Ultimate boundedness with respect to part of the variables of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notions of equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables for solutions with partly controlled initial conditions. We obtain sufficient conditions for the equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables of solutions with partly controlled initial conditions. We introduce the notions of equiboundedness and uniform boundedness with respect to part of the variables for solutions of systems with partly controlled initial conditions. We obtain sufficient conditions for the equiboundedness and uniform boundedness with respect to part of the variables of solutions with partly controlled initial conditions.     

The bound of supports of solutions for some classes of evolutionary systems and equations

We study the property of finiteness of a support for a system of evolutionary equations and for equations with nonstationary p-Laplacian and with convective terms. New bounds of the size of the support with regard for the convective terms are obtained.     

Generalization of the method of functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media

A new approach for constructing functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media is proposed. Solutions of the equations of motion in displacements and potentials, which express plane waves and waves from a point source, and also complex solutions of a general type are obtained and investigated. The problem of the reflection of plane waves from the boundary of a half-space is solved for comparison with earlier results [1]. The solutions obtained agree with the physical meaning of the problems and with the solutions for isotropic media.     

Quasiwidths and optimization of methods of mixed approximation of multidimensional singular integrals with kernels of hilbert type

We consider the problem of application of mixed methods to the construction of algorithms, optimal in accuracy, for the calculation of multidimensional singular integrals with Hilbert-type kernels. We propose a method for the optimization of cubature formulas for singular integrals with Hilbert-type kernels based on the theory of quasiwidths.     

An estimate of the error of the linearized equations of motion of an axisymmetric projectile

An estimate is obtained, by small-parameter methods, of the error of the solution for the system of equations of motion of a projectile in the atmosphere, linearized for small angles of attack with respect to the variables describing the angular oscillations of the axis of symmetry, compared with the solution of the initial non-linear system for the same initial conditions.     

Inversion of vector delay systems

We consider stationary linear vector systems with commensurable delays. We obtain sufficient conditions for the reducibility of such systems to canonical form with the extraction of null dynamics. A constructive algorithm for the reduction of a system to that form is presented. We suggest a method for estimating the unknown input for vector delay systems with given accuracy.     

The construction of dual variational principles of the theory of the seepage of an incompressible fluid in some deformable complex media

A method of constructing dual variational principles for problems of the seepage of an incompressible fluid in deformable media with complex rheology is presented. Dual variational principles of the fluid seepage in a typical deformable media with complex rheology [1, 2], namely, a Maxwell element, connected in parallel with a visco-plastic or elastic element, are constructed. The variational principles are derived from variational problems compiled for the constitutive relations for fluid and solid phases.     

Estimates of densities for Lévy processes with lower intensity of large jumps

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times.     

Removable singularities of solutions of elliptic equations

This paper is an overview of results devoted to metric conditions for removability of closed sets for solutions of homogeneous partial differential equations in various function classes. The author considers equations with a quasi-homogeneous semi-elliptic operator and with constant coefficients, linear second-order uniformly elliptic equations in the divergent form with real bounded measurable coefficients, quasilinear equations with the p-Laplacian, and the minimal surface equation. A number of results is published for the first time. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

Solution of one class of systems of dual summation equations for associated Legendre functions

We have obtained analytical solutions of one class of systems of dual summation equations for associated Legendre functions with fractional indices. Such equations appear in studying the interaction of vector electromagnetic fields with the circular edge of a conductive open cone in the low-frequency region. We have derived formulas for the reexpansion of Legendre functions, which are used for passage from summation equations to infinite systems of linear algebraic equations, containing convolution-type matrix operators. The operators inverse to them are applied for finding a solution in the required class of sequences. We give an example of determining the effect of interaction of TM- and TE-waves with the edge of a finite cone.     

A note on the approximability of the toughness of graphs

We show that, if NP≠ZPP, for any >0, the toughness of a graph with n vertices is not approximable in polynomial time within a factor of . We give a 4-approximation for graphs with toughness bounded by and we show that this result cannot be generalized to graphs with a bounded toughness. More exactly we prove that there is no constant approximation for graphs with bounded toughness, unless P=NP.     

Blow-up of solutions of some nonlinear wave equations

We analyze the problem of blow-up of global solutions of a semilinear wave equation with a potential and with a possible degeneration at infinity for nonnegative initial data with compact support. By using the nonlinear capacity method, we prove the theorem on the nonexistence of such a solution for a subcritical and the critical nonlinearity exponent.     

Asymptotic properties of sample quantiles of discrete distributions

The asymptotic distribution of sample quantiles in the classical definition is well-known to be normal for absolutely continuous distributions. However, this is no longer true for discrete distributions or samples with ties. We show that the definition of sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample quantiles from absolutely continuous and from discrete distributions. We demonstrate that the same asymptotic normal distribution result as for the classical sample quantiles holds at differentiable points, whereas a more general form arises for distributions whose cumulative distribution function has only one-sided differentiability. For discrete distributions with finite support, the same type of asymptotics holds and the sample quantiles based on mid-distribution functions either follow a normal or a two-piece normal distribution. We also calculate the exact distribution of these sample quantiles for the binomial and Poisson distributions. We illustrate the asymptotic results with simulations.     

Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.     

Classification of Certain Class of Ordinary Differential Equations of the First Order

We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer.     

Estimation of statistical characteristics of attainability sets of controllable systems

We study an expansion of the notion of invariance for sets with respect to controllable systems and differential inclusions. Namely, we study statistically invariant sets and statistical characteristics of attainability sets of controllable systems. We obtain a lower bound for the lower relative frequency of the absorption of the attainability set of a system by a given set and establish new sufficient conditions of the statistical invariance of the set with respect to the controllable system. We give examples of the calculation of statistical characteristics for the linear Cauchy problem and a linear controllable system with almost periodic coefficients.     

Life-span of Classical Solutions of Nonlinear Hyperbolic Systems

In this paper, we give a lower bound for the life-span of classical solutions to the Cauchy problem for first order nonlinear hyperbolic systems with small initial data, which is sharp, and give its application to the system of one-dimensional gas dynamics; for the Cauchy problem of the system of one-dimensional gas dynamics with a kind of small oscillatory initial data, we obtain a precise estimate for the life-span of classical solutions.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

Remark on minimization of depth of Boolean circuits

It is shown that Lozhkin’s method (1981) for minimization of the depth of formulas with a bounded number of changing types of elements in paths from input to output and Hoover-Klawe-Pippenger’s method (technical report in 1981, journal publication in 1984) for minimization of the depth of circuits with unbounded branching by insertion of trees from buffers with bounded branching of outputs for each buffer are dual to each other and can be proved by one and the same method.