Discrete Risk Model Revisited

In this paper, the fully discrete risk model is considered. Claim sizes are assumed to be integer-valued. A new method is employed to derive some explicit formulas of the Gerber-Shiu penalty function. Characteristic equations corresponding to recursive equations satisfied by Gerber-Shiu penalty function are analyzed and explicit expressions of the penalty function are then obtained. As a special case, the probability of ruin is obtained. National Natural Science Foundation of China(10571092,10271062)     

Archimedes and Pi - Revisited

Archimedes of Syracuse (287-212 B.C.) created the first known recursion process for estimating the value of π. The “classical method” of Archimedes is developed and programs for the TI-81 calculator are given for finding the upper and lower bounds. This classical method is compared with the Gregory-Machin infinite series method (after 1706). A Tl-81 program is given for this method.     

Vesicle Model with Bending Energy Revisited

The equations governing the conditions of mechanical equilibrium in fluid membranes subject to bending are revisited thanks to the principle of virtual work. The note proposes systematic tools to obtain the shape equation and the line condition instead of Christoffel symbols and the complex calculations they entail. The method seems adequate to investigate all problems involving surface energies.     

Blowup of solutions of the unsteady Prandtl's equation

We prove that for certain classes of compactly supported C^∞ initial data, smooth solutions of the unsteady Prandtl's equation blow up in finite time. © 1997 John Wiley & Sons, Inc.     

Remarks on the numerical treatment of Prandtl's boundary-layer equations

Summary The Crocco Transformation for the boundary-layer equations very successfully used by Oleinik and Nickel for theoretical purposes (existence of 2-D, uniqueness of 2-D and 3-D solutions) will be used here for the construction of numerical procedures. An approximate discretization of the transformed equations leads to a nicely structured system of nonlinear equations that can be devided into small parts to be solved one after the other (in the 3-D case even partially by parallel processing). The Jacobians of these parts are oftenM-matrices such that SOR iterations work. The inverse transformation can numerically be realized very simply.

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Zusammenfassung Die für theoretische Zwecke (Existenz von Lösungen im 2-D-Fall, Eindeutigkeit im 2-D- und 3-D-Fall) von Oleinik und Nickel so erfolgreich eingesetzte Crocco-Transformation der Grenzschichtgleichungen wird in der vorliegenden Arbeit zur Konstruktion numerischer Verfahren benutzt. Bei geeigneter Diskretisierung erhält man angenehm strukturierte Systeme nichtlinearer Gleichungen, die in sukzessiv aufzurufende und unabhängig zu lösende Teil-Systeme zerfallen (im 3-D-Fall können die Systeme teilweise parallel bearbeitet werden). Die Jacobi-Matrizen der Teil-Systeme sind oftmalsM-Matrizen, so daß SOR-Iterationsverfahren konvergieren. Die Rücktransformation erfordert numerisch nur geringen Aufwand.

     

Approximated Solution of a Nonlinear Singular Equation of Prandtl's Type

A nonlinear strongly singular integral equation, which can be reduced to a nonlinear singular integro-differential equation of Prandtl's type, is considered. A collocation method for solution is treated and the convergence of the approximated solution to the unique solution of the nonlinear integral equation is proved.     

Almost all graphs are rigid - Revisited

A graph is called asymmetric if it has the identity mapping as its only automorphism. In [P. Erdõs, A. Rényi, Asymmetric Graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315], P. Erdõs and A. Rényi have proven that almost all graphs are asymmetric. A graph is called rigid if it has the identity mapping as its only endomorphism, which is a stronger property than asymmetry. By adopting the approach of Erdõs and Rényi, it is shown that almost all graphs are rigid. A different proof of that result has already been published in [V. Koubek, V. Rödl, On the Minimum Order of Graphs with Given Semigroup, J. Combin. Theory Ser. B 36 (1984) 135–155] (as well as in [P. Hell, J. Nešetřil, Graphs and Homomorphisms, Oxford U. Press, Oxford, 2004]).     

A Shell Model for Optimal Mixing

What is the maximum mixing efficiency of an incompressible flow? To address this question we introduce a shell model—a reduced model mimicking the kinematics of advection and diffusion—to study the evolution of an initially inhomogeneous tracer concentration carried by a given incompressible fluid on a periodic spatial domain. We pose the mixing task as an optimization problem: Find the divergence-free velocity field (the control variable) that produces a well-mixed tracer concentration field (the state variable). We consider two alternative objectives: local-in-time optimization (maximize the instantaneous mixing rate) and global-in-time optimization (maximize mixing at a prescribed final time). Throughout, we use a shell-model analog of the \(H^{-1}\) mix-norm to measure mixing. In addition, lower bounds on the mix-norm are obtained and rule out perfect mixing in finite time in particular cases.     

On the global existence of solutions to the Prandtl's system

In this paper we establish a global existence of weak solutions to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik (J. Appl. Math. Mech. 30 (1966) 951) provided that the pressure is favourable. This generalizes the local well-posedness results due to Oleinik (1966; Mathematical Models in Boundary Layer Theory, Chapman & Hall, London, 1999). For the proof, we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations.     

Mixing Model Based on Parameterized Scalar Profiles

The composition fields in turbulent reacting flows are affected by turbulent transport (macromixing), molecular diffusion (micromixing), and chemical reactions. In the joint velocity-composition probability density function transport equation the highly nonlinear macromixing and chemical reaction terms appear in closed form. This is a considerable advantage over second moment closure methods. Micromixing on the other hand requires modelling and especially for turbulent combustion accurate mixing models are crucial. In this paper we present an approach to model the mixing of scalars, e.g. species mass fractions or temperature, based on considering one-dimensional parameterized scalar profiles (PSP). (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Information Criteria in Model Selection for Mixing Processes

We present information criteria for statistical model evaluation problems for stochastic processes. The emphasis is put on the use of the asymptotic expansion of the distribution of an estimator based on the conditional Kullback–Leibler divergence for stochastic processes. Asymptotic properties of information criteria and their improvement are discussed. An application to a diffusion process is presented.     

Turbulence Model Predictions for the Inhomogeneous Mixing Layer

The plane mixing layer between two streams of different densities and velocities is studied using a complete, phenomenological model of turbulence. The model is an extension of that described by Saffman [11]. The properties of similarity solutions are computed, and compared with experimental data. The qualitative dependence on velocity ratio, density ratio, compressibility and adverse pressure gradient is predicted correctly. The quantitative agreement is satisfactory.     

Parameterized Scalar Profile Mixing Model for Turbulent Combustion Simulations

The composition fields in turbulent reacting flows are affected by turbulent transport (macromixing), molecular diffusion (micromixing), and chemical reactions. In the joint velocity-composition probability density function transport equation the highly non-linear macromixing and chemical reaction terms appear in closed form. This is a considerable advantage over moment closure methods. Micromixing on the other hand requires modeling and especially for turbulent combustion accurate mixing models are crucial. Our approach to model the mixing of scalars, e.g. species mass fractions or temperature, is based on considering one-dimensional parameterized scalar profiles (PSP). Here, an extension of the PSP mixing model to inhomogeneous flows is presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Strong Mixing Data-Based Semiparametric Smooth Transition Regression Model

In this paper, we generalize the semiparametric smooth transition regression model proposed by Wang (2012a), to adapt for the strictly stationary strong mixing data and strong mixing data with deterministic trends. The unknown bounded smooth function embedded in the smooth transition function is estimated by series estimator, the consistency and asymptotic normality properties of estimators are proved employing nonlinear least square regression theory and series estimator approach. Variance matrix estimation and hypothesis testing problems are also discussed based on estimated standard errors. The new model is then used to study the annually inflation rates of China.     

ρ混合、φ混合、ψ混合线性模型M估计的强相合性

研究了 ρ混合、φ混合、ψ混合样本线性模型中回归参数M估计的强相合性 ,在条件不变的情况下 ,获得与独立情形一样的M估计是强相合的充分条件 ,推广了文 [1 ]定理 2 .     

A Multilevel Event History Model of Social Diffusion: Medical Innovation Revisited

This article presents a multilevel event history model of social diffusion and applies it to Coleman, Katz, and Menzel's (1966 Coleman , J. S. , Katz , E. , &; Menzel , H. ( 1966 ). Medical Innovation: A Diffusion Study . Indianapolis , IN : Bobbs-Merrill . [Google Scholar]) data on the adoption of tetracycline by physicians. The simplest form of a multilevel model allows a random intercept. In the present application of this simple model to the Medical Innovation data, structured for an event history analysis, the physicians are nested in city and time. Random intercepts capture effects of contextual conditions that are shared by event history cases with the same city–time status. The intercepts also reflect any baseline internal contagion effects, that is, the proportion of physicians in the city–time network who have adopted the drug at time t ? 1. Here, I show that Van den Bulte and Lilien's (2001 Van den Bulte , C. &; Lilien , G. L. ( 2001 ). Medical innovation revisited: social contagion versus marketing effort . American Journal of Sociology , 106 , 1409 – 1435 .[Crossref], [Web of Science ®] , [Google Scholar]) finding of an important contextual effect of drug firms' marketing effort is misleading. I also show that the social network in which physicians are situated significantly contributes to their adoptions, controlling for baseline internal contagion effects and individual-level characteristics of physicians, which have been emphasized in investigations of these data.     

Archimedes Revisited

Archimedes’ beautiful principle may be used to determine how an object floats in a vessel of liquid. This, in turn may be used to determine the global configuration associated with a floating object under the assumption that the surface of the liquid is planar. There are instances, however, in which capillary effects conspire to produce significantly nonplanar liquid interfaces and phenomena which is strikingly at odds with the predictions of the Archimedean principle. A most notable instance involves floatation of convex objects with density greater than that of the liquid. The effects of capillarity on liquid interface configurations invariably manifest themselves globally over the entire system, though the magnitude of such effects is often so small as to be overlooked. This phenomenon presents a major difficulty in obtaining an appropriate generalization of Archimedes’ principle which incorporates capillarity.     

An Urn Model for Population Mixing and the Phases Within

We introduce a generalization of the Ehrenfest urn model in which samples of independent identically distributed sizes with a general generating discrete distribution (the generator) are taken out of an urn containing white and blue balls (n in total). Each ball in the sample is repainted with the opposite color and the sample is replaced in the urn. We study the phases in the gradual change from the initial condition to the steady state for numerous cases where such a steady state exists. We look at the status of the urn after a number of draws. We identify a concept of linearity based on a combination of the generator and the number of draws, below which we consider the case to be sublinear and above which the case is superlinear. In a properly defined upper subphase of the sublinear phase the number of white balls is asymptotically normally distributed, with parameters that are influenced by the initial conditions and the generator. In the linear phase a different normal distribution applies, in which the influence of the initial conditions and the generator are attenuated. At the superlinear phase the mix is nearly perfect, with a nearly symmetrical normal distribution in which the effect of the initial conditions and the generator is obliterated. We give interpretations for how the results in different phases conjoin at the “seam lines.” The Gaussian results are obtained via martingale theory. We give a few concrete examples.     

基于强混合数据的半参数平滑转换回归模型

本文基于严平稳强混合数据和带确定性趋势的强混合数据序列,推广了文献[20]中提出的半参数平滑转换回归模型。对含于平滑转换函数中的未知光滑有界函数应用级数估计方法,并基于非线性最小二乘估计和级数估计理论证明了模型参数估计量的相合性和渐近正态性等大样本性质,简要讨论了其协方差矩阵的估计以及假设检验问题。最后,应用该模型重新研究了我国年度通货膨胀率的平滑转换结构。