A Note on the Order on the Set of Finite Subsets

Let T be a well-ordered set with an injective order-homomorphism ' : T ! Q . In this note we construct a simple extension OE of ' to P fin (T ), the set of all finite subsets of T , such that OE is an injective order-homomorphism wrt: the induced well-order on P fin (T ). We apply this construction to lexicographic orders on N n . Introduction Let (T; ) be a well-ordered set and let there be given a function ' : T ! Q such that a ! b =) '(a) ! '(b) for all a; b 2 T , which implies that ' is an order-isomorphism from T onto '(T ) ae Q . Here, ! is the strict part of . Now it is well-known that P fin (T ), the set of all finite subsets of T , is also a well-ordered set by means of the following definition of the induced order on P fin (T ): Definition 0.1. For ; 6= A 2 P fin (T ), let max(A) denote the unique maximal element of A. For A; B 2 P fin (T ); A 0 B is defined by recursion as follows: (i) If A = ;, then A 0 B (ii) If A 6= ;, then A 0 B iff B 6= ; and the ..

Anisotropic Adaption and Structure Detection

In this paper, we suggest a general approach of an anisotropic adaption procedure in the context of numerical schemes for hyperbolic conservation laws. This approach consists of coupling a structure detection tool from image processing with a new, non-iterative anisotropic adaption algorithm. Ingo Thomas Inst. f. Angew. Math. Bundesstr. 55 20146 Hamburg [email protected] August 20, 1999 Contents 1 Anisotropic Adaption 4 1.1 Representation of an anisotropic grid resolution by symmetric positive definite matrices . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Discrete interpretation of the anisotropic refinement information 4 1.3 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Isotropic Adaption 7 3 Anisotropic Adaption 8 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 First Step: Splitting of edges . . . . . . . . . . . . . . . ...

Reihe F Computational Fluid Dynamics and Data Analysis SHIFT GENERATED HAAR SPACES ON TRACK FIELDS

Dedicated to the memory of Walter Hengartner 1 Abstract. The general aim is to show that G(z): = 1/z2 is never a universal Haar space generator for all compact sets K in C. For many cases that was already shown in papers by Hengartner & Opfer [5, 2002], [6, 2005]. The remaining cases are those for which K is convex (different from ellipses) and K = K ◦ , where K ◦ is the interior of K and K ◦ is the closure of K ◦ and where the boundary of K is smooth. We show for several cases of compact, convex sets that G is not a 2-dimensional Haar space generator for K implying that it is not a universal Haar space generator for K. We will be guided by a model of a track field: a rectangle with two half disks attached on two opposite sides of the rectangle. We also show, that the above G is not a 3-dimensional Haar space generator for all regular polygons (with smoothed vertices). The definition of Haar spaces and Haar space generators will be given in the main text. The paper contains as a byproduct a

Reihe F Computational Fluid Dynamics and Data Analysis COMPUTING QUATERNIONIC ROOTS BY NEWTON’S METHOD (2006/12/23)

The final form may differ from this preprin

A Theory of Implicit Numerical Methods for Scalar Conservation Laws

We present a novel theory of implicit methods for scalar conservation laws which does to a large extent not rely on classical structures in numerical analysis, e.g. the notion of total variation stability is not employed. Thereby, for the first time rigorous theoretical results concerning the range of applicability of implicit schemes are given. A new proof of convergence is given relying on monotonicity of the considered methods and on a special discretization of the initial data. The theoretical results are confirmed by numerical tests