A review ofL=λW and extensions

A fundamental principle of queueing theory isL=W (Little's law), which states that the time-average or expected time-stationary number of customers in a system is equal to the product of the arrival rate and the customer-average or expected customer-stationary time each customer spends in the system. This principle is now well known and frequently applied. However, in recent years there have been extensions, such as H=G and the continuous, distributional, ordinal and central-limit-theorem versions, which show that theL=W relation, when viewed properly, has much more power than was previously realized. Moreover, connections have been established between H=G and other fundamental relations, such as the rate conservation law and PASTA (Poisson arrivals see time averages), which show that there is a much greater unity in the overall theory than was previously realized. This paper provides a review.This paper is dedicated to the memory of our colleague Professor Peter Franken (1937–1989), who contributed greatly to the subject of this paper and to queueing theory more generally.     

Rates of convergence of α-stable random motions

The rate-of-convergence problem in the central limit theorem is considered for -stable distributions on the noncommutative group of motions in ^d. The method used here is based on the theory of probability metrics.     

Interpretation of CALUX results in view of the EU maximal TEQ level in milk

Analyses of dioxins in food have become increasingly important since the European Commission has enforced maximal toxic equivalent concentration (TEQ) levels in various food and feed products. Screening methodologies are usually used to exempt those samples that are below the maximum permitted limit and that can, therefore, be released to the market. In addition, one needs to select those samples that require confirmation of their dioxin TEQ level. When bioassays are used as screening tools, the interpretation of the obtained results should consider the higher variability and uncertainty associated with them. This paper explores the use of CALUX data as quantitative screening results. The validation of the method for the polychlorinated dibenzo-p-dioxins (PCDD)/F TEQ determination in milk samples is described with emphasis on the decision limit (CC) and the precision of the method. The decision limit amounts to 4.53 pg TEQ/g fat. Repeatability and within-lab reproducibility coefficients of variation are below 30%. The newly introduced parameter CC^* of 1.47 pg TEQ/g fat delimits with CC a range of suspicious results. These data are not significantly different from the maximum limit of 3 pg TEQ/g fat and should be confirmed by a confirmatory analytical method such as HRGC–HRMS.     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

On the ν-dimensional one-component classical plasma: The thermodynamic limit problem revisited

We prove theH-stability property and the existence of the thermodynamic limit of the free energy density of the two-dimensional, one-component classical plasma. We give lower and upper bounds on the free energy density in any dimensionv and draw some consequences.     

A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation

We consider a shallow water equation of the Camassa–Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L ^p setting.     

On statistical properties of the lyapunov exponent of the generalized skew tent map

The statistical properties of the Lyapunov exponent of the chaotic generalized skew tent map is studied. Expressions of the mean and the variance of this Lyapunov exponent at each discrete time index are obtained. A sufficient condition for weakly mixing of the chaotic generalized skew tent map is derived, and the asymptotic distribution of its Lyapunov exponent is provided.     

A BGK‐penalization‐based asymptotic‐preserving scheme for the multispecies Boltzmann equation

An asymptotic‐preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK‐penalization‐based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625–7648), to its multispecies counterpart. For the multispecies Boltzmann equation, the new difficulties arise due to: (1) the breaking down of the conservation laws for each species and (2) different convergence rates to equilibria for different species in disparate masses systems. To resolve these issues, we find a suitable penalty function—the local Maxwellian that is based on the mean velocity and mean temperature and justify various asymptotic properties of this method. This AP scheme does not contain any nonlinear nonlocal implicit solver, yet it can capture the fluid dynamic limit with time step and mesh size independent of the Knudsen number. Numerical examples demonstrate the correct asymptotic‐behavior of the scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

钟摆模型极限环的存在性研究

钟摆系统是一类典型的分段光滑系统,结合Filippov系统刻画语言,解释了当钟摆无能量补充时,钟摆最终会停止在滑动集上的原因.利用数值模拟的方法,给出钟摆系统在有能量补充时,存在极限环的条件.最后,结合环域定理证明了一般的钟摆模型存在唯一稳定的极限环.