Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V^{(m + 1)} \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V^(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]     

The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

Integral points on quadrics with prime coordinates

Let f (x ₁, . . . , x _s ) be a regular indefinite integral quadratic form, and t an integer. Denote by V the affine quadric {x : f (x) = t}, and by V(\mathbb P){V(\mathbb {P})} the set of x ? V{{\bf x}\in V} whose coordinates are simultaneously prime. It is proved that, under suitable conditions, V(\mathbbP){V(\mathbb{P})} is Zariski dense in V as long as s ≥ 10.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

On existence and uniqueness of solution of stochastic differential equations with heredity

We consider the Cauchy problem for the stochastic differential equation with the heredity where x _t(s) = x(s)for s?(- ∞,t).Existence and uniqueness theorems for the problem (1),(2)are proved inthe case,when instead of the Lipschitz condition for the functions a(t,u) and b(t,u)on u someless restrictive conditions (Ousgood or Hölder type)are satisfied, and the operator(Fx)(t) = x(t)-f(t,x _t) is invertible.Similar questions were considered in[1-4]     

Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm

This paper deals with the initial value problem of the type

Kernel of a map of a shift along the orbits of continuous flows

Let F:M × \mathbbR \mathbb{R} → M be a continuous flow on a topological manifold M. For every subset V ì M V \subset M , we denote by P(V) the set of all continuous functions x:V ? \mathbbR \xi :V \to \mathbb{R} such that \textF( x,x(x) ) = x {\text{F}}\left( {x,\xi (x)} \right) = x for all x ? V x \in V . These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the corresponding periods (in general, not minimal). In this paper, the structure of P(V) is described for an arbitrary connected open subset V ì M V \subset M .     

Global stabilization of nonlinear systems by quadratic Lyapunov functions

The system x = A (t, x)x + B(t, x)u, where A(t, x) and B(t, x) are, respectively, n × n and n × m (m<n) continuous matrices whose elements are uniformly bounded for t ≽ t ₀ and x ∈ ℝ ⁿ , is considered. It is assumed that the system has relative degree q = n - m + 1, and the determinant of the matrix composed of the last m rows of the matrix B(t, x) is bounded away from zero for t ≽ t ₀ and x ∈ ℝ ⁿ . A special quadratic Lyapunov function with constant positive definite coefficient matrix H depending only on the range of variation of the coefficients in the matrices A(t, x) and B(t, x) is constructed and applied to obtain a control u(t, x) =7n ~B⋆ (t, x)H depending on a scalar parameter 7n under which the system is globally asymptotically stable provided that it is closed. Here, ~B (t, x) is the scalar matrix obtained from the matrix B(t, x) by setting the first n - m rows to zero.     

Positive solutions of non‐positone Dirichlet boundary value problems with singularities in the phase variables

The paper presents existence results for positive solutions of the differential equations x ″ + μh (x) = 0 and x ″ + μf (t, x) = 0 satisfying the Dirichlet boundary conditions. Here μ is a positive parameter and h and f are singular functions of non‐positone type. Examples are given to illustrate the main results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Problem of Erd?s Involving the Largest Prime Factor of n

Let P(n) denote the largest prime factor of an integer n (N(x) = x (2+O(?{log₂ x/logx} ) )ò₂^xr(logx/logt) [(logt)/(t²)] d t,N(x) = x \left(2+O\left(\sqrt{\log_{2}\,x/\!\log x}\,\right) \right)\int_2^x\rho(\log x/\!\log t) {\log t\over t^2} {\rm d} t,     

Nonconvex second-order differential inclusions with memory

We prove several existence theorems for the second-order differential inclusion of the form in the case whenF or bothG andF are maps with nonconvex values in an Euclidean or Hilbert space andF(t, T(t)x) is a memory term ([T(t)x]()=x(t+)).     

Quasi-periodic solutions for the general semilinear Duffing equations with asymmetric nonlinearity and oscillating potential

In this paper,we prove the existence of quasi-periodic solutions and the boundedness of all the solutions of the general semilinear quasi-periodic differential equation x′′+ax~+-bx~-=G_x(x,t)+f (t),where x~+=max{x,0},x~-=max{-x,0},a and b are two different positive constants,f(t) is C~(39) smooth in t,G(x,t)is C~(35) smooth in x and t,f (t) and G(x,t) are quasi-periodic in t with the Diophantine frequency ω=(ω_1,ω_2),and D_x~iD_t~jG(x,t) is bounded for 0≤i+j≤35.     

THE HAUSDORFF CENTRED MEASURE OF THE SYMMETRY CANTOR SETS

Let 0<λ≤1/3, K(λ) be the attractor of an iterated function system {ϕ₁ϕ₂} on the line, where ϕ₁(x)=λx, ϕ₁(x)=1-λ+λx,x∈[0,1]. We call K(λ) the symmetry Cantor sets. In this paper, we obtained the 0123 0132 V 3 exact Hausdorff Centred measure of K(λ).     

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)?F[x].Letx V → V be a linear operator. Let S_φ be the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S _φ to be a subspace.     

Exponential growth for the wave equation with compact time‐periodic positive potential

We prove the existence of smooth positive potentials V(t, x), periodic in time and with compact support in x, for which the Cauchy problem for the wave equation u_tt ? Δ_xu + V(t, x)u = 0 has solutions with exponentially growing global and local energy. Moreover, we show that there are resonances, z ∈ ?, |z| > 1, associated to V(t, x). © 2008 Wiley Periodicals, Inc.     

Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.     

Fundamental solution of the cauchy problem corresponding to the one-speed linear Boltzman equation for anisotropic media

We consider the fundamental solution E (t,x,s;s ₀) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t+c(s,grad _x)+γ)E(t,x,s;s ₀)=γν∫ f((s, s′))E(t,x,s′; s₀)ds′+Ωδ(t)δ(x)δ (s−s ₀) that is assumed to be valid for any (t,x)∈Rⁿ⁺¹; morevoer, for t<0 the condition E(t,x,s; s₀)=0 holds. By using the Fourier-laplace transform in space-time arguments, the problem reduces to the study of an integral equation in the variables. For 0<ν≤1, the uniqueness and existence of the solution of the original problem are proved for any fixeds in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic type (f(.)=1), the solution of the integral equation is given in explicit form. In the approximation of “small mean-free paths,” various weak limits of the solution are obtained with the help of a Tauberian-type theorem, for distributions. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 319–332. Translated by Yu. B. Yanushanets.     

An example of the Cauchy problem well posed in any Gevrey class

We study the Cauchy problem for second order hyperbolic operators in the Gevrey classes where H(t,x) is given by the limit of a finite sum of functions such as a(t)b(x) with a(t) ≥ 0, b(x) ≥ 0. As a result, for any given positive integer N, we give an example H(t,x) which depends not only on t but also on x such that the Cauchy problem for P is well posed in the Gevrey class of order N.