Large deviations of Markov chains with multiple time-scales

For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in Kang and Kurtz (2013) and Kang et al. (2014). We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which exhibit multiple time-scales, and provide explicit calculations for several relevant examples.     

Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry

This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.     

The Stochastic Theory of Fluvial Landsurfaces

A stochastic theory of fluvial landsurfaces is developed for transport-limited erosion, using well-established models for the water and sediment fluxes. The mathematical models and analysis are developed showing that some aspects of landsurface evolution can be described by Markovian stochastic processes. The landsurfaces are described by nondeterministic stochastic processes, characterized by a statistical quantity, the variogram, that exhibits characteristic scalings. Thus the landsurfaces are shown to be self-organized critical (SOC) systems, possessing both an initial transient state and a stationary state, characterized by respectively temporal and spatial scalings. The mathematical theory of SOC systems is developed and used to identify three stochastic processes that shape the surface. The SOC theory of landsurfaces reproduces established numerical results and measurements from digital elevation models (DEMs).     

The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations

Recently two articles studied scalings in biplot models, and concluded that these have little impact on the interpretation. In this article again scalings are studied for generalized biadditive models and correspondence analysis, that is, special cases of the general biplot family, but from a different perspective. The generalized biadditive models, but also correspondence analysis, are often used for Gaussian ordination. In Gaussian ordination one takes a distance perspective for the interpretation of the relationship between a row and a column category. It is shown that scalings—but also nonsingular transformations—have a major impact on this interpretation. So, depending on the perspective one takes, the inner product or distance perspective, scalings and transformations do have (distance) or do not have (inner-product) impact on the interpretation. If one is willing to go along with the assumption of the author that diagrams are in practice often interpreted by a distance rule, the findings in this article influence all biplot models.     

On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

We consider a bipolar kinetic model for charged media. In certain scalings the Debye length or the relaxation time are small. In addition different time scales are considered. These can be used in order to close the corresponding moment equations and leads to a (closed) set of macroscopic equations. We show three different scalings and obtain three completely different sets of macroscopic equations.     

Catalytic Discrete State Branching Models and Related Limit Theorems

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.      

Mixed-hybrid finite elements and streamline computation for the potential flow problem

An important class of problems in mathematical physics involves equations of the form ?? · (A??) = f. In a variety of problems it is desirable to obtain an accurate approximation of the flow quantity u = ?A??. Such an accurate approximation can be determined by the mixed finite element method. In this article the lowest-order mixed method is discussed in detail. The mixed finite element method results in a large system of linear equations with an indefinite coefficient matrix. This drawback can be circumvented by the hybridization technique, which leads to a symmetric positive-definite system. This system can be solved efficiently by the preconditioned conjugate gradient method. After approximating u by the lowest-order mixed finite element method, streamlines and residence times can be determined easily and accurately by computations at the element level.     

湍流运动论中的化学反应速率扰动

在分子与分子之间发生碰撞时,如果分子的动能足够大,当超过一定的阈值E时,分子碰撞就会发生化学反应.然而气体分子化学反应速率是受到一定因素影响的,诸如分子数密度n、温度T以及速度V.日本学者Sagara于上世纪70年代曾对分子数密度n和温度T对反应速率的变化做了深入的探究.主要是在此基础上利用二粒碰撞机理论对速度以及温度和速度的共同作用来研究反应速率的变化情况.     

On the M_t/M_t/K_t+M_t queue in heavy traffic

The focus of this paper is on the asymptotics of large-time numbers of customers in time-periodic Markovian many-server queues with customer abandonment in heavy traffic. Limit theorems are obtained for the periodic number-of-customers processes under the fluid and diffusion scalings. Other results concern limits for general time-dependent queues and for time-homogeneous queues in steady state.     

蒸汽沉淀化学反应方程混合元法的离散格式研究

蒸汽沉淀化学反应过程有着极其广泛的应用,其数学模型归结为一个包含流速场,温度场,压力场和气体溶质场的非线性偏微分方程组．用混合有限元方法研究蒸汽沉淀化学反应方程组,导出其半离散化和全离散化的混合元格式,并证明这些格式的解的存在性和收敛性(误差估计)．用混合元法处理究蒸汽沉淀化学反应方程组,可以同时求出流速场,温度场,压力场和气体溶质场的数值解. 因此该研究既具有重要的理论意义,又具有广泛的应用前景．     

Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Several choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two‐phase medium at the microscopic scale. This system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single‐phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial‐exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two‐scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings. Copyright © 2008 John Wiley & Sons, Ltd.     

Langevin equations for L′-Valued Gaussian processes and fluctuation limits of infinite particle systems

Summary L Gaussian processes of a certain class are shown to satisfy generalized Langevin equations. Examples are fluctuation limits of several infinite particle systems, in particular infinite particle branching Brownian motions with immigration under various scalings and the voter model with hydrodynamic scaling.Partially supported by CONACyT grants PCCBBNA 002042 and 140102 G203-006 (México) and a grant of the NSERC (Canada)     

Neutrino processes in an external magnetic field in the density matrix formalism

In the matrix density formalism for a charged spinor particle located in a constant uniform magnetic field, we develop a technique for calculating the reaction rate and the four-momentum carried away from a plasma by a neutrino in one-vertex neutrino processes. Using this technique, we reproduce results for the luminosity in processes of neutrino synchrotron emission by electrons (positrons) and of electron-positron annihilation producing the neutrino pair.     

Asymptotic analysis for a dynamic piezoelectric shallow shell

The paper deals with the asymptotic formulation and justification of a mechanical model for a dynamic piezoelastic shallow shell in Cartesian coordinates. Starting from the three‐dimensional dynamic piezoelastic problem and by an asymptotic approach, the authors study the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. In order to obtain a nontrivial limit problem by asymptotic analysis, we need different scalings on the mass density. The authors show that the transverse mechanical displacement field coupled with the in‐plane components solves an problem with new piezoelectric characteristics and also investigate the very popular case of cubic crystals and show that, for two‐dimensional shallow shells, the coupling piezoelectric effect disappears. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical solution of two-dimensional reaction-diffusion Brusselator system

In this paper, polynomial based differential quadrature method (DQM) is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system is known as the reaction-diffusion Brusselator system. The system arises in the modeling of certain chemical reaction-diffusion processes. In Brusselator system the reaction terms arise from the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The numerical results reported for three specific problems. Convergence and stability of the method is also examined numerically.     

Auto-ignition of diesel spray using the PDF-Eddy Break-Up model

This work presents the development and implementation of auto-ignition modelling for DI diesel engines by using the PDF-Eddy Break-Up (PDF-EBU) model. The key concept of this approach is to combine the chemical reaction rate dealing with low-temperature mode, and the turbulence reaction rate governing the high-temperature part by a reaction progress variable coupling function which represents the level of reaction. The average reaction rate here is evaluated by a probability density function (PDF) averaging approach. In order to assess the potential of this developed model, the well-known Shell ignition model is chosen to compare in auto-ignition analysis. In comparison, the PDF-EBU ignition model yields the ignition delay time in good agreement with the Shell ignition model prediction. However, the ignition kernel location predicted by the Shell model is slightly nearer injector than that by the PDF-EBU model leading to shorter lift-off length. As a result, the PDF-EBU ignition model developed here are fairly satisfactory in predicting the auto-ignition of diesel engines with the Shell ignition model.     

Applications of periodic unfolding on manifolds

Abstract

We show how the newly developed method of periodic unfolding on Riemannian manifolds can be applied to PDE problems: we consider the homogenization of an elliptic model problem. In the limit, we obtain a generalization of the well-known limit- and cell-problem. By constructing an equivalence relation of atlases, one can show the invariance of the limit problem with respect to this equivalence relation. This implies e.g. that the homogenization limit is independent of change of coordinates or scalings of the reference cell. These type of problems emerge for example when modeling surface diffusion and reactions in heterogeneous catalysts, or in processes involved in crystal formation.     

Convergence and conditioning of a Nyström method for Stokes flow in exterior three-dimensional domains

Convergence and conditioning results are presented for the lowest-order member of a family of Nyström methods for arbitrary, exterior, three-dimensional Stokes flow. The flow problem is formulated in terms of a recently introduced two-parameter, weakly singular boundary integral equation of the second kind. In contrast to methods based on product integration, coordinate transformation and singularity subtraction, the family of Nyström methods considered here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is designed to remove the weak singularity in the integral equation and provide control over the approximation error. Here we focus attention on the lowest-order method of the family, whose implementation is especially simple. We outline a convergence theorem for this method and illustrate it with various numerical examples. Our examples show that well-conditioned, accurate approximations can be obtained with reasonable meshes for a range of different geometries.     

Asymptotic Analysis of the Current-Voltage Curve of a pnpn Semiconductor Device

We consider the one-dimensional steady-state semiconductor deviceequations modelling a pnpn device. There are two relevant scalingsof the equations corresponding to small and large applied voltages.In both scalings, the semiconductor equations can be consideredas singularly perturbed. It turns out that the small-voltagescaling breaks down for current values between two saturationcurrents. In that interval, the large-voltage scaling has tobe employed. For both scalings, we derive the first-order termsof an asymptotic expansion and show that the reduced problemhas a solution. An example verifies that the current-voltagecurves obtained have the expected qualitative structure.     

A Least-Squares Mixed Finite Element for Quasi-Incompressible Elastodynamics

In the present work, a mixed finite element based on a modified least-squares formulation is proposed. Here, we consider the time-dependent equations for quasi-incompressible elastodynamics under small strain assumptions. The main goal is to obtain an accurate approximation of both displacements and stresses in particular for the lowest-order element. Basis for the element formulation is a weak form resulting from a least-squares method. The L₂-norm minimization of the time-discretized residuals of the given first-order system leads to a functional depending on approximations for displacements and stresses. By introducing a time-independent displacement test function, a weak form is derived. A numerical example concerning quasi-incompressible elasticity shows the performance of the approach for the lowest-order element RT₀P₁as. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)