Volume 65 , issue 4 ( 2013 ) back
 A note on generating functions of Ces\aro polynomials of several variables 425$-$430 Mohd Akhlaq Malik

Abstract

The present paper deals with certain generating functions of Ces\{a}ro polynomials of several variables.

Keywords: Ces\{a}ro polynomials of two and three variables; mixed bilateral generating functions; mixed trilateral generating functions.

MSC: 33C45

 On extension of Gabor transform to Boehmians 431$-$444 R. Roopkumar

Abstract

On the theory of windowed Fourier transform proposed in the article Wavelet transforms for integrable Boehmians, J. Math. Anal. Appl. 296 (2004) 473--478'' many conceptual mistakes are pointed out, and the windowed Fourier transform (Gabor transform) on $L^2({R})$ is extended to a suitable Boehmian space. The properties of the extended Gabor transform are also established.

Keywords: Boehmians; convolution; tempered distributions; ridgelet transform.

MSC: 44A15, 44A35, 42C40

 A common generalization of fuzzy primes 445$-$453 Naser Zamani and Zeinab Rezaei

Abstract

Let $R$ be a commutative ring with identity. Let $FI(R)$ be the set of all fuzzy ideals of $R$ and $\phi:FI(R)\rightarrow FI(R)\cup\{0_R\}$ be a function. We introduce the concept of fuzzy $\phi$-prime ideals of $R$ and study some of its properties. It will be shown that under additional conditions fuzzy $\phi$-primeness implies fuzzy primeness. We also prove that in the decomposable rings fuzzy $\phi_{(1)}$-primes and fuzzy primes coincide. The behavior of this concept with fuzzy localization and fuzzy quotient is also studied.

Keywords: Fuzzy $\phi$-prime ideals.

MSC: 08A72

 A generalized operator involving the $q$-hypergeometric function 454$-$465 Aabed Mohammed and Maslina Darus

Abstract

Motivated by the familiar $q$-hypergeometric functions, we introduce here a new general operator. By this operator, we define a subclass of analytic function. The class generalizes well known classes of starlike and convex functions. The integral means inequalities of this class are investigated. Also, we consider $p$-$\gamma$-neighborhood for functions in this class. Our result contains some interesting corollaries as its special cases.

Keywords: $q$-hypergeometric functions; $q$-derivative; $q$-shifted factorial; general operator; integral means inequalities; neighborhood.

MSC: 30C45

 A class of Cayley graphs induced by right solvable ward groupoids 466$-$469 Anil Kumar V.

Abstract

In this paper, we introduce a class of Cayley graphs induced by right solvable Ward groupoids. This class of Cayley graphs can be considered as a generalization of Cayley graphs induced by groups. Also, many graph properties are expressed in terms of algebraic properties. This did not attract much attention in the literature.

Keywords: Cayley graph; vertex-transitive graph; Hasse-diagram.

MSC: 05C25

 Entire functions and their derivatives share two finite sets 470$-$475 Chao Meng

Abstract

In this paper, we study the uniqueness of entire functions and prove two theorems which improve the result given by Fang [M.L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sci. Soc. 24 (2001), 7--16].

Keywords: Entire function; share set; uniqueness.

MSC: 30D35

 Closed subsets of star $\sigma$-compact spaces 476$-$480 Yan-Kui Song

Abstract

In this paper, we prove the following statements: (1) There exists a pseudocompact star $\sigma$-compact Tychonoff space having a regular-closed subspace which is not star $\sigma$-compact. (2) Assuming $2^{\aleph_0}=2^{\aleph_1}$, there exists a star countable (hence star $\sigma$-compact) normal space having a regular-closed subspace which is not star $\sigma$-compact.

Keywords: Pseudocompact space; normal space; star-Lindel{ö}f space; star $\sigma$-compact space.

MSC: 54D20, 54B10, 54D55

 Volterra type operators from weighted Hardy spaces to Bloch spaces 481$-$487 Xiangling Zhu

Abstract

Let $H(\Bbb D)$ denote the space of all analytic functions on the unit disk $\Bbb D$ of $\Bbb C$. In this paper we consider the following Volterra type operator $$J_g(f)(z)=ınt_0^zf(\xi)g'(\xi)\,d\xi,\quad fın H(\Bbb D),\; zın\Bbb D.$$ The boundedness and compactness of the operator $J_g$ from the weighted Hardy space to a Bloch space are studied.

Keywords: Volterra type operator; weighted Hardy space; Bloch space; boundedness; compactness.

MSC: 47B38, 32A18

 Equivalence relations of $n$-norms on a vector space 488$-$493 Tyas Rangga Kristiantoo, Raden Akbar Wibawa-Kusumah and Hendra Gunawan

Abstract

A vector space can be equipped with more than one $n$-norm. In such a case, an equivalence relation of $n$-norms is usually studied. Here we discuss and present some results on several equivalence relations of $n$-norms which may be defined on a vector space. In particular, our results correct an error that we found in B.S. Reddy, H. Dutta, [On equivalence of $n$-norms in $n$-normed spaces, preprint, {http://www.akamaiuniversity.us/PJST11\_1\_233.pdf}, March 7, 2011]. We also discuss an equivalence relation of $n$-norms on finite dimensional spaces.

Keywords: $n$-normed spaces; equivalence relations.

MSC: 46B05, 46B20, 46A99, 46B99

 A Liouville type theorem for $p$-harmonic functions on minimal submanifolds in $\Bbb R^{n+m}$ 494$-$498 Yingbo Han and Shuxiang Feng

Abstract

In this note, we prove that if an $n$-dimensional complete noncompact minimal submanifold $M$ in $R^{n+m}$ has sufficiently small total scalar curvature, and $u$ is a $p$-harmonic function on $M$ with $|du|^{2p-2}ın L^1(M)$, then $u$ is constant.

Keywords: Minimal submanifolds; $p$-harmonic function; Liouville type theorem.

MSC: 58E20, 53C42

 The $\chi^{2}$ fuzzy numbers defined by a modulus 499$-$510 N. Subramanian

Abstract

In this paper, we introduce the $\chi^{2}$ fuzzy numbers defined by a modulus, study some of their properties and inclusion results.

Keywords: Gai sequence; analytic sequence; modulus function; double sequences; completeness; solid space; symmetric space.

MSC: 40A05, 40C05, 40D05

 Rate of convergence of some neural network operators to the unit-univariate case, revisited 511$-$518 George A. Anastassiou

Abstract

This article deals with the determination of the rate of convergence to the unit of some neural network operators, namely, `the normalized bell and squashing type operators''. This is given through the modulus of continuity of the involved function or its derivative and that appears in the right-hand side of the associated Jackson type inequalities.

Keywords: Neural network approximation; bell and squashing functions; modulus of continuity.

MSC: 41A17, 41A25, 41A30, 41A36

 Almost metric versions of Zhong's variational principle 519$-$532 Mihai Turinici

Abstract

A refinement of Zhong's variational principle [Nonlinear Anal. 29 (1997), 1421--1431] is given, in the realm of almost metric structures. Applications to equilibrium points are also provided.

Keywords: Inf-proper lsc function; variational principle; maximal element; almost metric; Dependent Choice Principle; normal couple; equilibrium point.

MSC: 54F05, 47J20

 Characterization of $GCR$-lightlike warped product of indefinite Kenmotsu manifolds 533$-$546 Rakesh Kumar

Abstract

In this paper, we prove that there do not exist warped product $GCR$-lightlike submanifolds in the form $M = N_{\bot}\times_{\lambda}N_{\top}$ such that $N_{\bot}$ is an anti-invariant submanifold tangent to $V$ and $N_{\top}$ an invariant submanifold of $\overline{M}$, other than $GCR$-lightlike product in an indefinite Kenmotsu manifold. We also obtain some characterizations for a $GCR$-lightlike submanifold to be locally a $GCR$-lightlike warped product.

Keywords: $GCR$-lightlike submanifold, $GCR$-lightlike product, $GCR$-lightlike warped product submanifold, indefinite Kenmotsu manifold.

MSC: 53C40, 53C42, 53C50, 53C55

 The univalence of some integral operators using the Bessel functions 547$-$554 Nicoleta Ularu

Abstract

In this paper we will introduce some new integral operators using the generalized Bessel functions and analytic functions. For this operators we will prove the univalence condition.

Keywords: Bessel functions; integral operator; univalent functions; analytic functions.

MSC: 30C45

 On a new class of harmonic univalent functions 555$-$564 Waggas Galib Atshan and Abbas Kareem Wanas

Abstract

We define a new class of harmonic univalent functions of the form $f = h + \bar g$ in the open unit disk $U$. Also we study some properties of this class, like coefficient bounds, extreme points, convex combination, distortion bounds, integral operator and convolution property.

Keywords: Harmonic function; extreme points; convex combination; integral operator.

MSC: 30C45, 30C50