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## Some Approximation Problems in Statistics and Probability

P. 235-249.

Ulyanov V. V., Prokhorov Y.

We review the results about the accuracy of approximations for distributions of functionals of sums of independent random elements with values in a Hilbert space. Mainly we consider recent results for quadratic and almost quadratic forms motivated by asymptotic problems in mathematical statistics. Some of the results are optimal and could not be further improved without additional conditions.

### In book

Vol. 42: Limit Theorems in Probability, Statistics and Number Theory, In Honor of Friedrich Götze. , Berlin : Springer, 2013

Badanin A., Korotyaev E., Mathematische Nachrichten 2020 Vol. 293 P. 210-225

We consider fourth order ordinary differential operators on the half-line and on the
line, where the perturbation has compactly supported coefficients. The Fredholm
determinant for this operator is an analytic function in the whole complex plane without
zero.We describe the determinant at zero.We show that in the generic case it has
a pole of order 4 in the case ...

Added: February 5, 2021

Fonareva A. V., Gaydukov R. K., Russian Journal of Mathematical Physics 2021 Vol. 28 No. 2 P. 224-243

A subsonic flow of a viscous compressible fluid in a two-dimensional channel with small periodic or localized irregularities on the walls for large Reynolds numbers is considered. A formal asymptotic solution with double-deck structure of the boundary layer is constructed. A nontrivial time hierarchy is discovered in the decks. An analysis of the scales of irregularities at ...

Added: March 22, 2021

Ludmila I. Kuzmina, Osipov Y., Vetoshkin N., International Journal for Computational Civil and Structural Engineering 2018 Vol. 14 No. 1 P. 92-101

The filtration problem in a porous medium is an important part of underground hydromechanics. Filtration of suspensions and colloids determines the processes of strengthening the soil and creating waterproof walls in the ground while building the foundations of buildings and underground structures. It is assumed that the formation of a deposit is dominated by the ...

Added: July 1, 2018

Remizov I., Infinite Dimensional Analysis, Quantum Probability and Related Topics 2018 Vol. 21 No. 4 P. 1850025-1-1850025-35

A parabolic partial differential equation u 0 t (t, x) = Lu(t, x) is considered, where L is a linear second-order differential operator with time-independent (but dependent on x) coefficients. We assume that the spatial coordinate x belongs to a finite- or infinitedimensional real separable Hilbert space H. The aim of the paper is to ...

Added: October 5, 2018

V.L. Chernyshev, Minenkov D. S., Nazaikinskii V. E., Functional Analysis and Its Applications 2016 Vol. 50 No. 4 P. 291-307

We find the asymptotics of the element counting function for an additive arithmetic semigroup with exponential growth of the counting function of prime generators. ...

Added: October 18, 2015

Л.И. Кузьмина, Осипов Ю. В., Вестник Московского государственного строительного университета 2014 № 7 С. 34-40

A classic pursuit problem is studied in which two material points - a Pursuer and a Pursued - move in a plane at constant speeds. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at ...

Added: August 3, 2014

Grushin V. V., Математические заметки 2017 Т. 102 № 6 С. 816-827

Asymptotic formulas are obtained for a class of integrals that are Fourier transforms of rapidly oscillating functions. These formulas contain special functions and generalize the well-known method of stationary phase. ...

Added: January 13, 2018

Аникин А. Ю., Миненков Д. С., Математические заметки 2019 Т. 106 № 2 С. 163-171

В работе рассматривается линейная модель цунами и исследуется влияние параметров источника на заплеск. ...

Added: December 30, 2020

Protasov V., Mathematics of Computation 2017 Vol. 86 No. 305 P. 1499-1524

For an arbitrary set of nonnegative integers, we consider the Euler binary partition function which equals the total number of binary expansions of an integer with ``digits'' from . By applying the theory of subdivision schemes and refinement equations, the asymptotic behaviour of as is characterized. For all finite , we compute the lower and upper exponents of growth of , find when ...

Added: May 11, 2017

Chernyshev V.L., Minenkov D. S., Nazaikinskii V. E., Mathematical notes 2018 Vol. 104 No. 6 P. 939-942

This note is a continuation of the studies concerning the asymptotics of the integer counting function and the typical shape of elements in an arithmetical semigroup G with an exponential asymptotics of the prime counting function. Here we present a result for the probabilities of deviations of elements from the typical shape (which is given ...

Added: December 10, 2018

Veretennikov A., Da Prato G., Flandoli F. et al., / Cornell University. Series cond-mat "arxiv.org". 2014. No. arXiv:1404.5418.

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose non-linear drift parts are sums of the subdifferential of a convex function and a bounded part. This generalizes a classical result by one of the authors (Veretennikov) to infinite dimensions. ...

Added: October 22, 2014

Ludmila I. Kuzmina, Osipov Y., International Journal for Computational Civil and Structural Engineering 2013 Vol. 9 No. 3 P. 31-39

A classic pursuit problem is considered in which the pursuer is always moving towards the target. The shape of the mechanical trajectory is established, and the time of motion is calculated. An integral for the length of the pursuit curve is constructed, its asymptotics is calculated and compared with the result of the numerical computation. ...

Added: November 14, 2013

Baranov A., Belov Y., International Mathematics Research Notices 2011 No. 22 P. 5076-5108

Added: January 17, 2014

Danilov V., Rakhel M., Russian Journal of Mathematical Physics 2021 Vol. 28 No. 2 P. 179-187

In this paper, we show how to construct an asymptotic representation of the fundamental solution to the Cauchy problem for degenerate linear parabolic equations. ...

Added: June 6, 2021

Gaydukov R., Journal of Non-Newtonian Fluid Mechanics 2021

The asymptotic solution with double-deck structure was obtained in the problem of flow
of a non-Newtonian fluid with power-law rheological properties along a semi-infinite plate
with a small localized irregularity on the surface for large Reynolds numbers. The numerical
modeling of the flow in the region near the surface was performed for different fluid indexes.
The results of investigations ...

Added: November 14, 2021

Remizov I., Modeling and Analysis of Information Systems 2015 Vol. 22 No. 3 P. 337-355

A parabolic partial differential equation 𝑢′𝑡(𝑡, 𝑥) = 𝐿𝑢(𝑡, 𝑥) is considered, where 𝐿 is a linear second-order differential operator with time-independent coefficients, which may depend on 𝑥. We assume that the spatial coordinate 𝑥 belongs to a finite- or infinite-dimensional real separable Hilbert space 𝐻. Assuming the existence of a strongly continuous resolving semigroup ...

Added: October 30, 2018

Protasov V., Широков М. Е., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2019 Т. 489 № 5

Рассматриваются два преобразования функций на полупрямой и доказывается,что их композиция дает вогнутую мажоранту для любой неотрицательной функции.В частности, на классе неотрицательных вогнутых функций данная композиция есть тождественное преобразование. Указаны приложения этого результата в теории операторов в гильбертовом пространстве и в теории квантовых систем. Сформулирован ряд открытых задач. ...

Added: October 31, 2019

Danilov V., Albeverio S., Russian Journal of Mathematical Physics 2011 Vol. 18 No. 1 P. 10-25

The goal in this paper is to demonstrate a new method for constructing global-in-time approximate (asymptotic) solutions of (pseudodifferential) parabolic equations with a small parameter. We show that, in the leading term, such a solution can be constructed by using characteristics, more precisely, by using solutions of the corresponding Hamiltonian system and without using any ...

Added: April 12, 2012

Gaydukov R., European Journal of Mechanics - B/Fluids 2021 Vol. 89 P. 401-410

An asymptotic solution with double-deck boundary layer structure is constructed for the problem of an incompressible fluid flow along a semi-infinite plate with small localized or periodic (fast-oscillating) irregularities on the surface whose shape depends on time. Numerical simulation of flow in the near-plate region is presented for two types of the shape change: oscillations of the ...

Added: January 4, 2021

L.I. Kuzmina, Osipov Y. V., Вестник Московского государственного строительного университета 2017 No. 11 P. 1278-1283

Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck ...

Added: January 6, 2018

Grushin V. V., Наноструктуры. Математическая физика и моделирование 2012 Т. 7 № 2 С. 17-44

В работе получено асимптотическое разложение собственных значений оператора Лапласа с нулевыми данными Дирихле в бесконечных трубках, т. е. в бесконечных изогнутых цилиндрах с внутренним кручением при равномерном сжатии поперечных сечений, по малому параметру, характеризующему поперечные размеры трубки. Аналогичное разложение получено для оператора Шредингера с учетом магнитного поля для конечных изогнутых трубок. Предложен метод сведения задачи ...

Added: December 24, 2012

Gaydukov R., Fonareva A. V., European Journal of Mechanics - B/Fluids 2022 Vol. 94 P. 50-59

The problem of a rotating disk with slightly perturbed surface immersed in a viscous fluid is considered.
The asymptotic solutions with double-deck structure of the boundary layer are constructed for symmetric
periodic and localized types of irregularities on the disk surface for large Reynolds numbers. The paper
presents the results of numerical simulations of the flow near the ...

Added: November 14, 2021