Moshe Dubiner

Spectral methods on triangles and other domains

This article shows how to obtain multidimensional spectral methods as a warped product of one-dimensional spectral methods, thus generalizing direct (tensor) products. This generalization includes the fast Fourier transform. Applications are given for spectral approximation on a disk and on a triangle. The use of the disk spectral method for simulating the Navier-Stokes equations in a periodic pipe is detailed. The use of the triangle method in a spectral element scheme is discussed. The degree of approximation of the triangle method is computed in a new way, which favorably compares with the classical approximation estimates.

Faster tree pattern matching

Recently R. Kosaraju gave an O(nmo.75polylog(m)) step algorithm for tree pattern matching. We improve this result by designing a simple O(n6 polylog(m)) algorithm.

A lattice point problem and additive number theory

For every dimensiond≥1 there exists a constantc=c(d) such that for alln≥1, every set of at leastcn lattice points in thed-dimensional Euclidean space contains a subset of cardinality preciselyn whose centroid is also a lattice point. The proof combines techniques from additive number theory with results about the expansion properties of Cayley graphs with given eigenvalues.

Amplification by Read-Once Formulas

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulas of size

The theory of multi-dimensional polynomial approximation

O(n^{\alpha+2})

Asymptotic analysis of spectral methods

(where

How Do Read-Once Formulae Shrink?

\alpha=\log_{\sqrt{5}-1}2\simeq 3.27

Amplification and percolation (probabilistic Boolean functions)

) that amplify

The product of a convergent sequence of matrices

(\psi-\frac{1}{n},\psi+\frac{1}{n})

Amplification and Percolation

(where