mahaveeracharya was born in the first half of the 9th century A.D,. He is the author of ’Ganita sara samgraha’ that explains the methods of Indian mathematical system with Arithmetic, Geometry. Many solutions were given for the problems raised by the mathematicians earlier to him. The importance of borrowing numbers and short techniques of multiplications, the importance of L.C.M, zero and square are well discussed.

Bhaskaracharya-II (1114 A.D.) attained the name and fame as a great Mathematician and an eminent Astrologer. He wrote many books such as Siddhanta siromani, Bijaganitham, Lilavati, Karanakuthuhala, Muhurthapatala and Vivahapatala. Lilavati, Bijaganita (that are related to Mathematics) and some other books to Astrology. The satellite which was launched in to the space on 1979 June 7 th (named as Bhaskara – II) indicates a good reputation to this great Mathematician. Lilavathi is bearing 278 verses which narrates Arithmetic with suitable examples.

Ganita-koumudi and Bijaganitavatamsa are the two famous works written by an eminent scholar born in (14th century). They narrate Arithmatic and Algebra in a


53 simple style. Muniswara (1700 A.D), wrote two books entitled: Pati Ganita and a good commentary on Lilavati. His famous work Pati sarah enlightens the Arithmetic problems.

In the modern days a Sanskrit Mathematician Bapu Deva Sastri (1821 A.D.) wrote many books on Rekha ganitam, Trikonamiti, Mapanavadastatvaviveka pariksha, Ankaganitam. In 1860 A.D. Pandit Sudhakar Dwivedi an eminent scholar in (not only in Astronomy but also in Mathematics) wrote number of Books, best commentaries on Mathematics such as Goliya Rekha ganitam, Ganaka Tarangini, Lilavatyah sopapatti teeka, Bhaskariya Bija Ganita Teeka, Panca Siddhantika Prakasah and Arya Bhateeya Maha Siddhantasya Teeka.

By the above facts, and discussions made, one may conclude that the Sanskrit Scholars of our India, invented and propagated several fundamental conceptsof mathematics, particularly in the field of Arithmetic, Algebra, Astronomy and Geometry. Now it is possible for the Modern Mathematicians to dig out the mathematics of earlier Indian Sanskrit scholars to get the vast information stored in Sasthras. This may be useful for the present as well as future generations. The government may come forward to provide funding to such useful projects. Let us hope for the good of the future times.

References[1] Aryabhteeya (commentary by Sankarashukla , Indian National Science Academy, Delhi, 1976.[2] Samskrutasahitye Vignanam, Delhi Sanskrit Academy, New Delhi, 2000.[3] Some information browsed from the inter net. -------------------------------------------------------------------------------------------------------

A Note On 2-Quasi-total GraphsAuthors: Satyanarayana Bhavanari & D SrinivasuluDepartment of Mathematics, A.N.U, Nagarjuna Nagar-522 510, A.P., India.Department of Mathematics,NRI Institute of Technology, Agiripalli, Krishna(Dt.), A.P., India.-------------------------------------------------------------------------------------------------------Abstract: In this paper we considered Q2(G) (2-quasitotal graph) and it is proved that every triangle in Q2(G) contains an edge of G. Also proved that for a graph G the following conditions are equivalent: (i) )(GE =1, (ii) Q2(G) contains unique triangle;

Key Words: Total graph,1-quasi-total graphsAMS(2000)Subject Classification: 05C30,05C75.References:BONDY.J.A & MURTY U.S.R.[1]. “Graph Theory with Applications”, The macmillan press Ltd, (1976).SATYANARAYANA BH.&SYAMPRASAD K.[1]. “An Isomorphism Theorem on Directed Hypercubes of Dimension n”, Indian J.Pure&Appl.math34(10)(2003)1453-1457.[2]. “Discrete Mathematics &Graph Theory”, printice Hall of India Pvt. Ltd., New Delhi,2009.

Oral Presentation-------------------------

D. Srinivasulu



Some results on Fuzzy Ideals of M - Modules.---------------------------------------------------------------------------------------------------------------------Abstract: We consider Zero symmetric –near rings only. We collect some results related to relations between N-groups and M –Modules, definition of Fuzzy Ideal of M -Modules, some related fundamental results. Ackonwledgements: The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois). ---------------------------------------------------------------------------------------------------------------------

EFFICEINT USING GRAPHS IN COMPUTER SCIENCES.Authors: Mrs. S. Latha & J. PraveenaSt. Mary’s Group of Institutions, Chebrolu, Guntur A.P, India.Gudlavalleru Engineering College, Gudlavalleru, Krishna Dt., A.P., India.---------------------------------------------------------------------------------------------------------------------Graphs can be used to model many types of relations and process dynamics in computer science, physical, biological and social systems. Many problems of practical interest can be represented by graphs. Thegeneral theory of graphs has a wide range of applications in diverse fields. This paper explores different elements involved in graph theory including graph representations using computer systems and graph-theoretic data structures such as list structure and matrix structure. The emphasis of this paper is on graph applications in computer science.

---------------------------------------------------------------------------------------------------------------------

EFFICEINT USING GRAPHS IN COMPUTER SCIENCES

Authors: Mr. Y. Sankara Rao & Mrs. A. Latha St.Mary’s Group of Institutions, Chebrolu, Guntur A.P, India.--------------------------------------------------------------------------------------------------------------------- Mathematics plays a vital role in every walk of human life and in every system. You can find no system in which no mathematics were involved. In computer science, we use mathematics (algebra, in particular) for calculating the route to transmit the packets in fast and efficient manner using shortest routing methods. Routing is the process of selecting paths in a network along which we send the packets in a network traffic. Routing is performed in every kinds of network, including the telephone network (Circuit switching technique), electronic data networks (such as the Internet), and transportation networks. In this present work, a generalized algorithm has been developed to find the shortest path in a transmission line network. A network can be represented in the form of a graph to understand the network in a better way.

This algorithm has the ability to traverse a packet from a complex network with any complexsubstructures, with any number of nodes/edges. This algorithm has been developed to provide a route to the packet to traverse from the source node to the designation node. These algorithms are heuristic because they perform well only on some particular class of graphs. While their performance has been good in experimental studies, no theoretical bounds are known to support the experimental observations. Most of these algorithms have been motivated by finding paths in large road networks. Dijikstras’s algorithm is used to find the closest node from the source node.

Oral Presentation-----------------------------M. Babu Prasad

Oral Presentation------------------------------

J. Praveena

Oral Presentation-----------------------------

Mr. Y. Sankara Rao

Proceedings of the National Seminar on ALGEBRA College, Vijayawada, A.P., (October

and Mr Mohiddin Shaw Shaik.)

Proceedings of the National Seminar on ALGEBRA (200th Birth Anniversary Celebrations of Evariste GALOIS)October 25, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana

55

Birth Anniversary Celebrations of Evariste GALOIS), KBN Satyanarayana, Mr. VVN Suresh Kumar

Proceedings of the National Seminar on ALGEBRA (200th Birth Anniversary Celebrations of Evariste GALOIS), KBN College, Vijayawada, A.P., (October 25, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana, Mr. VVN

Suresh Kumar and Mr Mohiddin Shaw Shaik.) 56

On Seminearrings and their Ideals

P.Venu Gopala Rao, Department of Mathematics,

Andhra Loyola College (Autonomous), Vijayawada- 520 008.

E-mail:[email protected]

Abstract: A semiring is an algebraic system which is closed and associative under two operations usual addition and multiplication and satisfies both distributive laws. Semirings abound in the mathematical world around us. Indeed, the first mathematical structure we encounter, the set of natural numbers with the usual operations of addition and multiplication of integers is an example of a semiring. The algebraic systems with binary operations of addition and multiplication satisfying all the ring axioms except possibly one of the distributive laws and commutativity of addition are called Nearrings. A natural example of a Nearring is given by the set M (G) of all mappings of an additive group G (not necessarily abelian) into itself with addition and multiplication as composition of mappings.

In this Paper, the algebraic system seminearring is considered which is generalization of both a semiring and a nearing. A seminearring S is an algebraic system with two binary operations: usual addition and usual multiplication such that S forms a semigroup with respect to both the operations, and satisfies the right distributive law. A natural example of a seminearring is obtained by considering the operations usual addition and composition of mappings on a set of all mappings of an additive semigroup S into itself. We consider the s-ideal (left, right) of a seminearring defined by Javed Ahsan([1], [2]) and Weinert [7] and provide examples.We define and study the fuzzy point view of various analogue properties of ideals in seminearrings.

References

[1]. Javed Ahsan. “Seminear-rings Characterized by their s-ideals I”, Proceedings of Japan Academy, Series A, 101–103, 1995.

[2]. Javed Ahsan. “Seminear-rings Characterized by their s-ideals II”, Proceedings of Japan Academy, Series A, 111–113, 1995.

[3]. Jonathan S. Golan. “Semirings and their Applications”, Kluwer Academic Publishers,1999.

[4]. Pilz G. “Near-Rings: The theory and its Applications”, North-Holland Publishing Company, 1983. [5]. Van Hoorn, Willy. G and Van Rootoselaar.B. “Fundamental notions in the theory of seminear-rings”, Composition Math. , 18, 65–78, 1966. [6]. Weinert H.J. “Seminear-rings, Seminear-fields and their semigroup theoretical background” Semigroup Forum, vol.24, 231 – 254 ,1982. [7]. Weinert H.J and Hebisch. U. “Semirings- Algebraic theory and applications in Computer Science”, World Scientific Publishing Company Ltd., 1998. [8]. Xiang Yun Xie. “On prime, quasi-prime, weakly quasi prime fuzzy left ideals of semigroups”, Fuzzy Sets and Systems, vol.123, 239 – 249 , 2001.

Proceedings of the National Seminar on ALGEBRA College, Vijayawada, A.P., (October

and Mr Mohiddin Shaw Shaik.)

Proceedings of the National Seminar on ALGEBRA (200th Birth Anniversary Celebrations of Evariste GALOIS)October 25, 2011) (Editors: Prof. Dr Bhavanari Satyanarayana

57

Birth Anniversary Celebrations of Evariste GALOIS), KBN Satyanarayana, Mr. VVN Suresh Kumar



Nanotechnology, an emerging eraAuthors: Bhavanari Mallikarjun1, Kolla Ramasubramaniam1

& Udayagiri Harsha2

1. Centre for Nanotechnology Research, VIT University, Vellore, Tamilnadu2. MS student, Department of Computer science and Electrical EngineeringUniversity of Mussori, Kansas city, Mussori, USA---------------------------------------------------------------------------------------------------------------------Abstract: In this paper we discuss the introduction regarding Nanotechnology, what is

Nanotechnology Some of the applications of Nanotechnologies in the present world.

--------------------------------------------------------------------------------------------------------------------

DIVISIBLE SEMIGROUPSAuthors: N.V.Ramana Murty,Dept. of Mathematics, Andhra Loyola College, Vijayawada-520008---------------------------------------------------------------------------------------------------------------------Abstract The concept of divisibility has been studied in Abelian groups by many mathematicians and it

has enough literature in Abelian group theory. Similar to this concept, this paper makes an attempt to

study the concept of Divisibility in Semigroups. It has been discussed some important properties of

divisible semigroups and given some examples. Throughout this paper all semigroups are additive and

commutative with respect to addition.

Definition 1: A semigroup S is said to be divisible if for any element x in S and for any positive integer n,

there exists an element y in S such that ,ny x where ( )ny y y y n times

If the element y exists in S uniquely, then S is called “uniquely divisible”.

References1. Fuchs, L., Infinite Abelian Groups, Vol. I, Academic Press, New York (1970).2. Howie, John M., Fundamentals in Semigroup Theory, Oxford Science Publications, Clrendon (1995). 3. Tamura, T., Minimal Commutative Divisible Semigroups, Bull. Amer. Math. Soc., Vol. 69, No. 5 (1963), 713-716.

Oral Presentation----------------------------

Mr. Bhavanari MallikarjunVIT Univeristy, Tamilnadu

Oral Presentation----------------------------

Dr. N.V. Ramana Murthy



A Note on Inversive Localization in Noetherian Regular Delta Near Rings-------------------------------------------------------------------------------------------------------------------Abstract: In this paper we discuss some existing examples in torsion theory and obtain a general theorem on the relation between localizing and forming factor of Noetherian regular delta near rings.

Keywords : Near-ring, Semi Prime ideal , Prime ideal, Proper kernel Functor, Annihilator, Noetherian Near-ring, regular Near-ring, Noetherian Regular δ-near-ring.---------------------------------------------------------------------------------------------------------------------

A NOTE ON HAMILTONIAN PATH AND HAMILTONIAN CIRCUIT-----------------------------------------------------------------------------------------------------------------------------ABSTRACT: A Hamiltonian path is a spanning path in a graph, i.e. a path through every vertex. I discuss in this paper a sufficient condition of a Hamiltonian path and Complexity of the Hamiltonian circuit problem for planner graphs.----------------------------------------------------------------------------------------------------

A NOTE ON FUZZY IDEALS-----------------------------------------------------------------------------------------------------------------------------ABSTRACT: In this paper I would like to discuss about the fundamental concepts related to Fuzzy ideals in Gamma Near rings.----------------------------------------------------------------------------------------------------------------

Minimal Dominating Set of an Interval GraphsAuthors: A. Sudhakaraiah, E. Gnana Deepika, A.Sreenivasulu, V. Rama LathaDepartment of Mathematics, Sri Venkateswara University, Tirupati- 517502, Andhra Pradesh, India.-----------------------------------------------------------------------------------------------------------------------------ABSTRACT: Graph Theory is a Fascinating subject. One simple way of representing structure of system is to use graphs which are simply interval graphs consisting of points and lines. Among the various applications of the theory of domination, the most often discussed is a communication network. The problem is to select a smallest set of sites at which transmitters are placed so that every site in the network that does not have a transmitter, is joined by a direct communication link to the site, which has a transmitter. Then this problem reduces to that of finding a minimum dominating set in the graph corresponding to this network. Suppose communication network does not work due to link failure. In this paper we discuss minimal dominating set of interval graphs.


N V Nagendram

Oral Presentation------------------------------S. Venu Madava Sarma


B. Nanda Kumar




Nanotechnology: An Emerging EraAuthors: Mr. Bhavanari Mallikarjun, Mr. Kolla Rama SubramanyamM.Tech., StudentsCentre for Nanotechnology Research,VIT University, Vellore-632014TamilNadu, India.

Mr. Harsha WudayagiriM.S., StudentDepartment of Computer Science and Electrical Engineering,University of Missouri, Kansas CityUnited States of America.-------------------------------------------------------------------------------------------------------------------------------

Nanotechnology is a technology of using one nanometer size for the better standards of human with the help of technology. A nanometre is one billionth of a meter. Generally, the nanotechnology dealt for using 1nm to 100nm in present day. The electron in atom moves form 10-9 to 10-10 meters. Nanotechnology was first introduced in 1959 from the famous lecture “There’s a plenty of room at the bottom” delivered by Richard Feynman. Nanotechnology, the short gun marriage of chemistry and engineering in molecular manufacturing or more simply, builds things with one atom or molecule at a time with programmed nanoscopic robot arms.

Nanotechnology is the amalgamation of knowledge from chemistry, physics, biology, materials science, and various engineering fields. This technology proposes the construction of novel molecular devices possessing extraordinary devices by manipulating atoms individually and placing them exactly where needed to produce the desired result. Nanotechnology includes integration of nanoscale structures into larger architectures that could be used in industry, medicine, and environmental protection. Nanostructures are 0-Dimensional, 1-Dimensional, 2-Dimensional, 3-Dimensional. Example for 0-Dimensional Nanostructure is Quantum dots, for 1-Dimensional is Carbon Nanotubes, nanorods, nanowires. 2-Dimensional is Graphene, Nanobelts etc.

Nanostructures and its properties are fascinating and in this regard intensive research is going on with investigations on structural property relation with some diminishing dimensions. Nanostructures are hindered by the progress of the synthesis and characterization methods for these nanoscaled materials. Nanostructures are prepared by using different metal oxides, semi conducting materials with Physical Vapor Deposition and Chemical Vapor Deposition Techniques. There are different types techniques involved in these both techniques specified above.These nanostructures are like hierarchy development blocks for building the new devices and materials applications in fields of electronics, communication, photonics, solar power, sensing, thermoelectric power, biosensors for generation of power, robots, medicine, and various fields of engineering etc.,

Nanostructures applied in electronic devices such as i-pod, mobile, laptop, memory storage hard disks etc. Due to this the size of the electronic gadgets are manufactured to be consumer friendly with high performance and special specifications.

Nanoparticles are present in nature in the feather of a peacock, leaf of banana tree, butterfly wings, lotus flower leaf. Nanotechnology used in the field of biosciences is known as nanobiotechnology. The role of nanoparticles in medicine is referred to as nanomedicine. Nanomedicine involves the design manufacture, administration, and monitoring of drugs and diagnostic/therapeutic devices that use nanoparticles about 1-100 nanometers in size. The nanoparticles exhibit properties (strength, electrical conductivity, elasticity, colour etc.) that same materials do not have at micro or macro sizes.

Collaboration between Medicine and Nanotechnology that has given birth to the revolutionary concept of nanomedicine, which is the medical application of nanotechnology. The present era of Nanotechnology has reached to a stage where scientists are able to develop programmable and externally controllable complex machines that are built at molecular level which can work inside the patient’s body. Nanomedicine is the medical application of molecular nanotechnology (MNT) — a still-developing science dedicated to constructing microscopic

Survey Article____________________________

Presenter:Mr. Bhavanari Mallikarjun



biomechanical devices like nanorobots. Nanorobots would be programmed for specific biological tasks and injected into the blood in solution to work at the cellular level to do everything from repairing tissue, to cleaning arteries, attacking cancer cells and viruses like AIDS, and even reversing the aging process. Nanomedicine will therefore improve the efficacy of the drugs, help target the drug specifically to the desired area thus minimizing unnecessary side effects and toxicity. Overall, nano-medicine will help to improve quality of life. The idea to build a nanorobot comes from the fact that the body’s natural nanodevices

Airborne nanorobots could be programmed to rebuild the thinning ozone layer. Nanotechnology has the potential to benefit the environment through pollution treatment and remediation as any waste atoms could be recycled, since they could be kept under control. This would include improved detection and sensing, removal of the finest contaminants from air, water and soil, and creation of new industrial processes that reduce waste products and are eco friendly.

Nano-computer is a computer whose fundamental components measure only a few nanometers, thereby offering tremendous speed and density. Research is going on in this project by NCDT. The nanocomputer dream team (NCDT) is currently developing software to create a massive parallel supercomputer based on personal computers linked via internet.

Moving agriculture into greenhouses can recover most of the water used, by dehumidifying the exhaust air and treating and re-using runoff. Greenhouse agriculture requires less labor and far less land area than open-field agriculture, and provides greater independence from weather conditions including seasonal variations and droughts. When the glass surfaces are treated with our 'Nano Glass / Window Sealant SR', time and money on cleaning can be reduced. A large-scale move to greenhouse agriculture would reduce water use, land use, and weather-related food shortages. A new, more powerful industrial revolution capable of bringing wealth, health and education, without pollution, to every person on the planet is just around the corner.

Nano-materials are used in the solar cells to improve efficiency. Nanoparticles are used in the fabrics, which improves the life of the cloth and the color with better comfort. Nanotechnology applied in communication such as nanophotonics improves the speed of the transfer of digital information data.

Applications of nanotechnology will include the enhancement of agriculture productivity using nanoporous. Nanotechnology is used in the geotechnical engineering for better understanding of soil nature and structure. By addition of nanoparticles to soil as an external factor manipulates the soil at atomic or molecular level.

The risks of nanotechnology are: untraceable weapons of mass destruction, i.e., personal risk from criminal or terrorist use. There will be economic disruption from an abundance of cheap products, economic oppression from artificially inflated prices. Some environment damage by unregulated products i.e., by the increase of e-waste and social disruption from new products/lifestyles.

Nanotechnology offers great potential for benefits to human-kind. Consumer goods will become plentiful, inexpensive, smart and durable. Medicine will take a quantum leap forward. Space travel and colonization will become safe and affordable. For these and other reasons, global lifestyles will change radically and human behavior will be drastically impacted.

Acknowledgements: The author thank the authorities of KBN College, Vijayawada for their affection towards mathematics, and encouragement to conduct a National Seminar on Algebra in the College. He also thank Dr P. Krishna Murthy (Principal), Mr. V.V.N. Suresh Kumar (Head of the Department of Mathematics), and Prof. Dr Bhavanari Satyanarayana (Academic Secretary) for accepting to present this survey article at the One day National Seminar on 25th October 2011 (the 200 Birth day of Galois).

References[1] Bhavanari Mallikarjun and Wudayagiri Harsha “Nanorobots in Medicine”, in the book: Emerging Trends in Bio Medical Research (Editors: Dr. KRS Sambasiva Rao and Dr. RS. Prakasham) (ISBN 978-81-9088886 – 1-5) (2009) PP 155-171

[2] Some Websites in Internet.

ONE DAY NATIONAL SEMINAR ON ALGEBRA25TH OCTOBER 2011, KBN College, VIJAYAWADA.

PROGRAM

08:30 to 9.30 am Registration9.30 am to 10.30 am INAUGARAL FUNCTION.

Technical Session – 110.30 TO 11.00: INAUGARAL LECTURE by

PROF. DR P. ThrimurthyTopic: APPLICATIONS OF MATHEMATICS IN COMPUTER SCIENCE

11 TO 11.30: Key Note ADDRESS By :Prof. Dr. Bhavanari Satyanarayana

Topic: FINITE DIMENTION OF VECTOR SPACES AND MODULES

11.30 – 11.45 TEA-BREAK

Technical Session – 211:45 noon to 12:15 pm Invited talk by: Dr ATUL GAUR, DELHI UNIV. Topic: Multiplication Modules

12.15 PM TO 12.45 PM Invited talk by: DR V. AMBETHKAR, DELHI UNIV. Topic: Some Numerical Methods for Solving Navier-Stokes Equations

Technical Session – 3

12.45 TO 1.15PM SHORT TALKS 1. DR DASARI NAGARAJU, Manipal University Jaipur (Topic: Zero Square Rings) 2. SHAIK MOHIDDIN SHAH (Topic: Construction of the Prime Graph of a Ring)

1.15 PM TO 2 PM WORKING LUNCH


2 pm to 3 pm: (PARALLEL SESSIONS)

2 pm to 2.40 pm INVITED SHORT TALKS BY:

(i). DR Bavanari Satyanarayana, Nuzvid P.G Centre, Nuzvid. (ii). DR T.V. Pradeep Kumar, ANU College of Engineering

2 pm to 3 pm: PAPER READING SESSION (Chair Person: Dr Dasari Nagaraju)


3 PM to 3.45 pm : TEACHER SESSION (ORGANIZED BY AIMEd)

Discussion and Problem Solving Session (for School Teachers)

Chair Person: Sri Pokala Chandar, Executive Engineer, Warangal.Organized by: Mr. Ch. V. Narasimha Rao, Director, AIMEd, Vijayawada.

Speakers: Sri. G.V. Chalapathi Rao, Rtd. HM., Hon.President, Teachers Association, Narasaraopet

Mr. Khasim, Best Teacher Awardee 2011, GunturMrs. T. Madhavi Latha, M.Sc., M.Phil.,

Vice-Principal, APRSW School, Jangareddygudem.

Some more Teachers

3.45 TO 4 PM TEA BREAK

4 PM TO 4.45 VALIDECTORY FUNCTION& DISTRIBUTION OF CERTIFICATES FOR PARTICIPANTS.

Chief Guest: SRI POKALA CHANDAR, Executive Engineer, Warangal

Dr. BHAVANARI SCHOLARS / STUDENTS ASSOCIATION

Dr Kuncham Syam Prasad: He awarded Gold Medal for his first rank in M. Sc., Mathematics in 1994. He is a recipient of CSIR-Senior Research Fellowship. He got awarded M. Phil., (Graph Theory) in 1998 and Ph.D., (Algebra - Nearrings) in 2000 under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee). He published Nineteen research papers in reputed journals and presented research papers in thirteen National Conferences and six International

Conferences in which five of them were outside India: U.S.A (1999), Germany (2003), Taiwan (2005), Ukraine (2006), Austria (2007), Bankok (2008), Indonesia (2009). He also visited the Hungarian Academy of Sciences, Hungary for joint research work with Dr Bhavanari Satyanarayana (2003) and the National University of Singapore(2005) for Scientific Discussions. He authored nine books (UG/PG level). He is also a recipient of Best Research Paper Prize for the year 2000 by the Indian Mathematical Society for his research in Algebra. He received INSA Visiting Fellowship Award (2004) for the collaborative Research Work. Presently working as Associate Professor of Mathematics, Manipal University, Karnataka, India. E-mail: [email protected]

Dr. Tumurukota Venkata Pradeep Kumar: He got awarded M.Phil., (-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 Professorin 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 an 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 (AP SCIENTIST Awardee). 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 is now visiting Institute of Ring Theory, USA. E-mail: [email protected]

Dr Arava Venkata Vijaya Kumari: She completed her M.Sc., (Mathematics) from ANU with third rank. She got Awarded her Ph.D., (Nearrings) under the guidance of Dr Bhavanari Satyanrayana (AP SCIENTIST AWARDEE, by DST, New Delhi, 2009, Fellow-AP Akademy of Sciences, 2010). She published 3 research papers in National and International Journals. Presently Heading the Department of Mathematics, JMJ College, Tenali.

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 Six 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 (AP SCIENTIST Awardee). He authored two books at PG level. He published a research paper in USA. Presently working as a Lecturer in P.G. Department of P. B. Siddhartha College, Vijayawada. E-mail: [email protected]

Mr. K. S. Balamurugan: He got First Rank in B. Sc., and Second Rank in M. Sc., course. He awarded with M. Phil., (Ring Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee) 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 qualifications 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.

Dr. BHAVANARI SCHOLARS / STUDENTS ASSOCIATION

Mrs. Sk. Shakeera: She got M. Phil., degree (2007) in -ring theory under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee).Mr. D. Srinivasulu: He got his M. Phil., degree (Graph Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIENTIST Awardee).

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 Akademy of Sciences (2010), Siksha Rattan Puraskar (2011), Rajiv Gandhi Excellence Award (New Delhi, 2011)

International Awards: Glory of India & International Achievers’ Award (Thailand, March 26, 2011) Deputy Director General (IBC, England 2011) One of the “Top 100 Professionals – 2011” (by IBC, Cambridge, England) 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 29 General Articles in periodicals. Authored / Edited 38 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 65 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.

Name : Dr Bhavanari SatyanarayanaDesignation : Professor Date of Birth : 12-11-1957Place of Birth : Madugula (a Village in Palnadu region)Mother : B. Ansuryamma (Late)Father : B. Ramakotaiah (Retired Teacher) (Late)Elementary School Edu: : Reddypalem (Near pedakodamagundla), Adigoppula, Madugula of Palnadu.High School Education : St. Joseph’s Boys High School, Rentachintala, Guntur (Dt)Inter + B.Sc. : S.S.N. College, Narasaraopet, Guntur (Dt)M.Sc. + Ph.D : Acharya Nagarjuna University, Nagarjuna Nagar Ph: 0863-2232138 (R); Cell: 98480 59722.E-mail : [email protected], [email protected]

Resume of

Dr BHAVANARI SATYANARAYANAProfessor Ph: Cell: 98480 59722Department of Mathematics ® 0863 - 2232138Acharya Nagarjuna University e-mail: [email protected] Nagar – 522 510, A.P., India. _______________________________________________________________________________

1. Name : Bhavanari Satyanarayana

2. Address : Dr Bhavanari Satyanarayana,

Professor Dept. of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar - 522 510, A.P. INDIA.

3. Date of Birth & Age : 12th Nov., 1957, 54 Years.

4. Present Position : Professor

5. Academic Qualifications (from Degree level onwards):

Examination Year Class % of Marks Rank Name of the InstitutionB.Sc. 1974-77 FIRST 62.5 - SSN College, Narasaraopet,

(Andhra University)M.Sc. 1977-79 FIRST 75

DistinctionII Dept. Maths, Nagarjuna

Univ.B.Ed., 1985 First 60 Annamalai University,

TamilnaduM.Tech

(CS)2008-10 First

Class70

DistinctionAcharya Nagarjuna University (Dist. Education).

Ph.D. 1985 Nagarjuna University.(Title of Ph.D, dissertation : Contributions to Near-ring theory, Submitted in June 1984)

6. Teaching Experience : ( 27 Years at Acharya Nagarjuna University)

From To Cadre Level Degree/PG

Duration

01-10-85 17-05-86 Lecturer (Temp) PG 6 Months & 17 Days

21-05-86 20-11-86 Lecturer (Temp) PG 6 Months



18-11-87 20-01-88 Lecturer (Temp) PG 2 Months & 2 Days

20-01-88 30-08-88 Pool Officer (CSIR) PG 7 Months & 10 Days

30-08-88 30-08-93 Lecturer (Regular) PG 5 Years

30-08-93 26-07-98 Sr. Lecturer PG 4 Years & 11 Months

27-07-98 16-10-06 Associate Professor PG 8 Years & 2 Months

17-10-06 to date Professor PG 6 YEARS & 9 months

7. Attended the Refresher Courses for P.G. Teachers/Workshops

1. Workshop in Mathematics for P.G. Teachers, Oct 18-19, 2003, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur (Attended as a participant).

2. Workshop on Lattice Theory, July 25, 2004, Bapatla Engineering College, Bapatla, (Attended as a participant).

3. Refresher Course in Mathematics for Post Graduate Teachers, Dec 9-10, 2004, Vignan School of Post Graduate Studies, Guntur (Attended as a participant).

4. 3-Day Workshop on “Recent Trends in Mathematics Applied to Science and Technolgy”, April 20-22, 2006, Gudlavalleru Engineering College, Gudlavalleru(Attended as a participant and delivered a talk on ‘Graph Theory’ on 21-04-2006).

1. National Workshop on “Uniform Hyper Graphs and their Applications”, January 30th - February 5th, 2008, Sir CRR International Institute of Mathematics, Eluru (delivered an invited talk on ‘Some Results on Prinicipal Ideal Graph of a Ring’)

Enclosure Page 7

8. Research Experience : 33 (Thirty three) Years.

Scholarships/Fellowships received :

Name of the fellowship Duration Institution

Jr. Research Fellowship (CSIR) Jan 1980 - Jan 1982 (2 Yrs) CSIR (Placement at NU)

Sr. Research Fellowship (CSIR) Jan 1982 – Jan 1985 (3 Yrs) CSIR (Placement at NU)

Research Associateship (UGC) 1985 UGC (Placement at NU)

Pool Officer Jan 1988 – Aug. 1988 CSIR (Placement at NU)

INSA Visiting Scientist Award 2005

2005 INSA (to do joint research work with Dr. Kuncham Syam Prasad at Manipal University, Karnataka)

9. Reviews of work done :

(i). referee for one Research Paper submitted to J. of Indian Math.Society, India.

(ii). referee for more than ten Research papers submitted to Indian J. Pure & Appl. Mathematics, India.

(iii). referee for one research paper submitted to the Houston Journal of Mathematics, USA (2005).

(iv). referee for one research paper submitted to the IJMMS (International J. Mathematics & Mathematical Sciences), (2005).

(v). referee for one research paper submitted to the Journal “The Mathematics Student”, India (2005)

(vi). Referee for three research papers submitted to the Journal “Malesian J. Mathematics (2006, 2007)”

(vii). Referee for one research paper submitted to the Journal “South East Asian J Mathematics”, 2007.

(viii). adjudicator for a Ph.D., Thesis submitted to North Eastern Hill University, Shillong (1993 – 94)

(ix). adjudicator for a Ph.D., Thesis submitted to Manonmanian Sundarnar University, Tirunelveli (1999)

(x). adjudicator for a Ph.D., Thesis submitted to Manonmanian Sundarnar University, Tirunelveli (2001)

(xi) adjudicator for a Ph.D., Thesis submitted to Sri Padmavati Mahila Visvavidyalayam, Tirupati (2006).

(xii) adjudicator for a Ph.D., Thesis submitted to Gulbarga University, Karnataka, (2007).

(xiii) adjudicator for a Ph.D., Thesis submitted to Gauhati University, Guwahati, Assam (2007).

10. Memberships of Professional Societies/visiting Assignment etc.

Life Member of the Indian Mathematical Society (S-86-085). Life Member of the Ramanujan Mathematical Society. Life Member of the Allahabad Mathematical Society. Life Member (No: 141) of the AP Society for Mathematical

Sciences. Life Member (Registration No: L11680) of The Indian Science

Congress Association, Kolkata-17.

Member of the New York Academy of Sciences. Visited TIFR (Bombay) as a Visiting Fellow during1-31 May,1989. Selected Scientist under Indo-Hungarian Cultural Exchange

Programme to do joint research work with Prof. Richard Wiegandt at the A.Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary, June 05 to Sept.05, 2003.

Selected Sr.Scientist Under the Bilateral Exchange Programme of the Hungarian Academy of Sciences (Budapest), and INSA (New Delhi) to have scientific discussions with Prof. Richard Weigandt & Prof. Laszlo Marki at A. Renyi Institute of Mathematics, Budapest (Hungary), Aug 16 - Sep 05, 2005.

Visiting Professor at Walter Sisulu University, Umtata, South Africa, March 26-April 10, 2007.

Awards/honors received

National Awards:CSIR-JRF (1980-82), CSIR-SRF (1982-85),

UGC-Research Associateship (1985), CSIR-POOL OFFICER (1988),

INSA Visiting Scientist Award 2005,

ANU–Best Research Paper Award-2006,

AP Scientist Award 2009 (by DST, Centr. Govt., New Delhi, and APCOST, Hyderabad).

Fellow, AP Akademy of Sciences (2010),

Siksha Rattan Purskar (New Delhi, 20 - Jan - 2011).

Barat Vikas Rattan Award (New Delhi, 2011)

Rajiv Gandhi Excellence Award (26-August-2011).

Mother Teresa Excellence Award (25-11-2012)

International Awards:Glory of India of Award (Thailand, 26th March 2011)

International Achievers Award (Thailand, 26th March 2011).

One of the TOP 100 PROFESSIONALS (IBC, England, 2011),

Deputy Director General, IBC, England, 2011;

Great Mind of the 21st Century (ABI, USA, 2011)

Global Education Leadership Award, Thailand, Nov 2011

Fellow ABI, USA, 2011

The International Plato Award for Educational Achievement (IBC, England, 2012)

TEACHER DISTINCTION

(i) Nominated/elected to State Bodies

Fellow, AP Akademy of Sciences, Since 2010.Member of the Board of Studies in Mathematics (PG), Acharya Nagarjuna University, 1988 to present.

Chairman, Board of Studies in Mathematics (PG), Acharya Nagarjuna University, 2008 to 2011.Member of the Board of Studies in Mathematics (UG), Acharya Nagarjuna University, 2008 to 2011.

Member of Board of Studies in Mathematics (UG), Kakatiya University, Warangal, from 30-09-2008 to 2010 (two years).

Member of Board of Studies in Mathematics (PG), Kakatiya University, Warangal, from 30-09-2008 to 2010 (two years).

Member of the Text book development Committee, SCERT (State Council for Education, Research and Training), Hyderabad, 2009.

Member of Board of Studies in Mathematics (PG), Sri Venkateswara University, Tirupathi, 26-09-2010 to 12-08-2013 (three years).

Elected Executive Member (2007-2008), Andhra Pradesh Society for Mathematical Sciences, Hyderabad.

Elected Office Secretary (Dec. 2012-2014, for a period of two years), Andhra Pradesh Society for Mathematical Sciences, Hyderabad

(ii) Nominated/elected to National Bodies

Elected President (2005-2007) of the Association for Improvement of Maths Education, Vijayawada.


Selected Hon. President (2011 to date) of the Association for Improvement of Maths Education, Vijayawada.

TEACHER DISTINCTION

Member of the Vidyalaya Management Committee, Kendriya Vidyalaya, Nallapadu, Guntur

Member of the Draw Committee for student admissions in Vidyalaya Management Committee, Kendriya Vidyalaya, Nallapadu, Guntur, March, 2012

(iii) Nominated/elected to International Bodies

Dyputy Director General, IBC, England, 2011.

Fellow, ABI (American Biographical Institute), USA, 2012

(iv) Nominated/elected to Regional Bodies

Member of the Board of Studies (UG) in Mathematics, Andhra Layola College, Vijayawada.

Member of the Board of Studies (UG) in Mathematics, KBN College, Vijayawada, March 2010 – 2012 (two years).

Member of the Board of Studies (UG) in Mathematics, Machilipatnam, Krishna District, 2012 – 2014 (two years).

(v) Nominated/elected to Management Committees/Bodies

Member, Management Committee, Guptha College, Tenali

Member, Management Committee, VSR College, Tenali

(v) Nominated/elected to Selection Committees/Bodies

Member (Subject Expert), Selection Committee Meeting, Sri Vasavi Kanyaka Parameswari Arts, Science and Commerce College, Markapur, 28-01-2010.

Member to select the Core Group of Teachers to write the State Government School Text Books, SCERT, Hyderabad, 04-07-2009.

Judge, INSPIRE AWARDS PROGRAMME (Department of Science and Tech., New Delhi), for Guntur District, August 05-07, 2012.

TRAINED ABROADStay abroad on sabbatical leave for teaching / research / exchange Progremme on invitation.

Country visited Duration Purpose

Austria Aug. 21-27, 1988

To deliver a lecture at the international conference on General Algebra, Krems, Austria.

Hong Kong Aug. 14-18, 1990

To present a paper at the Asian Mathematical Society Conference, Hong Kong University, Hong Kong.

South Africa July 06-11, 1997

To deliver a talk at the International Conf. on the Theory of Radicals and Rings, University of Port Elizebeth, South Africa.

South Africa July 14-18, 1997

To present a talk at Inter. Conf on Near-Rings and Near-Fields, Univ. of Stellenbosch, South Africa.

Germany July 27 – Aug 03, 2003

To present a talk at 18th Inter. Conf. on Near-rings and Near-fields, Univ. der. Bundeshwer, Hamburg.

Hungary June 05 –Sept.05,2003

Selected Scientist under Indo-Hungarian Cultural Exchange Programme to do joint research work at A.Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest.

Taiwan July 31- Aug 06,2005

To present a talk at 19th International Conference on Near-rings and Near-fields, National Cheng Kung University, Taiwan.

Singapore Aug 6 - Aug 07, 2005

To Visit the National University of Singapore and to have Mathematical Discussions with the Mathematics Faculty.

Hungary Aug 16 - Sep 05, 2005

Selected Sr.Scientist Under the Bilateral Exchange Programme of the Hungarian Academy of Sciences (Budapest), and INSA (New Delhi) to visit A.Renyi Institute of Mathematics, Budapest, Hungary.

UkraineJuly 30 -

Aug 05, 2006

To deliver a talk at the International Conference on Radicals (ICOR-2006), Kyiv National Taras Shevchenko University, Ukraine

South Africa March 26 -April 10, 2007

Visiting Professor at Walter Sisulu University, Umtata, South Africa, to do joint research work with Dr. Lungisile Godloza, Walter Sisulu University.

Bangkok(Thailand) May 28-30,

2008

To deliver an invited talk on “Principal Ideal Graph of a Ring”, the International Conference on Algebra and Related Topics (ICART 2008), Chulalongkorn University, Bankok, Thailand, May 28-30, 2008.

Bangkok(Thailand) March 24-29,

2011

To receive two international awards: Glory of India Award; and International Achievers Award, in INDO-THAI FRIENDSHIP BANQUET, Bangkok, March 26, 2011

Mascut(Oman) Jan. 20 – 25,

2012

To deliver an invited talk at the International Conference, Sultan Qaboos University, Mascut. Also Chaired a Session

1

Books Authored/Edited

S.No Authors Title, Publisher Year1. Prof. K.V. Siviaiah

Prof. K. Satya Rao Dr. Bh. Satyanarayana & Dr. B. Satyanarayana

“Business Mathematics” (Quantitative Methods)(UG/PG level, B.Com/M.Com), Technical Publishers, Guntur, A.P.

1994

2. Editor:Dr. Bh. SatyanarayanaAuthors:Dr. B. Satyanarayana &Mr. G. V. Subba Rao

“Application Oriented Algebra (For beginners)”B.Ramakotaiah & Co., Madugula, Guntur Dt.

1994

3. Dr. Bh. Satyanarayana & Dr. K. Syam Prasad

“Modern Applied Algebra for beginners” (UG level) B. Ramakotaiah & Co., Madugula, Guntur Dt.

1995

4. Dr. Bh. SatyanarayanaMr. K. Syam PrasadMr. V. Dharma RaoMr. T. V. Pradeep Kumar &Smt. T. Madhavi Latha

“Introduction to Mathematics & Statistics”, (Quantitative Methods) (for M. A), Technical Publishers, Guntur

1996

5. V. KasaiahDr. Bh. SatyanarayanaDr. K. Syam Prasad & V.H. Leela

“Text book of Mathematics, B.C.A” (Bachelor of Computer Applications), Technical Publishers, Guntur

1999

6. Editors:Dr. Bh. Satyanarayana Dr. K. Syam Prasad Authors: T.V. Pradeep Kumar T. Madhavi Latha & A. Padmavathi

“Discrete Mathematics”, (for BCA/MCA)Maruthi Publishers, Guntur 1999

7. Dr. Bh. SatyanarayanaMr. K. Syam PrasadMr. T. V. Pradeep Kumar &Mr. M. Srinivasa Rao

“Algebra”B. Ramakotaiah & Co., Madugula, Guntur Dt.

1999

8. Dr. Bh. SatyanarayanaMr. K. Syam PrasadMr. T. V. Pradeep Kumar &Mr. M. Srinivasa Rao

“Topology”B. Ramakotaiah & Co., Madugula, Guntur Dt.

1999

9. Dr. Bh. SatyanarayanaDr. K. Syam Prasad & T.V. Pradeep Kumar

“Linear Programming”B. Ramakotaiah & Co., Madugula, Guntur Dt. 2000

10. Dr. Bh. SatyanarayanaDr. K. Syam Prasad & T.V. Pradeep Kumar

“Rings and Modules”B. Ramakotaiah & Co., Madugula, Guntur Dt. 2000

11. Dr. Bh. SatyanarayanaMr. K. Syam Prasad &Mr. T. V. Pradeep Kumar

“Analysis”B. Ramakotaiah & Co., Madugula, Guntur Dt.

2000

Contd…2

2

12. J. Venkateswarlu & Dr. Bh. Satyanarayana

“Quantitative Methods Part-I” (Telugu medium) (for I yr B.Com students) Jai Bharat Publishers, Guntur

2001

13. Dr. Bh. Satyanarayana J. Venkateswarlu & T.V Pradeep Kumar

“Quantitative Methods Part-II” (Telugu medium) (for II yr B.Com students) Jai Bharat Publishers, Guntur

2001

14. Dr. Bh. Satyanarayana &Dr. K. Syam Prasad

“Functional Analysis”Satyasri Maths Study Centre, Guntur

2001

15. Dr. Bh. SatyanarayanaDr. K. Syam Prasad &Smt. T. Madhavi Latha

“Measure Theory”Satyasri Maths Study Centre, Guntur

2001

16. Dr. Bh. Satyanarayana “Posets and Machines”Satyasri Maths study Centre, Guntur

2002

17. Dr. Bh. Satyanarayana “Lattices and Boolean Algebra”Satyasri Maths study Centre, Guntur

2002

18. Dr. Bh. Satyanarayana & Dr. K. Syam Prasad

“Graph Theory”Satyasri Maths study Centre, Guntur

2003

19. Editor:Dr. Bh. SatyanarayanaAuthor:Mr. J. L. Ram Prasad

“Galois Theory”Satyasri Maths Study Centre, Guntur

2004

20. Dr. Bh. SatyanarayanaDr. K. Syam Prasad &Mr. J. L. Ram Prasad

“Analytic Number Theory and Graph Theory”Centre for Distance Education, Acharya Nagarjuna University.

2004

21. Prof. L. Nagamuni ReddyProf. U. M. Swamy &Prof. Y. V. ReddyDr. Bh. Satyanarayana

“Algebra”Centre for Distance Education, Acharya Nagarjuna University.

2004

22. Prof. K. V. AchalapatiProf. V. GangadharProf. C. R. ReddyProf. Bh. Satyanarayana Prof. R. Sudarshan

“Quantitative Techniques” (for M. Com students)Dr. Ambedkar Open University, Hyderabad

2004

23. Dr Bhavanari Satyanarayana (one of the Concept Designers and Editors)

“Modern Applications using Discrete Mathematical Structures (Information Technology)”, Sikkim Manipal University, Directorate of Distance Education.Content Development by Dr Kuncham Syam Prasad & Mr. Deepak Shetty (Manipal University) Concept Design & Editing by Prof. Bhavanari Satyanarayana (ANU) & Mr. Kedukodi Babushri Srinivas (Manipal University), .

June 2007

24. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr K. Syam Prasad, Dr. T. Srinivas & Mr. T. V. Pradeep Kumar

“Discrete Mathematics” (for MCA students) Centre for Distance Education, Acharya Nagarjuna University, Andhra Pradesh.

2008

3

25. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr. Kuncham Syam Prasad, Dr T V Pradeep Kumar, Dr T. Srinivas

“Discrete Mathematics, Part 1”, (for B.Sc., Information Technology students), Centre for Distance Education, Acharya Nagarjuna University, Andhra Pradesh

2008

26. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr. Kuncham Syam Prasad, Dr T V Pradeep Kumar, Dr T. Srinivas

“Discrete Mathematics, Part 2”, (for B.Sc., Information Technology students), Centre for Distance Education, Acharya Nagarjuna University, Andhra Pradesh

2008

27. Prof. K. Chandan, Prof. GVSR Anjaneyulu, Prof. Dr. Bhavanari Satyanarayana, Smt. SVS Girija

“Operations Research, Computer Programming and Numerical Analysis” (for the students of III B.Sc Statistics, P-IV), Centre for distance Education, Acharya Nagarjuna University, Andhra Pradesh

2008

28. Dr. Bh. Satyanarayana &Dr. K. Syam Prasad

“Discrete Mathematics with Graph Theory” (for B. Tech., / B. E / M.Sc., (Maths) /MCA students), Prentice Hall of India, New Delhi (ISBN: 978-81-203-3842-5)

2009

29. Dr. Bhavanari Satyanarayana “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany (ISBN 978-3-639-22417-7)

2010

30. Dr. Satyanarayana Bhavanari & Mohiddin Shaw Sk.

“Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Muller, Germany (ISBN 978-3-639-23197-7)

2010

31. Dr Satyanarayana Bhavanari & Ram Prasad J L

“Fuzzy Prime Submodules”, VDM Verlag Dr Muller, Germany (ISBN 978-3-639-24355-0)

2010

32. Dr. Bhavanari Satyanarayana (one of the Editors)

“Ganitha Sastram”, Class –VI (in Telugu Language), Published by Government of Andhra Pradesh, Hyderabad

2010

33. Dr. Bhavanari Satyanarayana (one of the Editors)

“Mathematics, Class - VI” (English), Published by Government of Andhra Pradesh, Hyderabad

2010

34. Prof. Bhavanari Satyanarayana Dr. Eswaraiah Setty S., &Prof. Kuncham Syam Prasad

“Proceedings of the National Seminar on Present Trends in Mathematics and its Applications”, Sponsored by UGC, held at SGS College (Nov. 11-12, 2010), Jaggaiahpet, Andhra Pradesh, INDIA.

2010

35. Dr. Satyanarayana Bhavanari & Dr. Nagaraju Dasari

“Dimension and Graph Theoretic Aspects of Rings”, VDM Verlag Dr Muller, Germany(ISBN 978-3-639-30558-6)

2011

36. Dr. Syam Prasad Kuncham & Dr Satyanarayana Bhavanari

“Dimension of N-groups and Fuzzy ideals in Gamma Near-rings”, VDM Verlag Dr Muller, Germany (ISBN 978-3-639-36838-3)

2011

37. Dr. Bhavanari Satyanarayana (one of the editors),Dr. Vijaya Kumari A.V., &

“Proceedings of the National Seminar on Present Trends in Algebra and its Applications”,Sponsored by UGC held at JMJ College, Tenail,

2011

4

Dr. Mohiddin Shaw Shaik Andhra Pradesh, India. (July 11-12, 2011)38. Dr. Bhavanari Satyanarayana

(One of the Editors)Mr. V. V. N. Suresh Kumar,Dr. Mohiddin Shaw Shaik

“Proceedings of the One day National Seminar on Algebra”, (200th Birth Anniversary Celebrations of Evariste Galois) (in Collaboration with the Department of Mathematics, Acharya Nagarjuna University), Sponsored by KBN College Management, held at KBN College, Vijayawada, A.P., India, October 25, 2011

2011

39. N.P. Bali, Satyanarayana Bhavanari & Indrani Kelker

“Engineering Mathematics – I”, (JNTU -KAKINADA) Laxmipublications, New Delhi. (ISBN: 978-93-81159-21-7)

2012

40. Bhavanari SatyanarayanaKuncham Syam Prasad

“Near rings, Fuzzy Ideals and Graph Theory”, Chapman and Hall/CRC (Taylor and Francis), Newyork/England (ISBN 978-14-39873-10-6).

2013

Books authored/Edited and published

Single authored (reference books):

1. Dr.Bhavanari Satyanarayana , “Contributions to Near-ring Theory”, VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-639-22417-7).

Multi-authored (reference books) :

1. Dr. Bh. Satyanarayana & Dr. Kuncham Syam Prasad, Discrete Mathematics with Graph Theory (for B.Tech/B.Sc/M.Sc.,(Maths)) Printice Hall of India, New Delhi, 2009 (ISBN: 978-81-203-3842-5).

2. Dr. Satyanarayan Bhavanari & Mohiddin Shaw Sk, “Fuzzy Dimension of Modules over Rings”, VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-

639-23197-7).3. Dr Satyanarayan Bhavanari & Rama Prasad J L “Fuzzy Prime Submodules”,

VDM Verlag Dr Muller, Germany, 2010, (ISBN 978-3-639-24355-0)4. Dr Satyanarayan Bhavanari & Nagarju Dasari “Dimension and Graph Theoretic Aspects of Rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3- 639-30558-6)5. Satyanarayana Bhavanari and Syam Prasad Kuncham “Dimension of N-groups and Fuzzy ideals in Gamma Near-rings”, VDM Verlag Dr Muller, Germany, 2011, (ISBN 978-3-639-624851-7).6. Satyanarayana Bhavanari and Syam Prasad Kuncham “Near-rings, Fuzzy Ideals, and Graph Theory”, Chapman and Hall (Taylor and Francis), England/New York, 2013 (ISBN: 978-14-39873-10-6).

Single-authored (other than reference books):

1. Bhavanari Satyanarayana ‘POsets and Finite Machines’, (for B.Sc/M.Sc students), Satyasri Maths Study Centre, Guntur, 2002.

2. Bhavanari Satyanarayana ‘Lattices and Boolean Algebra,, (for B.Sc/M.Sc students), Satyasri Maths Study Centre, Guntur, 2002.


Multi-authored (other than reference books):

1. Prof. K.V. Siviaiah, Prof. K. Satya Rao, Dr. Bh. Satyanarayana & Dr. B. Satyanarayana Quantitative Methods, (for BBM/M.Com. students), Technical Publishers, Guntur, 1994.

2. Dr. Bh. Satyanarayana (Editor), Dr. B. Satyanarayana & Mr. G. V. Subba Rao Application Oriented Algebra, (for M.Sc. students), Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 1994.

3. Dr. Bh. Satyanarayana & Dr. K. Syam Prasad, Modern Applied Algebra, (for B.Sc. students), Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 1995.

4. Dr. Bh. Satyanarayana, Mr. K. Syam Prasad, Mr. V. Dharma Rao, Mr. T. V. Pradeep Kumar & Smt. T. Madhavi Latha Quantitative Methods(Introduction to Mathematics and Statistics) Telugu Medium (For M.A. Economics), Technical Publishers, Guntur, 1996.

5. Dr. Bh. Satyanarayana, Mr. K. Syam Prasad, Mr. T. V. Pradeep Kumar & Mr. M. Srinivasa Rao, Algebra (for B.Sc/M.Sc students), Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 1999.

6. Dr. Bh. Satyanarayana, Mr. K. Syam Prasad, Mr. T. V. Pradeep Kumar & Mr. M. Srinivasa Rao, Topology (for B.Sc/M.Sc students), Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 1999.

7. Dr. Bh. Satyanarayana & Dr. K. Syam Prasad (Editors); T.V. Pradeep, Kumar, T. Madhavi Latha & A. Padmavathi (Authors) Discrete Mathematics, (for BCA I Yr. students), Maruthi Publications, Guntur, 1999.

8. V. Kasaiah, Dr. Bh. Satyanarayana, Dr. K. Syam Prasad & V.H. Leela, A Text book of Mathematics, (for BCA I Yr. students), Technical Publishers, Guntur, 1999.

9. Dr. Bh. Satyanarayana, Mr. K. Syam Prasad & Mr. T. V. Pradeep KumarAnalysis (for M.Sc students), Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 2000.

10. Dr. Bh. Satyanarayana, Dr. K. Syam Prasad & T.V. Pradeep Kumar Rings and Modules, (for M.Sc students) , Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 2000.

11. Dr. Bh. Satyanarayana, Dr. K. Syam Prasad & T.V. Pradeep Kumar Linear Programming/Operation Research, (for B.Sc/M.Sc students), Bavanari Rama Kotaiah & Co., Madugula, Guntur(DT), 2000.

12. Dr. Bh. Satyanarayana, Dr. K. Syam Prasad & Smt. T. Madhavi LathaMeasure Theory, (for M.Sc students), Satyasri Maths Study Centre, Guntur, 2001.


13. Dr. Bh. Satyanarayana & Dr. K. Syam Prasad, Functional Analysis, (for M.Sc students), Satyasri Maths Study Centre, Guntur, 2001.

14. J. Venkateswarlu & Dr. Bh. Satyanarayana Quantitative Methods (Part-1) Telugu Medium, (for B.Com., I Yr. Students), Jai Bharat Publishers, Guntur, 2001.

15. Dr. Bh. Satyanarayana, J. Venkateswarlu & T.V Pradeep Kumar Quantitative Methods (Part-2) Telugu medium , (for B.Com., II Yr. Students), Jai Bharat Publishers, Guntur, 2001.

16. Dr. Bh. Satyanarayana & Dr. K. Syam Prasad, Graph Theory, (for M.Sc/MCA students), Satyasri Maths Study Centre, Guntur, 2003.

17. Dr. Bh. Satyanarayana (Editor), Mr. J. L. Ram Prasad Galois Theory, (for M.Sc students), Satyasri Maths Study Centre, Guntur, 2004.

18. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr. Kuncham Syam Prasad, Mr.J.L.Ram Prasad, Analytic Number Theory and Graph Theory, (for M.Sc Maths students), Centre for Distance Education, Acharya Nagarjuna University, 2004

19. Prof. Y.V. Reddy (Editor), Authors: Prof.Dr. Naga Muni Reddy, Prof.Dr. Madana Swamy, & Dr.Bhavanari Satyanarayana, Algebra, (for MSc Maths I year students), Centre for Distance Education, Acharya Nagarjuna University, Nagarjuna Nagar, AP, 2004.

20. Dr. Bh. Satyanarayana (one of the authors) Quantitative techniques, (for M.Com students), Dr. Ambedkar Open University, Hyderabad, 2004.

21. Dr Bhavanari Satyanarayana (one of the Concept Designers and Editors) Modern Applications using Discrete Mathematical Structures (Information Technology), Sikkim Manipal University, Directorate of Distance Education, June 2007.

Content Development by Dr Kuncham Syam Prasad & Mr. Deepak Shetty (Manipal University) Concept Design & Editing by Prof. Bhavanari Satyanarayana (ANU) & Mr. Kedukodi Babushri Srinivas (Manipal University), June 2007.

22. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr. Kuncham Syam Prasad, Dr T V Pradeep Kumar, Dr T. Srinivas Discrete Mathematics, (for M.C.A. students), Centre for Distance Education, Acharya Nagarjuna University, 2008

23. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr. Kuncham Syam Prasad, Dr T V Pradeep Kumar, Dr T. Srinivas Discrete Mathematics, Part 1,(for B.Sc., Information Technology students), Centre for Distance Education, Acharya Nagarjuna University, 2008


24. Dr. Bh. Satyanarayana (Editor and one of the authors), Dr. Kuncham Syam Prasad, Dr T V Pradeep Kumar, Dr T. Srinivas Discrete Mathematics, Part 2, (for B.Sc., Information Technology students), Centre for Distance Education, Acharya Nagarjuna University, 2008.

25. Prof. K. Chandan, Prof. GVSR Anjaneyulu, Dr. Bhavanari Satyanarayana, Smt. SVS Girija, Operations Research, Computer Programming and Numerical Analysis (for the students of III B.Sc Statistics, P-IV), Centre for distance Education, Acharya Nagarjuna University, 2008.

26. Dr. Bhavanari Satyanarayana is one of the Editors for the book “Ganitha Sastram, Class-VI (in Telugu Language)” Published by Government of Andhra Pradesh, Hyderabad, 2010.27. Dr. Bhavanari Satyanarayana is one of the Editors for the book, “Mathematics, Class-VI (English)” Published by Government of Andhra Pradesh, Hyderabad, 2010.

28. Dr. Bhavanari Satyanarayana is one of the Editors (Editors: Dr Eswaraiah Setty S., Prof. Bhavanari Satyanarayana, Prof. Kuncham Syam Prasad)“Proceedings of the National Seminar on Present Trends in Mathematics and its Applications”, Sponsored by UGC, held at SGS College, jaggaiahpet, A.P., India. , Nov. 11-12, 2010.

29. Dr. Bhavanari Satyanarayana is one of the Editors (Editors: Prof. Bhavanari Satyanarayana, Dr. Vijaya Kumari A.V., Dr. Mohiddin Shaw Shaik) “Proceedings of the National Seminar on Present Trends in Algebra and its Applications”, Sponsored by UGC, held at JMJ College, Tenali, A.P., India. , July 11-12, 2011.

30. Dr. Bhavanari Satyanarayana is one of the Editors (Editors: Prof. Bhavanari Satyanarayana, Mr. V.V.N. Suresh Kumar, Dr. Mohiddin Shaw Shaik) “Proceedings of the One day National Seminar on Algebra (200th Birth Anniversary Celebrations of Evariste Galois) (in collaboration with the Department of Mathematics, Acharya Nagarjuna University)”, Sponsored by KBN College Management, held at KBN College, Vijayawada, A.P., India, October 25, 2011.

31. N.P. Bali, Satyanarayana Bhavanari & Indrani Kelker, “Engineering Mathematics – I”, (JNTU - KAKINADA) Laxmipublications, New Delhi. (ISBN: 978-93-81159-21-7), 2012

RESEARCH AND CONSULTANCY - Research Guidance

M. Phil’s Awarded/Guided: 10

S.No Name of the Candidate

Title of the Dissertation Area Month/Year

1. M. B. V. Lokeswra Rao

On Primeness in Near-rings & Matrix Near-rings

Near-ringTheory

April1995

2. Kuncham Syam Prasad

An Isomorphism Theorem on Directed Hyper cubes of Dimension n

Graph Theory

May1998

3. T. V. Pradeep Kumar

On g1-γ- prime left Ideals and related Prime Radical in -rings

-rings May1998

4. Manchi Srinivasa Rao

On Complement Submodules of a Module with FGD and related graph

Module Theory

May2000

5. Dasari Nagaraju A Theorem on Modules with Finite Goldie Dimension

Module Theory

August2004

6. Shaik Mohiddin Shaw

On Fuzzy Dimension of a Module with DCC on Submodules.

Fuzzy ModuleTheory

April2005

7. J. L. Rama Prasad Prime Fuzzy Submodules Fuzzy ModuleTheory

May2005

8. K. S. Balamurugan

Rings with Finite Dimension with respect to two sided Ideals

Ring Theory

March2006

9. D. Srinivasulu Some Results on Types of Total Graphs

Graph Theory

January2007

10. Shaik Shakeera Some Results on m-Systems and g-Systems in -rings.

-ringTheory

September2007

RESEARCH AND CONSULTANCY - Research Guidance

Ph. D’s Awarded/Guided: 6

S.No Name of the Candidate

Title of the Dissertation Area Month/Yearof Award

1. Kuncham Syam Prasad

Contributions to Near-ring Theory - II

Near-ringTheory

May2000

2. T. V. Pradeep Kumar

Contributions to Near-ring Theory - III

Near-ringTheory 2008

3. D. Nagaraju Contributions to Ring Theory Ring Theory 20094. A.V. Vijaya

KumariContributions to Near-ring Theory - IV

Near-ringTheory

2009

5 Kedukodi Bahushri Srinivas

Fuzzy and Graph Theorietical Aspects of Near-rings (Contributions to Near-ring Theory - V) Manipal University

Near-ringTheory

18th

November 2009

6 Mohiddin Shaw Sk. Contributions to Ring Theory - II

Ring Theory

(iii) Students presently working for Ph.D: 2 (two)

Research Projects operated / under operation

(1) Major Research Projects (UGC Sponsored ): (3 in number)

Title of the Project Duration Agency Sponsored and Amount

Prime Radical and Decomposition Theory of

Near-ring Modules

1994 – 1997 UGC, New Delhi

Rs. 1,04,000/-

Finite Goldie Dimension in Modules over rings and near-

rings

2004 – 2007 UGC, New Delhi

Rs. 5,75,000/-

Finite Dimension and Fuzzy Dimension in Modules and

Generalized Fields

2009-2012 UGC, New Delhi

Rs.5,42,800/-

(3) Projects with funding from outside India (3 in number)

Area of the Research Project

Duration Agency/Institution Sponsored

Scientist with whom

the Collaborati

ve work done

Radical Theory of Near-rings

(Selected Scientist under the Indo-Hungarian Cultural

Exchange Program)

June 5 – Sept 5, 2003

Hungarian Academy Sciences,

Hungary, and UGC, New Delhi

Prof. Richard

Weigandt

Theory of Near-rings

(Selected Sr. Scientist under the Bilateral Exchange

Program)

August 16 –September 05,

2005

Hungarian Academy Sciences,

Hungary, and INSA, New Delhi

Prof. Richard

Weigandt

Finite Dimension in Rings and related systems

(Visiting Professor at Walter Sisulu University, South

Africa)

March 26 –April 10, 2007

Walter Sisulu University,

Umtata, South Africa

Dr Godloza Lungisile

(4). Number of projects having collaboration with other research Organizations. (3 in number)

(i) Collaboration research with Prof. Dr Sreenadh, Professor of Mathematics, Sri venkateswar University, Tirupathi, Andhra Pradesh, India.(ii) Collaboration research with Dr Kuncham Syam Prasad, Manipal University, Manipal, Karnataka.(iii) Collaboration research with Prof. Dr. T. Srinivas, Kakatiya University, A.P.

RESEARCH PUBLICATIONS 1

National Refereed Journals (Indian Journals)1984

1. Bhavanari Satyanarayana "Primary Decomposition in Noetherian Near rings", Indian J. Pure & Appl. Math. 15 (1984) 127-130.(Zbl 0860.16033).2 V. Sambasiva Rao & Bhavanari Satyanarayana "The Prime radical in Near-rings", Indian J. Pure & Appl. Math. 15 (1984) 361-364.(Zbl 0533.16020).19863. Y.V. Reddy & Bhavanari Satyanarayana "The f-prime Radical in Near-rings", Indian J.

Pure & Appl. Math. 17 (1986) 327-330.(Zbl 0588.16030) 19884 Y.V. Reddy & Bhavanari Satyanarayana "A note on N-groups", Indian J. Pure & Appl.

Math. 19(1988) 842-845.(Zbl 0661.16033)5. Y.V. Reddy & Bhavanari Satyanarayana "Finite Spanning Dimension in N-groups". The Mathematics Student, 56 (1988) 75-80.(Zbl 0708.16013).19896. Bhavanari Satyanarayana "The Injective Hull of a Module with FGD", Indian J. Pure & Appl. Math. 20 (1989) 874-883. (Zbl 0693.16016)19907. Bhavanari Satyanarayana "On Modules with Finite Goldie Dimension", J. Ramanujan Math. Society. 5 (1990) 61-75. (Zbl 0718.16016)19918. Bhavanari Satyanarayana "On Finite Spanning dimension in N-groups", Indian J. Pure & Appl. Maths. 22 (8) 633-636, August 1991.(Zbl 0748.16024).19939. BHAVANARI SATYANARAYANA & G. KOTESWARA RAO "ON A CLASS OF MODULES AND N-

GROUPS", JOURNAL OF INDIAN MATH. SOCIETY. 59 (1993) 39-44.(ZBL 0860.16034).199610. BHAVANARI SATYANARAYANA, M.B.V. LOKESWARA RAO AND K. SYAM PRASAD "A NOTE

ON PRIMNESS NEAR-RINGS AND MATRIX NEAR-RINGS", INDIAN J. PURE & APPL. MATH. 27(3)(1996) PP227-234. (ZBL 0858.16040).

199811. Bhavanari Satyanarayana and K. Syam Prasad "A Result on E-direct systems in N-groups",

Indian J. Pure & Appl. Math. 29 (1998) 285-287. (Zbl 0903.16026)

199912. Bhavanari Satyanarayana "A Note on -near-rings", Indian J. Mathematics (B.N. Prasad

Birth Centenary commemoration volume) 41(1999) 427-433.(Zbl 1033.16501)13. K. Syam Prasad, Bhavanari Satyanarayana "A Note on IFP N-groups", Proc. 6th Ramanujan

symposium on Algebra and its Applications, Ramanujan Institute for Adv. Study in mathematics (University of Madras), Feb 24-26, (1999). pp 62-65.

200014. Bhavanari Satyanarayana, T.V. Pradeep Kumar and M. Srinivasa Rao "On Prime left

ideals in -rings", Indian Journal of Pure & Appl. Mathematics 31 (2000) 687 - 693. (INSA, New Delhi).(Zbl 0989.16028)

15. Bhavanari Satyanarayana. & K. Syam Prasad "On Direct & Inverse Systems in N-groups", Indian J. Maths (BN Prasad Birth Commemoration Volume) 42 (2000) 183 - 192.

(Zbl 1033.16021)200316. BhavanariSatyanaraya & K.Syam Prasad "An Isomorphism theorem on Directed

Hypercubes of Dimensoin n", Indian Journal of Pure & Appl.Maths 34(10)(2003) 1453-1457. (Zbl. 1044.05509)


200417. Satyanarayana Bhavanari, Syam Prasad Kuncham, & Venkata Pradeep Kumar

Tumurukota "On IFP N-groups and Fuzzy IFP ideals", Indian Journal of Mathematics,46 (2004) 11-19. (Zbl. 1079.16035)

18. Satyanarayana Bhavanari, Godloza L., & Sk. Mohiddin Shaw "On Fuzzy Dimension of a Module with DCC on Submodules", Acharya Nagarjuna International Journal of Mathematics and Information Technology, 01 (2004) 13-32.

19. Satyanarayana Bhavanari "Modules over Gamma Near-rings", Acharya Nagarjuna International Journal of Mathematics and Information Technology, 01- 02 (2004) 109-120.

200820. Satyanarayana Bhavanari, Godloza L., and Vijaya Kumari A. V., “Some Dimension

Conditions in Near-rings with Finite Dimension”, Acta Ciencia Indica, 34 (2008) pp 1397-1404 (India). (Zbl. Pre. 05582404)

21. Satyanarayana Bhavanari, Godloza L and Vijaya Kumari A. V., “Finite Dimension in Near-rings”, Journal of AP Society for Mathematical Sciences, 1 (2) (2008) 62-80 (India).

200922. Satyanarayana Bhavanari, Mohiddin Shaw Sk., Eswaraiah Setty S., Babu Prasad M. “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)23. Satyanarayana Bhavanari “A Note on Completely Semiprime Ideals in Near-rings”, International Journal of Computational Mathematical Ideas, 1(3)(2009)107-112. (ISSN: 0974 – 8652).

201024. Satyanarayana Bhavanari, Nagaraju Dasari, Godloza Lungisile and S. Sreenadh, “Some Dimension Conditions in Rings with Finite Dimension”, The P.M.U. Journal of Humanities and Sciences,Vol.l, No.1, 2010. PP: 69-75 (ISSN 0976-1853).25. Satyanarayana Bhavanari, Pradeep Kumar T.V., Sreenadh Sridharamalle and Eswaraiah Setty Sriramula, “On Completely prime and Completely Semi-prime Ideals in -near-rings”, International Journal of Computational Mathematical Ideas, Vol-2, No.1&2, 2010. PP 22-27 (ISSN: 0974 – 8652). 26. Satyanarayana Bhavanari, Mohiddin shaw Sk, Mallikarjun Bhavanari and Venkata Pradeep Kumar Tumurukota, “ A Graph Related to the Ring of Integers Modulo n”, Acta Ciencia Indica, Vol.36 M, (2010) PP 699 – 706.27. Satyanarayana Bhavanari, Syam Prasad Kuncham and Nagaraju Dasari, “ Prime Graph of

a Ring”, Journal of Combinatories, Informations & Systems Sciences, 35 (2010).

201228. BHAVANARI SATYANARAYANA, D. SRINIVASULU, KUNCHAM SYAM PRASAD, AND

ESWARAIAH SETTY S “SOME RESULTS ON DEGREE OF VERTICES IN SEMITOTAL-BLOCK GRAPH AND TOTAL-BLOCK GRAPH”, INTERNATIONAL JOURNAL OF

COMPUTER APPLICATIONS, 50 (9) 2012, PP19-22, ISSN: 0975-8887


International Refereed Journals (outside India):

19821. Bhavanari Satyanarayana "Tertiary Decomposition in Noetherian N-groups",

Communations in Algebra, 10(1982) 1951-1963.(Zbl 0493.16026). 1983

2. Bhavanari Satyanarayana "A Note on -rings", Proceedings of the Japan Academy 59-A (1983)382-383.(Zbl 0543.16027).1985

3. Bhavanari Satyanarayana "On Modules with Finite Spanning Dimension" , Proceedings of the Japan Academy, 16-A(1985) 23-25.(Zbl 0561.16010).1987

4. Y.V. Reddy & Bhavanari Satyanarayana "A Note on Modules", Proc. of the Japan Academy, 63-A (1987) 208-211.(Zbl 0677.16015).1988

5. Bhavanari Satyanarayana "A Note on E-direct and S-inverse systems", Proc. of the Japan Academy, 64-A (1988) 292-295.(Zbl 0664.16022).6. Bhavanari Satyanarayana "A Note on g-prime ideals in gamma rings", Questions Mathematicae 12(1988) 415 - 423. (Zbl 0683.16028)19947. Bhavanari Satyanarayana "Modules with Finite Spanning Dimension", J. Austral, Math.

Society. (Series A) 57 (1994) 170-178. (Zbl 0821.16006).19998. Bhavanari Satyanarayana "The f-prime radical in -near-rings", South-East Asian Bulletin

of Mathematics 23 (1999) 507-511.(Zbl 0947.16033)20059. Bhavanari Satyanarayana & K. Syam Prasad "On Fuzzy Cosets of Gamma Nearrings",

Turkish J. Mathematics, 29 (2005) 11-22. (Zbl 1066.16057)10. K. Syam Prasad & Bhavanari Satyanarayana "Fuzzy Prime Ideal of a Gamma Nearring",

Soochow J. Mathematics, 31 (2005) 121-129. (Zbl. 1080.16051)11. Bhavanari Satyanarayana. & K. Syam Prasad "Linearly Independent Elements in N-groups

with Finite Goldie Dimension", Bulletin of the Korean Mathematical Society, 42 (2005), No.3, pp 433-441. (Zbl 1078.16058)

12. Satyanarayana Bhavanari, Syam Prasad Kuncham & Venkata Pradeep Kumar Tumurukota "On Fuzzy Dimension of N-groups with DCC on Ideals", East Asian Math. J. 21 (2005), No. 2 pp. 205-216. (Zbl. 1096.16021)

200613. Bhavanari Satyanarayana, K. Syam Prasad & D. Nagaraju "A Theorem on Modules with

Finite Goldie Dimension", Soochow J. Mathematics, 32(2) (2006) 311-315. (Zbl. 1113.16028)200714. Satyanarayana Bhavanari, Syam Prasad K., Pradeep Kumar T. V., and Srinivas T. “Some

Results on Fuzzy Cosets and Homomorphisms of N-groups”, East Asian Math. J. 23 (2007), No. 1, pp. 23-36. (Zbl 1141.16032)

15. Babushri Srinivas K, Satyanarayana Bh & Syam Prasad K. “C-Prime Fuzzy Ideals of Near-rings”, Soochow Journal of Mathematics, 33 (4) (2007), pp. 891-901. (Zbl. 1140.16020)

200816. Satyanarayana Bh, Nagaraju D, Balamurugan K. S & Godloza L “Finite Dimension in

Associative Rings”, Kyungpook Mathematical Journal, 48(2008), 37-43. (Zbl. 1158.16004)17. Satyanarayana Bhavanari, Godloza L., and Nagaraju D. “Ideals and Direct Product of Zero

Square Rings”, East Asian Mathematical Journal., 24(2008) 377-387


18. Babushri Srinivas Kedukodi, Syam Prasad Kuncham and Satyanarayana Bhavanari “Equiprime, 3-prime and c-prime fuzzy ideals of nearrings”. Soft Computing – A Fusion of Foundations, Methodologies and Applications, (Subject collection: Engineering), (Springer Link date: September 24. 2008) 13 (2009) 933 - 944. (Zbl. Pre. 05586573)

200919. Satyanarayana Bhavanari, Syam Prasad K & Ram Prasad J. L “Fuzzy Cosets of Prime Fuzzy Submodules”, Journal of Fuzzy Mathematics, 17 (3)(2009) 595-604. (USA).

201020. Satyanarayana Bhavanari, Syam Prasad K., and Babushri Srinivas K. “Graph of a Nearring with respect to an Ideal”, Communications in Algebra, Taylor & Francis, Taylor & Francis group 38 : 1957 - 1967, 2010 (UK) (ISSN: 0092-7872).21. Babushri Srinivas K , Syam Prasad K and Satyanarayana Bhavanari, “Reference Points and Roughness”, Information Sciences, 180 (2010) PP 3348 – 3361.

22. Satyanarayana Bhavanari, Godloza L., and Nagaraju D., “Some Results on m-Systems and g-Systems in Rings” , Southeast Asian Bulletin of Mathematics 34, (2010): 461-468. (China). (ISSN: 1476-8186).23. Bhavanari Satyanarayana, Godloza Lungisile, Munaga Babu Prasad and Kuncham syam Prasad, “Ideals and direct product of zero square nearrings”, International Journal of Algebra, Vol.4 (No.16) (2010) PP 777-789.

201124. Bhavanari Satyanarayana, D. Nagaraju, M. Babu Prasad and Sk. Mohiddin Shaw, “On

the dimension of the Quotient Ring R/K where K is a complement”, International Journal of Contemporary Advanced Mathematics (IJCM), Vol.1.Issue 2. (2011) PP 16-

22. (Malesia)25. Nagaraju D., Satyanarayana Bhavanari, Babu Prasad M., and Venkatachalam A “Some Results on Fuzzy Ideals of M gamma Modules”, South East Asian Bulletin of Mathematics, accepted for publication, 2011. 26. Satyanarayana Bhavanari, Godloza L., and Nagaraju D., “Fuzzy Ideals of Zero-square Rings”, Journal of Fuzzy Mathematics, 19 (No.3) (2011). (USA)27. Satyanarayana Bhavanari, Godloza L., and Nagaraju D., “Some results on Principal Ideal graph of a ring”, African Journal of Mathematics and Computer Science Research Vol.4 (6), 2011. PP 235-241.201328. S. KEDUKODI, S. P. KUNCHAM AND SATYANARAYANA BHAVANARI, “NEARRING

IDEALS, GRAPHS AND CLIQUES”, INTERNATIONAL MATHEMATICAL FORUM, 8 (2)(2013) 73-8329. Satyanarayana Bhavanari, Vijaya Kumari A. V., Godloza L., and Nagaraju D. “Fuzzy Ideals of Modules over -near-rings”, Italian Journal of Pure and Applied Mathematics (Italy), accepted for publication


RESEARCH PUBLICATIONS (Published in the Proceedings)

Proceedings of National Conferences (in India)1. Bhavanari Satyanarayana "On Essential E-irreducible submodules", Proc. 4th Ramanujan

symposium on Algebra and its Applications, University of Madras 1-3 Feb 1995, pp 127-129.2. K. Syam Prasad, Bhavanari Satyanarayana "A Note on IFP N-groups", Proc. 6th Ramanujan

symposium on Algebra and its Applications, Ramanujan Institute for Adv. Study in mathematics (University of Madras), Feb 24-26, (1999). pp 62-65.

3. Satyanarayana Bhavanari, “A Note on Semi-Prime Near-rings”, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications, November 11-12(2010). PP 92-95.4. T.V Pradeep Kumar, Satyanarayana Bhavanari, Kuncham Syam Prasad and Mohiddin Shaw Sk, “Some Results on Completely Semi-Prime Ideals in Gamma Near-rings”, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications, November 11- 12(2010). PP 101-105.5. Kuncham Syam Prasad, Satyanarayana Bhavanari and G.V. Subba Rao, “Generalized Fuzzy Ideals of Gamma Near-rings”, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications, November 11-12(2010). PP 111-118.6. Satyanarayana Bhavanari, Mohiddin Shaw Sk, and Venkata Vijaya Kumari Arava, “Prime Graph of an Integral Domain”, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications, November 11-12(2010). PP 124-134.7. Satyanarayana Bhavanari and Sk. Shakira, “Gamma Rings and m-systems”, Proceedings of the National Seminar on Present Trends in Mathematics and its Applications, November 11- 12(2010). PP 141-147.

PROCEEDINGS OF INTERNATIONAL CONFERENCE (HELD IN INDIA)

1. Bhavanari Satyanarayana "On Modules with FSD and property (p)", Proc. Ramanujan Centennial International Conference, Annamalainagar, 15-18, December 1987, pp137-140. (Zbl 0704.16019)

2. Satyanarayana Bhavanari, D.Nagaraju, Sk. Mohiddin Shaw and 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) January 11-13, (2010). (Publisher: Macmillan Research Series, 2010) PP.681-687. (India). (ISBN: 978 – 0230 – 32875 – 4).

3. Satyanarayana Bhavanari, Sk. Mohiddin Shaw, Mallikarjun Bhavanari and T.V.Pradeep Kumar , “ On a Graph related to the Ring of Integers Modulo n ”, Proceedings of the International Conference on Challenges and Applications of Mathematics in Science and Technology (CAMIST) January 11-13, (2010). (Publisher: Macmillan Research Series, 2010) PP.688-697. (India). (ISBN : 978 – 0230 – 32875 – 4).

PROCEEDINGS OF INTERNATIONAL CONFERENCE (HELD IN ABROAD)

1. Satyanaryana Bhavanari & Richard Wiegandt "On the f-prime Radical of Near- rings", in the book Nearrings and Nearfields (Edited by H. Kiechel, A. Kreuzer & M.J. 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. (Zbl. 1082.16049)

2. Bhavanari Satyanarayana. & K. Syam Prasad "On Finite Goldie Dimension of Mn(N)-group Nn " in the book Nearrings and Nearfields (Edited by H. Kiechel, A. Kreuzer & M.J. Thomsen) (Proc. 18th International Conference on Nearrings and Nearfields, Universitat Bundeswar, Hamburg, Germany, July 27-Aug 03, 2003) Springer Verlag, Netherlands, 2005, pp 301-310. (Zbl. 1078.16059)

MEMBERSHIP IN EDITORIAL BOARDS

1. Honorary Editor for the Mathematical Periodical: Ganitha Chandrika (Published by the Mathematics Association: AIMEd, Vijayawada, Andhra Pradesh).

2. Honorary Editor for the Mathematical Periodical: Ganitha Vahini (Published by theRamanujan Academy, Ramachandrapuram, E.Godavari DT, Andhra Pradesh).

3. Member Secretary and one of the Managing Editors, Research Journal: ANIJMIT (Acharya Nagarjuna International Journal of Mathematics and Information Technology), Acharya Nagarjuna University, Andhra Pradesh. (Since 2004)

4. Honorary Editor for the Journal IJCMI (International Journal of Computational Mathematical Ideas), Andhra Pradesh. (Since 2009).

5. Reviewer Member, Scientific Journals International, USA (Since 2009)

6. Associate Editor, Mind Share International Journal of Research and Development, World Education Network (Since 2011).

7. Editor, Journal of Mathematical Sciences and Applications, Science and Education Publishing, USA (Since 2012).

8. Member, Advisory Board, Journal of Statistics and Mathematics, Bio-info Publications, (ISSN:0976-8877(print), E-ISSN: 0976-8815), Impact Factor Value (ICV): 4.47, Marketed by ‘EBSCO Publishing, USA’ (Sicne 2013).

9. Editorial Board Member, International Journal of Mathematics and Statistics (IJMS), International Academy of Science, Engineering and Technology (IASET) (since 2013),

Seminars/Conferences/Symposia

INTERNATIONAL CONFERENCES ATTENDED/PRESENTED PAPERS/DELIVERED INVITED TALKS

1. International Symposium on Algebra and its applications, January 1-7, 1985, Nagarjuna University, Nagarjuna Nagar. (Presented a paper entitled “A Generalization of prime ideals in -Near-rings”).

2. International Conference on Mathematics with special emphasis on Number Theory (Ramanujan Centennial International Conference), Dec. 15-18, 1987, Annamalai University, Annamalai Nagar (Presented a paper entitled “On Modules with FSD and a property (p)”. The paper published in the proceedings).

3. International Conference on General Algebra, August, 21-27, 1988, Krems, Vienna, Austria. (Delivered a lecture “On Modules with Finite Goldie dimension and Finite Spanning dimension”).

4. Asian Mathematical Society Conference, August 14-18, 1990, Hong Kong University, Hong Kong. (Presented a paper entitled “Modules with Finite Spanning Dimension”).

5. International Conference on the Theory of Radicals and Rings, July 6-11, 1997, University of Port Elizebeth, Port Elizebeth, South Africa. (Delivered a talk entitled “The f-prime radical in gamma near-rings”).

6. International Conference on Near-rings and Near-fields, July 14-18, 1997, University of Stellenbosch, Stellenbosch, South Africa. (Presented a talk on “IFP N-groups”).

7. International Symposium on Mathematics and its Applications Nov. 13-15, 2002, University of Calcutta, Kolkata (Delivered an invited talk on “Finite Spanning dimension in Modules & N-groups”, and chaired a session).

8. 18th International Conference on Nearrings and Nearfields, July 27 – Aug 03, 2003, Universitat der Bundeshwer, Hamburg, Germany. (Presented a talk on FGD in N-groups).

9. 19th International Conference on Nearrings and Nearfields, July 31 – Aug 06, 2005, National Cheng Kung University, Taiwan. (Presented a talk on Independent elements of N-groups).

10. International Conference on Radicals (ICOR-2006), July 30 - Aug 05, 2006, Kyiv National Taras Shevchenko University, Ukraine (delivered a talk on The f-prime Radical of Near-rings).

11. Fourteenth Ramanujan Symposium International Conference on Non-Commutative Rings, Group Rings, Diagram Algebras and Applications, December 18-22, 2006, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai (presented a paper on Dimension of a Vector Space / Modules).

12. International Conference on Algebra and Related Topics (ICART 2008), Chulalongkorn University, Bankok, Thailand, May 28-30, 2008 (delivered an invited talk on “Principal Ideal Graph of a Ring”).

13. Indo-French Conference in Mathematics-2008, held at Institute of Mathematical Sciences and Chennai Mathematical Institute, Chennai, December 15-19, 2008 (attended the Conference).

14. ICM-2010 (INTERNATIONAL CONGRESS OF MATHEMATICIANS), Hyderabad, August 19-27, 2010 (Presented a short communication on “Prime graph of a Ring”).

15. International Conference on the Theory of Radicals, Rings and Modules, SULTAN QABOOS UNIVERSITY, Jan. 20-26, 2012. (Delivered an Invited talk and chaired a session in the Conference).

Enclosure Page 4


NATIONAL CONFERENCES ATTENDED/PRESENTED PAPERS/DELIVERED INVITED TALKS

1. 46th Conference of Indian Mathematical Society, Dec. 25-27, 1980, Bangalore University, Bangalore. (Presented the paper “The Prime Radical in Near-rings”).

2. 47th Conference of Indian Mathematical Society , Dec. 25-27, 1981, Mehta Research Institute, Allahabad. (Presented my paper entitled “Tertiary Decomposition in Noetherian N-groups”).

3. 49th Conference of Indian Mathematical Society, Dec. 27-29, 1983, IIT Madras. (Presented the paper entitled “On -Near-rings”).

4. National Symposium on Algebra and Applications, 5-9 March, 1984, M.I.T. Madras. (Presented my paper entitled “A Radical for M -Modules”).

5. 51st Annul Conference of Indian Mathematical Society, Dec. 28-30, 1985, University of Cochin, Cochin. (Presented a paper entitled “N-groups with Finite Goldie Dimension”).

6. 52nd Annul Conference of Indian Mathematical Society, Dec. 27-29, 1986 University of Rajastan, Jaipur. (Presented a paper entitled “On Modules with FGD”).

7. 3rd Annul Conference of the Ramanujan Mathematical Society, May 27-29, 1988, Manipal Institute of Technology, Manipal. (Presented a paper entitled “The Injective Hull of a Module with FGD”).

8. 54th Annul Conference of Indian Mathematical Society, Dec. 27-30, 1988, University of Poona, Pune. (Presented a paper entitled “A Note on the g-prime radical in gamma rings”).

9. Diamond Jubilee (60th) Conference of Indian Mathematical Society, University of Poona, Pune,27-30 Dec. 1994. (Presented a paper entitled “A Note on weakly prime left ideals in Near-rings and Matrix Near-rings”).

10. 4th Ramanujan Symposium (1994-95) on Algebra and its Applications, 1-3 Feb., 1995 at Ramanujan Institute for Advanced Study in Mathematics, Madras. (Presented a paper “On Modules Finite Dimension Conditions”).

11. 10th Annul Conference of Ramanjuan Mathematical Society, 25-27 May, 1995, Govt. P.G. College, Rishikesh. (U.P.). (Presented the paper entitled “On Essential E-irreducible submodules of a ring with FGD”).

12. 62nd Annul Conference of Indian Mathematical Society, 22-25 Dec. 1996, IIT Kanpur. (Presented a paper entitled “Prime Radical in -Near-rings”).

13. 13th Annul Conference of Ramanujan Mathematical Society, June 4-6, 1998, Manonmaniam Sundaranar University, Tirunelveli, (Tamil Nadu). (Presented a paper entitled “A Characterization of f-prime radical in -Near-rings”).

14. 6th Ramajunan Symposium on Algebra and its Applications, Feb.24-26, 1999; The Ramanujan Institute for Advanced Studies in Mathematics, University of Madras, Madras (Presented a paper entitled “On Minimal f-prime and f-semi prime ideals in -rings”).

15. Pure Mathematics Seminar 2001, Calcutta University, Feb. 20-21,2001. (Delivered a Lecture on ‘Finite Goldie Dimension in Modules’ on 20.02.2001, and chaired a session on 21.02.2001 at this conference)

16. Seminar on Engineering Mathematics, Sambhram Institute of Technology, Bangalore, November 5-6, 2004. (Delivered an invited talk on ‘Fuzzy Sets & Applications’)

17. Two-days National Seminar on ‘Role of Mathematics in Modern Computing World’, Dec.21-22, 2004, Sri Venkateswara College, Suryapet, Andhra Pradesh. (Delivered a lecture on Role of Mathematics in the Modern World).

18. National Seminar on “Fluid Mechanics – Its Applications to Science & Technology”, Feb 09-10, 2005, Osmania University, Hyderabad (Attended as a participant).

Enclosure Page 5


200619. National Conference on “Mathematics and its Applications” December 21-22, 2006, Sri

Venkateswara University, Tirupati. (Delivered an invited talk on ‘Dimension properties of Modules over Rings’).

20. National Seminar on “Algebra and its Applications”, December 27-28, 2006, Sri Padmavathi Mahila Visvavidyalayam, Tirupati (Delivered an invited talk on ‘Finite Goldie Dimension and Finite Spanning Dimension in Modules and N-groups’).

21. National Seminar on “Recent Trends in Algebra and Applications”, December 29-31, 2006, Andhra University, Visakhapatnam (Delivered an invited talk on ‘Generalization of the Concept Dimension of Vector Spaces to Modules’).

200722. Participated and Chaired a Session in the “XVI Congress of the Andhra Pradesh Society for

Mathematical Sciences”, 08th – 10th December, 2007, National Geophysical Research Institute, Hyderabad.

200823. Attend and delivered an invited talk at the “95th Indian Science Congress”, Andhra University,

Visakhapatnam, January 3-7, 2008 (Title: Fuzzy Ideals of Zero Square Rings). 24. National Seminar on “Mathematical Methods for Research”, February 8-9, 2008, A.V.C College

(Autonomous), Mannampandal, Mayiladuthurai, Tamilnadu (Delivered an invited talk on ‘Fuzzy Ideals in -near-rings’).

201025. National Seminar on “Present Trends in Mathematics and its Applications” (Sponsered by UGC),

November 11th and 12th, 2010 SGS College, Jaggaiahpet, Krishna Dt, A.P, (Delivered a talk on “Dimension in Vector Space”).

201126 National Seminar on “Present Trends in Algebra and its Applications”, July 11th and 12th, 2011

JMJ College, Tenali, Guntur Dt, A.P, (Delivered a talk on “Gamma Near-rings”).27. National Seminar on Mathematics/Algebra, 25 Oct. 2011 (200th Birth Day of Galois), KBN

College, Vijayawada, Andhra Predesh, (Delivered the Key Note Address: Dimension of Vector Spaces and Modules).

28 National Seminar, Erode, December 19, 2011 (Delivered the Inaugural Talk entitled”……”).

201229. National Conference on Mathematical Sciences (NCMS-2012) (Delivered an Invited talk on

“Gamma Near-rings”), School of Mathematical Sciences, North Maharastra University, Jalagon, India, 05 March 2012.

30. National Conference on Mathematical and Computational Sciences, July 5-6, 2012, (Delivered an invited talk on GAMMA NEAR-RINGS, and also Chaired a Session of the National Conference), Adikavi Nannaya University, Rajahmundry, Andhra Pradesh.

31. National Conference "Geometry, Algebra, Logic and Number Theory, Applications" (Delivered a Lecture on “Gamma Near-rings”)Department of Studies and Research in Mathematics, Tumkur University, Karnataka, India, 6th December, 2012.

32. XXI Congress of APSMS (AP Society for Mathematical Sciences), (Delivered Prof. A. Radha Krishna Endowment Lecture in Algebra entitledTHE PRIME GRAPH OF AN INTEGRAL DOMIAN), S.V. University, Tirupati, Andhra Pradesh, India, during 07-09 Dec 2012


33. National Work Shop on “Mathematics Flavour in Vedas and Shastras” (On the Occasion of 125th Birth Anniversary of Srinivasa Ramanujam) (Delivered an Invited Talk “SOME MATHEMATICAL CONCEPTS IN ANCINT SANSKRIT WORKS”), Sri Venkateswara Vedic Universtiy, Tirupati, Andhra Pradesh, India, during December 21-22, 2012

34. International Conference on Mathematical Sciences, (Delivered an invited talk on Gamma Near-rings), S. S. E. S. Amt’s Science College Nagpur, India, during December 28-31, 2012

201335. National Work Shop on Computer Applications based on Algebra, (Delivered an Invited

Talk on “The Prime Graph of an Integral Domain”), Conducted by the Department of Mathematics and the Department of Computer Science & Engineering, Manipal Universtiy, Karnataka, India, during January 03 – 05, 2013

36. National Seminar on Evariste Galois and Srinivasa Ramanujan (Sponsored by UGC) (Delivered an Invited Talk entitled: Prime Graph of an Integral domain), P.V.K.N. Govt. College, Chittoor, Andhra Pradesh, India, held during March 14-15, 2013.

37. National Seminar on Recent Developments in Mathematical Sciences (Sponsored by UGC under SAP) , Sri Venkateswara University, Tirupati, Andhra Pradesh, June 28, 2013. (Delivered an Invited Talk on FUZZY IDEALS OF GAMMA NEAR-RINGS).


Local/Regional Seminars/Conferences

1. Delivered A. Apparao Memorial Lecture on “Mathematics & its Applications” at Satavahana Mahila Kalasala, Vijayawada on 1st November, 2003 conducted by APAMT (Hyderabad) & AIMEd., (Vijayawada). Chief Guest: Prof. P. V. Arunachalam (Former Vice-Chancellor, Dravidian University, AP).

2. MEDHA 2005 – A Mathematics Seminar, Jan 20, 2005, Andhra Loyola College, Vijayawada(Delivered two lectures on “Fundamentals of Linear Programming” and “Graph Theory”).

3. One day Seminar on Mathematics, July 23, 2005, E.V.R.M. Degree & P.G. College, Kodad(Delivered two lectures on “Modern Mathematics & Applications” and “Creating Interest to Learn Mathematics”).

4. Delivered a lecture on “Generalization of dimension of Vector spaces to Modules” to a gathering of Research scholars and students of Mathematics, Sri Padmavathi Mahila University, Tirupathion 15-07-2006.

5. Delivered a lecture on “Development of Mathematics” to the students of B. Sc., / B. A (Mathematics), P. B. Siddhartha College of Arts & Sciences, Vijayawada on 22nd August, 2006.

6. Delivered a lecture on “Maths Applications” at S.K.R.B.R. College, Narasaraopet on 27th

October, 2006.7. Delivered extension lectures on “Graph Theory, Near-rings & Fuzzy Sets” at the Department of

Mathematics, Kakatiya University, Warangal on 13th & 14th February, 2007.8. Delivered a Guest Lecture on “Graph Theory” to M. Sc., Applied Mathematics students of

Kakatiya Institute of Technology, Warangal on 14th February, 2007.9. Delivered the Inaugural Lecture on “Mathematics- Its Significance” in the One Day Seminar on

Mathematics – Its Significance (including Ganitha Astavadhanam)”, at Smt. Gentela Sakuntalamma College, Jaggayyapet on 22nd November, 2007.

10. Delivered a Inaugural Lecture in the One Day Seminar on Mathematics (including Ganitha Astavadhanam)”, at Kakani Venkata Ratnam College, Nandigama on 23rd November, 2007.

2013.02.04 and 0511. Delivered a Lecture on “Applications of Mathematics and Graph Theory” in the Work Shop ‘Mathematical Applications of Engineering Disciplines’ (conducted by the ADITYA Institute of Technology and Management), Tekkali-532201, Srikakulam District, Andhra Pradesh, India during February 04-05, 2013.

2013.02.2412. Delivered a Lecture on “Importance of Mathematics” in the ‘One Day Seminar and Expo on Ancient Indian Mathematics’ (conducted by the Association for Improvement of Maths Education), Sisu Vidya Mandir High School, Vijayawada-11, Andhra Pradesh, India, on 2013.02.24..

Seminars/conferences organized

1. Prof. Bhavanari Satyanarayana is the Director of the National Seminar on Algebra & its Applications (NSAA- 2006) organized by the Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India, Jan 05-06, 2006.

2. Prof. Bhavanari Satyanarayana is the Academic Secretary of the National Seminar on Presents Trends in Mathematics and its Applications organized by the Department of Mathematics, (sponsored by UGC) held at SGS College, Jaggaiahpet, Andhra Pradesh, India, (in Collaboration with the Department of Mathematics, Acharya Nagarjuna University), November 11th and 12th, 2010.

3. Prof. Bhavanari Satyanarayana is the Academic Secretary of the National Seminar on “Presents Trends in Mathematics and its Applications” (sponsored by UGC) held in JMJ College, Tenali, Andhra Pradesh (in Collaboration with Department of Mathematics, Acharya Nagarjuna University), July 11th and 12th, 2011.

4. Prof. Bhavanari Satyanarayana is the Academic Secretary of the National Seminar on “National Seminar on Algebra, 25 Oct. 2011 (200th Birth Day of Galois) held in KBN College, Vijayawada, Andhra Pradesh (in Collaboration with Department of Mathematics, Acharya Nagarjuna University), October 25, 2011.

5. Also conducted four one-day national seminars in Vijayawada as the President of Association for Improvement of Maths Education during 2005-2009 (One seminar each year).

NATIONAL BUILDING ACTIVITIES

Written a series entitled “MATHEMATICS PROBLEMS/PUZZLES – FOR ALL, TO

THINK” (about 50 Saturdays) (named as PRAGNYA series) for the news paper VISALANDRA (a state level news paper).



Organized Regional Seminars and Talent Tests (for High School Students) every year as the President of AIMEd., (for four years).

Elected Executive Member (2007-2008), Andhra Pradesh Society for Mathematical Sciences, Hyderabad.

Member of the Text book development Committee, SCERT (State Council for Education, Research and Training), Hyderabad, 2009. Selected Hon. President (2011 to date) of the Association for Improvement of Maths Education, Vijayawada.

Elected Office Secretary (Dec. 2012-2014, for a period of two years), Andhra Pradesh Society for Mathematical Sciences, Hyderabad

Member of the Vidyalaya Management Committee, Kendriya Vidyalaya, Nallapadu, Guntur

Dyputy Director General, IBC, England, 2011.

Member of the Board of Studies (UG) in Mathematics, Andhra Layola College, Vijayawada.

Member of the Board of Studies (UG) in Mathematics, KBN College, Vijayawada, March 2010 – 2012 (two years).

Member of the Board of Studies (UG) in Mathematics, Machilipatnam, Krishna District, 2012 – 2014 (two years).

Member of Board of Studies in Mathematics (both PG and UG) Kakatiya University, Warangal, 2008-2010..

Member of Board of Studies in Mathematics (PG), Sri Venkateswara University, Tirupathi, 2010 - 2013 (three years).

40 books authored/edited, which provides/improves the knowledge of the readers and hence helps towards the education development of the society/world.

Administrative Experience/Assignments:

(i) Teacher Associate for N.U.P.G. Computerized Results-March 1999

(ii) Convener (Guntur District) for the State Level Talent Test in Mathematics for High School Students (in A.P) conducted (on 29-10-2000) by AIMEd,

(iii) Special Officer (i/c), N.U. P.G. Centre at Ongole, in November 2000

(iv) Co-coordinator, Refresher Course in Mathematics, N.U., Dec 1-21,2001

(v) Member Secretary of the Research Journal – Acharya Nagarjuna International Journal of Mathematics and Information Technology (Publisher: The Registrar, Acharya Nagarjuna University).

(vi) Principle Investigator of three Major Research Projects (Sponsored by UGC, New Delhi) (1994-1997) (2004-2007) (2009-2012).

(vii) Head of the Department, Department of Mathematics, Acharya Nagarjuna University (April 2005 – April 2007).

(viii) Chairman, Board of Studies (PG) in Mathematics, ANU (Feb 08 – August 2011)

(ix) Director of the National Seminar on Algebra & its Applications (NSAA- 2006) organized by the Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India, Jan 05-06, 2006.

(x) Selected member of Vidyalaya Management Committee (October 2005 -October 2008).

(xi) Academic Secretary of the National Seminar on Presents Trends in Mathematics and its Applications organized by the Department of Mathematics, (sponsored by UGC) SGS College, Jaggaiahpet, Andhra Pradesh, India, November 11th and 12th, 2010. (xii) Academic Secretary of the National Seminar on “Presents Trends in Mathematics and its Applications” organized by the Department of Mathematics, (sponsored by UGC) JMJ College, Tenali, Andhra Pradesh, India, July 11th and 12th, 2011.

OTHER ACADEMIC INFORMATION

ii) Completed P.G. Certificate course in Statistics (Part time course) of Nagarjuna University with First class and First rank. During 1984-85.

iii) Undergone the Intensive Course on Programming & Application of Electronic Computers, 21st session, Indian Statistical Institute, Calcutta during 13.01.1986 to 21.03.1986.

iv) Completed a Certificate course in Computer Programming (One year Correspondence Course) of Annamalai University during 1986-87.

v) Attended the Instructional Conference on Commutative Algebra and its Applications to Combinatorics, University of Bombay, 19th Dec 1988 to 6th January 1989.

Bio-Data of Editors

Prof. Dr Bhavanari Satyanarayana have 28 yrs Teaching experience in Acharya Nagarjuna Univ. Authored 36 books (including a book by Prentice Hall of India, New Delhi, and Five books by VDM Verlag Dr Muller, Germany). Published 66 Research papers (Algebra/Fuzzy Algebra/Graph Theory) in International Journals. He is a Member of several Editorial Boards, Mathematical Journals. He is an AP SCIENTIST–2009 Awardee, a Fellow AP Akademi of Sciences 2010. He received Shiksha Rattan Puraskar Award (IIFS, New Delhi, 2011), Glory of India Award and International Achievers Award (Indo-Thai Friendship Banquet, Thailand, 2011). Top 100 Professionals -2011 (International Biographical Centre, Cambridge, England). Rajiv Gandhi Excellence Award (Aug 2011); Deputy Director General (IBC, England, 2011). Collaborative Distance with Ein-stein is 5. Got Paul Erdos No. 3, Scientist UGC-HAS (Hungarian Academy of Sciences), 2003. Sr Scientist INSA–HAS 2005. Principal Investigator of 3 MAJOR Research Projects (UGC). He Introduced an algebraic system “Gamma near-ring”. Awarded Five Ph.D., and 10 M.Phil., Degrees under his supervision. Visiting Professor, Walter Sisulu University, South Africa (2011). Visited Austria (1988), Hongkong (1990), South Africa (1997), Germany (2003) Hungary (2003), Taiwan (2005), Singapore (2005), Hungary (2005), Ukraine (2006), South Africa (2007), and Thailand (2008, 2011) on official works (to deliver lectures/

Collaborative research work).

Mr V.V.N Suresh Kumar has 8 years of teaching experience in KBN

College, Vijayawada. He completed his M.Sc (Mathematics) from

Osmania University, Hyderabad. Presently Heading the Department

of Mathematics, KBN College, Vijayawada (Andhra Pradesh). He is

the Organising Secretary of the One-Day National Seminar on Alge-

bra (200th Birth Anniversary Celebrations of Evariste GALOIS), KBN

College, Vijayawada, 25, October, 2011.

Mr. Sk. Mohiddin Shaw completed his M. Phil., (Module Theory) under the guidance of Dr Bhavanari Satyanarayana (AP SCIEN-TIST 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 Nine Conferences/Seminars/ Workshops. He worked as a faculty in the ANU P.G Centre at Ongole for 5 years. He published Eight research papers in National and International

Journals. He is the co-author of the book “Fuzzy Dimension of Modules over rings” published by VDM Verlag Dr Muller, Germany, 2010. Authored/Edited four Books. Submitted thesis for Ph.D.,

(Title: Contributions to Ring Theory–II) under the guidance of Prof. Bhavanari Satanarayana

Thailand, March 26, 2011

153406929...

Education

Transcript of 153406929...