sendograph metric and relatively compact sets of fuzzy sets let k y be the family of all fuzzy subsets of an arbitrary metric space y which are upper semicontinuous normal with nonempty compact support the sendograph distance h between fuzzy sets is the hausdorff distance of their reduced hypographs called sendographs fan fuzzy sets and systems 143 2004 471 477 characterized compact sets of fuzzy numbers convex fuzzy subsets of the real line r with respect to sendograph metric and declared open the problem of finding a criterion for convex fuzzy subsets of r n in this paper relatively compact subsets of k y h are characterized see theorem 8 and consequently it is shown that fan s theorem holds for arbitrary metric spaces without any convexity assumption in the proofs a variational convergence called c convergence introduced by de giorgi franzoni atti accad naz lincei rend cl sc mat fis natur 58 8 1975 842 850 rend sem mat brescia 3 1979 63 101 and author s previous results are fundamental no isometrical embedding theorem is used co et al ann univ ferrara sez vii sc mat 44 1998 27 39 greco and moschen nonlinear anal tma to appear no isometrical embedding theorem is used

a metric space of subcopulas an approach via hausdorff distance abstract in this work we define a distance function on the set of bivariate subcopulas to generate a compact metric space moreover the copula space equipped with the uniform distance is essentially a metric subspace of this subcopula space we also characterize the convergence in this space and provide the interrelationship with the convergence of the distribution functions

semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

a cg fft approach to the solution of a stress velocity formulation of three dimensional elastic scattering problems in this paper we introduce a conjugate gradient fast fourier transform cg fft scheme for the numerical solution of the integral equation formulating three dimensional elastic scattering problems the formulation is in terms of the stress tensor and particle velocities as the unknown field variables in contrast with the formulation based on particle displacements this approach leads to integral representations that do not involve derivatives of the unknown fields thus resulting in simplified and more stable numerics the numerical procedure is based on suitable quadrature formulas that provide second order accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via ffts the scheme is further improved through the introduction of approximation based pre conditioners that are shown to accelerate the convergence of the cg iterations numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology

an efficient hybrid mlfma fft solver for the volume integral equation in case of sparse 3d inhomogeneous dielectric scatterers electromagnetic scattering problems involving inhomogeneous objects can be numerically solved by applying a method of moments discretization to the volume integral equation for electrically large problems the iterative solution of the resulting linear system is expensive both computationally and in memory use in this paper a hybrid mlfma fft method is presented which combines the fast fourier transform fft method and the high frequency multilevel fast multipole algorithm mlfma in order to reduce the cost of the matrix vector multiplications needed in the iterative solver the method represents the scatterers within a set of possibly disjoint identical cubic subdomains which are meshed using a uniform cubic grid this specific mesh allows for the application of ffts to calculate the near interactions in the mlfma and reduces the memory cost considerably since the aggregation and disaggregation matrices of the mlfma can be reused additional improvements to the general mlfma framework such as an extention of the fft interpolation scheme of sarvas et al from the scalar to the vectorial case in combination with a more economical representation of the radiation patterns on the lowest level in vector spherical harmonics are proposed and the choice of the subdomain size is discussed the hybrid method performs better in terms of speed and memory use on large sparse configurations than both the fft method and the hf mlfma separately and it has lower memory requirements on general large problems this is illustrated on a number of representative numerical test cases

characterizations of compact sets in fuzzy sets spaces with l_p metric abstract compactness criteria in fuzzy set spaces endowed with the l p metric have been studied for several decades total boundedness is a key feature of compactness in metric spaces however comparing existing compactness criteria in fuzzy set spaces endowed with the l p metric with the arzela ascoli theorem the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets currently compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star shapedness general fuzzy sets have become more important in both theory and applications therefore this paper presents characterizations of totally bounded sets relatively compact sets and compact sets in general fuzzy set spaces equipped with the l p metric but which do not have any assumptions of convexity or star shapedness subsets of these general sets include common fuzzy sets such as fuzzy numbers fuzzy star shaped numbers with respect to the origin fuzzy star shaped numbers and general fuzzy star shaped numbers existing compactness criteria are stated for fuzzy numbers space the space of fuzzy star shaped numbers with respect to the origin and the space of fuzzy star shaped numbers endowed with the l p metric respectively constructing completions of fuzzy set spaces with respect to the l p metric is a problem closely dependent on characterizing totally bounded sets based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star shapedness of fuzzy sets we show that the completions of fuzzy set spaces studied here can be obtained using the l p extension we also clarify relationships among the ten fuzzy set spaces studied here the five pairs of original spaces and their corresponding completions we show that the subspaces have parallel characterizations of totally bounded sets relatively compact sets and compact sets finally we discuss properties of the l p metric on fuzzy set space as an application of our results and review compactness criteria proposed in previous work

a modified gradient method for two dimensional problems in tomography abstract a method for reconstructing the complex index of refraction of a bounded two dimensional inhomogeneous object of known geometric configuration from measured scattered field data is presented this work is an extension of recent results on the direct scattering problem wherein the governing domain integral equation was solved iteratively by a successive over relaxation technique the relaxation parameter was chosen to minimize the residual error at each step convergence of this process was established for indices of refraction much larger than required for convergence of the born approximation for the inverse problem the same technique is applied except in this case both the index of refraction and the field are unknown iterative solutions for both unknowns are postulated with two relaxation parameters at each step they are determined by simultaneously minimizing the residual errors in satisfying the domain integral equation and matching the measured data this procedure retains the nonlinear relation between the two unknowns numerical results are presented for a number of representative two dimensional objects the algorithm is shown to be effective in cases where the iterative solution of the direct problem is rapidly convergent

a fuzzy logic system for channel equalization we present a new method for channel equalization using fuzzy logic the membership functions are derived from the training data set and a method to estimate the delay of the communication channel is proposed the performance of the fuzzy equalizer is compared with that of a transversal filter equalizer it is shown using simulations that the transversal filter requires a much larger training set to achieve the same error rate it is also shown using simulations that the fuzzy equalizer performs better in the presence of channel nonlinearities

fuzzy adaptive filters with application to nonlinear channel equalization two fuzzy adaptive filters are developed one uses a recursive least squares rls adaptation algorithm and the other uses a least mean square lms adaptation algorithm the rls fuzzy adaptive filter is constructed through the following four steps 1 define fuzzy sets in the filter input space rn whose membership functions cover u 2 construct a set of fuzzy if then rules which either come from human experts or are determined during the adaptation procedure by matching input output data pairs 3 construct a filter based on the set of rules and 4 update the free parameters of the filter using the rls algorithm the design procedure for the lms fuzzy adaptive filter is similar the most important advantage of the fuzzy adaptive filters is that linguistic information in the form of fuzzy if then rules and numerical information in the form of input output pairs can be combined in the filters in a uniform fashion the filters are applied to nonlinear communication channel equalization problems

a fast volume integral equation method for elastic wave propagation in a half space abstract a fast method for solving the volume integral equation is developed for scattering analysis of elastic wave propagation in a half space the proposed method applies the fast generalized fourier transform and inverse transform formulated in the present study to the krylov subspace method the amount of calculations required for the proposed method is o nlogn where n is the number of grid points used to model the elastic half space furthermore the mpi parallel algorithm for the generalized fourier transform is presented for further reduction of the cpu time numerical calculations are performed in order to examine the effects of the number of sampling grid points as well as their intervals on the solutions of the volume integral equation and the cpu time required for the analysis in addition comparisons of the proposed method with the previous method based on the trapezoidal approach touhei 2009 are also performed in order to discuss the properties of the solution of the present method

on the stress wave imaging of cavities in a semi infinite solid the problem of mapping underground cavities from surface seismic measurements is investigated within the framework of a regularized boundary integral equation bie method with the ground modeled as a uniform elastic half space the inverse analysis of elastic waves scattered by a three dimensional void is formulated as a task of minimizing the misfit between experimental observations and theoretical predictions for an assumed void geometry for an accurate treatment of the gradient search technique employed to solve the inverse problem sensitivities of the predictive bie model with respect to cavity parameters are evaluated semi analytically using an adjoint problem approach and a continuum kinematics description several key features of the formulation including the rigorous treatment of the radiation condition for semi infinite solids modeling of an illuminating seismic wave field and treatment of the prior information are highlighted a set of numerical examples with spherical and ellipsoidal cavity geometries is included to illustrate the performance of the method it is shown that the featured adjoint problem approach reduces the computational requirements by an order of magnitude relative to conventional finite difference estimates thus rendering the three dimensional elastic wave imaging of solids tractable for engineering applications

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

some results on the convergence of quasi copulas it is shown that pointwise convergence of a sequence a n n n of copulas to a copula a is equivalent 1 to the convergence of the corresponding endographs and 2 to the convergence of the corresponding upper or lower a levels for all but at most countably many a in 0 1 all with respect to the hausdorff metric examples are given that show that the countably many exceptions in 2 cannot be omitted it is furthermore shown that the main results also hold on the bigger class of quasi copulas

a fast volume integral equation method for the direct inverse problem in elastic wave scattering phenomena a fast method for solving the volume integral equation is introduced for the solution of forward and inverse multiple scattering problems in an elastic 3 d full space for both forward and inverse scattering analysis the volume integral equation in the wavenumber domain is used by means of the discrete fourier transform the volume integral equation in the wavenumber domain can be dealt with as a fredholm equation of the 2nd kind with respect to a non hermitian operator on a finite dimensional vector space the bi cgstab method is employed to construct the krylov subspace in the wavenumber domain the current procedure establishes a fast and simplified method without requiring the derivation of a coefficient matrix several numerical results validate the accuracy and effectiveness of the current method for both forward and inverse scattering analysis according to the numerical results the reconstruction of inhomogeneities of the wave field is successful even for multiple scattering of several cubes

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

equalization of nonlinear time varying channels using type 2 fuzzy adaptive filters presents a kind of adaptive filter type 2 fuzzy adaptive filter faf one that is realized using an unnormalized type 2 takagi sugeno kang tsk fuzzy logic system fls we apply this filter to equalization of a nonlinear time varying channel and demonstrate that it can implement the bayesian equalizer for such a channel has a simple structure and provides fast inference a clustering method is used to adaptively design the parameters of the faf two structures are used for the equalizer transversal equalizer te and decision feedback equalizer dfe a decision tree structure is used to implement the decision feedback equalizer in which each leaf of the tree is a type 2 faf this dfe vastly reduces computational complexity as compared to a te simulation results show that equalizers based on type 2 fafs perform much better than nearest neighbor classifiers nnc or equalizers based on type 1 fafs

joint measurability of mappings induced by a fuzzy random variable abstract the operation of taking the cut of a compact convex fuzzy set is shown to be jointly measurable with respect to both and the fuzzy set as a consequence a number of mappings on product spaces which are induced by a fuzzy random variable are shown to be jointly measurable some applications to the relationships between fuzzy random variables and other imprecise random elements are obtained finally a number of conditions are shown to be equivalent to being a fuzzy random variable at least in the case that the algebra in the sample space is complete and logical implications and equivalences between them are established in the general case

on the compactness of fuzzy numbers with sendograph metric in this paper by embedding the fuzzy number space with the sendograph metric into the product of two uniformly support bounded complete metric spaces we characterize compact sets of the fuzzy number space the main result is that with respect to the sendograph metric a closed set is compact if and only if it is uniformly support bounded and equi right continuous at 0

some complete metrics on spaces of fuzzy subsets the classes of the lp and lp metrics play an important role to develop a probability theory in fuzzy sample spaces all of these metrics are known to be separable but not complete the classes are closely related as for each lp metric there exists some lp metric which induces the same topology this paper deals with the completion of the lp and lp metrics we can also show that the relationship between the classes of lp and lp metrics still holds for the obtained respective classes of their completions

a hitchhiker s guide to quasi copulas abstract this paper presents a review of the concept of an n quasi copula starting from the characterization of functions that can be derived from operations on random variables we recall several other characterizations and properties that have been proven in the literature highlighting the differences between the bivariate case and the higher dimensional case additionally we recall several applications of n quasi copulas such as their role in the study of bounds on sets of n copulas we also recall various classes of n quasi copulas that have been studied in the literature finally we present some questions that have not yet been answered in the literature

extensions of subcopulas abstract in view of sklar s theorem the probability distribution function of every not necessarily continuous random vector can be uniquely decomposed in terms of the marginal distributions of its components and a suitable subcopula the study of such latter functions is therefore of interest for understanding the dependence information of non continuous variables here we investigate some analytical properties of the class of subcopulas including compactness with respect to a novel metric approximations and baire category results moreover under a suitable assumption we describe all possible extensions from a subcopula to a copula in any dimension

a linear sampling method for detecting fluctuations of the wavefield in an elastic half space abstract a linear sampling method for an elastic half space is developed to reconstruct fluctuations in the wavefield the starting point of the formulation is the near field equation that was also used by baganas et al 2006 instead of examining the norm of the solution of the near field equation we define a solvability index in order to obtain the spatial distribution of the amplitude of the solvability index and thus describe the location of the fluctuations a numerical method for the evaluation of the index is also provided for a simplified algorithm this method is based on a projection theorem for a hilbert space and a singular value decomposition numerical calculations were performed and the results validated the efficiency of the proposed method for reconstructing the fluctuations of a wavefield

generalized fourier transform and its application to the volume integral equation for elastic wave propagation in a half space in the present study a generalized fourier transform for time harmonic elastic wave propagation in a half space is developed the generalized fourier transform is obtained from the spectral representation of the operator derived from the elastic wave equation by means of the generalized fourier transform a volume integral equation method for the analysis of scattered elastic waves is presented the proposed method is based on the krylov subspace iteration technique during the iterative process the discrete generalized fourier transform is used where the derivation of a huge and dense matrix from the volume integral equation is not necessary

characterization of the sendograph convergence of fuzzy sets by means of their lp and levelwise convergence one particular way to define a metric on the family of all d dimensional fuzzy sets with non empty convex compact a cuts is the sendograph metric which measures the hausdorff distance of the sendographs of fuzzy sets as first result it will be proved that if a sequence of fuzzy sets converges to a limit fuzzy set with respect to the sendograph metric then it also converges with respect to the l p metric and that the converse is true if in addition the supports of the sequence converge to the support of the limit additionally convergence with respect to the sendograph metric will be shown to be equivalent to almost everywhere convergence of the a cuts of the sequence to the a cut of the limit plus the convergence of the supports of the sequence to the support of the limit already proved in the one dimensional setting by fan as second and stronger result it will be proved that analogous convergence interrelations also hold for the bigger class of fuzzy sets not necessarily having compact support if the hausdorff distance is replaced by a metrization of the fell topology this in particular implies that the two characterizations proved in the first step do not depend on the choice of the metric used in the definition of the hausdorff metric in addition it is demonstrated that the characterization of weakly compact sets with respect to the sendograph metric as stated by greco is a straightforward consequence of the abovementioned equivalences and an alternative definition of fuzzy random variables that is equivalent to the common definition is given

convergence of fuzzy sets with respect to the supremum metric abstract we characterize the convergence of fuzzy sets in the supremum metric given by the supremum of the hausdorff distances of the cuts of the fuzzy sets we do it by dividing this metric into its lower and upper quasi pseudometric parts this characterization is given in the more general context with no assumption on the fuzzy sets furthermore motivated from the theory of convex analysis we also provide some results about the behavior of the convergence in the supremum metric with respect to maximizers

a t norm embedding theorem for fuzzy sets it is well known that the class of upper semicontinuous normal convex fuzzy sets with compact supports can be embedded isometrically as a complete convex cone in a banach space we prove an analogous result for a subclass of fuzzy sets that is free from the normality limitation by exchanging the standard algebraic operations on fuzzy sets with operations based on strict t norms this allows us to investigate a new notion of fuzzy convexity that we call t convexity we show that the class of upper semicontinuous fuzzy t convex sets with nonempty compact supports can be embedded as a closed convex cone in a banach space this implies that fuzzy t convex sets satisfy the cancellation law we discuss a possible application of the embedding theorem in mathematical morphology

quasi copulas and signed measures we study the relationship between multivariate quasi copulas and measures that they may or may not induce on 0 1 n we first study the mass distribution of the pointwise best possible lower bound for the set of n quasi copulas for n 3 as a consequence we show that not every n quasi copula induces a signed measure on 0 1 n