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518
On the structure of spaces with Ricci curvature bounded below. I
 J. DIFFERENTIAL GEOM
, 1997
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Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media II: Nonlinear theory, preprint
, 2002
"... Communicated by G. Iooss Summary. Considered herein are a number of variants of the classical Boussinesq system and their higherorder generalizations. Such equations were first derived by Boussinesq to describe the twoway propagation of smallamplitude, long wavelength, gravity waves on the surfac ..."
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Cited by 94 (22 self)
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Communicated by G. Iooss Summary. Considered herein are a number of variants of the classical Boussinesq system and their higherorder generalizations. Such equations were first derived by Boussinesq to describe the twoway propagation of smallamplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of longcrested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a fourparameter family of Boussinesq systems from the twodimensional Euler equations for freesurface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.
A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow
 J. Comp. Phys
"... In this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies. The simulation methods rest essentially on the combination of: (a) Lagrangemultiplierbased fictitious domain methods which allow the fluid flow compu ..."
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Cited by 83 (4 self)
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In this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies. The simulation methods rest essentially on the combination of: (a) Lagrangemultiplierbased fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region. (b) Finite element approximations of the Navier–Stokes equations occurring in the global model. (c) Time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model. The above methodology is particularly well suited to the direct numerical simulation of particulate flow, such as the flow of mixtures of rigid solid particles and incompressible viscous fluids, possibly nonNewtonian. We conclude this article with the presentation of the results of various numerical experiments, including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions. c ° 2001 Academic Press Key Words: fictitious domain methods; finite element methods; distributed Lagrange multipliers; Navier–Stokes equations; particulate flow; liquid–solid mix
Linear Complementarity Systems
 SIAM J. Appl. Math
, 1997
"... We introduce a new class of dynamical systems called "linear complementarity systems." The time evolution of these systems consists of a series of continuous phases separated by "events" which cause a change in dynamics and possibly a jump in the state vector. The occurrence of e ..."
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Cited by 69 (23 self)
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We introduce a new class of dynamical systems called "linear complementarity systems." The time evolution of these systems consists of a series of continuous phases separated by "events" which cause a change in dynamics and possibly a jump in the state vector. The occurrence of events is governed by certain inequalities similar to those appearing in the Linear Complementarity Problem of mathematical programming. The framework we describe is suitable for certain situations in which both differential equations and inequalities play a role, for instance in mechanics, electrical networks, piecewise linear systems, and dynamic optimization. We present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions. 1 Introduction In many technical and economic applications one encounters systems of differential equations and inequalities. For a quick roundup of examples, one may think of the following: ...
Approximation error for quasiinterpolators and (multi)wavelet expansions
 APPL. COMPUT. HARMON. ANAL
, 1999
"... We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wa ..."
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Cited by 61 (23 self)
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We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wavelets, as well as finite elements and multiwavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multiwavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the twoscale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multiwavelet expansions of a given order L.
General properties of noncommutative field theories,” Nucl
 Phys. B
, 2000
"... In this paper we study general properties of noncommutative field theories obtained from the SeibergWitten limit of string theories in the presence of an external Bfield. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular we present a proo ..."
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Cited by 56 (3 self)
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In this paper we study general properties of noncommutative field theories obtained from the SeibergWitten limit of string theories in the presence of an external Bfield. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular we present a proof of the CPT theorem for theories with spacespace noncommutativity. We analyze as well questions associated to the spinstatistics connections, and show that noncommutative N = 4, U(1) gauge theory can be softly broken to N = 0 satisfying the axioms and providing an example where the Wilsonian low energy effective action can be constructed without UV/IR problems, after a judicious choice of soft breaking parameters is made. We also assess the phenomenological prospects of such a theory, which are in fact rather negative. 1
Gravitational Radiation from PostNewtonian Sources and Inspiralling Compact Binaries, Living Rev
 Rel
"... The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general postNewtonian sources. The exterior field of the source is investigated by means ..."
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Cited by 50 (0 self)
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The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general postNewtonian sources. The exterior field of the source is investigated by means of a combination of analytic postMinkowskian and multipolar approximations. The physical observables in the farzone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the nearzone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tailsoftails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third postNewtonian (3PN) order beyond the Newtonian acceleration. The gravitationalwave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument. 1 1
A Globally Convergent Successive Approximation Method for Severely Nonsmooth Equations
 SIAM J. Control Optim
, 1995
"... . This paper presents a globally convergent successive approximation method for solving F (x) = 0 where F is a continuous function. At each step of the method, F is approximated by a smooth function f k ; with k f k \Gamma F k! 0 as k ! 1. The direction \Gammaf 0 k (x k ) \Gamma1 F (x k ) is th ..."
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Cited by 47 (22 self)
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. This paper presents a globally convergent successive approximation method for solving F (x) = 0 where F is a continuous function. At each step of the method, F is approximated by a smooth function f k ; with k f k \Gamma F k! 0 as k ! 1. The direction \Gammaf 0 k (x k ) \Gamma1 F (x k ) is then used in a line search on a sum of squares objective. The approximate function f k can be constructed for nonsmooth equations arising from variational inequalities, maximal monotone operator problems, nonlinear complementarity problems and nonsmooth partial differential equations. Numerical examples are given to illustrate the method. Key words: Global convergence, successive approximation, integration convolution. AMS(MOS) subject classification. 90C30, 90C33 1. Introduction Let F : R n ! R n be a continuous, but not necessarily differentiable, function. We consider the system of nonlinear equations F (x) = 0; x 2 R n : (1) The recent literature of such nonsmooth equations inc...
Differentiation And The BalianLow Theorem
, 1995
"... The Balian–Low theorem (BLT) is a key result in timefrequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system {e 2πimbt g(t − na)}m,n∈Z with ab = 1 forms an orthonormal basis for L 2 (R), then t g(t) ..."
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Cited by 47 (19 self)
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The Balian–Low theorem (BLT) is a key result in timefrequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system {e 2πimbt g(t − na)}m,n∈Z with ab = 1 forms an orthonormal basis for L 2 (R), then t g(t)
Regularization, ScaleSpace, and Edge Detection Filters
 Journal of Mathematical Imaging and Vision
"... . Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them welldefined. Normally, a linear filtering is applied. This can be formulated in terms of scalespace, functional mini ..."
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Cited by 40 (8 self)
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. Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them welldefined. Normally, a linear filtering is applied. This can be formulated in terms of scalespace, functional minimization, or edge detection filters. The main emphasis of this paper is to connect these theories in order to gain insight in their similarities and differences. We take regularization (or functional minimization) as a starting point, and show that it boils down to Gaussian scalespace if we require scale invariance and a semigroup constraint to be satisfied. This regularization implies the minimization of a functional containing terms up to infinite order of differentiation. If the functional is truncated at second order, the CannyDeriche filter arises. 1 Introduction Given a digital signal in one or more dimensions, we want to define its derivatives in a wellposed way. This can...