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

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

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

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

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

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

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 $$J_g(f)(z)=\int_0^zf(\xi)g'(\xi)\,d\xi,\quad f\in H(\Bbb D),\; z\in\Bbb D.$$ 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

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

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 $|du|^{2p-2}\in L^1(M)$, then $u$ is constant.

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

MSC: 58E20, 53C42

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

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

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

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

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

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