a fast analysis of scattering from large scale finite periodic microstrip patch arrays arranged on a non orthogonal lattice using sub entire domain basis functions a sub entire domain sed basis function method which was first introduced for modeling large scale finite periodic pec structures in free space has been extended for fast characterization of electromagnetic scattering from an electrically large planar finite periodic microstrip patch array the microstrip array may have a nonrectangular layout and non orthogonal lattice configurations e g hexagons or quadrangles based on the mixed potential integral equation and utilizing the proposed sed basis function algorithm the original large scale finite periodic array of microstrip patches can be efficiently simulated by decomposing it into two problems with matrix equations of small dimensions the first is to construct the sed basis functions for the corresponding microstrip arrays with orthogonal non orthogonal lattices three kinds of the sed basis functions are constructed including those related to the edge patch elements the interior patch elements and the corner patch elements the second is to solve the system equation with significantly reduced problem dimension as compared to the original larger problem based on the obtained sed basis functions the reduced matrix equation of small size can be generated by the galerkin procedure and solved by use of the lu lower upper decomposition based direct solver which results in a fast solution the accuracy and efficiency of the developed algorithms are demonstrated by numerical tests that include the scattering from several large scale finite periodic arrays of microstrip patches with rectangular non orthogonal lattices

error estimates of quasi interpolation and its derivatives quasi interpolation of radial basis functions on finite grids is a very useful strategy in approximation theory and its applications a notable strongpoint of the strategy is to obtain directly the approximants without the need to solve any linear system of equations for radial basis functions with gaussian kernel there have been more studies on the interpolation and quasi interpolation on infinite grids this paper investigates the approximation by quasi interpolation operators with gaussian kernel on the compact interval the approximation errors for two classes of function with compact support sets are estimated furthermore the approximation errors of derivatives of the approximants to the corresponding derivatives of the approximated functions are estimated finally the numerical experiments are presented to confirm the accuracy of the approximations

fourier pseudospectral method with kepler mapping for travelling waves with discontinuous slope application to corner waves of the ostrovsky hunter equation and equatorial kelvin waves in the four mode approximation many species of travelling waves have a single branch of solutions which ends at finite amplitude with a singular wave whose slope is discontinuous a so called corner wave fourier pseudospectral methods converge exponentially fast with n the truncation of the series for the smooth waves but the error falls only as o 1 n for a wave with a slope discontinuity we show that the error rate can be accelerated to o 1 n3 by making the change of coordinate kepler map x z sin z unfortunately there is a subtlety the end in a singular solution bifurcation is possible only for infinite dimensional systems we show that the truncated fourier pseudospectral approximation has roots on both sides of the corner wave the bifurcation can be detected by i a step function like jump in the residual of the differential equation at the corner wave and ii by observing the slope of u x which becomes discontinuous only at the corner wave itself these concepts are illustrated for the equatorially trapped ocean kelvin wave in the so called four latitudinal mode approximation however corner waves arise in many species of waves and the concepts explained here are applicable to all

a two dimensional adaptive pseudo spectral method we develop a two dimensional adaptive pseudo spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions the method is based on introducing two dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated the method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel spatio temporal patterns are computed as a parameter related to the activation energy is increased above a critical value c the spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder together with a standing wave pattern in the direction of the azimuthal angle for small values of c the solutions exhibit a nearly sinusoidal dependence in both time and as is increased further relaxation oscillations in both time and occur beyond a critical value of stable time periodic solutions are no longer found and the solution exhibits a quasi periodic time dependence

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

chaotic motion of the n vortex problem on a sphere ii saddle centers in three degree of freedom hamiltonians we study the motion of n point vortices with n on a sphere in the presence of fixed pole vortices which are governed by a hamiltonian dynamical system with n degrees of freedom special attention is paid to the evolution of their polygonal ring configuration called the n ring in which they are equally spaced along a line of latitude of the sphere when the number of the point vortices is n 5n or 6n with n the system is reduced to a two degree of freedom hamiltonian with some saddle center equilibria one of which corresponds to the unstable n ring using a melnikov type method applicable to two degree of freedom hamiltonian systems with saddle center equilibria and a numerical method to compute stable and unstable manifolds we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle centers and hence chaotic motions occur especially the evolution of the unstable n ring is shown to be chaotic

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

boundary value problems with symmetry and their approximation by finite elements symmetry and related questions of group theory especially representation theory are well known popular topics among physicists this paper describes how these classical ideas can be put to use in the realm of finite element computations by substituting a family of independent problems on a reduced domain the symmetry cell for the original problem on the whole domain two specific difficulties appear one is how to set boundary conditions on symmetry planes representation theory gives the answer the other is how to proceed with the assembly of finite elements that constitute the symmetry cell to deal with this this paper introduces the concept of index of an element as part of the whole structure which is the number of its distinct symmetrical images the rule for assembly then becomes to perform as usual but to count each element as many times as its index

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

a fast analysis of scattering and radiation of large microstrip antenna arrays an accurate and efficient method that combines the precorrected fast fourier transform fft method and the discrete complex image method dcim is presented to characterize the scattering and radiation properties of arbitrarily shaped microstrip patch antennas in this method the mixed potential integral equation mpie is discretized in the spatial domain by means of the discrete complex image method the resultant system is solved iteratively using the generalized conjugate residual method gcr and the precorrected fft technique is used to speed up the matrix vector multiplication the precorrected fft eliminates the need to generate and store the usual square impedance matrix and thus leads to a significant reduction in memory requirement and computational cost numerical results are presented for arbitrarily shaped microstrip antenna arrays to demonstrate the accuracy and efficiency of this technique

the n vortex problem on a rotating sphere ii heterogeneous platonic solid equilibria we describe a new method of constructing point vortex equilibria on a sphere made up of n vortices with different strengths such equilibria called heterogeneous equilibria are obtained for the five platonic solid configurations hence for graphic 1 the method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant by symmetries inherent in the platonic solid configurations this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure for the tetrahedron n 4 and octahedron n 6 the degree of heterogeneity is 4 and 6 respectively hence we are free to choose each of the vortex strengths independently for the cube n 8 the degree of heterogeneity is 5 for the icosahedron n 12 it is 7 while for the dodecahedron n 20 it is 4 thus the entire set of equilibria based on the platonic solid configurations is obtained including substructures associated with each configuration constructed by taking different linear combinations of the basis elements 1 embed inline graphic 1 gif

a fourier error analysis for radial basis functions and the discrete singular convolution on an infinite uniform grid part 1 on an infinite grid with uniform spacing h the cardinal basis cj x h for many spectral methods consists of translates of a master cardinal function cj x h c x h j the cardinal basis satisfies the usual lagrange cardinal condition cj mh jm where jm is the kronecker delta function all such shift invariant subspace master cardinal functions are of localized sinc form in the sense that c x sinc x s x for a localizer function s which is smooth and analytic on the entire real axis and the whittaker cardinal function is sinc x sin x x the localized sinc approximation to a general f x is f localized sinc x h j f j h s x j h h sinc x jh h in contrast to most radial basis function applications matrix factorization is unnecessary we prove a general theorem for the fourier transform of the interpolation error for localized sinc bases for exponentially convergent radial basis functions rbfs gaussians inverse multiquadrics etc and the basis functions of the discrete singular convolution dsc the localizer function is known exactly or approximately this allows us to perform additional error analysis for these bases we show that the error is similar to that for sinc bases except that the localizer acts like a diffusion in fourier space smoothing the sinc error

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

a fast full wave analysis of scattering and radiation from large finite arrays of microstrip antennas a fast full wave analysis technique that can be used to analyze the scattering and radiation from large finite arrays of microstrip antennas is presented the technique discretizes the mixed potential integral equation mpie in the spatial domain by means of a full wave discrete complex image method the del operators on the green s functions are transferred from the singular kernel to the expansion and testing functions the resultant system of equations is solved using the biconjugate gradient bcg method in which the matrix vector product is evaluated efficiently using the fast fourier transform fft this results in an efficient and accurate computation of the scattering and radiation from finite arrays of microstrip antennas several numerical results are presented demonstrating the accuracy efficiency and capability of this technique

a mixed form thin stratified medium fast multipole algorithm for both low and mid frequency problems a mixed form thin stratified medium fast multipole algorithm is proposed for fast simulation of general microstrip structures at both low and mid frequencies the newly developed matrix friendly formula of layered medium green s function is applied in this algorithm for well separated interactions the path deformation technique is implemented to achieve a smoother and exponentially convergent integral the two dimensional addition theorem is then incorporated into the integrand to expedite the matrix vector product in our approach multipole expansion low frequency fast multiple algorithm as well as the plane wave expansion mid frequency fast multipole algorithm of the translational addition theorem are combined into a single multilevel tree to capture quasi static physics and wave physics simultaneously the outgoing wave is represented first in terms of multipole expansion at leafy levels and then switched to plane wave expansion automatically at higher levels this seamless connection makes the algorithm applicable in simulations where subwavelength interaction and wave physics both exist

review of shape representation and description techniques more and more images have been generated in digital form around the world there is a growing interest in 1nding images in large collections or from remote databases in order to 1nd an image the image has to be described or represented by certain features shape is an important visual feature of an image searching for images using shape features has attracted much attention there are many shape representation anddescription techniques in the literature in this paper we classify and review these important techniques we examine implementation procedures for each technique and discuss its advantages and disadvantages some recent research results are also included and discussed in this paper finally we identify some promising techniques for image retrieval according to standard principles

on choosing a radial basis function and a shape parameter when solving a convective pde on a sphere radial basis function rbf approximations have been used for some time to interpolate data on a sphere as well as on many other types of domains their ability to solve to spectral accuracy convection type pdes over a sphere has been demonstrated only very recently in such applications there are two main choices that have to be made i which type of radial function to use and ii what value to choose for their shape parameter denoted by e and with flat basis functions stretched out in the radial direction corresponding to e 0 the recent rbf qr algorithm has made it practical to compute stably also for small values of e results from solving a convective type pde on a sphere are compared here for many choices of radial functions over the complete range of e values from very large down to the limit of e 0 the results are analyzed with a methodology that has similarities to the customary fourier analysis in equispaced 1 d periodic settings in particular we find that high accuracy can be maintained also over very long time integrations we furthermore gain insights into why rbfs sometimes offer higher accuracy than spherical harmonics since the latter arise as an often non optimal special case of the former anticipated future application areas for rbf based methods in spherical geometries include weather and climate modeling

shape classification using the inner distance part structure and articulation are of fundamental importance in computer and human vision we propose using the inner distance to build shape descriptors that are robust to articulation and capture part structure the inner distance is defined as the length of the shortest path between landmark points within the shape silhouette we show that it is articulation insensitive and more effective at capturing part structures than the euclidean distance this suggests that the inner distance can be used as a replacement for the euclidean distance to build more accurate descriptors for complex shapes especially for those with articulated parts in addition texture information along the shortest path can be used to further improve shape classification with this idea we propose three approaches to using the inner distance the first method combines the inner distance and multidimensional scaling mds to build articulation invariant signatures for articulated shapes the second method uses the inner distance to build a new shape descriptor based on shape contexts the third one extends the second one by considering the texture information along shortest paths the proposed approaches have been tested on a variety of shape databases including an articulated shape data set mpeg7 ce shape 1 kimia silhouettes the eth 80 data set two leaf data sets and a human motion silhouette data set in all the experiments our methods demonstrate effective performance compared with other algorithms

a new class of oscillatory radial basis functions radial basis functions rbfs form a primary tool for multivariate interpolation and they are also receiving increased attention for solving pdes on irregular domains traditionally only nonoscillatory radial functions have been considered we find here that a certain class of oscillatory radial functions including gaussians as a special case leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat this flat limit is important in that it generalizes traditional spectral methods to completely general node layouts interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions such as multiquadrics and inverse multiquadratics

a quasi 3d thin stratified medium fast multipole algorithm for microstrip structures an accurate and efficient full wave simulation method is proposed for general microstrip structures it is called the quasi 3d thin stratified medium fast multipole algorithm tsm fma different from the 2d tsm fma method it is constructed on a vector wave function based dyadic green s function for layered medium dglm instead of the symmetric form dglm this new form of dglm is represented in terms of only two sommerfeld integrals and is suitable for developing fast algorithm in quasi 3d cases similar to the 2d tsm fma the path deformation technique and the multipole based acceleration is used to transform the sommerfeld integral and expedite the matrix vector multiplication the computational time per iteration and the memory requirement is o nlogn in the quasi 3d tsm fma it is suitable to perform a full wave analysis of a large microstrip array with less computational resource

quartic gaussian and inverse quartic gaussian radial basis functions the importance of a nonnegative fourier transform we catalogue the numerical properties of approximations using two novel types of radial basis functions f r the qg species is a basis of exponentials of quartic argument f x f r b f x a h j 1 na jexp a h 4 x x j 4 where the x j are the rbf centers and also the interpolation points we show that quartic gaussian rbfs fail at many discrete values of the shape parameter a we show through a detailed analysis that these singularities are directly related to zeros of q k the fourier transform of exp x 4 if we reverse the roles and take q x as the rbf all difficulties disappear because these iqg rbfs have a fourier transform which is nonnegative for all real k we explain that although the quartic gaussian exp x 4 is positive definite in the physics dynamical systems sense of being zero free and nonnegative it lacks the crucial property of being positive definition in the rbf analysis sense

a weak form of the conjugate gradient fft method for plate problems a number of electromagnetic field problems for planar structures can be formulated in terms of a hypersingular integral equation in which a grad div operator acts on a vector potential the vector potential is a convolution of the free space green s function and some surface current density over the domain of interest a weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only as a next step the vector potential is expanded in a sequence of subdomain basis functions and the grad div operator is integrated analytically over the plate domain only for the problem of electromagnetic scattering by a plate the method shows excellent numerical performance the numerical difficulties encountered in some previous conjugate gradient fast fourier transform cgfft methods have been eliminated

array decomposition method for the accurate analysis of finite arrays presented in this paper is a fast method to accurately model finite arrays of arbitrary three dimensional elements the proposed technique referred to as the array decomposition method adm exploits the repeating features of finite arrays and the free space green s function to assemble a nonsymmetric block toeplitz matrix system the toeplitz property is used to significantly reduce storage requirements and allows the fast fourier transform fft to be applied in accelerating the matrix vector product operations of the iterative solution process each element of the array is modeled using the finite element boundary integral fe bi technique for rigorous analysis consequently we demonstrate that the complete lu decomposition of the matrix system from a single array element can be used as a highly effective block diagonal preconditioner on the larger array matrix system this rigorous method is compared to the standard fe bi technique for several tapered slot antenna tsa arrays and is demonstrated to generate the same accuracy with a fraction of the storage and solution time

an extension of draghicescu s fast tree code algorithm to the vortex method on a sphere a fast and accurate algorithm to compute interactions between n point vortices and between n vortex blobs on a sphere is proposed it is an extension of the fast tree code algorithm developed by draghicescu for the vortex method in the plane when we choose numerical parameters in the fast algorithm suitably the computational cost of o n 2 is reduced to o n logn 4 and the approximation error decreases like o 1 n when n as demonstrated in the present article we also apply the fast method to long time evolution of two vortex sheets on the sphere to see the efficiency a key point is to describe the equation of motion for the n points in the three dimensional cartesian coordinates

the n vortex problem on a rotating sphere i multi frequency configurations the problem of n point vortices moving on a rotating unit sphere is considered through a sequence of linear coordinate transformations which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation we show how to reduce the problem to that on a non rotating sphere where the centre of vorticity vector is aligned with the z axis as a consequence we prove that integrability on the rotating sphere is the same as on the non rotating sphere namely the three vortex problem on the rotating sphere is integrable for all vortex strengths while the four vortex problem is integrable in the special case where the centre of vorticity is zero rigid multi frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained and necessary conditions for one and two frequency motions are derived examples including dipoles which exhibit global wobbling and tumbling dynamics rings and platonic solid configurations are shown to undergo either periodic or quasi periodic evolution on the rotating sphere depending on the ratio of the solid body rotational frequency to the rotational frequency associated with the rigid structure

efficient analysis of phased arrays of microstrip patches using a hybrid generalized forward backward method green s function technique with a dft based acceleration algorithm a hybrid method based on the combination of generalized forward backward method gfbm and green s function for the grounded dielectric slab together with the acceleration of the combination via a discrete fourier transform dft based algorithm is developed for the efficient and accurate analysis of electromagnetic radiation scattering from electrically large irregularly contoured two dimensional arrays consisting of finite number of probe fed microstrip patches in this method unknown current coefficients corresponding to a single patch are first solved by a conventional galerkin type hybrid method of moments mom green s function technique that uses the grounded dielectric slab s green s function because the current distribution on the microstrip patch can be expanded using an arbitrary number of subsectional basis functions the patch can have any shape the solution for the array currents is then found through gfbm where it sweeps the current computation element by element the computational complexity of this method which is originally being the total number of unknowns for each iteration is reduced to using a dft based acceleration algorithm making use of the fact that array elements are identical and the array is periodic numerical results in the form of array current distribution are given for various sized arrays of probe fed microstrip patches with elliptical and or circular boundaries and are compared with the conventional mom results to illustrate the efficiency and accuracy of the method

the near equivalence of five species of spectrally accurate radial basis functions rbfs radial basis function rbf interpolants have become popular in computer graphics neural networks and for solving partial differential equations in many fields of science and engineering in this article we compare five different species of rbfs gaussians hyperbolic secant sech s inverse quadratics multiquadrics and inverse multiquadrics we show that the corresponding cardinal functions for a uniform unbounded grid are all approximated by the same function c x 1 sin x sinh x for some constant which depends on the inverse width parameter shape parameter of the rbf and also on the rbf species the error in this approximation is exponentially small in 1 for sech s and inverse quadratics and exponentially small in 1 2 for gaussians the error is proportional to 4 for multiquadrics and inverse multiquadrics the error in all cases is small even for o 1 these results generalize to higher dimensions the gaussian rbf cardinal functions in any number of dimensions d are without approximation the tensor product of one dimensional gaussian cardinal functions c d x 1 x 2 x d j 1 d c x j for other rbf species we show that the two dimensional cardinal functions are well approximated by the products of one dimensional cardinal functions again the error goes to zero as 0 the near identity of the cardinal functions implies that all five species of rbf interpolants are almost the same despite the great differences in the rbf s themselves

exact polynomial reproduction for oscillatory radial basis functions on infinite lattices until now only nonoscillatory radial basis functions rbfs have been considered in the literature it has recently been shown that a certain family of oscillatory rbfs based on j bessel functions gives rise to nonsingular interpolation problems and seems to be the only class of functions not to diverge in the limit of flat basis functions for any node layout this paper proves another interesting feature of these functions exact polynomial reproduction of arbitrary order on an infinite lattice in n first a closed form expression is derived for calculating the expansion coefficients for any order polynomial in any dimension then a proof is given showing that the resulting interpolant using this class of oscillatory rbfs will give exact polynomial reproduction examples in one and two dimensions are presented it is specifically noted that such closed form expressions cannot be derived for other classes of rbfs due to the fact that j bessel rbfs reproduce polynomials via a different mechanism

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

description and discrimination of planar shapes using shape matrices a shape descriptor has been developed which can describe a shape independent of its translation rotation and scaling the description is in the form of a matrix and it is obtained by a polar quantization of a shape the quantization process takes into consideration not only a shape s outer geometry but its inner geometry as well the descriptor is information preserving and if the quantization parameters are selected properly it is possible to reconstruct an original shape from its description in this technique shape discrimination is possible by a simple exclusive or operation on shape descriptions

the fourier transform of the quartic gaussian exp ax4 hypergeometric functions power series steepest descent asymptotics and hyperasymptotics and extensions to exp ax2n the fourier transform of a quartic gaussian f k exp ikx exp x 4 dx is important in the theory of radial basis functions with f r exp r 4 we show that the transform is the exact sum of two generalized hypergeometric functions 0f 2 this implies that f k is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree using the method of steepest descent for integrals we derive the first five terms in an asymptotic series in powers of k 4 3 through a simple hyperasymptotic analysis we show that the magnitude of the error of the optimally truncated series superasymptotic error is oexp 0 818k 4 3 the lowest order approximation f k 2 7 6 p 3k 1 3exp 2 1 33 16 k 4 3 cos 3 3 22 1 3 16 k 4 3 p 6 shows that the transform f k is an exponentially decaying oscillation we show that the steepest descent method 2 27 bender and orszag 1978 miller 2006 26 33 lauwerier 1966 ursell 1965 can be applied to the transform of exp ax 2 n for general integer n and give the first two terms another theme is that modern computer technology has simplified the steepest descent method the steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function j t where z j t is the logarithm of the integrand and z is the large parameter here proportional to k 4 3 the task of inverting w t to obtain a series in powers of w for the metric factor dt dw after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete maple code

a 2d chebyshev differential operator for the elastic wave equation abstract this work analyses the performance of a two dimensional chebyshev differential operator for solving the elastic wave equation the technique allows the implementation of non periodic boundary conditions at the four boundaries of the numerical mesh which requires a special treatment of these conditions based on one dimensional characteristics in addition spatial grid adaptation by appropriate one dimensional coordinate mappings allows a more accurate modeling of complex media and reduction of the computational cost by controlling the minium grid spacing the examples illustrate the ability of the method to simulate rayleigh waves around a corner and adapt the mesh to the model geometry in addition a domain decomposition example shows how the boundary treatment handles wave propagation from one mesh to another mesh

shape classification using invariant features and contextual information in the bag of words model in this paper we describe a classification framework for binary shapes that have scale rotation and strong viewpoint variations to this end we develop several novel techniques first we employ the spectral magnitude of log polar transform as a local feature in the bag of words model second we incorporate contextual information in the bag of words model using a novel method to extract bi grams from the spatial co occurrence matrix third a novel metric termed weighted gain ratio is proposed to select a suitable codebook size in the bag of words model the proposed metric is generic and hence it can be used for any clustering quality evaluation task fourth a joint learning framework is proposed to learn features in a data driven manner and thus avoid manual fine tuning of the model parameters we test our shape classification system on the animal shapes dataset and significantly outperform state of the art methods in the literature highlightsa shape classifier tolerant to scale rotation and viewpoint changes is proposed the inclusion of the histogram of bi grams improves the bag of words model codebook selection and feature learning improve the classification accuracy we report best results on the animal shapes dataset using the proposed method

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

polynomials and potential theory for gaussian radial basis function interpolation we explore a connection between gaussian radial basis functions and polynomials using standard tools of potential theory we find that these radial functions are susceptible to the runge phenomenon not only in the limit of increasingly flat functions but also in the finite shape parameter case we show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices while not imposing restrictions on the shape parameter or number of points

structure and dynamics of modulated traveling waves in cellular flames we describe the formation and evolution of spatial and temporal patterns in cylindrical premixed flames we consider the cellular regime le 1 where the lewis number le is the ratio of thermal to mass diffusivity of a deficient component of the combustible mixture a transition from stationary axisymmetric flames to stationary cellular flames is predicted analytically if le is decreased below a critical value we present the results of numerical computations to show that as le is further decreased with all other parameters fixed traveling waves tws along the flame front arise via an infinite period bifurcation which breaks the reflection symmetry of the cellular array upon further decreasing le we find the development of different kinds of periodically modulated traveling waves mtws as well as a branch of quasiperiodically modulated traveling waves qpmtws these transitions are accompanied by the development of different spatial and temporal symmetries including period doublings and period halvings in appropriate coordinate systems we also observe the apparently chaotic temporal behavior of a disordered cellular pattern involving creation and annihilation of cells we analytically describe the stability of the tw solution near its onset using suitable phase amplitude equations within this framework one of the mtws can be identified as a localized wave traveling through an underlying stationary spatially periodic structure

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

multiresolution gray scale and rotation invariant texture classification with local binary patterns presents a theoretically very simple yet efficient multiresolution approach to gray scale and rotation invariant texture classification based on local binary patterns and nonparametric discrimination of sample and prototype distributions the method is based on recognizing that certain local binary patterns termed uniform are fundamental properties of local image texture and their occurrence histogram is proven to be a very powerful texture feature we derive a generalized gray scale and rotation invariant operator presentation that allows for detecting the uniform patterns for any quantization of the angular space and for any spatial resolution and presents a method for combining multiple operators for multiresolution analysis the proposed approach is very robust in terms of gray scale variations since the operator is by definition invariant against any monotonic transformation of the gray scale another advantage is computational simplicity as the operator can be realized with a few operations in a small neighborhood and a lookup table experimental results demonstrate that good discrimination can be achieved with the occurrence statistics of simple rotation invariant local binary patterns

motion of three point vortices on a sphere abstract we solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius r the system is more general than the corresponding one in the plane 1 3 5 19 28 33 34 37 38 and reduces to it in the limit r as long as the three vortices remain sufficiently close to each other during the course of their motion instead of using spherical coordinates we use cartesian coordinates in which the vector i points from the center of the sphere to the vortex with strength i an important conserved quantity is the center of vorticity vector c i i i which with no loss of generality we align with the z axis based on the size of this vector relative to the radius of the sphere we classify the motion into one of five types super radial radial sub radial degenerate or a limiting super radial case this categorization allows us to draw several conclusions about the qualitative motion of the vortices we then fully characterize all fixed and relative equilibria on the sphere for fixed equilibria the vortices must lie on great circles geodesics if the strengths are equal they form an equilateral triangle otherwise the triangle shape is specified once the strength of the vortices is given the relative equilibria are classified as either degenerate c 0 or non degenerate c 0 for each type the shape of the vortex triangle is described and the frequency of rotation is computed as in the planar problem it is possible to introduce trilinear coordinates and study the motion in a phase plane which allows us to locate all the equilibria as well as to characterize more complex relative dynamics we then describe self similar vortex collapse on the sphere stating necessary and sufficient conditions for collapse to occur computing the collapse times and vortex trajectories on the route towards collapse comparisons are made with the collapse formulas in the plane derived in 1 30

error saturation in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation one restriction is error saturation which occurs for many types of rbfs including gaussian rbfs of the form f x a h exp email protected 2 x h 2 in the limit h 0 for fixed a the error does not converge to zero but rather to e s a previous studies have theoretically determined the saturation error for gaussian rbf on an infinite uniform interval and for the same with a single point omitted the gap enormously increases e s a we show experimentally that the saturation error on the unit interval email protected 1 1 is about 0 06exp 0 47 a 2 email protected huge compared to the o email protected a 2 exp email protected 2 email protected 2 saturation error for a grid with one point omitted we show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points the saturation error can be avoided by choosing email protected 1 the flat limit but the condition number of the interpolation matrix explodes as o exp p 2 email protected 2 the best strategy is to choose the largest a which yields an acceptably small saturation error if the user chooses an error tolerance d then a o p t i m u m d 1 2log d 0 06

a combined full wave cg fft method for rigorous analysis of large microstrip antenna arrays an accurate and efficient technique is presented for the analysis of large microstrip antenna arrays the technique consists of an amalgamation of a spatially discrete scheme consisting of the cg fft method and the complex discrete image cdi technique the unique feature of this approach is the use of the spatially discrete cg fft for analyzing microstrip structures the aliasing and truncation errors are thoroughly eliminated in this approach in addition the grad div operators are transformed from singular green s functions to differentiable expansion and testing functions by using galerkin s procedure thereby improving the accuracy and the rate of convergence to show the accuracy and efficiency of this technique a number of microstrip arrays including a large microstrip reflectarray have been studied it is found that the simulations carried out using this technique are in very good agreement with measurements

shape matching and object recognition using shape contexts we present a novel approach to measuring similarity between shapes and exploit it for object recognition in our framework the measurement of similarity is preceded by 1 solving for correspondences between points on the two shapes 2 using the correspondences to estimate an aligning transform in order to solve the correspondence problem we attach a descriptor the shape context to each point the shape context at a reference point captures the distribution of the remaining points relative to it thus offering a globally discriminative characterization corresponding points on two similar shapes will have similar shape contexts enabling us to solve for correspondences as an optimal assignment problem given the point correspondences we estimate the transformation that best aligns the two shapes regularized thin plate splines provide a flexible class of transformation maps for this purpose the dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points together with a term measuring the magnitude of the aligning transform we treat recognition in a nearest neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image results are presented for silhouettes trademarks handwritten digits and the coil data set

six strategies for defeating the runge phenomenon in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation because it is meshless there is no canonical grid to act as a starting point for function adaptive grid modification uniform grids are therefore common with rbfs like polynomial interpolation which is the limit of rbf interpolation when the rbf functions are very wide the flat limit rbf interpolants are vulnerable to the runge phenomenon whether the grid is uniform or irregular the n point interpolant may diverge as n on the interval spanned by the interpolation points even for a function f x that is analytic everywhere on and near this interval in this work we discuss six strategies for defeating the runge phenomenon specializing to a uniform grid on a finite interval and one particular species of rbfs in one space dimension f x exp a 2 h 2 x 2 where h is the grid spacing three strategies fail but three are successful one good strategy is to vary a with the number of interpolation points n as n 1 4 unfortunately this gives both a subgeometric rate of convergence for the approximation error falling as exp qn for some constant q and a matrix condition number that grows as exp pn for some p 0 in order to explain why fixed a independent of n is not a convergent strategy we digress to discuss error saturation and show experimentally that the saturation error on the finite interval is about 0 6exp 0 47 a 2 f if a user chosen error tolerance d is acceptable then the optimum choice is a o p t i m u m d 1 2log d 0 06 the second good strategy is rbf extension which uses rbf functions with centers on a interval slightly larger than the target interval to approximate f x a third strategy is to split the interval into two boundary layers and a large middle interval and apply separate rbf approximations on each the slow decrease of a strategy is much cheaper but rbf extension is much more resistant to ill conditioning and therefore can achieve much lower errors the three interval method is the least accurate but is robust and inexpensive

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

deleted residuals the qr factored newton iteration and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary cnoidal and shock waves abstract solitary waves cnoidal waves and shock waves can be computed by solving a nonlinear eigenvalue problem which in discretized form is a system of nonlinear algebraic equations unfortunately many such systems are singular because the solution is not unique until one or more additional constraints are imposed for example if the waves are translationally invariant and u x is a solution then so also is u x for arbitrary to obtain a unique solution one must impose an additional condition to reduce the one parameter family of solutions by constraining we describe five methods for coping with such singular systems i reformulation of the problem ii deleting residuals iii keller s bordered matrix scheme iv qr factored overdetermined newton iteration and v pseudoinverse newton iteration we illustrate these ideas using the cnoidal waves of the korteweg de vries equation the traveling shocks of the korteweg de vries burgers equation and the weakly nonlocal solitary waves of the nonlinear equatorial beta plane equations finite difference fourier and rational chebyshev pseudospectral methods and spectrally upgraded finite differences are applied reformulation and deleting residuals are the cheapest strategies but the qr factored newton iteration is needed for the shock waves which lack the symmetry of the other two wave species

the arctan tan and kepler burgers mappings for periodic solutions with a shock front or internal boundary layer abstract many periodic solutions have internal regions of rapid change internal boundary layers shock waves and geophysical fronts are one class of examples a second class is composed of functions which decay rapidly away from a central peak or peaks spherical harmonics mathieu eigenfunctions prolate spheroidal wave functions and geophysical hough functions may all be locally approximated by hermite functions in the appropriate parameter range and decay exponentially fast outside a narrow subinterval similarly the large amplitude cnoidal waves of the korteweg devries equation are narrow isolated peaks which are well approximated by the sech 2 y form of the solitary wave in this article we show that a change of coordinate is a powerful tool for resolving such internal boundary layers in the first part we develop a general theory of mappings for the spherical harmonic cnoidal wave class of examples which decay rapidly away towards the edges of the spatial period the particular map y arc tan l tan x is a particularly effective choice four numerical examples show that this map and the fourier pseudospectral method are a good team in the second part we generalize the earlier theory to describe mappings which asymptote to a constant but non zero resolution at the ends of the periodicity interval we explain why the kepler burgers mapping is particularly suitable for shock and fronts

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

explicit parallel poisson integration of point vortices on the sphere point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics but also in practical applications for instance in aeronautics in spherical geometry the motion of vortex centres is governed by a dynamical system with a known poisson structure we construct poisson integration methods for these dynamics by splitting the hamiltonian into its constituent vortex pair terms from backward error analysis the method is formally known to provide solutions to a modified poisson system with the correct bracket but with a modified hamiltonian function different orderings of the pairwise interactions are considered and also used for the construction of higher order methods the energy and momentum conservation of the splitting schemes is demonstrated for several test cases for particular orderings of the pairwise interactions the schemes allow scalable parallelization this results in a linear as opposed to quadratic scaling of computation time with system size when scaling the number of processors accordingly

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

efficiently locating objects using the hausdorff distance the hausdorff distance is a measure defined between two point sets here representing a model and an image the hausdorff distance is reliable even when the image contains multiple objects noise spurious features and occlusions in the past it has been used to search images for instances of a model that has been translated or translated and scaled by finding transformations that bring a large number of model features close to image features and vice versa in this paper we apply it to the task of locating an affine transformation of a model in an image this corresponds to determining the pose of a planar object that has undergone weak perspective projection we develop a rasterised approach to the search and a number of techniques that allow us to locate quickly all transformations of the model that satisfy two quality criteria we can also efficiently locate only the best transformation we discuss an implementation of this approach and present some examples of its use

a fourier error analysis for radial basis functions on an infinite uniform grid part 2 spectral plus is special a frequent claim in the radial basis function rbf literature is that with a wise choice of the shape parameter a spectrally accurate rbfs can be spectral plus in the sense that the rbf approximation is orders of magnitude more accurate than the corresponding spectral method this is surprising because in the so called flat limit a 0 rbf methods provably reduce to a standard spectral method on an infinite uniform grid for example gaussian hyperbolic secant inverse quadratic and multiquadric and inverse multiquadric rbfs reduce to the classical sinc basis using the error theorem proved in part 1 we show that indeed it is possible for rbfs to be greatly superior to the sinc basis however rather special conditions are required for this superiority the generic case is that rbfs are less accurate than the sinc method for nonzero values of the shape parameter

non self similar partial and robust collapse of four point vortices on a sphere this paper gives numerical examples showing that non self similar collapse can occur in the motion of four point vortices on a sphere it is found when the four vortex problem is integrable in which the moment of vorticity vector is zero the non self similar collapse has significant properties it is partial in the sense that three of the four point vortices collapse to one point in finite time and the other one moves to the antipodal position to the collapse point moreover it is robust with respect to perturbation of the initial configuration as long as the system remains integrable

an accelerated dft mom for the analysis of large finite periodic antenna arrays a novel approach which utilizes a discrete fourier transform dft based preconditioner in an iterative moment method mom solution of the governing integral equation for the electromagnetic em problems of the radiation and scattering from large finite periodic antenna arrays is presented the preconditioner is primarily constructed via the block diagonal approximation of the transformed impedance matrix in the dft domain in which that matrix becomes sparse and nearly diagonal due to the properties of the dft basis functions the preconditioner in the original spatial domain is simply obtained by the inverse dft this preconditioner is shown to significantly improve the convergence rate of the iterative solution moreover it is rather simple to implement and requires relatively small extra computational cost the performance of the present approach is illustrated through numerical results

spectral implementation of an adaptive moving mesh method for phase field equations phase field simulations have been extensively applied to modeling microstructure evolution during various materials processes however large scale simulations of three dimensional 3d microstructures are still computationally expensive among recent efforts to develop advanced numerical algorithms the semi implicit fourier spectral method is found to be particularly efficient for systems involving long range interactions as it is able to utilize the fast fourier transforms fft on uniform grids in this paper we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy while maintaining a uniform grid in the computational domain for the spectral implementation this approach not only provides more accurate treatment at the interfaces requiring higher resolution but also retains the numerical efficiency of the semi implicit fourier spectral method numerical examples using the new adaptive moving mesh semi implicit fourier spectral method are presented for both two and three space dimensional microstructure simulations and they are compared with those obtained by other methods by maintaining a similar accuracy the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size

invariant dynamical systems embedded in the n vortex problem on a sphere with pole vortices abstract we are concerned with the system of n vortex points on a sphere with two fixed vortex points at the poles this article gives a reduction method of the system to invariant dynamical systems when all the vortex points have the same strength it is carried out by considering the invariant property of the system with respect to the shift and pole reversal transformations for which the polygonal ring configuration of the n vortex points at the line of latitude called the n ring remains unchanged we prove that there exists a 2 p dimensional invariant dynamical system reduced by the p shift transformation for an arbitrary factor p of n the p shift invariant system is equivalent to the p vortex points system generated by the averaged hamiltonian with the modified pole vortices it is also shown that the system can be reduced by the pole reversal transformation when the pole vortices are identical since the reduced dynamical systems are defined in the linear space spanned by the eigenvectors given in the linear stability analysis for the n ring we obtain the inclusion relation among the invariant reduced dynamical systems this allows us to decompose the system of a large number of vortex points into a collection of invariant reduced subsystems

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

spectral domain analysis of conducting patches of arbitrary geometry in multilayer media using the cg fft method a conjugate gradient fast fourier transform cg fft scheme for analyzing finite flat metallic patches in multilayer structures is presented rooftop and razor blade functions are considered as basis and testing functions respectively an equivalent periodic problem in both domains real and spectral is obtained and solved aliasing problems are avoided by performing a window on the green s function the spectral domain periodicity makes it feasible to take into account almost all the harmonics and to reduce the ripple in the computed current distributions nearly all the operations are performed in the spectral domain including green s function computations several results of convergence rates current distributions and radar cross section rcs values are given and compare favorably with measurements or results obtained by other methods

a rational spectral collocation method with adaptively transformed chebyshev grid points a spectral collocation method based on rational interpolants and adaptive grid points is presented the rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed chebyshev points the locations of the grid points are adapted to singularities of the underlying solution and the locations of these singularities are approximated by the locations of poles of chebyshev pade approximants numerical experiments on two time dependent problems one with finite time blow up and one with a moving front indicate that the method far outperforms the standard chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to 1 1

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

fourier coding of image boundaries a transform coding scheme for closed image boundaries on a plane is described the given boundary is approximated by a series of straight line segments depending on the shape the boundary is represented by the x y coordinates of the endpoints of the line segments or by the magnitude of the successive radii vectors that are equispaced in angle around the given boundary due to the circularity present in the data the discrete fourier transform is used to exactly decorrelate the finite boundary data by fitting a gaussian circular autoregressive model to represent the boundary data estimates of the variances of the fourier coefficients are obtained using the variances of the fourier coefficients and the max quantizer the coding scheme is implemented the scheme is illustrated by an example

streamline topologies for integrable vortex motion on a sphere we describe the instantaneous streamline patterns that occur on the surface of a two dimensional sphere in the presence of point vortices of general strength for cases in which the system is completely integrable after stating some general results based on the spherical topology we categorize all possible instantaneous streamline patterns and describe their stagnation point structure for the cases of two and three vortices it is found that for the case of two vortices the only non degenerate topologies that can arise are a figure eight lemniscate or a limacon which are homotopically equivalent for the case of three vortices there are 12 topologically distinct primitives from which an additional 23 patterns can be produced via continuous deformations on the sphere homotopies all possible streamline patterns that arise from three vortex motion can be obtained via linear superposition of the primitive topologies and their homotopic equivalents in this sense the primitives can be viewed as the building blocks for the general flow patterns the equations of motion and streamline patterns in the stereographic plane are obtained using the projected equations of motion in hamiltonian form we describe streamline patterns for three vortex fixed equilibria and relative equilibria as seen in both a fixed and a rotating frame of reference non equilibrium streamline patterns for a collapsing and special periodic solution are studied although to generally understand and classify the many bifurcations associated with non equilibrium patterns would require more extensive computation we conclude with a discussion of the relevance of the three vortex topological classification scheme and the bifurcation of streamline topologies for understanding global atmospheric weather patterns spherical isobars and large scale mixing phenomena 2000 elsevier science b v all rights reserved

scale based description and recognition of planar curves and two dimensional shapes the problem of finding a description at varying levels of detail for planar curves and matching two such descriptions is posed and solved in this paper a number of necessary criteria are imposed on any candidate solution method path based gaussian smoothing techniques are applied to the curve to find zeros of curvature at varying levels of detail the result is the generalized scale space image of a planar curve which is invariant under rotation uniform scaling and translation of the curve these properties make the scale space image suitable for matching the matching algorithm is a modification of the uniform cost algorithm and finds the lowest cost match of contours in the scale space images it is argued that this is preferable to matching in a so called stable scale of the curve because no such scale may exist for a given curve this technique is applied to register a landsat satellite image of the strait of georgia b c manually corrected for skew to a map containing the shorelines of an overlapping area