regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

on the computation of ground state and dynamics of schrodinger poisson slater system in this paper we deal with the computation of ground state and dynamics of the schrodinger poisson slater sps system to this end backward euler and time splitting pseudospectral methods are proposed for the nonlinear schrodinger equation with the nonlocal hartree potential approximated by solving a poisson equation the approximation approaches for the hartree potential include fast convolution algorithms which are accelerated by using fft in 1d and fast multipole method fmm in 2d and 3d and sine fourier pseudospectral methods the inconsistency in 0 mode in fourier pseudospectral approach is pointed out which results in a significant loss of high order of accuracy as expected for spectral methods numerical comparisons show that in 1d the fast convolution and sine pseudospectral approaches are compatible while in 3d the fast convolution approach based on fmm is second order accurate and the fourier pseudospectral approach is better than it from both efficiency and accuracy point of view among all these approaches the sine pseudospectral one is the best candidate in the numerics of the sps system finally we apply the backward euler and time splitting sine pseudospectral methods to study the ground state and dynamics of 3d sps system in different setups

combat modeling using fuzzy expected values abstract we demonstrate that current combat modeling techniques have serious limitations especially when the data that are the inputs to the model are not precise these limitations however can be overcome with the use of fuzzy mathematics specifically when we use the fuzzy expected value fev and the weighted fuzzy expected value wfev these concepts are needed when ambiguous vague and or ambivalent data are represented by coefficients of differential equations which represent the modeled situations also included is a program which demonstrates a particular differential equation that allows for a variety of coefficients and controlling variables

nonlinearity of the fuzzy integral we prove that the fuzzy integral has some linearity properties only for small classes of fuzzy measures we also show that an extension of this concept to bounded functions as proposed by kandel 2 5 is not well defined in general

dimension reduction of the schrodinger equation with coulomb and anisotropic confining potentials we consider dimension reduction for the three dimensional 3d schrodinger equation with the coulomb interaction and an anisotropic confining potential to lower dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model sam or surface density model sdm in two dimensions 2d and the line adiabatic model lam in one dimension 1d efficient and accurate numerical methods for computing ground states and dynamics of the sam sdm and lam models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower dimensional models they are applied to find numerically convergence and convergence rates for the dimension reduction from 3d to 2d and 3d to 1d in terms of ground state and dynamics which confirm some existing analytical results for the dimension reduction in the literature in particular we explain and demonstrate that the standard schrodinger poisson system in 2

fast and accurate evaluation of nonlocal coulomb and dipole dipole interactions via the nonuniform fft we present a fast and accurate algorithm for the evaluation of nonlocal long range coulomb and dipole dipole interactions in free space the governing potential is simply the convolution of an interaction kernel u mathbf x and a density function rho mathbf x psi mathbf x 2 for some complex valued wave function psi mathbf x permitting the formal use of fourier methods these are hampered by the fact that the fourier transform of the interaction kernel widehat u mathbf k has a singularity and or widehat rho mathbf k ne0 at the origin mathbf k bf 0 in fourier phase space thus accuracy is lost when using a uniform cartesian grid in mathbf k which would otherwise permit the use of the fft for evaluating the convolution here we make use of a high order discretization of the fourier integral accelerated by the nonuniform fast fourier transform nufft by adopting spherical and polar phase space discretizations in three and two dimensions respectively

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

coulomb interactions on planar structures inverting the square root of the laplacian we present an adaptive fast multipole method for inverting the square root of the laplacian in two dimensions solving this problem is the dominant computational cost in many applications arising in electrical engineering geophysical fluid dynamics and the study of thin films it corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad tree the amount of work grows linearly with the number of mesh points

fuzzy integral representation abstract in this paper we give representation results for nonlinear functionals in arbitrary spaces as fuzzy integrals the main assumption is that of comonotonic maxitivity i e i f g if ig for comonotonic f g under the more restrictive condition of stochastic dominance we prove a more specific representation as a fuzzy integral with respect to a distorted probability measure we discuss the particular cases of possibility measures and upper fuzzy expectations some partial randon nikodym results are proved for both the choquet and the fuzzy integrals finally an example is shown where the fuzzy integral representation is useful

on the use of fuzzy sets in histogram equalization abstract the main objective of image enhancement is to generate a new image such that the new image is more suitable for an application than the original image one of the methods for image enhancement is histogram equalization in this paper an algorithm for histogram equalization is presented based on probability then a new algorithm based on fuzzy sets is presented the new method denoted by fuzzy histogram equalization fhe generates images which are sharper than the images produced by the classical approach of histogram equalization

numerical methods for calculating the fuzzy integral abstract a general procedure for calculating the fuzzy integral fi is developed for monotonic and single maxima functions in the case of monotonic functions an iterative algorithm based on the standard iteration method sim is shown to converge to the fi independently of the initial value an alternative newton type procedure can be applied for both monotonic and single maxima functions provided that an appropriate initial approximation is used

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

toward a general theory of fuzzy variables abstract a general framework for a theory is presented that encompasses both statistical uncertainty which falls within the province of probability theory and nonstatistical uncertainty which relates to the concept of a fuzzy set and possibility theory l a zadeh j fuzzy sets 1 1978 3 28 the concept of a fuzzy integral is used to define the expected value of a random variable properties of the fuzzy expectation are stated and a mean value theorem for the fuzzy integral is proved comparisons between the fuzzy and the lebesgue integral are presented after a new concept of dependence is formulated various convergence concepts are defined and their relationships are studied by using a chebyshev like inequality for the fuzzy integral the possibility of using this theory in bayesian estimation with fuzzy prior information is explored

free boundary regularity in the optimal partial transport problem in the optimal partial transport problem one is asked to transport a fraction 0 m min f l1 g l1 of the mass of f f onto g g while minimizing a transportation cost if f and g are bounded away from zero and infinity on strictly convex domains and respectively and if the cost is quadratic then away from the free boundaries of the active regions are shown to be cloc1 hypersurfaces up to a possible singular set this improves and generalizes a result of caffarelli and mccann 2010 6 and solves a problem discussed by figalli 2010 8 remark 4 15 moreover a method is developed to estimate the hausdorff dimension of the singular set assuming and to be uniformly convex domains with c1 1 boundaries we prove that the singular set is hn 2 finite in the general case and hn 2 finite if and are separated by a hyperplane

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

the use of weighted fuzzy expected value wfev in fuzzy expert systems abstract the use of the fuzzy expected value fev as a typical grade of membership within a fuzzy set may occasionally generate improper results in these cases we replace it by a new quantity the weighted fuzzy expected value wfev and demonstrate its improved performance the applicability of this concept to decision making in fuzzy intelligent systems is discussed and illustrated

the strongly confined schrodinger poisson system for the transport of electrons in a nanowire we study the limit of the three dimensional schrodinger poisson system with a singular perturbation to model a quantum electron gas that is strongly confined near an axis for well prepared data which are polarized on the ground space of the transversal hamiltonian the resulting model is the cubic defocusing nonlinear schrodinger equation our main tool is a refined analysis of the poisson kernel when acting on strongly confined densities in that direction an appropriate scaling of the initial data is required to avoid divergent integrals when the gas concentrates on the axis

most typical values for fuzzy sets abstract the classical definitions of expected value and standard deviation lead to quantities which generally fail to represent any typical feature of a given data set whenever it consists of more than one cluster the use of the fuzzy expected value fev as a typical grade of membership within a fuzzy set may occasionally also lead to improper results other quantities like the clustering fuzzy expected value cfev hardly deviate from the set average both conceptually and numerically and cannot represent a typical value of the given data in this work a new quantity the most typical value mtv for n dimensional sets is defined a given fuzzy set in r n is first clustered and replaced by a finite set of clusters each represented by the cluster center and its size this set of ordered pairs is then replaced by a single vector the most typical value of the fuzzy set

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

accurate and efficient computation of nonlocal potentials based on gaussian sum approximation abstract we introduce an accurate and efficient method for the numerical evaluation of nonlocal potentials including the 3d 2d coulomb 2d poisson and 3d dipole dipole potentials our method is based on a gaussian sum approximation of the singular convolution kernel combined with a taylor expansion of the density starting from the convolution formulation of the nonlocal potential for smooth and fast decaying densities we make a full use of the fourier pseudospectral plane wave approximation of the density and a separable gaussian sum approximation of the kernel in an interval where the singularity the origin is excluded the potential is separated into a regular integral and a near field singular correction integral the first is computed with the fourier pseudospectral method while the latter is well resolved utilizing a low order taylor expansion of the density both parts are accelerated by fast fourier transforms fft the method is accurate 14 16 digits efficient o n log n complexity low in storage easily adaptable to other different kernels applicable for anisotropic densities and highly parallelizable

the bounded convergence in measure theorem for nonlinear integral functionals in this paper we introduce a new notion of the perturbation of nonlinear functionals to formulate a functional form of the convergence theorems for nonlinear integrals in nonadditive measure theory as its direct consequences we obtain the bounded convergence in measure theorems for typical nonlinear integrals which show that the autocontinuity of a nonadditive measure is equivalent to the validity of the bounded convergence in measure theorems for the choquet the sugeno and the shilkret integrals as well as their symmetric and asymmetric extensions

properties of the fuzzy expected value and the fuzzy expected interval abstract the evaluation of the fuzzy expected value fev requires a complete knowledge about the domain of the evaluation and the distribution of the population in that domain since it is not always possible to assume a complete knowledge about the domain it is necessary to find some relaxations to the restrictions involving the evaluation of fev in this paper some solutions to this problem are proposed via the concept of the fuzzy expected interval fei

measuring properties of fuzzy sets a general technique and its use in fuzzy query evaluation abstract a fuzzy set can be viewed as a weighted max combination or as a convex combination of a family of nested ordinary sets the latter point of view allows the interpretation of a fuzzy set as a consonant random set then any numerical set function applied to a fuzzy set leads to a result which is a random number the expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set function this approach is very general and can be used for various kinds of measurements cardinality confidence measures attached to the occurrence of events diameter or perimeter of a fuzzy region distance between two sets etc also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed the weighted max combination view which leads to another type of scalar evaluation is also briefly discussed in this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property this kind of query can be managed by languages like sql in case of a crisp set of items when this set is fuzzily specified a scalar answer can still be provided as well as a fuzzy valued or interval valued one if we want to keep track of the effect of the fuzziness of the query the case where the data base contains precise information but also the more general case of fuzzy information data bases are considered

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

on the regularity of the free boundary in the optimal partial transport problem for general cost functions abstract this paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions more specifically we prove that a c 1 cost implies a locally lipschitz free boundary as an application we address a problem discussed by caffarelli and mccann 1 regarding cost functions satisfying the ma trudinger wang condition a3 if the non negative source density is in some l p r n space for p n 1 2 and the positive target density is bounded away from zero then the free boundary is a semiconvex c loc 1 hypersurface furthermore we show that a locally lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the riemannian setting

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

the optimal partial transport problem given two densities f and g we consider the problem of transporting a fraction m in 0 min f l 1 g l 1 of the mass of f onto g minimizing a transportation cost if the cost per unit of mass is given by x y 2 we will see that uniqueness of solutions holds for m in f wedge g l 1 min f l 1 g l 1 this extends the result of caffarelli and mccann in ann math in print where the authors consider two densities with disjoint supports the free boundaries of the active regions are shown to be n 1 rectifiable provided the supports of f and g have lipschitz boundaries and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex moreover assuming f and g supported on two bounded strictly convex sets omega lambda subset mathbb r n and bounded away from zero and infinity on their respective supports c 0 alpha rm loc regularity of the optimal transport map and local c1 regularity of the free boundaries away from omega cap lambda are shown finally the optimal transport map extends to a global homeomorphism between the active regions

efficient numerical methods for computing ground states and dynamics of dipolar bose einstein condensates new efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar bose einstein condensates becs described by a three dimensional 3d gross pitaevskii equation gpe with a dipolar interaction potential due to the high singularity in the dipolar interaction potential it brings significant difficulties in mathematical analysis and numerical simulations of dipolar becs in this paper by decoupling the two body dipolar interaction potential into short range or local and long range interactions or repulsive and attractive interactions the gpe for dipolar becs is reformulated as a gross pitaevskii poisson type system based on this new mathematical formulation we prove rigorously existence and uniqueness as well as nonexistence of the ground states and discuss the existence of global weak solution and finite time blow up of the dynamics in different parameter regimes of dipolar becs in addition a backward euler sine pseudospectral method is presented for computing the ground states and a time splitting sine pseudospectral method is proposed for computing the dynamics of dipolar becs due to the adoption of new mathematical formulation our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem extensive numerical examples in 3d are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar becs

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

characterization and comparison of sugeno and choquet integrals abstract starting from the concept of equiordered functions we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure sugeno and choquet integrals the characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals pointing out their formal similarities and their conceptual differences

short note the type 3 nonuniform fft and its applications the nonequispaced or nonuniform fast fourier transform nufft arises in a variety of application areas including imaging processing and the numerical solution of partial differential equations in its most general form it takes as input an irregular sampling of a function and seeks to compute its fourier transform at a nonuniform sampling of frequency locations this is sometimes referred to as the nufft of type 3 like the fast fourier transform the amount of work required is of the order o nlogn where n denotes the number of sampling points in both the physical and spectral domains in this short note we present the essential ideas underlying the algorithm in simple terms we also illustrate its utility with application to problems in magnetic resonance imagin and heat flow

optimal partial mass transportation and obstacle monge kantorovich equation abstract optimal partial mass transport which is a variant of the optimal transport problem consists in transporting effectively a prescribed amount of mass from a source to a target the problem was first studied by caffarelli and mccann 2010 6 and figalli 2010 12 with a particular attention to the quadratic cost our aim here is to study the optimal partial mass transport problem with finsler distance costs including the monge cost given by the euclidian distance our approach is different and our results do not follow from previous works among our results we introduce a pde of monge kantorovich type with a double obstacle to characterize active submeasures kantorovich potential and optimal flow for the optimal partial transport problem this new pde enables us to study the uniqueness and monotonicity results for the active submeasures another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper 14 igbida and nguyen 2018

fuzzy measures assuming their values in the set of fuzzy numbers abstract fuzzy valued fuzzy measures are defined in an axiomatic way extending a result of klement j math anal appl 75 1980 330 339 it is shown that they can be characterized by a suitable family of ordinary measures and markov kernels

interactive clustering and merging with a new fuzzy expected value abstract major problems exist in both crisp and fuzzy clustering algorithms the fuzzy c means type of algorithms use weights determined by a power m of inverse distances that remains fixed over all iterations and over all clusters even though smaller clusters should have a larger m our method uses a different distance for each cluster that changes over the early iterations to fit the clusters comparisons show improved results we also address other perplexing problems in clustering i find the optimal number k of clusters ii assess the validity of a given clustering iii prevent the selection of seed vectors as initial prototypes from affecting the clustering iv prevent the order of merging from affecting the clustering and v permit the clusters to form more natural shapes rather than forcing them into normed balls of the distance function we employ a relatively large number k of uniformly randomly distributed seeds and then thin them to leave fewer uniformly distributed seeds next the main loop iterates by assigning the feature vectors and computing new fuzzy prototypes our fuzzy merging then merges any clusters that are too close to each other we use a modified xie bene validity measure as the goodness of clustering measure for multiple values of k in a user interaction approach where the user selects two parameters for eliminating clusters and merging clusters after viewing the results thus far the algorithm is compared with the fuzzy c means on the iris data and on the wisconsin breast cancer data

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

fuzzy expectation and energy states in fuzzy media abstract this paper is concerned with energy levels and the density of states in fuzzy crystals the determination of eigenvalue spectrum or the density of states for a particle in a fuzzy crystal is obtained by using the concepts developed in fuzzy statistics various analytical results are found using the fuzzy expectation these results can be applied to a variety of fields such as decision making under uncertainty pattern analysis and quantum mechanics

properties of the fuzzy expected value and the fuzzy expected interval in fuzzy environment abstract the fuzzy expected value fev over some domain measures the typical value of some object in that domain the evaluation of fev requires a complete knowledge about the domain of the evaluation and the distribution of the population in that domain 1 since it is not always possible to assume a complete knowledge about the domain it is necessary to find some relaxations to the restrictions involving the evaluation of fev in this paper some solutions to this problem are proposed via the concept of the fuzzy expected interval fei

restored fuzzy measures in expert decision making the paper deals with problems of fuzzy measure restoration from insufficient data on a finite set the proposed approach is constructed in the class of second order choquct capacities fuzzy sets and systems 31 1989 23 reports of the enlarged session of i vekua inst appl math 14 1999 31 ann inst fourier 5 1953 131 when the fuzzy weights of singletons are known this essentially concerns certain frequency distributions where the nature of additivity is doubtful because of the fuzzy nature of data distribution this is the indisputable condition for the introduction of a fuzzy measure but the insufficient one for its construction measures of specificity indices of uncertainty and estimators of approximations are calculated some approximation properties are proved using the approach of a kaufman s experton theory les expertons hermes paris 1987 the unique fuzzy subset fuzzy object is constructed from the associated probabilities fuzzy sets and systems 31 1989 23 of the restored fuzzy measure the application of the constructed method is illustrated by an example of the processing of certain insufficient statistical data of the georgian language as a result it becomes possible to derive estimates for the georgian language on the whole

a nonlinear optimization procedure for generalized gaussian quadratures we present a new nonlinear optimization procedure for the computation of generalized gaussian quadratures for a broad class of square integrable functions on intervals while some of the components of this algorithm have been previously published we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works the new algorithm successfully computes generalized gaussian quadratures in a number of instances in which the previous algorithms fail four applications of our scheme to computational physics are presented the construction of discrete plane wave expansions for the helmholtz green s function the design of linear array antennae the computation of a quadrature for the discretization of laplace boundary integral equations on certain domains with corners and the construction of quadratures for the discretization of laplace and helmholtz boundary integral equations on smooth surfaces

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

the weighted fuzzy expected interval abstract the use of the fuzzy expected value fev as a typical grade of membership within a fuzzy set may occasionally generate improper results in these cases we replace it by the weighted fuzzy expected value wfev and demonstrate its improved performance however when the information available is fuzzy or incomplete neither the fev nor the wfev is applicable we will explain the concepts of the fev and the wfev and show how the wfev can be modified and still be used to compute a typical value by introducing a new concept the weighted fuzzy expected interval wfei

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

weak convergence of nonadditive measures based on nonlinear integral functionals abstract in this paper we formulate a general portmanteau theorem for a perturbative nonlinear integral functional and discuss the uniformity of weak convergence of nonadditive measures based on such a functional as their direct consequences it turns out that levy convergence of nonadditive measures coincides with every one of three types of weak convergence that is weak choquet weak sugeno and weak shilkret convergence and they are uniform on every bounded subset of lipschitz functions those results are applied when discussing the metrizability of the levy topology on the space of nonadditive measures and defining the fortet mourier type metrics on a uniformly equi autocontinuous set of nonadditive measures

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

spectral accuracy in fast ewald based methods for particle simulations a spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented we follow the established framework of fft based ewald summation but obtain a method with an important decoupling of errors it is shown for the proposed method that the error due to frequency domain truncation can be separated from the approximation error added by the fast method this has the significance that the truncation of the underlying ewald sum prescribes the size of the grid used in the fft based fast method which clearly is the minimal grid both errors are of exponential squared order and the latter can be controlled independently of the grid size we compare numerically to the established spme method by essmann et al and see that the memory required can be reduced by orders of magnitude we also benchmark efficiency i e error as a function of computing time against the spme method which indicates that our method is competitive analytical error estimates are proven and used to select parameters with a great degree of reliability and ease

computing the ground state and dynamics of the nonlinear schrodinger equation with nonlocal interactions via the nonuniform fft we present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear schrodinger equation nlse with nonlocal interactions based on a fast and accurate evaluation of the long range interactions via the nonuniform fast fourier transform nufft we begin with a review of the fast and accurate nufft based method in 29 for nonlocal interactions where the singularity of the fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates we then extend the method to compute other nonlocal interactions whose fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform many of these interactions do not decay at infinity in the physical space which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain the performance of our method against other existing methods is illustrated numerically with particular attention on the effect of the size of the computational domain in the physical space finally to study the ground state and dynamics of the nlse we propose efficient and accurate numerical methods by combining the nufft method for potential evaluation with the normalized gradient flow using backward euler fourier pseudospectral discretization and time splitting fourier pseudospectral method respectively extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the nlse with various nonlocal interactions numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency