Schéma d'Euler discret pour diffusion multidimensionnelle tuée

We are interested in the approximation of the law of a multidimensional diffusion killed as it leaves an open set D, when the diffusion is approximated by its discrete Euler scheme, with discretization step N⁻¹ T. We show that the error on E_x[1_{T< τ} ƒ(X_T)] (where τ = inft > 0: X_t ∉ D) is of order N^−1/2, under some conditions on ƒ near the boundary of D. This rate is intrinsic to the problem of discrete killing time.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

On coverings in the lattice of all group topologies of arbitrary Abelian groups

The remainder of the completion of a topological abelian group (G, τ₀) contains a nonzero element of prime order if and only if G admits a Hausdorff group topology τ₁ that precedes the given topology and is such that (G, τ₀) has no base of closed zero neighborhoods in (G, τ₁).     

On J. Nagata's universal spaces

The main result of this paper is the following extension of an embedding theorem by Nagata: given a sequence of zero-dimensional sets X₁,X₂,… in a metrizable space X of weight τ⩾ℵ₀, the set of homeomorphic embeddings h of X into S(τ)^ℵ₀, satisfying h(X_n)⊂K_n-1(τ) for n= 1,2,…, is dense in the function space of all continuous mappings of X into S(τ)^ℵ₀, where K_n(τ) is the n-dimensional universal Nagata's space in the countable product of the star-space S(τ) of weight τ. This seems to be a new result even in the separable case τ=χ₀ and provides in particular an answer to a question asked by Kuratowski (see Remark 2.6 for the details).     

Network delay in a model with inputs of functional elements

We consider a model of delays in networks of functional elements in an arbitrary finite complete basis B, where the delays of basis elements are arbitrary positive real numbers for each input and each input set of variables going to the remaining inputs. This model estimates the delays in a multiplexer function of nth order asymptotically as τ_B n ± O(logn), where τ_B is a constant depending only on the basis B. On the basis of these estimates and within this model, asymptotic estimates of the form τ_B n ± O(logn) are obtained for the corresponding Shannon function, i.e., for the delay of the “worst” Boolean algebra function of given n variables.     

Polarized light transfer above a reflecting surface

The existence and uniqueness are established for the solution of the equation of transfer of polarized light in a homogeneous atmosphere of finite optical thickness, assuming reflection by the planetary surface. A general L_p-space formulation is adopted. The boundary value problem is first written as a vector-valued integral equation. Using monotonicity properties of the spectral radii of the integral operators involved as well as recent half-range completeness results for kinetic equations with reflective boundary conditions, the present results follow as a corollary.     

Structure des algebres de valuation discrete

We define a topology τ_e, on anyC-algebra of discrete valuation, generalizing the topology of coefficientwise convergence on C[[X]] studied by G. R. Allan. We give a necessary and sufficient condition for τ_e to be complete and prove that the completion provides an algebra of discrete valuation. We also prove that if aC-algebra of discrete valuation is Fréchet andm-convex for τ_e then it is isomorphic to (C[[X]], τ_e) and then τ_e is the uniqueF-algebra topology in A. We prove that a commutative, unital Fréchet l.m.c.a. that is aC-algebra of valuation is in fact aC-algebra of discrete valuation and so is embeddable in (C[[X]], τ_e). Whence a result of H. Bouloussa.     

Asymptotic analysis and asymptotic domain decomposition for an integral equation of the radiative transfer type

We consider an integral equation of the radiative transfer type stated in the interval [0,τ₀] with the length τ₀1. We construct an asymptotic solution of the problem and we give a method transforming this problem to some similar problems set in the interval with the length dτ₀. Error estimates are proved.     

A mixed problem for the Laplace operator in a domain with moderately close holes

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ω_ε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by u_ε the solution of a mixed problem for the Laplace equation in Ω_ε. We describe what happens to u_ε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion.     

FBN Modules

For M ∈ R-Mod and τ ∈M-tors, we define the concept of fully τ-bounded module as a generalization of the concept of fully τ-bounded ring. We prove that for a τ-noetherian module M with local τ _M -Gabriel correspondence, which is a progenerator of σ[M] and with τ is FIS-invariant, then M is fully τ-bounded. Also, we show that if M is τ-noetherian and fully τ-bounded, then M has local τ _M -Gabriel correspondence.     

The Skorokhod problem in a time-dependent interval

We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness of the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. Under the assumption that the first time τ$\tau$ when the moving boundaries touch after time zero is strictly positive, we derive two sets of conditions on the moving boundaries. We show that the variation of the local time of the associated reflected Brownian motion on [0,τ]$[0,\tau ]$ is finite under the first set of conditions and infinite under the second set of conditions. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.     

On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology τ_F on 2 ^X has as a subbase all sets of the form {A ∈ 2 ^X :A ∩V ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2 ^X :A ?W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τ_F in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.     

Renormalization of the τ-functions for integrable systems: A model problem

We introduce a renormalization procedure for the τ-function of integrable systems. We illustrate the procedure using the supercritical Toda shock problem as a model problem. We start with a finite chain and take the limit of the solution as the number of particles N → ∞. This results in a new formula for the τ-function for the problem with an infinite chain. We apply the renormalized formula to rederive leading-order effects of the supercritical Toda shock problem. © 1998 John Wiley & Sons, Inc.     

Homogenization of the Signorini boundary‐value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ω_ε which is the union of a domain Ω₀ and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω₀ and an appropriate postprocessing. The equations in Ω₀ finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Decomposition of representations induced from uniform subgroups and the “Mackey machine”

Let G be a separable locally compact group with a discrete cocompact subgroup Γ; let N be a normal subgroup of G, and suppose that Γ₀ = Γ ∩ N is cocompact in N. Let ? be a finite-dimensional representation of Γ, and let τ = Ind_{Γ → G ?}. We investigate the problem of determining the decomposition of τ, given the decomposition of the corresponding representation and certain other facts (essentially those used in the “Mackey machine”). The last section is devoted to a reciprocity theorem for τ.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup (T(s)) _s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common element z of the set of the equilibrium points and the set of common fixed points of (T(s)) _s≥0. Such common element z is the unique solution of a variational inequality, which is the optimality condition for a minimization problem.     

Properties of stationary states of delay equations with large delay and applications to laser dynamics

We consider properties of periodic solutions of the differential‐delay system, which models a laser with optical feedback. In particular, we describe a set of multipliers for these solutions in the limit of large delay. As a preliminary result, we obtain conditions for stability of an equilibrium of a generic differential‐delay system with fixed large delay τ. We also show a connection between characteristic roots of the equilibrium and multipliers of the mapping obtained via the formal limit τ→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

Delays of schemes in a model that considers the input values of functional elements

A delay model for schemes of functional elements in arbitrary finite complete basis B is studied; in the model, delays of the basic element are given by random positive real numbers for each input and each input set of variables entering other inputs. Asymptotic estimates in the form τ_B n ± O(logn), where τ_B is a constant that depends only on basis B, are obtained for the delay of the multiplex function of order n. Based on these estimates, asymptotic estimates of the form τ_B n ± O(logn) for the corresponding Shannon function, i.e., for the delay of the worst function of logic algebra that depends on given n variables, are established.