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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 [...]

This issue of the journal is devoted to the proceedings of the second Northwest Number Theory and Smarandache Problems Conference held in Yan'an during March 18-20, 2006. The organizers were [...]

Using the principle least action, we investigate the existence solution for p(x)-Laplacian equation with no flux boundary condition in a bounded domain [OMEGA] [subset or equal to] [R.sup.N], where no flux boundary condition is given in the following: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Keywords Variable exponent Sobolev spaces, p(x)-Laplacian, no flux boundary, the principle least action., [section] 1.Introduction In recent years there has been an increasing interest in the study of various mathematics problem with variable exponent, see the papers [1,2,3,5]. The existence of solutions of [...]

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 [...]

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 [...]

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 = [...]

Abstract In this paper, we use the elementary method to study the properties of the Smarandache additive sequence, and give several interesting identities for some infinite series involving the Smarandache [...]

Abstract The aim of this short paper is to establish an asymptotic formula for the sum [ZIGMA]/n≤ x (n)(1), where I(n) is the multiplicative function such that I([p.sup.a) = [p.sup.a]/ [...]

Abstract Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Where l, q are fixed numbers. This paper uses the hyperbolic summation method and the exponential pair theory to obtain the asymptotic [...]

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 [[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]. [...]

Abstract The main purpose of this paper is using the elementary method to study the mean value properties of a new function for n, and give a sharp asymptotic formula [...]

Abstract In this paper, we use the elementary method to study the properties of the Smarandache multiplicative sequence, and proved that some infinite series involving the Smarandache multiplicative sequence are [...]

This issue of the journal is devoted to the proceedings of the third International Conference on Number Theory and Smarandache Problems held in Weinan during March 23-25, 2007. The organizers [...]

Abstract In this paper, we define a new arithmetical function S[L.sup.*](n), which is related with the famous F.Smarandache LCM function SL(n). Then we studied the properties of S[L.sup.*] (n), and [...]

Let a(n) denote the number of nonisomorphic Abelian groups with n elements. In this paper, we shall establish a short internal result of [a.sup.2](n). Keywords Nonisomorphic Abelian groups, convolution method, short interval., [section] 1. Introduction We define a(n) to be the number of nonisomorphic Abelian groups with n elements. The properties of a(n) were investigated by many authors. P. Erdos and G. [...]

Let n > 1 be an integer, gcd(a, b) denote the greatest commom divisor of a and b. A(n) denotes the arithmetic mean of gcd(1,n), ..., gcd(n,n). In this paper, we shall establish a short interval result for the function A(n). Keywords Gcd-Sum function, short interval, convolution method., [section]1. Introduction The gcd-sum function (pillai's function) is defined by P(n) = [n.summation over k=1] gcd(k,n), (1) where gcd(a, b) denotes the greatest commom divisor of a and b. Pillai [...]

The Smarandache friendly numbers have been defined by Murthy [1]. This paper finds the Smarandache friendly numbers by solving the associated Pell's equation. Keywords Smarandache friendly numbers, pell's equation, recurrence relation., [section]1. Introduction Murthy [1] defines the Smarandache friendly numbers as follows : Definition 1.1. A pair of positive integers (m, n) (with n > m) is called the Smarandache friendly [...]

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 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 [...]

The main purpose of this paper is using the elementary method to study the hybrid mean value properties of the Smarandache kn-digital sequence and Smarandache function, and give an interesting asymptotic formula for it. Keywords Smarandache kn-digital sequence, Smarandache function, hybrid mean value, as -ymptotic formula, elementary method., [section]1. Introduction For any positive integer k, the famous Smarandache kn-digital sequence a(k, n) is defined as all positive integers which can be partitioned into two groups such that the [...]

The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 if [b.sub.i]\[a.sub.n] for every i [member of] {1, 2, ..., s}. The exponential convolution of aithmetic functions is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The inverse of the constant function with respect to [mid dot] is called the exponential analogue of the Mobius function, which is denoted by p,(e). The aim of this paper is to establish a short interval result for the function [[mu].sup.(e)]. Keywords The exponential divisor function, generalized divisor function, short interval., [section]1. Introduction Let n > 1 be an integer of canonical from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential [...]

In this paper, we study the exponential stability of linear non-autonomous systems with multiple delays. Using Lyapunov-like function, we find sufficient conditions for the exponential stability in terms of the solution of a Riccati differential equation. Our results are illustrated with numerical examples. Keywords Exponential stability, time-delay, Lyapunov function, Riccati equation., [section]1. Introduction The topic of Lyapunov stability of linear systems has been an interesting research area in the past decades. An integral part of the stability analysis of differential equations [...]

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 [...]

Murthy [1] introduced the Smarandache bisymmetric arithmetic determinant sequence. In this paper, we derive the sum of the first n terms of the sequence. Keywords The Smarandache bisymmetric arithmetic determinant sequence, the n - th term, the sum of the first n terms., § 1. Introduction and results The Smarandache bisymmetric arithmetic determinant sequence (SBADS), introduced by Murthy [1], is defined as follows. Definition 1. The Smarandache bisymmetric arithmetic determinant sequence, {SBADS(n)}, is [...]

Abstract This paper gives an alternative approach to the solution of the LP problem with equality constraints. In this scheme, the constraint system of linear equations is first solved by [...]

Abstract It is proved that product of interval number and triangular fuzzy number is a trapezoidal fuzzy number. This paper transformed a multiple attribute decision making problem with attribute weight [...]

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, as generalizations of the number sequences with alternate common differences, two types of special sequence of numbers are discussed. One is the periodic sequences of numbers [...]

Abstract For any positive integer n, we define the Smarandache multiplicative function U(n) as follows: V(1) = 1. If n > 1 and n = [p.sup.α1].sub.1] [p.sup.α2].sub.2] ... [p.sup.αs].sub.s] denotes [...]

Abstract Let [p.sub.1] and [p.sub.2] are two primes with [p.sub.2] < [p.sub.2] and [p.sub.2] = [p.sub.2] - 2, we call such pairs of primes are twin primes. About the elementary [...]

Abstract In this paper, The dual Wegl-Heisenberg frame's structure is discussed and then explicit structure of the dual Wegl-Heisenberg frame is given. Under the small perturbation of φ, some changes [...]

Abstract Let D ≠ 1 be a positive non-square integer. Let a be a positive integer. In this paper is considered the Pell's equation [x.sup.2] - D[y.sup.2] = ± a [...]

Abstract In this paper we shall study several kinds of mean values of the multiplicative arithmetic function D(n) by the convolution method. Keywords Arithmetic function, mean value, convolution method. §1. [...]

Abstract In this paper, we obtain a successive linear programming algorithm for solving the semidefinite programming (SDP) relaxation of the binary quadratic programming. Combining with a randomized method of Goemans [...]

Abstract In this paper, we consider an impulsive model of mutualism of Lotka-Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist. The species grow in a [...]

Abstract In this paper, we establish some recurrence formulas for generalized Euler numbers. Keywords Euler numbers, generalized Euler numbers, Bernoulli numbers, recurrence formula. § 1. Introduction and results For an [...]

Abstract The main purpose of this (lamer is using the elementary methods to study the solutions of an equation involving the Smarandache-type multiplicative function, and give its all positive integer [...]

Abstract The main purpose of this paper is to calculate the value of the series [+∞.summation over (n=1)] [(-1).sup.n]/[n.sup.α], [a.sup.β].sub.κ](n), where [a.sub.κ](n) is the k-power complement number of any positive [...]

Abstract In this paper, we prove some identities. In particular, we determine the stirling number of the second kind [s.sub.2](n, 4), when n ≥ 4 is given. Then we discuss [...]

Abstract In this paper, we use analytic method to study the mean value properties of Smarandache-Type Multiplicative Functions [K.sub.m](n), and give its asymptotic formula. Finally, the convolution method is used [...]

Abstract In this paper, we use the elementary method to study the solutions of an equation involving the Smarandache dual function [[bar.s].sub.k] (n), and give its all solutions. Keywords Smarandache [...]

Abstract The main purpose of this paper is using the elementary method to study the LCM Sequence, and give an asymptotic formula about this sequence. Keywords Smarandache LCM Sequence, limitation. [...]

Abstract The main purpose of this paper is using elementary method and an analytic method to study the asymptotic properties of the odd sieve sequence, and give two interesting asymptotic [...]

In this paper, the Smarandache bisymmetric geometric determinant sequence was defined and the formula for its n-th term was obtained. Keywords Smarandache bisymmetric geometric determinant sequence., [section]1. Introduction and preliminaries Majumdar [1] gave the formula for n-th term of the following sequences: Smarandache cyclic natural determinant sequence, Smarandache cyclic arithmetic determinant sequence, Smarandache bisymmetric natural determinant [...]

This issue of the journal is devoted to the proceedings of the fourth International Con- ference on Number Theory and Smarandache Problems held in Xianyang during March 22-24, 2008. The [...]

Abstract For any positive integer n, let S(n) denotes the Smarandache function, then S(n) is defined the smallest m ∈ [N.sup.+], where n|m!. In this paper, we study the mean [...]

Abstract In this paper, we study the hybrid mean value of some Smarandache-type multiplicative functions and the Mangoldt function, and give two asymptotic formulae. Keywords: Smarandache-type multiplicative functions; Mangoldt function; [...]

Abstract The main purpose of this paper is using the elementary method to study the asymptotic properties of the Smarandache factorial sequence, and give an interesting asymptotic formula. Keywords: Factorial [...]