We investigate linear independence of integer translates of a finite number of com-pactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some ℓ p space (1 ≤ p ≤ ∞) and we are interested in bounding their ℓ p-norms in terms of the L p-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give nec-essary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

We consider the motion of the interface separating an inviscid, incompressible, irrotational fluid from a region of zero density in three-dimensional space; we assume that the fluid region is below the vacuum, the fluid is under the influence of gravity and the surface tension is zero. Assume that the density of mass of the fluid is one,

We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoenberg 's positive definite functions on the m-sphere. The wavelets are intrinsically defined on the m-sphere, and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We will discuss decomposition, reconstruction, and localization for these wavelets. In the special case of the 2-sphere, we derive an uncertainty principle that expresses the trade-off between localization and the presence of high harmonics---or high frequencies---in expansions in spherical harmonics. We discuss the application of this principle to the wavelets that we construct. I. Introduction Geophyiscal or meteorological data collected over the surface of the earth via satellites or ground stations will invariably come from scattered sites. Synthesizing and analyzing such data is the motivation for the work that is pr...

In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the recent advent of methods of "perfect simulation": it helps to establish the "price of perfection" for such methods. In general one can always achieve efficient coupling if the coupling is allowed to "cheat" (if each component's behaviour is affected by future behaviour of the other component), but the situation is more interesting if the coupling is required to be co-adapted. We present an informal heuristic for the existence of an efficient coupling, and justify the heuristic by proving rigorous results and examples in the contexts of finite reversible Markov chains and of reflecting Brownian motion in planar domains. Keywords: DIFFUSION, CHEN-OPTIMAL COUPLING, CO-ADAPTED COUPLING,...

We construct a counterexample to the “hot spots ” conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain. 1.

Abstract. For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the Laplace-Beltrami operator on Riemannian manifolds for p> 2. 1.

. In this paper we study one-dimensional Cahn-Morral systems, which are the multicomponent analogues of the Cahn-Hilliard model for phase separation and coarsening in binary mixtures. In particular, we examine solutions that start with initial data close to the preferred phases except at finitely many transition points where the data has sharp transition layers, and we show that such solutions may evolve exponentially slowly; i.e., if # is the interaction length then there exists a constant C such that in exp(C/#) units of time the change in such a solution is o(1). This corresponds to extremely slow coarsening of a multicomponent mixture after it has undergone fine-grained decomposition. Key words. Cahn-Hilliard equation, phase separation, transition layers, metastability AMS subject classifications. 35B30, 35B25, 35K55 1. Introduction. One of the leading continuum models for the dynamics of phase separation and coarsening in a binary mixture is the Cahn-Hilliard equation, which in t...

Given a bounded polyhedral domain Ω ⊂ R3, we construct a sequence of tetrahedralizations (i.e., meshes) T ′ k that provides quasi-optimal rates of convergence with respect to the dimension of the aproximation space for the Poisson problem with data f ∈ Hm−1 (Ω), m ≥ 2. More precisely, let Sk be the Finite Element space of continuous, piecewise polynomials of degree m ≥ 2 on T ′ k and let uk ∈ Sk be the finite element approximation of the solution u of the Poisson problem −∆u = f, u = 0 on the boundary, then �u − uk � H 1 (Ω) ≤ C dim(Sk) −m/3 �f � H m−1 (Ω) , with C independent of k and f. Our method relies on the a priori estimate �u�D ≤ C�f � H m−1 (Ω) in cer-tain anisotropic weighted Sobolev spaces D = D m+1 a+1 (Ω), with a> 0 small and determined by Ω. The weight is the distance to the set of singular boundary points (i.e., edges). The main feature of our mesh refinement is that a segment AB in T ′ k will be divided into two segments AC and CB in T ′ k+1 as follows: |AC | = |CB | if A and B are equally singular and |AC | = κ|AB | if A is more singular than B. We can chose κ ≤ 2−m/a. This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct T ′ k.

Abstract. The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin’s theorem on semisimple Frobenius manifolds. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singularity theory and the Barannikov–Kontsevich construction. For homogeneous polynomials which give Calabi–Yau hypersurfaces certain Frobenius submanifolds in both constructions are isomorphic. 1.

In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O(h 2) accuracy even if the truncation error is O(h) at the interface, while O(h 2) in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O(h 2 log (1/h)). Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O(h 2).