a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

fast and robust fuzzy c means clustering algorithms incorporating local information for image segmentation fuzzy c means fcm algorithms with spatial constraints fcm s have been proven effective for image segmentation however they still have the following disadvantages 1 although the introduction of local spatial information to the corresponding objective functions enhances their insensitiveness to noise to some extent they still lack enough robustness to noise and outliers especially in absence of prior knowledge of the noise 2 in their objective functions there exists a crucial parameter a used to balance between robustness to noise and effectiveness of preserving the details of the image it is selected generally through experience and 3 the time of segmenting an image is dependent on the image size and hence the larger the size of the image the more the segmentation time in this paper by incorporating local spatial and gray information together a novel fast and robust fcm framework for image segmentation i e fast generalized fuzzy c means fgfcm clustering algorithms is proposed fgfcm can mitigate the disadvantages of fcm s and at the same time enhances the clustering performance furthermore fgfcm not only includes many existing algorithms such as fast fcm and enhanced fcm as its special cases but also can derive other new algorithms such as fgfcm s1 and fgfcm s2 proposed in the rest of this paper the major characteristics of fgfcm are 1 to use a new factor s i j as a local both spatial and gray similarity measure aiming to guarantee both noise immunity and detail preserving for image and meanwhile remove the empirically adjusted parameter a 2 fast clustering or segmenting image the segmenting time is only dependent on the number of the gray levels q rather than the size n q of the image and consequently its computational complexity is reduced from o nci 1 to o qci 2 where c is the number of the clusters i 1 and i 2

on the ubiquity of the cauchy distribution in spectral problems we consider the distribution of the values at real points of random functions which belong to the herglotz pick hp class of analytic mappings of the upper half plane into itself it is shown that under mild stationarity assumptions the individual values of hp functions with singular spectra have a cauchy type distribution the statement applies to the diagonal matrix elements of random operators and holds regardless of the presence or not of level repulsion i e applies to both random matrix and poisson type spectra

recovering exponential accuracy from collocation point values of smooth functions with end point singularities gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity the truncated spectral series has large oscillations near the jump and the overshoot does not decay as the number of terms in the truncated series increases there is therefore no convergence in the maximum norm and convergence in smooth regions away from the discontinuity is also slow in gottlieb and shu 1995 5 a methodology is proposed to completely overcome this difficulty in the context of spectral collocation methods resulting in the recovery of exponential accuracy from collocation point values of a piecewise analytic function in this paper we extend this methodology to handle spectral collocation methods for functions which are analytic in the open interval but have singularities at end points with this extension we are able to obtain exponential accuracy from collocation point values of such functions similar to gottlieb and shu 1995 5 the proof is constructive and uses the gegenbauer polynomials c n l x the result implies that the gibbs phenomenon can be overcome for smooth functions with endpoint singularities

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

an averaging method for the fourier approximation to discontinuous functions both the truncated fourier integral and the truncated fourier series approximations for a discontinuous function bring about the inevitable oscillating error say the gibbs phenomenon most basic filtering methods for the gibbs phenomenon like the fejer averaging method and the lanczos averaging method have a disadvantage that the rise time is very slow even though the filtering effect is prominent away from the discontinuity in this paper we propose a new averaging method of polynomial type which improves the rise time of the existing method the present method can be regarded as a generalization of the traditional lanczos averaging method by several numerical examples we show the efficiency of the present method

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

a modified fuzzy c means algorithm for bias field estimation and segmentation of mri data we present a novel algorithm for fuzzy segmentation of magnetic resonance imaging mri data and estimation of intensity inhomogeneities using fuzzy logic mri intensity inhomogeneities can be attributed to imperfections in the radio frequency coils or to problems associated with the acquisition sequences the result is a slowly varying shading artifact over the image that can produce errors with conventional intensity based classification our algorithm is formulated by modifying the objective function of the standard fuzzy c means fcm algorithm to compensate for such inhomogeneities and to allow the labeling of a pixel voxel to be influenced by the labels in its immediate neighborhood the neighborhood effect acts as a regularizer and biases the solution toward piecewise homogeneous labelings such a regularization is useful in segmenting scans corrupted by salt and pepper noise experimental results on both synthetic images and mr data are given to demonstrate the effectiveness and efficiency of the proposed algorithm

spectral simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

a fast spectral solver for a 3d helmholtz equation we present a fast solver for the helmholtz equation delta u pm lambda 2 u f in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows us to reduce the errors associated with the gibbs phenomenon and achieve any prescribed rate of convergence the algorithm requires o n3 log n operations where n is the number of grid points in each direction we solve a dirichlet boundary problem for the helmholtz equation we also extend the method to the solution of mixed problems where dirichlet boundary conditions are specified on some faces and neumann boundary conditions are specified on other faces high order accuracy is achieved by a comparatively small number of points for example for the accuracy of 10 8 the resolution of only 16 32 points in each direction is necessary

an investigation of resolution of 2 variate gibbs phenomenon abstract in this paper we consider an implementation approach of a method for the resolution of the gibbs phenomenon for f x y which has been discussed extensively by jung and shizgal j h jung b d shizgal generalization of the inverse polynomial reconstruction method in the resolution of the gibbs phenomenon j comput appl math 172 2004 131 151 j h jung b d shizgal inverse polynomial reconstruction of two dimensional fourier images j sci comput 25 3 2005 367 399 b d shizgal j h jung towards the resolution of the gibbs phenomenoa j comput appl math 161 2003 41 65 here we go through some details of applying that technique and also we try to work out its application on some time dependent partial differential equations numerical experiments confirm the efficiency of the method

time domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method the acoustical wave propagator awp scheme involves an effective time domain calculation of sound propagation using the combination of chebyshev polynomial expansion and the fourier pseudospectral method the accuracy of this scheme degrades when the media has discontinuities due to the well known gibbs phenomenon in this paper several issues concerning awp are addressed including an analysis of the effect of gibbs phenomenon on the accuracy a mapped pseudospectral method is proposed wherein the grid points are redistributed with the emphasis across the media discontinuities by a pre determined smooth mapping curve then the spatial derivatives are calculated through a modified fourier pseudospectral method using this method the influence of the gibbs phenomenon is effectively alleviated while the computational efficiency of awp is still maintained the superiority of this improved awp scheme is illustrated by three one dimensional 1 d numerical examples

fcm based model selection algorithms for determining the number of clusters clustering is an important research topic that has practical applications in many fields it has been demonstrated that fuzzy clustering using algorithms such as the fuzzy c means fcm has clear advantages over crisp and probabilistic clustering methods like most clustering algorithms however fcm and its derivatives need the number of clusters in the given data set as one of their initializing parameters the main goal of this paper is to develop an effective fuzzy algorithm for automatically determining the number of clusters after a brief review of the relevant literature we present a new algorithm for determining the number of clusters in a given data set and a new validity index for measuring the goodness of clustering experimental results and comparisons are given to illustrate the performance of the new algorithm

gibbs phenomenon for fourier partial sums on zp abstract it is of interest to know whether the gibbs phenomenon occurs on a local field a p adic heaviside function on the group of p adic integers is defined as an analogy of real variable heaviside function and it is also shown that there exists the gibbs phenomenon with an undershoot of at least 1 p 1 as an approximate level as a consequence of the theorem we see that the fourier partial sum for a heaviside function does not converge at the point of discontinuity

a strategy for choosing gegenbauer reconstruction parameters for numerical stability the gegenbauer reconstruction method was first proposed in 1992 but in early studies no attempts were made to optimize the relevant parameters of this method these parameters were allowed to grow proportionally with the number of nodes which in many cases resulted in exponential convergence for a selected range of the proportionality constants early studies also made clear that very large error bounds could be expected if these key parameters were not chosen carefully subsequent studies then pointed out that although unrelated to the method s analytically predictable domains of poor accuracy round off errors could also sabotage the method s accuracy the challenge of successfully implementing a gegenbauer reconstruction then rests on understanding the performance trade offs we can expect when choosing the key parameters in accordance with different objectives in this study we propose a new strategy for choosing optimal parameters in the chebyshev gegenbauer reconstruction method specifically to achieve numerical stability this strategy is based on asymptotic analysis as well as minimization problems in one and two dimensions the effectiveness of our approach which could also be applied to a wider selection of polynomials is then illustrated with results from numerical experiments

models of intermediate spectral statistics based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the anderson model at the transition point a simple theoretical model where such statistics can be obtained analytically is briefly discussed a few models with similar behavior are considered in particular we analytically found the eigenvalue statistics of a poisson distributed matrix perturbed by a rank one matrix which is a good model for spectral statistics of a singular billiard

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

resolution properties of the fourier method for discontinuous waves in this paper we discuss the wave resolution properties of the fourier approximations of a wave function with discontinuities it is well known that a minimum of two points per wave is needed to resolve a periodic wave function using fourier expansions for chebyshev approximations of a wave function a minimum of points per wave is needed see numerical analysis of spectral methods theory and applications by d gottlieb and s orszag here we obtain an estimate for the minimum number of points per wave to resolve a discontinuous wave based on its fourier coefficients in our recent work on overcoming the gibbs phenomenon we have shown that the fourier coefficients of a discontinuous function contain enough information to reconstruct with exponential accuracy the coefficients of a rapidly converging gegenbauer expansion we therefore study the resolution properties of a gegenbauer expansion where both the number of terms and the order increase

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

fuzzy constrained lp norm inversion of direct current resistivity data abstractwe evaluate the use of a fuzzy c means clustering procedure to improve an inverted 2d resistivity model within the iterative error minimization procedure the algorithm is coded in matlab language for the lp norm inversion of 2d direct current resistivity data and is referred to as fuzzy constrained inversion fci two additional input parameters are required to be provided by the interpreter 1 the number of geologic units in the model i e the number of clusters and 2 the mean resistivity values of each geologic unit i e cluster center values of the geologic units the efficacy of our approach is evaluated by tests carried on the synthetic and field electrical resistivity tomography ert data inversion results from the fci algorithm are presented for conventional l1 and l2 norm minimization techniques fci indicates improvement over conventional inversion approaches in differentiating the geologic units if a proper number of the geologic units is provided to the algorithm inappro

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

partially supervised clustering for image segmentation all clustering algorithms process unlabeled data and consequently suffer from two problems p1 choosing and validating the correct number of clusters and p2 insuring that algorithmic labels correspond to meaningful physical labels clustering algorithms such as hard and fuzzy c means based on optimizing sums of squared errors objective functions suffer from a third problem p3 a tendency to recommend solutions that equalize cluster populations the semi supervised c means algorithms introduced in this paper attempt to overcome these problems domains where a few data from each clas can be labeled segmentation of magnetic resonance images is a problem of this type and we use it to illustrate the new algorithm our examples show that the semi supervised approach provides mri segmentations that are superior to ordinary fuzzy c means and to the crisp k nearest neighbor rule and further that the new method ameliorates p1 p3

feature selection based on a modified fuzzy c means algorithm with supervision in this paper we propose a new approach to feature selection based on a modified fuzzy c means algorithm with supervision mfcms mfcms completes the unsupervised learning of classical fuzzy c means with labeled patterns the labeled patterns allow mfcms to accurately model the shape of each cluster and consequently to highlight the features which result to be particularly effective to characterize a cluster these features are distinguished by a low variance of their values for the patterns with a high membership degree to the cluster if with respect to these features the distance between the prototype of the cluster and the prototypes of the other clusters is high then these features have the property of discriminating between the cluster and the other clusters to take these two aspects into account for each cluster and each feature we introduce a purposely defined index the higher the value of the index the higher the discrimination capability of the feature for the cluster we execute mfcms on the training set considering all patterns as labeled then we retain the features which are associated at least for one cluster with an index larger than a threshold we applied mfcms to several real world pattern classification benchmarks we used the well known k nearest neighbors as learning algorithm we show that feature selection performed by mfcms achieved an improvement in generalization on all data sets

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

validity guided re clustering with applications to image segmentation when clustering algorithms are applied to image segmentation the goal is to solve a classification problem however these algorithms do not directly optimize classification duality as a result they are susceptible to two problems 1 the criterion they optimize may not be a good estimator of true classification quality and 2 they often admit many suboptimal solutions this paper introduces an algorithm that uses cluster validity to mitigate problems 1 and 2 the validity guided re clustering vgc algorithm uses cluster validity information to guide a fuzzy re clustering process toward better solutions it starts with a partition generated by a soft or fuzzy clustering algorithm then it iteratively alters the partition by applying novel split and merge operations to the clusters partition modifications that result in improved partition validity are retained vgc is tested on both synthetic and real world data for magnetic resonance image mri segmentation evaluations by radiologists show that vgc outperforms the unsupervised fuzzy c means algorithm and vgc s performance approaches that of the supervised k nearest neighbors algorithm

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

lagrange interpolation for continuous piecewise smooth functions this note is devoted to lagrange interpolation for continuous piecewise smooth functions a new family of interpolatory functions with explicit approximation error bounds is obtained we apply the theory to the classical lagrange interpolation

accuracy of high order and spectral methods for hyperbolic conservation laws with discontinuous solutions higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions on the other hand higher order methods have been applied to hyperbolic problems with less success as higher order approx imations of discontinuous solutions suffer from the gibbs phenomenon we extend past work and show that spectral methods yield spectral convergence of moments even when applied to problems with discontinuous solutions besides spectral fourier methods for periodic domains we also prove high order convergence for adjoint consistent non periodic numerical methods exemplified by the discontinuous galerkin finite element method

analysis and application of fourier gegenbauer method to stiff differential equations the fourier gegenbauer fg method introduced by gottlieb shu solomonoff and vandeven icase report 92 4 hampton va 1992 is aimed at removing the gibbs phenomenon that is recovering the point values of a nonperiodic function from its fourier coefficients in this paper we discuss some numerical aspects of the fg method related to its em pseudospectral implementation in particular we analyze the behavior of the gegenbauer series with a moderate several hundred number of terms suitable for computations we also demonstrate the ability of the fg method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles bearing on the previous analysis we suggest a high order spectral fourier method for the solution of nonperiodic differential equations it includes a polynomial subtraction technique to accelerate the convergence of the fourier series and the fg algorithm to evaluate derivatives on the boundaries of nonperiodic functions the present hybrid fourier gegenbauer hfg method possesses better resolution properties than the original fg method the precision of this method is demonstrated by solving stiff elliptic problems with steep solutions

minkowski metric feature weighting and anomalous cluster initializing in k means clustering this paper represents another step in overcoming a drawback of k means its lack of defense against noisy features using feature weights in the criterion the weighted k means method by huang et al 2008 2004 2005 5 7 is extended to the corresponding minkowski metric for measuring distances under minkowski metric the feature weights become intuitively appealing feature rescaling factors in a conventional k means criterion to see how this can be used in addressing another issue of k means the initial setting a method to initialize k means with anomalous clusters is adapted the minkowski metric based method is experimentally validated on datasets from the uci machine learning repository and generated sets of gaussian clusters both as they are and with additional uniform random noise features and appears to be competitive in comparison with other k means based feature weighting algorithms

generalized wasserstein distance and its application to transport equations with source in this article we generalize the wasserstein distance to measures with different masses we study the properties of this distance in particular we show that it metrizes weak convergence for tight sequences we use this generalized wasserstein distance to study a transport equation with a source in which both the vector field and the source depend on the measure itself we prove the existence and uniqueness of the solution to the cauchy problem when the vector field and the source are lipschitzian with respect to the generalized wasserstein distance

on weighting clustering recent papers and patents in iterative unsupervised learning have emphasized a new trend in clustering it basically consists of penalizing solutions via weights on the instance points somehow making clustering move toward the hardest points to cluster the motivations come principally from an analogy with powerful supervised classification methods known as boosting algorithms however interest in this analogy has so far been mainly borne out from experimental studies only this paper is to the best of our knowledge the first attempt at its formalization more precisely we handle clustering as a constrained minimization of a bregman divergence weight modifications rely on the local variations of the expected complete log likelihoods theoretical results show benefits resembling those of boosting algorithms and bring modified weighted versions of clustering algorithms such as k means fuzzy c means expectation maximization em and k harmonic means experiments are provided for all these algorithms with a readily available code they display the advantages that subtle data reweighting may bring to clustering

value distribution of the eigenfunctions and spectral determinants of quantum star graphs we compute the value distributions of the eigenfunctions and spectral determinant of the schrodinger operator on families of star graphs the values of the spectral determinant are shown to have a cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related this is in contrast to the spectral determinants of random matrices for which the logarithm is known to satisfy a gaussian limit distribution the value distribution of the eigenfunctions also differs from the corresponding random matrix result we argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of seba type billiards

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise analytic function we continue the investigation of overcoming gibbs phenomenon i e obtaining exponential accuracy at all points including at the discontinuities themselves from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function we show that if we are given the first n expansion coefficients of an l 2 function f x in terms of either the trigonometrical polynomials or the chebyshev or legendre polynomials we can construct an exponentially convergent approximation to the point values of f x in any sub interval in which it is analytic

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

density weighted fuzzy c means clustering in this short paper a unified framework for performing density weighted fuzzy c means fcm clustering of feature and relational datasets is presented the proposed approach consists of reducing the original dataset to a smaller one assigning each selected datum a weight reflecting the number of nearby data clustering the weighted reduced dataset using a weighted version of the feature or relational data fcm algorithm and if desired extending the reduced data results back to the original dataset several methods are given for each of the tasks of data subset selection weight assignment and extension of the weighted clustering results the newly proposed weighted version of the non euclidean relational fcm algorithm is proved to produce the identical results as its feature data analog for a certain type of relational data artificial and real data examples are used to demonstrate and contrast various instances of this general approach

spectrally accurate solution of nonperiodic differential equations by the fourier gegenbauer method it is well known that the fourier partial sum of an analytic nonperiodic function supported on a finite interval converges slowly inside the interval and exhibits o 1 spurious oscillations near the boundaries the gibbs phenomenon an effective algorithm which allows one to completely overcome the gibbs phenomenon was developed in j comput appl math 43 1992 pp 81 92 the basic concept of this approach consists of the re expansion of the fourier partial sums into the rapidly convergent gegenbauer series in this paper we extend the fourier gegenbauer f g method of j comput appl math 43 1992 pp 81 92 to the evaluation of the spatial derivatives of a piecewise analytic function also we apply this method to the solution of nonperiodic boundary value problems although the derivatives of a discontinuous function are not in l2 the exponential convergence of the truncated gegenbauer series can be proved and the rate of convergence can be estimated the solution of differential equations is accomplished in two steps first a particular solution with arbitrary boundary conditions is constructed using the f g method this particular solution is then corrected to satisfy the prescribed boundary conditions of the problem by adding a proper linear combination of homogeneous solutions for boundary layer problems the intermediate particular solution has steep profiles near the boundaries these steep regions introduce a large error into the final solution which presumably has a smooth profile on the whole interval a method which compensates for this loss of accuracy by using the appropriately constructed homogeneous solutions is proposed

wave chaos in quantum systems with point interaction we study perturbations of the quantized version 0 of integrable hamiltonian systems by point interactions we relate the eigenvalues of to the zeros of a certain meromorphic function assuming the eigenvalues of 0 are poisson distributed we get detailed information on the joint distribution of the zeros of and give bounds on the probability density for the spacings of eigenvalues of our results confirm the wave chaos phenomenon as different from the quantum chaos phenomenon predicted by random matrix theory

the resolution of the gibbs phenomenon for spliced functions in one and two dimensions abstract in this paper we study approximation methods for analytic functions that have been spliced into nonintersecting subdomains we assume that we are given the first 2n 1 fourier coefficients for the functions in each subdomain the objective is to approximate the spliced function in each subdomain and then to glue the approximations together in order to recover the original function in the full domain the fourier partial sum approximation in each subdomain yields poor results as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain thus once we glue the subdomain approximations back together the approximation for the function in the full domain will exhibit oscillations throughout the entire domain recently methods have been developed that successfully eliminate the gibbs phenomenon for analytic but nonperiodic functions in one dimension these methods are based on the knowledge of the first 2n 1 fourier coefficients and use either the gegenbauer polynomials gottlieb et al or the bernoulli polynomials abarbanel gottlieb cai et al and eckhoff we propose a way to accurately reconstruct a spliced function in a full domain by extending the current methods to eliminate the gibbs phenomenon in each nonintersecting subdomain and then gluing the approximations back together we solve this problem in both one and two dimensions in the one dimensional case we provide two alternative options the bernoulli method and the gegenbauer method as well as a new hybrid method the gegenbauer bernoulli method in the two dimensional case we prove for the very first time exponential convergence of the gegenbauer method and then we apply it to solve the spliced function problem

speeding up fuzzy c means using a hierarchical data organisation to control the precision of membership calculation we examine the run time behaviour of conventional fuzzy c means implementations investigating into fcm termination conditions and membership update equations we derive an approximative fcm that yields the same results as a conventional implementation within a given precision we incorporate additional information about the data set by reorganizing the set as a tree our modification leads to an fcm algorithm with a significantly different run time behaviour the gain of using the modified implementation increases with an increasing number of data objects and especially an increasing number of clusters but is also sensitive to the chosen fuzzifier

a spatially constrained fuzzy hyper prototype clustering algorithm we present in this paper a fuzzy clustering algorithm which can handle spatially constraint problems often encountered in pattern recognition the proposed method is based on the notions of hyperplanes the fuzzy c means and spatial constraints by adding a spatial regularizer into the fuzzy hyperplane based objective function the proposed method can take into account additionally important information of inherently spatial data experimental results have demonstrated that the proposed algorithm achieves superior results to some other popular fuzzy clustering models and has potential for cluster analysis in spatial domain

trouble with gegenbauer reconstruction for defeating gibbs phenomenon runge phenomenon in the diagonal limit of gegenbauer polynomial approximations to defeat gibbs phenomenon in fourier and chebyshev series gottlieb et al d gottlieb c w shu a solomonoff h vandeven on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function j comput appl math 43 1992 81 98 developed a gegenbauer reconstruction the partial sums of the fourier or other spectral series are reexpanded as a series of gegenbauer polynomials c n m x recovering spectral accuracy even in the presence of shock waves or other discontinuities to achieve a rate of convergence which is exponential in n however gegenbauer reconstruction requires increasing the order m of the polynomials linearly with the truncation n of the series m bn for some constant b 0 when the order m is fixed it is well known that the gegenbauer series converges as n everywhere on x 1 1 if f x the function being expanded is analytic on the interval but what happens in the diagonal limit in which m n tend to infinity simultaneously we show that singularities of f x off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x 1 1 gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation n this off axis singularity constraint is likely to impair the effectiveness of the reconstruction in some applications

single point iterative weighted fuzzy c means clustering algorithm for remote sensing image segmentation in this paper a remote sensing image segmentation procedure that utilizes a single point iterative weighted fuzzy c means clustering algorithm is proposed based upon the prior information this method can solve the fuzzy c means algorithm s problem that the clustering quality is greatly affected by the data distributing and the stochastic initializing the centrals of clustering after the probability statistics of original data the weights of data attribute are designed to adjust original samples to the uniform distribution and added in the process of cyclic iteration which could be suitable for the character of fuzzy c means algorithm so as to improve the precision furthermore appropriate initial clustering centers adjacent to the actual final clustering centers can be found by the proposed single point adjustment method which could promote the convergence speed of the overall iterative process and drastically reduce the calculation time otherwise the modified algorithm is updated from multidimensional data analysis to color images clustering moreover with the comparison experiments of the uci data sets public berkeley segmentation dataset and the actual remote sensing data the real validity of proposed algorithm is proved

correction to on cluster validity for the fuzzy c means model correspondence validation of partitions produced by the fuzzy c means fcm clustering algorithm was discussed by pal and bezdek 1 two tables in 1 contain erroneous values this errata reports the correct values and notes that the conclusions drawn in 1 based on these simulations remain unchanged

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws