Spectral measure of heavy tailed band and covariance random matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $\mu$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^\sigma$ whose (i,j) entry is $\sigma(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $\alpha$-stable law, $0<\alpha<2$, and $\sigma$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^\sigma$ and with appropriate rescaling $a_N$, we prove that the distribution $\mu$ of $a_N^{-1}Y_N^\sigma + D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.Comment: 31 pages, minor modifications, mainly in the regularity argument for Theorem 1.3. To appear in Communications in Mathematical Physic

Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes to 0.Comment: 64 pages, no figure

A class of Poisson-Nijenhuis structures on a tangent bundle

Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis structure from a given type (1,1) tensor field J on Q. It is argued that the complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to co-tangent manifold of Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case that Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.Comment: 22 page

Моделирование микроклимата отапливаемых помещений

The paper presents main theoretical principles that serve as a basis for the developed numerical model of processes pertaining to conjugated heat exchange in the heated buildings. Information on functional capabilities of the developed programme, results of calculations and comparison of the obtained data with the conditions of heat comfort are given in the paper. It has been established that an application of simulation processes pertaining to energy and substance transfer at the design stage is considered as a rather efficient method for provision of the required parameters of a micro-climate and optimization of heat consumption by heated buildings.Излагаются основные теоретические положения, лежащие в основе разработанной численной модели процессов сопряженного теплообмена в отапливаемых помещениях. Приводятся сведения о функциональных возможностях разработанной программы, результаты расчетов и сравнение полученных данных с условиями теплового комфорта. Установлено, что применение на этапе проектирования моделирования процессов переноса энергии и вещества является эффективным способом обеспечения требуемых параметров микроклимата и оптимизации теплопотребления отапливаемыми помещениями

On a Camassa-Holm type equation with two dependent variables

We consider a generalization of the Camassa Holm (CH) equation with two dependent variables, called CH2, introduced by Liu and Zhang. We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the same algebra underlying the NLS hierarchy. We study the structural properties of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. We finally sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.Comment: 22 pages, 2 figures. A few typos correcte

РАЗВИТИЕ СТРУИ НАД НАГРЕТОЙ ГОРИЗОНТАЛЬНОЙ ПОВЕРХНОСТЬЮ

The development of thermal contaminated air stream formed above the surface of lengthy heated plate is investigated on the basis of the mathematical model of heat and mass transfer processes. The numerical simulation results performed for the purpose of finding a way to prevent the contaminants outflow from the hood in the presence of external influence are given. Three variants of the hood side panels lengthening are considered: on either side; on the side of the fresh air supply; on the side opposite to the location of air distributors. Comparison of the effects of using these variants of hood design allowed determining that the hood panel located on the side of the coming air flow gives the greatest leakage reduction. It is found out that lengthening of the hood side panels does not ensure full localization of the contaminated convective stream.Several variants of using the screen located on the side of the lateral air flow were investigated in the course of work. Best result is observed under conditions of application of the screen symmetrically located relative to spacing between the hood and the plate. The analysis of local values of temperature and velocity fields in this case shows that losses of heated contaminated air to the workshop space are practically absent. It is found out that the convective stream can be most efficiently localized under the condition of hood-symmetric distribution of supply air. Influence of the open gate on convective stream formation in the workshop space with heat emission is considered. The shift of upward flow to the opposite side relative to the gate is observed. The radiant heat release from the board is of great importance in the temperature field formation of closely located areas of floor and walls surfacesНа основании математической модели процессов переноса теплоты и массы исследуется развитие тепловой загрязненной воздушной струи, формируемой над поверхностью протяженной нагретой плиты. Приводятся результаты численных расчетов, выполненных с целью поиска способа исключить выход загрязняющих веществ за пределы вытяжного зонта при наличии внешних воздействий. Рассматриваются три варианта удлинения свесов зонта: с обеих сторон, со стороны раздачи приточного воздуха, со стороны, противоположной расположению воздухораспределителей. Сравнение последствий применения указанных вариантов конструктивного решения зонта позволило установить, что наибольшее снижение утечек дает свес, расположенный со стороны набегающего потока воздуха. Установлено, что удлинение боковых свесов зонта не обеспечивает полной локализации загрязненной конвективной струи.В ходе работы исследованы несколько вариантов использования экранов, расположенных со стороны бокового потока воздуха. Наилучший результат наблюдается в условиях применения экрана, симметрично расположенного относительно зазора между зонтом и плитой. Анализ локальных значений полей температуры и скорости в данном случае показывает, что утечка нагретого загрязненного воздуха в пространство цеха практически отсутствует. Установлено, что тепловая струя наиболее эффективно может быть локализована при условии симметричного относительно зонта распределения приточного воздуха. Рассматривается влияние открытых ворот на формирование конвективной струи в пространстве цеха с тепловыделениями. Отмечается смещение восходящей струи в противоположную от ворот сторону. Поступление лучистой теплоты от плиты имеет большое значение в формировании поля температуры близко расположенных участков поверхностей пола и стен

Gaudin Models and Bending Flows: a Geometrical Point of View

In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the 'bending flows' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.Comment: 27 pages, LaTeX with amsmath and amssym

ДОПУСТИМЫЕ ПРЕДЕЛЫ КОРРЕКТИРОВКИ ОТОПИТЕЛЬНОЙ НАГРУЗКИ ПРИ ОПТИМИЗАЦИИ РЕЖИМОВ РАБОТЫ ТЕПЛОИСТОЧНИКОВ

Numerical investigations of temperature fields in the apartments of a multi-dwelling building has been carried out on the basis of the developed mathematical model and algorithm of its realization. The investigations have  made it possible to evaluate dynamics of changes in air temperature under unsteady conditions due to increase or reduction heating device capacity with regard to standardized values. The calculation results have demonstrated that a permissible period for capacity change accounts for not more than 6 hours. At the same time the rate of capacity change is up to 40 %. The obtained results can be practically used in the intelligent heating supply systems with the purpose to minimize consumption of fuel and power resourcesНа основе разработанной математической модели и алгоритма ее реализации выполнено численное исследование температурных полей жилого помещения многоквартирного дома, позволившее оценить динамику изменения температуры воздуха в нестационарных условиях в результате повышения или снижения мощности отопительных приборов относительно нормируемой величины. Результаты расчетов показали, что для панельных домов допустимый период изменения мощности составляет не более 6 ч. Диапазон изменения мощности при этом – до 40 %. Практическое применение полученные результаты могут найти в интеллектуальных системах теплоснабжения минимизации потребления топливно-энергетических ресурсов

Minimal Change Disease Is Associated With Endothelial Glycocalyx Degradation and Endothelial Activation

Minimal change disease (MCD) is considered a podocyte disorder triggered by unknown circulating factors. Here, we hypothesized that the endothelial cell (EC) is also involved in MCD