Volume 52 , issue 3$-$4 ( 2000 ) | back |

On splitting rings for Azumaya skew group rings | 63$-$69 |

**Abstract**

Let $B$ be a ring with 1, $G$ an automorphism group of $B$ of order $n$ for some integer $n$, $B\ast G$ the skew group ring over $B$ with a free basis $\{g\mid gın G\}$, $B^G$ the set of elements in $B$ fixed under $G$, and $\overline G$ the inner automorphism group of $B\ast G$ induced by~$G$. It is shown that when the center $C$ of $B$ is a $G$-Galois algebra over $C^G$ with Galois group $G|_C\cong G$ or $B$ is a $G$-Galois extension of $B^G$ and $n^{-1}ın B$, then, $B\ast G$ is an Azumaya algebra if and only if so is $(B\ast G)^{\overline G}$, and some splitting rings of $B\ast G$, $(B\ast G)^{\overline G}$ and $B$ are shown to be the same.

**Keywords:** Skew group rings, Azumaya algebras, Galois extensions, splitting
rings.

**MSC:** 16S30; 16W20

Convexity and reflexivity | 71$-$78 |

**Abstract**

In recent years some papers have appeared containing generalizations of the concept of convexity with the help of the notion of measure of noncompactness. Furthermore, some authors have introduced the concept of near uniform convexity by means of the Hausdorff measure and of weak near uniform convexity by means of the De Blasi measure. In this work we present a generalization of these concepts by means of a general set quantity.

**Keywords:** The Hausdorff and the De Blasi
measures, nearly and locally nearly uniformly convex space, set quantity,
reflexive Banach space.

**MSC:** 46A25; 46B10

Conditions for existence of periodic solutions of nonlinear differential equations of the third order | 79$-$82 |

**Abstract**

Using Schauder's fixed point theorem sufficient conditions are found for existence of periodic solutions of equations of the third order.

**Keywords:** Periodic solution, nonlinear differential equation of third order.

**MSC:** 34C25

Relationships between usual and approximate inverse systems | 83$-$97 |

**Abstract**

We shall prove that if $\bold X = \{X_{a}, p_{ab}, A\}$ is an approximate inverse system of compact non-metric spaces with surjective bonding mappings $p_{ab}$ such that each $X_{a}$ is a limit of a usual $\tau $-directed inverse system $X(a)=\{X_{(a,\gamma)}$, $f_{(a,\gamma)(a,\delta)}$, $\Gamma_{a}\}$ of metric compact spaces, then there exist: 1) a usual $\tau$-directed inverse system $X_{D} = \{X_{d}, F_{de}, D\}$ whose inverse limit $X_{D}$ is homeomorphic to $X=\lim\bold X$, 2) every $X_{d}$ is a limit of an approximate inverse system $\{X_{(a,\gamma_{a})}$, $g_{(a,\gamma _{a})(b,\gamma_{b})},A\}$ of compact metric spaces $X_{(a,\gamma_{a})}$, 3) if the mappings $p_{ab}$ and $f_{(a,\gamma)(a,\delta)}$ are monotone, then $g_{(a,\gamma_{a})(b,\gamma_{b})}$ and $F_{de}$ are monotone.

**Keywords:** Approximate inverse system, usual inverse system.

**MSC:** 54C10; 54F15, 54B35

Bases in sequence spaces and expansion of a function in a series of power series | 99$-$112 |

**Abstract**

In this paper we establish a relation between the existence of a basis and the solution of an infinite linear system. Then we study the expansion of a function in a series of power series.

**Keywords:** Sequence spaces, basis.

**MSC:** 46A15

Differences of decreasing slowly varying functions | 113$-$118 |

**Abstract**

The class of slowly varying functions is not closed under subtraction. In previous papers we found some subclasses of nondecreasing slowly varying functions, characterized by the good decomposition property, which (under additional conditions) are closed under subtraction. The main result of this paper is that the assumption that functions are nondecreasing is essential because only nondecreasing functions can have the good decomposition property.

**Keywords:** Slowly varying function.

**MSC:** 26A12

A note on subparacompact spaces | 119$-$123 |

**Abstract**

Some results on subparacompact spaces have only been obtained for the class of regular spaces. The aim of this note is to obtain some properties of subparacompact spaces, and also some applications to generalized metric spaces, without requiring the regularity of the spaces involved.

**Keywords:** Subparacompact space, semi-stratifiable space, (strong)
$\Sigma$- [$\Sigma^*$-] space, isocompact space.

**MSC:** 54D20; 54E20, 54E18, 54C10