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159
On linear independence of integer translates of a finite number of functions
 Proc. Edinburgh Math. Soc
, 1992
"... We investigate linear independence of integer translates of a finite number of compactly 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 som ..."
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Cited by 77 (32 self)
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We investigate linear independence of integer translates of a finite number of compactly 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 pnorms in terms of the L pnorm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary 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.
Wellposedness in sobolev spaces of the full water wave problem in 3d
 J. Amer. Math. Soc
, 1997
"... We consider the motion of the interface separating an inviscid, incompressible, irrotational fluid from a region of zero density in threedimensional 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 t ..."
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Cited by 74 (0 self)
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We consider the motion of the interface separating an inviscid, incompressible, irrotational fluid from a region of zero density in threedimensional 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,
Infinitedimensional linear systems with unbounded control and observation: A functional analytic approach
 Transactions of the American Mathematical Society
, 1987
"... ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and ..."
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Cited by 59 (1 self)
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ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail. 1. Introduction. For
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 44 (6 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
On the “hot spots” conjecture of
 J. Rauch
, 1999
"... 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 ma ..."
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Cited by 34 (17 self)
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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.
Efficient Markovian couplings: examples and counterexamples
, 1999
"... 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 ..."
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Cited by 33 (18 self)
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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 coadapted. 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, CHENOPTIMAL COUPLING, COADAPTED COUPLING,...
Nonstationary Wavelets on the mSphere for Scattered Data
, 1996
"... 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 msphere. The wavelets are intrinsically defined on the msphere, and are independent of the choice of coordinate ..."
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Cited by 33 (5 self)
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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 msphere. The wavelets are intrinsically defined on the msphere, 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 2sphere, we derive an uncertainty principle that expresses the tradeoff between localization and the presence of high harmonicsor high frequenciesin 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...
Asymptotically flat initial data with prescribed regularity at infinity
 Comm. Math. Phys
, 2001
"... We prove the existence of a large class of asymptotically flat initial data with nonvanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at spacelike infinity in terms of powers of a radial coordinate. 1 1 ..."
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Cited by 29 (8 self)
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We prove the existence of a large class of asymptotically flat initial data with nonvanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at spacelike infinity in terms of powers of a radial coordinate. 1 1
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction.
, 1998
"... We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective ..."
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Cited by 23 (5 self)
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We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective computational identification procedure. 1. Introduction 2. The electrostatic problem 3. An energy estimate 4. Some additional preliminary estimates 5. An asymptotic formula for the voltage potential 6. Properties of the polarization tensor 7. The continuous dependence of the inhomogeneities 8. Computational results. 9. References 1 Introduction The nondestructive inspection technique known as electrical impedance imaging has recently received considerable attention in the mathematical as well as in the engineering literature [2, 4, 10, 14, 17]. Using this technique one seeks to determine information about the internal conductivity (or impedance) profile of an object based on boundary i...
Improving the rate of convergence of ‘high order finite elements’ on polyhedra II: Mesh refinements and interpolation
 NUMERISCHE MATEMATHIK
, 2006
"... Given a bounded polyhedral domain Ω ⊂ R3, we construct a sequence of tetrahedralizations (i.e., meshes) T ′ k that provides quasioptimal 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 ..."
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Cited by 21 (13 self)
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Given a bounded polyhedral domain Ω ⊂ R3, we construct a sequence of tetrahedralizations (i.e., meshes) T ′ k that provides quasioptimal 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 certain 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  = κAB  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.