on stability in possibilistic linear equality systems with lipschitzian fuzzy numbers abstract linear equality systems with fuzzy parameters and crisp variables defined by the extension principle are called possibilistic linear equality systems the study focuses on the problem of stability with respect to perturbations of fuzzy parameters of the solution in these systems

convex fuzzy mapping and operations of convex fuzzy mappings convex and concave fuzzy mappings are considered in this paper based on the seminal work of nanda and kar on the definition of convex fuzzy mapping the operations of convex fuzzy mapping are established some of the important concepts such as positively homogeneous infimal convolution right scalar multiplication and convex hull are introduced and the corresponding theorems are demonstrated by using the parametric representation of fuzzy numbers

applications of fuzzy chaos to fuzzy simulation in this paper we wish to use fuzzy chaos to generate fuzzy simulation we first discuss how fuzzy chaos can produce a chaotic sequence of fuzzy numbers we apply this technique to fuzzily simulate a solution to a simple fuzzy optimization problem other applications of fuzzy simulation to fuzzy mathematical programming fuzzy max flow fuzzy regression and tuning a fuzzy controller are also presented

fuzzy plane geometry i points and lines abstract we introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line in fuzzy plane geometry we consider the fuzzy distance between fuzzy points and show it is a weak fuzzy metric we study various definitions of a fuzzy line develop their basic properties and investigate parallel fuzzy lines

a new i vector topology generated by a fuzzy norm this paper is devoted to the research for a topological structure for i vector on the fuzzy normed linear space in the sense of felbin by using a natural method a new i vector topology on the fuzzy normed linear space is constructed we study some of its properties and prove that the fuzzy normed linear space is a locally convex i topological vector space with respect to the i topology under a proper condition we give a necessity and sufficienct condition for a fuzzy normed linear space to be hausdorff and weakly hausdorff with respect to the i vector topology respectively in addition we also discuss the relations between this i vector topology and other two known i vector topologies on the fuzzy normed linear space lastly we give some canonical examples of fuzzy normed linear spaces using this i vector topology

spanning properties for fuzzy matroids abstract in this paper we investigate the fuzzy analogs of some well known properties of the span of subsets of crisp matroids

benchmarking in linguistic importance weighted aggregations in this work we concentrate on the issue of weighted aggregations and provide a possibilistic approach to the process of importance weighted transformation when both the importances interpreted as benchmarks and the ratings are given by symmetric triangular fuzzy numbers we will show that using the possibilistic approach i small changes in the membership function of the importances can cause only small variations in the weighted aggregate ii the weighted aggregate of fuzzy ratings remains stable under small changes in the nonfuzzy importances iii the weighted aggregate of crisp ratings still remains stable under small changes in the crisp importances whenever we use a continuous implication operator for the importance weighted transformation we illustrate the proposed method by some simple examples

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

fuzzy plane geometry crisp plane geometry starts with points then lines and parallel lines circles triangles rectangles etc in fuzzy plane geometry we will do the same our fuzzy points lines circles etc will all be fuzzy subsets of r r we assume the standard xy rectangular coordinate system in the plane since fuzzy subsets of r r will be surfaces in r 3 we can not easily present graphs of their membership functions however cuts of fuzzy subsets of r r will be crisp subsets of the plane using an xy coordinate system we may draw pictures of cuts of fuzzy points in r r also for fuzzy lines etc in this way we can see what the membership functions of fuzzy subsets of r r look like

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

platform type fuzzy number and separability of fuzzy number space abstract in this paper we introduce the notions of the rectangular fuzzy set and step type fuzzy number it is shown that the family of all platform type fuzzy numbers e t 1 is dense in fuzzy numbers space e 1 about l p metric d p and the family of all rational type step fuzzy numbers e qg 1 e qg 1 e t 1 and e qg 1 is a countable set is dense in e t 1 d p further as corollaries we obtain that e 1 d p is a separable metric space and several separable metric spaces about p mean symmetric difference metric d p

fuzzy input output controllers are universal approximators abstract in this paper we show that a fuzzy input output controller can approximate to any degree of accuracy a continuous fuzzy system this generalizes the known result that fuzzy controllers crisp input output can approximate continuous functions

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

some remarks on distances between fuzzy numbers in this paper we consider different approaches to assigning distances between fuzzy numbers a pseudo metric on the set of fuzzy numbers arising from the idea of the value of a fuzzy number is described and some of its topological properties are noted reducing functions are used to define a family of metrics on the space of fuzzy numbers some convergent properties for these metrics are illustrated finally a fuzzy distance between fuzzy numbers is introduced and its basic properties are studied

eigen fuzzy number sets in this paper we extend certain results of sanchez resolution of eigen fuzzy sets equations fuzzy sets and systems 1 1978 69 75 to the context of fuzzy numbers a slight modification of the dubois prade definition of fuzzy numbers is used

representations with fuzzy darts abstract fuzzy information is organized by partitioning the fuzzy interval 0 1 into layers of fuzzy strata at each level fuzzy information is approximated by a fuzzy dart information at all levels is combined and recorded as an n tuple of fuzzy darts from this perspective fuzzy numbers with continuous boundaries are approximated by pn fuzzy numbers then within the framework of fuzzy level sets the problem of comparing pn fuzzy numbers is addressed for applications differentiably smooth fuzzy maps which are either expansive or contractive are characterized by their derivatives finally anti fuzzy numbers are introduced and discussed

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

the topological structure of fuzzy sets with endograph metric for a non degenerate convex subset y of the n dimensional euclidean space r n let k y be the family of all fuzzy sets of r n which are upper semicontinuous fuzzy convex and normal with compact supports contained in y we show that the space k y with the topology of endograph metric is homeomorphic to the hilbert cube q 1 1 w iff y is compact and the space k y is homeomorphic to x n q sup x n 1 iff y is non compact and locally compact

the topology and dynamics of the hyperspaces of normal fuzzy sets and their inverse limit spaces given a compact and connected metric space continuum x we study topological and dynamical properties of the hyperspace of normal fuzzy sets f1 x equipped with the hausdorff endograph or sendograph metric among the many results we show that it is contractible path connected locally contractible locally path connected locally simply connected and locally connected for the endograph metric the hyperspace f1 x is a continuum and then for a topological graph x we show how using the inverse limit approach of barge and martin the inverse limit of a fuzzy dynamical system on x can be realized as an attractor of a fuzzy dynamical system on a manifold

analysis of heat conduction in solids by space time finite element method a functional in the form of a convolution of the temperature variation in the space time domain has been derived it has been used as the basis for a finite element formulation of heat conduction in solids numerical illustrations have indicated that the space time finite element algorithm provides a more rapid convergence to the exact solutions than the usual finite element analysis with discretization in space only

a streamline upwind full galerkin method for space time convection dominated transport problems an original space time finite element approach for the solution of the diffusion convection equation is proposed in this paper a slight manipulation of the differential equation suggests that transient transport problems may in fact be seen as steady state space time transport problems accurately and easily soluble by the standard galerkin technique however concerning convective transport involving sharp fronts or coarse discretization it is shown that implementation of dissipation along space time trajectories significantly improves the solutions classical comparative test problems are run to establish the performances of this method and to show the limits of the more sophisticated petrov and taylor galerkin schemes evocation of a possible space time anisotropy generated by usual finite difference time stepping procedures as well as comparative analysis of amplification matrices help to understand the accuracy and the robustness of the proposed approach

boolean fuzzy sets and possibility measures abstract b possibility and b necessity measures are defined as the qualitative analogues of the well known 0 1 measures their connection to b fuzzy sets boolean fuzzy sets is established it is also proved that b fuzzy sets define b possibility and b necessity measures by their membership function we also introduce the possibility of a b fuzzy set and prove that it is the extension of the possibility measure defined on crisp sets this extension can be defined in two ways which are proved to be equivalent

topological properties of the spaces of regular fuzzy sets abstract by identifying the space f 0 x of regular fuzzy sets with a subspace t 0 x of some compact set valued function space we define a uniform topology on f 0 x and discuss several topological properties such as first countability separability connectedness etc when x is a reflexive banach space f 0 x is imbedded into a locally convex topological vector space and a definition of integration and differentiation for f 0 x valued functions is given

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

comparison of topological and uniform structures for fuzzy numbers and the fixed point problem abstract the goal of this paper is to compare certain topological and uniform structures on sets of fuzzy numbers since fuzzy sets can be viewed as a kind of subsets they can be endowed with uniform structures in ways similar to those used for classic subsets the hausdorff construction is one such approach the structures we study are the myope structure the uniform convergence of the cuts in the sense of hausdorff and a structure induced by a metric defined by goetschel and voxman we show how these structures are related to fixed point problems for some classes of fuzzy numbers

dynamic topologies and their applications to crisp topologies fuzzifications of crisp topologies and fuzzy topologies on the crisp real line abstract we introduce and develop topologies whose members are dynamic sets a la zeleny this development spawns the notions of phase and co phase spaces fuzzy dual dynamic dual and co fuzzy dual with the following results being obtained result a the notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of hutton gantner steinlage warren klein and lowen result b the dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic result c the fuzzy real lines re interpreted as fuzzy duals of r with the usual euclidean topology t induce on r via the notion of co fuzzy dual a class of canonical stratified fuzzy topologies co r l e g if l is a demorgan algebra and lc 1 then the compactness of a r is equivalent to the ordinary compactness of a w r t t and if l is a hutton algebra r l co r l is metrizable in the sense of hutton and erceg with erceg metric d l such that d l p r 1 p s 1 r s where pr is the crisp point or impulse with support r i e d l extends the euclidean metric on r

a representation of fuzzy numbers abstract each fuzzy number can be written in a unique way as the sum of a symmetric and a skew fuzzy number these two kinds of fuzzy numbers correspond bijectively to certain functions on the unit interval that are of bounded variation these bijections give rise to a representation of fuzzy numbers by pairs of functions of bounded variation as application mathematical structures on the set of functions of bounded variation are transported to that of fuzzy numbers making the set of fuzzy numbers into a complete metric space a commutative semiring and a commutative residuated lattice

geometric compatibility modification abstract compatibility modification is a rule based inference strategy that uses the similarity of the input with the antecedent of a rule to modify the consequent existing compatibility modification inference techniques have employed a set theoretic assessment of compatibility in this paper a distance based compatibility measure is derived from a generalization of the dissemblance index for fuzzy sets this measure is then used to develop an inference technique based on geometric compatibility this geometric approach is compared with two other distance based inference techniques linear rule interpolation and bound dependent linear revision

convex and strongly convex fuzzy sets abstract properties of the support and of the core of convex and strongly convex fuzzy sets are considered the convex and strongly convex fuzzy sets in the real line are characterized by means of the piece wise monotonic functions

a characterization of the extension principle abstract we provide an alternative characterization of the extension principle of fuzzy sets this characterization is based upon using relations to represent mappings we apply this new characterization to develop an extension for non deterministic mappings

generalisation of the goetschel voxman embedding abstract goetschel and voxman 3 gave a mapping from the space of crisp fuzzy numbers of the real line into l 2 their embedding is a homeomorphism and its range is convex using their construction and introducing a new metric we can extend their embedding to a subspace of the metric space of normal fuzzy convex upper semicontinuous and compactly supported fuzzy numbers of scr n which have their supports contained in a fixed compact cube of scr n if we consider the case of goetschel and voxman i e the fuzzy numbers of the real line the extra properties of our mapping are we do not need the crispness and both the embedding and its inverse are lipschitz continuous with respect to different metrics

sparse and low rank tensor estimation via cubic sketchings in this paper we propose a general framework for sparse and low rank tensor estimation from cubic sketchings a two stage non convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability the non asymptotic analysis sheds light on an interplay between optimization error and statistical error the proposed procedure is shown to be rate optimal under certain conditions as a technical by product novel high order concentration inequalities are derived for studying high moment sub gaussian tensors an interesting tensor formulation illustrates the potential application to high order interaction pursuit in high dimensional linear regression

equations with fuzzy numbers abstract assuming that is any operation defined in the real number set it can be extended to the fuzzy sets by zadeh s extension principle in this paper we study which conditions have to be verified by the cuts of the fuzzy numbers f and g in order that the equation f x g has a solution by the way we study some properties of the function h g f considered by e sanchez

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

generalized t norm structures abstract this is a continuation of the work in drossos and navara eufit 96 aachen germany 1996 in this paper we investigate the possibility of defining a quantitative structure non idempotent on the top of a usual qualitative structure these generalized monoidal structures are very much related to the notion of t norms and t conorms defined on the closed interval 0 1 we would then like to define an appropriate generalization of the notion of t norm as a generalized monoidal structure which will reflect a minimum of properties of t norms defined on 0 1 considered as absolutely basic

a natural topology for fuzzy numbers a natural topology for fuzzy numbers is proposed by representing a fuzzy number as a pair of distribution functions the relationship between fuzzy numbers and the hutton unit interval is also investigated

suitability in fuzzy topological spaces abstract an l fuzzy topological space is said to be suitable if it possesses a nontrivial crisp closed subset basic properties of and sufficient conditions for suitable spaces are derived characterizations of the suitable subspaces of the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are obtained suitability is l fuzzy productive nondegenerate 1 hausdorff spaces are suitable the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are not suitable and no suitable subspace of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line is a fuzzy retract of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line respectively without restrictions there cannot be a fuzzy extension theorem

on fuzzy number lattice r abstract we give a systematic development of fuzzy number lattice theory many of our results generalize to number lattice over the lattice algebra our first main result is that the real number chain r is a homomorphic image of the fuzzy number lattice r that is r r and the congruence class r is a component of r thus we can be specialized to describe the structure of the fuzzy number lattice r next we study the structure and representation of the congruence class

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

some remarks on the topological properties of fuzzy numbers in r n abstract the space of normal upper semicontinuous fuzzy convex and compactly supported fuzzy numbers in r b are considered endowed with different metrics generated by the hausdorff metric a subspace is introduced and properties of the metrics restricted to the subspace are investigated

l metric on the space of l fuzzy numbers abstract in this article a natural l topology is constructed on the set of l fuzzy numbers which is called the standard l topology it is proved that the standard l topology can be induced by an l metric the space of trapezoidal fuzzy numbers can be regarded as a subspace of the space of l fuzzy numbers an l metric on the set of trapezoidal fuzzy numbers is presented

tensor completion based on nuclear norm minimization for 5d seismic data reconstruction abstractmany standard seismic data processing and imaging techniques require regularly sampled data prestack seismic data are multidimensional signals that can be represented via low rank fourth order tensors in the frequency space f x domain we propose to adopt tensor completion strategies to recover unrecorded observations and to improve the signal to noise ratio of prestack seismic volumes tensor completion can be posed as an inverse problem and solved by minimizing a convex objective function the objective function contains two terms a data misfit and a nuclear norm the data misfit measures the proximity of the reconstructed seismic data to the observations the nuclear norm constraints the reconstructed data to be a low rank tensor in essence we solve the prestack seismic reconstruction problem via low rank tensor completion the cost function of the problem is minimized using the alternating direction method of multipliers we present synthetic examples to illustrate the behavior of the al

contributions to fuzzy analysis in some sense fuzzy analysis deviates highly from classical analysis needing its own specific notions the main reason which causes this situation consists in the different algebraic and structure theoretic properties of the usual real numbers and fuzzy numbers the paper which is an extended version of 4 is devoted to this problem presenting some of these specific notions in particular examples of proper fuzzy normed vector spaces and proper fuzzy normed algebras are given

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

difference distance and similarity as a basis for fuzzy decision support based on prototypical decision classes the use of prototypical decision classes as a basis for fuzzy decision support is outlined it is shown that given certain assumptions about the structure of both numeric and non numeric linguistic variables trapezoidal fuzzy sets can be used to model linguistic terms in particular the notion of a fuzzy index is introduced to model sets of linguistic terms for which there is no formal measurement scale the similarity between prototypes and input cases is measured as a function of the difference or distance between fuzzy sets the difference measure is itself a fuzzy set to reflect the underlying uncertainties in the original linguistic model similarity profiles for different decision classes are output as trapezoidal fuzzy sets the approach can be applied to the rapid development of decision aids using standard business software

adaptive fuzzy interpolation fuzzy interpolative reasoning strengthens the power of fuzzy inference by the enhancement of the robustness of fuzzy systems and the reduction of the systems complexity however after a series of interpolations it is possible that multiple object values for a common variable are inferred leading to inconsistency in interpolated results such inconsistencies may result from defective interpolated rules or incorrect interpolative transformations this paper presents a novel approach for identification and correction of defective rules in interpolative transformations thereby removing the inconsistencies in particular an assumption based truth maintenance system atms is used to record dependences between interpolations and the underlying technique that the classical general diagnostic engine gde employs for fault localization is adapted to isolate possible faulty interpolated rules and their associated interpolative transformations from this an algorithm is introduced to allow for the modification of the original linear interpolation to become first order piecewise linear the approach is applied to a realistic problem which predicates the diarrheal disease rates in remote villages to demonstrate the potential of this study

a lattice of continuities for fuzzy topological spaces abstract several continuities are defined for fuzzy topological spaces and characterizations of these continuities are obtained it is shown that these continuities form a lattice ordered by implication whose supremum is fuzzy continuity and whose infimum is the continuity of general topology

b fuzzy probabilities abstract we treat b fuzzy probabilities in nonstandard frameworks by the methods based on boolean valued models and especially boolean powers these formal methods apart from the difficulties associated with the formal model itself give a simple and powerful theory for fuzzy probability spaces and fuzzy random variables essentially everything is reduced to a transfer principle however one can also work using classical methods of proofs we first give the necessary background on nonstandard frameworks within these frameworks we develop boolean fuzzy probabilities and fuzzy random variables

fuzzy equivalence and the resulting topology abstract in this paper we show how a topology can be generated by a fuzzy binary relation of indifference i e a fuzzy equivalence relation on a set of objects this construction generalizes the classical construction of the topology of metric spaces thus showing that the idea of fuzziness is already present in classical mathematical objects an application related to poincare s conception of continuum is also given

on the level convergence of a sequence of fuzzy numbers abstract in this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space e 1 established by wu and wu j math anal appl 210 1997 499 and improve the expressions of the supremum and infimum as a straightforward corollary of this result we obtain a necessary and sufficient condition under which for a bounded sequence of fuzzy numbers u n the pair of functions sup n u n and sup n u n can determine a fuzzy number secondly we give a necessary and sufficient condition for a sequence of fuzzy numbers u n to be levelwise convergent in e 1 and generalize some important theorems in real number spaces to fuzzy number spaces finally we prove the existence of supremum and infimum for the level continuous fuzzy valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained

boolean fuzzy sets abstract the classical theory of fuzzy sets has been developed within the cantorian framework as a simple generalization of the indicator or characteristic function and thus as a generalization of the membership property only paying no attention to the equality relation the nonstandard approach to fuzzy sets on the contrary is based on a non cantorian framework which essentially amounts to viewing the eternally constant cantorian platonic universe through two observers the absolute cantorian and the local non cantorian 5 the construction of such a theory is based on a generalization of both equality and membership relations and technically is based on the theory of boolean powers and a boolean generalization of nonstandard analysis based on them 5 6 8 15 for nonstandard analysis see 1 13 in this paper which is a continuation of 5 we give some new results for the basic concepts of fuzzy sets from our point of view along with interpretations which are more interesting to fuzzy theorists

fuzzy preuniform structures and the structures they induce 2 applications in this second article we use the methodology and results we have introduced in the first part in this text we principally develop applications of what we have called the stabilization mechanism in the case of uniform spaces in fact we identify the set of the fuzzy subsets of a set e with the quotient set for an equivalence relation of a larger family we endow this family with uniform and topological structures and we study the quotient structures induced on the set of the fuzzy subsets of e we especially detail the case where e is a classic uniform space when e is a locally compact metric and separable space we can generate the classic closed convergence structure on the family of the closed subsets of e and the myope convergence structure on the family of the compact subsets of e we extend these structures to the case of fuzzy subsets and we obtain results which are generally close to the corresponding classic ones we think that a fuzzy mapping is a concept which is close to the concept of correspondence and then the continuity of fuzzy mappings must be defined in a way similar to those used for correspondences so we define the upperand lowerhemi continuity of fuzzy mappings finally we apply these results to the case where e is a linear space and more especially where e r we consider the fuzzy vectors on r and the corresponding extended operations we give continuity results for these operations and applications to fuzzy real analysis we have quoted some linked works for instance 5 71 in this article we use the notations and conventions of the first article

incoherent tensor norms and their applications in higher order tensor completion in this paper we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion we introduce a class of tensor norms by allowing for different levels of coherence which allows us to leverage the incoherence of a tensor in particular we show that a k th order tensor of multilinear rank r and dimension d times cdots times d can be recovered perfectly from as few as o r k 1 2 d 3 2 r k 1 d log d 2 uniformly sampled entries through an appropriate incoherent nuclear norm minimization our results demonstrate some key differences between completing a matrix and a higher order tensor they not only point to potential room for improvement over the usual nuclear norm minimization but also highlight the importance of explicitly accounting for incoherence when dealing with higher order tensors although our focus is primarily on the theoretical guarantees for nuclear norm minimization such insights may prove useful for understanding performance of other related methods and developing improved practical algorithms

fuzzy plane geometry ii circles and polygons abstract this paper continues our research in fuzzy plane geometry previously we studied fuzzy points and fuzzy lines now we investigate fuzzy circles and fuzzy polygons we show that the fuzzy area of a fuzzy circle or a fuzzy polygon is a fuzzy number we also argue that the fuzzy perimeter of a fuzzy circle or a fuzzy polygon is a fuzzy number

extended lattice ordered structures for l fuzzy sets and l fuzzy numbers abstract an l fuzzy set a is a mapping of a set x into another set l the set l is called the true set of a and x is called the universe of a it has been discussed that a better structure for the truth set l is a complete lattice ordered monoid then operations on l can be directly extended to operate on lx and a complete lattice ordered monoid can be obtained in this paper we consider operations on lx which are extended from operations on x by the extension principle we will obtain a similar complete lattice ordered monoid constituted by extended operations moreover a notion of l fuzzy numbers is proposed then extended operations on l fuzzy numbers are discussed and a distributively lattice ordered structure is developed for l fuzzy numbers

an experimental study on fuzzy distances for skull face overlay in craniofacial superimposition abstract skull face overlay is the most time consuming and error prone stage in craniofacial superimposition an important skeleton based forensic identification technique this task focuses on achieving the best possible overlay of an unknown skull found and a single ante mortem image of a candidate missing person the process is influenced by some sources of uncertainty since two objects of different nature are involved i e a skull and a face in previous works we have developed a computer aided craniofacial superimposition system aimed to assist forensic anthropologists in obtaining the best possible skull and face overlay the system has successfully allowed us to reduce the processing time simplify the forensic anthropologists work and make the process more objective and reproducible our approach is based on automatically overlaying a skull three dimensional model onto a facial photograph by minimizing the distance between two subsets of corresponding cranial and facial landmarks the proposed method properly deals with the inherent uncertainty sources to the skull face overlay process by considering fuzzy sets to model imprecise landmark location and imprecise cranial and facial landmarks spatial correspondence resulting from the presence of soft tissues in the face accordingly our methodology requires computing two kinds of distance metrics between a point and a fuzzy set and between two fuzzy sets this contribution is devoted to study the performance and influence of the most significant and suitable fuzzy distances proposed in the specialized literature as well as other new ones proposed on our skull face overlay system in particular we have tested the behavior of our automatic method when considering eight different distance measurements the system performance has been objectively evaluated considering 18 case studies resulting from a ground truth dataset following a rigorous statistical experimental setup the fact that the choice of a good distance metric is crucial to our method has been demonstrated since it significantly affects the quality of the final solutions it has been shown that our skull face overlay approach presents the best performance using the weighted mean distance in most of the cases and that the results are both more accurate and robust than the other studied metrics

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

the space time sinc gallerkin method for parabolic problems this paper contains the formulation of a space time sinc galerkin method for the numerical solution of the parabolic partial differential equation in one space dimension the space time adjective means that the galerkin technique is employed simultaneously in time and space salient features of the method include exponential rate of convergence ease of assembly of the discrete system a global approximation and the ability to handle singular problems two methods of solution for the discrete system are offered and numerical results for test problems selected from the literature are included

separation axioms and the fuzzy real lines we show that for each lattice l which is completely distributive and equipped with an order reversing involution the canonical topology of the fuzzy real line l is uniformizable in the sense of hutton furthermore this uniformity induces a pseudometric which induces the canonical topology consequently l satisfies the higher order separation axioms of hutton and sarkar more generally we answer most of the open questions concerning l and its subspaces with respect to each of the separation axiom schemes of hutton and sarkar answer some of these same questions for the stratified fuzzy real line c l and its subspaces and analyze c l and its subspaces with respect to the separation axioms of rodabaugh

learning tensors from partial binary measurements we generalize the 1 bit matrix completion problem to higher order tensors consider a rank r order d tensor t in mathbb r n times cdots times mathbb r n with bounded entries we show that when r o 1 such a tensor can be estimated efficiently from only m o r nd binary measurements this shows that the sample complexity of recovering a low rank tensor from 1 bit measurements of a subset of its entries is roughly the same as recovering it from unquantized measurements a result that had been known only in the matrix case i e when d 2 by using a certain atomic m norm as a convex proxy for rank we allow for approximately low rank tensors and give error bounds based on the m norm of the tensor we prove that the theoretical bounds are optimal both in the m norm bound and the size n moreover we show the advantage of directly using the low rank tensor structure rather than matricization both theoretically and numerically specifically we show how the 1 bit measurement model can be used for context aware recommender systems

on the solution of diffusion convection equations by the space time finite element method a functional has been developed for the finite element solution of diffusion convection problems this functional is suitable for the application of the variational principle on discretization schemes in the space time domain this algorithm has shown to be computationally efficient over the conventional finite element discretization in the space domain alone numerical examples on one dimensional energy transport have been included to illustrate the merit of this technique

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

the problem of completeness for p mean symmetric difference metric in this paper by applying lebesgue s measure on the symmetric difference of sets the p mean symmetric difference metric d p is established to measure the difference between fuzzy numbers the basic properties of cauchy sequences of e1 d p are discussed and we show e1 k d p is complete where e1 k a a e1anda0 k k is a closed interval a0 is the closure of support set of a further an example where the space e1 d p is not complete is given

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

study on fuzzy optimization methods based on principal operation and inequity degree fuzzy optimization is a well known optimization problem in artificial intelligence manufacturing and management so establishing general and operable fuzzy optimization methods are important in both theory and application in this paper by distinguishing principal indices and secondary indices we give a method for comparing fuzzy information based on synthesizing effect and an operation for achieving fuzzy optimization based on a principal indices transformation further we propose an axiomatic system for fuzzy inequity degree based on the essence of constraint and give an instructive metric method for fuzzy inequity degree then by combining with genetic algorithm we give some fuzzy optimization methods based on principal operation and inequity degree denoted by bpo id fga for short finally we consider the convergence of our algorithm using the theory of markov chains and analyze its performance through two concrete examples all these indicate that bpo id fga can effectively merge decision preferences into the optimization process and that it also possesses better global convergence so it can be applied to many fuzzy optimization problems

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

a novel visual codebook model based on fuzzy geometry for large scale image classification the codebook model has been developed as an effective means for image classification however the inherent operation of assigning visual words to image feature vectors in traditional codebook approaches causes serious ambiguities in image classification in particular the nearest word may not be the best fit to a feature and multiple words may be equally appropriate for one specific feature to resolve these ambiguities we propose a novel visual codebook model based on the n dimensional fuzzy geometry n d fg theory where all visual words and features are modeled as fuzzy points in the n d fg space and appropriate uncertainty is introduced to each fuzzy point to enhance the representation capacity this n d fg codebook model not only inherits advantages from the fuzzy set theory but also facilitates the analysis and determination of the relationship between visual words and features in geometric form by explicitly taking into account the ambiguities we propose a novel measure of similarity between the visual words and fuzzy features following the proposed codebook model and the novel similarity measure we develop two useful image classification algorithms by modifying popular image coding algorithms i e spm and llc finally experimental results demonstrate that the classification accuracy of the proposed algorithms is dramatically improved for a standard large scale image database for example with a codebook size of 256 the proposed algorithms achieve similar performance as traditional algorithms with a codebook size of 1024 indicating that the proposed algorithms reduce the computational cost by 75 while achieving almost identical classification accuracy to traditional algorithms thus the proposed algorithms represent a more efficient and appropriate scheme for big image data this paper aims to overcome the drawbacks of traditional visual words fuzzy visual word and fuzzy feature are defined using n dimensional fuzzy geometry a new similarity measure between fuzzy features and fuzzy visual words is designed two modified image classification frameworks based on fuzzy codebook are proposed experimental results demonstrated their advantages against traditional algorithms

lipschitz functions and fuzzy number approximations we prove that some important properties of convex subsets of vector topological spaces remain valid in the space of fuzzy numbers endowed with the euclidean distance we use these results to obtain a characterization of fuzzy number valued lipschitz functions this fact helps us to find the best lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity introduced in a recent paper finally applications in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the value and ambiguity in the case when the direct formula is not applicable and an estimation for the defect of additivity of the trapezoidal approximation preserving the value and ambiguity are given

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

extending lattice structures on essaie d etendre des operations d une structure reticulee et de definir une sous classe d ensembles flous sur lesquels les extensions induisent un treillis

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism

can fuzzy neural nets approximate continuous fuzzy functions abstract we first argue that regular fuzzy neural nets based on standard fuzzy arithmetic and the extension principle cannot be universal approximators this negative result is surprising since regular neural nets are universal approximators we then argue that hybrid fuzzy neural nets not necessarily based on standard fuzzy arithmetic can be universal approximators