Volume 65 , issue 4 ( 2013 ) | back |

A note on generating functions of Ces\`aro polynomials of several variables | 425$-$430 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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 |

**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