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We consider a linear nonstationary system of ordinary differential equations with interval coefficients which is not solvable with respect to the derivative of the unknown vector-valued function for any matrix coefficients in a given interval family. We obtain conditions sufficient for the preservation of the internal structure of the system. Under conditions guaranteeing the structure preservation, we obtain sufficient and necessary robust stability conditions. A variable rank of matrix coefficients and an arbitrary high unsolvability index are allowed. Bibliography: 14 titles., UDC 517.9 1 Introduction We consider the system of ordinary differential equations A(t)x'(t)+ B(t)x(t) = 0, t [member of] I = [0, +[infinity]), (1.1) where A(t) and B(t) are given [...]

We study two game families with total payoffs that are defined either as the Cesaro average (the long run average game family) or Abel average (the discounting game family) of the running costs. We study value functions for all sufficiently small discounts and for all sufficiently large planning horizons (asymptotic approach), investigate a robust strategy that provides a near-optimal total payoff in this case (uniform approach). Assuming the Dynamic Programming Principle, we prove the corresponding Tauberian theorems without requiring the boundedness of the running cost. Bibliography: 11 titles., UDC 517.9 In dynamic optimization, most often, the potential infinity of the planning horizon is emulated by considering the running cost averaged with respect to the uniform or exponential distributions [...]

We develop an algorithmic support for some classes of nonlinear optimal control problems. We propose several algorithms based on the first- and second order necessary conditions for solving optimal control problems with constraints on control. Efficiency of the algorithms is confirmed by solving various test and applied optimal control problems. Bibliography: 11 titles. Illustrations: 3 figures., UDC 517.9 Application of general methods for solving real problems requires a lot of computational effort and is rarely successful. The complexity involved in the solutions is attributed either to [...]

We consider a semilinear differential-algebraic system of partial differential equations of index (k, 0). We numerically solve this system by applying the spline- collocation method based on splitting the matrix pencil. The method has high accuracy coinciding with the smallest order of the approximating spline. Bibliography: 8 titles., UDC 517.9 In [1], the author considered some problems arising while numerical solving differential-algebraic systems of partial differential equations of index greater than 1. The index of a system, in [...]

We consider a class of multiple sums involving odd powers of natural numbers. Such sums appear while considering the continuous limit of the integrable hierarchy of evolution equations associated with the Itoh-Narita-Bogoyavlenskii lattice. We discuss the problem of constructing polynomials that allow us to calculate the values of the corresponding sums. Bibliography: 13 titles., UDC 517.9 1 Introduction In this paper, we consider multiple sums of the form [mathematical expression not reproducible], (1.1) where [B.sub.k,kn] := {[q.sub.j] : 1 [less than or equal to] [...]

An optimal control problem for a system of one dimensional hyperbolic conservation laws is considered. The system is controlled through the boundary condition, which depends on the state of a nonlinear control system. It is proven that the problem has an optimal solution; moreover, it admits [epsilon]-optimal solutions generated by bang-bang controls. Bibliography: 8 titles. Illustrations: 1 figure., UDC 517.9 1 Introduction In this paper, we analyze the control system [mathematical expression not reproducible], (1.1) composed of a system of hyperbolic conservation laws and a system of ordinary [...]

We study analytical synthesis of control systems for nonlinear objects of high dimension. We propose to apply a synergetic approach owing to which it is not necessary to linearize the model and it is possible to synthesize regulators of high dimension by successively solving problems of small dimension. For an example of a nonlinear multidimensional control object we consider an unmanned aerial vehicle of helicopter type. Bibliography: 8 titles. Illustrations: 9 figures., UDC 517.935.4:681.5.013 The paper is devoted to analytical synthesis of control systems for nonlinear objects of high dimension. The known methods for constructing optimal regulators (cf., for example, [1]-[4]) face [...]

We consider a nonstationary system of differential-algebraic equations, i.e., the first order ordinary differential equations with variable coefficients and identically singular matrix at the derivative of the unknown vector-valued function. We construct the structural form for the perturbed system and obtain sufficient conditions for the robust (complete, differential, R-) controllability of such systems of index 1 and 2. Bibliography: 17 titles., UDC 517.922,517.977.1,517.926.4 1 Introduction We consider the system of differential equations, called differential-algebraic equations A(t)x'(t) + B(t)x(t) + U(t)u(t) = 0, t [member of] I = [0, +[infinity]), (1.1) where [...]

For a nonlinear parabolic heat equation we construct a heat wave type solution composed of the zero and nonnegative solutions joined continuously along the wave front. We prove the existence and uniqueness of an analytic solution to the problem with a given wave front in the cases of plane, circular, and spherical symmetry. The solution is constructed in the form of a characteristic series with recurrently defined coefficients. In the case of a power source, we show that the original problem can be reduced to the Cauchy problem for a second order ordinary differential equation and the solution is invariant. We present numerical results verified by using the constructed analytic solutions. Bibliography: 11 titles. Illustrations: 2 figures., UDC 517.95 1 Introduction In this paper, we construct a heat wave type solution (a filtration wave) to the nonlinear heat equation (also called the porous medium equation; cf., for [...]

We study a multi-point correlation function of the six-vertex model on the square lattice with domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes the probability of observing ferroelectric order around all vertices of any Ferrers diagram [lambda] at the top left corner of the lattice. For the free fermion model, we derive and compare explicit formulas for this correlation function in two cases: when the diagram X has a square or a triangular shape. We find a connection between our formulas and the t-function of the sixth Painleve equation. Bibliography: 25 titles., 1. Introduction One of the most interesting properties of the six-vertex model with domain wall boundary conditions [1-3] is the existence of a limit shape, or phase separation phenomena [4,5]; [...]

We present the Lagrangian and Hamiltonian of the Born-Infeld model in Cartesian and light cone variables. Using the auto-Backlund transformation, we construct new solutions of the corresponding nonlinear equation. In particular, a 'dressed' Barbashov-Chernikov solution is obtained. Bibliography: 18 titles., UDC 517.9 The Born-Infeld (BI) model [1] was suggested in the early 1930s as a nonlinear generalization of Maxwell's electrodynamics (a detailed 'hot on the trail' discussion of its advantages [...]

We suggest an asymptotic approach to renormalization in the case of dimensional regularization. As an example, the quantum Yang-Mills theory in the four-dimensional space-time is considered. A formula for the renormalized effective action is derived by using the asymptotic behavior of the bare coupling constant. Then we discuss the dimensional transmutation, the process of renormalization, and the properties of the coupling constant. Bibliography: 15 titles., 1. Introduction The appearance of divergent integrals in such models as [[phi].sup.4], quantum electrodynamics, quantum chromodynamics (see [4-6] or [7]) has resulted in the development of renormalization approaches, which currently [...]

We discuss the construction of the SL(2, C)-invariant R-operator acting in the tensor product of two principal series representations of SL(2, C) and satisfying the Yang-Baxter equation. We present a closed-form expression for this R-operator as a multiple two-dimensional propagator-type Feynman integral, which can be reduced to a double integral of Mellin-Barnes type. The obtained R-operator can be interpreted as a Boltzmann weight of the corresponding SOS model. All necessary formulas concerning principal series representations of SL(2, C) are presented. Bibliography: 22 titles., UDC 517.9 1. Unitary series representations of the group SL(2, C) Let us remind the reader of the most important formulas concerning unitary series representations of SL(2, C), see [1-3]. [...]

Determinant and higher-loop terms, usually treated by the Pauli-Villars and higher covariant derivatives methods, in the background field method can hardly be regularized simultaneously. At the same time, we observe that introducing a scalar multiplier in front of the quadratic form, which is equivalent to changing the measure in the functional integral, influences only the determinant part of the effective action. This allows one to choose the integration measure and the function in the regularized propagator in such a way as to make all terms in the expansion finite. Bibliography: 15 titles., UDC 517.9 Introduction Originally introduced in [1,2], the background field method significantly simplifies the calculation of the effective action and the [beta]-function in quantum field models. In the general case, [...]

We discuss a connection between the XXZ Heisenberg spin chain in the limiting case of zero anisotropy and some aspects of enumerative combinatorics. The representation of the Bethe wave functions in terms of Schur functions allows us to apply the theory of symmetric functions to calculating correlation functions. We provide a combinatorial derivation of the dynamical correlation functions of the projection operator in terms of nests of self-avoiding lattice paths. Bibliography: 26 titles., UDC 517.9 1. Introduction The theory of random walks, being one of the classical directions of enumerative combinatorics [1], has been successfully applied in various fields: in the theory of [...]

We consider the set of closable homogeneous extensions of the quadratic form of the Laplace operator generated by interaction with two point-like sources. We show that this set consists of the trivial (maximal) extension, one point, and a subset equivalent to the Riemann sphere [S.sup.2]. Bibliography: 7 titles., UDC 517.9 Introduction Quadratic forms defined by the expression [mathematical expression not reproducible] (1) where {[x.sup.n]} are N points in general position in the space [R.sup.3], can be regarded as [...]

We consider continuous-time random walks on multidimensional symplectic lattices. It is shown that the generating functions of random walks and the transition amplitudes of continuous-time quantum walks can be expressed through dynamical correlation functions of an exactly solvable model describing strongly correlated bosons on a chain, the so-called phase model. The number of random lattice paths with a fixed number of steps connecting the starting and ending points on the multidimensionasl lattice is expressed through solutions of the Bethe equations of the phase model. Its asymptotics is obtained in the limit of a large number of steps. Bibliography: 31 titles., UDC 517.9 1. Introduction Classical random walks on a one-dimensional lattice have been intensively investigated for many years in mathematics and physics [1-6]. Quantum walks are unitary processes that describe [...]

We consider the problem of constructing birational Darboux coordinates on nilpotent coadjoint orbits of the complex Lie groups SO(N, C) and Sp(N, C). The nilpotent case is the most difficult one. Difficulties arise if the Jordan form of matrices from the orbit under consideration contains Jordan blocks of sizes of different parity. The desired coordinates have been found on orbits consisting of matrices with 1 x 1 and 2 x 2 Jordan blocks. Explicit formulas for them are given in the paper. Bibliography: 8 titles., UDC 517.9 1. Introduction. Notation Any coadjoint orbit of a Lie group equipped with the canonical Lie-Poison-Kirillov-Kostant two-form is a symplectic manifold. The classical Darboux theorem states that there are [...]

Let z [member of] C be a complex variable, and let h [member of] (0, 1) and p [member of] C be parameters. For the equation [mathematical expression not reproducible], solutions having the minimal possible growth simultaneously as Im z [right arrow] [infinity] and as Im z [right arrow] -[infinity] are studied. In particular, it is shown that they satisfy one more difference equation [mathematical expression not reproducible]. Bibliography: 13 titles., UDC 517 1. INTRODUCTION In this paper we begin a systematic study of solutions of the difference equation [mathematical expression not reproducible] (1.1) on the complex plane. Here z is [...]

A waveguide occupies a strip in [R.sup.2] having two identical narrows of small diameter [epsilon]. An electron wave function satisfies the Helmholtz equation with the homogeneous Dirichlet boundary condition. The energy of electrons may be rather high, i.e., any (fixed) number of waves can propagate in the strip far from the narrows. As [epsilon] [right arrow] 0, a neighborhood of a narrow is assumed to transform into a neighborhood of the common vertex of two vertical angles. The part of the waveguide between the narrows as [epsilon] = 0 is called the resonator. An asymptotics of the transmission coefficient is obtained in the waveguide as [epsilon] [right arrow] 0. Near a degenerate eigenvalue of the resonator, the leading term of the asymptotics has two sharp peaks. Positions and shapes of the resonant peaks are described. Bibliography: 10 titles., UDC 517.958 1. INTRODUCTION Recently, the resonant tunneling in quantum waveguides of variable cross-section has actively been studied [1, 5] in connection with applications in electronics. The obtained results are [...]

This paper is a complement to the article 'Scattering amplitudes in a neighborhood of limit rays in short-wave diffraction by elongated bodies of revolution'. It contains discussions of some points of the article, which worth of more detailed considerations, such as the influence of the integration limits on the computation result of scattering amplitudes and the estimation of permissible values of scattering angle intervals as functions of parameters of the problems. Bibliography: 4 titles., UDC 517.9 In paper [1] we obtained formulas for scattering amplitudes of the plane wave for smooth elongated bodies of revolution (axially symmetric case) in the direction of limit rays [...]

In the paper, diffraction problems of a plane wave by smooth, convex, and elongated bodies of revolution are considered within the framework of short-wave approximation (axially symmetric cases). The scattering amplitudes are calculated in the direction of limit rays, and the influence of the elongation of the scatterers on the amplitudes behavior is investigated. Mathematical techniques of our approach are based on the Green's formulas in the exterior of the scatterers and numerical calculations of the wave field current in the boundary layers in the vicinity of the light-shadow zone. It is established that the elongation of axially symmetric bodies relatively weakly affects the scattering amplitudes of the short-wave asymptotics. The main contribution to the amplitudes is made by the solution of the 2D diffraction problem by a convex, smooth curve in the cross section of the scatterers by a plane containing the rotation axis. Bibliography: 8 titles., UDC 517.9 1. INTRODUCTION In this paper we consider the problem of diffraction of a plane wave by smooth elongated bodies of revolution (axially symmetric case) within the framework of [...]

The inverse dynamic problem for the wave equation with a potential on a real line is considered. The forward initial-boundary value problem is set up with the help of boundary triplets. As an inverse data, an analog of the response operator (dynamic Dirichlet-to-Neumann map) is used. Equations of the inverse problem are derived; also, a relationship between the dynamic inverse problem and the spectral inverse problem from a matrix-valued measure is pointed out. Bibliography: 16 titles., UDC 517 1. INTRODUCTION For a potential q [member of] [C.sup.2](R) [intersection] [L.sub.1](R) we consider an operator H in [L.sub.2](R) given by [mathematical expression not reproducible]. Then [mathematical expression not [...]

An acoustic incident plane wave completely illuminates a narrow convex cone satisfying the impedance boundary condition on its surface. The wave field at far distances from the vertex of the cone and in some close neighborhood of the cone's surface is asymptotically computed. Bibliography: 9 titles., UDC 517 1. INTRODUCTION An incident plane wave propagates from infinity and completely illuminates the surface of a convex cone with the impedace boundary condition on its surface. If the [...]

Modified Kirchhoff transmission conditions in a simple one-dimensional model of a branching artery developed by the authors, allow one to describe an anomaly of its bifurcation node, congenital or acquired due to trauma or disease of a vessel wall. The pathology of the blood flow through the damaged node and the methods of determining the aneurysm parameters from the data measured at the peripheral parts of the circulatory system by solving inverse problems are discussed. Bibliography: 41 titles., UDC 517.958, 539.3(5), 531.3-324 1. Preamble. The absolute majority of one-dimensional models of various physical phenomena and processes inside junctions of thin cylinders, e.g., acoustic pipes, vessels and channels filled [...]

This paper continues a series of publications on the shortwave diffraction of the plane wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich-Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles., UDC 517.9 We consider a shortwave diffraction of a plane incident wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures [...]

A system of equations describing the motion of fluids of Maxwell type is considered: [mathematical expression not reproducible]. Here K(t) is an exponential series [mathematical expression not reproducible]. The existence of a weak solution for the initial boundary value problem v(x, 0) = [v.sub.0](x), v x n[|.sub.[partial derivative][OMEGA]] = 0, rot v [|.sub.[partial derivative][OMEGA]] = 0 is proved. Bibliography: 11 titles., UDC 517 1. INTRODUCTION Consider a system of equations that describes Maxwell flows. This system has the form [mathematical expression not reproducible], (1) div v = 0. (2) The kernel [...]

The waveguide considered coincides with a strip having two narrows of width [epsilon]. An electron wave function satisfies the Dirichlet boundajry value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator, and conditions for the electron resonant tunneling may occur. In the paper, asymptotic formulas as [epsilon] [right arrow] 0 for characteristics of the resonant tunneling are used. The asymptotic results are compared with the numerical ones obtained by approximate calculation of the scattering matrix for energies in the interval between the second and third thresholds. The comparison allows us to state an interval of e, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than that considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-sections. Bibliography: 3 titles., UDC 517 1. INTRODUCTION In [1] (see also [2]) an asymptotic theory of the resonant tunneling was developed in two-dimensional quantum waveguides having a resonator formed by two narrows of [...]

An integral representation of solutions of the wave equation obtained earlier is studied. The integrand contains weighted localized solutions of the wave equation that depend on parameters, which are variables of integration. Dependent on parameters, a family of localized solutions is constructed from one solution by means of transformations of shift, scaling, and the Lorentz transform. Sufficient conditions are derived, which ensure the pointwise convergence of the obtained improper integral in the space of parameters. The convergence of this integral in [L.sub.2] norm is proved as well. Bibliography: 22 titles., UDC 517.958, 517.444, 512.815.8, 517.986.6 1. Introduction The paper considers the integral representation of solutions of the wave equation found in [1-3]. That integral representation is obtained by means of [...]

In this research, we study the scattering of electromagnetic waves by a dielectric impediment in 2D geometry. The impediment is determined by an inhomogeneous component of the refractive index in the Helmholtz equation. It is assumed that the characteristic gauge of one of the two impediment sizes is much lesser than the length of waves generated by a monochromatic point source. Nevertheless, the structure of the impediment is taken into consideration in the process of calculating the scattered field. The scattered field is defined by a derived model integral equation the unique solvability of which is proved. Bibliography: 5 titles., UDC 517 1. Statement of the problem In this paper we present a systematic approach to solving a class of problems modelling the scattering of electromagnetic waves on a dielectric [...]

The question of influence of the spectral neighborhood of an accumulative point of bound energies of a pair subsystem on the structure of eigenfunctions of the continuous spectrum for a system of three charged quantum particles is studied. The unified contribution of pair high-excited states are separated in the coordinate asymptotics of such functions. Bibliography: 20 titles., UDC 517.9 1. Introduction The diffraction approach in quantum few-body scattering problems was suggested earlier in [1,2] for a system of three one-dimensional neutral quantum particles and later developed in [...]

The well-known fact following from the Holmgren-John-Tataru uniqueness theorem is a local approximate boundary [L.sub.2]-controllability of the dynamical system governed by the wave equation. Generalizing this result, we establish the controllability in certain classes of differentiable functions in the domains filled up with waves. Bibliography: 10 titles., UDC 517 1. INTRODUCTION The paper deals with a local approximate boundary controllability of dynamical systems governed by the wave equation. This property means that the states of the system [...]

The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrodinger operator in the scattering problem of three one-dimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed. Bibliography: 15 titles., UDC 517.9 1. Introduction We suggest a new approach for deriving coordinate asymptotics of the resolvent kernel of the Schrodinger operator on the continuous spectrum for the case of several [...]

It is demonstrated that the heuristic formulas presented by V. S. Buldyrev for description of an interference head wave do not contradict the locality principle. Bibliography: 3 titles., UDC 517.9 1. INTRODUCTION In paper [1], V. S. Buldyrev considered a plane problem of diffraction on the interface of two media. The incident wave was generated by a high-frequency [...]

We study linear images of a symmetric convex body C [subset or equal to] [R.sup.N] under an n x N Gaussian random matrix G, where N [greater than or equal to] n. Special cases include common models of Gaussian random polytopes and zonotopes. We focus on the intrinsic volumes of GC and study the expectation, variance, small and large deviations from the mean, small ball probabilities, and higher moments. We discuss how the geometry of C, quantified through several different global parameters, affects such concentration properties. When n = 1, G is simply a 1 x N row vector, and our analysis reduces to Gaussian concentration for norms. For matrices of higher rank and for natural families of convex bodies [C.sub.N] [subset or equal to] [R.sup.N], with N [right arrow] [infinity], we obtain new asymptotic results and take first steps to compare with the asymptotic theory. Bibliography: 44 titles., 1. Introduction In this paper, we study random convex sets which arise as linear images of Gaussian matrices. Specifically, let G = G(n, N) be an n x N random [...]

We consider the convergence of the empirical spectral measures of random N x N unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and uniform measure on the unit circle is of order log N/N, both in expectation and almost surely. This implies, in particular, that the convergence happens more slowly for Kolmogorov distance than for the [L.sub.1]-Kantorovich distance. The proof relies on the determinantal structure of the eigenvalue process. Bibliography: 16 titles., Let U [member of] U (N) be a random matrix distributed according to Haar measure. Denote the eigenvalues of U by [mathematical expression not reproducible], and let [[mu].sub.N] denote the [...]

Let [X.sub.1], ..., [X.sub.n] be independent nonnegative random variables with cumulative distribution functions [F.sub.1], [F.sub.2], ..., [F.sub.n] satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector ([X.sub.1] ..., [X.sub.n]) is equivalent to the quantile of order (k - 1/2)/n with respect to the averaged distribution F = 1/n [n.summation over (i=1)] [F.sub.i]. Bibliography: 12 titles., 1. Introduction The goal of this note is to provide sharp estimates for order statistics of independent, not necessarily identically distributed random variables whose distributions satisfy certain (rather mild) conditions. [...]

Let [X.sub.1], ..., [X.sub.n] be independent random variables having as common distribution the standard Gaussian measure [mu] on [R.sup.2] and let [mathematical expression not reproducible] be the associated empirical measure. We show that 1/C log n/n [less than or equal to] E ([W.sup.2.sub.2] ([[mu].sub.n], [mu])) [less than or equal to] C [(log n).sup.2]/n for some numerical constant C > 0, where [W.sub.2] is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan. Bibliography: 39 titles., To the memory of Professor V. N. Sudakov 1. Introduction Given [x.sub.1], ..., [x.sub.n] and [y.sub.1], ..., [y.sub.n] in [R.sup.d] and p [greater than or equal to] 1, the optimal [...]

To any n-dimensional random vector X we may associate its [L.sub.p]-centroid body [Z.sub.p](X) and the corresponding norm. We formulate a conjecture concerning the bound on the [Z.sub.p](X)-norm of X and show that it holds under some additional symmetry assumptions. We also relate our conjecture to estimates of covering numbers and Sudakov-type minoration bounds. Bibliography: 11 titles., 1. Introduction. Formulation of the problem Let p [greater than or equal to] 2 and let X = ([X.sub.1], ..., [X.sub.n]) be a random vector in [R.sup.n] such that E[absolute [...]

The aim of this paper is to survey properties of Gaussian approximation numbers. We state the basic relations between these numbers and other s-numbers as, e.g., entropy, approximation, or Kolmogorov numbers. Furthermore, we fill a gap and prove new two-sided estimates in the case of operators with values in a K-convex Banach space. In the final section, we apply relations between Gaussian and other s-numbers to the d-dimensional integration operator defined on [L.sub.2][[0,1].sup.d]. Bibliography: 28 titles., Dedicated to the memory of Vladimir Nikolaevich Sudakov 1. Introduction Basic results of R. M. Dudley in 1967 and of V. N. Sudakov in 1969 relate regularity properties of a [...]

The main problem of the paper looks as follows. A functional parameter [theta] [member of] [THETA] [subset] [L.sub.2](- -[infinity], [infinity]) is observed in Gaussian noise. The problem is to estimate the value F([theta]) of a given function F. A construction of asymptotically efficient estimates for F([theta]) is suggested under the condition that [THETA] admits approximations by subspaces [H.sub.T] [subset] [L.sub.2] with reproducing kernels [K.sub.T](t, s), [K.sub.T](t, t) [less than or equal to] T. Bibliography: 10 titles., 1. Introduction. Main result 1.1. Consider the following general problem of statistical estimation. We have a statistical experiment {X, [??], {[P.sub.[theta]], [theta] [member of] [THETA]}} generated by an observation [X.sub.[epsilon]] [...]

We study the existence theory of solutions of the kinetic Bohmian equation, a nonlinear Vlasov-type equation proposed for the phase-space formulation of Bohmian mechanics. Our main idea is to interpret the kinetic Bohmian equation as a Hamiltonian system defined on an appropriate Poisson manifold built on a Wasserstein space. We start by presenting an existence theory for stationary solutions of the kinetic Bohmian equation. Afterwards, we develop an approximative version of our Hamiltonian system in order to study its associated flow. We then prove the existence of solutions of our approximative version. Finally, we present some convergence results for the approximative system; our aim is to show that, in the limit, the approximative solution satisfies the kinetic Bohmian equation in a weak sense. Bibliography: 24 titles., 1. Introduction In this paper, we study the existence theory of solutions of the kinetic Bohmian equation [15,16], [[partial derivative].sub.t][beta] + [upsilon] * [[nabla].sub.x][beta] - [[nabla].sub.x] (V - 1/2 [[DELTA].sub.x] [...]

Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and, in particular, by its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on integrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure. Bibliography: 18 titles., 1. Introduction In the late eighties, Talagrand conjectured that the 'convolution by a biased coin' on the hypercube [{-1,1}.sup.n] satisfies some refined hypercontractivity property. We refer to Problems 1 and [...]

After setting geometric notions, we revisit an exponential functional which has arisen in several contexts, with special attention to a set of geometric parameters and associated inequalities. Bibliography: 32 titles., 1. Introduction It is an honor and pleasure to contribute to this volume. V. N. Sudakov's work has had a great influence on my own interests. In that spirit, what [...]

The paper presents a general duality theory for vector measure spaces taking its origin in author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications. Bibliography: 23 titles., To the memory of V. N. Sudakov 1. Introduction The axiomatics of vector measure spaces discussed in this paper was conceived and partially described in author's papers written in the [...]

This paper is concerned with the study of conditions under which the Monge and Kantorovich problems with a continuous cost function on a product of two completely regular spaces and two given atomless Radon measures-projections on these spaces have equal values of the corresponding infima. Bibliography: 41 titles., Dedicated to the memory of Vladimir Nikolaevich Sudakov Our paper is concerned with the study of conditions under which the Monge and Kantorovich problems with a continuous cost function on [...]

We derive a general upper bound on the distance of the standard normal law to typical distributions in V. N. Sudakov's theorem (in terms of the weighted total variation). Bibliography: 22 titles., Dedicated to the memory of Vladimir Nikolaevich Sudakov 1. Introduction Let X = ([X.sub.1], ..., [X.sub.n]) be a random vector in [R.sup.n] with finite second moment and let [S.sup.n-1] = [...]

We study deviation probabilities for the number of high positioned particles in branching Brownian motion and confirm a conjecture of Derrida and Shi. We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton-Watson processes. Bibliography: 15 titles., Dedicated to the memory of Professor V. N. Sudakov 1. Introduction Consider the model of one-dimensional Branching Brownian Motion (BBM): Initially a particle starts at the origin and performs standard [...]

The paper contains a survey of the current state of a constructive part in dessin d 'enfants theory. Namely, it is devoted to actual establishing the correspondence between the Belyi pairs and their combinatorial-topological representations. This correspondence is established in terms of equivalence of categories, and all relevant categories are introduced. Several connections with arithmetic are discussed. One of the sections presents a possible generalization of the theory, in which three branch points, allowed for the Belyi functions, are replaced by four. Several directions for further research are presented. Bibliography: 80 titles., UDC 512.772.7 0. Introduction We agree (more or less) in this conference on which kinds of embedded graphs deserve our attention. Some of us prefer to paint vertices, or edges, [...]

Rational functions on algebraic curves, which have a single zero and a single pole, are considered. A pair consisting of such a function and a curve is called an Abel pair; a special case of an Abel pair is a Belyi pair. In the present paper, moduli spaces of Abel pairs for curves of genus one are studied. In particular, a number of Belyi pairs over the fields C and is computed. This approach could be fruitfully used in studying Hurwitz spaces and modular curves for fields of finite chasracteristics. Bibliography: 7 titles., UDC 512.772.7, 515.179.25 1. Introduction In the present paper, we consider algebraic curves and rational function on them with the divisor of a certain combinatorial type. More specifically, we study [...]