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

neural coding higher order temporal patterns in the neurostatistics of cell assemblies recent advances in the technology of multiunit recordings make it possible to test hebb s hypothesis that neurons do not function in isolation but are organized in assemblies this has created the need for statistical approaches to detecting the presence of spatiotemporal patterns of more than two neurons in neuron spike train data we mention three possible measures for the presence of higher order patterns of neural activation coef ficients of log linear models connected cumulants and redundancies and present arguments in favor of the coefficients of loglinear models we present test statistics for detecting the presence of higher order interactions in spike train data by parameterizing these interactions in terms of coefficients of log linear models we also present a bayesian approach for inferring the existence or absence of interactions and estimating their strength the two methods the frequentist and the bayesian one are shown to be consistent in the sense that interactions that are detected by either method also tend to be detected by the other a heuristic for the analysis of temporal patterns is also proposed finally a bayesian test is presented that establishes stochastic differences between recorded segments of data the methods are applied to experimental data and synthetic data drawn from our statistical models our experimental data are drawn from multiunit recordings in the prefrontal cortex of behaving monkeys the somatosensory cortex of anesthetized rats and multiunit recordings in the visual cortex of behaving monkeys

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

galerkin finite element methods with symmetric pressure stabilization for the transient stokes equations stability and convergence analysis we consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient stokes equation the analysis is valid for a class of symmetric pressure stabilization operators but also for standard inf sup stable velocity pressure spaces without stabilization provided the initial data are chosen as a specific method dependent ritz projection we get unconditional stability and optimal convergence for both pressure and velocity approximations in natural norms for arbitrary interpolations of the initial data a condition between the space and time discretization parameters has to be verified in order to guarantee pressure stability

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

bayesian estimation of stimulus responses in poisson spike trains a bayesian method is developed for estimating neural responses to stimuli using likelihood functions incorporating the assumption that spike trains follow either pure poisson statistics or poisson statistics with a refractory period the bayesian and standard estimates of the mean and variance of responses are similar and asymptotically converge as the size of the data sample increases however the bayesian estimate of the variance of the variance is much lower this allows the bayesian method to provide more precise interval estimates of responses sensitivity of the bayesian method to the poisson assumption was tested by conducting simulations perturbing the poisson spike trains with noise this did not affect bayesian estimates of mean and variance to a significant degree indicating that the bayesian method is robust the bayesian estimates were less affected by the presence of noise than estimates provided by the standard method

a note on fuzzy rational choice functions abstract this short note gives a counterexample showing that kim s theorem on characterization of fuzzy rational choice function 1 is incorrect and it gives a proper characterization of such choice functions

decoding poisson spike trains by gaussian filtering the temporal waveform of neural activity is commonly estimated by low pass filtering spike train data through convolution with a gaussian kernel however the criteria for selecting the gaussian width are not well understood given an ensemble of poisson spike trains generated by an instantaneous firing rate function t the problem was to recover an optimal estimate of t by gaussian filtering we provide equations describing the optimal value of using an error minimization criterion and examine how the optimal varies within a parameter space defining the statistics of inhomogeneous poisson spike trains the process was studied both analytically and through simulations the rate functions t were randomly generated with the three parameters defining spike statistics being the mean of t the variance of t and the exponent of the fourier amplitude spectrum 1 f of t the value of opt followed a power law as a function of the pooled mean interspike interval i opt aib where a was inversely related to the coefficient of variation cv of t and b was inversely related to the fourier spectrum exponent besides applications for data analysis optimal recovery of an analog signal waveform t from spike trains may also be useful in understanding neural signal processing in vivo

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

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

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

a characterization of relatively compact sets of fuzzy sets abstract a purely topological characterization of relatively compact sets is given for the metric space k y d of upper semicontinuous compact supported normal fuzzy subsets of a metric space y the considered metric d is that of the distance between fuzzy subsets which is the supremum of the hausdorff distances of the corresponding level sets in the given proof the compactness of a variational convergence which was introduced by de giorgi and franzoni is fundamental

endographic approach on supremum and infimum of fuzzy numbers in this paper it is shown that supremum and infimum of fuzzy numbers can be characterized by membership function method via endograph metric more directly than by using the levelwise metric it is proved that the endograph metric is approximative with respect to orders on fuzzy number spaces also the endograph metric is computable the result in this paper shows that the fuzzy number valued integrals of the kind based on concept like riemann sum in calculus are computable

on fuzzifications of discrete dynamical systems let x denote a locally compact metric space and f x x be a continuous map in the 1970s zadeh presented an extension principle helping us to fuzzify the dynamical system x f i e to obtain a map f for the space of fuzzy sets on x we extend an idea mentioned in p diamond a pokrovskii chaos entropy and a generalized extension principle fuzzy sets syst 61 1994 277 283 to generalize zadeh s original extension principle in this paper we study basic properties of so called g fuzzifications such as their continuity properties we also show that for any g fuzzification i a uniformly convergent sequence of uniformly continuous maps on x induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and ii a conjugacy resp a semi conjugacy between two discrete dynamical systems can be extended to a conjugacy resp a semi conjugacy between fuzzified dynamical systems throughout this paper we consider different topological structures in the space of fuzzy sets namely the sendograph endograph and levelwise topologies

stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection abstract we consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation free forcing in the momentum equation we provide analysis that suggests for such problems obtaining accurate solutions necessitates either the use of pointwise divergence free finite elements such as scott vogelius or heavy grad div stabilization of weakly divergence free elements the theory is demonstrated in numerical experiments for a benchmark natural convection problem where large irrotational forcing occurs with high rayleigh numbers

a note on the sendograph metric of fuzzy numbers in this paper by studying the attainable properties of sendograph metric in fuzzy number space we generalize monotone convergence theorem and nested theorem of intervals from the space of real numbers to the space of fuzzy numbers

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

characterizations of endograph metric and γ convergence on fuzzy sets abstract this paper is devoted to the relationships and properties of the endograph metric and the convergence the main contents can be divided into three closely related parts firstly on the class of upper semi continuous fuzzy sets with bounded cuts we find that an endograph metric convergent sequence is exactly a convergent sequence satisfying the condition that the union of cuts of all its elements is a bounded set in r m for each 0 secondly based on investigations of level characterizations of fuzzy sets themselves we present level characterizations level decomposition properties of the endograph metric and the convergence it is worth mentioning that using the condition and the level characterizations given above we discover the fact the endograph metric and the convergence are compatible on a large class of general fuzzy sets which do not have any assumptions of normality convexity or star shapedness its subsets include common particular fuzzy sets such as fuzzy numbers compact and noncompact fuzzy star shaped numbers compact and noncompact and general fuzzy star shaped numbers compact and noncompact thirdly on the basis of the conclusions presented above we study various subspaces of the space of upper semi continuous fuzzy sets with bounded cuts equipped with the endograph metric we present characterizations of total boundedness relative compactness and compactness in these fuzzy set spaces and clarify relationships among these fuzzy set spaces it is pointed out that the fuzzy set spaces of noncompact type are exactly the completions of their compact counterparts under the endograph metric

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

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

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

some notes on the characterization of compact sets of fuzzy sets with lp metric in 1990 diamond gave a characterization of compact sets in the space of fuzzy star shaped numbers with respect to the origin with l p metric for 1

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

some notes on zadeh s extensions this paper shows that the zadeh s extensions of sendograph metric continuous fuzzy valued functions are sendograph metric continuous fuzzy functions

a peano theorem for fuzzy differential equations with evolving membership grade abstract the peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations the difficulty appears to be due to the standard use of the supremum metric defined by the supremum over the hausdorff metric between the level sets of the fuzzy sets another may have been the classical formulation of fuzzy differential equations in terms of the hukuhara derivative of the level sets here a peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions with aubin s morphological equations which allow non local set evolution a major difference from previous publications is the use of the endograph metric end essentially the hausdorff metric between the endographs in r n 0 1 of fuzzy sets instead of the supremum metric another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations the result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity compact supports or even normality

analysis of the pressure stabilized petrov galerkin method for the evolutionary stokes equations avoiding time step restrictions optimal error estimates for the pressure stabilized petrov galerkin method for the evolutionary stokes equations are proved in the case of regular solutions without restriction on the length of the time step these results clarify the instability of the discrete pressure for small time steps reported in the literature first the limit situation of the continuous in time discretization is considered and second the numerical analysis for the backward euler scheme is carried out the main results are applicable to higher order finite elements the analytical results are strongly based on an appropriate approach for computing the initial velocity which is suggested in this paper numerical studies confirm the theoretical results showing in particular that this instability does not occur for the proposed initial condition