Bounds on the solution of the time-varying linear matrix differential equation P(t) = AH(t) P(t) + P(t) A(t) + Q(t)

^** Email: pagilla{at}ceat.okstate.edu We derive upper and lower bounds for the trace of the solutionof the time-varying linear matrix differential equation (t)= A^H(t) P(t) + P(t) A(t) + Q(t). A practical numerical exampleis given to verify the bounds. The bounds obtained are usefulsince the considered equation is encountered in a number ofapplications in systems and control theory.     

Using History to teach mathematics

The questions involved in the connection of the History of Mathematics with its Pedagory were addressed at two recent conferences, one in Seville (Spain), the other in Braga (Portugal). This volume collects together all the relevant papers presented at these two venues. This review provides brief accounts of their contents.     

Holomorphic solutions to functional differential equations

The pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics, biology, and problems arising in industry. This equation is characterized by a linear functional argument. Heard (1973) [10] considered a generalization of this equation that included a nonlinear functional argument. His work focussed on the asymptotic behaviour of solutions for a real variable x as x→∞. In this paper, we revisit Heard's equation, but study it in the complex plane. Using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary. We consider solutions that are holomorphic on the Julia set of the nonlinear argument. We show that the solutions are either constant or have a singularity at the origin. There is a special case of Heard's equation that includes only the derivative and the functional term. For this case we construct solutions to the equation and illustrate the general results using classical complex analysis.     

STABILITY OF SINGULAR UNCERTAIN DIFFERENTIAL SYSTEMS WITH MULTIPLE TIME-VARYING DELAYS

The stability of singular uncertain differential systems with multiple time-varying delays is discussed, and many results on this are obtained.     

Delay-independent stability of Euler method for nonlinear one-dimensional diffusion equation with constant delay

This paper is concerned with delay-independent asymptotic stability of a numerical process that arises after discretization of a nonlinear one-dimensional diffusion equation with a constant delay by the Euler method. Explicit sufficient and necessary conditions for the Euler method to be asymptotically stable for all delays are derived. An additional restriction on spatial stepsize is required to preserve the asymptotic stability due to the presence of the delay. A numerical experiment is implemented to confirm the results.      

Stability,stabilization and duality for linear time-varying systems

In this article, we study stability properties of linear continuous time-varying systems. Based on a time-varying version of the Lyapunov stability theorem, we obtain stabilizability, stability and duality properties of associated systems.     

Using the Hill cipher to teach cryptographic principles

The Hill cipher is the simplest example of a block cipher, which takes a block of plaintext as input, and returns a block of ciphertext as output. Although it is insecure by modern standards, its simplicity means that it is well suited for the teaching of such concepts as encryption modes, and properties of cryptographic hash functions. Although these topics are central to modern cryptography, it is hard to find good simple examples of their use. The conceptual and computational simplicity of the Hill cipher means that students can experiment with these topics, see them in action, and obtain a better understanding that would be possible from a theoretical discussion alone. In this article, we define the Hill cipher and demonstrate its use with different modes of encryption, and also show how cryptographic hash functions can be both designed and broken. Finally, we look at some pedagogical considerations.     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.     

From an iteration formula to Poincaré's Isochronous Center Theorem for holomorphic vector fields

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincaré, for the condition under which a singularity of a holomorphic vector field is an isochronous center.

     

Using video to teach future teachers to learn from teaching

Video is commonly used in teacher preparation programs. Teacher educators use video for various purposes. In this study, we describe the Learning to Learn from Mathematics Teaching project. In this project, video is used to develop pre-service teachers’ (PSTs) orientations, knowledge and skills for analyzing and reflecting on mathematics teaching in ways that generate knowledge for improvement. We discuss the ways we have used video in a course aimed at developing elementary PSTs’ abilities to learn from teaching. In addition, we report on a study that investigated PSTs’ changes in lesson analysis abilities as a result of participating in the course.     

常系数非齐次线性微分方程组的初等解法

利用初等变换将常系数非齐次线性微分方程组化为由若干个相互独立的高阶常系数非齐次线性微分方程组成的方程组,再利用高阶常系数齐次线性微分方程的特征根法和非齐次方程的待定系数法求该方程组的基本解组及特解,最后通过初等变换求原方程组的基本解组及特解,从而可求出其通解.     

A classification of one-dimensional affine rank three graphs

The rank three subgroups of a one-dimensional affine group over a finite field were classified in 1978 by Foulser and Kallaher. Although one can use their results for a classification of corresponding rank three graphs, the author did not find such a classification in a literature. The goal of this note is to present such a classification. It turned out that graph classification is much simpler than the group one. More precisely, it is shown that the graphs in the title are either the Paley graphs or one of the graphs constructed by Van Lint and Schrijver or by Peisert. Our approach is based on elementary group theory and does not use the classification of rank three affine groups.     

分数阶时变广义系统稳定性分析

基于Caputo分数阶导数,研究了分数阶时变广义线性系统和分数阶时变广义非线性系统的稳定性问题.首先利用相关不等式,给出了一个时变广义线性系统无脉冲且稳定的充分条件.然后,通过慢子系统来判断快子系统的变化,并利用Riccati方程,建立了分数阶时变广义非线性系统是渐近稳定的判定准则.最后,给出了算例和Simulink仿真结果,以说明结论的正确性.     

时滞速度反馈对强迫自持系统动力学行为的影响

研究强迫自持振动系统因时滞反馈产生的主共振解及其分岔．通过对强迫非自治系统的时滞反馈控制,得到所要研究的数学模型．讨论对应的线性化系统使平凡平衡态失稳出现周期解的稳定性临界条件．特别关注主共振及分岔．结果表明,稳定的主共振解随着时滞的变化周期性地出现在系统中．同时,也给出了不稳定的主共振关于时滞变化的区域,在理论方面给出了系统出现概周期运动的时滞区域．数据模拟证实了理论结果．     

Using dynamic graphs to reveal student reasoning

Using dynamic graphs, future secondary mathematics teachers were able to represent and communicate their understanding of a brief mathematical investigation in a way that a symbolic proof of the problem could not. Four different student work samples are discussed.     

Using a virtual population to authentically teach epidemiology and biostatistics

Epidemiology is the study of the distribution of disease in human populations. This means that authentically teaching primary data collection in epidemiology is difficult as students cannot easily access suitable human populations. Using an action research methodology, this paper studied the use of a virtual human population (called The Island) to enable students to experience many features of authentic primary data collection in epidemiological research. The Island was used in a course introducing epidemiology and biostatistics for students in non-quantitative disciplines. This paper discusses how The Island was introduced into the course, and then evaluates the change. Students were highly engaged, and students and teaching staff responded favourably to the use of The Island, with 70% of students agreeing or strongly agreeing that The Island was easy to use, and 64% agreeing or strongly agreeing that the use of a virtual population was beneficial to their understanding of epidemiology.     

On Differential Equations with Retarded Argument

We study the limit properties of solutions to one class of systems of differential equations as the number of equations and some parameters tend to infinity. We establish a close connection between solutions to these systems and differential equations with retarded argument. Our convergence theorems provide a new method for approximation of solutions to nonlinear differential equations with retarded argument.Original Russian Text Copyright © 2005 Demidenko G. V. and Likhoshvai V. A.The authors were supported by the Russian Foundation for Basic Research (Grants 03-01-00328, 03-04-48506, 03-04-48829, 03-07-96833, 04-01-00458, 05-04-49068, and 05-07-90274), the Integration Grant of the Siberian Branch of the Russian Academy of Sciences (No. 119 and 148), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8273), and NSF:FIBR (Grant EF-0330786).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 538–552, May–June, 2005.