trigonometric hermite wavelet approximation for the integral equations of second kind with weakly singular kernel this paper is concerned with a trigonometric hermite wavelet galerkin method for the fredholm integral equations with weakly singular kernel the kernel function of this integral equation considered here includes two parts a weakly singular kernel part and a smooth kernel part the approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric hermite wavelet constructed by e quak trigonometric wavelets for hermite interpolation math comp 65 1996 683 722 are developed the use of trigonometric hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix it is shown that we only need to compute and store o n entries for the weakly singular kernel representation matrix with dimensions n 2 which can reduce the whole computational cost and storage expense the computational schemes of the resulting matrix elements are provided for the weakly singular kernel function furthermore the convergence analysis is developed for the trigonometric wavelet method in this paper

a two dimensional wavelet packet transform for matrix compression of integral equations with highly oscillatory kernel we examine the use of wavelet packets for the fast solution of integral equations with a highly oscillatory kernel the redundancy of the wavelet packet transform allows the selection of a basis tailored to the problem at hand it is shown that a well chosen wavelet packet basis is better suited to compress the discretized system than wavelets the complexity of the matrix vector product in an iterative solution method is then substantially reduced a two dimensional wavelet packet transform is derived and compared with a number of one dimensional transforms that were presented earlier in literature by means of some numerical experiments we illustrate the improved efficiency of the two dimensional approach

a study of wavelets for the solution of electromagnetic integral equations the use of wavelet basis functions for the efficient solution of electromagnetic integral equations is studied it has previously been demonstrated that the use of wavelets for expansion and testing functions produces a sparse moment method matrix here this effect is examined and analyzed in terms of the radiation receiving characteristics of the wavelet basis functions the limitations of wavelets as an efficient solution technique are discussed and a comparison is made to other fast algorithms

a note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel this paper analyzes the effect of large wave numbers on the wavelet method for integral equations arising in electromagnetic applications it is shown that the compression of the stiffness matrix deteriorates with increasing wave number a characteristic that has been reported before in the literature here however the exact dependence on the wave number is calculated analytically for the two dimensional helmholtz problem

plane wave interpolation in direct collocation boundary element method for radiation and wave scattering numerical aspects and applications the classical boundary element formulation for the helmholtz equation is rehearsed and its limitations with respect to the number of variables needed to model a wavelength are explained a new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves propagating in a range of directions this is termed the plane wave basis boundary element method the modifications needed to the classical procedures in terms of integration of the element matrices and location of collocation points are described the well known singular value decomposition solution technique which is adopted here for the solution of the system matrix equation in its complex form is briefly outlined the conditioning of the system matrix is analysed for a simple radiation problem the corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions the chief method is adopted to enhance the quality of the solution particularly in the vicinity of irregular frequencies the plane wave basis boundary element method is then applied to two problems scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem in both cases results are compared with analytical solutions the results clearly demonstrate that the new method is considerably more efficient than the classical approach for a given number of degrees of freedom the frequency for which accurate results can be obtained using the new technique can be up to three or four times higher than that of the classical method this makes the method a powerful new addition to our tools for tackling high frequency radiation and scattering problems

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

on the structure of inhomogeneous quantum groups we investigate inhomogeneous quantum groups g built from a quantum group h and translations the corresponding commutation relations contain inhomogeneous terms under certain conditions which are satisfied in our study of quantum poincare groups 12 we prove that our construction has correct size find the r matrices and the analogues of minkowski space for g

on a class of predefined wavelet packet bases for efficient representation of electromagnetic integral equations a general wavelet packet tree is proposed to design predefined wavelet packet pwp bases for the efficient representation of electrodynamic integral equations the wavelet packet decomposition tree is constructed by zooming in along the spectral oscillatory frequency of the free space green s function numerical results show that for typical two dimensional 2 d scatterers the number of above threshold elements in the pwp based moment matrix is on the order of o n sup 1 3 and tends to grow at a rate of o n spl middot log n for large scale problems therefore the complexity of solving the moment equations can be reduced accordingly furthermore it is shown that the elements of the moment matrix based on the pwp bases can be computed directly at approximately the same complexity as the fast wavelet transform approach consequently with on the fly thresholding of the matrix elements the o n sup 2 memory bottleneck in the formation of the pwp based moment matrix can be circumvented

fast solution of electromagnetic integral equations using adaptive wavelet packet transform the adaptive wavelet packet transform is applied to sparsify the moment matrices for the fast solution of electromagnetic integral equations in the algorithm a cost function is employed to adaptively select the optimal wavelet packet expansion testing functions to achieve the maximum sparsity possible in the resulting transformed system the search for the best wavelet packet basis and the moment matrix transformation are implemented by repeated two channel filtering of the original moment matrix with a pair of quadrature filters it is found that the sparsified matrix has above threshold elements that grow only as o n sup 1 4 for typical scatterers consequently the operations to solve the transformed moment equation using the conjugate gradient method scales as o n sup 1 4 the additional computational cost for carrying out the adaptive wavelet packet transform is evaluated and discussed

wavelet packets for fast solution of electromagnetic integral equations this paper considers the problem of wavelet sparsification of matrices arising in the numerical solution of electromagnetic integral equations by the method of moments scattering of plane waves from two dimensional 2 d cylinders is computed numerically using a constant number of test functions per wavelength discrete wavelet packet dwp similarity transformations and thresholding are applied to system matrices to obtain sparsity if thresholds are selected to keep the relative residual error constant the matrix sparsity is of order o n sup p with p 2 this stands in contrast with o n sup 2 sparsities obtained with standard wavelet transformations numerical tests also show that the dwp method yields faster matrix vector multiplication than some fast multipole algorithms

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

composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions the computation of wavelet coefficients of a function typically requires the computation of a large number of integrals these integrals represent the inner product of that function with a wavelet function on different scales or with the corresponding scaling function on a fine scale we develop quadrature rules for those integrals that converge fast for piecewise smooth and singular functions they do not require the evaluation of the scaling function and the convergence does not depend on the smoothness of that function the analysis and computation is based completely on the filter coefficients that define the scaling function an application is presented from the field of electromagnetics involving the inner product of a singular function with two dimensional tensor product wavelets

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

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

two weight norm inequalities for commutators of potential type integral operators abstract we give a condition which is sufficient for the two weight p q inequalities for commutators of potential type integral operators

lie algebraic structure from inhomogeneous hopf algebras we construct the vector space dual to the space of right invariant differential forms constructed from a first order differential calculus on inhomogeneous quantum groups we show that this vector space is equipped with a structure of a hopf algebra which closes on a noncommutative lie algebra satisfying a jacobi identity

differential calculus onisoq n quantum poincare algebra and q gravity we present a general method to deform the inhomogeneous algebras of thebn cn dn type and find the corresponding bicovariant differential calculus the method is based on a projection frombn 1 cn 1 dn 1 for example we obtain the bicovariant inhomogeneousq algebraisoq n as a consistent projection of the bicovariant q algebrasoq n 2 this projection works for particular multiparametric deformations ofso n 2 the so called minimal deformations the case ofisoq 4 is studied in detail a real form corresponding to a lorentz signature exists only for one of the minimal deformations depending on one parameterq the quantum poincare lie algebra is given explicitly it has 10 generators no dilatations and contains theclassical lorentz algebra only the commutation relations involving the momenta depend onq finally we discuss aq deformation of gravity based on the gauging of thisq poincare algebra the lagrangian generalizes the usual einstein cartan lagrangian

the brst operator of quantum symmetries the quantum analogs of donaldson invariants we construct the brst operator of quantum symmetries and show that the nilpotency of this operator can either be derived from the hopf axiom structure of the quantum group symmetries or from the jacobi identity of their quantum lie algebra we extend this brst operator to the topological transformations and we investigate the properties of invariant polynomials of curvatures from which we derive the descent equations for donaldson invariants