Representation of the density functions of a multidimensional strictly stable distributions by series of generalized functions

В статье рассматриваются многомерные строго устойчивые распределения. Как известно, функции плотности этих законов не представляются в явном виде за исключением известных законов Гаусса и Коши. Отправным пунктом для исследований являются характеристические функции. Имеется несколько различных форм их представления. В статье выбирается форма, предложенная в [1]. Применение обратного преобразования Фурье совместно с суммированием интегралов по Абелю позволило получить разложения функций плотности многомерных устойчивых распределений (см.[1], [12]). Основным результатом статьи являются представления этих функций с помощью рядов обобщенных функций над пространством Лизоркина. Они позволяют определить порядок убывания главного члена разложения на бесконечности для любого радиального направления. Кроме того, выведенные формулы дают возможность увидеть структуру формирования слагаемых в разложениях. В следствии приводятся примеры для различных случаев носителей спектральной меры многомерных устойчивых законов. The article discusses multidimensional strictly stable distributions. As is known, the density functions of these laws are not represented in closed form, with the exception of the well-known laws of Gauss and Cauchy. Characteristic functions are the starting point for research. There are several different forms of their presentation. The article chooses the form proposed in [1]. The application of the inverse Fourier transform together with the Abel summation of the integrals made it possible to obtain expansions of the density functions of multidimensional stable distributions (see [1], [12]). The main result of the article is the representation of these functions using series of generalized functions over the Lizorkin space. They make it possible to determine the order of decay of the principal term of the expansion at infinity for any radial direction. In addition, the derived formulas make it possible to see the structure of the formation of terms in expansions. In the corollary, examples are given for various cases of the support of the spectral measure of multidimensional stable laws.

Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part I: using the Schwarz alternating method

The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results.

Refinement of the Maxwell Formula for Fiber-Reinforced Composites

The effective properties of the fiber-reinforced composite materials with fibers of square cross-section are investigated. The novel formula for the effective coefficient of thermal conductivity refining the classical Maxwell formula (MF) is derived. The methods of asymptotic homogenization, boundary shape perturbation and Schwarz alternating process are applied. It is shown that the principal term of the asymptotic expansion of the refined formula in powers of small size of inclusions coincides with the classical MF. The corrections to the MF are obtained for different values of geometrical and physical properties of the constituents of the composite material. The analytical and numerical analyses are carried out and illustrated graphically. In particular, the derived refined formula and the MF are compared for the limiting values of the geometric dimensions and physical properties of the composite. It is shown that the refined formula is applicable for the inclusions with any conductivity in the entire range of the geometric sizes of inclusions, including the limiting cases of inclusions with zero thermal conductivity and maximally large inclusions.

Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

We define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

THE EMOTIONAL COMPONENT OF THE PRESENT-DAY STUDENTS' PSYCHICALLY HEALTH: THE QUESTION ANALYSIS

The investigation of the problem of the emotional component of the present-day students' psychological health is topical in some circumstances. Firstly, the dialectical character of the emotions and feelings correlation, the role of personal feelings in psychological health's general characteristic. The article also elaborates on the necessity of including the spheres of value and motivation into the overall theoretical and methodological context of discussing the outlined problem. The article also presents the results of empiric studies aimed at investigating the internal organization of the notion of psychological health in students of local universities. The results suggest that students give an exceptional value to the motivational dimension of psychological health. Also, the propositions for the future research in the field of psychological health are outlined. The article also provides a sufficient theoretical review of the notion of psychological health in literary sources. The role of quantitative and qualitative emotional characteristics in personal alteration are important nowadays. The idea of the following theoretical constructs such as "psychological resource" and "psychological health" is investigated. Such principal term as students' age period in personal skills development, that is defined as an ability "to operate his/her resources" is analyzed. The fundamental psychological researches confirm the importance of the role of the person emotional readiness in his/her development as an "integrated education" which involves not only the psychological health, but also the mental health and spirituality among other components. Such aspect of the problem is investigated in the context of a behavior-oriented approach, where the components are the following: person action that correlate with the phenomenal-emotional manifestation, that is operationalized as the transition of moral sense in philosophy and spiritual dimension.

On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals

In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and {f\in F(\cdot,\cdot,u)} such that {u(\cdot,0)=u_{0}} and \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

Could Suicide Really Be a Fundamental Human Right?

Could suicide be not just a right, but a fundamental right, rooted in dignity? A linguistic triple threat complicates this question: problems about “rights,” problems about “dignity,” and problems about what counts as suicide. For example, Thich Nhat Hanh’s insistence that the self-immolations of Buddhist monks and nuns in Vietnam are not suicides provides one sort of challenge; Valerius Maximus’s account of the self-elected death of a 90-year-old woman of Cea in good health and ample wealth, another. Linguistic variation also complicates the question of rights: English’s principal term for suicide, “suicide,” has strongly negative connotations; of German’s four major terms, one has comparatively positive connotations, giving German speakers greater linguistic flexibility than English speakers have. Because background intuitions, practices, and linguistic resources are so variable, establishing that there is a right to suicide and if so, whether it is a fundamental one, is a challenging task.

ENERGY INVARIANTS IN THEORY OF ELASTOPLASTIC CRACKS

The paper considers a problem on a rectilinear crack in hardening elastoplastic material with load which is applied at infinity under plane-strain deformation conditions. While distributing J-integral in this case it is necessary to take into account specific characteristics associated with strain potential for environments with nonholonomic state equations. While considering a problem on a crack in elastoplastic material a principal term of asymptotic expansion in crack tip vicinity has an unknown singularity index in addition to an indefinite multiplier. It has been shown for steel 12X18H9T that while having invariance of energy integral it is possible to trace a singularity index for a principal term of stresses. The paper presents dependences of crack length compared to permissible Griffith’s length in accordance with the applied load which is associated with yield strength. Conceptions of J-integrals have been described for solution of a quasi-static problem. The developed approach can be used to formulate a criterion for destruction of elastoplastic material containing a rectilinear crack. The obtained theoretical dependences pertaining to determination of structure limit state characteristics have permitted to make a motivated selection of geometric parameters with due account of material strength properties. Results of the investigations can be used while preparing recommendations for development of structures with prescribed properties. The given approach makes most sense to be applied for determination of critical forces and critical value of crack length for elastoplastic material.