interpolation of electromagnetic radiated fields over a plane by nonuniform samples an iterative algorithm is developed which allows the interpolation of two dimensional band limited functions from nonuniformly distributed samples such a scheme is based on optimal sampling expansions of central type and is applied to the interpolation of electromagnetic fields over a plane the algorithm can be applied whenever the average sample s density is higher than the nyquist one and there is a one to one correspondence between the nonuniform samples and a lattice of the uniform ones which associates at each uniform sampling point the nearest nonuniform sampling point as compared with the optimal linear estimation algorithm the procedure is computationally much more efficient many numerical simulations have assessed the rapid convergence of the method as well as its stability with respect to both absolute and relative errors in the data

quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

overlapping radial basis function interpolants for spectrally accurate approximation of functions of eigenvalues with application to buckling of composite plates abstract many physical simulations involve eigenvalue computations natural frequencies stability engineering optimization requires many evaluations of such eigenvalues leading to excessive computational times especially for global optimization techniques accurate approximations of eigenvalues with respect to optimization variables that often are natural parameters of the physical systems are then required to alleviate the computational burden dependence of the critical eigenvalue with respect to these natural parameters is often complex part of the reason is that the critical eigenvalue is the minimum of several eigenvalues resulting in a loss of differentiability for physical parameters where the critical eigenvalue becomes multiple this discontinuous derivative prevents from accurate approximation whenever the approximation model is smooth such as most of the standard approximation techniques kriging artificial neural network in this work we present an original strategy that takes into account this discontinuous behavior by dividing the input space through the clustering of the gradient space radial basis interpolants are then constructed over each region and a patch region is defined that encompasses the non differentiable region this allows us to retrieve over the whole domain of definition excellent convergence rates and numerical experiments over realistic numerical simulation show that it is possible to achieve spectrally accurate approximations

near field antenna measurements using nonideal measurement locations we introduce a near field to far field transformation algorithm that relaxes the usual restriction that data points be located on a plane rectangular grid computational complexity is o nlog n where n is the number of data points this algorithm allows efficient processing of near field data with known probe position errors also the algorithm is applicable to other measurement approaches such as plane polar scanning where data are collected intentionally on a nonrectangular grid

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

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

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

on the interpolation of electromagnetic near field without prior knowledge of the radiating source a method is proposed to interpolate the electromagnetic near field when no information on the radiating source is available in the absence of a priori knowledge general properties of the electromagnetic field are exploited to estimate the field namely the minimum complexity of the field and the continuity of the first derivatives these properties are enforced by minimizing the nuclear norm and using the thin plate spline interpolation results respectively the proposed procedure is validated experimentally by interpolating the planar electrical near field radiated by three antennas the quality of the interpolation and its robustness to noise are investigated despite its simplicity the interpolation method is able to properly estimate the near field from a random coarse sampling of 2 lambda the quality of the near field interpolation is also confirmed by deriving the far field these promising results pave the way for the development of fast antenna measurement procedures

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

singular localization for quantum groups at generic q abstract we quantize parabolic flag manifolds and describe categories of quantum d modules on them at a singular central character we compute global sections of generators for these categories for any q c for generic q we prove a singular version of beilinson bernstein localization for a quantized enveloping algebra

representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples it is shown that the electromagnetic em field radiated or scattered by bounded sources can be accurately represented over a substantially arbitrary surface by a finite number of samples even when the observation domain is unbounded the number of required samples is nonredundant and essentially coincident with the number of degrees of freedom of the field this result relies on the extraction of a proper phase factor from the field expression and on the use of appropriate coordinates to parameterize the domain it is demonstrated that the number of degrees of freedom is independent of the observation domain and depends only on the source geometry the case of spheroidal sources and observation domains with rotational symmetry is analyzed in detail and the particular cases of spherical and planar sources are explicitly considered for these geometries precise and fast sampling algorithms of central type are presented which allow an efficient recovery of em fields from a nonredundant finite number of samples such algorithms are stable with respect to random errors affecting the data

antenna coupling and near field sampling in plane polar coordinates a probe corrected vector transmission formula and a rigorous sampling reconstruction theorem for near field antenna measurements in plane polar coordinates are derived from three fundamental theorems of antenna theory the mutual coupling function between two antennas satisfies the homogeneous wave equation a receiving antenna can be represented as a differentiator of the incident field and the mutual coupling function is virtually bandlimited the rigorous sampling equations are applied to compute the far fields of a circular aperture antenna sampled in the near field at half wavelength radial spacing

the sampling window corresp the concept of the sampling window is introduced for the central interpolation of finite energy band limited functions the sampling window does not increase the rate of convergence of the truncation error series as do various convergence factors but does significantly reduce the truncation error bound

preconditioned conjugate gradients radial basis functions and toeplitz matrices abstract radial basis functions provide highly useful and flexible interpolants to multivariate functions further they are beginning to be used in the numerical solution of partial differential equations unfortunately their construction requires the solution of a dense linear system therefore much attention has been given to iterative methods in this paper we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data thus our method applies to the corresponding toeplitz matrices the number of iterations required to achieve a given tolerance is independent of the number of variables

phaseless antenna measurement on non redundant sample points via leith upatnieks holography a scheme for reducing the number of sample points in phaseless acquisition based on leith upatnieks holography is presented an appropriate choice of the phase of the reference field enables to drastically decrease the sampling rate by acquiring the so called reduced field instead of the real radiated field the reference phase changes are enabled by applying the recent developments on the synthesis of the reference wave moreover optimal sampling interpolation can be applied to the retrieved field so that the real field can be computed at any point on the acquisition surface the method is illustrated by means of multiple simulations for observation domains such as straight lines azimuthal circles as well as for a plane polar acquisition scheme

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

phaseless bi polar planar near field measurements and diagnostics of array antennas antenna near field measurements typically require very accurate measurement of the near field phase there are applications where an accurate phase measurement may not be practically achievable phaseless measurements are beginning to emerge as an alternative microwave antenna measurements technique when phase cannot be directly measured there are many important aspects for successful implementation of a phaseless measurement algorithm this paper presents appropriate phaseless measurement requirements and a phase retrieval algorithm tailored for the bi polar planar near field antenna measurement technique two amplitude measurements and a squared amplitude optimal sampling interpolation method are integrated with an iterative fourier procedure to first retrieve the phase information and then construct both the far field pattern and diagnostic characteristics of the antenna under test in order to critically examine the methodologies developed in this paper phaseless measurement results for two different array antennas are presented and compared to results obtained when the near field amplitude and phase are directly measured

localization for quantum groups at a root of unity let c be the field of complex numbers and fix q g c let g be a semi simple lie algebra over c and let g be the corresponding simply connected algebraic group let uq be the quantized enveloping algebra of q let oq be the algebra of quantized functions on g let oq b be the quotient hopf algebra of oq corresponding to a borel subgroup b of g in the paper bk we defined categories of equivariant quantum o modules and pg modules on the quantum flag variety of g we proved that the beilinson bernstein localization theorem holds at a generic q namely the global section functor gives an equivalence between categories of modules and vq iaod es on the quantum flag variety thus one can translate questions about the represen tation theory of quantum groups to the study of the noncommutative geometry of the quantum flag variety in this paper we prove that a derived version of this theorem holds at the root of unity case using this equivalence we get that one can understand the representation theory of quantum groups at roots of unity through the now commutative geometry of the springer fibers we now recall the main results in bk the constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section we defined an equivariant sheaf of quasi coherent modules over the quantum flag variety to be a left oq mod e equipped with a right q b comod ile structure satisfying certain compatibility conditions such objects form a category denoted ai q gq it contains certain line bundles oq for in the weight lattice we proved that oq x is ample for 0 holds for every q this implies that the category m bq gq is a proj category in the sense of serre

the ramification of centres lie algebras in positive characteristic and quantised enveloping algebras let h be a hopf algebra over the field k which is a finite module over a central affine sub hopf algebra r examples include enveloping algebras u mathfrak g of finite dimensional k lie algebras mathfrak g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity the ramification behaviour of the maximal ideals of z h with respect to the subalgebra r is studied and the conclusions are then applied to the cases of classical and quantised enveloping algebras in the case of u mathfrak g for mathfrak g semisimple a conjecture of humphreys 28 on the block structure of u mathfrak g is confirmed in the case of u epsilon mathfrak g for mathfrak g semisimple and epsilon an odd root of unity we obtain a quantum analogue of a result of mirkovic and rumynin 35 and we fully describe the factor algebras lying over the regular sheet 9 the blocks of u epsilon mathfrak g are determined and a necessary condition which may also be sufficient for a baby verma u epsilon mathfrak g module to be simple is obtained

the truncation error in the application of sampling series to electromagnetic problems truncation error interpolation methods used in the evaluation and representation of scattered fields are discussed three main points are considered the choice of the sampling point lattice the choice of the type of sampling expansion and the determination of the number of terms of the sampling series that must be retained to insure a negligible truncation error the aim is to provide a clear reference frame for the applications which allow the optimal choice for the reconstruction algorithm the cases of bounded and square integrable band limited functions are separately dealt with in order to exploit the properties of each method a general analysis of the multidimensional truncation error is carried out no particular assumption on the sampling expansion is made except for that concerning the use of a rectangular sampling lattice simple and effective expressions relating the multidimensional error to the corresponding one dimensional bounds are derived the one dimensional error is then analyzed in detail with reference to the most relevant case of central interpolation by very simple methods error bounds for all the sampling expansions explicitly considered in the literature are derived in a systematic way expressions providing the values of the sampling rate and number of retained samples which minimize the overall computer time and memory requirement are derived some numerical examples are presented and briefly discussed

minimal length elements of coxeter groups let w be a coxeter group and wj a finite parabolic subgroup we present here a geometric new proof of x he s result on minimal length elements of an orbit of wj which acts on w in a certain way

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

strictly transversal slices to conjugacy classes in algebraic groups abstract we show that for every conjugacy class o in a connected semisimple algebraic group g over an algebraically closed field of characteristic good for g one can find a special transversal slice to the set of conjugacy classes in g such that o intersects and dim o codim

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

sampling in plane polar coordinates a number of details are clarified regarding the recently derived sampling reconstruction theorem for near field scanning in plane polar coordinates the rigorous sampling reconstruction theorem is applied to the near field measurement of a circular aperture test antenna offset from the plane polar axis of rotation so that a large number of angular modes are necessary to represent the fields of the test antenna an algorithm is described for computing accurately and rapidly the required zeros of bessel functions of arbitrary integer order

quantum flag varieties equivariant quantum d modules and localization of quantum groups abstract let o q g be the algebra of quantized functions on an algebraic group g and o q b its quotient algebra corresponding to a borel subgroup b of g we define the category of sheaves on the quantum flag variety of g to be the o q b equivariant o q g modules and prove that this is a proj category we construct a category of equivariant quantum d modules on this quantized flag variety and prove the beilinson bernstein s localization theorem for this category in the case when q is transcendental

advanced field interpolation from plane polar samples experimental verification a new and advanced sampling technique far field interpolation with a nonredundant number of samples on plane polar geometry has been experimentally validated for cases of both complex and only amplitude measurements experimental results show good stability of the interpolation algorithm with respect to noise and the lack of samples in the case of a limited scanning area

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

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

optimal time domain field interpolation from plane polar samples new time domain sampling series for representing radiated fields on a plane in terms of plane polar samples are presented the representations make it possible for the advanced sampling field interpolations introduced in the frequency domain to be generalized to the time domain thus enabling minimum redundance when using either a cardinal sampling scheme or a central interpolation one numerical analysis is then conducted to prove the effectiveness of the new representations

optimal reconstruction of bandlimited bounded signals a formula is presented for restoring bounded bandlimited signals from a finite number of samples the restoration is linear in the samples used and the error bound of the method is the theoretical minimum

on the degrees of freedom of scattered fields starting from the observation that fields differing less than a prescribed error cannot be resolved as distinct entities the degrees of freedom of the scattered field are introduced and then computed the degrees of freedom are shown to be practically equal to the nyquist number appropriate to the effective spatial bandwidth of the scattered field and to the extension of the observation domain accordingly a finite number of elements of information can be used to determine the scattered field to a prescribed approximation error it is also shown that the field representation can be made in terms of field values and simple sampling functions provided that a marginal increase in the approximation error is tolerated the results not only completely justify the use of sampling interpolation for representing scattering fields but also demonstrate that such representation is practically an optimal one an algorithm for the reconstruction of scattered fields given the maximum allowed error is then produced

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

the beilinson bernstein correspondence for quantized enveloping algebras theory of the quantized flag manifold as a quasi scheme non commutative scheme has been developed by lunts rosenberg 15 they have formulated an analogue of the beilinson bernstein correspondence using the q differential operators introduced in their earlier paper 14 in this paper we shall establish its modified version using a class of q differential operators which is possibly smaller than the one in 14

smoothing technique and its applications in semidefinite optimization in this paper we extend the smoothing technique nesterov in math program 103 1 127 152 2005 nesterov in unconstrained convex mimimization in relative scale 2003 onto the problems of semidefinite optimization for that we develop a simple framework for estimating a lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices using this technique we can justify the lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices we analyze the efficiency of the special gradient type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix which depends linearly on the design variables we show that in the first case the number of iterations of the method is bounded by o 1 epsilon where epsilon is the required absolute accuracy of the problem in the second case the number of iterations is bounded by 4 delta sqrt 1 delta r ln r where is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality thus the latter method is a fully polynomial approximation scheme

minimal length elements in some double cosets of coxeter groups abstract we study the minimal length elements in some double cosets of coxeter groups and use them to study lusztig s g stable pieces and the generalization of g stable pieces introduced by lu and yakimov we also use them to study the minimal length elements in a conjugacy class of a finite coxeter group and prove a conjecture in m geck s kim g pfeiffer minimal length elements in twisted conjugacy classes of finite coxeter groups j algebra 229 2 2000 570 600

localization of modules for a semisimple lie algebra in prime characteristic with an appendix by r bezrukavnikov and s riche computation for sl 3 we show that on the level of derived categories representations of the lie algebra of a semisimple algebraic group over a field of finite characteristic with a given generalized regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding generalized springer fiber the first step is to observe that the derived functor of global sections provides an equivalence between the derived category of d modules with no divided powers on the flag variety and the appropriate derived category of modules over the corresponding lie algebra thus the derived version of the beilinson bernstein localization theorem holds in sufficiently large positive characteristic next one finds that for any smooth variety this algebra of differential operators is an azumaya algebra on the cotangent bundle in the case of the flag variety it splits on springer fibers and this allows us to pass from d modules to coherent sheaves the argument also generalizes to twisted d modules as an application we prove lusztig s conjecture on the number of irreducible modules with a fixed central character we also give a formula for behavior of dimension of a module under translation functors and reprove the kac weisfeiler conjecture the sequel to this paper bmr2 treats singular infinitesimal characters

geometric and homological properties of affine deligne lusztig varieties this paper studies ane deligne lusztig varieties x w b in the ane ag variety of a quasi split tamely ramied group we describe the geometric structure of x w b for a minimal length element w in the conjugacy class of an extended ane weyl group we then provide a reduction method that relates the structure of x w b for arbitrary elements w in the extended ane weyl group to those associated with minimal length elements based on this reduction we establish a connection between the dimension of ane deligne lusztig varieties and the degree of the class polynomial of ane

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

optimal interpolation of radiated fields over a sphere an optimal sampling interpolation algorithm is developed that allows the accurate recovery of scattered or radiated fields over a sphere from a minimum number of samples using the concept of the field equivalent spatial bandwidth a central interpolation scheme is developed to compute the field in theta phi coordinates starting from its samples the maximum allowable sample spacing and error upper bounds are also rigorously derived several simulated examples of pattern reconstruction are presented for both the cases of field and power pattern interpolation the interpolation error as a function of the retained sample number has been also evaluated and compared with the theoretical upper bounds the algorithm stability versus randomly distributed errors added to the exact samples is demonstrated

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

representations of quantum groups at roots of unity whittaker vectors and q w algebras abstract let u e g be the standard simply connected version of the drinfeld jumbo quantum group at an odd m th root of unity e the center of u e g contains a huge commutative subalgebra isomorphic to the algebra z g of regular functions on a finite covering of a big cell in a complex connected simply connected algebraic group g with lie algebra g let v be a finite dimensional representation of u e g on which z g acts according to a non trivial character g given by evaluation of regular functions at g g then v is a representation of the finite dimensional algebra u g u e g u e g ker g we show that in this case under certain restrictions on m u g contains a subalgebra u g m of dimension m 1 2 dim o where o is the conjugacy class of g and u g m has a one dimensional representation c g we also prove that if v is not trivial then the space of whittaker vectors hom u g m c g v is not trivial and the algebra w g end u g u g u g m c g naturally acts on it which gives rise to a schur type duality between representations of the algebra u g and of the algebra w g called a q w algebra

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

fast and accurate near field far field transformation by sampling interpolation of plane polar measurements an optimal sampling interpolation algorithm which allows the accurate recovery of plane rectangular near field samples from the knowledge of the plane polar ones is developed this enables the standard near field far field nf ff transformation which takes full advantage of the fast fourier transform fft algorithm to be applied to plane polar scanning the maximum allowable sample spacing is also rigorously derived and it is shown that it can be significantly greater than lambda 2 as the measurement place moves away from the source this allows a remarkable reduction of both measurement time and memory storage requirements the sampling approach is compared with that based on the bivariate lagrange interpolation bli method the sampling reconstruction agrees with the exact results significantly better than the bli in spite of the significantly lower number of required measurements