Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11 T. Donchev ( 2008 ). Approximation of the solution set of impulsive systems . In Lecture Notes in Comput. Sci . ( et al. . eds.). Vol. 4818 , Springer-Verlag , Berlin , pp. 309 – 316 . [Google Scholar]] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.     

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time‐dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge‐Kutta‐Fehlberg‐type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed‐up, memory compression, and errors in different norms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Inhomogeneous diophantine approximation with prime constraints

We study the problem of inhomogeneous diophantine approximation under certain primality restrictions.     

Divisibility problems for function fields

We investigate three combinatorial problems considered by Erd?s, Rivat, Sárközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.     

Charge qubit purification by an electronic feedback loop

We propose the manipulation of an isolated qubit by a simple instantaneous closed-loop feedback scheme in which a time-dependent electronic detector current is directly back-coupled into qubit parameters. As a specific detector model, we employ a capacitively coupled single-electron transistor. We demonstrate the stabilization of pure delocalized qubit states above a critical detector-qubit coupling. This electronic purification is independent of the initial qubit state and is accomplished after few electron jumps through the detector. Our simple scheme can be used for the efficient and robust initialization of solid-state qubits in quantum computational algorithms at arbitrary temperatures.     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system