A least-squares approach based on a discrete minus one inner product for first order systems

The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in (the Sobolev space of order minus one on ). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.

     

-bounds for spectral multipliers on graphs

We prove that certain spectral multipliers associated with the discrete Laplacian on graphs satisfying the doubling volume property and the Poincaré inequality are bounded on the Hardy space .

     

A compactification of the moduli space of polynomials

In this paper, we introduce a compactification of the moduli space of polynomial maps with a fixed degree such that the map from it to defined by using the elementary symmetric functions of multipliers at fixed points is a continuous surjection.

     

Wandering vectors for irrational rotation unitary systems

An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for wandering vector multipliers, and strengthen this by showing that for an irrational rotation unitary system , every unitary operator in is a wandering vector multiplier. Moreover, we show that there is a class of wandering vector multipliers, induced in a natural way by pairs of characters of the integer group , which fail to factor even as the product of a unitary in and a unitary in . Incomplete maximal wandering subspaces are also considered, and some questions are raised.

     

Multigrid and multilevel methods for nonconforming elements

In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the -cycle with any number of smoothing steps and close-to-optimal condition number estimates for -cycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for -cycles with a sufficiently large number of smoothing steps and variable -cycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming rotated discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory.

     

Herz-Schur multipliers and weakly almost periodic functions on locally compact groups

For a locally compact group and , let be the Herz-Figà-Talamanca algebra and the Herz-Schur multipliers of , and the multipliers of . Let be the algebra of continuous weakly almost periodic functions on . In this paper, we show that (1), if is a noncompact nilpotent group or a noncompact [IN]-group, then contains a linear isometric copy of ; (2), for a noncommutative free group is a proper subset of .

     

Contiguous relations, continued fractions and orthogonality

We examine a special linear combination of balanced very-well-poised basic hypergeometric series that is known to satisfy a transformation. We call this and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's -analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new -series identities are also obtained. One is an important three-term transformation for 's which generalizes all the known two- and three-term transformations. Others are new and unexpected quadratic identities for these very-well-poised 's.

     

A Markov dilation for self-adjoint Schur multipliers

We give a formula for Markov dilation in the sense of Anantha- raman-Delaroche for real positive Schur multipliers on .

     

The Molchanov-Vainberg Laplacian

It is well known that the Green function of the standard discrete Laplacian on ,

exhibits a pathological behavior in dimension . In particular, the estimate

fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian,

and conjectured that the estimate

holds for all . In this paper we prove this conjecture.

     

Univariate modified Fourier methods for second order boundary value problems

We develop and analyse a new spectral-Galerkin method for the numerical solution of linear, second order differential equations with homogeneous Neumann boundary conditions. The basis functions for this method are the eigenfunctions of the Laplace operator subject to these boundary conditions. Due to this property this method has a number of beneficial features, including an condition number and the availability of an optimal, diagonal preconditioner. This method offers a uniform convergence rate of , however we show that by the inclusion of an additional 2M basis functions, this figure can be increased to for any positive integer M.      

An iterative substructuring method for Maxwell's equations in two dimensions

Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner. For conforming finite element approximations of , it is known that the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures, and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure. In this paper, the same result is established for similar iterative methods for low-order Nédélec finite elements, which approximate in two dimensions. Results of numerical experiments are also provided.

     

Robust norm equivalencies for diffusion problems

Additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem. These solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition. To obtain a robust iterative scheme, it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem.

In this paper we present such a robust norm equivalence for the model problem with a scalar diffusion coefficient in . Our estimates involve only very weak information about , and the results are applicable for a large class of diffusion coefficients. Namely, we require to be in the Muckenhoupt class , a function class well-studied in harmonic analysis.

The presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems.

     

Wavelet decompositions of Fourier multipliers

We show that in terms of its weak topology, the space of Fourier multipliers for , , can be decomposed by band-limited wavelets belonging to the Schwartz class.

     

Weak type estimates for cone multipliers on

We consider operators associated with the Fourier multipliers and show that is of weak type on , , for the critical value .

     

The Laplacian MASA in a free group factor

The Laplacian (or radial) masa in a free group factor is generated by the sum of the generators and their inverses. We show that such a masa is strongly singular and has Popa invariant . This is achieved by proving that the conditional expectation onto is an asymptotic homomorphism. We also obtain similar results for the free product of discrete groups, each of which contains an element of infinite order.

     

Length spectrum in rank one symmetric space is not arithmetic

In this paper we show that a nonelementary nonparabolic group in a real semisimple Lie group of rank one has the property that the set of translation lengths of hyperbolic elements is not contained in any discrete subgroup of .

     

Discrete threshold growth dynamics are omnivorous for box neighborhoods

In the discrete threshold model for crystal growth in the plane we begin with some set of seed crystals and observe crystal growth over time by generating a sequence of subsets of by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site's prescribed neighborhood. The growth dynamics generated by this model are said to be omnivorous if finite and imply . In this paper we prove that the dynamics are omnivorous when the neighborhood is a box (i.e. when, for some fixed , the neighborhood of is . This result has important implications in the study of the first passage time when is chosen randomly with a sparse Bernoulli density and in the study of the limiting shape to which converges.

     

Toeplitz -algebras on ordered groups and their ideals of finite elements

Let be a discrete abelian group and an ordered group. Denote by the minimal quasily ordered group containing . In this paper, we show that the ideal of finite elements is exactly the kernel of the natural morphism between these two Toeplitz -algebras. When is countable, we show that if the direct sum of -groups , then .

     

Block combinatorics

In this paper we extend the block combinatorics partition theorems of Hindman and Milliken-Taylor in the setting of the recursive system of the block Schreier families , consisting of families defined for every countable ordinal . Results contain (a) a block partition Ramsey theorem for every countable ordinal (Hindman's Theorem corresponding to , and the Milliken-Taylor Theorem to a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.