the discrete wavelet transform for a symmetric anti symmetric multiwavelet family on the interval the chui lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval this requires in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets four new functions of each type at each edge of the interval i e one more than required just to preserve the approximation order the criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions and a specific choice is examined in applications to both noiseless and noisy data these are accomplished through the development of interval basis extensions for both an earlier multiwavelet projection based prefilter postfilter combination and a multiwavelet vector thresholding algorithm

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

towards the conservation laws and lie symmetries for the khokhlov zabolotskaya equation in three dimensions the authors have obtained the explicit structure for the generating function of lie symmetries for the khokhlov zabolotskaya equation describing the propagation of a sound beam in a non linear medium in the absence of a canonical framework they have derived the set of conservation laws through the technique of differential forms and prolongation such conservation laws are of utmost importance for the analysis of a sound beam in the medium

solution of atomic orbitals in an interpolating wavelet basis the schrodinger equation of hydrogenic atoms and the hartree fock equations of some many electron atoms are solved using interpolating wavelets as basis functions the nonstandard operator form is used to compute operators in basis sets including multiple resolution levels we introduce an algorithm for converting matrices from nonstandard operator form to standard operator form we also consider the different components of the hamiltonian and fock operators separately and derive analytic formulas for their evaluation extension to many electron atoms is done within the hartree fock formalism convergence of atomic parameters such as orbital eigenvalues with respect to the number of resolution levels is inspected numerically for hydrogenlike atoms ions and some light many electron atoms helium lithium beryllium neon sodium magnesium and argon

the application of multiwavelet filterbanks to image processing multiwavelets are a new addition to the body of wavelet theory realizable as matrix valued filterbanks leading to wavelet bases multiwavelets offer simultaneous orthogonality symmetry and short support which is not possible with scalar two channel wavelet systems after reviewing this theory we examine the use of multiwavelets in a filterbank setting for discrete time signal and image processing multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank we describe two methods repeated row and approximation deapproximation for obtaining such a vector input stream from a one dimensional 1 d signal algorithms for symmetric extension of signals at boundaries are then developed and naturally integrated with approximation based preprocessing we describe an additional algorithm for multiwavelet processing of two dimensional 2 d signals two rows at a time and develop a new family of multiwavelets the constrained pairs that is well suited to this approach this suite of novel techniques is then applied to two basic signal processing problems denoising via wavelet shrinkage and data compression after developing the approach via model problems in one dimension we apply multiwavelet processing to images frequently obtaining performance superior to the comparable scalar wavelet transform

integrable systems without the painleve property we examine whether the painleve property is a necessary condition for the integrability of nonlinear ordinary differential equations we show that for a large class of linearizable systems this is not the case in the discrete domain we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non painleve continuous linearizable systems we find that while these discrete systems are themselves linearizable they possess non confined singularities

symmetries of the khokhlov zabolotskaya equation the group of lie symmetries of the khokhlov zabolotskaya equation in two and three space dimensions is given

four dimensional lattice gauge theory with ribbon categories and the crane yetter state sum lattice gauge theory in four dimensional euclidean space time is generalized to ribbon categories which replace the category of representations of the gauge group this provides a framework in which the gauge group becomes a quantum group while space time is still given by the classical lattice at the technical level this construction generalizes the spin foam model dual to lattice gauge theory and defines the partition function for a given triangulation of a closed and oriented piecewise linear four manifold this definition encompasses both the standard formulation of d 4 pure yang mills theory on a lattice and the crane yetter invariant of four manifolds the construction also implies that certain classes of spin foam models formulated using ribbon categories are well defined even if they do not correspond to a topological quantum field theory

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

multiwavelet moments and projection prefilters an efficient projection procedure is derived for use of orthogonal multiwavelets in the analysis of discrete data sequences a family of simple prefilters corresponding to numerical quadrature evaluation of the projection integrals provides exact results for locally polynomial data the full approximation order of the multiwavelet basis can thus always be enabled for nonpolynomial signals the prefilters provide approximations to the coefficients of the multiwavelet series whose convergence accelerates quickly with increase in sampling rate comparison is also made with previous time invariant multiwavelet prefilters

wavelet solutions for the dirichlet problem a modified classical penalty method for solving a dirichlet boundary value problem is presented this new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains it is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast approximate calculation of boundary integral the compact support and orthonormality of the basis are essential for representing the boundary measure numerically and therefore for implementing this methodology

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

symmetric daubechies wavelets and numerical solution of nls equations recent work has shown that wavelet based numerical schemes are at least as effective and accurate as standard methods and may allow an easy implementation of a spacetime adaptive grid up to now wavelets which have been used for such studies are the classical ones real daubechies wavelets splines shannon and meyer wavelets etc and were applied to diffusion type equations the present work differs in two points firstly for the first time we use a new set of complex symmetric wavelets which have been found recently the advantage of this set is that unlike classical wavelets they are simultaneously orthogonal compactly supported and symmetric secondly we apply these wavelets to the physically meaningful cubic and quintic nonlinear schrodinger equations the most common method to simulate these models numerically is the symmetrized split step fourier method for the first time we propose and study a new way of implementing a global spacetime adaptive discretization in this numerical scheme based on the interpolation properties of complex symmetric scaling functions second we propose a locally adaptive split step wavelet method

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

comment on towards the conservation laws and lie symmetries for the khokhlov zabolotskaya equation in three dimensions the authors comment on the lie point symmetries for the khokhlov zabolotskaya equation as calculated by roy chowdhury and nasker 1986 and demonstrate that their result for the coefficients of the vector field is correct but incomplete

infinite dimensional symmetries of two dimensional generalized burgers equations the conditions for a class of generalized burgers equations which a priori involve nine arbitrary functions of one or two variables to allow an infinite dimensional symmetry algebra are determined although this algebra can involve up to two arbitrary functions of time it does not allow a virasoro subalgebra this result reconfirms a long standing fact that variable coefficient generalizations of a nonintegrable equation should be expected to remain as such

wavelet and finite element solutions for the neumann problem using fictitious domains this paper presents a new fictitious domain formulation for the solution of a strongly elliptic boundary value problem with neumann boundary conditions for a bounded domain in a finite dimensional euclidean space with a smooth possibly only lipschitz boundary this extends the domain to a larger rectangular domain with periodic boundary conditions for which fast solvers are available the extended solution converges on the original domain in the appropriate function spaces as the penalty parameter approaches zero both wavelet galerkin and finite elements numerical approximation schemes are developed using this methodology the convergence rates of both schemes are comparable and the use of finite elements requires a parameterization of the boundary while the wavelet galerkin method admits an implicit description of the boundary in terms of a wavelet representation of the boundary measure defined as the distributional gradient of the characteristic function of the interior the accuracy of both methods is investigated and compared both theoretically and for numerical test cases the conclusion is that the methods are comparable and that the wavelet method allows the use of more general boundaries which are not explicitly parametrized which would be of greater advantage in higher dimensions the numerical tests are carried out in two dimensions

exact duality transformations for sigma models and gauge theories we present an exact duality transformation in the framework of statistical mechanics for various lattice models with non abelian global or local symmetries the transformation applies to sigma models with variables in a compact lie group g with global g g symmetry the chiral model and with variables in coset spaces g h and a global g symmetry for example the nonlinear o n or rpn models in any dimension d 1 it is also available for lattice gauge theories with local gauge symmetry in dimensions d 2 and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory the duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model transformations are available for the partition function for expectation values of fundamental variables correlators and generalized wilson loops and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition

symmetries and invariant solutions of the two dimensional variable coefficient burgers equation we discuss symmetries and reductions of the two dimensional burgers equation with variable coefficient we classify one dimensional and two dimensional subalgebras of the burgers symmetry algebra which is infinite dimensional into conjugacy classes under the adjoint action of the symmetry group invariance under one dimensional subalgebras provides reductions to lower dimensional partial differential equations further reductions of these equations to second order ordinary differential equations are obtained through invariance under two dimensional subalgebras the reduced odes are then analysed and shown that they belong to the polynomial class of second order equations which can be linearized only for particular values of parameters figuring in the coefficient

wavelets and the numerical solution of partial differential equations we present a new numerical method for the solution of partial differential equations in nonseparable domains the method uses a wavelet galerkin solver with a nontrivial adaptation of the standard capacitance matrix method the numerical solutions exhibit spectral convergence with regard to the order of the compactly supported daubechies wavelet basis furthermore the rate of convergence is found to be independent of the geometry we solve the helmholtz equation since for variations in the parameter the solutions have qualitative properties that well illustrate the applications of our method

wavelet galerkin discretization of hyperbolic equations the relative merits of the wavelet galerkin solution of hyperbolic partial differential equations typical of geophysical problems are quantitatively and qualitatively compared to traditional finite difference and fourier pseudo spectral methods the wavelet galerkin solution presented here is found to be a viable alternative to the two conventional techniques

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

continuum limits and exact finite size scaling functions for one dimensional o n invariant spin models we solve exactly the general one dimensionalo n invariant spin model taking values in the spheresn 1 with nearest neighbor interactions in finite volume with periodic boundary conditions by an expansion in hyperspherical harmonics the possible continuum limits are discussed for a general one parameter family of interactions and an infinite number of universality classes is found for these classes we compute the finite size scaling functions and the leading corrections to finite size scaling a special two parameter family of interactions which includes the mixed isovector isotensor model is also treated and no additional universality classes appear in the appendices we give new formulae for the clebsch gordan coefficients and 6 j symbols of theo n group and some new generalizations of the poisson summation formula these may be of independent interest

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

quadrature prefilters for the discrete wavelet transform discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data prefilters based on lagrange interpolants are derived here for biorthogonal compact support wavelet systems providing exact subspace projection in cases of local polynomial smoothness the resulting divergence acceleration in a nonpolynomial test case is examined irregular sampling rates are also accommodated

galerkin wavelet methods for two point boundary value problems anti derivatives of wavelets are used for the numerical solution of differential equations optimal error estimates are obtained in the applications to two point boundary value problems of second order the orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems numerical examples are given

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

wavelet galerkin algorithms for boundary integral equations the implementation of a fast wavelet based galerkin discretization of second kind integral equations on piecewise smooth surfaces gamma subset ir 3 is described it allows meshes consisting of triangles as well as quadrilaterals the algorithm generates a sparse approximate stiffness matrix with cn o n log n 2 nonvanishing entries in o n log n 4 operations where n is the number of degrees of freedom on the boundary while essentially retaining the asymptotic convergence rate of the full galerkin scheme a new proof of the matrix compression estimates is given based on derivative free kernel estimates the condition number of the sparse stiffness matrices is bounded independently of the meshwidth the data structure containing the compressed stiffness matrix is described in detail it requires o cn memory and can be set up in o cn operations numerical experiments show that the asymptotic performance estimates apply for moderate n problems with n 106 degrees of freedom were computed in core on a workstation the impact of various parameters in the compression scheme on the performance and the accuracy of the algorithm is studied

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

exact solutions of the two dimensional burgers equation soliton like solutions of the two dimensional burgers equation are obtained some new solutions from soliton like solutions are also presented

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

group invariant solutions of a nonlinear acoustics model based on a recent classification of subalgebras of the symmetry algebra of the zabolotskaya khokhlov equation all similarity reductions of this equation into ordinary differential equations are obtained large classes of group invariant solutions of the equation are also determined and some properties of the reduced equations and exact solutions are discussed

the computation of wavelet galerkin approximation on a bounded interval this paper describes exact evaluations of various finite integrals whose integrands involve products of daubechies compactly supported wavelets and their derivatives and or integrals these finite integrals play an essential role in the wavelet galerkin approximation of differential or integral equations on a bounded interval

numerical solution of the schrodinger equation in a wavelet basis for hydrogen like atoms an iterative method is proposed to solve the schrodinger eigenvalue problem in a wavelet framework orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix this representation called the nonstandard ns form is obtained by means of the beylkin coifman rokhlin bcr algorithm and simplifies the numerical calculations problems due to the one dimensional mathematical model and to the discretization process receive special attention

adaptive multiwavelet initialization the multiwavelet concept uses a set of scaling functions and a set of wavelets to generate an orthogonal multichannel multiresolution pyramid decomposition for finite energy signals the multiple scaling functions extract the lowpass information of the input data set and the multiwavelets extract the bandpass information when the lowpass filters associated with the scaling functions are cascaded a multiresolution pyramid decomposition reconstruction system is formed with each pyramid convolution operator having several inputs and several outputs however there is only one data set available to initialize this process this paper addresses the question of how to best modify the data set using prefilters so that its decomposition contains the most useful information for the chosen application the best prefilters are determined by the minimization of the energy of preselected decomposition components the resulting decomposition is therefore signal adaptive and under appropriate conditions perfect reconstruction of the input data set can be achieved with a proper postfiltering the detailed discussions in the paper are focussed on the two wavelet and two scaling function case the general multiwavelet case is addressed at the end of the paper a compression example is provided to demonstrate the performance of the optimally initialized multiwavelet method and to compare it with a single wavelet method and another multiwavelet initialization method proposed by other authors

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

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

higher order wavelet reconstruction differentiation filters and gibbs phenomena an orthogonal wavelet basis is characterized by its approximation order which relates to the ability of the basis to represent general smooth functions on a given scale it is known though perhaps not widely known that there are ways of exceeding the approximation order i e achieving higher order error in the discretized wavelet transform and its inverse the focus here is on the development of a practical formulation to accomplish this first for 1d smooth functions then for 1d functions with discontinuities and then for multidimensional here 2d functions with discontinuities it is shown how to transcend both the wavelet approximation order and the 2d gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet basis grid

a wavelet galerkin scheme for analysis of large scale problems on simple domains a wavelet galerkin scheme tailored to address the numerical solution of large scale boundary value problems defined on domains of simple geometry is presented the variation of parameters e g material properties within the domain is arbitrary but the method is specifically designed to solve problems where parameters vary in raster like fashion boundary conditions are imposed via lagrange multipliers using a fictitious domain approach a preconditioner specially designed for this problem is developed to guarantee that convergence of conjugate gradient algorithms is quick and insensitive to problem size the strategy is applied to the solution of steady state heat conduction problems in 2 d but it can be generalized without conceptual changes to 3 d problems and to problems in linear elasticity copyright 1999 john wiley sons ltd

analysis of the symmetry group and exact solutions of the dispersionless kp equation in n 1 dimensions the lie algebra of the symmetry group of the n 1 dimensional generalization of the dispersionless kadomtsev petviashvili equation is obtained and identified as a semi direct sum of a finite dimensional simple lie algebra and an infinite dimensional nilpotent subalgebra group transformation properties of solutions under the subalgebra sl 2 r are presented known explicit analytic solutions in the literature are shown to be actually group invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group

multiscale computation with interpolating wavelets multiresolution analyses based upon interpolets interpolating scaling functions introduced by deslauriers and dubuc are particularly well suited to physical applications because they allowexactrecovery of the multiresolution representation of a function from its sample values on afiniteset of points in space we present a detailed study of the application of wavelet concepts to physical problems expressed in such bases the manuscript describes algorithms for the associated transforms which for properly constructed grids of variable resolution compute correctly without having to introduce extra grid points we demonstrate that for the application of local homogeneous operators in such bases the nonstandard multiply of beylkin coifman and rokhlin also proceeds exactly for inhomogeneous grids of appropriate form to obtain less stringent conditions on the grids we generalize the nonstandard multiply so that communication may proceed between nonadjacent levels the manuscript concludes with timing comparisons against nai x0308 ve algorithms and an illustration of the scale independence of the convergence rate of the conjugate gradient solution of poisson s equation using a simple preconditioning

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

compactly supported wavelets and boundary conditions abstract wavelet bases subject to various homogeneous boundary conditions on an interval are constructed they are obtained from daubechies compactly supported wavelets which are restricted to the interval following an idea by y meyer and then suitably modified the sobolev spaces arising from homogeneous boundary problems are characterized in terms of wavelet coefficients in these bases

projection based prefiltering for multiwavelet transforms we introduce a method for initializing the multiwavelet decomposition algorithm the initialization procedure is the orthogonal projection of the input signal into the space defined by the multiscaling function the approach will always have a solution places no restrictions on the input except that it be contained within l sub 2 and can be implemented in a fast algorithm we present the details of our approach and compare it with another proposed method of prefiltering

painleve analysis of the two dimensional burgers equation a painleve analysis of the two dimensional burgers equation is carried out and used to obtain a restricted backlund transformation that maps a subclass of the solutions of the 1 2d burgers equation onto a linear heat like equation alternatively the backlund transformation can be expressed as a map onto the derivative of the one dimensional burgers equation in appropriate dependent and independent variables the singularity analysis also yields a further class of solutions obtained by solving a schwarzian differential equation

a new prefilter design for discrete multiwavelet transforms in conventional wavelet transforms prefiltering is not necessary due to the lowpass property of a scaling function this is no longer true for multiwavelet transforms a few research papers on the design of prefilters have appeared but the existing prefilters are usually not orthogonal which often causes problems in coding moreover the condition on the prefilters was imposed based on the first step discrete multiwavelet decomposition we propose a new prefilter design that combines the ideas of the conventional wavelet transforms and multiwavelet transforms the prefilters are orthogonal but nonmaximally decimated they are derived from a very natural calculation of multiwavelet transform coefficients in this new prefilter design multiple step discrete multiwavelet decomposition is taken into account our numerical examples by taking care of the redundant prefiltering indicate that the energy compaction ratio with the geronimo hardin massopust 1994 2 wavelet transform and our new prefiltering is better than the one with daubechies d sub 4 wavelet transform

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed