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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. [...]

Smarandache LCM function and LCM ratio are already defined in [1]. This paper gives some additional properties and obtains interesting results regarding the figurate numbers. In addition, the various sequaences thus obtained are also discussed with graphs and their interpretations. Keywords Smarandache LCM Function, Smarandache LCM ratio., 1. Introduction Definition 1.1. Smarandache LCM Function is defined as SL(n) = k, where SL:N → N (1) n divides the least common multiple of 1, 2, 3, ... , [...]

In this paper, the natural partial order on semiabundant semigroups is firstly defined. And then, some properties and two examples of semiabundant semigroups which are not abundant semigroups are given. Finally one of the main resulsts is proved: if the natural partial order on semiabundant semigroup is compatible with the multiplication, then the semigroup is locally semiadequate. Keywords Green's-relation, semiabundant semigroup, natural partial order., § 1. Introduction As we know, constructions and many properties on regular and abundant semigroups have been described in terms of their natural partial orders (see [1]-[3]). The natural partial [...]

This paper first introduces the fuzzy number valued Choquet integral on crisp sets with respect to fuzzy number valued fuzzy measure, Its properties are obtained, Then it introduces the fuzzy number valued Choquet integral on fuzzy sets with respect to fuzzy number valued fuzzy measure, All these are generalizations of Choquet integral. Keywords Fuzzy number, fuzzy measure, fuzzy integral., § 1. Introduction After the formulation of fuzzy integral by Sugeno, various generalizations of fuzzy integral were introduced and investigated. Fuzzy number fuzzy integral (FNFI) were defined by various authors [...]

Let a(n) denote the number of non-isomorphic finite abelian groups with n elements and ω(n) = (a * a * a)(n). In this paper we shall study the mean value [summation over (n ≤ [ω.sup.2])](n). Keywords Divisor problem, finite abelian groups, § 1. Introduction Let a(n) denote the number of non-isomorphic finite abelian groups with n elements. This is a well-known multiplicative function, such that a([p.sup.α]) = P(α), where P(α) is [...]

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 [...]

The demand of resources in emergency systems often is uncertain and multiple. It has an important practical significance for research on the scheduling decision in multi-resource emergency systems. For the certain resource demand, the rich research results have achieved. The uncertain demand of single resource is discussed [7]. In this paper, for the uncertain demand of multi-resource, a weight coefficient is defined and a corresponding membership function is established for each resource by the policy-maker. A fuzzy mathematical programming model is established and an algorithm is proposed, which we find the maximum satisfactory degree of time restraint under the lowest satisfactory degree of resource demand. An example illustrates that the model is reasonable and the obtained algorithm is effective. Keywords Emergency systems, fuzzy number, satisfactory degree, scheduling decision in multi-resource, § 1. Introduction The scheduling problem of resources in emergency systems has become a hot topic. For the emergency-time is a certain number or interval number and the different type [...]

In this paper we discuss new separation axioms using v-- open sets. Keywords v--[T.sub.i]--, v--[R.sub.i]--and v--[C.sub.i]--spaces, § 1. Introduction Norman Levine [6] introduced the concept of semi open sets in topological spaces. After the introduction of semi open sets by Norman Levine [6] various authors have [...]

Vyawahare and Purohit [1] introduced the near pseudo Smarandache function, K(n). In this paper, we derive some more recurrence formulas satisfied by K(n). We also derive some new series, and give an expression for the sum of the first n terms of the sequence {K(n)}. Keywords The near pseudo Smarandache function, recurrence formulas, the sum of the first n terms., § 1. Introduction Vyawahare and Purohit [1] introduced a new function, called the near pseudo Smarandache function, and denoted by K(n), is defined as follows. Definition 1.1. The near pseudo [...]

Let d(n) denote the Dirichlet divisor function and a(n) denote the number of nonisomorphic abelian groups with n elements, l ≥ 1 is a fixed integer. In this paper we shall study the hybrid mean value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and obtain its asymptotic formula. Keywords Divisor problem, finite abelian groups., § 1. Introduction Let a(n) denote the number of non-isomorphic abelian groups with n elements. This is a well-known multiplicative function such that for any prime p we have a([p.sup.α]) [...]

A [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] graphs be a graph abtained from [P.sub.k] whose every vertex [υ.sub.i](i = 1, 2, ..., k) attached one cycle [C.sub.i](i = 1, 2, ..., k). In this paper, we determine the lower and the higher bound for the Merrifield--simmons index in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] graphs in terms of the order k, and characterize the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] graphs with the smallest and the largest Merrifield-simmons index. Keywords [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] graphs, σ-index or Merrifield-Simmons index., § 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 υ ∈ V, [N.sub.G](υ) = [...]

In this paper, we use the elementary method to study the convergence of the Smarandache alternate consecutive, reverse Fibonacci sequence and Smarandache multiple sequence. Keywords Convergence, Smarandache sequences, elementary method., § 1. Introduction and results For any positive integer n, the Smarandache alternate consecutive and reverse Fibonacci sequence a(n) is defined as follows: a(1) = 1, a(2) = 11, a(3) [...]

In this paper, a new class of generalized set-valued quasi variational inclusions in Banach spaces are studied, which include many variational inclusions studied by others in recent years. Using the resolvent operator technique, we establish the equivalence between the generalized set-valued quasi variational inclusions in Banach spaces and the fixed point problems, suggest and analyze a class of resolvent dynamical systems for quasi variational inclusion in Banach spaces. We show that the trajectory of the solution of the dynamics system converges globally exponentially to the unique solution of quasi variational inclusions in Banach spaces. Our results can be considered as a significant extension of previously known results. Keywords Set-valued quasi variational inclusion, dynamical systems, Banach space., § 1. Introductions In recent years, variational inequalities have been extended and generalized in different directions by using novel and innovative techniques and ideas both for their own sake and [...]

For any positive integer n, Pseudo-Smarandache-Squarefree function [Z.sub.w](n) is defined as [Z.sub.w](n) = min{m : n|[m.sup.n], m ∈ N}. Smarandache function S(n) is defined as S(n) = min{m : n|m!, m ∈ N}. The main purpose of this paper is using the elementary methods to study the mean value properties of the Pseudo-Smarandache-Squarefree function and Smarandache function, and give two sharper asymptotic formulas for it. Keywords Pseudo-Smarandache-Squarefree function [Z.sub.w](n), Smarandache function S(n), me an value, asymptotic formula., § 1. Introduction and result For any positive integer n, the famous Smarandache function S(n) is defined as S(n) = Min{m : n|m!, m ∈ N}, Pseudo-Smarandache-Squarefree function [Z.sub.w](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 Murthy [1] introduced the Smarandache Bisymmetric Determinant Natural Sequence. In this paper, we derive the sum of the first n terms of the sequence. Keywords The Smarandache bisymmetric determinant [...]

Abstract In this paper, a system of generalized set-valued variational inclusions in Banach spaces is introduced and studied, which includes many variational inclusions studied by others in recent years. By [...]

Abstract The main purpose of this paper is using the elementary methods to study the relationship between the continued fractions and Dirichlet L-functions, and give some interesting identities. Keywords Continued [...]

Abstract In this paper, we use the elementary methods to study some divisibility of the Smarandache numerical carpet, and obtain two interesting congruent theorems. Keywords Divisibility, Smarandache numerical carpet, elementary [...]

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 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 In this paper, a system of differential equation whose solution gives the components on the Frenet axis in [E.sup.3] is established by means of Frenet equations. In view of [...]

Abstract In this paper, we continue studying the properties of (γ, γ')-operations on topological spaces initiated by J. Umehara, H. Maki and T. Noiri [12] and further investigated by S. [...]

Abstract This paper considers some of the properties of the lattice cubic y = [x.sup.3] * We show that the area of any triangle inscribed in the lattice cubic is [...]

Abstract In this paper we investigate the relation for the Bernoulli numbers of higher order and the Stirling numbers of the first kind, and establish an computational formula for the [...]

Abstract In this paper, defining a vector product in Semi-Euclidean space [E.sup.4.sub.2], we present a method to calculate Frenet apparatus of Partially Null curves. Thereafter, in the same space, using [...]

Abstract For any positive integer n, let φ (n) and S(n) be the Euler function and the Smarandache function respectively. In this paper, we use the properties and the curve [...]

Abstract For orthogonal projective matrix R, i.e., [R.sup.2] = R and [R.sup.T] = R, we say that A is generalized Hermitian matrix, if RAR = A*. In this paper, we [...]

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 [...]

In this paper we introduce a new subclass [S.sup.*] (α,β, ξ, v) consisting of functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] which are analytic and q-valent in the punctured unit [...]