Algebraic Approaches to Stability Analysis of Biological Systems

In this paper, we improve and extend the approach of Wang and Xia for stability analysis of biological systems by making use of Gr?bner bases, (CAD-based) quantifier elimination, and discriminant varieties, as well as the stability criterion of Liénard and Chipart, and showing how to analyze the stability of Hopf bifurcation points. The stability and bifurcations for a class of self-assembling micelle systems with chemical sinks are analyzed in detail. We provide experimental results with comparisons for 15 biological models taken from the literature.      

A Generalization of Algebraic Stability for Runge--Kutta Methods

The theory of algebraic stability does not give detailed informationabout the stability properties of explicit methods. This theoryis generalized by considering a monotonic and bounded test problem.The results give additional information about the stabilityproperties of explicit and implicit methods and show how particularexplicit methods may be selected.     

求解延迟微分代数方程的多步Runge-Kutta方法的渐近稳定性

延迟微分代数方程(DDAEs)广泛出现于科学与工程应用领域．本文将多步Runge-Kutta方法应用于求解线性常系数延迟微分代数方程，讨论了该方法的渐近稳定性．数值试验表明该方法对求解DDAEs是有效的．     

Algebraic and Discrete Velte Decompositions

Starting from an algebraic equivalent of the Velte decomposition of the function space (H₀ ¹) ^d (where d = 2, 3) into three orthogonal subspaces, we consider finite difference and finite element classes for the approximate solution of the first-kind Stokes problem in two and three dimensions which admit discrete Velte decompositions.This revised version was published online in October 2005 with corrections to the Cover Date.     

奇异离散线性系统的开关切换控制

In this paper, we study the stability of discrete linear singular systems by switching controller. Using some recent results on multiple-Lyapunov function technique, we obtain two sufficient conditions of linear singular systems.     

Stability Analysis for Discrete Linear Multipass Processes with Non-Unit Memory

A new set of stability conditions for discrete linear multipassprocesses with non-unit memory is derived and used to developcomputationally feasible stability tests. This results in asystematic procedure which tests these conditions in a particularorder, with termination if the one just tested does not hold.The use of this procedure within the area of control systemsdesign is discussed.     

Wegner Estimate for Discrete Alloy-type Models

We study discrete alloy-type random Schrödinger operators on ${\ell^2(\mathbb{Z}^d)}We study discrete alloy-type random Schr?dinger operators on l²(\mathbbZ^d){\ell^2(\mathbb{Z}^d)} . Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.     

Qualitative Analysis of One-Step Algebraic Models with Tridiagonal Toeplitz Matrices

One-step algebraic models with symmetric and positive definite (SPD) tridiagonal Toeplitz matrices are introduced and their qualitative properties are studied. Theoretical results are applied to the numerical solution of parabolic differential equations, with illustrations by numerical examples. Possible extensions as well as arising open problems are discussed in concluding remarks.     

On Discrete Zariski-Dense Subgroups of Algebraic Groups

We investigate under which conditions an algebraic group G defined over a locally compact field k admitr a subgroup Γ? G(k) which is dense in the Zariski topology, but discerte in the topology induced by the locally compact topology on k. For non—solvable groups we provide a complete answer.     

New Multivalue Methods for Differential Algebraic Equations

Multivalue methods are slightly different from the general linear methods John Butcher proposed over 30 years ago. Multivalue methods capable of solving differential algebraic equations have not been developed. In this paper, we have constructed three new multivalue methods for solving DAEs of index 1, 2 or 3, which include multistep methods and multistage methods as special cases. The concept of stiff accuracy will be introduced and convergence results will be given based on the stage order of the methods. These new methods have the diagonal implicit property and thus are cheap to implement and will have order 2 or more for both the differential and algebraic components. We have implemented these methods with fixed step size and they are shown to be very successful on a variety of problems. Some numerical experiments with these methods are presented.     

Using Discrete and Continuous Models to Solve Nanoporous Flow Optimization Problems

We consider using a discrete network model in combination with continuous nonlinear optimization models to solve the problem of optimizing channels in nanoporous materials. The problem and the hierarchical optimization algorithm are described in [2]. A key feature of the model is the fact that we use the edges of the finite element grid as the locations of the channels. The focus here is on the use of the discrete model within that algorithm. We develop several approximations to the relevant flow and a greedy algorithm for quickly generating a "good" tree connecting all of the nodes in the finite-element mesh to a designated root node. We also consider Metropolis-Hastings (MH) improvements to the greedy result. We consider both a regular triangulation and a Delaunay triangulation of the region, and present some numerical results.     

Algebraic Criteria of Absolute Stability

New algebraic criteria of absolute stability are obtained on the basis of a polynomial form of the frequency condition for absolute stability (Popov inequality). The analysis of the inequality is reduced to an analysis of the local properties of polynomials and a rational function on the negative real semiaxis. As a result, the stability criteria do not require the plotting of any sort of curves and can be expressed on the basis of algorithms of the Routh–Hurwitz type and calculations of the negative real roots of polynomials.     

单支方法关于多延迟Pantograph方程的稳定性分析

本文主要讨论了单支方法关于多延迟pantograph方程解的存在唯一性及渐近稳定性 ,同时将非线性多延迟系统数值解的有关结论推广到多延迟pantograph方程 ,证明了单支方法对于ODEs的A(α) 稳定等价于单支方法关于多延迟pantograph方程NGPk(α) 稳定 .     

Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems

In this paper we are interested in the existence of infinitely many solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, we determine unbounded intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. Finally, the attained solutions are positive when the nonlinearity is supposed to be nonnegative thanks to a discrete maximum principle.     

Using Statistical Models for Mathematical Programming Sensitivity Analysis

When exploring the potential payoffs from energy R & D, it is necessary to perform benefit-cost calculations over a wide range of future scenarios. Repeated runs of large-scale mathematical programming models are often required. This paper describes how simple statistical models might be used to eliminate at least some of the costly computer runs. The approach is illustrated in the context of a recent study on the effect of breeder timing on electricity generating costs.     

Using Discrete Distributions to Approximate General Service Time Distributions in Queueing Models

In solving discrete time queueing models by numerical techniques, the computational requirements (computer memory and time) are a practical limitation and are particularly dependent on the number of discrete time intervals required in the discrete distribution chosen to match the general service distribution. This paper shows that the minimum number of points required for matching to the first two moments depends on the size of the discrete interval relative to the mean and also on the coefficient of variation. Equations and graphs are provided that will enable the OR practitioner to select the discrete distribution to be used as an approximation. Additionally, it is concluded that discrete time modelling, using these approximations to model service time, now provides a practical means to model both steady-state measures and transient behaviour of M/G/c, M(t)/G/c and M(t)/G/c(t) queueing systems on a personal computer.     

以代数方法探讨结构系统再设计问题

利用矩阵修改理论探讨结构系统再设计问题,以等惯性转换求解动态劲度矩阵的隐根,并导出将特征值定位的计算方法;继而在隐根为已知下探讨隐向量的特质及解法,并确认修改后结构的振型必须区分成驻留性与非驻留性自然频率等两种状况处理.