In this paper and in [12], [13], we study the structure of spaces, Y, which arepointed Gromov-Hausdor limits of sequences, f(M n i �pi)g, of complete, connected Riemannian manifolds whose Ricci curvatures

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 shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets 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 multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale 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 multi-wavelet expansions of a given order L.

In this paper we study general properties of noncommutative field theories obtained from the Seiberg-Witten limit of string theories in the presence of an external B-field. 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 space-space noncommutativity. We analyze as well questions associated to the spin-statistics 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

. 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...

. The Balian--Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system fe 2ßimbt g(t \Gamma na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then `Z 1 \Gamma1 jt g(t)j 2 dt ' `Z 1 \Gamma1 jfl g(fl)j 2 dfl ' = +1: The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g 0 ) (fl) = 2ßifl g(fl), the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form fe 2ßibm t g(t \Gamma an )g such that f(an ; bm )g has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems fe 2ßimbt g(t \Gamma na)g that form exact frames, and a new proof of the BLT for exact frame...

. 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 well-defined. Normally, a linear filtering is applied. This can be formulated in terms of scale-space, 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 scale-space if we require scale invariance and a semi-group 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 Canny-Deriche filter arises. 1 Introduction Given a digital signal in one or more dimensions, we want to define its derivatives in a well-posed way. This can...

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: ...

We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the Connes-Lott action [15] previously computed by Gayral [23] for symplectic Θ.