The Becker-Döring equations at large times and their connection with the LSW theory of coarsening

A systematic procedure is given for obtaining the asymptotic late-time behavior of the Becker-Döring equations describing the time evolution of a population of clusters of particles. In lowest order of approximation, the distribution of the sizes of the largest clusters satisfies the equations of the Lifshitz-Slyozov-Wagner theory of coarsening.     

Asymptotics of the 1/2-dimension,Discrete Airy Equation and its Approximating Differential Equation

A discrete finite difference model is constructed for the Airy equation using a nonstandard scheme formulated by Mickens and Ramadhani. The method of dominant balance is then applied to obtain a first-order difference equation for the solution that increases sufficiently fast as k→∞. We then calculate the corresponding approximating differential equation and obtain its exact solution as well as its “exact” discrete finite difference representation. The application of various symmetry operations allows the determination of the related rapidly decreasing solution and the oscillatory solutions for negative values of x k>=hk, where h=?x.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

Non-standard asymptotics in an inhomogeneous gamma process

Nonhomogeneous Poisson process (NHPP) is a commonly used stochastic model that is utilized to describe the pattern of repeated occurrence of certain events or conditions. Aninhomogeneous gamma process evolves as a generalization to NHPP, where the observed failure epochs correspond to every successive κ-th event of the underlying Poisson process, κ being an unknown parameter to be estimated from the data. This article focuses on a special class of inhomogeneous gamma process, calledmodulated power law process (MPLP) that assumes the Weibull form of the intensity function. The traditional power law process is a popular stochastic formulation of certain empirical relationships between the time to failure and the cumulative number of failures, often observed in industrial experiments. The MPLP retains this underlying physical basis and provides a more flexible modeling environment potentially leading to a better fit to the failure data at hand. In this paper, we investigate inference issues related to MPLP. The maximum likelihood estimators (MLE’s) of the model parameters are not in closed form and enjoy the curious property that they are asymptotically normal with a singular variance-covariance matrix. Consequently, the derivation of the large-sample results requires non-standard modifications of the usual arguments. We also propose a set of simple closed-form estimators that are asymptotically equivalent to the MLE’s. Extensive simulation results are carried out to supplement the theoretical findings. Finally, we implement our inference results to a failure dataset arising from a repairable system.     

Asymptotics for a Population Size Estimator of a Partially Uncatchable Population

We present the asymptotic distribution for an estimator of the population size for the case of s partially catchable populations. Our approach is useful for capture–recapture studies with photo-identification data where part of the population does not have any distinctive characteristic which allows unique identification of the individuals. This work represents an extension of Theorem 4 in Sanathanan (1972, The Annals of Mathematical Statistics, 43, 142–152).     

A unified set of proposals for control and design of high speed data networks

In this paper we articulate our philosophy and approach to the design and control of high speed data networks. The object is to put into perspective and to explain the coordination of various isolated pieces of detailed technical analyses that have been reported in several recent papers. In the process we summarize what we have learnt in our recent work and, also, we give indications of the direction of our future work. Our scheme integrates feedback and open loop control. The feedback control is exercised by sliding windows; access controllers regulate bursty sources. All our design proposals are rooted in asymptotic analyses; the justification for asymptotics comes from the largeness of the parameters, such as propagation delay, speed, window size, buffer size, and the number of virtual circuits. This analysis makes a strong case for operating in a specific moderate usage regime, and adaptive dynamic windowing algorithms are given that make this happen; moreover, when in this regime, buffers may be sized aggressively small without jeopardizing performance and the simplicity of the retransmission protocol. The topics in the paper are: model of communication, results on the steady-state behavior of the basic model, access control, small buffers and retransmission protocols, dynamic adaptive windows, bursty sources, and contrast with previous work.I. Mitrani's work was done while AT&T Bell Laboratories.     

半线性奇摄动反应扩散方程初边值问题解的渐近性(英文)

本文运用边界层以及角点层函数法构造了一类半线性奇摄动反应扩散方程初边值问题解的渐近展开式 ,并用微分不等式方法证明了该展式达到任一精度的一致有效性     

Growth of heat trace and heat content asymptotic coefficients

We show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly. In the real analytic context, however, this is not true and we establish universal bounds on their growth.     

The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as . We investigate the case , ν∈N^?, which encompasses the gravitational fluctuations, ν=4, and the electromagnetic waves, ν=2. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as , and we get global Lp estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L²−L^∞ estimates in suitable weighted Sobolev spaces.