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The paper deals with the limit behavior of a compound Poisson process with switching between a finite number of sequences of i.i.d. random variables. The switching is provided by Bernoulli’s random variables. Under suitable normalization, the limit process is a Brownian motion with switching variance.

We prove a limit theorem on the convergence of mathematical expectations of functionals of sums of independent random variables to the Cauchy problem solution for the nonstationary Schr¨odinger equation with a symmetric fractional derivative operator of order $$ \upalpha\ \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) $$ in the righthand side.

We analyze the stability of elastic systems composed of the shells of revolution with variable curvature and complex structures in the field of conservative axisymmetric loads of different nature. Within the framework of classical and refined theories of shells, we determine the limit and bifurcation critical values of the acting loads based on the geometrically nonlinear statement of the problem and a criterion of dynamic stability. To solve the corresponding nonlinear and eigenvalue problems, we propose to use a numerical-analytic approach based on their rational reduction to one-dimensional linear boundaryvalue problems in the meridional coordinate and their numerical solution by the discrete-orthogonalization method. We present test examples that confirm the applicability of the proposed procedure to the analyzed class of problems. The limit and bifurcation values of the critical loads in the shell system are analyzed depending on its geometric parameters.

We consider a nonlinear nonautonomous system of two ordinary differential equations with a stable fixed point and assume that the non-Hamiltonian part of the system tends to zero at infinity. We examine the asymptotic behavior of a two-parameter family of solutions that start from a neighborhood of the stable equilibrium. The proposed construction of asymptotic solutions is based on the averaging method and the transition in the original system to new dependent variables, one of which is the angle of the limit Hamiltonian system, and the other is the Lyapunov function for the complete system.

We study the influence of decaying perturbations on autonomous oscillatory systems in a plane under the assumption that the perturbations preserve the equilibrium state of the limit system, oscillate with asymptotically constant frequency, and satisfy the nonresonance condition. We discuss the long-term behavior of the perturbed trajectories in a neighborhood of the equilibrium state. We describe conditions on the perturbation parameters that guarantee preservation or loss of stability of the equilibrium. The results are illustrated by an example of decaying perturbations of the Duffing oscillator.

In the present paper, the enumeration of a certain class of directed lattice paths is based on the analysis of dynamical correlation functions of the exactly solvable XX0 model. This model is the zero anisotropy limit of one of the basic models of the theory of integrable systems, the XXZ Heisenberg magnet. It is demonstrated that the considered correlation functions under different boundary conditions are the exponential generating functions of various types of paths, in particular, Dyck and Motzkin paths.

The six-vertex model on a finite square lattice with the so-called partial domain wall boundary conditions is considered. For the case of rational Boltzmann weights, the polarization on the free boundary of the lattice is computed. For the finite lattice the result is given in terms of a ratio of determinants. In the limit, where the side of the lattice with free boundary tends to infinity (the limit of a semi-infinite lattice), they are simplified and can be evaluated in a closed form.

We study the models of multicomponent discrete dynamical conflict systems with attractive interaction characterized by a positive value called the attractor index. The existence of limit equilibrium states in these systems is proved and their description in terms of the attractor index is given. An explicit relationship between the limit state of the system and the attractor index is established. A series of concrete examples is presented. They illustrate the dynamics of the system for different attractor indices.

By using an analog of the δc -model, we reduce the problem of stressed state and limit equilibrium of an inhomogeneous shell of revolution weakened by an internal crack of any configuration with plastic strains developed on the continuation of the crack in the form of a narrow strip to an elastic problem. The indicated elastic problem is then reduced to a system of singular integral equations with unknown limits of integration and discontinuous functions on the right-hand sides. We propose an algorithm for the numerical solution of these systems with regard for the conditions of plasticity of thin shells and the conditions of boundedness for stresses. The effects of loading, geometric parameters, and mechanical characteristics on the crack-opening displacement and the sizes of plastic zones are investigated for cylindrical and spherical shells made of functionally graded materials and containing internal parabolic cracks.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

We consider an initial-boundary value problem for a partial differential equation with mixed third derivative and small parameter. We establish the existence and uniqueness of a solution to the limit problem. The solution is found in an analytic form. We study the behavior of the solution to the prelimit problem for small parameter.

We consider continuous-time branching random walks on multidimensional lattices with birth and death of particles at a finite number of lattice points. Such processes are used in numerous applications, in particular, in statistical physics, population dynamics, and chemical kinetics. In the last decade, for various models of branching random walks, a series of limit theorems about the behavior of the process for large times has been obtained. However, it is almost impossible to analyze analytically branching random walks on finite time intervals; so in this paper we present an algorithm for simulating branching random walks and examples of its numerical realization.

We consider a fractal equation of the integral form with Bessel fractional integrodifferential operator and a positive parameter. In a part of the initial hyperplane, where the limit value has good properties, we establish the property of local strengthening of the convergence of regular solutions with generalized limit values.

For the second-order nonlinear ordinary differential equation $$ {u}_{xx}^{\hbox{'}\hbox{'}}={u}^5-{tu}^3-x, $$ we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5 − tu3 − x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2 →∞.

The literature contains quite a few theorems on subdividing the volume of a convex body by a system of cones and on the possibility to circumscribe the body about a polytope of one type or another. See R. N. Karasev, “Topological methods in combinatorial geometry,” Russian Math. Surveys, 63, No. 6, 1031–1078 (2008) for a survey of similar results. In the following, we also prove theorems of this kind. As a limit case, we obtain well-known theorems on inscribed polytopes.

The Fejer integrals for finite measures on the real line and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating, in fact, with the same formulas (by integrating of the Fejer kernels). Thus this ergodic theorem is a statement about the asymptotic of the growth of the Fejer integrals at zero point of the spectral measure of corresponding dynamical system. It gives a possibility to rework well-known estimates of convergence rates in the von Neumann ergodic theorem into the estimates of the Fejer integrals at a point for finite measures: for example, natural criteria of polynomial growth and polynomial decay of these integrals are obtained. And vice versa, numerous known estimates of the deviations of Fejer integrals at a point allow to obtain new estimates of convergence rates in this ergodic theorem.

A modification of service system model introduced by I. Kaj and M. S. Taqqu is considered. This model describes the dynamics in time and space of various system workloads created by a set of service processes. In the model under consideration, two types of resource having its own distribution are used. Such a model can be identified with the presence of two operators of the resource. At the time of the active operator failure, one can switch to another operator whose resource has distribution workloads different from the first operator. A limit theorem on the convergence of finite-dimensional distributions of the integral workload process with two types of resource to Brownian motion with switching variance is proved.

For mappings with unbounded characteristics we prove theorems on removal of isolated singularities on Riemannian manifolds. We prove that if a mapping satisfies certain inequality of absolute values and its quasiconformity characteristic has a majorant of finite average oscillation at a fixed singular point, then it has a limit at this point.

We study gradient-like flows with no heteroclinic intersections on an n-dimensional (n ≥ 3) sphere from the point of view of topological conjugacy. We prove that the topological conjugacy class of such a flow is completely determined by the bicolor tree corresponding to the frame of separatrices of codimension 1. We show that for such flows the notions of topological equivalence and topological conjugacy coincide (which is not the case if there are limit cycles and connections.

We establish a series of sufficient conditions for the realization of a sure event as the limit (in time) state of a multicomponent dynamical system with attractive interaction. The sure event is characterized by the state of a system with finitely many positions when all coordinates of the distribution are equal to zero, except a single fixed coordinate equal to 1. The sure event can be interpreted as the state of consensus in social networks and, hence, the obtained results can be used in voter and opinion-formation models.

We study the problem of limit equilibrium of a long cylindrical shell with longitudinal crack subjected to the action of a load that varies with time according to the exponential law. For the case of symmetric loading of the crack, we construct a system of singular integral equations. We also study the influence of the rate of changes in the load on the value of the force intensity factor in the vicinity of the crack ends.

The paper contains some comments to the limit fractional Poisson distributions introduced in the preceding works of V.V. Uchaikin and his colleagues. Here, some numerical data on the distribution are tabulated, the link with the Levy-subordinator is ascertained, and a Monte Carlo algorithm of simulation is proposed.

We consider the most general class of multipoint boundary-value problems for systems of linear ordinary differential equations of the first order whose solutions belong to a given Sobolev space $$ {W}_p^n $$n ϵ ℕ, 1 ≤ p ≤ ∞: Sufficient constructive conditions under which the solutions of these problems are continuous with respect to the parameter e for e = 0 in the space $$ {W}_p^n $$ are established.

We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, and generalized Mittag–Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag–Leffler mixing distribution. Based on these representations, we prove some limit theorems for a wide class of statistics constructed from samples with random sized including, e.g., random sums of independent random variables with finite variances, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is by far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.

We consider a population of particles with unit lifetime. Dying, each particle produces offspring whose size depends on the random environment specifying the reproduction law of all particles of the given generation and on the number of relatives of the particle. We study the asymptotic behavior of the survival probability of the population up to a distant moment n under some restrictions on the properties of the environment and family ties and prove a limit theorem for the number of particles in such processes given the respective populations survive for a long time.

In some non-homogeneous generalized allocation schemes we formulate conditions under which the number of given value cells from the first K cells converges to a Gaussian random variable. This result is applied to the study of the type I error and the type II error for analogs of the empty boxes test.

The paper is concerned with special questions on the behavior of zeros of sequences of Bernstein polynomials. For a piecewise linear generating function, computer mathematics machinery was used to find the rules controlling the limit behavior of zeros as the number of the Bernstein polynomial unboundedly increases. New problems for theoretical investigations are formulated.

The paper deals with the limit behavior of a compound Poisson process with switching. The switching is provided by Bernoulli’s random variables. Under a suitable normalization, the limit process is a Brownian motion with switching variance.

A service system model introduced by I. Kaj and M. S. Taqqu is considered. We prove a limit theorem on the convergence of finite-dimensional distributions of the integral workload process with a multidimensional resource to the corresponding distributions of a multidimensional stable process.

High-frequency diffraction by a contour with a jump of curvature is addressed. The outgoing wavefield on the limit ray is studied in the framework of ray theory.

In this paper, we show that an operator introduced by A. A. Tsvetkov enjoys all the necessary properties of a Q-operator. It is shown that the Q-operator of the XXX spin chain with spin l turns into Tsvetkov’s operator in the continuous limit as l→∞. Bibliography: 18 titles.

Necessary and sufficient conditions for the validity of the central limit theorem for densities are considered with respect to the norms in Orlicz spaces. The obtained characterization unites several results due to Gnedenko and Kolmogorov (uniform local limit theorem), Prokhorov (convergence in total variation) and Barron (entropic central limit theorem).

Let G be a graded graph with levels V0, V1, . . .. Fix m and choose a vertex v in Vn where n ≥ m. Consider the uniform measure on the paths from V0 to v. Each such path has a unique vertex at the level Vm, so a measure $$ {\nu}_v^m $$ on Vm is induced. It is natural to expect that these measures have a limit as the vertex v goes to infinity in some “regular” way. We prove this (and compute the limit) for the Young and Schur graphs, for which regularity is understood as follows: the fraction of boxes contained in the first row and the first column goes to 0. For the Young graph, this was essentially proved by Vershik and Kerov in 1981; our proof is more straightforward and elementary.

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.

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.

On the basis of the initial-boundary-value Lord–Shulman problem of thermopiezoelectricity, we formulate the corresponding variational problem in terms of the vector of elastic displacements, electric potential, temperature increment, and the vector of heat fluxes. By using the energy balance equation of the variational problem, we establish sufficient conditions for the regularity of input data of the problem and prove the uniqueness of its solution. To prove the existence of the general solution to the problem, we use the procedure of Galerkin semidiscretization in spatial variables and show that the limit of the sequence of its approximations is a solution of the variational problem of Lord–Shulman thermopiezoelectricity. This fact allows us to construct a reasonable procedure for the determination of approximate solutions to this problem.

Extreme values are considered in samples with random size that have a mixed Poisson distribution that is generated by a doubly stochastic Poisson process. Some inequalities are proved relating the distributions and moments of extrema with those of the leading process (the mixing distribution). Limit theorems are proved for the distributions of max-compound Cox processes, and limit distributions are described. An important particular case of the negative binomial distribution of a sample size corresponding to the case where the Cox process is led by a gamma Levy process is considered, explaining a possible genesis of tempered asymptotic models.

This paper gives an overview of the existing approaches for modelling high quantiles of dependent spatial data and apply the methods to the bivariate circular case. We also adapt the bootstrap to the situation at hand. Since any data set one might use is finite, the interest lies in estimating a continuous curve as its upper limit (or quantile function). This can be obtained by either a kernel type regression or a fitted parametric model. We also introduce a new, more realistic correction formula for a nonparametric method for estimating the pointwise maximum (called frontier in this setup [8]). An additional common challenge in real-life applications is the dependence among subsequent observations. The theoretical results about the GPD limit of the exceedances beyond high threshold remain valid under mixing-type conditions, called D(un) in the extreme-value literature. However, if one intends to use the bootstrap-based reliability estimators, then they need to be adjusted — e.g., by the block-bootstrap approach in [15]. We estimate the reliability of the estimators by a suitable application of the m out of n bootstrap, which turned out to be suitable for high-quantile estimation. We illustrate the introduced methods for simulated as well as real enzymes data.

In this paper, a multi-channel queueing network with input flow of a general structure is considered. The multi-dimensional service process is introduced as the number of customers at network nodes. In the heavy-traffic regime, a functional limit theorem of diffusion approximation type is proved under the condition that the input flows converge to their limits in the uniform topology. A limit Gaussian process is constructed and its correlation characteristics are represented explicitly via the network parameters. A network with nonhomogeneous Poisson input flow is studied as a particular case of the general model, and a correspondent Gaussian limit process is built.

We study equilibrium of a two-layer construction of elastic and rigid layers with a crack along the line joining the layers. We consider the limit problem as the rigid layer size tends to zero and the optimal control problem where the cost functional is the derivative of the energy functional with respect to the crack length and the control parameter characterizes the rigid layer size.

We consider problems governing a contact between an elastic plate with a thin elastic inclusion and a thin elastic obstacle and study the equilibrium of the plate with or without cuts. We discuss various statements and establish the existence of a solution. We analyze the limit problem as the rigidity parameter of the elastic inclusion tends to infinity.

We study operators that project a vector-valued function υ ∈ W1,2(Ω, ℝd) to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that the divergence is equal to zero almost everywhere presents the first (narrowest) limit case while the integral condition of zero mean divergence generates the other (widest) case. Estimates of the distance between υ and the respective projection on such a subspace are important for analysis of various mathematical models related to incompressible media problems (especially in the context of a posteriori error estimates. We establish different forms of such estimates, which contain only local constants associated with the stability (LBB) inequalities for subdomains. The approach developed in the paper also yields two-sided bounds of the inf-sup (LBB) constant.

We establish a connection between the limit characteristics of minimizing sequences of the energy functional of a two-phase elastic medium and the derivative of the equilibrium energy with respect to the temperature.

This paper is a further development of the results of [20] where, based on the negative binomial model for the duration of wet periods measured in days [16], an asymptotic approximation was proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Fr´echet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor–Fisher distribution. Here we extend this result to the mth extremes, m ∈ ℕ, and sample quantiles. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution [17] and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions [10]. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution of the maximum precipitation is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions this limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. It is also shown that the limit distribution of sample quantiles is the well-known Student distribution. Several methods are proposed for the estimation of the parameters of the asymptotic distributions. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability (“Bayesian”) models.

Adaptive algorithms for constructing spline-wavelet decompositions of matrix flows from a linear space of matrices over a normed field are presented. The algorithms suggested provides for an a priori prescribed estimate of the deviation of the basic flow from the initial one. Comparative bounds of the volumes of data in the basic flow for various irregularity characteristics of the initial flow are obtained in the cases of pseudo-equidistant and adaptive grids. Limit characteristics of the above-mentioned volumes are given in the cases where the initial flow is generated by differentiable functions.

We establish the unique solvability of the equilibrium problem for a two-layer body. One layer contains a crack, whereas the second one is glued along its edge to the first layer in such a way that to cover one of the crack ends. In the case, where the second layer is a rigid plate, we show that the problem with a rigid patch is the limit of problems with elastic patches as the rigidity parameter tends to infinity. We also study the optimal control problem with the exterior forces acting on both layers taken for the control function.

A new method of probabilistic approximation of solution to the Cauchy problem for the unperturbed Schrodinger equation by expectations of functionals of a random walk is suggested. In contrast to earlier papers of the authors, the existence of exponential moment for each step of the random walk is not assumed.

We consider the action of the quadratic form of the Laplace operator and its extensions in subspaces of linear combinations of the “transverse” and “longitudinal” functions with the fixed orbital momentum with respect to the coordinate origin. In the statement of the problem, it is required that the extensions obtained, after the transfer back to the space of vector functions, can be represented as simple limit expressions with two coefficients. We study the behavior of these coefficients with respect to the initial choice of the linear subspace. Bibliography: 5 titles.

We analyze a class of explicitly solvable models for the problems of redistribution of the conflict space between two alternative opponents. The existence of equilibrium state is proved for a complex nonlinear system whose time evolution is generated by the conflict interaction between its components. Explicit formulas are obtained for the limit compromise distributions in terms of the densities of probability measures. We consider a number of specific model examples of the dynamics of redistribution of the conflict territory and formation of an equilibrium (compromise) distribution of the space. We propose an interpretation of the results for the case of social and territorial conflicts.