В статье доказываются две тауберовых теоремы для преобразования Лапласа медленно меняющихся с остатком функций и рассматриваются их приложения к суммам значений неотрицательных мультипликативных функций, связанных с проблемой Вирзинга, поставленной им в 1967 г. в работе [1].

Получено описание некоторого класса экспоненциальных мономов на локально компактных абелевых группах. We describe the structure of some class of exponential monomials on some locally compact abelian groups. The main result of the paper is the next theorem. Let G' and G be locally compact abelian groups, a: G'->G be a continuous surjective homomorphism and H be a kernel of a. If a is a an open maps from G' to G then any exponential monomial F(t) on the group G', which satisfy the condition F(t+h)=F(t) for all h in H, t in G, can be presented in the form F(t)=f(a(t)) for some exponential monomial f(x) on the group G.

В работе вводятся понятия относительных размерностей Реньи для покрытий, упаковок и разбиений. а так же устанавливаются некоторые связи между ними. Recently, many authors are discussing the use of methods of fractal geometry [5] to compare the distributions of the various measures. However, in practical applications, comparison of distributions by comparing the calculated multifractal spectra can be difficult. It often happens that a completely different distributions of measures can give very imperceptible differences in the spectra. To solve this problem, some authors [4,10] propose to use different methods of direct comparison of distributions. These methods are generalizations of the classical multifractal analysis developed in the works L. Olsen [9], K.-S. Lo and S.-M. Ngai [8] and others. Based on the idea of multifractal analysis [9] and the mutual multifractal analysis [1,2] we propose to introduce new concepts of relative Renyi dimensions for coverings, packings and partitions, as well as we establish some connection between them. It should be noted that these dimension proved mathematically rigorous new analogues "new relative multifractal spectrum of dimensions" proposed for purely practical purposes, R. Dansereau and W. Kinser [6].

В статье рассматривается тело, являющиеся пересечением шара и прямого произведения квадрата на прямую (бесконечный параллелепипед), причем диаметр шара лежит на оси симметрии параллелепипеда. Вычисляются объем и площадь поверхности этого тела. In the paper we study the solid being a model of the new phase nucleus for a phase change reaction. The solid is the intersection of the ball of a given radius R and an infinite parallelepiped, i.e. the cartesian product of the square with a given side a and a line. Such model appears, e.g., when describing dehydriding of activated alane: numerous nuclei of new metal phase appear and grow as hemispheres, but later they intersect being cut off by planes. Their total surface increases, reaching the maximal value that exceeds the initial total surface area S 0 of the old phase, then reduces, asymptotically tending to S 0. This property can explain the higher dehydriding rate (which depends on the surface area of the new phase) in the middle of the dehydriding reaction. We calculate volume, surface area, and some other quantities for this solid as functions of R and a. They are expressed via non-trivial integrals as elementary functions. Using these quantities, we present the conservative mathematical model describing the dehydriding reaction. Also we discuss properties of the obtained functions and the constructed model.