Limits of elliptic hypergeometric integrals

In Ann. Math., to appear, 2008, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level. The author was supported in part by NSF Grant No. DMS-0401387.     

Exact Buffer Overflow Calculations for Queues via Martingales

Let _n be the first time a queueing process like the queue length or workload exceeds a level n. For the M/M/1 queue length process, the mean _n and the Laplace transform e^-sn is derived in closed form using a martingale introduced in Kella and Whitt (1992). For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in Asmussen and Kella (2000) to derive sets of linear equations determining the same quantities. Numerical illustrations are presented in the framework of M/M/1 and MMPP/M/1 with an application to performance evaluation of telecommunication systems with long-range dependent properties in the packet arrival process. Different approximations that are obtained from asymptotic theory are compared with exact numerical results.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

On Short Time Asymptotic Behavior of Some Symmetric Diffusions on General State Spaces

For conservative symmetric diffusions on a general state space (X,m), the short time asymptotic behavior of tlog _X 1 _A T _t 1 _B dm is investigated, where T _t is the associated semigroup and A and B are measurable subsets of X. It is proved that the superior limit is dominated by the inferior limit up to some absolute constant. When ₂ of the associated Dirichlet form is lower bounded, it is shown that the limit exists for any A and B, and is described by the intrinsic metric between them. Applications to infinite-dimensional spaces and fractals are given.     

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

Sturm-liouville operators with singular potentials

This paper deals with Sturm-Liouville operators generated on a finite interval and on the whole axis by the differential expressionl(y)=−y ^" +q(x)y, whereq(x) is a distribution of first order, such that . The minimal and maximal operators corresponding to potentials of this type on a finite interval are constructed. All self-adjoint extensions of the minimal operator are described and the asymptotics of the eigenvalues of these extensions is found. It is proved that the constructed operator coincides with the norm resolvent limit of the Sturm-Liouville operators generated by smooth potentialsq _n , provided that the condition holds. The convergence of the spectra of these operators to the spectrum of the limit operator is also proved. Similar results are obtained in the case of the whole axis. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 897–912, December, 1999.     

关于多指标变量大数律尾概率级数的完全收敛性

本文对ｉｄ多指标随机变量序列｛Ｘｋ;ｋ∈Ｎｄ｝（ｄ≥２）的部分和Ｓｎ＝∑ｋ≤ｎＸｋ及Ｈ（ｔ）↑＋∞,（ｔ→＋∞）,提出并讨论了Порохоров的３个问题（ｄ≥２）,并讨论了多指标随机变量和的完全收敛性．     

Jacobi matrices with absolutely continuous spectrum

Let be a Jacobi matrix defined in as , where is a unilateral weighted shift with nonzero weights such that Define the seqences: If and , then has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution is also given.

     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.