Volume 48 , issue 3$-$4 ( 1996 ) | back |

On the impossibility of one ruler-and compass construction | 73$-$75 |

**Abstract**

We study the problem to determine all integers $n\geq4$ for which there exists the ruler-and-compass construction of the cyclic polygon whose sides are equal to $n$ given segments. The impossibility of such construction for $n=5$ is proved.

**Keywords:** Ruler-and-compass construction, cyclic polygon.

**MSC:** 51M15

A basis property of families of the Mittag-Leffler functions | 77$-$82 |

**Abstract**

In this paper we study basis properties of the Mittag-Leffler functions $\{y(x,\lambda_k)\}$ where $$ y(x,\lambda)=|x|^{\omega/2}E_1(ix\lambda;1+\omega/2),\quad\omega\in(-1,1), $$ $E_1(z;\alpha)=\sum\limits_{k=0}^{\infty}\dfrac{z^k}{\Gamma(\alpha+k)}$ and $\{\lambda_k\}$ is a sequence of complex numbers. In the case $\omega=0$ this system is reduced to the exponential system.

**Keywords:** Basis property, Riesz basis, Mittag-Leffler functions.

**MSC:** 30B60

Characterization of regular semirings | 83$-$85 |

**Abstract**

The main purpose of this paper is to establish some necessary and sufficient conditions for a semiring to be regular, in terms of its $k$-ideals.

**Keywords:** Regular semiring, $k$-ideal.

**MSC:** 16Y60

On $P_{\Sigma}$ and weakly $P_{\Sigma}$ spaces | 87$-$93 |

**Abstract**

In this paper, we point out that properties $P_{\Sigma}$ due to Wang [18] and strongly $s$-regular due to Ganster [5] are equivalent to each other. We further study these spaces and weakly-$P_{\Sigma}$ spaces defined by the second author [15].

**Keywords:** $P_{\Sigma}$, weakly-$P_{\Sigma}$, strongly
$s$-regular, $s^*$-regular, $s$-regular, almost-regular, regular closed,
semi-open, almost-continuous, almost-open.

**MSC:** 54D10, 54B05 54C08

Some inequalities for entire functions of exponential type | 95$-$98 |

**Abstract**

In this paper we give simple proofs of some inequalities for entire functions of exponential type.

**Keywords:** Entire function of exponential type, Bernstein's inequality.

**MSC:** 30D15

Asymptotic behaviour of a sequence defined by iteration | 99$-$105 |

**Abstract**

One considers some results in [1] and [2] concerning asymptotic behaviour of recurrently defined sequences and one gives more general and more precise form to them, especially by obtaining new terms in asymptotic expansion.

**Keywords:** Asymptotic behaviour of sequences.

**MSC:** 40A05

The unique extremal QC mapping and uniqueness of Hahn-Banach extensions | 107$-$112 |

**Abstract**

Let $\chi$ be an essentialy bounded complex valued measurable function defined on the unit dise $\Delta$, and let $\Lambda_\chi$ be the corrensponding linear functional on the space $\Cal B$ of analytic $L^1$-integrable functions. An outline of proof of main steps of the following is given: If $|\chi|$ is a constant function in $\Delta$, then the uniqueness of Hahn-Banach extension of $\Lambda_\chi$ from $B$ to $L^1$, when $\|\Lambda_\chi\|=\|\chi\|_\infty$, implies that $\chi$ is the unique complex dilatation. We give a short review of some related results.

**Keywords:** Hahn-Banach extension, complex dilatation, extremal QC mapping.

**MSC:** 32G15, 30C60, 30C70, 30C75