diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions the three dimensional problem of diffraction of a skew incident plane wave by a wedge with anisotropic impedance boundary conditions is explicitly solved by the probabilistic random walk method the problem is formulated in terms of two certain components of the electric and magnetic fields which satisfy independent helmholtz equations but are coupled through the first order boundary conditions the solution is represented as a superposition of the geometric waves that are completely determined by elementary methods and of the waves diffracted by the apex of the wedge the diffracted field is explicitly represented as the mathematical expectation computed over the trajectories of a two state random motion which runs in a complex space and switches states under the control of stochastic equations determined by the problem s geometry and by the boundary conditions

explicit solvability of a class of maliuzhinets equations with matrix coefficients a coupled system of maliuzhinets functional equations for a pair of unknown functions is studied it is shown that the system can be solved in an explicit form for a specific but non trivial class of its matrix coefficients two examples arising in electromagnetic diffraction theory in a wedge shaped region are briefly discussed

rayleigh wave scattering by two adhering elastic wedges we consider the problem of wave propagation in a composite bi material elastic wedge with traction free faces the wedge is composed of two distinct isotropic elastic solids which are perfectly bonded along their common face the angles of each wedge are essentially arbitrary satisfying 0 up to 2 2 1 2 2 up to 360 in particular we solve the problem of reflection transmission and diffraction of an incoming rayleigh wave on one of the wedge faces the problem is treated by the method of sommerfeld integrals whose spectral functions are defined through singular integral equations contour integration and analytic continuation we first follow the maliuzhinetz method and derive functional equations for the spectral functions we convert the maliuzhinetz functional equations into well posed singular integral equations which are shown to be suitable for effective inversion the solution of the integral equations explicitly determines the sommerfeld amplitudes inside some strips of the complex plane then explicit recursive formulas analytically continue the amplitudes to a larger region of the complex plane in a finite number of steps which correspond to multiple reflections from the boundaries the integral equations admit clear physical interpretation in terms of fermat9s reflection principle and their right hand sides are directly related to the boundary values of incident waves it is shown how our previous solutions on homogeneous wedges are recovered from the bi material wedge case we also show how several degenerate cases such as a liquid solid wedge and a liquid liquid wedge can be treated in appropriate limits finally the integral equations are solved numerically for particular material combinations and wedge angles to show the usefulness of the solution the results are plotted graphically and shown to be in general agreement with relevant special cases where prior results are available

arbitrarily oriented perfectly conducting wedge over a dielectric half space diffraction and total far field complex scattering targets are often made by structures constituted by wedges and penetrable substrates which may interact at near field in this paper we describe a complete procedure to study this problem with possible developments in radar technologies like gpr antenna development or electromagnetic compatibility tips near substrates the diffraction of an incident plane wave by a perfectly conducting pec wedge over a dielectric half space is studied using generalized wiener hopf equations gwhes and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf wh factorization to integral equations ies the mathematical aspects are described in a unified and consistent theory for angular and layered region problems the proposed procedure is valid for the general case and the paper focuses on e polarization at normal incidence the solutions are given in terms of geometrical uniform theory of diffraction gtd utd diffraction coefficients and total far fields for engineering applications this paper presents several numerical test cases that show the validity of the proposed method

a general approach to the systematic derivation of so 3 shift operator relations i theory a new technique is established for the construction of relations between products consisting of two three or more so 3 shift operators within the framework of the lie algebras of so 3 tensor operators constructed out of an underlying single boson structure

diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory a new class of canonical cases an original mathematical approach for the diffraction of an electromagnetic skew incident wave by a wedge allows us to reduce such a problem to new elementary functional equations using this method we present an exact analytical solution with closed form expressions for a class of wedge of any angle with a certain type of anisotropic boundary condition which does not require any field component to vanish on the faces we thereby consider the important case where the principal axes of anisotropy are along and normal to the edge the relative impedance matrix attached to each face being diagonal with its determinant equal to unity

shift operator techniques for the classification of multipole phonon states xi properties of mixed type quadratic product operators in r 7 expressions connecting r 3 scalar and nonscalar product operators of the type p jl kokl and okl jp jl are constructed within the group r 7

shift operator techniques for the classification of multipole phonon states xii o0l eigenstate and eigenvalue determination in r 7 on account of previously derived relations between quadratic shift operator products in the group g2 part of the eigenvalue spectrum of the scalar shift operator p0l is derived in closed form the corresponding eigenstates which are closely related to the octupole phonon states are defined in terms of angular momentum lowering shift operator actions upon the maximum angular momentum state

generalized taylor operators and polynomial chains for hermite subdivision schemes hermite subdivision schemes act on vector valued data that is not only considered as functions values of a vector valued function from mathbb r to mathbb r r but as evaluations of r consecutive derivatives of a function this intuition leads to a mild form of level dependence of the scheme previously we have proved that a property called spectral condition or sum rule implies a factorization in terms of a generalized difference operator that gives rise to a difference scheme whose contractivity governs the convergence of the scheme but many convergent hermite schemes for example those based on cardinal splines do not satisfy the spectral condition in this paper we generalize the property in a way that preserves all the above advantages the associated factorizations and convergence theory based on these results we can include the case of cardinal splines in a systematic way and are also able to construct new types of convergent hermite subdivision schemes

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

on the spectra of so 3 scalars in the enveloping algebra of su 3 formulas are given that make it possible to calculate the eigenvalues of the two independent so 3 scalars o0l and q0l in the su 3 enveloping algebra

on construction of multivariate parseval wavelet frames abstract a new method for the construction of compactly supported parseval wavelet frames in l 2 r d with any preassigned approximation order n for arbitrary matrix dilation m is proposed the number of wavelet functions generating a frame constructed in this way is less or equal to d 1 det m d the method is algorithmic and the algorithm is simple to use the number of generating wavelet functions can be reduced to det m for a large class of matrices m

from frame like wavelets to wavelet frames keeping approximation properties and symmetry abstract for a given symmetric refinable mask obeying the sum rule of order n an explicit method is suggested for the construction of mutually symmetric almost frame like wavelet system providing approximation order n a transformation based on the lifting scheme is described that allows to improve almost frame like wavelets up to dual wavelet frames and preserve other properties a direct method for the construction of dual wavelet frames providing approximation order n and mutual symmetry properties is also discussed for an abelian symmetry group h a technique providing the h symmetry property for each wavelet function is given for the above three methods

wiener hopf analysis of the em diffraction by an impedance discontinuity in a planar surface and by an impedance half plane the wiener hopf technique is used to solve two canonical problems the first problem considered is the electromagnetic em diffraction by a planar surface with an impedance discontinuity two part surface of an arbitrarily polarized plane wave obliquely incident to the axis of the two dimensional objects the second problem considers the em diffraction by a half plane with equal impedances on both sides the solutions obtained are cast in a matrix notation which is useful for diffraction problems the exact formal solutions are expressed in terms of integrals which can be asymptotically evaluated uniform asymptotic expressions are obtained where the presence of the geometrical optics go poles as well as the surface wave poles near the saddle point are fully taken into account several numerical examples are presented and it is shown that the solutions are continuous across the shadow boundaries of the go and surface wave fields

construction of hermite subdivision schemes reproducing polynomials abstract the aim of this study is to present a new class of quasi interpolatory hermite subdivision schemes of order two with tension parameters this class extends and unifies some of well known hermite subdivision schemes including the interpolatory hermite schemes acting on a function and the associated first derivative values each scheme in this class reproduces polynomials up to a certain degree depending on the size of stencil this is desirable property since the reproduction of polynomials up to degree d leads to the approximation order d 1 the smoothness analysis has been performed by using the factorization framework of subdivision operators lastly we present some numerical examples to demonstrate the performance of the proposed hermite schemes

on the solution of two coupled wiener hopf equations vector wiener hopi equations previously solved in closed form by using different techniques involve matrices having a particular structure a general procedure is given to factorize these matrices it leads to an approach which even if sometimes cumbersome is uniform and allows the solution of new problems in addition a new method is proposed which transforms simultaneous wiener hopf equations in the form of fredholm integral equation of the second kind

design of hermite subdivision schemes aided by spectral radius optimization we present a method for constructing multivariate refinable hermite interpolants and their associated subdivision algorithms based on a combination of analytical and numerical approaches as the limit of a linear iterative procedure the critical l2 sobolev smoothness of a refinable hermite interpolant is given by the spectral radius of a matrix dependent upon the refinement mask the design question is given certain constraints support size symmetry type refinement pattern etc how can one choose the refinement mask so that the resulting refinable function has optimal smoothness this question naturally gives rise to a spectral radius optimization problem in general the objective function is not convex and may not be differentiable or even lipschitz at a local minimizer nonetheless a recently developed robust solver for nonsmooth optimization problems may be applied to find local minimizers of the spectral radius objective function in fact we find that in specific cases that are of particular interest in the present context the objective function is smooth at local minimizers and may be accurately minimized by standard techniques we present two necessary mathematical tricks that make the method practical i compression of matrix size based on symmetry and ii efficient computation of gradients of the objective function we conclude by reporting some computational results

multivariate refinable hermite interpolant we introduce a general definition of refinable hermite interpolants and investigate their general properties we study also a notion of symmetry of these refinable interpolants results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable hermite interpolants the theory developed here is constructive and yields an easy to use construction method for multivariate refinable hermite interpolants using this method several new refinable hermite interpolants with respect to dierent dilation matrices and symmetry groups are constructed and analyzed some of the hermite interpolants constructed here are related to well known spline interpolation schemes developed in the computer aided geometric design community e g the powell sabin scheme we make

wiener hopf solution for impenetrable wedges at skew incidence a new wiener hopf approach for the solution of impenetrable wedges at skew incidence is presented mathematical aspects are described in a unified and consistent theory for angular region problems solutions are obtained using analytical and numerical analytical approaches several numerical tests from the scientific literature validate the new technique and new solutions for anisotropic surface impedance wedges are solved at skew incidence the solutions are presented considering the geometrical and uniform theory of diffraction coefficients total fields and possible surface wave contributions

a general approach to the systematic derivation of so 3 shift operator relations ii applications the technique reported on in the preceding paper is applied to construct shift operator relations in the su 3 and so 5 lie algebras comments are made concerning the integrity basis for so 3 scalar operators appearing in their respective enveloping algebras

a general multisegment artificial neural network architecture for the efficient evaluation of electromagnetic plane wave wedge diffraction a multisegment artificial neural network ann is proposed as an interpolation technique for the evaluation of the electromagnetic field diffracted at the edge of anisotropic impedance wedges under plane wave illumination at oblique incidence multisegmentation is needed as the high frequency wedge diffracted field is characterized by a number of discontinuities at the shadow boundaries of the geometrical optics and surface wave fields the proposed approach is applied as a test case to the problem of an anisotropic impedance right angled wedge illuminated by a skewly incident plane wave some exact analytical solutions valid for specific surface impedance tensors are used to obtain numerical data for the ann training phase as well as to show the interpolation capabilities of the implemented ann nevertheless the proposed ann structure is general and can be trained with data obtained from other available solutions analytical perturbative numerical valid for more general wedge configurations eventually leading to a single software tool encompassing all of them and providing accurate approximations of the wedge diffracted field in a relatively short time comparable to that of a closed form analytical solution

triangular 3 subdivision schemes the regular case the paper deals with the investigation of triangular 3 subdivision schemes in the stationary shift invariant setting in section 2 we collect the available theory on refinable functions subdivision surfaces with emphasis on their sobolev and holder smoothness families of interpolatory and approximating 3 subdivision schemes are investigated in section 3 some dual 3 subdivision schemes which are related to vector valued refinable functions are also analyzed for this purpose we have developed matlab routines for numerically investigating properties of vector subdivision schemes

electromagnetic diffraction by arbitrary angle impedance wedges the problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi an

optimal interpolatory subdivision schemes in multidimensional spaces we analyze the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces in particular we provide a general construction of bivariate interpolatory refinement masks such that the corresponding fundamental and refinable functions attain the optimal approximation order and smoothness order in addition these interpolatory refinement masks are minimally supported and enjoy full symmetry several examples are explicitly computed

diffraction of a plane wave by a circular cone with an impedance boundary condition we consider the boundary value problem for the helmholtz equation in a circular cone with an impedance boundary condition on its face a new approach for its solution is proposed the scheme of solution includes applying the kontorovich lebedev transform derivation of a second order difference equation in a strip of a complex variable and reduction of the latter to an integral equation of the convolution type with variable coefficients it is also shown that the equation is equivalent to a 2 2 matrix riemann hilbert problem with a discontinuous coefficient we analyze the behavior of the solution of the integral equation at the ends of the contour and construct an approximate solution using a collocation method the diffraction coefficient is found in terms of the solution of the integral equation numerical results for the diffraction coefficient and comparative analysis of the results for the impedance cone with the limiting cases of the acoustically soft and hard cones are reported a full high f

extended hermite subdivision schemes subdivision schemes are efficient tools for building curves and surfaces for vector subdivision schemes it is not so straightforward to prove more than the hlder regularity of the limit function on the other hand hermite subdivision schemes produce function vectors that consist of derivatives of a certain function so that the notion of convergence automatically includes regularity of the limit in this paper we establish an equivalence between a spectral condition and operator factorizations then we study how such schemes with smooth limit functions can be extended into ones with higher regularity we conclude by pointing out how this new approach applies to cardinal splines

bi frames with 4 fold axial symmetry for quadrilateral surface multiresolution processing when bivariate filter banks and wavelets are used for surface multiresolution processing it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry this symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices recently lifting scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing if biorthogonal wavelets have certain smoothness then the analysis or synthesis scaling function or both have big supports in general in particular when the synthesis low pass filter is a commonly used scheme such as loop s scheme or catmull clark s scheme the corresponding analysis low pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness big supports of scaling functions or in other words big templates of multiresolution algorithms are undesirable for surface processing on the other hand a frame provides flexibility for the construction of basis systems this paper concerns the construction of wavelet or affine bi frames with high symmetry in this paper we study the construction of wavelet bi frames with 4 fold symmetry for quadrilateral surface multiresolution processing with both the dyadic and 2 refinements considered the constructed bi frames have 4 framelets or frame generators for the dyadic refinement and 2 framelets for the 2 refinement namely with either the dyadic or 2 refinement a frame system constructed in this paper has only one more generator than a wavelet system the constructed bi frames have better smoothness and smaller supports than biorthogonal wavelets furthermore all the frame algorithms considered in this paper are given by templates so that one can easily implement them

nonscalar extension of shift operator techniques for su 3 in an o 3 basis iii shift operators of second degree in the tensor components shift operators qkl 2 k 2 of second degree in the tensor components q 2 2 are constructed relations connecting quadratic shift operator products of the type ojl k qkl or qkl j ojl and of the type qjl k qkl are derived the usefulness of these relations is demonstrated by the example of the ool and qol eigenvalue calculation for various irreducible respresentations p q of su 3

quincunx fundamental refinable functions and quincunx biorthogonal wavelets we analyze the approximation and smoothness properties of quincunx fundamental refinable functions in particular we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in r2 their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order in addition these examples are minimally supported with symmetry for two special families of such quincunx interpolatory masks we prove that their symbols are nonnegative finally a general way of constructing quincunx biorthogonal wavelets is presented several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed

multidimensional interpolatory subdivision schemes this paper presents a general construction of multidimensional interpolatory subdivision schemes in particular we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three direction box spline br r r of equal multiplicities the resulting mask for the interpolatory subdivision exhibits all the symmetries of the three direction box spline and with this increased symmetry comes increased smoothness several examples are computed for r 2 8 regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness

adaptive multiresolution analysis structures and shearlet systems in this paper we first introduce the concept of an adaptive multiresolution analysis amra structure which is a variant of the classical mra structure well suited to the main goal of a fast flexible decomposition strategy adapted to the data at each decomposition level we then study this novel methodology for the general case of affine like systems and derive a unitary extension principle uep for filter design finally we apply our results to the directional representation system of shearlets this leads to a comprehensive theory for fast decomposition algorithms associated with shearlet systems which encompasses tight shearlet frames with spatially compactly supported generators within such an amra structure also shearlet like systems associated with parabolic scaling and unimodular shear matrices are studied within this framework

the shift operator technique for so 7 in an su 2 3 basis i theory shift operators for the chain of lie groups so 7 su 2 3 are constructed and several properties are examined quite a number of relations connecting quadratic products of shift operators are built up finally the connection between matrix elements of the shift operators and reduced matrix elements of the tensor operator which forms part of the so 7 generator basis is explained

su 2 su 2 shift operators and representations of so 5 su 2 su 2 shift operators analogous to the su 2 shift operators developed and used by the author for the classification and analysis of representations of lie algebras in an su 2 or so 3 basis are obtained for the su 2 su 2 lie algebra in the case where one has an additional set of operators forming an irreducible four dimensional tensor representation of su 2 su 2 the shift operators obtained are used to treat the representations of so 5 in an su 2 su 2 basis

wiener hopf splitting of a 2 x 2 matrix explicit expressions for the wiener hopf factors of a 2 x 2 matrix are determined in terms of a single scalar examples of the application to various matrices are given

diffraction by a highly contrast transparent wedge the problem of a plane wave diffraction by a highly contrast transparent wedge of an arbitrary opening is studied with the aid of the sommerfeld integrals it is reduced to a system of coupled maliuzhinets equations by use of the theory of s integrals the system of functional equations is transformed to a system of linear equations in a banach space in the case of the high contrast of a material inside the wedge in comparison with that in the wedge s exterior the linear equations are solved by means of perturbation theory convergence of the corresponding neumann series is proved singularities of the integrands in the sommerfeld integrals are investigated application of the steepest descent method leads to the determination of the reflected transmitted and diffracted waves expressions for the diffraction coefficients are also represented

on one approach to an electromagnetic diffraction problem in a wedge shaped region an electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional equations by means of sommerfeld integrals anisotropic impedance boundary conditions are satisfied on the wedge s faces this system of functional equations is solved by regular perturbation method it is shown that for weak anisotropy the solution is presented by converging series which are neumann recurrent series for linear equations with contracting operators in a general case the problem is reduced to linear equations with compact operators the wave field asymptotic is computed for the region outside the vicinity of the edge of the wedge

nonscalar extension of shift operator techniques for su 3 in an o 3 basis ii applications in a preceding paper relations have been derived which connect nonscalar quadratic shift operator products here the extreme usefulness of these relations is demonstrated by the example of the o0l eigenvalue calculation for the cases l p i i 0 1 2 3 and 4 where p q is any su 3 representation for the first time a case of threefold l degeneracy is completely solved in a pure analytical way

generalized interpolating refinable function vectors interpolating scalar refinable functions with compact support are of interest in several applications such as sampling theory signal processing computer graphics and numerical algorithms in this paper we shall generalize the notion of interpolating scalar refinable functions to compactly supported interpolating d refinable function vectors with any multiplicity r and dilation factor d more precisely we are interested in a d refinable function vector f f 1 f r t such that f is an rx1 column vector of compactly supported continuous functions with the following interpolation property f mr k email protected email protected 1 m email protected z m 0 r 1 1 r where d 0 1 and d k 0 for k 0 now for any function f email protected c the function f can be interpolated and approximated by email protected email protected email protected email protected email protected zf 1r k f k email protected email protected z f k f 1r k f r 1r k f k since f is interpolating email protected k r f k r for all email protected z that is email protected agrees with f on r 1z moreover for r 2 or d 2 such interpolating refinable function vectors can have the additional orthogonality property r email protected d k k for all k k z and 1

exact evaluation of the maliuzhinets half plane function the maliuzhinets half plane function psipi z has been used for many applications but its closed form has not been known in this communication we exactly evaluate a known integral representation of psipi z to formulate an exact closed form of the function which is expressed in terms of the dilogarithm function the exact formulation is numerically verified by comparing it with the results obtained by numerical integration of a integral representation of general maliuzhinets function a closed form of psipi 2 z is also obtained since it can be expressed in terms of psipi z

a generalized taylor factorization for hermite subdivision schemes in a recent paper we investigated factorization properties of hermite subdivision schemes by means of the so called taylor factorization this decomposition is based on a spectral condition which is satisfied for example by all interpolatory hermite schemes nevertheless there exist examples of hermite schemes especially some based on cardinal splines which fail the spectral condition for these schemes and others we provide the concept of a generalized taylor factorization and show how it can be used to obtain convergence criteria for the hermite scheme by means of factorization and contractivity

exact solution to diffraction problem by wedges with a class of anisotropic impedance faces oblique incidence of a plane electromagnetic wave this paper studies diffraction of an obliquely incident arbitrarily polarized plane electromagnetic wave by an anisotropic impedance wedge with an opening angle 2 spl phi between 0 and 2 spl pi and presents a closed form exact solution to a class of impedance wedge faces and the related uniform asymptotic solution uas on use of a unitary similarity transform the boundary conditions on the wedge faces is brought into a form which makes the exactly soluble class of impedance faces evident the exact solution is found with help of the sommerfeld malyuzhinets 1896 1958 technique a generalized malyuzhinets function spl chi sub spl phi and the so called s integrals a standard procedure yields therefrom the uas the exact solution agrees with known analytical results in special cases and the numerical results of uas are confirmed by that of parabolic equation method pem

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

two dimensional diffraction by a wedge with impedance boundary conditions the two dimensional problem of diffraction by a wedge with impedance boundary conditions on its faces is explicitly solved in a form that admits effective numerical simulation by simple perfectly scalable algorithms with unlimited capability for parallel processing the solution is represented as a superposition of the geometric field that is completely determined by elementary ray analysis and of the waves diffracted by the tip of the wedge the diffracted field is explicitly represented as a mathematical expectation of a specified functional on trajectories of a random motion determined by the configurations of the problem and by the boundary conditions the numerical results confirm the efficiency of this approach

a uniform gtd formulation for the diffraction by a wedge with impedance faces a uniform high frequency solution is presented for the diffraction by a wedge with impedance faces illuminated by a plane wave perpendicularly incident on its edge arbitrary uniform isotropic impedance boundary conditions may be imposed on the faces of the wedge and both the transverse electric te and transverse magnetic tm cases are considered this solution is formulated in terms of a diffraction coefficient which has the same structure as that of the uniform geometrical theory of diffraction utd for a perfectly conducting wedge its extension to the present case is achieved by introducing suitable multiplying factors which have been derived from an asymptotic evaluation of the exact solution given by maliuzhinets when the field point is located on the surface near the edge a more accurate asymptotic evaluation is employed to obtain a high frequency expression for the diffracted field which is suitable for several specific applications the formulation described in this paper may provide a useful rigorous basis to search for a more numerically efficient but yet accurate approximation

compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas in this paper for any d d dilation matrix m we demonstrate in a constructive way that we can construct compactly supported tight m wavelet frames and orthonormal m wavelet bases in l2 rd of exponential decay which are derived from compactly supported m refinable functions such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments this paper improves several results in battle comm math phys 110 1987 601 bownik j fourier anal appl 7 2001 489 grochenig and ron proc amer math soc 126 1998 1101 lemarie j math pures appl 67 1988 227 and strichartz constr approx 9 1993 327

analysis of non stationary hermite subdivision schemes reproducing exponential polynomials abstract the aim of this paper is to study the convergence and smoothness of non stationary hermite subdivision schemes of order 2 in conti et al 2017 provided sufficient conditions for the convergence of a non stationary hermite subdivision scheme that reproduces a set of functions including exponential polynomials the analysis has been focused on the non stationary hermite scheme with the order 3 but the case of 2 which is practically most useful is yet to be investigated in this regard the first goal of this paper is to fill the gap we analyze the convergence of non stationary hermite subdivision schemes of order 2 next we provide a tool which allows us to estimate the smoothness of a non stationary hermite scheme by developing a novel factorization framework of non stationary vector subdivision operators using the proposed non stationary factorization framework we estimate the smoothness of the non stationary hermite subdivision schemes the non stationary interpolatory hermite scheme proposed by conti et al 2015 and a new class of non stationary dual hermite subdivision schemes of de rham type

on a class of shape preserving refinable functions with dilation 3 we analyze the properties of a class of shape preserving refinable functions with dilation m 3 we give an algorithm to construct totally positive bases with optimal shape preserving properties on a finite interval bernstein type bases on 0 1 are also treated moreover semiorthogonal wavelets associated with these refinable functions are constructed finally a detailed example is described

noninterpolatory hermite subdivision schemes bivariate interpolatory hermite subdivision schemes have recently been applied to build free form subdivision surfaces it is well known to geometric modelling practitioners that interpolatory schemes typically lead to unfair surfaces surfaces with unwanted wiggles or undulations and noninterpolatory a k a approximating in the cagd community schemes are much preferred in geometric modelling applications in this article we introduce analyze and construct noninterpolatory hermite subdivision schemes a class of vector subdivision schemes which can be applied to iteratively refine hermite data in a not necessarily interpolatory fashion we also study symmetry properties of such subdivision schemes which are crucial for application in free form subdivision surfaces a key step in our mathematical analysis of hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme namely the subdivision scheme is of hermite type to an algebraic condition on the subdivision mask the latter algebraic condition can then be used in a computational framework to construct specific schemes

numerical study of diffraction and slope diffraction at anisotropic impedance wedges by the method of parabolic equation space waves the method of parabolic equation pe has been successfully applied to the numerical determination of diffraction slope diffraction and multiple diffraction coefficients of scalar impedance wedges illuminated by a line source as a continuation this paper studies for the first time to the authors knowledge another important canonical problem for the uniform geometrical theory of diffraction utd namely diffraction and slope diffraction of an incident cylindrical wave at wedges with anisotropic impedance surfaces by using the same method for the diffracted fields the exact helmholtz equation is asymptotically approximated by the corresponding parabolic one it is proved that the sufficient conditions for the unique solution of the helmholtz equation also guarantee the uniqueness of the solution of the parabolic one the latter is then efficiently solved by using crank nicholson finite difference fd scheme due to the lack of exact solutions the pe results were compared to uniform asymptotic theory of diffraction uat ones for weak anisotropy and in this case very good agreement has been achieved the diffraction and slope diffraction behavior dependent upon the measure of the weakness of the anisotropy has been demonstrated by several examples

nonscalar extension of shift operator techniques for su 3 in an o 3 basis i theory a set of relations is set up which connect quadratic products of the shift operators okl k 0 1 2 which are nonscalar with respect to the o 3 subgroup of su 3 the usefulness of these relations is illustrated by the calculation of the eigenvalues of the scalar shift operator o0l for various irreducible representations p q of su 3

totally symmetric irreducible representations of the group so 6 in the principal so 3 subgroup basis explicit matrix elements are found for the generators of the group so 6 in an arbitrary totally symmetric irreducible representation using the physical principal so 3 subgroup in the chain so 6 so 5 so 3 the internal one missing label problem is solved through the definition of intrinsic states associated to the su 2 su 2 subgroup in the chain so 6 so 5 su 2 su 2 and out of which is projected a complete set of states in the physical basis by integrations over the physical rotation group manifold the matrix elements of the so 6 generators in the su 2 su 2 basis are themselves obtained by the intermediate use of an su 2 su 2 u 1 basis the latter group being a subgroup of so 6 but not of so 5

solution of a second order difference equation using the bilinear relations of riemann a recently proposed technique to solve a class of second order functional difference equations arising in electromagnetic diffraction theory is further investigated by applying it to a case of intermediate complexity the proposed approach is conceptually simple and relies on first obtaining well defined branched solutions to a pair of associated first order difference equations the construction of these branched expressions leads to an equation system whose solution requires relationships akin to riemann s bilinear relations for differentials of the first and third kinds their derivation necessitates the application of cauchy s theorem on riemann surfaces of in this particular instance genera one and three branch free solutions of the second order difference equation are then obtained by taking appropriate linear combinations of the branched solutions of the first order equations analysis and computation demonstrate that the resulting expressions have the desired analytical properties and recover known solutions in the appropriate limit

the leading asymptotic term for the scattering diagram in the problem of diffraction by a narrow circular impedance cone the leading term for the scattering diagram in the scalar problem concerning the diffraction of a plane wave by a narrow circular impedance cone is obtained although up to now the problem cannot be solved in an explicit form its reduction to a non oscillating integral equation has been recently developed it is used here in order to determine formal asymptotics of the scattering diagram by means of a perturbation method

a utd solution for the scattering by a wedge with anisotropic impedance faces skew incidence case asymptotic expressions for the fields scattered by an anisotropic impedance wedge at oblique incidence are derived in the context of the uniform geometrical theory of diffraction utd they are obtained by resorting to a perturbative approach considering the normal incidence case as the imperturbed configuration we observe that the limits of applicability of this approximate analytical solution extend far beyond those of standard perturbative approaches allowing us to account for deviations from the normal incidence case of 20 spl deg to 30 spl deg

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

symmetry property and construction of wavelets with a general dilation matrix in this note we are interested in the symmetry property of a refinable function with a general dilation matrix we investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has a certain kind of symmetry given two dilation matrices which produce the same lattice we demonstrate that if a mask has a certain kind of symmetry then its associated refinable functions with respect to the two dilation matrices are the same therefore the two corresponding derived wavelet systems are essentially the same finally we illustrate that for any dilation matrix orthogonal masks as well as interpolatory masks having nonnegative symbols can be easily constructed with any preassigned order of sum rules by employing a linear transform without solving any equations the method in this note on constructing masks with certain desirable properties is simple painless and general examples of quincunx wavelets and wavelets with respect to the checkerboard lattice are presented to illustrate the general theory

diffraction of an electric polarized wave by a dielectric wedge the problems of diffraction of an e polarized plane wave by an inhomogeneous imperfectly conducting dielectric wedge or a homogeneous dielectric wedge with vanishing conductivity are formulated as integral equations conditions for existence and uniqueness of the solutions of the integral equations are found the integral equations are solved by iterative methods an explicit expression for the far field is found the series expansion of the far field near the edge of a homogeneous wedge converges very fast for relative permittivities in the interval from one to ten the physical optical approximation of the far field is poor