on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

visual assessment of clustering tendency for rectangular dissimilarity matrices we have an m times n matrix d and assume that its entries correspond to pair wise dissimilarities between m row objects or and n column objects oc which taken together as a union comprise a set o of n m n objects this paper develops a new visual approach that applies to four different cluster assessment problems associated with o the problems are the assessment of cluster tendency pi amongst the row objects or p2 amongst the column objects oc p3 amongst the union of the row and column objects or u oc and p4 amongst the union of the row and column objects that contain at least one object of each type co clusters the basis of the method is to regard d as a subset of known values that is part of a larger unknown n times n dissimilarity matrix and then impute the missing values from d this results in estimates for three square matrices dr dc druc that can be visually assessed for clustering tendency using the previous vat or svat algorithms the output from assessment of druc ultimately leads to a rectangular covat image which exhibits clustering tendencies in d five examples are given to illustrate the new method two important points i because vat is scalable by svat to data sets of arbitrary size and because covat depends explicitly and only on vat this new approach is immediately scalable to say the scovat model which works for even very large unloadable data sets without alteration and ii vat svat and covat are autonomous parameter free models no hidden values are needed to make them work

reconstruction from radon projections and orthogonal expansion on a ball the relation between radon transform and orthogonal expansions of a function on the unit ball in is exploited a compact formula for the partial sums of the expansion is given in terms of the radon transform which leads to algorithms for image reconstruction from radon data the relation between orthogonal expansion and the singular value decomposition of the radon transform is also exploited

the interior radon transform the interior radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured this problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the x ray dose the corresponding operator mathcal r i is investigated as a mapping between weighted l 2 spaces the main result is a singular value decomposition svd for this operator for functions of unbounded support in mathbb r 2 the proof is based on the construction of intertwining differential operators the techniques used are unified in the sense that svds for other radon transforms with rotational symmetry can easily be derived in the same way consistency conditions are also obtained for mathcal r i for functions of bounded and unbounded support many of the results generalize to higher dimensions they are stated whenever they follow directly from the two dimensional case

visual stability analysis for model selection in graded possibilistic clustering abstract when compared to standard clustering fuzzy clustering provides more flexible and powerful data representation most fuzzy methods require setting some parameters as is the case for the graded possibilistic c means clustering method which has two parameters in addition to number of centroids however for this model selection task there is no well established criterion available building on our own previous work on fuzzy clustering similarity indexes we introduce a technique to evaluate the stability of clusterings by using the fuzzy jaccard index and use this procedure to select the most suitable values of parameters the experiments indicate that the procedure is effective for fuzzy and possibilistic clustering

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

local algorithms in exterior tomography exterior tomographic data are taken over lines outside a central region and such data occur in the industrial nondestructive evaluation of large objects such as rockets we explain using microlocal analysis which singularities are well reconstructed from exterior data and we explain how this phenomenon is reflected in the singular value decomposition for the exterior transform e t quinto singular value decompositions and inversion methods for the exterior radon transform and a spherical transform j math anal appl 95 1983 437 448 we extend lambda tomography to exterior data and to limited angle exterior data the algorithm is tested on industrial data from perceptics inc

optimization for limited angle tomography in medical image processing this paper aims to reduce the problems of incomplete data in computed tomography which happens frequently in medical image process and analysis e g when the high density region of objects can only be penetrated by x rays at a limited angular range as the projection data are available only in an angular range the incomplete data problem can be attributed to the limited angle problem which is an ill posed inverse problem image reconstruction based on total variation tv reduces the problem and gives better performance on edge preserving reconstruction however the artificial parameter can only be determined through considerable experimentation in this paper an effective tv objective function is proposed to reduce the inverse problem in the limited angle tomography this novel objective function provides a robust and effective reconstruction without any artificial parameter in the iterative processes using the tv as a multiplicative constraint the results demonstrate that this reconstruction strategy outperforms some previous ones

proximity based frameworks for generating embeddings from multi output data this paper is about supervised and semi supervised dimensionality reduction dr by generating spectral embeddings from multi output data based on the pairwise proximity information two flexible and generic frameworks are proposed to achieve supervised dr sdr for multilabel classification one is able to extend any existing single label sdr to multilabel via sample duplication referred to as mesd the other is a multilabel design framework that tackles the sdr problem by computing weight proximity matrices based on simultaneous feature and label information referred to as mope as a generalization of many current techniques a diverse set of different schemes for label based proximity calculation as well as a mechanism for combining label based and feature based weight information by considering information importance and prioritization are proposed for mope additionally we summarize many current spectral methods for unsupervised dr udr single multilabel sdr and semi supervised dr ssdr and express them under a common template representation as a general guide to researchers in the field we also propose a general framework for achieving ssdr by combining existing sdr and udr models and also a procedure of reducing the computational cost via learning with a target set of relation features the effectiveness of our proposed methodologies is demonstrated with experiments with document collections for multilabel text categorization from the natural language processing domain

multiple solutions for a neumann problem involving the p x laplacian abstract the aim of this paper is to obtain weak solutions for a neumann problem involving the p x laplacian the main tool used for obtaining our multiplicity result is a recent revised form of a three critical points theorem established by ricceri

multiplicity results for a differential inclusion problem with non standard growth abstract in this paper we examine the multiplicity of solutions of a differential inclusion problem involving p x laplacian of the type p p x u v x u p x 2 u j 1 x u x j 2 x u x in u n 0 on by using the nonsmooth version of ricceri variational principle we get three critical points of the corresponding energy motreanu panagiotopoulos type functional which are the solutions of p

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

multiplicity theorems for the dirichlet problem involving the p laplacian abstract in this paper we establish some results on the existence of at least three weak solutions for dirichlet problems involving the p laplacian by a variational approach

existence and multiplicity of solutions for a neumann problem involving the p x laplace operator abstract in this paper we study the following nonlinear neumann boundary value problem div u p x 2 u u p x 2 u f x u x t r u v 0 x t r where r n is a bounded domain with smooth boundary u v is the outer unit normal derivative on 0 is a real number p is a continuous function on with inf x p x 1 f r r is a continuous function using the three critical point theorem due to ricceri under the appropriate assumptions on f we establish the existence of at least three solutions of this problem some known results are generalized

mercer kernel based clustering in feature space the article presents a method for both the unsupervised partitioning of a sample of data and the estimation of the possible number of inherent clusters which generate the data this work exploits the notion that performing a nonlinear data transformation into some high dimensional feature space increases the probability of the linear separability of the patterns within the transformed space and therefore simplifies the associated data structure it is shown that the eigenvectors of a kernel matrix which defines the implicit mapping provides a means to estimate the number of clusters inherent within the data and a computationally simple iterative procedure is presented for the subsequent feature space partitioning of the data

a cluster validity index combining an overlap measure and a separation measure based on fuzzy aggregation operators since a clustering algorithm can produce as many partitions as desired one needs to assess their quality in order to select the partition that most represents the structure in the data if there is any this is the rationale for the cluster validity cv problem and indices this paper presents a cv index that helps to find the optimal number of clusters of data from partitions generated by a fuzzy clustering algorithm such as the fuzzy c means fcm or its derivatives given a fuzzy partition this new index uses a measure of multiple cluster overlap and a separation measure for each data point both based on an aggregation operation of membership degrees experimental results on artificial and benchmark datasets are given to demonstrate the performance of the proposed index as compared with traditional and recent indices

the solution of transient electromagnetic inverse source problems using time domain tlm method the electromagnetic inverse source problem isp for homogeneous background medium is investigated numerically using the transmission line matrix tlm method by transforming all electromagnetic sources and fields into their equivalent link impulses within a tlm computational domain a discrete linear inversion formulation is developed our formulation solves for the unknown source distribution inside a closed region using the knowledge of the near field data on its boundary unlike the conventional frequency domain treatments both our source solution and the field measurements are in the time domain the nonuniqueness of the isp is addressed by additionally imposing smoothness constraints that minimize the first order time and spatial derivatives of the source distribution our algorithm is illustrated through a number of numerical examples it is also shown that our algorithm is robust against the noisy field measurements

infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous neumann conditions abstract the aim of this paper is to establish the existence of an unbounded sequence of weak solutions for a class of differential equations with p x laplacian and subject to small perturbations of nonhomogeneous neumann conditions the approach is based on variational methods

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

on a perturbed p x laplacian problem in bounded and unbounded domains abstract in this paper we investigate the existence of multiple solutions for a quasilinear elliptic equation involving the p x laplacian in which the nonlinearity depends on three real parameters the solutions we look for are defined in a domain of r n which we allow to be possibly unbounded our approach relies upon variational arguments

existence and multiplicity of weak solutions for elliptic dirichlet problems with variable exponent abstract under appropriate growth conditions on the nonlinearity the existence of multiple solutions for nonlinear elliptic dirichlet problems with variable exponent is established the approach is based on variational methods some applications and examples illustrate the obtained results

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

relational generalizations of cluster validity indices numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another one such scheme clustering looks for self similar groups of objects to use clustering algorithms an investigator must often have a priori knowledge of the number of clusters i e c to search for in the data moreover it is often convenient to have ways to rank the returned results either for a single value of c a range of c s different clustering methods or any combination thereof however the task of assessing the quality of the results so that c may be determined objectively is currently ill defined for object object relationships to bridge this gap we generalize three well known validity indices the modified hubert s gamma xie beni and the generalized dunn s indices to relational data in doing so we develop a framework to convert many other validity indices to a relational form numerical examples on 12 datasets samples from four normal mixtures four real world object datasets and four real world pure relational datasets using the relational duals of the hard fuzzy and possibilistic c means cluster algorithms are offered to illustrate and evaluate the new indices

fuzzy pca guided robust k means clustering this paper proposes a new approach to robust clustering in which a robust k means partition is derived by using a noise rejection mechanism based on the noise clustering approach the responsibility weight of each sample for the k means process is estimated by considering the noise degree of the sample and cluster indicators are calculated in a fuzzy principal component analysis pca guided manner where fuzzy pca guided robust k means is performed by considering responsibility weights of samples then the proposed method achieves cluster core estimation in a deterministic way the validity of the derived cluster cores is visually assessed through distance sensitive ordering which considers responsibility weights of samples numerical experiments demonstrate that the proposed method is useful for capturing cluster cores by rejecting noise samples and we can easily assess cluster validity by using cluster crossing curves

the approximate inverse in action with an application to computerized tomography the approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind yet in some applications the computation of a crucial ingredient the reconstruction kernel is time consuming and instable it may even happen that the kernel does not exist for a particular semidiscrete system to cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator the results are applied to the reconstruction problem in 2d computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm

incomplete data problems in x ray computerized tomography i singular value decomposition of the limited angle transform etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons x assistee par ordinateur decomposition de la transformation de radon

relational duals of cluster validity functions for the c means family clustering aims to identify groups of similar objects to evaluate the results of cluster algorithms an investigator uses cluster validity indices while the theory of cluster validity is well established for vector object data little effort has been made to extend it to relationship based data as such this paper proposes a theory of reformulation for object data validity indices so that they can be used to rank the results produced by the relational means clustering algorithms more specifically we create a class of relational validity indices which is called dual relational indices that are guaranteed under certain but easily met constraints to produce the same results and hence the same cluster counts as their object data counterparts

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

the ill conditioned nature of the limited angle tomography problem the instability of inverting the limited angle radon transform is studied by constructing the singular value decomposition of this operator connections are established with previously known series expansions and with the discrete prolate spheroidal wave functions mollification methods are shown to delay but not prevent the onset of instability as the angular range of projections is decreased

biorthogonal decompositions of the radon transform the concept of singular value decompositions is a valuable tool in the examination of ill posed inverse problems af g such as the inversion of the radon transform a singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators a a a a in this paper we consider a new approach which generalizes this concept by application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator with respect to the radon transform for functions with support in the unit ball we apply the system of appell polynomials which is a natural generalization of the univariate system of gegenbauer ultraspherical polynomials to the multivariate case the corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions vice versa by application of our decompositions we are able to prove new properties of the appell polynomials

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

inversion of k plane transforms and applications in computer tomography the mathematics behind computerized tomography ct is based on the study of the parallel beam transform p and the divergent beam transform d both of these map a function f in mathbb r n into a function defined on the set of all lines in mathbb r n by integrating f along these lines the parallel and divergent k plane transforms are defined in a similar fashion by integration over k planes i e translates of k dimensional subspaces and are also denoted p and d respectively a related transform is the spherical k plane transform s which maps a function f on the sphere s n 1 into its integrals over k dimensional great circlesthis paper discusses the properties of the k plane transforms p d and s and their inverses emphasizing relationships and similarities between the operators and their relation to ct some new results are included most notable are more general conditions under which the inversion formulas hold

clustering ellipses for anomaly detection comparing clustering and merging ellipsoids are problems that arise in various applications e g anomaly detection in wireless sensor networks and motif based patterned fabrics we develop a theory underlying three measures of similarity that can be used to find groups of similar ellipsoids in p space clusters of ellipsoids are suggested by dark blocks along the diagonal of a reordered dissimilarity image rdi the rdi is built with the recursive ivat algorithm using any of the three dis similarity measures as input and performs two functions i it is used to visually assess and estimate the number of possible clusters in the data and ii it offers a means for comparing the three similarity measures finally we apply the single linkage and clodd clustering algorithms to three two dimensional data sets using each of the three dissimilarity matrices as input two data sets are synthetic and the third is a set of real wsn data that has one known second order node anomaly we conclude that focal distance is the best measure of elliptical similarity ivat images are a reliable basis for estimating cluster structures in sets of ellipsoids and single linkage can successfully extract the indicated clusters