Abstract

This paper deals with the existence and asymptotic analysis of positive continuous solutions for the following nonlinear polyharmonic boundary value problem: \begin{align*} \begin{cases} (-\Delta)^{m}u= b(x)u^{p}, \ \ \text{in} \ D, & \\ \lim\limits_{|x| \rightarrow 1 } \big(|x|^{2}-1\big)^{1-m} u(x) =0, \\ \lim\limits_{|x| \rightarrow \infty } (|x|^{2}-1\big)^{1-m} u(x) = 0. \end{cases} \end{align*} Here $m$ is an integer greater than 2, $p \in(-1,1)$, $ D$ is the complementary of the closed unit ball of $\mathbb{R}^{n} $ with $n >2m$, and the function $b$ is nonnegative and continuous on $D$, satisfying some appropriate assumptions related to Karamata regular variation theory.

Keywords: Polyharmonic equation; positive solutions; Kato class; Karamata classes; Green function; Schauder's fixed point theorem.

MSC: 31B30, 35B09, 35B40, 35J30, 35J40

Abstract

This study is on an innovative technique of integral-type operators that adopt the Baskakov basis function in recursion form and the Szász basis function, accentuating how well they approximate integrable functions. The study addresses the challenge of achieving more accurate function approximation, and mainly contributes to improving the theoretical aspects of the proposed operators. We examine the convergence properties of the proposed operators by employing Peetre's $K$-functional, second-order modulus of smoothness, and modulus of continuity. Additionally, we derive the Voronovskaja-type asymptotic formula and establish approximation results in weighted spaces. Finally, we show that the proposed operators significantly enhance the approximation accuracy through various examples and graphs.

Keywords: Modified Baskakov operators; Szász operators; point-wise convergence; modulus of continuity; asymptotic formula; weighted spaces.

MSC: 26A15, 41A25, 41A35

Abstract

A specialization semilattice is a semilattice together with a coarser preorder satisfying a compatibility condition. We show that the category of specialization semilattices is isomorphic to the category of semilattices with a congruence, hence equivalent to the category of semilattice epimorphisms.
Guided by the above example, we recall an ``internal'' characterization of surjective homomorphisms between general relational systems.

Keywords: Specialization semilattice; specialization poset; epimorphism; semilattice with a congruence; compatible preorder; arrow category of epimorphisms.

MSC: 06A12, 54A05, 18A25

Abstract

Let $R$ be a commutative Noetherian ring, $\frak{b}$ be an ideal of $R$, $M$ an $R$-module and let $d$ be a non-negative integer. We introduce a general $d$-transform functor $T_{d,\frak{b}}(M,-)$ and its right derived functors $T^i_{d,\frak{b}}(M,-)$, $i\in\mathbb{N}_0$, on the category of $R$-modules and study their various properties. The connection of these functors with some kind of generalized local cohomology functors $H^i_{d,\frak{b}}(M,-)$ is discussed. When both $M$ and $N$ are finitely generated, some finiteness results on $T^i_{d,\frak{b}}(M,N)$ and $H^i_{d,\frak{b}}(M,N)$ are concluded. Then, we study how the depth and dimension of certain subsets of $\mathrm{Spec}(R)$ affect the behavior and vanishing of these modules.

Keywords: Local cohomology; generalized local cohomology; projective dimension; $d$-transform; injective modules.

MSC: 13C05, 13C12, 16D70

Abstract

In this article, we will study the equivariant cohomology theory for actions of compact group (not necessarily Lie group) on compact spaces. We will present a somewhat more general and shorter proof of the localization theorem, known as the Borel-Hsiang-Quillen localization theorem, which was generalized by Özkurt and Onat to actions of finite-dimensional compact groups on compact connected spaces. In particular, we will apply this to the problem of the existence of equivariant maps between topological transformation groups.

Keywords: Equivariant cohomology; Borel-Hsiang-Quillen localization theorem; Borsuk-Ulam type theorem.

MSC: 55N25, 54H15, 55N91, 22C05

Abstract

It is well known that if the Borel fibration of a compact abelian connected group action on a compact space is totally nonhomologous to zero, then the action must have fixed points. But this is usually not true. In this article, we will give an example that are both TNHZ and the action has no fixed point. Further, we will examine paracompact spaces with TNHZ for compact abelian connected group actions without fixed points, and provide certain necessary conditions for the existence of such actions.

Keywords: TNHZ; totally disconnected; compact group; solenoidal subgroup.

MSC: 55N91, 57S10, 22C05

Abstract

In this paper, we examine the recently introduced concept of equidistant dimension $\mathrm{eqdim}(G)$ for Hamming graphs $H_{r,k}$. For hypercubes $Q_r=H_{r,2}$, exact values have been derived for $r\not\equiv 0 (\!\!\!\mod 4)$. Finally, the exact value for $\mathrm{eqdim}(H_{2,k})$ has been derived. We have shown that for Hamming graphs $H_{2,k}$, the equidistant dimension remains constant when $k\ge 5$, whereas for hypercubes, it grows linearly with the order of the graph.

Keywords: Distance-equalizer set; equidistant dimension; hypercubes; Hamming graphs.

MSC: 05C12, 05C69

Abstract

The concept of subordination is attributed to Lindelöf (1909), but Littlewood and Rogosinski introduced the terms and discovered the basic relations. Two of the earliest first-order differential subordinations were given in 1935 by Goluzin and in 1947 by Robinson, but it was Miller and Mocanu who worked out the problem in detail. Different types of differential subordination were considered in the early articles of the originators of this theory, and then by other mathematicians. In this paper we introduce and explore differential subordinations related to generalized Pythagorean means i.e. arithmetic, geometric and harmonic means. By transferring the obtained results to the ground of analytic functions we obtain new concepts in geometric functions theory.

Keywords: Differential subordination; Pythagorean means; generalized harmonic mean; generalized geometric mean; generalized arithmetic mean; convex function.

MSC: 30C80, 30C45