scaling functions and wavelets with vanishing moments abstract the authors investigate conditions equivalent to the vanishing of moments of wavelets in a multiresolution analysis

approximation using scattered shifts of a multivariate function the approximation of a general d variate function f by the shifts r d of a fixed functionoccurs in many applications such as data fit ting neural networks and learning theory when hz d is a dilate of the integer lattice there is a rather complete understanding of the approximation problem 6 18 using fourier techniques however in most applications the center set is either given or can be chosen with complete freedom in both of these cases the shift invariant setting is too restrictive this paper studies the approximation problem in the caseis arbitrary it establishes approximation theorems whose error bounds reflect the local density of the points in two different settings are analyzed the first is when the setis prescribed in advance in this case the theorems of this paper show that in analogy with the classical univariate spline ap proximation improved approximation occurs in regions where the density is high the second setting corresponds to the problem of non linear approximation in that setting the setcan be chosen using information about the target function f we discuss how to best make these choices and give estimates for the approximation error

adaptive biorthogonal spline schemes for advection reaction equations in this paper based on eulerian lagrangian localized adjoint method ellam we use biorthogonal spline wavelets to develop numerical schemes for multidimensional advection reaction equations the derived schemes produce accurate numerical solutions even if large time steps are used these schemes are explicit but unconditionally stable they also have the ability to carry out adaptive compression while preserving the total mass numerical experiments including observations on their convergence rates are presented to show the strong potential of these methods

optimal space time adaptive wavelet methods for degenerate parabolic pdes we analyze parabolic pdes with certain type of weakly singular or degenerate time dependent coefficients and prove existence and uniqueness of weak solutions in an appropriate sense a localization of the pdes to a bounded spatial domain is justified for the numerical solution a space time wavelet discretization is employed an optimality result for the iterative solution of the arising systems can be obtained finally applications to fractional brownian motion models in option pricing are presented

computation of decaying turbulence in an adaptive wavelet basis abstract the paper presents computations of decaying two dimensional turbulence in an adaptive wavelet basis at each time step the vorticity is represented by an adaptively selected set of wavelet functions which adjusts to the instantaneous distribution of vorticity essential features are the use of operator adapted test functions and the adaptive evaluation of the convection term the results of this new algorithm are compared to a classical fourier method and a fourier method supplemented with wavelet compression in each time step they show that turbulent flows with a multitude of spatial scales can be computed with a reduced number of degrees of freedom the investigation of diverse spectral and statistical criteria validates the wavelet approach

non self dual yang mills connections with nonzero chern number we prove the existence of non self dual yang mills connections on su 2 bundles over the standard four sphere specifically on all bundles with second chern number not equal to 1

the topology of instanton moduli spaces i the atiyah jones conjecture in this paper we study the global geometry and topology of the moduli spaces of based su 2 instantons over the 4 sphere s4 these instanton moduli spaces have a rich history and have been analyzed from many points of view originally these spaces which we denote by mk were defined as solution spaces modulo gauge equivalence to certain partial differential equations namely the self duality equations associated to the yang mills functional in su 2 gauge theory they have been successfully studied from this point of view by taubes t1 t2 uhlenbeck u and others an important alternative approach was initiated by ward w who related instantons to certain holomorphic bundles on cp3 and was continued by atiyah and ward in aw this allowed the classification of instantons on 4 in terms of quaternionic linear algebra by atiyah drinfeld hitchin and manin adhm this holomorphic approach was further extended by donaldson d who showed that these bundles were determined by their restriction to a cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in c2r2 hurtubise hul then exploited this fact to study the moduli spaces mk as did atiyah a to show that mk arise naturally in the theory of holomorphic maps into loop groups this latter approach was continued by gravesen g atiyah and jones aj obtained the first results and formulated the foundational questions on the global topology of these moduli spaces recall that an element of mk is a based gauge equivalence class of a connection on the principal su 2 bundle over 4 denoted by pk with second chern class k satisfying the self duality equations there is a natural forgetful map to the based equivalence classes of all connections in pk atiyah and jones aj showed that this target space which we denote by bk is homotopy equivalent

an adaptive multi scale basis method for modelling discrete time nonlinear dynamical systems based on locally applicable autoregressive moving average with exogenous armax models this paper proposes two new adaptive modelling schemes for discrete time non linear dynamic systems based on the concept of a multiscale which originates in approximation theory a general result on the convergence of modelling based on the multiscale basis functions with a least square estimated is derived with the advantage of the multiresolution nature of the multiscale basis the new modelling scheme can be used to capture both the global and local characteristics of a non linear system over distinct scales the algorithms also provide a trade off between the model structure model size and the modelling error one of which emphasizes the modelling error with variable model structure whilst the other stresses a desired model size with better modelling error data based modelling examples are used to demonstrate the effectiveness of the multiscale modelling schemes with different multiscale basis

small coupling limit and multiple solutions to the dirichlet problem for yang mills connections in four dimensions ii in this paper we complete the proof of the existence of multiple solutions and in particular non minimal ones to the e dirichlet problem obtained as a variational problem for the su 2 e yang mills functional this is equivalent to proving the existence of multiple solutions to the dirichlet problem for the su 2 yang mills functional with small boundary data in the first paper of this series this non compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function je q which is essentially the yang mills functional evaluated on the approximate solutions constructed via a gluing technique in the present paper we establish a morse theory for je q by means of ljusternik schnirelmann theory thus complete the proofs of theorems 1 3 given by isobe and marini small coupling limit and multiple solutions to the dirichlet problem for yang mills connections in 4 dimensions part i j math phys 53 063706 2012 10 1063 1 4728211

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

riesz bases of wavelets and applications to numerical solutions of elliptic equations we investigate riesz bases of wavelets in sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order first we study bi cubic splines on the unit square with homogeneous boundary conditions the approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation second we develop a fairly general theory for riesz bases of hilbert spaces equipped with induced norms under the guidance of the general theory we are able to construct wavelet bases for sobolev spaces on the unit square the condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded third we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient finally we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case

multiscale decompositions on bounded domains a construction of multiscale decompositions relative to domains rd is given multiscale spaces are constructed on which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported stable bases

multiscale approximation of vortex patches we propose an adaptive scheme based on multiscale decomposition into interpolatory wavelet bases to simulate the evolution of vortex patches associated with solutions of the two dimensional euler equations this scheme is based on fast algorithms that relate the numerically significant wavelet coefficients with the irregular sampling of the boundary of the patch on an adaptive grid this method allows analysis of the evolution of the singularities on this boundary at a low computational cost

empirical bayes approach to improve wavelet thresholding for image noise reduction wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients this is basically a local procedure because wavelet coefficients characterize the local regularity of a function although a wavelet transform has decorrelating properties structures in images like edges are never decorrelated completely and these structures appear in the wavelet coefficients a classification based on a local criterion like coefficient magnitude is not the perfect method to distinguish important uncorrupted coefficients from coefficients dominated by noise we therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a bayesian framework the local characterization is incorporated into the conditional model whereas the prior model describes only configurations not coefficient values more precisely local characterization favors co

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

yang mills fields which are not self dual the purpose of this paper is to prove the existence of a new family of non self dual finite energy solutions to the yang mills equations on euclidean four space withsu 2 as a gauge group the approach is that of equivariant geometry attention is restricted to a special class of fields those that satisfy a certain kind of rotational symmetry for which it is proved that 1 a solution to the yang mills equations exists among them and 2 no solution to the self duality equations exists among them the first assertion is proved by an application of the direct method of the calculus of variations existence and regularity of minimizers and the second assertion by studying the symmetry properties of the linearized self duality equations the same technique yields a new family of non self dual solutions on the complex projective plane

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

a symmetric family of yang mills fields we examine a family of finite energyso 3 yang mills connections overs4 indexed by two real parameters this family includes both smooth connections when both parameters are odd integers and connections with a holonomy singularity around 1 or 2 copies ofrp2 these singular ym connections interpolate between the smooth solutions depending on the parameters the curvature may be self dual anti self dual or neither for the anti self dual connections we compute the formal dimension of the moduli space for the non self dual connections we examine the second variation of the yang mills functional and count the negative and zero eigenvalues each component of the non self dual moduli space appears to consist only of conformal copies of a single solution

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

cryptanalysis of short rsa secret exponents a cryptanalytic attack on the use of short rsa secret exponents is described the attack makes use of an algorithm based on continued fractions that finds the numerator and denominator of a fraction in polynomial time when a close enough estimate of the fraction is known the public exponent e and the modulus pq can be used to create an estimate of a fraction that involves the secret exponent d the algorithm based on continued fractions uses this estimate to discover sufficiently short secret exponents for a typical case where e pq gcd p 1 q 1 is small and p and q have approximately the same number of bits this attack will discover secret exponents with up to approximately one quarter as may bits as the modulus ways to combat this attack ways to improve it and two open problems are described this attack poses no threat to the normal case of rsa where the secret exponent is approximately the same size as the modulus this is because the attack uses information provided by the public exponent and in the normal case the public exponent can be chosen almost independently of the modulus

energy gap for yang mills connections i four dimensional closed riemannian manifolds we extend an l2l2 energy gap result due to min oo 28 theorem 2 and parker 30 proposition 2 2 for yang mills connections on principal g bundles p over closed connected four dimensional oriented smooth manifolds x from the case of positive riemannian metrics to the more general case of good riemannian metrics including metrics that are generic and where the topologies of p and x obey certain mild conditions and the compact lie group g is su 2 su 2 or so 3 so 3

on the importance of combining wavelet based nonlinear approximation with coding strategies this paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory contrasting linear and nonlinear approximation spaces we show that there are interesting classes of functions random processes which are much more compactly represented by wavelet based nonlinear approximation these classes include locally smooth signals that have singularities and provide a model for many signals encountered in practice in particular for images however we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate distortion sense based on this observation we construct compression techniques and formulate the family of functions stochastic processes for which they provide efficient descriptions in a rate distortion sense we show that this family invariably leads to besov spaces yielding a natural relationship among besov smoothness linear nonlinear approximation order and compression performance in a rate distortion sense the designed compression techniques show similarities to modern high performance transform codecs allowing us to establish relevant rate distortion estimates and identify performance limits

a framework for morse theory for the yang mills functional on explique comment la theorie de morse pour la fonctionnelle de yang mills peut se retrouver en examinant la restriction de la fonctionnelle a un ensemble denombrable de varietes non compactes de dimension finie

triangular wavelet based finite elements via multivalued scaling equations this paper develops a class of triangular finite elements based on l2 rn orthonormal compactly supported wavelets as in the case of recently derived tensor product wavelet elements the triangular elements are defined by deriving an algebraic eigenvalue problem characterizing the entries of the elemental matrices these entries are uniquely defined by normalization conditions derived from the polynomial reproducing properties of the wavelet scaling functions the work herein generalizes the development of the tensor product wavelet elements in that the geometry of the triangular elemental domain is characterized using a multivalued scaling equation the multivalued scaling function is used to represent the self similar tiling of rn onto itself by triangular domains

on the equivalence of brushlet and wavelet bases abstract we prove that the meyer wavelet basis and a class of brushlet systems associated with exponential type partitions of the frequency axis form a family of equivalent unconditional bases for the besov and triebel lizorkin function spaces this equivalence is then used to obtain new results on nonlinear approximation with brushlets in triebel lizorkin spaces

nonlinear functionals of wavelet expansions adaptive reconstruction and fast evaluation summary this paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process in particular evaluation covers here the computation of inner products of such expressions with wavelets which arise for instance in the context of galerkin or petrov galerkin schemes for the solution of differential equations the central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion i e the number of nonzero wavelet coefficients as little as possible the following issues are addressed first motivated by previous treatments of the subject we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of besov norms the principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets the analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal we conclude with a simple numerical example

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

on the stability in weak topology of the set of global solutions to the navier stokes equations let x be a suitable function space and let mathcal g subset x be the set of divergence free vector fields generating a global smooth solution to the incompressible homogeneous three dimensional navier stokes equations we prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of mathcal g belongs to mathcal g if n is large enough provided the convergence holds anisotropically in frequency space typically this excludes self similar type convergence anisotropy appears as an important qualitative feature in the analysis of the navier stokes equations it is also shown that initial data which do not belong to mathcal g hence which produce a solution blowing up in finite time cannot have a strong anisotropy in their frequency support

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

singular sobolev connections with holonomy we consider local sobolev connections on su 2 bundles over the complement in r of a smoothly embedded compact 2 manifold finite action implies that a holonomy condition is satisfied and we obtain an a priori estimate for the connection 1 form in terms of curvature and the flat connection carrying the holonomy the a priori estimate classifies the possible singularities in these connections by the set of flat connections in a certain case this leads to smoothness and extendability results let n be a full 4 dimensional neighborhood of the singular set s the objects of study are connections d d a defined on su 2 bundles over x n s we assume that a hloc x and that the action is finite i e the curvature f da a a a is in l n the following holonomy condition was first stated by cliff taubes choose coordinates r 0 u v with u v coordinates on s and r 9 coordinates in a plane normal to s fixing u and v and denoting by ae the 6 component of a the initial value problem for an su 2 valued function

symmetric instantons and the adhm construction we construct allsu 2 yang mills instantons ons 4 that admit a certain symmetry quadrupole symmetry this is accomplished by an equivariant version of the adhm monad classification of instantons this work is part of an attempt to better understand the structure of non self dual yang mills connections with the same symmetry

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k

from projection pursuit and cart to adaptive discriminant analysis while many efforts have been put into the development of nonlinear approximation theory and its applications to signal and image compression encoding and denoising there seems to be very few theoretical developments of adaptive discriminant representations in the area of feature extraction selection and signal classification in this paper we try to advocate the idea that such developments and efforts are worthwhile based on the theoretical study of a data driven discriminant analysis method on a simple yet instructive example we consider the problem of classifying a signal drawn from a mixture of two classes using its projections onto low dimensional subspaces unlike the linear discriminant analysis lda strategy which selects subspaces that do not depend on the observed signal we consider an adaptive sequential selection of projections in the spirit of nonlinear approximation and classification and regression trees cart at each step the subspace is enlarged in a direction that maximizes the mutual information with the unknown class we derive explicit characterizations of this adaptive discriminant analysis ada strategy in two situations when the two classes are gaussian with the same covariance matrix but different means the adaptive subspaces are actually nonadaptive and can be computed with an algorithm similar to orthonormal matching pursuit when the classes are centered gaussians with different covariances the adaptive subspaces are spanned by eigen vectors of an operator given by the covariance matrices just as could be predicted by regular lda however we prove that the order of observation of the components along these eigen vectors actually depends on the observed signal numerical experiments on synthetic data illustrate how data dependent features can be used to outperform lda on a classification task and we discuss how our results could be applied in practice

the existence of nonminimal solutions to the yang mills equation with group rm su 2 on s 2 times s 2 and s 1 times s 3

non self dual yang mills connections with quadrupole symmetry we prove the existence of non self dual yang mills connections onsu 2 bundles over the four sphere specifically on all bundles with second chern number not equal 1 we study connections equivariant under ansu 2 symmetry group to reduce the effective dimensionality from four to one and then use variational techniques the existence of non self dualsu 2 ym connections on the trivial bundle second chern number equals zero has already been established by sibner sibner and uhlenbeck via different methods

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

automatic generation of rbf networks using wavelets abstract learning can be viewed as mapping from an input space to an output space examples of these mappings are used to construct a continuous function that approximates the given data and generalizes for intermediate instances radial basis function rbf networks are used to formulate this approximating function a novel method is introduced that automatically constructs a generalized radial basis function grbf network for a given mapping and error bound this network is shown to be the smallest network within the error bound for the given mapping the integral wavelet transform is used to determine the parameters of the network simple one dimensional examples are used to demonstrate how the network constructed using the transform is superior to that constructed using standard ad hoc optimization techniques the paper concludes with the automatic generation of grbf networks for a multi dimensional problem namely real time 3d object recognition and pose estimation the results of this application are favorable

solutions to yang mills equations that are not self dual abstract the yang mills functional for connections on principle su 2 bundles over s4 is studied critical points of the functional satisfy a system of second order partial differential equations the yang mills equations if in particular the critical point is a minimum it satisfies a first order system the self dual or anti self dual equations here we exhibit an infinite number of finite action nonminimal unstable critical points they are obtained by constructing a topologically nontrivial loop of connections to which min max theory is applied the construction exploits the fundamental relationship between certain invariant instantons on s4 and magnetic monopoles on h3 this result settles a question in gauge field theory that has been open for many years

grey enterprise input output analysis a system whose information is partially known and partially unknown to the investigator is named a grey system due to various inevitable noises contained in the process of data collection managers of enterprise usually have to deal with poor information and make decisions under the influence of uncertainty in this article we propose grey enterprise input output analysis the value and the physical types of grey models are established in particular we present a detailed way of calculating the matrix covered set of the inverse of grey triangular matrix the proposed method makes the management of enterprises more practically possible when the available data contains uncertainty at the end we look at a case study to show the practical feasibility of our work

approximation algorithms for wavelet transform coding of data streams this paper addresses the problem of finding a b term wavelet representation of a given discrete function fepsirn whose distance from is minimized the problem is well understood when we seek to minimize the euclidean distance between f and its representation the first known algorithms for finding provably approximate representations minimizing general lp distances including linfin under a wide variety of compactly supported wavelet bases are presented in this paper for the haar basis a polynomial time approximation scheme is demonstrated these algorithms are applicable in the one pass sublinear space data stream model of computation they generalize naturally to multiple dimensions and weighted norms a universal representation that provides a provable approximation guarantee under all mu norms simultaneously and the first approximation algorithms for bit budget versions of the problem known as adaptive quantization are also presented further it is shown that the algorithms presented here can be used to select a basis from a tree structured dictionary of bases and find a b term representation of the given function that provably approximates its best dictionary basis representation

adaptive transient solution of nonuniform multiconductor transmission lines using wavelets this paper presents a highly adaptive algorithm for the transient simulation of nonuniform interconnects loaded with arbitrary nonlinear and dynamic terminations the discretization of the governing equations is obtained through a weak formulation using biorthogonal wavelet bases as trial and test functions it is shown how the multiresolution properties of wavelets lead to very sparse approximations of the voltages and currents in typical transient analyzes a simple yet effective time space adaptive algorithm capable of selecting the minimal number of unknowns at each time iteration is described numerical results show the high degree of adaptivity of the proposed scheme

wavelet methods for inverting the radon transform with noisy data because the radon transform is a smoothing transform any noise in the radon data becomes magnified when the inverse radon transform is applied among the methods used to deal with this problem is the wavelet vaguelette decomposition wvd coupled with wavelet shrinkage as introduced by donoho 1995 we extend several results of donoho and others here first we introduce a new sufficient condition on wavelets to generate a wvd for a general homogeneous operator whose class includes the radon transform we show that a variant of donoho s method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a besov norm as the smoothing functional we give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise corrupted data we conduct tomographic reconstruction computations that support the hypothesis that near optimal shrinkage parameters can be derived if one can estimate only two besov space parameters about an image f both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than kolaczyk s 1996 variant of donoho s method and the classical filtered backprojection method

wavelet adaptive method for second order elliptic problems boundary conditions and domain decomposition summary wavelet methods allow to combine high order accuracy multilevel preconditioning techniques and adaptive approximation in order to solve efficiently elliptic operator equations one of the main difficulty in this context is the efficient treatment of non homogeneous boundary conditions in this paper we propose a strategy that allows to append such conditions in the setting of space refinement i e adaptive discretizations of second order problems our method is based on the use of compatible multiscale decompositions for both the domain and its boundary and on the possibility of characterizing various function spaces from the numerical properties of these decompositions in particular this allows the construction of a lifting operator which is stable for a certain range of smoothness classes and preserves the compression of the solution in the wavelet basis an explicit construction of the wavelet bases and the lifting is proposed on fairly general domains based on c 0 conforming domain decomposition techniques

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

classification of singular sobolev connections by their holonomy for a connection on a principalsu 2 bundle over a base space with a codimension two singular set a limit holonomy condition is stated in dimension four finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy if there is no holonomy then a codimension two removable singularity theorem is obtained

a construction of interpolating wavelets on invariant sets we introduce the concept of a refinable set relative to a family of contractive mappings on a metric space and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure the notion of a refinable set parallels that of a refinable function which is the basis of wavelet construction the interpolation points we recursively generate from a refinable set by a set theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction we then use this recursive structure for the points to construct multiscale interpolants several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included

on besov regularity of solutions to nonlinear elliptic partial differential equations abstract in this paper we study the regularity of the solutions of some nonlinear elliptic equations in kondratiev spaces on certain domains of polyhedral type general embedding theorems between kondratiev spaces and besov spaces will allow to avoid drawbacks to the standard sobolev regularity theory for those nonsmooth domains this will give us the opportunity to derive optimal rates for certain nonlinear approximation schemes

the atiyah jones conjecture the purpose of this note is to announce our proof of the atiyah jones conjecture concerning the topology of the moduli spaces of based su 2 instantons over s 4 full details and proofs appear in our paper bhmm1

stationary points of the yang mills action we examine the structure of a recently discovered set of non self dual solutions of the yang mills equations these solutions have a symmetry that reduces the ym equations to a set of ode s the distinct solutions are indexed by two postive odd integers we develop a scheme to approximate on a computer the solutions for small values of the indexing integers and present some numerical results we then analyze the asymptotic behavior of the solutions as the indexing integers become large

local linear independence of refinable vectors of functions this paper is devoted to a study of local linear independence of refinable vectors of functions a vector of functions 1 r c ir r is said to be refinable if it satisfies the vector refinement equation x z s a 2x where a is a finitely supported sequence of r r matrices called the refinement mask a complete characterization for the local linear independence of the shifts of 1 r is given strictly in terms of the mask several examples are provided to illustrate the general theory this investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets local linear independence of refinable vectors of functions

on a variational norm tailored to variable basis approximation schemes a variational norm associated with sets of computational units and used in function approximation learning from data and infinite dimensional optimization is investigated for sets gk obtained by varying a vector y of parameters in a fixed structure computational unit k y e g the set of gaussians with free centers and widths upper and lower bounds on the gk variation norms of functions having certain integral representations are given in terms of the 1 norms of the weighting functions in such representations families of functions for which the two norms are equal are described

multiscale modeling and estimation of motion fields for video coding we present a systematic approach to forward motion compensated predictive video coding the first step is the definition of a flexible model that compactly represents motion fields the inhomogeneity and spatial coherence properties of motion fields are captured using linear multiscale models one possible design is based on linear finite elements and yields a multiscale extension of the triangle motion compensation tmc method the second step is the choice of a computational technique that identifies the coefficients of the linear model we study a modified optical flow technique and minimize a cost function closely related to horn and schunck s 1981 criterion the cost function balances accuracy and complexity of the motion compensated predictor and is viewed as a measure of goodness of the motion field it determines not only the coefficients of the model but also the quantization method we formulate the estimation and quantization problems jointly as a discrete optimization problem and solve it using a fast multiscale relaxation algorithm a hierarchical extension of the algorithm allows proper handling of large displacements simulations on a variety of video sequences have produced improvements over tmc and over the half pel accuracy full search block matching algorithm in excess of 0 5 db in average the results are visually superior as well in particular the reconstructed video is entirely free of blocking artifacts

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

regular families of kernels for nonlinear approximation abstract this article studies sufficient conditions on families of approximating kernels which provide n term approximation errors from an associated nonlinear approximation space which match the best known orders of n term wavelet expansion these conditions provide a framework which encompasses some notable approximation kernels including splines cardinal functions and many radial basis functions such as the gaussians and general multiquadrics examples of such kernels are given to justify the criteria additionally the techniques involved allow for some new results on n term greedy interpolation of sobolev functions via radial basis functions

bounds on rates of variable basis and neural network approximation the tightness of bounds on rates of approximation by feedforward neural networks is investigated in a more general context of nonlinear approximation by variable basis functions tight bounds on the worst case error in approximation by linear combinations of n elements of an orthonormal variable basis are derived

optimized general sparse grid approximation spaces for operator equations this paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order based on the framework of tensor product biorthogonal wavelet bases and stable subspace splittings we construct operator adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled specifically for operators of positive order their dimension is o 2 j independent of the dimension n of the problem compared to o 2 jn for the full grid space also for operators of negative order the overall cost is significantly in favor of the new approximation spaces we give cost estimates for the case of continuous linear information we show these results in a constructive manner by proposing a galerkin method together with optimal preconditioning the theory covers elliptic boundary value problems as well as boundary integral equations

a novel cryptosystem based on grey system theory and genetic algorithm abstract in this paper we first attempt to introduce the biological concepts of genetic factor and genetic variation into the study of a kaleidoscope queer in brief k q cryptosystem the novel k q cryptosystem including the genetic factor gm x x can be applied to ensure the security of an information system the main contributions include the k q cryptosystem concept with genetic factor the existence theorem of era k q cryptosystem with genetic factor the limited embedded theorem of k q cryptosystem with genetic factor and the genetic variability theorem of k q cryptosystem

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

integral operators on sparse grids in this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions the spaces are based on biorthogonal anisotropic tensor product wavelets we introduce sparse grid hyperbolic cross approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured furthermore we introduce compression schemes for the corresponding discretizations numerical examples for the laplace equation with dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results