Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier

In this paper, we prove an analogue of the uniqueness theorems of Führer [15] and Amann and Weiss [1] to cover the degree of Fredholm operators of index zero constructed by Fitzpatrick, Pejsachowicz and Rabier [13], whose range of applicability is substantially wider than for the most classical degrees of Brouwer [5] and Leray–Schauder [22]. A crucial step towards the axiomatization of this degree is provided by the generalized algebraic multiplicity of Esquinas and López-Gómez [8, 9, 25], $$\chi $$ χ , and the axiomatization theorem of Mora-Corral [28, 32]. The latest result facilitates the axiomatization of the parity of Fitzpatrick and Pejsachowicz [12], $$\sigma (\cdot ,[a,b])$$ σ ( · , [ a , b ] ) , which provides the key step for establishing the uniqueness of the degree for Fredholm maps.

An analytical and numerical study of capillary menisci

В работе рассмотрены математические модели осесимметричных капиллярных менисков — лежащая и висящая капли, развернутый мениск, учитывающие размерную зависимость поверхностного натяжения. Доказаны теоремы существования и единственности решений задач, описывающих равновесные поверхности менисков. Разработаны и протестированы эффективные численные методы, предназначенные для приближенного расчета профилей менисков. На языке «Wolfram Language» написана компьютерная программа, с помощью которой проведены масштабные вычислительные эксперименты по выявлению степени и характера влияния параметров моделей на равновесную форму каждого из рассматриваемых менисков In the current paper we consider the mathematical models of axisymmetric capillary menisci — sessile and pendant drops, rolled out meniscus, taking into account the size dependence of surface tension. Existence and uniqueness theorems for solutions of problems describing equilibrium meniscus surfaces are proved. Effective numerical methods have been developed and tested for the approximate calculation of meniscus profiles. A computer program is written in the Wolfram Language, with the help of which large-scale computational experiments were carried out to reveal the degree and nature of the influence of the model parameters on the equilibrium shape of each type of menisci.

PS-hollow representations of modules over commutative rings

Let [Formula: see text] be a commutative ring and [Formula: see text] a nonzero [Formula: see text]-module. We introduce the class of pseudo-strongly[Formula: see text]PS[Formula: see text]-hollow submodules of [Formula: see text]. Inspired by the theory of modules with secondary representations, we investigate modules which can be written as finite sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of minimal PS-hollow strongly representations of modules over Artinian rings.

Inverse problems of determining an order of fractional Riemann-Liouville time-fractional derivative for the subdiffusion equation in RN

An initial-boundary value problem for the subdiffusion equation with an elliptic operator A(D) in RN is studied in the article. Existence and uniqueness theorems for the problem under study are proved by the Fourier method. Considering the order of the Riemann-Liouville time-fractional derivative as an unknown parameter, an inverse problem of determining this parameter is investigated. Likewise, the initial-boundary value problem was considered in the case of replacing the operator A(D) with its power Aσ.Then, existence and uniqueness theorems were proved for the solution of the inverse problem of determining the order of the fractional derivative and the power σ.

Proving Fixed Point Theorems Employing Fuzzy $(\sigma,\mathcal{Z})$-Contractive Type Mappings

In this article, the concept of fuzzy $(\sigma,\mathcal{Z})$-contractive mapping has been introduced in fuzzy metric spaces which is an improvement over the corresponding concept recently introduced by Shukla et al. [Fuzzy Sets and system. 350 (2018) 85--94]. Thereafter, we utilized our newly introduced concept to prove some existence and uniqueness theorems in $\mathcal{M}$-complete fuzzy metric spaces. Our results extend and generalize the corresponding results of Shukla et al.. Moreover, an example is adopted to exhibit the utility of newly obtained results.

Problem of Statics of the Linear Thermoelasticity of the Microstretch Materials with Microtemperatures for a Half-space

We consider the statics case of the theory of linear thermoelasticity with microtemperatures and microstrech materials. The representation formula of differential equations obtained in the paper is expressed by the means of four harmonic and four metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrate an application of these formulas to the III type boundary value problem for a half-space. Uniqueness theorems are proved. Solutions are obtained in quadratures. 2010 Mathematics Subject Classification. 74A15, 74B10, 74F20.