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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 [...]

[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 presents the projective synchronization of chaos systems by designing tracking controller based on Lyapunov stability theory. Frist, this method is implemented in synchronization of a simple system, then we realize the synchronization of Lu hyper-chaotic system. Numerical simulations show the united synchronization method works well. Keywords Projective synchronization, tracking control, chaotic system, Hyper-chaotic system., § 1. Introduction Synchronization is a fundamental phenomenon that enables coherent behavior in coupled systems. In 1990, pecora and carroll proposed a successful method to synchronize two identical chaotic systems [...]

In this paper we introduce an AC-algebra as an algebra (X, *, 0) with a binary operation * which satisfies the conditions; x * (y * z) = (x * y) * z, x * y = y * x and x * y = 0 if and only if x = y for all x, y, z in X. We show that an AC-algebra is an Abelian group and that a subset of an AC-algebra is a subalgebra if and only if it is an ideal. We also present an algebraic structure called a quotient AC-algebra relative to a subalgebra of the AC-algebra and show that the quotient algebra is also an AC-algebra. Moreover,we define functions on an AC-algebra and prove that a set of finite compositions of these functions is also an AC-algebra and show the properties of this set. Keywords AC-algebra, quotient AC-algebra, functions on AC-algebra., § 1. Introduction By an algebra X = (X, *, 0) we mean a non-empty set X together with a binary operation * and some distinguished element 0. Kondo [1] [...]

In this paper, the author studies a risk model. Under this model the two claim processes are correlated. Claim occurrence relate to Poisson and Erlang processes. The formulae is derived for the distribution of the surplus immediately before ruin, for the distribution of the surplus after ruin and the joint distribution of the surplus immediately before and after ruin. The asymptotic property of these ruin functions is also discussed. Keywords Double risk model, Poisson process, Erlang(2) process, correlated aggregate claim, ruin function., [section] 1. Introduction Resently, many authors studied various correlated aggregate claims models. In this model, the two claim number processes are correlated. Ambagaspitiya(1998) considered a general method of constructing a [...]

In this paper we shall study the mean value of the exponential divisor function involving a negative r-th power by the convolution method. This gives a comparison with the result of Laszlo Toth. Keywords The exponential divisor function, the general divisor function, Euler product, convolution method., [section] 1. Introduction The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [b.sub.i] | [a.sub.i] for every i [...]

The upper and lower chromatic number of uniform mixed hypergraphs have inevitable relation with C-hyperedge and D-hyperedge. In general, the increase in the C-hyperedge will increase lower chromatic number XH, increasing D-hyperedge will decrease upper chromatic number [bar.X]H. This papers take complete uniform mixed hypergraphs for example, revealed further the relationship of C-hyperedge with between the upper chromatic number and lower chromatic number. We give a few conclusions for mixed hypergraph [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whose some C-hyperedge are deleted. Keywords A complete uniform mixed hypergraph, polychromatic C-hyperedges, the minimum, upper chromatic number, lower chromatic number., § 1. Lemma and the basic concepts Definition 1.1. [1] Let X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]} be a finite set, C = {[C.sub.1], [C.sub.2], ..., [C.sub.1]}, D = {[D.sub.1], [...]

Let A be a subalgebra of B(H). We say that a linear mapping [psi] from A into itself is a multiplicative mapping at Z(Z [member of] A) if [psi](ST) = [psi](S)[psi](T) for any S, T [member of] A with ST = Z. Let H be an infinite dimensional complex Hilbert space, and let [psi] be a surjective linear map on B(H). In this paper, we prove that if [psi] is a multiplicative mapping at I and continuous in the weak operator topology, then [psi] is an automorphism. We also prove that if [psi] is a weak continuous multiplicative mapping at any invertible operator with [psi](I) = I then [psi] is an automorphism. Keywords Operator algebra, multiplicative mappings at unit operator, automorphism., [section] 1. Introduction and preliminaries Let H be an infinite dimensional complex Hilbert space. We denote by B(H) the algebra of all bounded linear operators on H. The purpose of [...]

The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every i [member of] 1, 2, ...; s. Let [[tau].sup.(e)](n) denote the number of exponential divisors of n, where [[tau].sup.(e)](1) = 1 by convention. In this paper we shall establish a short interval result for r-th power of the function [[tau].sup.(e)], where r [greater than or equal to] 1 is a fixed integer. Keywords The exponential divisor function, generalized divisor function, short interval., [section] 1. Introduction The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [b.sub.i]|[a.sub.i] for every i [member of] [...]

In this paper, some complete monotonicity properties of functions related to the gamma function and Barnes G-function are proved. Keywords Gamma function, double gamma function, Barnes G-function, complete monotonicity, asymptotic expansion., [section] 1. Introduction A function f is said to be completely monotonic over (a, b), where -[infinity] [less than or equal to] a < b [less than or equal to] [...]

For any positive integer n, the Pseudo-Smarandache function Z(n) is defined as Z(n) = min {m : m [member of] N, n | m (m+1)/2}. And the F. Smarandache function SL(n) is defined as the smallest positive integer k such that n [absolute value of [1, 2, ..., k]], where [1, 2, ..., k] denotes the least common multiple of 1, 2, ..., k. In this paper, we use the elementary methods and congruences to study the solvability of the equation Z(n) + SL(n) = n, and give its all positive integer solutions. Keywords Pseudo-Smarandache function, F. Smarandache LCM function, equation, solution, congruences., [section] 1. Introduction and results For any positive integer n [greater than or equal to] 1, the Pseudo-Smarandache function Z(n) is defined as the smallest positive integer m such that [...]

Let [t.sup.(e)] (n) denote the number of e-squarefree e-divisor of n. The aim of the present paper is to establish an asymptotic formula for the mean value of the function ([t.sup.(e)])r, where r [greater than or equal to] 1 is a fixed integer. Keywords E-squarefree e-divisor function, the generalized divisor function, convolution method., [section] 1. Introduction An integer d = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an e-divisor of n =[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 if [b.sub.i]|[a.sub.i] for [...]

Let n > 1 be an integer, [??](n) denote the number of regular integers m(mod n) such that 1 [less than or equal to] m [less than or equal to] n. In this paper we shall investigate the mean value of the function log [??](n) by the convolution method. Keywords Regular integers (mod n), mean value, convolution method, Perron's formula., [section] 1. Introduction and main results Let n > 1 be an integer. Consider the integers m for which there exists an integer x such that [m.sup.2]x = m(mod n). [...]

The homothetic motion in 2-dimensional Euclidean space [E.sup.2], the relation between the velocities of this motion and geometric results for the pole curves were studied in ref [1]. In this paper a canonical relative system of any plane with respect to other planes are given. Therefore in a homothetic motion E/E', Euler-Savary formula giving the relation between the curvature of trajectory curves drawn in the fixed plane E' by the points of the moving plane E is obtained. In the special case of homothetic scale h identically equal to 1, we get the Euler-Savary formula which was given by Muller [5]. Finally some geometrical results are reached using Euler-Savary formula. Keywords Euler-Savary formula, homothetic motion, kinematics., § 1. Introduction This study deals with instantaneous geometric plane kinematics. Which is the study, for a certain instant during a continuous motion, of the differential geometric properties? So if [...]

An integer a is called regular (mod n) if there is an integer x such that [a.sup.2]x [equivalent to] a(mod n). Let [??](n) denote the number of regular integers a(mod n) such that 1 [less than or equal to] a [less than or equal to] n, [phi](n) is the Euler function. In this paper we investigate the mean value of the function [([??](n)/[phi](n)).sup.r], where r [greater than or equal to] 1 is a fixed integer. Keywords Regular integers (mod n), Euler's function, average order, convolution method, Euler product., [section] 1. Introduction Let n > 1 be an integer. Consider the integers a for which there exists an integer x such that [a.sup.2]x = a(mod n). Properties of these [...]

Orthogonal polynomials play a very important role in analysis, mainly because functions belonging to very general classes can be expanded in series of orthogonal polynomials. The main purpose of this paper is using the elementary method and the properties of power series to study the calculating problem of one kind summation involving the Laguerre polynomials, and give some interesting identities. Keywords Laguerre polynomials, power series, generating function, elementary method, orthogonal polynomials, identity., [section] 1. Introduction For any real number x, the famous Laguerre polynomials [L.sub.n](x) are defined by the coefficients of the power series expansion of the function 1/1-t [e.sup.-xt/1-t]. That is, [...]

The main purpose of this paper is using the elementary method and the properties of the power series to study an integral calculating problem involving the Hermite polynomials, then give an interesting identity. Keywords Hermite polynomials, integral calculation, identity, elementary method., [sections]1. Introduction For any real number x, the polynomial solutions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Hermite equations [d.sup.2]y/[dx.sup.2] - 2x dy/dx + 2ny = 0 (n = [...]

Let R be a 2-torsion-free prime ring and J anonzero Jordan ideal and a subring of R. For a ([sigma], [tau])-derivation d : R [right arrow] R, we prove the following results: (1) If d is a ([sigma], [sigma]]) derivation which acts as a homomorphism or as an anti-homomorphism on J, then either d = 0 on R or J [subset or equal to] Z(R); (2) If F is a generalized ([sigma], [sigma]])-derivation which acts as a homomorphism or as an anti- homomorphism on J, then either d = 0 on R or J [subset or equal to] Z(R); (3) If d is a ([sigma], [tau])-derivation which acts as a homomorphism on J, then d = 0 on R. Keywords Fuzzy Prime rings, derivations, ([sigma], [tau])-derivations, generalized derivations, generalized ([sigma], [tau])-derivations., [sections]1. Introduction Throughout the present paper, R will denote an associvative ring with center Z(R). We will write for all x,y [member of] R, [x, y] = xy - yx [...]

For any positive integer n, the Smarandache dual function S** (n) is defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The main purpose of this paper is using the elementary methods to study the convergent properties of an infinity series involving S**(n), and give an interesting limit formula for it. Keywords The Smarandache dual function, limit problem, elementary method., [sections]1. Introduction and Results For any positive integer n, the Smarandache dual function S**(n) is defined as the greatest positive integer 2m - 1 such that (2m - 1)1! divide [...]

The upper and lower chromatic number of uniform mixed hypergraphs and C-hyperedge and D-hyperedge contact with the inevitable. In general, the increase in the C-hyperedge will increase lower chromatic number [sub.[chi]H], an increase D-hyperedge will decrease upper chromatic number [[bar.[chi].H]. In this paper the relationship between C-hyperedge with the upper chromatic number and lower chromatic number and some conclusions with respect to mixed hypergraph are given. Keywords A complete uniform mixed hypergraph, chromatic, upper chromatic, Smarandache conclusions., [sections]1. Lemma and the basic concepts Definition 1.1. [1] W Let X = {[x.sub.1], [x.sub.2], ... , [x.sub.n]} be a finite set, C = {[C.sub.1], [C.sub.2], ... ,[C.sub.l]}, D = [...]

In this paper (i,j) semi compact and pairwise semi compact; (i, j) semi Lindeloff and pairwise semi Lindeloff Bitopological spaces are defined and their basic properties are studied. Keywords Bitopological space, semi open set, (i, j) semi compact, pairwise semi compact, (i, j) semi Lindeloff, pairwise semi Lindeloff Bitopological space., [sections]1. Introduction J.C. Kelly[2] introduced the concept of Bitopological spaces in 1963 which paved way to the theory of Bitopological spaces. After him many Authors defined different version of Bitopo- [...]

In this paper, we introduce the notion of n-sequential closure operators and use these operators to construct classes of n-Frechet spaces as generalizations of the class of Frechet spaces. Then we get some necessary and sufficiency conditions for a sequential space to be a Frechet space. These results are generalizations of the preceding one of W. C. Hong in [5]. Keywords Frechet, n-Frechet, sequential, sequential closure, n-sequential closure., [section]1. Introduction and results M. Venkataraman poses the following problem in [8]. Problem 1.1. Characterize "the class of topological spaces which can be specified completely by knowledge of their convergent [...]

In this paper, we study homomorphisms of KU-algebras and investigate its properties. Moreover, some consequences of the relations between quotient KU-algebras and isomorphisms are shown. Keywords Homomorphism, isomorphism, ideal, congruence, KU-algebras., [section]1. Introduction In [3], C. Prabpayak and U. Leerawat studied ideals and congruences of BCC- algebras ([1],[2]) and introduced a new algebraic structure which is called KU-algebras. They gave the [...]

In this paper, we consider Jensen type mapping in the setting of generalized random normed spaces. We generalize a Hyers-Ulam stability result in the framework of classical normed spaces. Keywords Jensen type mapping, generalized random normed space., [section]1. Introduction In 1941 D.H. Hyers [5] solved this stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. In 1951 D.G. Bourgin [2] was the [...]

This paper gives a generalization of Beta-nacci and Fibonacci sequences and the general solution obtained is given in terms of Beta-Fibonacci numbers. Keywords Beta-nacci sequence, Fibonacci sequence., [section]1. Preliminaries and introduction The Fibonacci sequence, say [{[F.sub.n]}.sup.[infinity].sub.n=0] is defined recurrently by [F.sub.n] = [F.sub.n-1] + [F.sub.n-2], for all n [greater than or equal to] 2, (1) with initial [...]

The main purpose of this paper it to studied the mean value properties of the Smarandache Superior m-th power part sequence ssmp(n) and the Smarandache Inferior m-th power part sequence simp(n), and give several interesting asymptotic formula for them. Keywords Smarandache Superior m-th power part sequence, Smarandache Inferior m-th power part sequences, mean value, asymptotic formula., [section]1. Introduction and Results For any positive integer n, the Smarandache Superior m-th power part sequence SSMP(n) is defined as the smallest m-th power greater than or equal to n. [...]

Let n be any positive integer, [P.sub.d](n) denotes the product of all positive divisors of n. The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of a new arithmetical function S ([P.sub.d](n)), and give an interesting asymptotic formula for it. Keywords Smarandach function, Divisor product sequences, Composite function, mean value, Asymptotic formula., [section]1. Introduction For any positive integer n, the famous F.Smarandache function S(n) is defined as the smallest positive integer m such that n divide m!. That is, S(n) = min{m [...]

In this paper, Smarandache U-liberal semigroup structure is given. It is shown that a semigroup S is Smarandache U-liberal semigroup if and only if it is a strong semilattice of some rectangular monoids. Consequently, some corresponding results on normal orthocryptou semigroups and normal orthocryptogroups are generalized and extended. Keywords Smarandache U-liberal semigroup, U-semiabundant semigroups, normal band, rectangular monoid, strong semilattice., [Section] 1. Introduction and preliminaries In order to generalize regular semigroups, new Green's relations, namely, the Green's *-relations on a semigroup S have been introduced in [1] and [2] as [...]

The aim of this paper is to study Smarandache #-rpp semigroups whose idempotents satisfy permutation identities. After some properties are obtained, the weak spined product structure of such semigroups is established. Keywords Smarandache #-rpp semigroups, normal band, weak spined product., [section]1. Introduction Similar to rpp rings, a semigroup S is called an rpp semigroup if for any a [member of] S, a[S.sup.1] regarded as a right [S.sup.1] system is projective. [...]

Given generalized reflection matrices R, S, i.e., R * = R, [R.sup.2] = I, S * = S, [S.sup.2] = I, a complex matrix A is said to be generalized reflexive (or anti-reflexive), if RAS = A (or RAS = -A). In this paper, Smarandache iterative algorithm is proposed to solve matrix equation AXB + CYD = F with generalized reflexive matrix dual (X,Y). For any initial iterative matrix pair ([X.sub.1],[Y.sub.1]), we show that a solution of this equation can be obtained within finite iteration steps in the absence of roundoff errors. Some numerical examples illustrate the feasibility and efficiency of this algorithm. Keywords Matrix equation, generalized reflexive matrices, Smarandache iterative algorithm, least-norm solution, optimal approximation., §1. Introduction Let [C.sup.m x n.sub.k] be the set of all m x n complex matrices with rank k, and U[C.sup.nxn] the set of all unitary matrices in [C.sup.n x [...]

In this paper Smarandache v-connectedness and Smarandache locally v-connectedness in topological space are introduced, obtained some of its basic properties and interrelations are verified with other types of connectedness. Keywords Smarandache v-connected and Smarandache locally v-connected spaces, [section]1. Introduction After the introduction of semi open sets by Norman Levine various authors have turned their attentions to this concept and it becomes the primary aim of many mathematicians [...]

In this paper, the chaotic characteristic of the Sprott N system with parameter is studied in theoretical analysis and numerical simulations, and abundant dynamical behavior is presented. When the parameter β = 1.9, the system exist chaotic attractor. The abundance dynamical behavior of the system is presented by the global bifurcation graph and the Lyapunov exponent. In order to realize the full state hybrid projective synchronization (FSHPS) of the system, Smarandache controller is design. Numerical simulations show that the controller works well. Keywords Chaos, Bifurcation, The Sprott N system with parameter, FSHPS., §1. Introduction Since synchronization of chaotic systems was first introduced by Fujisaka and Yamada [1] and Pecora and Carroll [2]. Due to the importance and applications of coupled systems, ranging [...]

In this paper, we mainly discuss the Smarandache relation on U-semiabundant semigroups. Keywords U-semiabundant semigroups, Green relations, Natural partial orders., [section]1. Introduction In generalizing regular semigroups, a generalized Green relation [L.sup.U] was introduced by M. V. Lawson [6] on a semigroup S as follows: Let E be the set of [...]

A set D C V of vertices in a graph G is a dominating set if every vertex v in V - D is adjacent to a vertex in D. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G). A decomposition of a graph G is a collection Ψ of edge disjoint subgraphs [G.sub.1], [G.sub.2], ... , [G.sub.n] of G such that every edge of G is in exactly one [G.sub.i]. If γ([G.sub.i]) = 1 for each i, then Ψ is called an 1-equidomination cover of a graph G. The minimum cardinality of an 1-equidomination cover of G is called the 1-equidomination covering number of a graph G and is denoted by [γ.sub.e.sup[(1)]] (G). In this paper we initiate a study on this parameter. Keywords 1-equidomination cover, 1-equidomination covering number, minimum 1-equidomination cover., 1. Introduction By a graph, we mean a finite, undirected, non-trivial, connected graph without loops and multiple edges. For terms not defined here we refer to Harary [5]. The theory [...]

In this paper a failure model of the mixed Inverse Weibull distributions(MIWD) is considered and its estimators for all unknown parameters based on type-II and type-I censored data are obtained by means of ENI-Algorithm. Some simulations suggest that EM algorithm is effective to our model. Keywords mixed Inverse Weibull distribution, failure model, EM-algorithm, censored data., [section]1. Introduction Mixture models play a important role in many applicable fields, such as medicine, psychology research, cluster analysis, life testing and reliability analysis and so on. Mixture distributions have [...]

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. In this paper, we determine the lower bound for the Merrifield-simmons index in (n, n - + 1)-graphs in terms of the order n, and characterize the (n, n + 1)-graph with the smallest Merrifield-simmons index. Keywords (n, n + 1)-graphs, [sigma]-index, Merrifield-Simmons index., [section]1. Introduction Let G = (V, E) be a simple connected graph with the vertex set V (G) and the edge set E(G). For any v [member of] V, [N.sub.G](v) [...]

For any positive integer n, we define the arithmetical function G(n) as G(1) = 0. If n > 1 and n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] be the prime power factorization of n, then G(n) = [[alpha].sub.1]/[p.sub.1] + [[alpha].sub.2]/[p.sub.2] + ... + [[alpha].sub.k]/[p.sub.k]. The main purpose of this paper is using the elementary method and the prime distribution theory to study the mean value properties of G(n) in Smarandache divisor product sequences {[p.sub.d](n)} and {[q.sub.d](n)}, and give two sharper asymptotic formulae for them. Keywords Additive function, Smarandache divisor product sequences, mean value, elementary method, asymptotic formula., [section]1. Introduction and results In elementary number theory, we call an arithmetical function f (n) as an additive function, if for any positive integers m, n with (m, n) = [...]

In this paper we prove that the "intermediate point" [xi] of Lagrange mean value theorem is a function, furthermore, we study its monotonicity, continuity and derivable property. For application, we give an example to show the incorrect for a proof method in course of calculus teaching. Keywords Lagrange mean value theorem, "intermediate point" [xi], monotonicity, continuity, derivable property, [section]1. Introduction Lagrange mean value theorem is one of most important theorems in calculus. It is an important tool to study the property of function. It sets up a "bridge" [...]

This paper introduces the concept of fuzzy number valued Lebesgue outer measure. A non-negative countably subadditive function m* on the power set of a set X by means of a given additive function on an algebra of subsets of X, and a new collection of measurable sets E are constructed , where E satisfy the relation m* (A) = m* (A [intersection] E) + m* (A [intersection] (X \E)), for any subset A of X . Keywords Fuzzy number, fuzzy outer measure., [section]1. Introduction There are articles in the literature associated with fuzzy outer measure [3]. We construct fuzzy number valued outer measures and measurable sets that is similar to that of [...]

For any positive integer n, let SP(n) denotes the Smarandache power sequence. And for any Smarandache sequence a(n), the Smarandache totient function St(n) is defined as [phi](a(n)), where [phi](n) is the Euler totient function. The main purpose of this paper is using the elementary and analytic method to study the convergence of the function [S.sub.1]/[S.sub.2], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and give an interesting limit Theorem. Keywords Smarandache power function, Smarandache totient function, convergence., 1. Introduction and results For any positive integer n, the Smarandache power function SP(n) is defined as the smallest positive integer m such that n | [m.sup.m], where m and [...]

For any positive integer n, we define the arithmetical function F(n) as F(1) = 0. If n > 1 and n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] be the prime power factorization of n, then F(n) = alpl + [[alpha].sub.2][p.sub.2] + ... + [[alpha].sub.k] [p.sub.k]. Let S(n) be the Smarandache function. The main purpose of this paper is using the elementary method and the prime distribution theory to study the mean value properties of [(F(n) - S(n)).sup.2], and give a sharper asymptotic formula for it. Keywords Additive function, Smarandache function, Mean square value, Elementary method, Asymptotic formula., [section]1. Introduction and result Let [Florin] (n) be an arithmetical function, we call [Florin] (n) as an additive function, if for any positive integers m, n with (m, n) = [...]

In this paper [nu]-compactness and [nu]-Lindeloffness in topological space are introduced, obtained some of its basic properties and interrelations are verified with other types of compactness and Lindeloffness. Keywords [nu]-compact, [nu]-Lindeloff spaces, 1. Introduction After the introduction of semi open sets by Norman Levine various authors have turned their attentions to this concept and it becomes the primary aim of many mathematicians [...]

In this paper, we study the interior structures of Girard quantale and the cyclic dualizing elements of Girard quantale. some equivalent descriptions for Girard quantale are given and an example which shows that the cyclic dualizing element is not unique is given. Keywords Complete lattice, quantale, Girard quantale, cyclic dualizing element., §1. Preliminaries Quantales were introduced by C.J. Mulvey in [1] with the purpose of studying the spectrum of C*-algebras and the foundations of quantum mechanics. The study of such partially [...]

A bitopologiacl space (X,[T.sub.P],[T.sub.Q]) generated by probabilistic-quasi-pseudo-metric P is quasi-pseudo-metrizable if and only if there exists a quasi-pseudo-metric which induces those two topologies. In this paper, we consider a problem of quasi-pseudo-metrization of PgpM-spaces. Keywords Probabilistic-quasi-pseudo-metric-space (PgpM-space), quasi-pseudo-metrization, bitopological space (X,[T.sub.P],[T.sub.Q]), quasi-uniform structure., [section]1. Introduction A topological space (X,[tau]) is said to be metrizable if there exists a metric d : [X.sup.2] [??] which generates a topology [T.sub.d] equivalent to [tau]. For PgpM-spaces [...]

In this paper, we will introduce some kind of algebras which is called KU-algebras. We define ideals and study congruences on KU-algebras, and also investigate some related properties. Keywords ideals, congruences, KU-algebras, [section]1. Preliminaries Several authors ([1],[2],[3],[4],[5]) introduced some structures of algebras such as BCK, BCC and BCI-algebras. In [2], C.S. Hoo introduced the notions of filters and commutative ideals in BCI-algebras. [...]