Search

We consider the problem of shadow in the Lobachevsky space. This problem can be treated as the problem of finding conditions guaranteeing that points belong to the generalized convex hull of a family of sets. We determine the limit values of the parameters for which the same configurations of balls guarantee that a point belongs to the generalized convex hull of balls in the Euclidean and hyperbolic spaces. Parallel with families of balls, we consider families of horoballs, as well as certain combinations of balls and horoballs.

We establish the rate of convergence to the exponential distribution in the general limit theorem for the extreme values of regenerative processes. We also suggest some applications of this result to the birth and death processes and to the queue length processes.

We consider the most general class of multipoint boundary-value problems for systems of linear ordinary differential equations of any order whose solutions belong to a given Sobolev space $$ {W}_p^{n+r} $$ , with n ≥ 0, r ≥ 1, and 1 ≤ p ≤ ∞. We establish constructive sufficient conditions under which the solutions of the analyzed problems are continuous with respect to the parameter e for e = 0 in the space $$ {W}_p^{n+r} $$ .

We introduce a concept of A-strongly statistical convergence for sequences of complex numbers, where A = (ank)n,k∈ℕ is an infinite matrix with nonnegative entries. A sequence (xn) is called strongly convergent to L if $$ \underset{n\to \infty }{\lim }{\sum}_{k=1}^{\infty }{a}_{nk}\left|{x}_k\right.-L\left|=0\right. $$ in the ordinary sense. In the definition of A-strongly statistical limit, we use the statistical limit instead of the ordinary limit with a convenient density. We study some densities and show that the (ank)-strongly statistical limit is an ( $$ {a}_{m_nk} $$ )-strong limit, where the density of the set {mn ∈ ℕ : n ∈ ℕ} is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence (rn) is dense positive provided the limit superior of a subsequence ( $$ {r}_{m_n} $$ ) is positive for all (mn) with density equal to 1. We show that the dense positivity of (rn) is a necessary and sufficient condition for the uniqueness of A-strongly statistical limit, where A = (ank) and rn = $$ {\sum}_{k=1}^{\infty }{a}_{nk} $$ . Furthermore, necessary conditions for the regularity, linearity and multiplicativity of the A-strongly statistical limit are established.

We construct a model of conflict dynamical system whose limit states are associated with singular distributions. It is proved that a criterion for the appearance of point spectrum in the limit distribution is the strategy with fixed priority. In all other cases, the limit distributions are pure singular continuous.

For mappings in metric spaces satisfying one inequality for the modulus of families of curves, we establish the property of lightness of the limit mapping. It is shown that the uniform limit of these mappings is a light mapping, whenever the majorant responsible for the distortion of the families of curves is of finite mean oscillation at any point. In addition, for one class of homeomorphisms of metric spaces, we prove theorems on the equicontinuity of the families of inverse mappings.

Let C0 be a curve in a disk D = {|z| < 1} tangential to a circle at the point z = 1 and let Cθ be the result of rotation of this curve by an angle θ about the origin z = 0. We construct a bounded function u(z) three-harmonic in D with zero normal derivatives $$ \frac{\partial u}{\partial n}\mathrm{and}\frac{\partial^2u}{\partial {r}^2} $$ on the boundary such that the limit along Cθ does not exist for all θ, 0 ≤ θ ≤ 2π.

We present the definition of weighted pseudoinverse matrices with nondegenerate weights of indefinite sign and study these matrices. The theorems on existence and uniqueness for these matrices are proved. The weighted pseudoinverse matrices with indefinite weights are represented in terms of the coefficients of characteristic polynomials of symmetrized matrices. The decompositions of weighted pseudoinverse matrices into matrix power series and products and their limit representations are obtained. We also propose regularized iterative methods for the determination of these matrices.

We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equations of motion convergent to equilibrium in the infinite-time limit. These results are applied to the Coulomb systems of three point charges with singular equilibrium in a line.

We study the conditions of weak convergence of the maximum of sums of independent random processes in the spaces C [0, 1] and Lp and present examples of applications to the analysis of statistics of the type ω2.

We study the uniform limit with respect to a parameter for the solutions of a sequence of general boundaryvalue problems for systems of linear ordinary differential equations of any order on a finite interval. An essential generalization of the Kiguradze theorem (1987) for these problems is obtained.

On a finite interval, we consider a system of nonlinear ordinary differential equations with singularity at the left endpoint of the interval. The definition of weighted limit solution is introduced and its attracting property is established.

We find periodic and bounded (for positive time) solutions of the planar Coulomb equations of motion for two and three identical negative charges in the field of two equal fixed positive charges. The analyzed systems have equilibrium configurations and the constructed bounded solutions converge to these configurations in the infinite-time limit. The periodic solutions are obtained with the help of the Lyapunov center theorem.

We consider infinite systems of stochastic differential equations used to describe the motion of interacting particles in random media. It is assumed that the mass of each particle approaches zero and the density of particles infinitely increases in a proper way. It is proved that the sequence of corresponding measure-valued processes converges in distribution. We also prove the existence and uniqueness of a strong solution for the limit equation.

We study the local behavior of closed-open discrete mappings of the Orlicz–Sobolev classes in $$ {\mathbb{R}}^n $$ , n ≥ 3. It is proved that the indicated mappings have continuous extensions to an isolated boundary point x 0 of a domain D \ {x 0}, whenever its inner dilatation has FMO (finite mean oscillation) at this point and, in addition, the limit sets of f at x 0 and on ∂D are disjoint. Another sufficient condition for the possibility of continuous extension can be formulated as a condition of divergence of a certain integral.

We propose a new approach to the application of the Korolyuk potential method for the investigation of limit functionals for processes with independent increments. The formulas for the joint distribution of functionals related to crossing a level by the process are obtained and their asymptotic analysis is performed. The possibility of crossing a level by the process in a continuous way is also investigated.

It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W(t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process. Відомо, що майже всі ви6іркові траєкторії броунівського руху чи вінєрівського процесу {W(t) мають нескінченно багато нульових перетинів в інтервалі (0, δ) при δ > 0. За умови Каца телеграфний процес слабко збігається до вінерівського процесу. В роботі оцінюється число перетинів рівня для телеграфного процесу. Переходячи до границі за умови Каца, ми також отримуємо оцінку перетинів рівня для вінерівського процесу.

We study the operators L n y = −(p n y′)′+q n y, n ∈ ℤ+ , given on a finite interval with various boundary conditions. It is assumed that the function q n is a derivative (in a sense of distributions) of Q n and 1/p n , Q n /p n , and $$ {Q}_n^2/{p}_n $$ are integrable complex-valued functions. The sufficient conditions for the uniform convergence of Green functions G n of the operators L n on the square as n → ∞ to G 0 are established. It is proved that every G 0 is the limit of Green functions of the operators L n with smooth coefficients. If p 0 > 0 and Q 0(t) ∈ ℝ, then they can be chosen so that p n > 0 and q n are real-valued and have compact supports.

We study the limit behavior of a sequence of Markov processes whose distributions outside any neighborhood of a “singular” point are attracted to a certain probability law. In any neighborhood of this point, the limit behavior can be irregular. As an example of application of the general result, we consider a symmetric random walk with unit jumps perturbed in the neighborhood of the origin. The invariance principle is established for the standard time and space scaling. The limit process is a skew Brownian motion.

For a two-dimensional continued fraction another generalization of the Worpitzky theorem is proved and the limit sets are proposed for Worpitzky-like theorems in the case where the element sets of the twodimensional continued fraction are replaced by their boundaries.

We study a subcritical branching process with inhomogeneous immigration in the case where the mean value and variance of immigration are regularly varying at infinity. We show that a properly normalized subcritical process with immigration weakly approaches a deterministic process and prove the limit theorem for the fluctuation of this process.

We study the limit with respect to a parameter in the uniform norm for the solutions of general boundary-value problems for systems of linear ordinary differential equations of the first order. A generalization of the Kiguradze theorem (1987) to this class of problems is obtained. The conditions imposed on the asymptotic behavior of the coefficients of systems are weakened as much as possible. Sufficient conditions for the uniform convergence of Green matrices to the Green matrix of the limit boundary-value problem are also established.

The problem of estimation of a distribution function is considered in the case where the observer has access only to a part of the indicator random values. Some basic asymptotic properties of the constructed estimates are studied. The limit theorems are proved for continuous functionals related to the estimation of F^n(x) in the space C[a, 1 - a], 0

We study fading random evolutions in multidimensional spaces. By reducing multidimensional cases to the one-dimensional case, we calculate the limit distributions of fading evolutions for some semi-Markov media.

The problem of large deviations for random evolutions with independent increments is solved in the scheme of asymptotically small diffusion by passing to the limit in the nonlinear (exponential) generator of semigroups by using the solution of the problem of singular perturbation for a reducibly invertible operator.

We show that the best summability factors of functions that satisfy the reverse Holder inequality in limit cases can be obtained from the nonlimit case by passing to the limit.

We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.

We consider weighted Sobolev spaces correlated with a sequence of n-dimensional domains. We prove a theorem on the choice of a subsequence Γ-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated.

We investigate conditions for the weak convergence of the maximum of sums of independent random processes in the space L p and present several applications to the asymptotic analysis of certain ω 2-type statistics.

The truncated local limit theorem is proved for difference approximations of multidimensional diffusions. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities uniformly convergent to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain nontrivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times for multidimensional diffusions are presented.

We investigate a differential equation in a Hilbert space that describes vibrations of the Euler-Bernoulli elastic beam with feedback control. The relative compactness of positive semitrajectories of the considered equation is proved. Constructing a Lyapunov functional in explicit form and using the invariance principle, we obtain representations of limit sets.

For one class of systems of nonlinear functional differential equations, we study asymptotic properties of their solutions continuously differentiable and bounded for t ≥ T > 0 together with their first derivatives.

We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to the space variables. The case of arbitrarily many space variables is considered. We establish conditions for the existence and uniqueness of a solution of this problem independent of the behavior of the solution as |x| → + ∞. The indicated classes of existence and uniqueness are the spaces of locally integrable functions, and, furthermore, the dimension of the domain does not limit the order of nonlinearity.

We consider a queuing system of the type Mθ/G/1/b in which the input stream is regulated by a certain threshold level. The asymptotic properties of the first busy period and the number of calls served during this period are studied.

For the integrable generalized model of superconductivity a solution of the Richardson equations for a spectrum of model is studied. For the case of narrow band the solution is presented in terms of the generalized Laguerre or Jacobi polynomials. In asymptotic limit, when the Richardson equations are transformed to an integral singular equation, the properties of an integration contour are discussed and a spectral density is calculated. Conditions for appearance of gaps in the spectrum are considered. Для інтегровної узагальненої моделі надпровідності досліджено розв'язання рівнянь Річардсона для спектра моделі. У випадку вузької зони розв'язок подано в термінах узагальнених поліномів Лагерра та Якобі. В асимптотичному випадку, коли рівняння Річардсона трансформуються в інтегральне сингулярне рівняння, з'ясовано властивості контура інтегрування та розраховано спектральну щільність. Розглянуто умови появи щілин у спектрі.

In this paper, we study the lower and upper bounds for solutions of the limit problem for the plane vacuum diode in the magnetic field in the statement by N. Ben Abdallah, P. Degond, and F. Mehats. This problem was finally set by physicists in the late 1980s and was extensively studied by numerous mathematicians in the 1990s.

We investigate boundary functionals of a semicontinuous process with independent increments on an interval with two reflecting boundaries. We determine the transition and ergodic distributions of the process, as well as the distributions of boundary functionals of the process, namely, the time of first hitting the upper (lower) boundary, the number of hittings of the boundaries, the number of intersections of the interval, and the total sojourn time of the process on the boundaries and inside the interval. We also present a limit theorem for the ergodic distribution of the process and asymptotic formulas for the mean values of the distributions considered.

For a sequence of independent random elements belonging to an ideal of order-bounded elements of a Banach lattice, we investigate the asymptotic relative stability of extremal values, the law of large numbers for the pth powers, and the central limit theorem.

We consider a multidimensional free-boundary problem for a parabolic equation that arises in combustion theory. We prove the existence of a global classical solution. The idea of the method is as follows: first, we perform the differential–difference approximation of the problem and establish its solvability; then we prove uniform estimates and perform a limit transition.

For essentially nonlinear differential systems with the limit matrix of coefficients of the first-approximation system, we establish sufficient conditions for functional polystability, which generalizes the notion of exponential polystability.

We prove inequalities of the Landau–Kolmogorov–Hormander type for the uniform norms (on some subinterval) of positive and negative parts of intermediate derivatives of functions defined on a finite interval. By using the limit transition, we obtain a new proof or the well-known Hormander result.

On the basis of the limit theorem for quadratic variation, we construct a consistent estimator for parameters of a homogeneous Gaussian random field of a certain class. We find confidence ellipsoids for estimators of this sort.

Possible limit laws are studied for the multivariate conditional distribution of a subset of components of the sum of independent identically distributed random vectors under the condition that other components belong to the domain of large deviations. It is assumed that the considered distribution is absolutely continuous and belongs to the domain of attraction of the normal law but possesses “heavy tails.” The approach suggested is based on the local theorem for large deviations.

We obtain Gaussian and non-Gaussian distributions of estimates of regression coefficients of a long-memory time series.

For a modified risk process with instantaneous reflection at the pointB>0 under which the precess considered

We establish sufficient conditions for the Lyapunov instability of the trivial solution of a nonautonomous equation of thenth order in the case where its limit characteristic equation has a multiple zero root. The instability is determined by nonlinear terms.

We consider the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of a system of hard spheres. A new concept of averages over states of stochastic systems is introduced, in which the contribution of the hypersurfaces on which stochastic point particles interact is taken into account. We give a rigorous derivation of the infinitesimal operators of the semigroups of evolution operators. Визначена стохастична динаміка, яка є границею Больцмана — Греда від гамільтоиової динаміки системи пружних куль. Введено нову концепцію середніх від спостережуваних за станами стохастичиих систем. В ньому враховуються вклади від гіперповерхоиь, на яких взаємодіють точкові стохастичні частки. Дано строге визначення інфінітезімальних операторов для півгрупи еволюційних операторів.

By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem.

We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time.