the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

the petrov galerkin and iterated petrov galerkin methods for second kind integral equations we develop in this paper a theoretical framework for the analysis of convergence for the petrov galerkin method and superconvergence for the iterated petrov galerkin method for fredholm integral equations of the second kind as important approaches to the analysis we introduce notions of the generalized best approximation and the regular pair of trial space sequence and test space sequence in hilbert spaces we characterize the regular pair in terms of the angle of two space sequences or the generalized best approximation projections several specific constructions of the petrov galerkin elements for equations of both one dimension and two dimensions are presented and the convergence of the petrov galerkin method and the iterated petrov galerkin method using these elements is proved

asymptotics of orthogonal polynomial s entropy this is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree this study is motivated primarily by quantum mechanics where the wave functions and the densities of the states of solvable quantum mechanical systems are expressed by means of orthogonal polynomials moreover the uncertainty principle lying in the ground of quantum mechanics is best formulated by means of position and momentum entropies in this sense the behavior for large values of the degree is intimately connected with the information characteristics of high energy states but the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials we describe some results obtained in the last 15 years as well as sketch the ideas behind their proofs

multiwavelets for second kind integral equations we consider a galerkin method for an elliptic pseudodifferential operator of order zero on a two dimensional manifold we use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis using this basis we can compress the stiffness matrix from n2 to o n log n nonzero entries and still obtain up to log n terms the same convergence rates as for the exact galerkin method

a fast multiscale galerkin method for solving second order linear fredholm integro differential equation with dirichlet boundary conditions abstract in this paper a fast multiscale galerkin method is developed for solving second order linear fredholm integro differential equation with dirichlet boundary conditions the method is based on a matrix truncation strategy which leads to generating coefficient matrix rapidly we prove that the method is stable and has an optimal convergence order and nearly linear computational complexity up to a logarithmic factor numerical examples are presented to illustrate its computational efficiency approximation accuracy and theoretical results and to compare the computed results with those of the original multiscale galerkin method proposed recently by the same authors

fast accurate fuzzy clustering through data reduction clustering is a useful approach in image segmentation data mining and other pattern recognition problems for which unlabeled data exist fuzzy clustering using fuzzy c means or variants of it can provide a data partition that is both better and more meaningful than hard clustering approaches the clustering process can be quite slow when there are many objects or patterns to be clustered this paper discusses the algorithm brfcm which is able to reduce the number of distinct patterns which must be clustered without adversely affecting the partition quality the reduction is done by aggregating similar examples and then using a weighted exemplar in the clustering process the reduction in the amount of clustering data allows a partition of the data to be produced faster the algorithm is applied to the problem of segmenting 32 magnetic resonance images into different tissue types and the problem of segmenting 172 infrared images into trees grass and target average speed ups of as much as 59 290 times a traditional implementation of fuzzy c means were obtained using brfcm while producing partitions that are equivalent to those produced by fuzzy c means

robust fuzzy clustering of relational data popular relational data clustering algorithms relational dual of fuzzy c means rfcm non euclidean rfcm nerfcm both by hathaway et al and fanny by kaufman and rousseeuw are examined a new algorithm which is a generalization of fanny called the fuzzy relational data clustering frc algorithm is introduced having an identical objective functional as rfcm however the frc does not have the restriction of rfcm which is that the relational data is derived from euclidean distance as the measure of dissimilarity between the objects and it also does not have limitations of fanny including the use of a fixed membership exponent or a fuzzifier exponent m the frc algorithm is further improved by incorporating the concept of dave s object data noise clustering nc algorithm done by proposing a concept of noise dissimilarity next based on the constrained minimization which includes an inequality constraint for the memberships and corresponding kuhn tucker conditions a noise resistant frc algorithm is derived which works well for all types of non euclidean dissimilarity data thus it is shown that the extra computations for data expansion spl beta spread transformation required by the nerfcm algorithm are not necessary this new algorithm is called robust non euclidean fuzzy relational data clustering robust ne frc and its robustness is demonstrated through several numerical examples advantages of this new algorithm are faster convergence robustness against outliers and ability to handle all kinds of relational data including non euclidean the paper also presents a new and better interpretation of the noise class

fast collocation methods for second kind integral equations in this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels for this purpose we construct multiscale interpolating functions and collocation functionals having vanishing moments moreover we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms an optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated the stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed

discrete entropy of generalized jacobi polynomials abstract given a sequence of orthonormal polynomials on r p n n 0 with p n of degree n we define the discrete probability distribution n x n 1 x n n x with n j x j 0 n 1 p j 2 x 1 p j 1 2 x j 1 n in this paper we study the asymptotic behavior as n of the shannon entropy s n x j 1 n n j x log n j x x 1 1 when the orthogonality weight is 1 x 1 x h x 1 and where h is real analytic and positive on 1 1 we show that the limit lim n s n x log n exists for all x 1 1 but its value depends on the rationality of arccos x for the particular case of the chebyshev polynomials of the first and second kinds we compare our asymptotic result with the explicit formulas for s n j n where j n are the zeros of p n obtained previously in 2

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

harmonic wavelets towards the solution of nonlinear pde in this paper harmonic wavelets which are analytically defined and band limited are studied together with their differentiable properties harmonic wavelets were recently applied to the solution of evolution problems and more generally to describe evolution operators in order to consider the evolution of a solitary profile and to focus on the localization property of wavelets it seems to be more expedient to make use of functions with limited compact support either in space or in frequency the connection coefficients of harmonic wavelets are explicitly computed in the following at any order and characterized by some recursive formulas in particular they are functionally and finitely defined by a simple formula for any order of the basis derivatives

robust fuzzy filter design for a class of nonlinear stochastic systems this paper describes the robust hinfin fuzzy filtering design for a class of nonlinear stochastic systems the system dynamic is modelled by itocirc type stochastic differential equations for general nonlinear stochastic systems the hinfin filter can be obtained by solving a second order nonlinear hamilton jacobi inequality in general it is difficult to solve the second order nonlinear hamilton jacobi inequality in this paper using fuzzy approach takagi sugeno t s fuzzy model the hinfin fuzzy filtering design for the nonlinear stochastic systems can be given via solving linear matrix inequalities lmis instead of a second order hamilton jacobi inequality when the worst case fuzzy disturbance is considered a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error the near minimum variance fuzzy filtering problem under the worst case fuzzy disturbance is also characterized in terms of linear matrix inequality problem lmip which can be efficiently solved by the convex optimization techniques simulation examples are provided to illustrate the design procedure and expected performance

self adaptive neuro fuzzy inference systems for classification applications this paper presents a self adaptive neuro fuzzy inference system sanfis that is capable of self adapting and self organizing its internal structure to acquire a parsimonious rule base for interpreting the embedded knowledge of a system from the given training data set a connectionist topology of fuzzy basis functions with their universal approximation capability is served as a fundamental sanfis architecture that provides an elasticity to be extended to all existing fuzzy models whose consequent could be fuzzy term sets fuzzy singletons or functions of linear combination of input variables without a priori knowledge of the distribution of the training data set a novel mapping constrained agglomerative clustering algorithm is devised to reveal the true cluster configuration in a single pass for an initial sanfis construction estimating the location and variance of each cluster subsequently a fast recursive linear nonlinear least squares algorithm is performed to further accelerate the learning convergence and improve the system performance good generalization capability fast learning convergence and compact comprehensible knowledge representation summarize the strength of sanfis computer simulations for the iris wisconsin breast cancer and wine classifications show that sanfis achieves significant improvements in terms of learning convergence higher accuracy in recognition and a parsimonious architecture

fuzzy perceptron neural networks for classifiers with numerical data and linguistic rules as inputs this paper presents a novel learning algorithm of fuzzy perceptron neural networks fpnns for classifiers that utilize expert knowledge represented by fuzzy if then rules as well as numerical data as inputs the conventional linear perceptron network is extended to a second order one which is much more flexible for defining a discriminant function in order to handle fuzzy numbers in neural networks level sets of fuzzy input vectors are incorporated into perceptron neural learning at different levels of the input fuzzy numbers updating the weight vector depends on the minimum of the output of the fuzzy perceptron neural network and the corresponding nonfuzzy target output that indicates the correct class of the fuzzy input vector this minimum is computed efficiently by employing the modified vertex method moreover the fuzzy pocket algorithm is introduced into our fuzzy perceptron learning scheme to solve the nonseparable problems simulation results demonstrate the effectiveness of the proposed fpnn model

fuzzy analysis of statistical evidence bayesian classifiers are effective methods for pattern classification although their assumptions on the belief structure among attributes are not always justified in this paper we introduce a new classification method based on the possibility measure which does not require a precise belief model and in a sense it includes the bayesian classifiers as special cases this new classification method uses the fuzzy operators to aggregate attributes information evidence and it is referred to as fuzzy analysis of statistical evidence fase fase has several nice properties it is noise tolerant it can handle missing values with ease and it can extract statistical patterns from the data and represent them by knowledge of beliefs which in turn are propositions for an expert system thus from pattern classification to expert systems fase provides a linkage from inductive reasoning to deductive reasoning

wavelet galerkin method for eigenvalue problem of a compact integral operator abstract we consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the galerkin method using wavelet bases by truncating the galerkin operator we obtain a sparse representation of a matrix eigenvalue problem we prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders o n 2 nr and o nr respectively where n denotes the norm of the partition and r denotes the order of the wavelet basis functions by iterating the eigenvectors we show that the error bounds for the eigenvectors are of the order o n 2 nr we illustrate our results with numerical results

locally adaptive metric nearest neighbor classification nearest neighbor classification assumes locally constant class conditional probabilities this assumption becomes invalid in high dimensions with finite samples due to the curse of dimensionality severe bias can be introduced under these conditions when using the nearest neighbor rule we propose a locally adaptive nearest neighbor classification method to try to minimize bias we use a chi squared distance analysis to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations neighborhoods are elongated along less relevant feature dimensions and constricted along most influential ones as a result the class conditional probabilities are smoother in the modified neighborhoods whereby better classification performance can be achieved the efficacy of our method is validated and compared against other techniques using both simulated and real world data

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

galerkin method with new quadratic spline wavelets for integral and integro differential equations abstract the paper is concerned with the wavelet galerkin method for the numerical solution of fredholm linear integral equations and second order integro differential equations we propose a construction of a quadratic spline wavelet basis on the unit interval such that the wavelets have three vanishing moments and the shortest support among such wavelets we prove that this basis is a riesz basis in the space l 2 0 1 we adapt the basis to homogeneous dirichlet boundary conditions and using a tensor product we construct a wavelet basis on the hyperrectangle we use the wavelet galerkin method with the constructed bases for solving integral and integro differential equations and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices we present numerical examples and compare the results with the galerkin method using other quadratic spline wavelet bases and other methods

multiresolution analysis of lp space on compact set abstract in this paper we define multiresolution analysis mra in function space l p for p 1 where is a compact subset in r n and is a probability measure defined on we establish a general framework for construction of mra in l p and its conjugate space l q with p 1 q 1 1 p q 1

multiresolution of lp spaces multiresolution analysis plays a major role in wavelet theory in this paper multiresolution of lp spaces is studied let s be a shift invariant subspace of lp rs 1 p generated by a finite number of functions with compact support and let sk be the 2k dilate of s for each integer k z it is shown that the intersection of sk k z is always trivial it is more difficult to deal with the problem whether the union of sk k z is dense in lp rs the case p 1 or 2 can be solved by wiener s density theorem under the assumption that s is refinable it is proved in this paper that the union of sk k z is dense in lp rs provided s 1 2 or 2 p the same is true for p if l rs is replaced by c0 rs counterexamples are given to demonstrate that for s 3 and 1 p 2 the aforementioned results on density are no longer true in general

leave one out procedures for nonparametric error estimates the use of nonparametric error estimates may lead to biased results if the kernel covariances are estimated from the same data as are used to form the error estimate if additional design samples are available one may eliminate this bias by estimating the class covariances using an independent set of data if however additional samples are not available one may resort to leave one out type estimates of the kernel for parzen estimates or metric for nearest neighbor estimates for every sample being tested the authors present an efficient algorithm for computation of these leave one out type estimates that requires little additional computational burden over procedures currently in use the presentation is applicable to both parzen and k nearest neighbor k nn type estimates experimental results demonstrating the efficiency of the algorithm are provided

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

fuzzy interpolation and extrapolation a practical approach fuzzy interpolation does not only help to reduce the complexity of fuzzy models but also makes inference in sparse rule based systems possible it has been successfully applied to systems control but limited work exists for its applications to tasks like prediction and classification almost all fuzzy interpolation techniques in the literature make strong assumptions that there are two closest adjacent rules available to the observation and that such rules must flank the observation for each attribute also some interpolation approaches cannot handle fuzzy sets whose membership functions involve vertical slopes to avoid such limitations and develop a more practical approach this paper extends the work of huang and shen the result enables both interpolation and extrapolation which involve multiple fuzzy rules with each rule consisting of multiple antecedents two realistic applications namely truck backer upper control and computer activity prediction are provided in this paper to demonstrate the utility of the extended approach experiment based comparisons to the most commonly used mamdani fuzzy reasoning mechanism and to other existing fuzzy interpolation techniques are given to show the significance and potential of this research

pointwise convergence of wavelet expansions in this note we announce that under general hypotheses wavelet type ex pansions of functions in lp p in one or more dimensions converge pointwise almost everywhere and identify the lebesgue set of a function as a set of full measure on which they converge it is shown that unlike the fourier sum mation kernel wavelet summation kernels pj are bounded by radial decreasing l convolution kernels as a corollary it follows that best l spline approxi mations on uniform meshes converge pointwise almost everywhere moreover summation of wavelet expansions is partially insensitive to order of summation we also give necessary and su cient conditions for given rates of conver gence of wavelet expansions in the sup norm such expansions have order of convergence s if and only if the basic wavelet is in the homogeneous sobolev space h s d h we also present equivalent necessary and su cient conditions on the scaling function the above results hold in one and in multiple dimen sions

an integrated approach to fuzzy learning vector quantization and fuzzy c means clustering derives an interpretation for a family of competitive learning algorithms and investigates their relationship to fuzzy c means and fuzzy learning vector quantization these algorithms map a set of feature vectors into a set of prototypes associated with a competitive network that performs unsupervised learning derivation of the new algorithms is accomplished by minimizing an average generalized distance between the feature vectors and prototypes using gradient descent a close relationship between the resulting algorithms and fuzzy c means is revealed by investigating the functionals involved it is also shown that the fuzzy c means and fuzzy learning vector quantization algorithms are related to the proposed algorithms if the learning rate at each iteration is selected to satisfy a certain condition

fuzzy clustering with volume prototypes and adaptive cluster merging two extensions to objective function based fuzzy clustering are proposed first the point prototypes are extended to hypervolumes whose size can be fixed or can be determined automatically from the data being clustered it is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function second a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization starting with an overestimation of the number of clusters in the data similar clusters are merged in order to obtain a suitable partitioning an adaptive threshold for merging is proposed the extensions proposed are applied to gustafson kessel and fuzzy c means algorithms and the resulting extended algorithm is given the properties of the new algorithm are illustrated by various examples

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

fuzzy versus nonfuzzy in combining classifiers designed by boosting boosting is recognized as one of the most successful techniques for generating classifier ensembles typically the classifier outputs are combined by the weighted majority vote the purpose of this study is to demonstrate the advantages of some fuzzy combination methods for ensembles of classifiers designed by boosting we ran two fold cross validation experiments on six benchmark data sets to compare the fuzzy and nonfuzzy combination methods on the fuzzy side we used the fuzzy integral and the decision templates with different similarity measures on the nonfuzzy side we tried the weighted majority vote as well as simple combiners such as the majority vote minimum maximum average product and the naive bayes combination in our experiments the fuzzy combination methods performed consistently better than the nonfuzzy methods the weighted majority vote showed a stable performance though slightly inferior to the performance of the fuzzy combiners

on the solution of differential equations with fuzzy spline wavelets fuzzy systems built with spline wavelets can approximate any function in a general hilbert space since wavelets are used extensively for the efficient solution of various types of differential equations it is demonstrated that fuzzy spline wavelets can be used for the solution of the same type of problems the advantage in using fuzzy spline wavelets for the solution of such problems is that the solution would enjoy the excellent numerical and computational characteristics of the fast wavelet transform while retaining the explanatory power of fuzzy system the method is demonstrated with the feedforward control of a flexible robotic arm

comments on robust stabilization of a class of uncertain nonlinear systems via fuzzy control quadratic stabilizability h sup spl infin control theory and linear matrix inequalities this paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems first a stability condition for takagi and sugeno s fuzzy model is given in terms of lyapunov stability theory next new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization h sup spl infin control theory and linear matrix inequalities the derived stability conditions are used to analyze the stability of takagi and sugeno s fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems the design method employs the so called parallel distributed compensation important issues for the stability analysis and design are remarked finally three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented

bayes error estimation using parzen and k nn procedures the use of k nearest neighbor k nn and parzen density estimates to obtain estimates of the bayes error is investigated under limited design set conditions by drawing analogies between the k nn and parzen procedures new procedures are suggested and experimental results are given which indicate that these procedures yield a significant improvement over the conventional k nn and parzen procedures we show that by varying the decision threshold many of the biases associated with the k nn or parzen density estimates may be compensated and successful error estimation may be performed in spite of these biases experimental results are given which demonstrate the effect of kernel size and shape parzen the size of k k nn and the number of samples in the design set

a characteristic point based fuzzy inference classifier by a closeness matrix in this paper a characteristic point based fuzzy inference classifier cpfic is proposed to perform two class classification through fuzzy interpolation a subset of classified samples can be taken as representatives of all samples they are called characteristic points cps a closeness matrix representing the closeness of two samples in a same class is proposed in selecting cps by solving a number of constrained minimizations the cpfic is systematically built experiments were conducted on four classification problems with known bayes errors two benchmark classification problems and a real world application used in our research the cpfic performs well in accuracy evaluations in all the seven experiments the summarizing abilities from the cps into the linguistic descriptions of the fuzzy rule bases were also demonstrated in these examples

on the comparison of trilinear cubic spline and fuzzy interpolation methods in the high accuracy measurements this paper provides a comparison between a novel technique used for the pose error measurements and compensations of robots based on a fuzzy error interpolation method and some other popular interpolation methods a traditional robot calibration implements either model or modeless methods the measurement and compensation of pose errors in a modeless method moves the robot s end effector to the target poses in the robot workspace and measures the target position and orientation errors using some interpolation techniques in terms of the premeasured neighboring pose errors around the target pose for the measurement purpose a stereo camera or other measurement devices such as a coordinate measurement machine cmm or a laser tracking system lts can be used to measure the pose errors of the robot s end effector at predened grid points on a cubic lattice by the use of the proposed fuzzy error interpolation technique the accuracy of the pose error compensation can be improved in comparison with other interpolation methods which is conrmed by the simulation results given in this paper a comparison study among most popular interpolation methods used in modeless robot calibrations such as trilinear cubic spline and the fuzzy error interpolation technique is also made and discussed via simulations the simulation results show that more accurate measurement and compensation results can be achieved using the fuzzy error interpolation technique compared with its trilinear and cubic spline counterparts

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

reducing the time complexity of the fuzzy c means algorithm in this paper we present an efficient implementation of the fuzzy c means clustering algorithm the original algorithm alternates between estimating centers of the clusters and the fuzzy membership of the data points the size of the membership matrix is on the order of the original data set a prohibitive size if this technique is to be applied to very large data sets with many clusters our implementation eliminates the storage of this data structure by combining the two updates into a single update of the cluster centers this change significantly affects the asymptotic runtime as the new algorithm is linear with respect to the number of clusters while the original is quadratic elimination of the membership matrix also reduces the overhead associated with repeatedly accessing a large data structure empirical evidence is presented to quantify the savings achieved by this new method

wavelet galerkin methods for second kind integral equations we use vector valued multiwavelets on compact sets to develop a galerkin method for systems of integral equations of the second kind we propose a compression strategy for the coefficient matrix of the linear system obtained from this method and show that the compressed scheme preserves almost optimal convergence rate of the original scheme and yields a sparse matrix with a bounded condition number

tsallis and renyi divergences of generalized jacobi polynomials abstract we consider some discrete probability distribution n x n 1 x n 2 x n n x we show that under suitable conditions the sequence of tsallis divergence d t n x n and the sequence of renyi divergence d r n x n are convergent for any x 1 1

iterated fast multiscale galerkin methods for eigen problems of compact integral operators an iterated fast multiscale galerkin method is developed for solving the eigen problem of integral operators with weakly singular kernels we propose a theoretical framework for analysis of the convergence of these methods and show the fast multiscale galerkin method obtain the optimal convergence order for eigenvectors and superconvergence order for eigenvalues while the computational complexity for coefficient matrix is almost optimal the iterated fast multiscale galerkin method can improve the convergence for eigenvectors and exhibit superconvergence through the iteration technique numerical examples are presented to illustrate the theoretical estimates for the error of these methods

characterization of distributions having a value at a point in the sense of robinson abstract we characterize schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by a robinson they appear to be continuous functions in a neighborhood of the point this characterization improves a result by p loeb which assumes the everywhere existence of point values

segmenting images corrupted by correlated noise image segmentation is fundamental to many image analysis problems it aims to partition a digital image into a set of nonoverlapping homogeneous regions the main contribution of this paper is the development of a new segmentation procedure which is designed to segment images corrupted by correlated noise this new segmentation procedure is based on rissanen s minimum description length mdl principle and consists of two components 1 an mdl based criterion in which the best segmentation is defined as its minimizer and 2 a merging algorithm which attempts to locate this minimizer the performance of this procedure is illustrated via a simulation study with promising results

robustness design of nonlinear dynamic systems via fuzzy linear control this study introduces a fuzzy linear control design method for nonlinear systems with optimal h sup spl infin robustness performance first the takagi and sugeno fuzzy linear model 1985 is employed to approximate a nonlinear system next based on the fuzzy linear model a fuzzy controller is developed to stabilize the nonlinear system and at the same time the effect of external disturbance on control performance is attenuated to a minimum level thus based on the fuzzy linear model h sup spl infin performance design can be achieved in nonlinear control systems in the proposed fuzzy linear control method the fuzzy linear model provides rough control to approximate the nonlinear control system while the h sup spl infin scheme provides precise control to achieve the optimal robustness performance linear matrix inequality lmi techniques are employed to solve this robust fuzzy control problem in the case that state variables are unavailable a fuzzy observer based h sup spl infin control is also proposed to achieve a robust optimization design for nonlinear systems a simulation example is given to illustrate the performance of the proposed design method

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

numerical approximation of 2d fredholm integral eigenvalue problems by orthogonal wavelets we investigate the numerical approximation of two dimensional second kind fredholm integral eigenvalue problems by the galerkin method with the cohen daubechies vial cdv wavelet family this choice provides us orthogonal bases for bounded domains avoiding the need of periodization or domain truncation the cdv family is indexed by the number of vanishing moments which drives the regularity of the basis we generate the galerkin basis from tensorized scaling functions and employ weighted gaussian quadratures derived from refinement equations numerical experiments address the relative computational cost of this approach with respect to the haar basis and the relationship between convergence rate and number of vanishing moments

a construction of interpolating wavelets on invariant sets we introduce the concept of a refinable set relative to a family of contractive mappings on a metric space and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure the notion of a refinable set parallels that of a refinable function which is the basis of wavelet construction the interpolation points we recursively generate from a refinable set by a set theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction we then use this recursive structure for the points to construct multiscale interpolants several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included

fuzzy solutions to partial differential equations adaptive approach a new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations pdes is presented the design objective is to find a fuzzy solution to satisfy precisely the pdes with boundary conditions according to the adaptive scheme of fuzzy logic systems a fuzzy solution with adjustable parameters for the pde is first described then a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function in addition an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived furthermore the convergence of error equations in mesh points is also discussed from the energy perspective in this paper we show that the proposed method can solve a variety of pdes encountered in engineering comparisons are also made with solutions obtained by the finite element method

wavelet expansions and asymptotic behavior of distributions abstract we develop a distribution wavelet expansion theory for the space of highly time frequency localized test functions over the real line s 0 r s r and its dual space s 0 r namely the quotient of the space of tempered distributions modulo polynomials we prove that the wavelet expansions of tempered distributions converge in s 0 r a characterization of boundedness and convergence in s 0 r is obtained in terms of wavelet coefficients our results are then applied to study local and non local asymptotic properties of schwartz distributions via wavelet expansions we provide abelian and tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients

novel determination of differential equation solutions universal approximation method in a conventional approach to numerical computation finite difference and finite element methods are usually implemented to determine the solution of a set of differential equations des this paper presents a novel approach to solve des by applying the universal approximation method through an artificial intelligence utility in a simple way in this proposed method neural network model nnm and fuzzy linguistic model flm are applied as universal approximators for any nonlinear continuous functions with this outstanding capability the solutions of des can be approximated by the appropriate nnm or flm within an arbitrary accuracy the adjustable parameters of such nnm and flm are determined by implementing the optimization algorithm this systematic search yields sub optimal adjustable parameters of nnm and flm with the satisfactory conditions and with the minimum residual errors of the governing equations subject to the constraints of boundary conditions of des the simulation results are investigated for the viability of efficiently determining the solutions of the ordinary and partial nonlinear des

multiscale approximation of the solution of weakly singular second kind fredholm integral equation in legendre multiwavelet basis numerical solution of fredholm integral equation of second kind with weakly singular kernel is obtained in this paper by employing legendre multi wavelet basis the low and high pass filters for two scale relations involving legendre multiwavelets having four or five vanishing moments of their wavelets have been derived and are used in the evaluation of integrals for the multiscale representation of the integral operator explicit expressions for the elements of the matrix associated with the multiscale representation are given an estimate for the holder exponent of the solution of the integral equation at any point in its domain is obtained a number of examples is provided to illustrate the efficiency of the method developed here

multiresolution analysis on local fields abstract in this paper we first establish the fundamental framework of multiresolution on local field in wavelet analysis accurately orthonormal systems consisting of integer translations u k k p of a single function l2 k are characterized theorems 1 and 2 the concept of multiresolution analysis on local field definition 3 and construction of corresponding wavelet vectors theorem 3 are given the definition of integral periodicity of a function on local field definition 4 is also given an example is finally presented

wavelet approximations for first kind boundary integral equations on polygons an elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation its galerkin discretization with n degrees of freedom on the boundary with spline wavelets as basis functions is analyzed a truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from o n 2 to o n log n entries the condition numbers are bounded independently of the meshwidth it is proved that the compressed scheme thus obtained yields in o n log n 2 operations approximate solutions with the same asymptotic convergence rates as the full galerkin scheme in the boundary energy norm as well as in interior points numerical examples show the asymptotic error analysis to be valid already for moderate values of n

a fast multiscale solver for modified hammerstein equations abstract this paper presents a fast solver called the multilevel augmentation method for modified nonlinear hammerstein equations when we utilize the method to solve a large scale problem most of components of the solution can be computed directly and lower frequency components can be obtained by solving a fixed low order algebraic nonlinear system the advantage of using the algorithm to modified equations is that it leads to reduce the cost of numerical integrations greatly the optimal error estimate of the method is established and the nearly linear computational complexity is proved finally numerical examples are presented to confirm the theoretical results and illustrate the efficiency of the method

adaptive fuzzy segmentation of magnetic resonance images an algorithm is presented for the fuzzy segmentation of two dimensional 2 d and three dimensional 3 d multispectral magnetic resonance mr images that have been corrupted by intensity inhomogeneities also known as shading artifacts the algorithm is an extension of the 2 d adaptive fuzzy c means algorithm 2 d afcm presented in previous work by the authors this algorithm models the intensity inhomogeneities as a gain field that causes image intensities to smoothly and slowly vary through the image space it iteratively adapts to the intensity inhomogeneities and is completely automated in this paper the authors fully generalize 2 d afcm to three dimensional 3 d multispectral images because of the potential size of 3 d image data they also describe a new faster multigrid based algorithm for its implementation they show using simulated mr data that 3 d afcm yields lower error rates than both the standard fuzzy c means fcm algorithm and two other competing methods when segmenting corrupted images its efficacy is further demonstrated using real 3 d scalar and multispectral mr brain images

fuzzy spline interpolation with optimal property in parametric form in this paper we propose a new fuzzy interpolation procedure that uses cubic splines with an optimal property obtained in parametric form this fuzzy interpolation operator is also a linear operator of uniform approximation that preserves the monotonicity moreover by integrating this fuzzy cubic spline we obtain the trapezoidal quadrature rule for fuzzy henstock integrable functions

fuzzy transforms theory and applications the technique of direct and inverse fuzzy f transforms of three different types is introduced and approximating properties of the inverse f transforms are described all three types of the direct f transform are transformations from a function space to a finite dimensional vector space the first ordinary f transform is constructed on the basis of the ordinary algebra of reals while the other two types of the f transform are constructed on the basis of residuated lattice the core idea of the technique of f transforms is a fuzzy partition of a universe into fuzzy subsets factors clusters granules etc we claim that for a sufficient representation of a function defined on this universe we may consider its average values over fuzzy subsets from the partition thus a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values a number of theorems establishing best approximation properties of the inverse f transforms are proved in fact three types of the inverse f transform are the best approximations in average from below and from above respectively as one of many possible applications we present a method of image compression and reconstruction on the basis of the f transform

wavelet collocation method and multilevel augmentation method for hammerstein equations a wavelet collocation method for nonlinear hammerstein equations is formulated a sparsity in the jacobian matrix is obtained which gives rise to a fast algorithm for nonlinear solvers such as the newton s method and the quasi newton method a fast multilevel augmentation method is developed on a transformed nonlinear equation which gives an additional saving in computational time

analysis and efficient implementation of a linguistic fuzzy c means the paper is concerned with a linguistic fuzzy c means fcm algorithm with vectors of fuzzy numbers as inputs this algorithm is based on the extension principle and the decomposition theorem it turns out that using the extension principle to extend the capability of the standard membership update equation to deal with a linguistic vector has a huge computational complexity in order to cope with this problem an efficient method based on fuzzy arithmetic and optimization has been developed and analyzed we also carefully examine and prove that the algorithm behaves in a way similar to the fcm in the degenerate linguistic case synthetic data sets and the iris data set have been used to illustrate the behavior of this linguistic version of the fcm

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