Abstract

In this paper we introduce and study some stronger forms of transitivity like total transitivity, weakly mixing for maps on $G$-spaces. We obtain their relationship with the earlier defined notion of strongly mixing for maps on $G$-spaces. We also study in detail $G$-minimal maps on $G$-spaces.

Keywords: Topological transitivity; topological mixing; $G$-space; pseudoequivariant map.

MSC: 54H20, 37B05

Abstract

We present here a strong mixed fractional calculus theory for Banach space valued functions of generalized Canavati type. Then we establish several mixed fractional Bochner integral inequalities of various types.

Keywords: Right and left fractional derivative; right and left fractional Taylor's formula; Banach space valued functions; integral inequalities; Bochner integral.

MSC: 26A33, 26D10, 26D15, 46B25

Abstract

Recently Andrews, Lewis and Lovejoy introduced the partition functions $PD(n)$ defined by the number of partitions of $n$ with designated summands and they found several modulo 3 and 4. In this paper, we find several congruences modulo 3 and 4 for $PBD_{3}(n)$, which represent the number of 3-regular bi-partitions of $n$ with designated summands. For example, for each \quad $n\ge1$ and $\alpha\geq0$ \quad $PBD_{3}(4\cdot3^{\alpha+2}n+10\cdot3^{\alpha+1})\equiv 0 \pmod{3}$.

Keywords: Partitions; designated summands; congruences.

MSC: 05A17, 11P83

Abstract

Chaatit, Mascioni, and Rosenthal defined finite Baire index for a bounded real-valued function $f$ on a separable metric space, denoted by $i(f)$, and proved that for any bounded functions $f$ and $g$ of finite Baire index, $i(h)\leq i(f)+i(g)$, where $h$ is any of the functions $f+g$, $fg$, $f\vee g$, $f\wedge g$. In this paper, we prove that the result is optimal in the following sense : for each $n,k<\omega$, there exist functions $f,g$ such that $i(f)=n$, $i(g)=k$, and $i(h)=i(f)+i(g)$.

Keywords: Finite Baire index; oscillation index; Baire-1 functions.

MSC: 26A21, 54C30, 03E15

Abstract

In this paper, we will develop a new method to study coupled fixed points of a mapping $T:X \times X \to X$, where $(X, d)$ is a special class of $b$-metric spaces endowed with a graph. We will prove some general fixed point theorems which enable us to extend some old results in fixed point theory. Moreover, we will extend Edelstein's fixed point theorem for two variable mappings in $\varepsilon$-chainable $b$-metric spaces.

Keywords: Coupled fixed point; connected graph; Picard operator; $b$-metric space.

MSC: 47H10, 54M15, 54C10

Abstract

A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central quotient is isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ or $D_{2m}$, where $p$ is any prime integer and $D_{2m}$ is the dihedral group of order $2m$.

Keywords: Integral graph; commuting graph; spectrum of a graph.

MSC: 05C25, 05C50, 20D60