Volume 58 , issue 1$-$2 ( 2006 ) | back |

Operational quantities derived from the minimum modulus | 1$-$5 |

**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 |

**Abstract**

The main purpose of this paper is to introduce a class $TS_H^*(\gamma)$ ($\gamma\in C\setminus\{0\}$) 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 |

**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 $$ P_n^{(\alpha)}(x):=\sum_{m\le n}c_mp_mx^m\sim c_nP_n(x) \quad (n\to\infty), $$ to hold for each $x\ge A$ and an arbitrary regularly varying sequence $\{c_n\}$ of index $\alpha\in R$.

**Keywords:** Regular variation, polynomials, asymptotic behavior.

**MSC:** 26A12

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

**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 |

**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 \in 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 |

**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 |

**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 |

**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 |

**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