Asymptotic relation for the density of a multidimensional random evolution with rare poisson switchings

A symmetric random evolution X(t) = (X ₁ (t), …, X _m (t)) controlled by a homogeneous Poisson process with parameter λ > 0 is considered in the Euclidean space ℝ ^m , m ≥ 2. We obtain an asymptotic relation for the transition density p(x, t), t > 0, of the process X(t) as λ → 0 and describe the behavior of p(x, t) near the boundary of the diffusion domain in spaces of different dimensions. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1631 – 1641, December, 2008.     

On the Erlang Loss Model with Time Dependent Input

We consider the M(t)/M(t)/m/m queue, where the arrival rate λ(t) and service rate μ(t) are arbitrary (smooth) functions of time. Letting p_n(t) be the probability that n servers are occupied at time t (0≤ n≤ m, t > 0), we study this distribution asymptotically, for m→∞ with a comparably large arrival rate λ(t) = O(m) (with μ(t) = O(1)). We use singular perturbation techniques to solve the forward equation for p_n(t) asymptotically. Particular attention is paid to computing the mean number of occupied servers and the blocking probability p_m(t). The analysis involves several different space-time ranges, as well as different initial conditions (we assume that at t = 0 exactly n₀ servers are occupied, 0≤ n₀≤ m). Numerical studies back up the asymptotic analysis. AMS subject classification: 60K25,34E10 Supported in part by NSF grants DMS-99-71656 and DMS-02-02815     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

布朗局部时主值滞后增量的几个结果

Let W be a standard Brownian motion,and define Y(t) =∫ods/W(s) as Cauchy‘s principal value related to the local time of W. We study some limit results on lag increments of Y(t) and obtain various results all of which are related to earlier work by Hanson and Russo in 1983.     

Pappus type theorems for hypersurfaces in a space form

In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM _λ ⁿ of sectional curvature λ. LetP ₀ be a totally geodesic hypersurface ofM _λ ⁿ throughc(0) and orthogonal toc(t). LetC ₀ be a hypersurface ofP ₀. LetC be the hypersurface ofM _λ ⁿ obtained by a motion ofC ₀ alongc(t). We shall denote it byC ^PorC ^Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC ₀, then volume(C) ≥ volume(C),^P),and the equality holds whenC ₀ is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion). Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052.     

Uniform convergence of wavelet solution to the sideways heat equation

We consider the problem uxx（x, t） = ut（x, t）, 0 ≤ x 〈 1, t ≥ 0, where the Cauchy data g（t） is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g（t） can produce a big alteration on its solution （if it exists）. We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space Vj. In the previous papers, the theoretical results concerning the error estimate are L2-norm and the solutions aren＇t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ [0, 1].     

On the set of <Emphasis Type="Italic">t</Emphasis>-linked overrings of an integral domain

Let R be an integral domain with quotient field L. An overring T of R is t-linked over R if I ⁻¹ = R implies that (T : IT)  =  T for each finitely generated ideal I of R. Let O _t (R) denotes the set of all t-linked overrings of R and O(R) the set of all overrings of R. The purpose of this paper is to study some finiteness conditions on the set O _t (R). Particularly, we prove that if O _t (R) is finite, then so is O(R) and O _t (R) = O(R), and if each chain of t-linked overrings of R is finite, then each chain of overrings of R is finite. This yields that the t-linked approach is more efficient than the Gilmer’s treatment (Proc Am Math Soc 131:2337–2346, 2002). We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain, etc. We establish a connection between O _t (R) and the set of all strongly divisorial ideals of R and we conclude by a characterization of domains R that are t-linked under all their overrings. This work was funded by KFUPM under Project # FT/18-2005.     

Small Time One-Sided LIL Behavior for Lévy Processes at Zero

We specify a function b ₀(t) in terms of the Lévy triplet such that lim sup  _t→0 X _t /b ₀(t)∈[1,1.8] a.s. iff ò₀¹[` \varPi ]⁽⁺⁾(b₀(t)) dt < ￥\int_{0}^{1}\overline{ \varPi }^{(+)}(b_{0}(t))\,dt<\infty for any Lévy process X with unbounded variation and a Brownian component σ=0. We show with an example that there are cases where lim sup  _t→0 X _t /b(t)=1 a.s. but b(t) is not asymptotically equivalent to b ₀(t) as t tends to 0. We achieve this by introducing an integral criterion which checks whether lim sup  _t→0 X _t /b(t) is 0, infinity, or a finite positive value for b(t) satisfying very mild conditions and any Lévy process.     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence

We establish an estimate for the rate of convergence of a solution of an ordinary stochastic differential equation of order p ≥ 2 with a small parameter in the coefficient of the leading derivative to a solution of a stochastic equation of order p − 1 in the metric ρ(X, Y) = (sup_0≤t≤T M|X(t) − Y(t)|²)^1/2 __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1587–1601, December, 2006.     

On the transversal Helly numbers of disjoint and overlapping disks

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t_k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T(k) has a line transversal. We determine exact values of t₃ and t₄, and give general lower and upper bounds on the sequence t_k, showing that t_k = O(1/k) as k → ∞. In honour of Helge Tverberg’s seventieth birthday Received: 9 June 2005     

Smooth solutions to some differential-difference equations of neutral type

The paper is devoted to the scalar linear differential-difference equation of neutral type

Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Let ϕ_t(x), x ∈ ℝ₊ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic measure-valued process μ_t = μ ○ ϕ _t ⁻¹ that is the image of a certain absolutely continuous measure μ under random mapping ϕ_t(·). We prove that the restriction of the Hausdorff measure H ^d−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component is absolutely continuous with respect to H ^d−1. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006.     

Large deviations and continuum limit in the 2D Ising model

Summary. We study the 2D Ising model in a rectangular box Λ _L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ _t∈ΛL σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m ^* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {|∑ _t∈ΛL σ(t) −m|Λ _L ||≤|Λ _L |L ^{− c} }, with 0<c<1/4 and −m ^*<m<m ^*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type. Received: 17 October 1996 / In revised form: 7 March 1997     

Implicit Euler scheme for an abstract evolution inequality

For a triple {V, H, V*} of Hilbert spaces, we consider an evolution inclusion of the form u′(t)+A(t)u(t)+δϕ(t, u(t)) ∋ f(t), u(0) = u0, t ∈ (0, T ], where A(t) and ϕ(t, ·), t ∈ [0, T], are a family of nonlinear operators from V to V * and a family of convex lower semicontinuous functionals with common effective domain D(ϕ) ⊂ V. We indicate conditions on the data under which there exists a unique solution of the problem in the space H ¹(0, T; V)∩W _∞¹ (0, T;H) and the implicit Euler method has first-order accuracy in the energy norm.     

Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

We consider the periodic boundary-value problem u _tt − u _xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u ⁰(x, t) + ũ(x, t), where u ⁰(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.     

Center conditions II: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). We introduce a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1). We show that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Let C _t = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C ¹-family of circles in the plane such that lim _t→0+ C _t = {a}, lim _t→1− C _t = {b}, a ≠ b, and |c′(t)|² + |r′(t)|² ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation ̅c′(t)w ² + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.     

Stochastic Equations with Time-Dependent Drift Driven by Levy Processes

The stochastic equation dX _t =dS _t +a(t,X _t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X ₀=x ₀∈ℝ when (ℛe  ψ(ξ))⁻¹=o(|ξ|⁻¹) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.