Some Remarks on p-Adic Analytic Groups

A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and p^e exactly divides |F|, then a Z_pF-latticeM is determined up to isomorphism by its finite quotient M/p^e+1M.If M is a free Z_p-module of rank d, this is equivalent to sayingthat representations of F in GL_d(Z_p) are determined up to equivalenceby their images modulo p^e+1. 1991 Mathematics Subject Classification20E18, 22E20.     

Preconditioners for Incompressible Navier-Stokes Solvers

<正>In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's method.It is shown that block preconditioners form an excellent approach for the solution,however if the grids are not to fine preconditioning with a Saddle point ILU matrix(SILU) may be an attractive alternative. The applicability of all methods to stabilized elements is investigated.In case of the stand-alone Stokes equations special preconditioners increase the efficiency considerably.     

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety. Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000     

Local n-connectivity of decomposition spaces

A corollary of the main result of this paper is the following Theorem. Suppose f: X → Y is a closed surjection of metrizable spaces whose point inverses are LC^{n + 1}-divisors (n ? 1). If Y is complete and f is homology n-stable, then Y is LC^{n + 1}provided X is LC^{n + 1}.Intuitively, f is homology n-stable if the ?ech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LCⁿ.     

Isomorphic Steiner symmetrization of p-convex sets

In this paper we show that given a \(p\) -convex set \(K \subset \mathbb{R }^n\) , there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean ball. That is, if \(|K| = |D_n| = \kappa _n\) , we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\) , where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.     

Selecting distances in arrangements of hyperplanes spanned by points

In this paper we consider a problem of distance selection in the arrangement of hyperplanes induced by n given points. Given a set of n points in d-dimensional space and a number k, , determine the hyperplane that is spanned by d points and at distance ranked by k from the origin. For the planar case we present an O(nlog²n) runtime algorithm using parametric search partly different from the usual approach [N. Megiddo, J. ACM 30 (1983) 852]. We establish a connection between this problem in 3-d and the well-known 3SUM problem using an auxiliary problem of counting the number of vertices in the arrangement of n planes that lie between two sheets of a hyperboloid. We show that the 3-d problem is almost 3SUM-hard and solve it by an O(n²log²n) runtime algorithm. We generalize these results to the d-dimensional (d4) space and consider also a problem of enumerating distances.