Asymptotics of Brownian integrals and pressure: Bose-Einstein statistics

The paper studies the asymptotics of the Brownian integrals with paths restricted to a bounded domain of ? ^v , when the domain is dilated to infinity. The framework is that of the Bose-Einstein statistics with paths observed within random time intervals which are integer multiplies of some fixed β > 0. The three first terms of the asymptotics are found explicitly via the functional integrals. In the case of a gas of interacting Brownian loops an expression for the volume term of the asymptotics of the log-partition function is found and the correction term is proved to by order be the boundary area of the domain.     

On the Gerber-Shiu discounted penalty function for subexponential claims

We obtain the asymptotics of the Gerber-Shiu discounted penalty function in the classical Lundberg model. We cosider claims from a class of subexponential distributions and find the asymptotics as the initial surplus x tends to infinity. The main term of the discounted penalty function ψ(x, δ) has different expressions in the cases where the interest rate δ > 0 and where δ = 0. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 598–605, October–December, 2006.     

Efficient Algorithms for Approximating Polygonal Chains

We consider the problem of approximating a polygonal chain C by another polygonal chain C' whose vertices are constrained to be a subset of the set of vertices of C . The goal is to minimize the number of vertices needed in the approximation C' . Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ɛ \geq 0 , compute an approximation of C , among all approximations whose error is at most ɛ , that has the smallest number of vertices. We present an O(n ^{4/3 + δ} ) -time algorithm to solve this problem, for any δ > 0; the constant of proportionality in the running time depends on δ . (2) Given a polygonal chain C and an integer k , compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n ^{4/3 + δ} ) , to solve this problem. Received September 17, 1998, and in revised form July 8, 1999.     

Scattering theory for perturbed stratified media

We study the operatorH = -c ² x,y)Μx,y)∇ · Μ ^-1 (x,y)∇, wherec andΜ are perturbations of functionsc ₀(y) andΜ ₀(y) which depend only on the one-dimensional variabley. In particular, we study the spatial asymptotics of lim_ε↺0(H - (λ +iε)²)^-1 applied to functions which have compact support or are otherwise well-behaved at infinity and relate the scattering matrix to the asymptotics of the generalized eigenfunctions. We then prove a trace formula for the operatorH in terms of the scattering phase, and, in a very special situation, use the trace formula to find spectral asymptotics forH. Partially supported by an NSF Postdoctoral Fellowship and the University of Missouri Research Board.     

S-polar sets of super-brownian motions and solutions of nonlinear differential equations

This paper gives probabilistic expressions of the minimal and maximal positive solutions of the partial differential equation -1/2△v(x) γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 < α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.     

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

Asymptotic behavior of positive solutions to emden–fowler type equations

We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same asymptotics as solutions to the model equation Δu = −x| ^p |u| ^σ−1 u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles.     

Construction of nonlinear regulators in economic growth models

We prove an inequality for the second moduli of continuity of continuous functions. Applying this inequality, we construct a nonnegative nonincreasing continuous function ω on [0, +g8) that vanishes at zero and is such that the function ω(δ)/δ ² decreases on (0, +g8) while ω is not asymptotically (as δ → 0) equivalent to the second modulus of continuity of any continuous function.     

On the best constants in Markov-type inequalities involving Laguerre norms with different weights

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial is taken in L ² on the half-line with the weight t ^α e ^−t and the derivative is measured in L ² on the half-line with the weight t ^β e ^−t . Under an additional assumption on the difference β − α, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.     

Boundary Value Problems with Compatible Boundary Conditions

If Y is a subset of the space ℝⁿ × ℝⁿ, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝⁿ into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P _δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.     

Convergence Properties of a Self-adaptive Levenberg-Marquardt Algorithm Under Local Error Bound Condition

We propose a new self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear equations F(x) = 0. The Levenberg-Marquardt parameter is chosen as the product of ‖F_k‖^δ with δ being a positive constant, and some function of the ratio between the actual reduction and predicted reduction of the merit function. Under the local error bound condition which is weaker than the nonsingularity, we show that the Levenberg-Marquardt method converges superlinearly to the solution for δ∈ (0, 1), while quadratically for δ∈ [1, 2]. Numerical results show that the new algorithm performs very well for the nonlinear equations with high rank deficiency. This work is supported by Chinese NSFC grants 10401023 and 10501013, Research Grants for Young Teachers of Shanghai Jiao Tong University, and E-Institute of Shanghai Municipal Education Commission, N. E03004.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines

We present an overlapping domain decomposition technique for solving the hypersingular integral equation on the sphere with spherical splines. We prove that the condition number of the additive Schwarz operator is bounded by O(H/δ), where H is the size of the coarse mesh and δ is the overlap size, which is chosen to be proportional to the size of the fine mesh. In the case that the degree of the splines is even, a better bound O(1 + log²(H/δ)) is proved. The method is illustrated by numerical experiments on different point sets including those taken from magsat satellite data.     

Wavelet characterization of Hörmander symbol classS ρ,δ m and applications

In this paper, we characterize the symbol in Hormander symbol classS _ρ ^m ,δ (m ∈ R, ρ, δ ≥ 0) by its wavelet coefficients. Consequently, we analyse the kerneldistribution property for the symbol in the symbol classS _ρ ^m ,δ (m ∈R, ρ > 0, δ≥ 0) which is more general than known results ; for non-regular symbol operators, we establish sharp L²-continuity which is better than Calderón and Vaillancourt’s result, and establishL ^p (1 ≤p ≤∞) continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator’s continuity on the basis of the wavelets coefficients in phase space.     

Cantor-Bendixson degrees and convexity in ℝ2

We present an ordinal rank, δ³, which refines the standard classification of non-convexity among closed planar sets. The class of closed planar sets falls into a hierarchy of order type ω₁ + 1 when ordered by δ-rank. The rank δ³ (S) of a setS is defined by means of topological complexity of 3-cliques in the set. A 3-clique in a setS is a subset ofS all of whose unordered 3-tuples fail to have their convex hull inS. Similarly, δⁿ (S) is defined for alln>1. The classification cannot be done using δ², which considers only 2-cliques (known in the literature also as “visually independent subsets”), and in dimension 3 or higher the analogous classification is not valid.     

Artinian property of constants of algebraicq-skew derivations

Let δ denote aq-skew σ-derivation of an algebraR andR ^(δ)={r εR│δ(r)=0} stand for the subalgebra of invariants. We prove thatR ^(δ) is left artinian iffR is left artinian providedR is semiprime and the action of δ onR is algebraic. This research was supported by the grant 190/R97/R98 in the frame of French-Polish joint projects and by Polish KBN grant no. 2 PO3A 039 14. We would like to thank all three universities for their hospitality.     

On certain arithmetic functions involving the greatest common divisor

The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ²(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.     

Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity

The initial-boundary value problem in a domain on a straight line that is unbounded in x is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter ɛ², where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as x → ∞ at a rate of O(x ²). This causes the unbounded growth of the solution at infinity at a rate of O(Ψ(x)), where Ψ(x) = x ² + 1. The initialboundary function is piecewise smooth. When ɛ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as x → ∞ even for smooth solutions when ɛ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem and its grid approximations is considered in the weighted maximum norm ∥·∥ ^w with the weighting function Ψ⁻¹(x); in this norm, the solution of the initial-boundary value problem is ɛ-uniformly bounded. Using the method of special grids that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge ɛ-uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy problem with the right-hand side and the initial function growing as O(Ψ(x)) that converge ɛ-uniformly in the weighted norm are also considered.