Volume 74 , issue 1 ( 2022 ) | back |

ON ALGEBROID FUNCTIONS WITH UNIFORM SCHWARZIAN DERIVATIVE | 1$-$14 |

**Abstract**

The question of determining under which conditions the Schwarzian derivative of an algebroid function turns out to be a uniform meromorphic function in the plane is considered. In order to do this the behaviour of the Schwarzian derivative of an algebroid function $w(z)$ around a ramification point is analyzed. It is concluded that in case of a uniform Schwarzian derivative $S_{w}(z)$, this meromorphic function presents a pole of order two at the projection of the ramification point, with a rational coefficient $\gamma_{-2}$, where $0<\gamma_{-2}<1.$ A class of analytic algebroid functions with uniform Schwarzian derivative is presented and the question arises whether it contains all analytic algebroid functions with this property.

**Keywords:** Schwarzian derivative; algebroid function; Möbius transformation; ramification point.

**MSC:** 30B40, 30F99

HARNACK ESTIMATES FOR THE POROUS MEDIUM EQUATION WITH POTENTIAL UNDER GEOMETRIC FLOW | 15$-$25 |

**Abstract**

Let $(M, g(t))$, $tın[0,T)$ 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

NEARNESS STRUCTURE ON TEXTURE SPACES | 26$-$34 |

**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

THE BOREL MAPPING OVER SOME QUASIANALYTIC LOCAL RINGS | 35$-$41 |

**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 $fın\hat{\mathcal{{E}}}_{n}(M)$ and $A \cup B =\mathbb{N}^{n}$ a partition of $\mathbb{N}^{n}$, when $G=\sum_{wın A}a_{w}x^{w}$ and $H=\sum_{wın B}a_{w}x^{w}$ are formal power series with $f=G+H$, then $Gın\hat{\mathcal{{E}}}_{n}(M)$ and $Hın\hat{\mathcal{{E}}}_{n}(M)$. 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

FRACTIONAL INTEGRAL INEQUALITIES OF VARIABLE ORDER ON SPHERICAL SHELL | 42$-$55 |

**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

OPEN-POINT AND BI-POINT OPEN TOPOLOGIES ON CONTINUOUS FUNCTIONS BETWEEN TOPOLOGICAL (SPACES) GROUPS | 56$-$70 |

**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

HYPERGEOMETRIC REPRESENTATIONS OF GELFOND'S CONSTANT AND ITS GENERALISATIONS | 71$-$77 |

**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