The hierarchical basis multigrid method

Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

H-extension of graphs

We consider the problem of constructing a graphG* from a collection of isomorphic copies of a graphG in such a way that for every two copies ofG, either no vertices or a section graph isomorphic to a graphH is identified. It is shown that ifG can be partitioned into vertex-disjoint copies ofH, thenG* can be made to have at most |H| orbits. A condition onG so thatG* can be vertextransitive is also included.     

Modular inclusion,the Hawking temperature,and quantum field theory in curved spacetime

A recent result by Borchers connecting geometric modular action, modular inclusion and spectrum condition, is applied in quantum field theory on spacetimes with a bifurcate Killing horizon (these are generalizations of black-hole spacetimes, comprising the familiar black-hole spacetime models). Within this framework, we give sufficient, model-independent conditions ensuring that the temperature of thermal equilibrium quantum states is the Hawking temperature.     

Embeddings of 2-spheres in 4-manifolds

We prove that the action of the orthogonal group of an integral, unimodular lattice whose rank and signature satisfy a certain indefiniteness condition acts transitively on pairs of mutually orthogonal elements with the same isometric invariants.     

Crowns,cutsets, and valuable partial orders

We consider the problem of recognizing those partial orders which admit a valuation: this is a linear-algebraic condition which arises naturally in an algebraic/geometric context. We show that a partial order has at most one valuation (which is integer-valued) and present various structural conditions which are either necessary or sufficient for a partial order to be valuable. The first main result is a reduction theorem which allows us to restrict attention to those partial orders which do not have a bounded cutset. We use this and a theorem of Kelly and Rival to prove the second main result: that every contraction of a bounded partial order is fibre-valuable if and only if the partial order is a dismantlable lattice. This result has a geometric interpretation.This research was supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392.     

A time- and spaceadaptive algorithm for the linear time-dependent Schrödinger equation

Summary. We prove an a posteriori error estimate for the linear time-dependent Schr?dinger equation in . From this, we derive a residual based local error estimator that allows us to adjust the mesh and the time step size in order to obtain a numerical solution with a prescribed accuracy. As a special feature, the error estimator controls localization and size of the finite computational domain in each time step. An algorithm is described to compute this solution and numerical results in one space dimension are included. Received March 17, 1995     

Potential theory of hyperfinite Dirichlet forms

In this paper, we are going to study the capacity theory and exceptionality of hyperfinite Dirichlet forms. We shall introduce positive measures of hyperfinite energy integrals and associated theory. Fukushima's decomposition theorem will be established on the basis of discussing hyperfinite additive functionals and hyperfinite measures. We shall study the properties of internal multiplicative functionals, subordinate semigroups and subprocesses. Moreover, we shall discuss transformation of hyperfinite Dirichlet forms.Research was supported by the National Natural Science Foundation of P.R. China, No. 18901004. The support from the position of Wissenschaftliche Hilfskraft of Ruhr-University Bochum under Prof. Sergio Albeverio is also acknowledged.     

The cascadic multigrid method for elliptic problems

Summary. The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, whichused the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method. Received November 12, 1994 / Revised version received October 12, 1995     

Crystal algebra

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U _q (sl(2)) and give a presentation by generators and relations for the case of U _q (sl(n)).