Zusammenfassung

In Kapitel 2 wurden wichtige mathematische Objekte behandelt: Funktionen. Die dort beschriebenen Dinge haben Sie sich inzwischen sicherlich noch einmal etwas verinnerlicht. Das ist auch gut so, denn ich möchte mich jetzt mit den analytischen Eigenschaften von reellwertigen Funktionen beschäftigen. Damit meine ich, dass ich gemeinsam mit Ihnen klären möchte, wie man natürliche Fragen der folgenden Art beantworten kann: Wie kann man die Steigung einer Funktion mathematisch beschreiben? An welchen Stellen werden die Werte einer Funktion am größten? Wie kann der Graph einer Funktion gezeichnet werden? oWie grß ist die Fläche, die der Graph einer Funktion mit der x-Achse umschließt?

The main purpose of this paper is to investigate dynamical systems $${F : \mathbb{R}^2 \rightarrow \mathbb{R}^2}$$ of the form F(x, y) = (f(x, y), x). We assume that $${f : \mathbb{R}^2 \rightarrow \mathbb{R}}$$ is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x₀, y₀), such that the orbit $$ O({\rm{x}}_0) = \{{\rm{x}}_0, {\rm{x}}_1 = F({\rm{x}}_0), {\rm{x}}_2 = F({\rm{x}}_1), . . . \}, $$ is bounded, O(x₀) converges provided that the set of fixed point of F is totally disconnected and F does not admit periodic orbits of prime period two. The obtained result is used to show that all aperiodic orbits can be removed from the dynamics of the map H of Hénon. The goal can be achieved by perturbing H so that the perturbed map H₁ does not have any periodic point of prime period two.

Multiple scattering of waves is one of the important research topics in scientific and industrial fields. A number of numerical methods have been developed to compute waves scattered by several obstacles, e.g., acoustic waves scattered by schools of fish, water waves by ocean structures, and elastic waves by particles in composite materials.

We establish the order of approximation by Riesz means of the Fourier series in a multiplicative system of a class of functions with given majorant of the sequence of best approximations. In some cases, approximations by Riesz means and best approximations are considered in a specific space, but, in other cases, approximations by Riesz means are considered in spaces with a stronger norm.

We obtain a sufficient condition on the number of points of a Chebyshev alternance for the uniqueness of a simple partial fraction of degree n of best uniform approximation of a real-valued function on a segment of the real axis. We discuss the case where this number is greater than n + 1 and some aspects concerning approximations of constants. Bibliography: 6 titles.

Discrete gate sizing is a critical optimization in VLSI circuit design. Given a set of available gate sizes, discrete gate sizing problem asks to assign a size to each gate such that the delay of a combinational circuit is minimized while the cost constraint is satisfied. It is one of the most studied problems in VLSI computer-aided design. Despite this, all of the existing techniques are heuristics with no performance guarantee. This limits the understanding of the discrete gate sizing problem in theory.

This paper designs the first fully polynomial time approximation scheme (FPTAS) for the delay driven discrete gate sizing problem. The proposed approximation scheme involves a level based dynamic programming algorithm which handles the specific structures of a discrete gate sizing problem and adopts an efficient oracle query procedure. It can approximate the optimal gate sizing solution within a factor of (1+ε) in O(n^1+cm^3c/ε^c) time for 0<ε<1 and in O(n^1+cm^3c) time for ε≥1, where n is the number of gates, m is the maximum number of gate sizes for any gate, and c is the maximum number of gates per level. The FPTAS needs the assumption that c is a constant and thus it is an approximation algorithm for the restricted discrete gate sizing problem.

A class of discontinuous Galerkin methods is studied for the time discretisation of the initial-value problem for a nonlinear first-order evolution equation that is governed by a monotone, coercive, and hemicontinuous operator. The numerical solution is shown to converge towards the weak solution of the original problem. Furthermore, well-posedness of the time-discrete problem as well as a priori error estimates for sufficiently smooth exact solutions are studied.