Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

Wolfgang Hackbusch ^{In this paper, we discuss the application of hierarchical matrixtechniques to the solution of Helmholtz problems with largewave number in 2D. We consider the Brakhage–Werner integralformulation of the problem discretized by the Galerkin boundary-elementmethod. The dense n x n Galerkin matrix arising from this approachis represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats.A well-known multipole expansion is used to construct the 2-matrix. We present a new approach to dealingwith the numerical instability problems of this expansion: theparts of the matrix that can cause problems are approximatedin a stable way by an -matrix. Algebraic recompression methods are used to reducethe storage and the complexity of arithmetical operations ofthe -matrix.Further, an approximate LU decomposition of such a recompressed-matrix is aneffective preconditioner. We prove that the construction ofthe matrices as well as the matrix-vector product can be performedin almost linear time in the number of unknowns. Numerical experimentsfor scattering problems in 2D are presented, where the linearsystems are solved by a preconditioned iterative method.}     

The nucleophilic substitution reaction for [18F]fluoride-ion on the series of N6-benzoyl-2",3"-isopropylideneadenosine-5"-sulfonates

The reaction of the nca [¹⁸F]fluoride ion was investigated toward a series of N⁶-benzoyl 2",3"-isopropylidene-adenosine-5"-sulfonates including the methane-(mesylate), p-toluene-(tosylate), p-nitro-benzene-(nosylate)- and 2,2,2-trifluoro-ethane-sulfonate (tresylate) derivatives under usual nucleophilic substitution conditions. In these reactions cyclisation of the title compounds was observed whilst the radiofluorination took place only with low yield. The fluorine-18 uptake was found to be 1.17% for mesylate, 1.46% for tosylate, 0.99% for nosylate and 0.40% for tresylate under the conditions applied.     

Covers in hypergraphs

Let α(H) denote the stability number of a hypergraphH. The covering number ?(H) is defined as the minimal number of edges fromH to cover its vertex setV(H). The main result is the following extension of König’s wellknown theorem: If α(H′)≧|V(H′)|/2 holds for every section hypergraphH′ ofH then ?(H)≦α(H). This theorem is applied to obtain upper bounds on certain covering numbers of graphs and hypergraphs. In par ticular, we prove a conjecture of B. Bollobás involving the hypergraph Turán numbers.     

Vertex Coloring of Graphs by Total 2-Weightings

An assignment of weights to the edges and the vertices of a graph is a vertex-coloring total weighting if adjacent vertices have different total weight sums. Of interest in this paper are vertex-coloring total weightings with weight set of cardinality two, a problem motivated by the conjecture that every graph has such a weighting using the weights 1 and 2. Here we prove the existence of such weightings for certain families of graphs using any two different real weights. A related problem where all vertices have unique weight sums is also discussed.