Fully adaptive multiresolution finite volume schemes for conservation laws

The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

The Orevkov invariant of an affine plane curve

We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.

     

Schedule-an expert-like system for machine scheduling

The construction of an expert-like system for machine scheduling called SCHEDULE is presented. Essential parts of SCHEDULE were developed by students in a laboratory course Operations Research on Microcomputers at the University of Karlsruhe, Germany. SCHEDULE consists of the components data base, knowledge base, inference engine, explanation facility, dialog component, and knowledge acquisition component. The knowledge base contains an algorithm base for solving different types of scheduling problems. To establish the rules of the knowledge base the well-known three-field classification of deterministic machine scheduling problems and the concept of the reduction digraph are exploited. Experiences gained during building and demonstrating SCHEDULE are reported.     

On Mercury(I) Oxo Compounds — Quasi‐Relativistic Computational and Experimental Studies

About 60 molecular species composed of up to 10 mercury atoms and of oxygen atoms and/or of some other elements or groups (such as halogen, OH₂, OH, H, alkali, NO₃) have been investigated quantum chemically. Different density functional approaches and the ab initio SCF‐MP2 method were applied, comparing different basis sets and different atomic core sizes. It is important not to treat the Hg 5s, p, d as inactive core shells, and to use sufficiently many polarization functions. The shape of the 〉O‐Hg‐Hg‐O〈 units is not favorable concerning the formation of lattices composed of Hg^I, O and OH only. Despite its bulkiness, the OHgHgO units can easily come into contact with each other and then disproportionate. This is prevented in the so‐called ternary M‐Hg^I oxides by the embedded oxometallate (oxoacidic) anions. Furthermore, the Hg^I and Hg^II oxide bond energies are less favorable towards the stability of Hg^I oxo compounds, as compared to Hg halidic or oxoacidic compounds. Both points are not promising concerning the search for Hg^I oxides/hydroxides, although the preparation of such compounds, including spacer groups, by topochemical reactions can still not be excluded. So far, experimental efforts towards the synthesis of such a new class of compounds have only demonstrated that Hg^II is strictly preferred over Hg^I in the formation of solids of binary Hg‐O or ternary A‐Hg‐O composition (A = electropositive metal such as alkali, in contrast to M = transition or semi‐metal). This is so even if compounds containing ‘electron rich Hg^δ— atoms’ (i.e. A‐Hg amalgams) are oxidized under mild conditions.