Abstract

Let $(M, g(t))$, $t\in[0,T)$ be a closed Riemannian $n$-manifold whose Riemannian metric $g(t)$ evolves by the geometric flow $\frac{\partial }{\partial t} g_{ij}=-2S_{ij}$, where $S_{ij}(t)$ is a symmetric two-tensor on $(M,g(t))$. We discuss differential Harnack estimates for positive solution to the porous medium equation with potential, $\frac{\partial u}{\partial t}=\Delta u^{p}+S u$, where $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$, on time-dependent Riemannian metric evolving by the above geometric flow.

Keywords: Harnack estimates; geometric flow; porous medium equation.

MSC: 53C21, 53C44, 58J35

Abstract

Textures are point-set setting for fuzzy sets, and they provide a framework for the complement-free mathematical concepts. This paper is the first of a series of two papers on the theory of nearness spaces. This paper aims to give a new perspective for nearness structure from the textural point of view. It is proved that nearness spaces are embeddable into texture space which is connected with nearness structure.

Keywords: Nearness; symmetric topological space; cover; texture; fuzzy sets.

MSC: 54E17, 54E05A15, 06D72, 54E15

Abstract

Let $M={(M_{j})}_{j}$ be an increasing sequence of positive real numbers with $M_{0}=1$ such that the sequence $M_{j+1}/M_j$ increases and let $\mathcal{{E}}_{n}(M)$ be the Denjoy-Carleman class associated to this sequence. Let $\hat{\mathcal{{E}}}_{n}(M)$ denote the Taylor expansion at the origin of all elements that belong to the ring $\mathcal{{E}}_{n}(M)$. We say that $\hat{\mathcal{{E}}}_{n}(M)$ satisfies the splitting property if for each $f\in\hat{\mathcal{{E}}}_{n}(M)$ and $A \cup B =\mathbb{N}^{n}$ a partition of $\mathbb{N}^{n}$, when $G=\sum_{w\in A}a_{w}x^{w}$ and $H=\sum_{w\in B}a_{w}x^{w}$ are formal power series with $f=G+H$, then $G\in\hat{\mathcal{{E}}}_{n}(M)$ and $H\in\hat{\mathcal{{E}}}_{n}(M)$. Our first goal is to show that if the Borel mapping $^\wedge:\mathcal{{E}}_{1}(M)\rightarrow\mathbb{R}[[x_1]]$ is a homeomorphism onto its range for the inductive topologies, then the ring $\mathcal{{E}}_{1}(M)$ coincides with the ring of real analytic germs. Secondly, we will give a negative answer to the splitting property for the quasianalytic local rings $\mathcal{{E}}_{n}(M)$. In the last section, we will show that the ring of smooth germs that are definable in the polynomially bounded o-minimal structure of the real field expanded by all restricted functions in some Denjoy-Carleman rings does not satisfy the splitting property in general.

Keywords: Denjoy-Carleman rings; splitting property; Borel mapping; quasianalyticity.

MSC: 26E10, 03C64

Abstract

Here left and right Riemann{-}Liouville generalized fractional radial integral operators of variable order over a spherical shell are introduced, as well as left and right weighted Caputo type generalized fractional radial derivatives of variable order over a spherical shell. After proving continuity of these operators, we establish a series of left and right fractional integral inequalities of variable order over the spherical shell of Opial and Hardy types. Extreme cases are met.

Keywords: Fractional Opial and Hardy type inequalities; generalized fractional operators of variable order; spherical shell.

MSC: 26A33, 26D10, 26D15

Abstract

In this paper, we study the notions of point-open topology $C_p(X,H)$, open-point topology $C_h(X,H)~[\text{resp.}~C_h(G,H)]$ and bi-point-open topology $C_{ph}(X,H)~[\text{resp.}$ $C_{ph}(G,H)]$ on $C(X,H)~[\text{resp.} C(G,H)]$, the set of all continuous functions from a topological space $X$ (topological group $G)$ to a topological group $H$. In this setting, we study the countability, separation axioms and metrizability. The equivalent conditions are given so that the space $C_h(G,H)$ is a zero-dimensional topological group. Further, if $G$ is $H^{\star\star}$-regular, then $C_h(G,H)$ is Hausdorff if and only if $G$ is discrete. It is shown that under certain conditions the topological groups $C_p(X,H)$, $C_h(X,H)$ and $C_{ph}(X,H)$ are $\omega$-narrow. Sufficient conditions are given for the topological spaces $C_p(X,H)$, $C_h(X,H)$ and $C_{ph}(X,H)$ to be discretely selective and to have a disjoint shrinking.

Keywords: Point-open topology; open-point topology; bi-point-open topology; topological group; zero dimensional; $\omega$-narrow; disjoint shrinking; discrete selection.

MSC: 54C35, 54A10, 54C05, 54D10, 54D15, 54E35, 54H11

Abstract

The aim of this note is to provide a natural extension and generalisation of the well-known Gelfond constant $e^\pi$ using a hypergeometric function approach. An extension is also found for the square root of this constant. Several known mathematical constants are also deduced in hypergeometric form from our newly introduced constant.

Keywords: Gelfond's constant; hypergeometric function; Gauss summation theorem.

MSC: 11Y60, 33B10, 33C05, 33C20