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Abstract In reference [2], we used the elementary method to study the mean value properties of a new arithmetical function, and obtained two mean value formulae for it, but there [...]

In this paper we establish an identity involving logarithmic derivative of theta function by the theory of elliptic functions. Using these identities we re-deduced some Ramanujan's modular identities from elementary approach, and also found many other new interesting identities. Keywords Theta function, elliptic function, logarithmic derivative. 2000 Mathematics Subject Classification: Primary 26A48,33B15; Secondary 26A51, 26D07,26D10, [section]1. Introduction Assume throughout this paper that q = [e.sup.[pi]x[tau]], where ℑ[tau]>0. As usual, the classical Jacobi theta functions are defined as follows, [[theta].sub.1](z|[tau]) = -i[q.sup.1/4][[infinity].summation over (n = -[infinity])][(-1).sup.n][q.sup.n(n [...]

Let G be a(p, q) graph. Let f be a map from V(G) to {1, 2, ..., p}. For each edge xy, assign the label [absolute value of f(x) - f(y)]. f is called a difference cordial labeling if f is a one to one map and [absolute value of [e.sub.f](0) - [e.sub.f](1)] ≤ 1 where [e.sub.f](1) and [e.sub.f](0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph. In this paper we investigate the difference cordial labeling behavior of subdivision of some graphs. Keywords Subdivision, web, wheel, book, corona. 2000 Mathematics Subject Classification: 05C78, §1. Introduction and preliminaries Let G = (V, E) be (p, q) graph. In this paper we have considered only simple and undirected graphs. The number of vertices of G [...]

In the paper [1], the following problems have been proposed. Is it true that every connected bicritical graph has a minimum dominating set containing any two specified vertices of the graphs? Is it true if G is a connected bicritical graph, then [gamma](G) = i(G) , where i(G) is the independent domination number? We disprove the second problem and show the truth of the first problem for a certain family of graphs. Furthermore this family of graphs is characterized with respect to bicriticality, diameter, vertex connectivity and edge connectivity. Keywords Domination number, bicritical, diameter, connectivity, circulant graph., [section]1. Introduction and preliminaries In this paper, we concerned only with undirected simple graphs (loops and multiple edges are not allowed). All notations on graphs that are not defined here [...]

In continuation of earlier work [14], we further investigate several characterizations of Fuzzy Baire spaces. Keywords Fuzzy dense, Fuzzy nowhere dense, Fuzzy first category, Fuzzy second category, totally Fuzzy second category, Fuzzy [F.sub.[sigma]]-set, Fuzzy [G.sub.[delta]]-set, Fuzzy nodec, Fuzzy regular, Fuzzy Baire spaces., [section]1. Introduction The concepts of Fuzzy sets and Fuzzy set operations were first introduced by L. A. Zadeh in his classical paper [15] in the year 1965. Thereafter the paper [...]

In this paper we prove some results towards classifying Smarandache groupoids which are in Z*(n) and not in Z(n) when n is even and n is odd. Keywords Groupoids, Smarandache groupoids., [section]1. Introduction and preliminaries In [3] and [4], W. B. Kandasamy defined new classes of Smarandache groupoids using [Z.sub.n]. In this paper we prove some theorems for construction of Smarandache [...]

and definitions and statement of results In this paper, we give the proof of a simple Theorem which simultaneously implies the Sophie Germain primes conjecture and the Mersenne primes conjecture, by using only elementary combinatoric, elementary arithmetic congruences, elementary logic, induction and reasoning by reduction to absurd. In addition we also show that the previous two conjectures that we solved elementary were an immediate consequence of the Fermat's last conjecture. Moreover, our paper clearly shows that the Mersenne primes conjecture and the Sophie Germain primes conjecture were only simple arithmetic conjectures, so that strong investigations used in the past to try to solve the previous two conjectures were clearly not welcome. we recall that a Mersenne prime (see [1],[4],[6],[9],[10]) is a prime of the form [M.sub.m] = [2.sup.m] - 1, where m is prime, for example M13 and [M.sub.19] are Mersenne prime. Mersenne primes are known for some integers > [M.sub.19] and it is conjectured that there are infinitely many Mersenne primes. We recall, (see [2]), that a prime h is called a Sophie Germain prime, if both h and 2h +1 are prime; the first few Sophie Germain primes are 2, 3, 5,11, 23, 29,41, ... and it is easy to check that 233 is a Sophie Germain prime. Sophie Germain primes are known for some integers > 233 and it is conjectured that there are infinitely many couples of the form (h, 2h + 1), where h and 2h + 1 are prime, the Sophie Germain primes conjecture. Finally, we recall ([3],[4],[5],[7],[8],[11]) that the Fermat's last conjecture solved by A. Wiles in a paper of at least 105 pages long (see [11]), and resolved by Ikorong Nemron in a detailled paper of only 19 pages long (see [8]) states that when n is an integer [greater than or equal to] 3, the equation [x.sup.n] + [y.sup.n] = [z.sup.n] has no non-zero integer solutions for x, y and z, in other words, no three integers of the form x [greater than or equal to] 1, y [greater than or equal to] 1 and z [greater than or equal to] 1 can satisfy the equation [x.sup.n] + [y.sup.n] = [z.sup.n]. Keywords Sophie Germain primes, Mersenne primes. 2000 AMS Classification : 05XX and 11XX., [section]1. Denotations and simple properties For every integer n [greater than or equal to] 2, we define M(n), [m.sub.n] and [m.sub.n,1]; H(n), [h.sub.n] and [h.sub.n,1] as follow: M(n) = {x;1 [...]

In this paper the concept of soft fuzzy T'-prefilter, soft fuzzy T'-ultrafilter, soft fuzzy prime T'-prefilter are introduced. The concept of soft fuzzy [F.sup.*] space and soft fuzzy normal family are discussed. The concept of soft fuzzy C space is established. Also the process of structure compactification of soft fuzzy C space is established. Keywords Soft fuzzy T'-prefilter, soft fuzzy T'-ultrafilter, soft fuzzy prime T'-prefilter, soft fuzzy [F.sup.*] space, soft fuzzy normal family, soft fuzzy C structure, soft fuzzy C space, soft fuzz -y C structure prefilter. 2000 Mathematics Subject Classification: 54A40, 03E72., [section]1. Introduction Zadeh [6] introduced the fundamental concepts of fuzzy sets in his classical paper. Fuzzy sets have applications in many fields such as information [3] and control [4]. In [...]

The purpose of this paper is to introduce the concept of b-connectedness in smooth fuzzy topological space by using r-fuzzy b-open sets and study some of their fundamental properties. Further, several characterizations of these spaces are established. Keywords Fuzzy b-connectedness, fuzzy super b-connectedness, fuzzy strongly b-connectedness, fuzzy weakly b-connectedness, fuzzy upper* b-continuous, fuzzy lower* b-continuous. 2000 Mathematics Subject Classifacition: 54A40-03E72., [section]1. Introduction and preliminaries The concept of the fuzzy set was introduced by Zadeh [8] in his classical paper. Fuzzy sets have applications in many fields such as information [4] [...]

In this paper, we investigate the influence of some subgroups of Sylow subgroups with semi cover-avoiding property and S-supplementation on the structure of finite groups. Some conditions of p-nilpotency are obtained and some recent results are generalized. Keywords SCAP, S-supplemented, p-nilpotent, subgroup., [section] 1. Introduction Throughout the paper, all groups are finite. We use conventional notions and notation, as in Huppert [1]. G always denotes a group, [absolute value of G] is [...]

The objective of this paper is to present several inequalities between the angles of a triangle and the sides based on Jordan's inequality, 2[pi] / x < sin x < x, for every x [member of] (0,[pi] / 2). Keywords Geometric inequality, triangle. Mathematics Subject Classification (2010): 26D15, 51M16., [section]1. Introduction In this paper, we will study the inequalities of type f(a,b,c,A,B,C,r,s,R,[DELTA]) [greater than or equal to] 0, in an acute triangle ABC, where a,b,c are the lenghts of [...]

RC-graphs are special type of graphs. This paper deals with the Eigen values of adjacency matrices and other related matrices of RC-graphs. Many interesting results on eigen values apart from eigen values of general graphs are obtained for RC-graphs. Several directions are initiated towards possible future study. Keywords Relative character graph, adjacency matrix, Frobenious group, Borel subgroup, Dihedral group., §1. Introduction In this paper, we consider the adjacency matrix for RC-graphs and the eigen values are taken into account and a handful of results are obtained. An RC-graph is [...]

A near-ring N is said to be [S.sub.1] if for every a [member of] N, there exists x [member of] N* = N - {0} such that axa = xa. Closely following this, we introduce in this paper the concept of [S.sub.2] near-rings. A near-ring N is said to be an [S.sub.2] near-ring if, for every a [member of] N, there exists x [member of] N* such that axa = ax. Further by generalizing this, we introduce strong [S.sub.2] near-rings i.e., near-rings in which aba = ab for all a,b [member of] N. We discuss some of their properties, obtain a characterisation and also a structure theorem. Keywords [S.sub.2] near-rings, strong [S.sub.2] near-rings, zero divisors., [section]1. Introduction Throughout this paper N stands for a right near-ring (N, +, *) and '0' denotes the identity element of (N, +). A subgroup (M, +) of (N, +) [...]

In this paper we discuss the probability of strings generated by various forms of probabilistic finite state automata in terms of extended probability transition function. On the same line, the ambiguity of probabilistic deterministic finite state automata is also discussed. Keywords Probabilistic finite state automata, extended probability transition function, ambiguous probabilistic finite state automata., §1. Introduction As in the classical automata theory, there are deterministic and non-deterministic finite state automata deals with probabilistic processes [5]. Over the years, there are papers analysing the parsing [...]

In this paper, we study minimal surfaces for simply connected immersed minimal surfaces in the special three-dimensional Kenmotsuu manifold K with [eta]-parallel ricci tensor. We consider the Riemannian left invariant metric and use some results of Levi-Civita connection. Keywords Weierstrass representation, Kenmotsu manifold, minimal surface., [section]1. Introduction For standard terminology and notion in graph theory we refer the reader to Harary [4]; the non-standard will be given in this paper as and when The study [...]

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G,σ) (S = (G,μ)) where G = (V,E) is a graph called underlying graph of S and σ : E → ([bar.e].sub.1],[bar.e].sub.2], ..., [bar.e].sub.k]) (μ : V →([bar.e].sub.1], [bar.e].sub.2], ..., [bar.e].sub.k])) is a function, where each [bar.e].sub.i] ∈ {+, -}. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signed graph or a marked graph. E. Prisner [9] in his book Graph Dynamics Defines the t-path step operator on the class of finite graphs. Given a graph G and a positive integer t, the t-path step graph [(G).sub.t] of G is formed by taking a copy of the vertex set V(G) of G, joining two vertices u and v in the copy by a single edge e = uv whenever there exists a u = v path of length t in G. Analogously, one can Define the Smarandachely t-path step signed graph [(S).sub.t] = ([(G).sub.t],σ') of a signed graph S = ([G, σ) is a signed graph whose underlying graph is [(G).sub.t] called t-path step graph and sign of any edge e = uv in [(S).sub.t] is μ(u)μ(v). It is shown that for any signed graph S, its [(S).sub.t] is balanced. We then give structural characterization of Smarandachely t-path step signed graphs. Further, in this paper we characterize signed graphs which are switching equivalent to their Smarandachely 3-path step signed graphs. Keywords Smarandachely k-signed graphs, Smarandachely k-marked graphs, signed graphs, marked graphs, balance, switching, Smarandachely t-path step signed graphs, negation., §1. Introduction For standard terminology and notion in graph theory we refer the reader to Harary[4]; the non-standard will be given in this paper as and when required. We treat [...]

Let G(V, E) be a graph with p vertices and q edges. For every assignment f : V (G) → {0,1, 2, 3, ..., q}, an induced edge labeling f * : E(G) → {1, 2, 3 ..., q} is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every edge uv ∈ E(G). If f * (E) = {1, 2, ..., q}, then we say that f is a mean labeling of G. If a graph G admits a mean labeling, then G is called a mean graph. In this paper we study the meanness of the splitting graph of the path [P.sub.n] and [C.sub.2n] (n ≥ 2), meanness of some duplicate graphs, meanness of Armed crown, meanness of Bi-armed crown and mean labeling of cyclic snakes. Keywords Mean labeling, splitting graphs, duplicate graphs, armed crown, cyclic snake., Introduction Throughout this paper, by a graph we mean a finite, undirected, simple graph. Let G(V, E) be a graph with p vertices and q edges. For notations and terminology [...]

The theorems of continuous dependence of bounded [PHI]--variation solutions on parameters for Kurzweil equations are established through making use of the functions of bounded [PHI]-variation were introduced by Musiclak and Orlicz in paper [2]. The results are essential generalization of continuous dependence of bounded variation solutions on parameters for Kurzweil equations in paper [6]. Keywords Kurzweil equation, bounded [PHI]--variation solution, continuous dependence on parameter., [section] 1. Introduction Kurzweil generalized ordinary differential equation was introduced in 1957 by Kurzweil [1], the functions of bounded [PHI]-variation were introduced by Musiclak and Orlicz in paper [2]. The [...]

The paper mainly gives Harnark extention principle of Henstock integral on banach-valued function, promotes Harnark extention principle of Henstock integral on Real-valued function. Keywords Henstock integral, Harnark extention principle., [section] 1. Introductions and basic definitions Throughout this paper X denotes a real Banach Space and A partition of [a, b] is a finite collection of interval-point pairs {I, [xi]} [...]

In this paper we will establish some new methods for solving third and fourth degree equations. Keywords Cardano method, Ferrari method, equation., [section] 1. Introduction In this paper, using new transformations, we will introduce new elementary methods for solving third and fourth degree equations, which essentially differ from Cardano's method and Ferrari's [...]

In this paper, the concepts of fuzzy [??] - [T.sub.i] (i = 0,1, 2,1/2) spaces, fuzzy [??]-normal spaces and fuzzy g-regular spaces in the sense of Sostak [7] are introduced. Also, some interesting properties and characterizations of them are investigated. Keywords Fuzzy [??]-Ti (i = 0,1, 2,1/2) spaces, fuzzy [??]-normal spaces, fuzzy [??]-regular spaces., [sections]1. Introduction and Preliminaries The concept of fuzzy set was introduced by Zadeh [10] in his classical paper. Fuzzy sets have applications in many fields such as information [6] and [...]

In this paper we investigate generalized Bernoulli and Euler polynomials. In section 1, we enter multiplication theorem for generalized Bernoulli and Euler numbers. Aslo in section 2, we introduce a matrix representation of [B.sup.-1.sub.n] and [E.sup.-1.sub.n] . Furthermore in section 3 we consider Euler Maclaurin summation formula for [alpha]- Bernoulli numbers and in section 4 we give a relation between associated stirling numbers and [B.sub.n](a, b). Also in end section of this paper we consider a new method for representation of Apostol Bernoulli numbers. Keywords Bernoulli numbers, Euler polynomials, Stirling numbers of the second kind, generating functions., Short review of classical approaches Bernoulli numbers were first introduced by Jacques Bernoulli (1654-1705), in the second part of his treatise published in 1713, Ars conjectandi , at the time, [...]

[T.sub.i], i = 1, 2 be measurable transformations which define bounded composition operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [L.sup.2] of a [sigma]-finite measure space. Denote their respective Radon-Nikodym derivatives by [h.sub.i], i = 1,2. The main result of the paper is that, if [h.sub.i] * [T.sub.i] = [h.sub.j] a.e., for i,j = 1, 2, then for each of the positive integers m,n,p the operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are quasinormal. As a consequence, we see that the sufficient condition established in our paper [5] for quasinormality of a composition operator is actually sufficient for all powers to be quasinormal. Keywords Quasinormal operators, powers of quasinormal composition operators., [sections]1. Preliminaries Let (X, [SIGMA], [lambda]) be a sigma-finite measure space and let T be a measurable transformation from X into itself. Set [L.sup.2] = [L.sup.2](X, [SIGMA], [lambda]). The equation [...]

This paper deals with the author's initiation of signed domination functions in Relative character Graphs (RC-graphs). Given any finite group G and a subgroup H, one can construct a finite, simple, undirected graph [Gamma](G, H). We introduce two canonical functions f and [f.sup.*] from the vertex set V = IrrG into the set {-1, 1} satisfying the neighborhood condition. Then f or f * is called a signed domination function. Those RC-graphs for which for [f.sup.*] is a domination function is called an RC-signed dominated graph. In this paper we prove that all abelian groups (and subgroups), groups giving rise to [K.sub.n], the Frobenius groups with certain special properties are RC-signed dominated. We also study those RC-graphs [Gamma] for which a comparison can be made between a minimum dominating RC-graph number [γ.sub.R]([Gamma]) and the signed domination number [γ.sub.S]([Gamma]) (In some cases these number coincide). Finally certain bounds for are [γ.sub.R]([Gamma](G,H)) + [γ.sub.R]([bar.[Gamma]](G,H)) obtained in line with similar results for arbitrary graphs obtained by Haas and Wexler. Keywords Irreducible characters, RC-graphs, signed domination function., §1. Introduction Around the year 2000, T. Gnanaseelan, a Ph.D. scholar of Prof. A.V. Jeyakumar, M.K. University constructed a finite, simple, undirected graph for any finite group G and any [...]

In this paper based on the group characters there are certain graphs constructed. The properties are analysed with examples. Keywords IrrG, RC-graphs, [Core.sub.G]H, 1. Introduction This paper is the outcome of the author in throwing some more light on the relative character Graphs [Gamma](G, H) (RC-graphs) originally studied by T. Gnanaseelan in his [...]

This paper gives the expression of quintic supported spline wavelets and then discusses the expansion of the integrand with the quintic supported spline-wavelet. As the main result a new useful numerical integration formula is obtained by our analyzing and proving. Furthermore, a numerical example is given at the end of this paper. Keywords supported spline-wavelet, numerical integration, interpolation with wavelets., § 1. Introduction of the quintic supported spline wavelets In 1992, C. K. Chui and J. Z. Wang have constructed the supported spline wavelets with B-spline as scaling function, and [...]

Abstract In a recent paper, Zhang and Zhang [1] introduced the concept of sequences of numbers with alternate common diferences. In this paper, we extend the idea to sequences in [...]

Abstract In this paper, we introduce a new arithmetic function [τ.sub.sp] (n) which we called the simple divisor function. The main purpose of this paper is to study the asymptotic [...]

In this paper, the notion of Cartesian product of fuzzy SU-ideals and Cartesian product of anti fuzzy SU-ideals on SU-algebra are introduced and some of their properties are investigated. Keywords Cartesian product of fuzzy SU-ideal, Cartesian product of anti fuzzy SU-ideal. 2000 Mathematics Subject Classification: 03E72, 08A72, [section]1. Introduction and preliminaries In 1965, Zadeh defined fuzzy subset of a non-empty set as a collection of objects with grade of membership in continum, with each object being assigned [...]

In this paper we have developed some indefinite integrals involving inverse Hyperbolic function. The results represent here are assumed to be new. Keywords Hypergeometric function, elliptic integral. 2000 Mathematics Subject Classification: 33D50,33D60,33C75,33E05, [section]1. Introduction and preliminaries In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio [...]

This paper deal with the existence and uniqueness of positive solution for the integral boundary value problem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The discussion is based on the method of lower and upper solutions and maximal principle. Keywords Lower and upper solution, fixed point, maximal principle, positive solutions. 2000 Mathematics Subject Classification: 34B15,34B18, [section]1. Introduction We consider the integral boundary value problem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1) where f : [0, 1] x [0, + [infinity]) [right arrow] [0, + [infinity]) [...]

The objective of this paper is to introduce certain subclass of analytic functions and obtain an upper bound to the second Hankel functional[absolute value of ([a.sup.2][a.sub.4] - [a.sup.3]/2)] for the functions belonging to this class, using Toeplitz determinants. The result presented here include two known results as their special cases. Keywords Analytic function, starlike and convex functions with respect to symmetric points, convolution, upper bound, second Hankel functional, positive real function, Toeplitz determinants. 2010 Mathematics Subject Classification: 30C45,30C50, [section]1. Introduction and preliminaries Let A denote the class of functions f of the form f(z) = z + [[infinity].summation over (n = 2)][a.sup.n][z.sup.n] (1) in the open unit disc [...]

The aim of this paper is to introduce the notions of [[lambda].sub.J]-Closed sets in generalized topological spaces and supplement their properties. We also investigate their characterizations. Keywords J-closed set, J-open set, gJA-closed set, J[lambda]g-closed set, [[lambda].sub.J]-closed set. 2000 Mathematics Subject Classification: 54A05,54A10, [section]1. Introduction and preliminaries A. Csaszar [1] has introduced and studied generalized open sets of a set X denned in terms of monotonic functions [gamma] : P(X) [right arrow] P(X). [...]

In this paper, the eigenvalues and the upper bounds for spectral norm and Euclidean norm of right circulant matrices with sine and cosine sequences were derived. Keywords Circulant matrix, sine and cosine sequences., [section]1. Introduction and preliminaries In [1] and [2] the formulae for the determinant, eigenvalues, Euclidean norm, spectral norm and inverse of the right circulant matrices with arithmetic and geometric sequences [...]

In this paper, we characterise [k.sup.*]-paranormal composition operators and [k.sup.*]-paranormal weighted composition operators and their adjoints in [L.sup.2] spaces. Keywords [k.sup.*]-paranormal operators, composition operators, weighted composition operators, aluthge transformation., [section]1. Introduction and Preliminaries Let B(H) be the Banach Algebra of all bounded linear operators on a non-zero complex Hilbert space H. By an operator, we mean an element from [...]

Let [[tau].sup.(e)](n) denote the number of exponential divisor of n. Similar to the generalization from d(n) to [d.sup.k](n), we extended [[tau].sup.(e)](n) to [([[tau].sup.(e).sub.k](n)).sup.k - 1] The aim of this paper is to establish a short interval result for the function [([[tau].sup.(e).sub.3](n)).sup.2] This enriches the properties of the exponential divisor function. Keywords The exponential divisor function, the generalized divisor function, short interval. 2000 Mathematics Subject Classification: llE45, [section]1. Introduction In 1972, M. V. Subbarao [3] established the definition of exponential divisor: Let n > 1 be an integer of canonical from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [...]

The I-total coloring of a graph G is an assignment of some colors to its vertices and edges such that no two adjacent vertices receive the same color and no two adjacent edges receive the same color. Under an I-total coloring of G, the color set of a vertex x of G is the set of all colors which are assigned to vertex x or the edges incident to x. An I-total coloring is called adjacent vertex distinguishing if any two adjacent vertices have different color sets. The minimum number of colors required in an adjacent vertex-distinguishing Itotal coloring is called adjacent vertex-distinguishing I-total chromatic number. The adjacent vertex-distinguishing I-total chromatic number of Ladder graph is discussed in this paper. Keywords Ladder graph, adjacent vertex-distinguishing, I-total coloring, adjacent vertex-dist -inguishing, I-total chromatic number., [section]1. Introduction and preliminaries Coloring is a important research area of graph theory. Some new colorings of graphs are produced from applied areas of computer science, information science and light [...]

Basing on the priori assumption principle, the main goal of this paper is to propose the rational expansion method that can be used to handle the exact solution of the nonlinear partial differential equation. It is extension of tanh function method. As an application, three classes of exact solutions to Klein-Gordon-Schrodinger equation are obtained. Keywords Klein-Gordon-Schrodinger equation, exact solution, tanh function method, rational expansion method., §1. Introduction The investigation of exploring the exact traveling wave solutions of nonlinear mathematical physics equations plays an important role in the study of the solitary wave solution. Up to [...]

In this paper we have established two indefinite integrals involving Lucas polynomials, Fibonacci polynomials, Gegenbauer polynomials and Hermite polynomials. The results represent here are assumed to be new. Keywords Gegenbauer polynomials, hermite polynomials; Fibbonacci polynomials, Lucas polynomials, polylogarithm function, hypergeometric function. 2000 Mathematics Subject Classification: 33C05,33C45,33C15,33D50,33D60, [section]1. Introduction and preliminaries Definition 1.1. Gegenbauer polynomials or ultraspherical polynomials [C.sup.[alpha].sub.n](x) are orthogonal polynomials on the interval [-1,1] with respect to the weight function [(1-[x.sup.2]).sup.[alpha]-1/2]. They generalize Legendre polynomials [...]

In this paper, we introduced Samarandache-2-algebraic structure of Boolean-near-ring namely Smarandache-Boolean-near-ring. A Samarandache-2-algebraic structure on a set N means a weak algebraic structure [S.sub.1] on N such that there exists a proper subset M of N, which is embedded with a stronger algebraic structure [S.sub.2], stronger algebraic structure means satisfying more axioms, that is [S.sub.1] [much less than] [S.sub.2], by proper subset one understands a subset different from the empty set, from the unit element if any, from the whole set [3]. We define Smarandache-Boolean-near-ring and obtain the some of its characterization through Boolean-ring with sub-direct sum structure. For basic concept of near-ring we refer to G. Pilz [11]. Keywords Boolean-ring, Boolean-near-ring, Smarandache-Boolean-near-ring, compatibility, maximal set, idempotent and uni-element., [section]1. Preliminaries Definition 1.1. A (Left) near ring A is a system with two Binary operations, addition and multiplication, such that (i) the elements of A form a group (A, [...]

Suppose that S is a locally inverse semigroup with an inverse transversal S[degrees] In this paper, we first introduce the CS[degrees]-cone of S, which modify the definitions of cone in [3] and S[degrees]-cone in [4]. By an CS[degrees]-cone, we can construct an amenable partial order on S. Conversely, every amenable partial order on S can be constructed in this way. It is easily verified that the set E(S[degrees]) of all idempotent elements of S[degrees] is the smallest CS[degrees]-cone of S. Also, we show that the amenable partial order constructed by E(S[degrees]) is equal to the natural partial order on S and so the natural partial order on S is the smallest amenable partial order. Keywords Locally inverse semigroup, amenable partial order, inverse transversal, order-preserving bijection. 2000 Mathematics Subject Classification: 20M10,06B10, [section]1. Introduction and preliminaries A semigroup S is said to be a partially ordered semigroup, or to be partially ordered, if it admits a compatible ordering [less than or equal [...]

The purpose of this paper is to introduce the concepts of an intuitionistic fuzzy grill, intuitionistic fuzzy G structure space, intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] set and intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] open set. The concepts of an intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] continuous function, intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] space, i = 0, 1, 2 and intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-co-closed graphs are defined. Some interesting properties are established. Keywords Intuitionistic fuzzy grill, intuitionistic fuzzy G structure space, intuitionistic fuz -zy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] set and intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] open set, intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exterior, intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] continuous function, intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] space, i = 0,1,2 and intuitionistic fuzzy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-co-closed graphs. 2010 Mathematics Subject Classification: 54A40,03E72, §1. Introduction and preliminaries The concept of fuzzy set s was introduced by Zadeh [7] and later Atanassov [1] generalized the idea to intuitionistic fuzzy sets. On the other hand, [...]

In this paper, we investigate the outcome when the trigonometric functions sin x, cos x, [e.sup.x] and the hyperbolic functions sinh x and cosh x are evaluated on circulant matrices. Keywords circulant matrix, trigonometric functions, hyperbolic functions. 2000 Mathematics Subject Classification: 54Dl5,54DlO, [section]1. Introduction Given any sequence of numbers, [c.sub.0], [c.sub.1], ..., [c.sub.n - 1], we can form circulant matrices. From [2], circulant matrices have four types: the right circulant, the left [...]

In this paper the concept of intuitionistic fuzzy pre-β-irresolute functions are introduced and studied. Besides giving characterizations and properties of this function, preservation of some intuitionistic fuzzy topological structure under this function are also given. We also study relationship between this function with other existing functions. Keywords Intuitionistic Fuzzy β-open set, intuitionistic fuzzy pre-α- irresolute, intuitionistic Fuzzy pre-β-irresolute. 2000 Mathematics Subject Classification: 54A40, 03E72, §1. Introduction Ever since the introduction of Fuzzy sets by L. A. Zadeh [14], the Fuzzy concept has invaded almost all branches of mathematics. The concept of Fuzzy topological spaces [...]

In this paper by using vg-open sets, I define almost vg-normality and mild vg-normality also we continue the study of further properties of vg-normality. We show that these three axioms are regular open hereditary. I also define the class of almost vg-irresolute mappings and show that vg-normality is invariant under almost vg-irresolute M-vg-open continuous surjection. Keywords vg-open, semiopen, semipreopen, almost normal, midly normal, M-vg-closed, M-vg-open, rc-continuous. 2000 Mathematics Subject Classification: 54D15,54D10, §1. Introduction In 1967, A. Wilansky has introduced the concept of US spaces. In 1968, C. E. Aull studied some separation axioms between [T.sub.1] and [T.sup.2] spaces, namely, [S.sub.1] and [...]

In this paper we discuss new separation axioms using vg-open sets. Keywords [vg.sub.i] spaces. 2000 Mathematics Subject Classification: 54D1O, 54D15, §1. Introduction Norman Levine introduced generalized closed sets, K. Balachandaran and P. Sundaram studied generalized continuous functions and generalized homeomorphism. V. K. Sharma studied generalized separation axioms. Following V. K. [...]

The main object of present paper is the establishment of two summation formulae associated with the contiguous relation and hypergeometric function. Keywords Contiguous relation, recurrence relation, Legendre's duplication formula. 2000 Mathematics Subject Classification: 33C05,33C20,33Dl5,33D50,33D60, [section]1. Introduction Generalized Gaussian hypergeometric function of one variable : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1) or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2) where the parameters [b.sup.1], [b.sup.2], [...]

In this paper, the Schur harmonic convexity of Stolarsky extended mean values are discussed. Keywords Stolarsky, harmonic, concavity, convexity. 2000 Mathematics Subject Classification: 26Dl5, [section]1. Introduction and preliminaries In literature the importance and applications of means and its inequality to science and technology is explored by eminent researchers see [3]. In [20-28], we studied [...]

In this paper, we present a generalized criterion for starlike univalent functions. We show that our result unifies some known results of starlike functions. Using the dual concept of differential subordination and superordination, we find some sandwich-type results regarding starlike univalent functions. Mathematica 7.0 is used to plot the images of the unit disk under certain functions. Keywords differential subordination, differential superordination, starlike function., §1. Introduction Let H be the class of functions analytic in the open unit disk E = {z : [absolute value of z] < 1} and for a ∈ C [...]

In this paper Type-1 connectedness and Type-2 connectedness in topology of intuitionistic fuzzy rough sets are defined and properties of two types of connectedness are studied. Keywords Fuzzy subsets, rough sets, instuitionistic fuzzy sets, intuitionistic fuzzy rough sets, fuzzy topology, Type-1 connectedness, Type-2 connectedness. 2000 Mathematics Subject Classification: 54A40, [section]1. Introduction After introducing the idea of fuzzy subsets by Loth Zadeh [22] different variations/generalisations of fuzzy subsets were made by several authors. Pawlak [20] introduced the idea of rough [...]