多维区域中非线性偏微分方程的修正Laguerre谱与拟谱方法

研究多维区域中非线性偏微分方程的谱与拟谱方法．建立了修正Laguerre正交逼近与插值结果，这些结果对于建立和分析无界区域中的数值方法起着重要的作用．作为结果的一个应用，研究了二维无界区域中的Logistic方程的修正Laguerre谱格式，证明了它的稳定性和收敛性．数值试验结果表明所提出方法具有很高的精度，与理论分析结果完全吻合．     

非线性差分方程的多重周期解

用变分方法得到一类非线性差分方程多重周期解的存在性．我们的结果推广了Cai，Yu和Guo[Comput．Math．Appl．，52（2006），1630-1647]的结果，并且这里给出的证明显著地简化了．     

基于分散型和扩散型随机序对几种寿命分布类的刻画

本文基于分散型和(或)扩散型随机序，利用随机变量的剩余寿命给出了寿命分布类ILR，IFR，DMRL和IFR(2)及其对偶类的刻画．作为主要结果的一个应用，用k／n-系统的剩余寿命对IFR和DMRLS及其对偶类进行了刻画．这些结果拓宽和加强了文献中已有的结果．     

高阶退化Bernoulli数和多项式

本文研究了高阶退化Berrioulli数和多项式的两个显明公式，得到了一个包含高阶Bemoulli数和Stirling数的恒等式，并推广了F．H．Howard，S．Shirai和K．I．Sato的结果。     

Krawtchouk多项式零点的渐近展开及其误差限

Krawtchouk多项式在现代物理学中有着广泛应用．基于Li和Wong的结果，利用Airy函数改进了Krawtchouk多项式的渐近展开式，而且得到了一个一致有效的渐近展开式．进一步，利用A的函数零点的性质，推导出了Krawtchouk多项式零点的渐近展开式，并讨论了其相应的误差限．同时还给出了Krawtchouk多项式和其零点的渐近性态，它优于Li和Wong的结果．     

空间同宿环和异宿环的稳定性

关于平面同（异）宿环的稳定性已有不少文献讨论过，但关于空间同（异）宿环的稳定性尚没有任何结果．本文在可定义回复映射的条件下给出了同（异）宿环在其部分邻域中是渐近稳定的判据．这些结果在某种意义下是平面系统相应结果的推广，包括并推广了［２］，［３］的结果．本文最后讨论了Ｌｏｒｅｎｚ系统同宿环和三种群竞争系统异宿环的稳定性，所得结果和Ｓｐａｒｒｏｗ与Ｍａｙ等的数值结果相吻合．     

微极各向同性弹性板中的Rayleigh-Lamb波

研究了无应力作用条件下，均匀、各向同性、圆柱形微极结构弹性板中波的传播．导出了对称和斜对称模式下波传播的特征方程．对短波这一极端情况，无应力圆板中对称和斜对称模态波的特征方程退化为Pmyle曲表面波频率方程．并得到薄板的计算结果．给出了位移和微转动分量，并绘制了相应图形．给出了若干特殊情况的研究结果及对称和斜对称模态特征方程的图示．     

局部对称流形中具常平均曲率的完备超曲面

本文研究了局部对称流形中具常平均曲率的完备超曲面，得到了这种超曲面的一个特征定理，推广了H．Alencar和M．do Carmo以及N．X．Shui和G．Q．Wu的结果。     

一致凸Banach空间中渐近非扩张映像的Ishikawa迭代格式之强收敛定理

通过使用新的分析技巧，建立了渐近非扩张映像的Ishikawa迭代格式之强收敛定理．所得结果改进了Schu，Rhoades，Liu和Xue以及其他作者的相关结果．     

关于Nielson Zeta函数的有理性

Ａ．Ｌ．Ｆｅｌ＇ｓｈｔｙｎ和Ｖ．Ｂ．Ｐｉｌｙｕｇｉｎａ定义了Ｎｉｅｌｓｏｎｚｅｔａ函数并证明关于其有理性的一些结果，本文改进了他们的一些结果。     

A new approach to Kac's theorem on representations of valued quivers

By using the Ringel-Hall algebra approach we first present a new proof of Kac's theorem for species over a finite field. The method used here is quite different from earlier techniques. In [SV] Sevenhant and Van den Bergh have discovered an interesting relation between a conjecture of Kac on representations of quivers and the structure of the Ringel-Hall algebras in some special cases. We second show that this relation still holds for species. The research was supported in part by the NSF of China and the TRAPOYP.     

Effects of pulsatile flow and straight length on low-density lipoprotein transport in a curved arterial wall

Abnormal accumulation of macromolecules such as low-density lipoproteins (LDLs) in the arterial wall causes narrowing and blockage of vessels, which leads to atherosclerosis. Effects of pulsatile nature of blood flows as well as the initial length on transport of the LDL species in the arterial boundary layer region are analyzed numerically in the present work. The set of governing equations consisting of continuity, Navier-Stokes, and species transport is solved using a projection method based on the second-order central difference discretization. The obtained results are in excellent agreement with the pertinent data. The computational results imply that the flow field and concentration distribution are time dependent but the variation of the filtration velocity can be ignored. The LDL concentration boundary layer thickness decreases in the outer part and increases in the inner part for both with or without straight length. Presence of initial straight length generates about 26% growth in the boundary layer thickness, although its effect on the LDL surface concentration (LSC) is negligible. The maximum LSC is related to the regions with minimum wall shear stress in the inner part of the curved artery, which have more potential for formation of atherosclerosis. A new numerical correlation between the LSC and boundary layer thickness is proposed and examined.     

On the effects of migration and spatial heterogeneity on single and multiple species

We first investigate in a logistic model the effects of migration and spatial heterogeneity of the environment on the total population size at equilibrium of a single species. Our study shows that (i) the total population size is maximized at some intermediate migration rate, and hence is a non-monotone function of the migration rate; (ii) heterogeneity of the environment increases the population size. In the second part of this paper, these findings are applied to ecological invasions. For a two-species Lotka-Volterra competition model with migration, we show that (i) without migration, the invading species eliminates the resident species at every point of the habitat, whereas when migration is present, for certain ranges of migration rates the invader may be eliminated when it is rare; and (ii) without migration, the two species can coexist at every point of the habitat, whereas when migration is present, for some ranges of migration rates one of the species is extinguished for all initial conditions.     

Electrolysis and isoelectric focusing

This paper consists of two parts. In the first part, the authorsprove the existence of steady-state solutions for a three-specieselectrolyte. The species are subject to both dissociation-associationreactions inside the electrolyte and electrochemical reactionsat the boundary electrodes. This is a common occurrence in electrolysis.In the second part, the authors investigate how to use thismodel to describe isoelectric focusing, which is a common techniqueused to separate large protein molecules. In particular, theisoelectric focusing point for a particular type of proteinmolecule is calculated using formal perturbation analysis.     

Permanence under strong aggressions is possible

This paper analyzes the limiting behavior of the positive solutions of a general class of sublinear elliptic weighted mixed boundary value problems as the amplitude of the positive part of the lower order terms of the differential operator blows up to infinity. The main result establishes that the positive solutions approximate zero within the support of the positive part of the potential, whereas they stabilize to the positive solution of a certain elliptic mixed boundary value problem on its complement. Further, we use this result for deriving some general principles in competing species dynamics. Precisely, we shall show that in the presence of a refuge region two competing species must coexist if their reproduction rates are sufficiently large, independently of the strength of the competition. It should be emphasized that the abstract theory developed here allows measuring how large the reproduction rates should be for being permanent, providing us, simultaneously, with the limiting behavior of each of the species separately. Basically, when the pressure from the competitor grows the tackled species concentrates within its refuge. The results of this paper are substantial extensions of some pioneer results found by one of the authors in [16, Section 4]. The main ingredients in deriving the main results of this paper are the continuous dependence of the principal eigenvalue with respect to a general class of perturbations of the domain around its Dirichlet boundary – very recent result coming from [6] – and the continuous dependence of the positive solutions of the sublinear problem – coming from [7].     

Quadratic harvesting dominated optimal strategy for a stochastic single-species model

A stochastic population model with the mixed harvesting strategy is formulated and studied in this paper. Sufficient and necessary conditions for survival of the species are derived firstly. Then, based on the ergodic stationary distribution, the optimal strategy is identified. Results show that the linear harvesting effort threatens to the survival of the species; the quadratic harvesting strategy occupies an absolute advantage in the harvesting and excludes the linear part out of the optimal harvesting strategy. It''s interest to see all these occur only in the random environments. Computer simulations are carried out to support the obtained results.     

A model of sex‐based social stability

For some time theoretical biologists have been discussing the origin of sexes, which seems somewhat puzzling as there are intrinsic inefficiencies for a species to produce males and females, rather than hermaphrodites. We propose that part of the explanation of this may be the additional social stability that sexes offer. We use notions from cooperative game theory to show that for hermaphrodites, it may be impossible for a social group to find a stable set of partnerships. The existence of such a stable set of partnerships is, however, guaranteed for species with both males and females. © 2004 Wiley Periodicals, Inc. Complexity 9: 52–56, 2004     

An asymptotic analysis of two-phase fluid mixing

We study the motion of the slightly compressible multi-phase flow model proposed by Chen, Glimm, Sharp and Zhang. The interface velocity and constitutive law are analyzed by derivation of the exact quantity. Using singular perturbation theory, a formal asymptotic expansion is derived for the solution of the compressible equations. An asymptotic analysis in the incompressible limit, for slightly compressible flows supplies important new information to resolve nonuniqueness of the pressure difference between the two fluid species of the incompressible flow equations.Supported in part by the U.S. Department of Energy DE-FG02-90ER25084, DE-FG02-989ER2536, DE-FG03-98DP00206, the National Science Foundation DMS-0102480, the Army Research Office grant DAAD19-01-0642 and Los Alamos National Laboratory.Supported in part by the Clay Mathematics Institute.     

An assessment of two models for the subgrid scale tensor in the rational LES model

In this paper, we develop computational methods for a three-dimensional model of competition for light between phytoplankton species. The competing phytoplankton populations are exposed to both horizontal and vertical mixing. The vertical light-dependence of phytoplankton photosynthesis implies that the three-dimensional model is formulated in terms of integro-partial differential equations that require an efficient numerical solution technique.Due to the stiffness of the discretized system we select an implicit integration method. However, the resulting implicit relations are extremely expensive to solve, caused by the strong coupling of the components. This coupling originates from the three spatial dimensions, the interaction of the various species and the integral term. To reduce the amount of work in the linear algebra part, we use an Approximate Matrix Factorization technique.The performance of the complete algorithm is demonstrated on the basis of two test examples. It turns out that unconditional stability (i.e., A-stability) is a very useful property for this application.     

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.