Stability and bifurcations analysis of a competition model with piecewise constant arguments

In this paper, we investigate local and global asymptotic stability of a positive equilibrium point of system of differential equations where t ≥ 0, the parameters r₁, k₁, α₁, α₂, r₂, k₂, and d₁ are positive, and [t] denotes the integer part of t ∈ [0, ∞ ). x(t) and y(t) represent population density for related species. Sufficient conditions are obtained for the local and global stability of the positive equilibrium point of the corresponding difference system. We show through numerical simulations that periodic solutions arise through Neimark–Sacker bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [t]

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences     

一阶非线性具偏差变元脉冲微分方程

给出了具有时滞和时超的一阶非线性脉冲微分方程所有解为振动的充分条件,所得结论包含了线性情形作为其推论。     

The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning

The study aims to explore the structural aspects of generic examples, to get better insight into what makes them potentially opaque for learners. We have analyzed 27 written arguments, for which student teachers (grades 1–10) were asked to use a generic example to prove a given statement in multiplication. Using Toulmin’s framework, we developed five categories of arguments based on their structure: examples, empirical arguments, leap arguments, embedded arguments, and other arguments. Also, we conclude that none of the student teachers provided arguments that we recognize as complete generic examples. The results bring us to a discussion about features of generic examples making them difficult to come to grips with, having implications for how teacher educators can support student teachers’ learning to prove. From this, we propose a definition of generic examples that attends to the criteria suggested in previous research, yet, emphasizing their structural nature.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

On the global attractivity for a logistic equation with piecewise constant arguments

In this paper, we consider the following logistic equation with piecewise constant arguments:

     

Runge–Kutta Methods for Systems of Differential Equation with Piecewise Continuous Arguments: Convergence and Stability

This work deals with the convergence and stability of Runge–Kutta methods for systems of differential equation with piecewise continuous arguments x^′(t) = Px(t)+Qx([t+1∕2]) under two cases for coe?cient matrix. First, when P and Q are complex matrices, the su?cient condition under which the analytic solution is asymptotically stable is given. It is proven that the Runge–Kutta methods are convergent with order p. Moreover, the su?cient condition under which the analytical stability region is contained in the numerical stability region is obtained. Second, when P and Q are commutable Hermitian matrices, using the theory of characteristic, the necessary and su?cient conditions under which the analytic solution and the numerical solution are asymptotically stable are presented, respectively. Furthermore, whether the Runge–Kutta methods preserve the stability of analytic solution are investigated by the theory of Padé approximation and order star. To demonstrate the theoretical results, some numerical experiments are adopted.     

Measure-Valued Differentiation for Markov Chains

This paper addresses the problem of sensitivity analysis for finite-horizon performance measures of general Markov chains. We derive closed-form expressions and associated unbiased gradient estimators for the derivatives of finite products of Markov kernels by measure-valued differentiation (MVD). In the MVD setting, the derivatives of Markov kernels, called -derivatives, are defined with respect to a class of performance functions such that, for any performance measure , the derivative of the integral of g with respect to the one-step transition probability of the Markov chain exists. The MVD approach (i) yields results that can be applied to performance functions out of a predefined class, (ii) allows for a product rule of differentiation, that is, analyzing the derivative of the transition kernel immediately yields finite-horizon results, (iii) provides an operator language approach to the differentiation of Markov chains and (iv) clearly identifies the trade-off between the generality of the performance classes that can be analyzed and the generality of the classes of measures (Markov kernels). The -derivative of a measure can be interpreted in terms of various (unbiased) gradient estimators and the product rule for -differentiation yields a product-rule for various gradient estimators. Part of this work was done while the first author was with EURANDOM, Eindhoven, Netherlands, where he was supported by Deutsche Forschungsgemeinschaft under Grant He3139/1-1. The work of the second author was partially supported by NSERC and FCAR grants of the Government of Canada and Québec.     

伴有边界摄动的三阶拟线性向量微分方程边值问题的奇摄动

本文研究伴有边界摄动的三阶拟线性向量微分方程边值问题的奇摄动,在适当的假设下,利用对角化技巧和不动点原理证明了摄动问题解的存在唯一性并给出解的任意阶的一致有效的渐近展开式和余项的估计。