Class Of Step Functions Research Articles

A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations

We consider a variational problem associated with the minimal speed of pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1-u)u$, $x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$ is the so-called minimal speed of pulsating traveling waves. In this paper, we study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$ under some constraints. We prove the existence of the maximizer under a certain assumption of the constraint and derive the Euler--Lagrange equation which the maximizer satisfies under $L^2$ constraint $\int_0^L b(x)^2dx=\beta$. The limit problems of the solution of this Euler--Lagrange equation as $L\rightarrow0$ and as $\beta\rightarrow0$ are also considered. Moreover, we also consider the variational problem in a certain class of step functions under $L^p$ constraint $\int_0^L b(x)^pdx=\beta$ when $L$ or $\beta$ tends to infinity.

On randomization of Halton quasi-random sequences

The problem of estimating the error of quasi-Monte Carlo methods by means of randomization is considered. The well known Koksma–Hlawka inequality enables one to estimate asymptotics for the error, but it is not useful in computational practice, since computation of the quantities occurring in it, the variation of the function and the discrepancy of the sequence, is an extremely timeconsuming and impractical process. For this reason, there were numerous attempts to solve the problem mentioned above by the probability theory methods. A common approach is to shift randomly the points of quasi-random sequence. There are known cases of the practical use of this approach, but theoretically it is scantily studied. In this paper, it is shown that the estimates obtained this way are upper estimates. A connection with the theory of cubature formulas with one random node is established. The case of Halton sequences is considered in detail. The van der Corput transformation of a sequence of natural numbers is studied, and the Halton points are constructed with its help. It is shown that the cubature formula with one free node corresponding to the Halton sequence is exact for some class of step functions. This class is explicitly described. The obtained results enable one to use these sequences more effectively for calculating integrals and finding extrema and can serve as a starting point for further theoretical studies in the field of quasi-random sequences.

Search algorithm of optimum control actions on dynamic object

In the state-space (SS) of nonlinear non-stationary dynamic objects are investigated numerical algorithms of their optimum control with restrictions on phase coordinates. The control actions limit by a class of step functions, as positive and negative impulses generating in the SS binary trees. The dynamic process is interpreted as growth binary tree. In accordance with growth binary tree, it the knots get into various areas of the SS (clusters). The purpose of control is the get, during growth binary tree, one or several knots in a specific cluster (target set). In the present work is considered the new numerical search method of optimum control with adaptation to restrictions of an external environment, called by a method of binary trees.

Solution of the general KdV equation in the class of step functions

In this work, we deduce laws of evolution of the scattering data for the Sturm—Liouville operator with a potential that is a solution of the general Korteweg-de Vries equation and general Korteweg-de Vries equation with a source in the class of step functions. Bibliography: 19 titles.

Solving Optimal Control Problems on the Class of Step Functions

Linear optimal control problems with multipoint non-separated conditions with a quadratic performance criterion of control are analyzed. An approach with a violation of intermediate conditions is proposed. The original problem is reduced to the classical quadratic programming problem with the dimension determined by the number of intermediate points.

Orthogonal Transforms in Bases of Slant Step Functions. I. Constructing Complete Sets of Orthogonal Slant Step Functions

The author introduces a new class of step functions and defines them in terms of the Rademacher functions. Two complete systems of orthogonal slant step functions are constructed and their characteristics are analyzed. The relationship between the basis functions of these systems and the system of the Walsh functions is established and their orthonormality is proved. It is shown that the proposed systems of slant step functions can be efficiently used to code video signal.

An empirical Bayes approach to the estimation of the incidence curve of HIV infection.

A new approach is proposed for estimating the incidence curve of HIV infection and obtaining short term prediction of AIDS incidence. It is based on the method of back calculation which utilizes the fact that AIDS incidence is generated from HIV infection incidence by convolution with the incubation period distribution, but avoids the difficulties associated with approximating the infection incidence curve by a class of step functions. Instead, the infection incidence is modelled as a simple stochastic epidemic process which ensures smoothness of the estimate. We first derive the distribution of AIDS incidence knowing infection incidence. The best linear estimators of infection incidence as well as future AIDS incidence are given; a smoothed past AIDS incidence is also obtained. The parameters of the stochastic infection process can be estimated by maximum likelihood using a normal approximation to the marginal distribution of the AIDS incidence. This approach is applied to AIDS incidence data in the United States up to mid 1987. We find that the epidemic started in mid 1976, peaked at the end of 1983 and dropped afterwards.

Bayes Linear Estimators of the Intensity Function of the Nonstationary Poisson Process

Abstract A Bayesian approach is taken to the problem of estimating the intensity function of a nonstationary Poisson process. The intensity function, Λ(t), − T < t < T, is a priori a second-order stochastic process. Estimators are found which minimize among, respectively, the class of step functions, the class of moving averages, and a class of linear estimators. An approximation for large T to the last estimator is found. These estimators are compared in a numerical example involving the wildcat oilwell discovery process in Alberta for the years 1953–1971.