Solving of second order nonlinear PDE problems by using artificial controls with controlled error

In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region inR ². We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error inL ₁ is controlled.     

Coherent States for the Legendre Oscillator

Two families of coherent states (the eigenstates of the annihilation operator and the Klauder–Gazeau temporally stable coherent states) are constructed for the Legendre oscillator defined in this paper. Bibliography: 14 titles.     

Theory of Linear Dissipative Scattering Systems with Discrete Time and with Space States of the Class π k

An arbitrary operator-valued function from the generalized Shur class is realized as the transfer function of a minimal optimal and minimal *-optimal dissipative scattering system with the Pontryagin state space. These results generalize Arov's results for the case of the Hilbert state space. Bibliography: 11 titles.     

Axially Symmetric Boundary Value Problem Describing the Distribution of Charges in Semiconductors

We prove that bifurcation solutions to the stationary problem exist and can be extended with respect to parameter. The problem under consideration has a solution with the so-called interior transition layer phenomenon. In the nonstationary case, we establish the existence and uniqueness of a solution for any t > 0. Under certain assumptions, the nonstationary problem determines a dynamical system in some compact set. Bibliography: 9 titles.     

Computation of the Morse Spectrum

The Morse spectrum is a limit set of Lyapunov exponents of periodic pseudo-trajectories. This notion is especially important in the case where a dynamical system has infinitely many periodic trajectories of large period. A method for estimating the Morse spectrum was suggested by the first author in J. Math. Anal. Appl., 252 (2000). This method is based on ideas of symbolic dynamics which reduces the study of a dynamical system to the study of a certain graph, called a symbolic image. Within the framework of this method, the computation of the Morse spectrum is connected with searching simple closed paths and extracting contours with suitable characteristics. However, under iterations of the symbolic image, the number of such paths sharply increases, which leads to huge expenses of memory and time. We suggest an algorithm for constructing contours with the maximal and minimal mean values. This algorithm is based on a special version of the simplex method. Numerical tests are also described. Bibliography: 13 titles. Illustrations: 3 figures.     

Discrete mathematics in primary and secondary schools in the United States

This article provides a status report on discrete mathematics in America's schools, including an overview of publications and programs that have had major impact. It discusses why discrete mathematics should be introduced in the schools and the authors' efforts to advocate, facilitate, and support the adoption of discrete mathematics topics in the schools. Their perspective is that discrete mathematics should be viewed not only as a collection of new and interesting mathematical topics, but, more importantly, as a vehicle for providing teachers with a new way to think about traditional mathematical topics and new strategies for engaging their students in the study of mathematics.     

Numerical Modeling of Solitons in a Low Birefringent Optical Fiber

A numerical investigation, based on the split-step Fourier transform algorithm of all optical switching of solitons in a low birefringent optical fiber is presented. The numerical algorithm is described in detail. This revised version was published online in August 2006 with corrections to the Cover Date.