The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

若干新的p进制Hardy-Littlewood-Pólya型不等式

本文建立了若干新的具最佳常数因子的p进制Hardy-Littlewood-Pólya型不等式，同时也给出了它们的等价形式以及一些特殊结果.     

An algorithm for fitting circular arcs to data using the <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">1</Emphasis></Subscript> norm

There are many applications of fitting circular arcs to data. We have for example, system control, using a computer controlled cutting machine, approximating hulls of boats, drawing and image techniques. Out of these applications comes the least squares norm to be the most commonly used criterion. This paper examines how the l ₁ norm is used which seems to be more appropriate than the use of least squares in the context of wild points in the data. An algorithm and different methods to determine the starting points are developed. However, numerical examples are given to help illustrate these methods.      

A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces

In this paper, we study the existence of solutions and approximation of the solutions by using a Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces. The results presented in this paper generalize the corresponding results of J. Li [J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004) 115-126].     

模糊线性系统的扰动分析

使用谱范数分析了模糊线性系统在三种情形下的扰动: (1)右端模糊向量有扰动, 系数矩阵不变; (2)系数矩阵有扰动,右端模糊向量不变; (3)系数矩阵和右端模糊向量都有扰动,并通过数值实例验证给出的扰动界的估计.     

On the generation of updates for quasi-Newton methods

We present a unified technique for updating approximations to Jacobian or Hessian matrices when any linear structure can be imposed. The updates are derived by variational means, where an operator-weighted Frobenius norm is used, and are finally expressed as solutions of linear equations and/or unconstrained extrema. A certain behavior of the solutions is discussed for certain perturbations of the operator and the constraints. Multiple secant relations are then considered. For the nonsparse case, an explicit family of updates is obtained including Broyden, DFP, and BFGS. For the case where some of the matrix elements are prescribed, explicit solutions are obtained if certain conditions are satisfied. When symmetry is assumed, we show, in addition, the connection with the DFP and BFGS updates.This work was partially supported by a grant from Control Data     

Banach空间中的拟范均值

在算术平均值和几何平均值的基础上,给出了一种新的平均值—拟算数平均值以及拟几何平均值,并将它们引入到Banach空间进行了讨论.给出了一些定义并在其基础上得到了这几类平均值在Banach空间中的几个有用的性质.     

Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.     

Computing the minimal covering set

We present the first polynomial-time algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set–the minimal upward covering set and the minimal downward covering set–unless P equals NP. Finally, we observe a strong relationship between von Neumann–Morgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.     

Strong convergence of iterative sequences for nonexpansive mappings

An iterative algorithm is proposed for finding a fixed point of a nonexpansive self-mapping of a closed convex subset of a Banach space with a uniformly Gâteaux differentiable norm, and an estimate of the convergence speed also is given.