Volume 58 , issue 1$-$2 ( 2006 ) back
 Operational quantities derived from the minimum modulus 1$-$5 Manuel Gonzáles and Antonio Martin\'on

Abstract

The minimum modulus $\gamma(T)$ of an operator $T$ is useful in perturbation theory because it characterizes the operators with closed range. Here we study the operational quantities derived from $\gamma(T)$. We show that the behavior of some of these quantities depends largely on whether the null space of $T$ is finite dimensional or infinite dimensional.

Keywords: Minimum modulus, perturbation theory.

MSC: 47A53

 Harmonic functions starlike of the complex order 7$-$11 Sibel Yal\c cin and Metin Öztürk

Abstract

The main purpose of this paper is to introduce a class $TS_H^*(\gamma)$ ($\gammaın C\setminus\{0\}$) of functions which are harmonic in the unit disc. We give necessary and sufficient conditions for the functions to be in $TS_H^*(\g)$.

Keywords: Harmonic functions, starlike functions.

MSC: 30C45, 30C50, 31A05

 On Hadamard type polynomial convolutions with regularly varying sequences 13$-$17 Slavko Simić

Abstract

For a sequence of polynomials $P_n(x):=\sum_{m\le n}p_mx^m$, $n\ge 1$, we give a necessary and sufficient condition for the asymptotic equivalence $$P_n^{(\alpha)}(x):=\sum_{m\le n}c_mp_mx^m\sim c_nP_n(x) \quad (n\toınfty),$$ to hold for each $x\ge A$ and an arbitrary regularly varying sequence $\{c_n\}$ of index $\alphaın R$.

Keywords: Regular variation, polynomials, asymptotic behavior.

MSC: 26A12

 Direct and inverse theorems for Sz\^asz-Lupas type operators in simultaneous approximation 19$-$29 Naokant Deo

Abstract

In this paper we give the direct and inverse theorems for Sz\^{a}sz-Lupas operators and study the simultaneous approximation for a new modification of the Sz\^{a}sz operators with the weight function of Lupas operators.

Keywords: Linear positive operators, linear combination, integral modulus of smoothness, Steklow means.

MSC: 41A35

 Finite groups admitting some coprime operator groups 31$-$37 Enrico Jabara

Abstract

Let $G$ be a finite group, with a finite operator group $A$, satisfying the following conditions: (1)~$(\vert G \vert, \vert A \vert)=1$; (2)~there exists a natural number $m$ such that for any $\alpha, \beta ın A^{\sharp}$ we have: $[\,C_G(\alpha),\underbrace{C_G(\beta),\dots,C_G(\beta)}_{m}\,]=\{1\}$; (3)~$A$ is not cyclic. We prove the following: (1)~If the exponent $n$ of $A$ is square-free, then $G$ is nilpotent and its class is bounded by a function depending only on $m$ and $\lambda(n)$ ($=n$). (2)~If $Z(A)=\{1\}$ and $A$ has exponent $n$, then $G$ is nilpotent and its class is bounded by a function depending only on $m$ and $\lambda(n)$.

Keywords: Finite groups, operator group, Frobenius group.

MSC: 20D45

 On pseudo-$BCI$ ideals of pseudo-$BCI$ algebras 39$-$46 Young Bae Jun, Hee Sik Kim and J. Neggers

Abstract

The notions of pseudo-atoms, pseudo-$BCI$ ideals and pseudo-$BCI$ homomorphisms in pseudo-$BCI$ algebras are introduced. Characterizations of a pseudo-$BCI$ ideal are displayed, and conditions for a subset to be a pseudo-$BCI$ ideal are given. The concept of a $\diamond$-medial pseudo-$BCI$ algebra is also introduced, and its characterization is provided. We show that every pseudo-$BCI$ homomorphic image and preimage of a pseudo-$BCI$ ideal is also a pseudo-$BCI$ ideal.

Keywords: Pseudo-$BCK/BCI$-algebra, pseudo-atom, pseudo-$BCI$ ideal, pseudo-$BCI$ homomorphism.

MSC: 06F35, 03G25

 On the periods of 2-step general Fibonacci sequences in dihedral groups 47$-$56 Erdal Karaduman and Hüseyin Aydin

Abstract

In this paper, we investigate the simply periodic cases of 2-step general Fibonacci sequences in dihedral groups $D_n$, and we also find the period of the sequences if the sequences are simply periodic.

Keywords: Fibonacci sequences, period, dihedral group

MSC: 11B39

 Open covers and function spaces 57$-$70 Vladimir Pavlović and Bruno A. Pansera

Abstract

We investigate some closure properties of the space $C(X)$ of the continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology and the pointwise convergence topology.

Keywords: Reznichenko property, Pytkeev property, $k$-cover, countable (strong) fan tightness, groupability, compact-open topology, pointwise convergence topology, selection principles.

MSC: 54C35, 54D20

 Covering of curves, gonality, and scrolar invariants 71$-$75 E. Ballico

Abstract

Let $f\: X \to Y$ be a degree $k$ covering of smooth and connected projective curves with $p_a(Y)>0$. Here we continue the study of the Brill-Noether theory of divisors on $X$.

Keywords: Covering of curves; gonality; scrollar invariant; hyperelliptic curve; Brill-Noether theory; ruled surface.

MSC: 14H51, 14H50