parametrization of compactly supported orthonormal wavelets the parametrization of two channel reconstruction quadrature mirror filter banks is extended to obtain a single characterization of discrete orthonormal wavelets in particular it is shown that discrete orthonormal wavelets of support less than 2n 1 where n is an even integer are parametrized by n 1 real parameters each taking values in 0 2 pi a parametrization of discrete orthonormal wavelets with a number of vanishing moments greater than one is also given

wavelets and the poincare half plane a square integrable signal of positive energy is transformed into an analytic function in the upper half plane on which sl 2 r acts it is shown that this analytic function is determined by its scalar products with the discrete family of functions obtained by acting with sl 2 z on a cyclic vector provided that the spin of the representation is less than 3

finitized conformal spectrum of the ising model on the cylinder and torus abstract the spectrum of the critical ising model on a lattice with cylindrical and toroidal boundary conditions is calculated by commuting transfer matrix methods using a simple truncation procedure we obtain the natural finitizations of the conformal spectra recently proposed by melzer these finitizations imply polynomial identities which in the large lattice limit give rise to the rogers ramanujan identities for the c 1 2 virasoro characters

functions analytic on the half plane as quantum mechanical states a transform between the state space of one dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half plane is constructed it gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple generalizations to higher dimensions are also discussed

paths and partitions combinatorial descriptions of the parafermionic states the zk parafermionic conformal field theories despite the relative complexity of their modes algebra offer the simplest context for the study of the bases of states and their different combinatorial representations three bases are known the classic one is given by strings of the fundamental parafermionic operators whose sequences of modes are in correspondence with restricted partitions with parts at distance k 1 differing at least by 2 another basis is expressed in terms of the ordered modes of the k 1 different parafermionic fields which are in correspondence with the so called multiple partitions both types of partitions have a natural bressoud path representation finally a third basis formulated in terms of different paths is inherited from the solution of the restricted solid on solid model of andrews baxter forrester the aim of this work is to review in a unified and pedagogical exposition these four different combinatorial representations of the states of the zk parafermionic models

a wavelet or lifting scheme based imputation method summary the paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis our method incorporates a bayesian mixture prior for these wavelet coefficients into a gibbs sampler to generate a complete posterior distribution for the variable of interest intuitively the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites we demonstrate the strong performance of our estimator in both one and two dimensional imputation problems where we also compare its application with the standard imputation techniques of kriging and thin plate splines

exponential bases paley wiener spaces and applications abstract we investigate the connection between translation bases for paley wiener spaces and exponential fourier bases for a domain we apply these results to the characterization of vector valued time frequency translations of a paley wiener window signal

a wavelet transform for the poincare group it is noted that certain reducible cyclic representations of the poincare group are square integrable in the sense of wavelet theory the corresponding wavelet transform and resolution of the identity are constructed an inversion formula for an irreducible representation is obtained by taking limits

biorthogonal spline type wavelets let f be an orthonormal scaling function with approximation degree p 1 and let b nbe the b spline of order n then spline type scaling functions defined by n b n n 1 2 possess higher approximation order p n 1 and compact support the corresponding biorthogonal wavelet functions are also constructed this technique is extended to the case of biorthogonal scaling function system as an application of the method supplied in this paper one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function in particular if both the method and the lifting scheme of sweldens see 1 are applied then all pairs of biorthogonal spline type scaling functions shown in references 2 and 3 can be constructed from the haar scaling function

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

nonuniform multiresolution analyses and spectral pairs abstract a generalization of the notion of multiresolution analysis based on the theory of spectral pairs is considered in contrast to the standard setting the associated subspacev0ofl2 r has as an orthonormal basis a collection of translates of the scaling function of the form x where 0 r n 2z n 1 is an integer andris an odd integer with 1 r 2n 1 such thatrandnare relatively prime andzis the set of all integers furthermore the corresponding dilation factor is 2n the case wheren 1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2 a necessary and sufficient condition for the existence of associated wavelets which is always satisfied whenn 1 or 2 is obtained and is shown to always hold if the fourier transform of is a constant multiple of the characteristic function of a set

information theoretic wavelet noise removal for inverse elastic wave scattering theory a discussion of noise removal in ultrasound elastic wave scattering for nondestructive evaluation is given the methods used in this paper include a useful suboptimal wiener filter information theory and orthonormal wavelets the multiresolution analysis mra due to mallat is the key wavelet feature used here whereas fourier transforms have a translational symmetry wavelets have a dilation or affine symmetry which consists of the semi direct product of a translation with a change of scale of the variable the mra describes the scale change features of orthonormal wavelet families first an empirical method of noise removal from scattered elastic waves using wavelets is shown to markedly improve the l 1 and l 2 error norms this suggests that the wavelet scale can act as dial to tune out noise maximization of the kullback liebler information is also shown to provide a scale dependent noise removal technique that supports but does not prove the intuition that certain small energy coefficients that are retained contain large information content the wavelet mra thereby locates islands of information in the phase space of the signal it is conjectured that this method holds more generally

fermionic solution of the andrews baxter forrester model i unification of tba and ctm methods the problem of computing the one dimensional configuration sums of the abf model in regime iii is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules we thus obtain a newfermionic method to compute the local height probabilities of the model combined with the originalbosonic approach of andrews baxter and forrester we obtain a new proof of some of melzer s polynomial identities in the infinite limit these identities yield rogers ramanujan type identities for the virasoro characters l 1 r l r q as conjectured by the stony brook group as a result of our work the corner transfer matrix and thermodynamic bethe ansatz approaches to solvable lattice models are unified

families of biorthogonal wavelets several families of biorthogonal wavelet bases are constructed with various properties in particular for a given filter fn of finite length 2n 1 a parametric family of dual filters fvn of length 2n 1 is constructed the parametric nature of the dual filters makes it possible to design the optimum dual filter fv0n corresponding to a fixed filter fn

development of cfl free explicit schemes for multidimensional advection reaction equations we combine an eulerian lagrangian approach and multiresolution analysis to develop unconditionally stable explicit multilevel methods for multidimensional linear hyperbolic equations the derived schemes generate accurate numerical solutions even if large time steps are used furthermore these schemes have the capability of carrying out adaptive compression without introducing mass balance error computational results are presented to show the strong potential of the numerical methods developed

q trinomial identities we obtain connection coefficients between q binomial and q trinomial coefficients using these one can transform q binomial identities into q trinomial identities and back again to demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of berkovich mccoy and pearce which have recently arisen in their study of the 2 1 and 1 5 perturbations of minimal conformal field theory

local height probabilities in a composite andrews baxter forrester model we study the local height probabilities in a composite height model derived from the restricted solid on solid model introduced by andrews baxter and forrester and their connection with conformal field theory characters the obtained conformal field theories also describe the critical behavior of the model at two different critical points in addition at criticality the model is equivalent to a one dimensional chain of anyons subject to competing two and three body interactions the anyonic chain interpretation provided the original motivation to introduce the composite height model and by obtaining the critical behavior of the composite height model the critical behavior of the anyonic chains is established as well depending on the overall sign of the hamiltonian this critical behavior is described by a diagonal coset model generalizing the minimal models for one sign and by fateev zamolodchikov parafermions for the other

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

proof of the flohr grabow koehn conjectures for characters of logarithmic conformal field theory in a recent paper flohr grabow and koehn conjectured that the characters of the logarithmic conformal field theory ck 1 admit fermionic representations labelled by the lie algebra dk in this paper we provide a simple analytic proof of this conjecture

false theta functions and companions to capparelli s identities abstract capparelli conjectured two modular identities for partitions whose parts satisfy certain gap conditions which were motivated by the calculation of characters for the standard modules of certain affine lie algebras and by vertex operator theory these identities were subsequently proved and refined by andrews who related them to jacobi theta functions and also by alladi andrews gordon capparelli and tamba xie in this paper we prove two new companions to capparelli s identities where the evaluations are expressed in terms of jacobi theta functions and false theta functions

multiresolution representation in unstructured meshes in this paper we describe techniques to represent data which originate from discretization of functions in unstructured meshes in terms of their local scale components to do so we consider a nested sequence of discretization which corresponds to increasing levels of resolution and we define the scales as the difference in information between any two successive levels we obtain data compression by eliminating scale coefficients which are sufficiently small this capability for data compression can be used to reduce the cost of numerical schemes by solving for the more compact representation of the numerical solution in terms of its significant scale coefficients

fermionic solution of the andrews baxter forrester model ii proof of melzer s polynomial identities we compute the one dimensional configuration sums of the abf model using the fermionic technique introduced in part i of this paper combined with the results of andrews baxter and forrester we prove polynomial identities for finitizations of the virasoro characters chi b a r 1 r q as conjectured by melzer in the thermodynamic limit these identities reproduce rogers ramanujan type identities for the unitary minimal virasoro characters conjectured by the stony brook group we also present a list of additional virasoro character identities which follow from our proof of melzer s identities and application of bailey s lemma

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

particles in rsos paths we introduce a new representation of the paths of the forrester baxter restricted solid on solid rsos models which represents the states of the irreducible modules of the minimal models this representation is obtained by transforming the rsos paths for the cases p 2p 1 to new paths for which horizontal edges are allowed at certain heights these new paths are much simpler in that their weight is nothing but the sum of the position of the peaks this description paves the way for the interpretation of the rsos paths in terms of fermi type charged particles out of which the fermionic characters could be obtained constructively the derivation of the fermionic character for p 2 and p kp 1 is outlined finally the particles of the rsos paths are put in relation to the kinks and the breathers of the restricted sine gordon model

a nonlocal operator basis from the path representation of the cal m k 1 hbox k 2 and the cal m k 1 hbox 2k 3 minimal models we reinterpret a path describing a state in an irreducible module of the unitary minimal model in terms of a string of charged operators acting on the module s ground state path each such operator acts non locally on a path the path characteristics are then translated into a set of conditions on sequences of operators that provide an operator basis as an application we re derive the vacuum finite fermionic character by constructing the generating function of these basis states these results generalize directly to the models the close relatives of the unitary models in terms of path description

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

exceptional structure of the dilute a 3 model e 8 and e 7 rogers ramanujan identities the dilute a3 lattice model in regime 2 is in the universality class of the ising model in a magnetic field here we directly establish the existence of an e8 structure in the dilute a3 model in this regime by expressing the one dimensional configuration sums in terms of fermionic sums which involve the e8 root system explicitly in the thermodynamic limit these polynomial identities yield a proof of the e8 rogers ramanujan identity recently conjectured by kedem et al 1993 the polynomial identities also apply to regime 3 which is obtained by transforming the modular parameter by q to 1 q in this case we find an a1xe7 structure and prove a rogers ramanujan identity of a1xe7 type finally in the critical q to 1 limit we give some intriguing expressions for the number of l step paths on the a3 dynkin diagram with tadpoles in terms of the e8 cartan matrix all our findings confirm the e8 and e7 structure of the dilute a3 model found recently by means of the thermodynamic bethe ansatz

periodic quasi orthogonal spline bases and applications to least squares curve fitting of digital images presents a new covariant basis dubbed the quasi orthogonal q spline basis for the space of n degree periodic uniform splines with k knots this basis is obtained analogously to the b spline basis by scaling and periodically translating a single spline function of bounded support the construction hinges on an important theorem involving the asymptotic behavior in the dimension of the inverse of banded toeplitz matrices the authors show that the gram matrix for this basis is nearly diagonal hence the name quasi orthogonal the new basis is applied to the problem of approximating closed digital curves in 2d images by least squares fitting since the new spline basis is almost orthogonal the least squares solution can be approximated by decimating a convolution between a resolution dependent kernel and the given data the approximating curve is expressed as a linear combination of the new spline functions and new control points another convolution maps these control points to the classical b spline control points a generalization of the result has relevance to the solution of regularized fitting problems

joint similarities and parameterizations for naimark complementary frames abstract it is known that the naimark complementary frames for a given frame are not necessarily unique up to the similarity in this paper we introduce the concept of joint complementary frame pairs for a given dual frame pair and prove that they are unique up to the joint similarity as an application we give a necessary and sufficient condition under which two naimark complementary frames are similar for different pairs of dual frames we present an operator parameterization for their joint complementary frame pairs

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

a load identification method based on wavelet multi resolution analysis abstract load identification as a kind of indirect identification method uses system characteristic and responses to calculate loads a method based on wavelet multi resolution analysis is proposed in this paper by wavelet decomposition and transform at certain resolutions the proposed method transforms the convolution relation between responses and loads in time domain into the linear multiplicative relation between system responses and wavelet responses in the wavelet domain loads can be identified as long as the linear multiplicative relation is solved qualitative and quantitative rules are proposed for selecting parameters that affect the accuracy of the proposed method and are illustrated via numerical investigations the method is illustrated by a multi input multi output numerical simulation a multi input multi output laboratory experiment is performed to compare the proposed method with the frequency method on the identification ability

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

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

refinement of operator valued reproducing kernels this paper studies the construction of a refinement kernel for a given operator valued reproducing kernel such that the vector valued reproducing kernel hilbert space of the refinement kernel contains that of the given kernel as a subspace the study is motivated from the need of updating the current operator valued reproducing kernel in multi task learning when underfitting or overfitting occurs numerical simulations confirm that the established refinement kernel method is able to meet this need various characterizations are provided based on feature maps and vector valued integral representations of operator valued reproducing kernels concrete examples of refining translation invariant and finite hilbert schmidt operator valued reproducing kernels are provided other examples include refinement of hessian of scalar valued translation invariant kernels and transformation kernels existence and properties of operator valued reproducing kernels preserved during the refinement process are also investigated

an independence result on cotorsion theories over valuation domains it is shown that over suitable valuation domains r with field of quotients q the cotorsion theory gk generated by k q r coincides with the cotorsion theory c cogenerated by the fuchs divisible module provided that godel s axiom of constructibility v l is assumed on the other hand assuming martin s axiom and the negation of the continuum hypothesis it is proved that the cotorsion theory gk is strictly smaller than c by exhibiting a strongly 1 k free divisible module m of projective dimension 2 such that ext1r m k 0 applications to whitehead modules are derived

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

algorithms for designing wavelets to match a specified signal algorithms for designing a mother wavelet spl psi x such that it matches a signal of interest and such that the family of wavelets 2 sup j 2 spl psi 2 sup j x k forms an orthonormal riesz basis of l sup 2 spl rscr are developed the algorithms are based on a closed form solution for finding the scaling function spectrum from the wavelet spectrum many applications require wavelets that are matched to a signal of interest most current design techniques however do not design the wavelet directly they either build a composite wavelet from a library of previously designed wavelets modify the bases in an existing multiresolution analysis or design a scaling function that generates a multiresolution analysis with some desired properties in this paper two sets of equations are developed that allow us to design the wavelet directly from the signal of interest both sets impose bandlimitedness resulting in closed form solutions the first set derives expressions for continuous matched wavelet spectrum amplitudes the second set of equations provides a direct discrete algorithm for calculating close approximations to the optimal complex wavelet spectrum the discrete solution for the matched wavelet spectrum amplitude is identical to that of the continuous solution at the sampled frequencies an interesting byproduct of this work is the result that meyer s spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary specific examples are given which demonstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals

towards a multiresolution approach to linear control we develop a multiresolution approximation framework for linear control we construct a multiresolution analysis of the set of input functions of a linear system the approximation of an input u at a scale j is defined as the input uj of minimal energy such that the trajectories of the system associated with u and uj coincide on a grid of step length 2 j we propose a set of wavelet functions which generate this multiresolution analysis these functions called control theoretic wavelets satisfy useful properties for the representation of control inputs of a linear system as an example of application of our multiresolution approximation framework we propose a method for efficient encoding of control inputs with regard to several criteria

q trinomial coefficients and the dilute a model we express the configuration sums of the dilute model in regimes and in terms of q trinomial coefficients we then extend previous results to give expressions for all phases and regimes for odd l for finite system size

wavelet transform based adaptive filters analysis and new results in this paper the wavelet transform is used in an adaptive filtering structure the coefficients of the adaptive filter are updated by the help of the least mean square lms algorithm first the wavelet transform based adaptive filter wtaf is described and it is analyzed for its wiener optimal solution then the performance of the wtaf is studied by the help of learning curves for three different convergence factors 1 constant convergence factor 2 time varying convergence factor and 3 exponentially weighted convergence factor the exponentially weighted convergence factor is proposed to introduce scale based variation to the weight update equation it is shown for two different sets of data that the rate of convergence increases significantly for all three wtaf structures as compared to that of time domain lms the high convergence rates of the wtaf give us reason to expect that it will perform well in tracking rapid changes in a signal

local reproducible smoothing without shrinkage a simple local smoothing filter is defined for curves or surfaces combining the advantages of gaussian smoothing and fourier curve description unlike gaussian filters the filter described has no shrinkage problem repeated application of the filter does not yield a curve or surface smaller than the original but simply reproduces the approximate result that would have been obtained by a single application at the largest scale unlike fourier description the filter is local in space the wavelet transform of y meyer 1989 is also shown to have these properties

polynomial identities indices and duality for the n 1 superconformal model sm 2 4v we prove polynomial identities for then 1 superconformal modelsm 2 4v which generalize and extend the known fermi bose character identities our proof uses theq trinomial coefficients of andrews and baxter on the bosonic side and a recently introduced very general method of producing recursion relations forq series on the fermionic side we use these polynomials to demonstrate a dual relation underq q 1 betweensm 2 4v andm 2v 1 4v we also introduce a genralization of the witten index which is expressible in terms of the rogers false theta functions

n 2 supersymmetry and bailey pairs abstract we demonstrate that the bailey pair formulation of rogers ramanujan identities unifies the calculations of the characters of n 1 and n 2 supersymmetric conformal field theories with the counterpart theory with no supersymmetry we illustrate this construction for the m 3 4 ising model where the bailey pairs have been given by slater we then present the general unitary case we demonstrate that the model m p p 1 is derived from m p 1 p by a bailey renormalization flow and conclude by obtaining the n 1 model sm p p 2 and the unitary n 2 model with central charge c 3 1 2 p

multi resolution multi scale topology optimization a new paradigm abstract the purpose of this work is to present a new concept multi resolution multi scale topology optimization the key idea of the present strategy is that design optimization should be performed progressively from low to high resolution not at a single resolution level to achieve the multi resolution strategy design optimization is formulated in a wavelet based variable space not in a direct density variable space the major advantages of the multi resolution design optimization include 1 topologically simple and close to the global optimum structures may be obtained without any explicit constraint and 2 the convergence is not sensitive to mathematical programming methods for the efficient numerical implementation of the multi resolution approach the side constraints imposed on the direct density variables are removed by mapping the density variables into intermediate variables these intermediate variables are then wavelet transformed to new design variables it is addressed that the present multi resolution topology optimization can resolve major numerical instability problems such as mesh dependencies and local minima the usefulness of the multi scale nature of the wavelets in the present multi resolution multi scale optimization formulation is also discussed

adaptive learning for event modeling and characterization adaptive learning of specific patterns or events of interest has been an area of significant research for various applications in the last two decades in developing diagnostic evaluation and safety monitoring applications of a propulsion system it is critical to detect characterize and model events of interest it is a challenging task since the detection system should allow adaptive characterization of potential events of interest and correlate them to learn new models for future detection for online health monitoring and diagnostic evaluation in this paper a novel framework is established using a hierarchical adaptive clustering approach with fuzzy membership functions to characterize specific events of interest from the measured and processed features raw engine measurement data is first analyzed using the wavelet transform to provide features for localization of frequency information for use in the classification system a method combining hierarchical and fuzzy k means clustering is then applied to a set of selected measurements and computed features to determine the events of interest during engine operations experimental results have shown that the proposed approach is effective and computationally efficient to detect characterize and model new events of interest from data collected through continuous operations

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling