a fusion approach for managing multi granularity linguistic term sets in decision making the aim of this paper is to present a fusion approach of multi granularity linguistic information for managing information assessed in different linguistic term sets multi granularity linguistic term sets together with its application in a decision making problem with multiple information sources assuming that the linguistic performance values given to the alternatives by the different sources are represented in linguistic term sets with different granularity and or semantic in this context a decision process based on two steps is proposed with a view to obtaining the solution set of alternatives first the fusion of the multi granularity linguistic performance values is carried out in order to obtain collective performance evaluations in this step on the one hand the multi granularity linguistic information is made uniform using a linguistic term set as the uniform representation base the basic linguistic term set on the other hand the collective performance evaluations of the alternatives are obtained by means of an aggregation operator being fuzzy sets on the basic linguistic term set second the choice of the best alternative s from the collective performance evaluations is performed to do that a fuzzy preference relation is computed from the collective performance evaluations using a ranking method of pairs of fuzzy sets in the setting of possibility theory applied to fuzzy sets on the basic linguistic term set then a choice degree may be applied on the preference relation in order to rank the alternatives

some ideas about quantitative convergence of collision models to their mean field limit we consider a stochastic n particle model for the spatially homogeneous boltzmann evolution and prove its convergence to the associated boltzmann equation when n with non asymptotic estimates for any time t 0 we bound the distance between the empirical measure of the particle system and the measure given by the boltzmann evolution in a relevant hilbert space the control got is gaussian i e we prove that the distance is bigger than xn 1 2 with a probability of type o e x 2 the two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of lipschitz continuity for the boltzmann collision kernel we study more extensively the case where our hilbert space is the homogeneous negative sobolev space smash dot h s then we are only able to give bounds for maxwellian models however numerical computations tend to show that our results are useful in practice

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

a phase transition behavior for brownian motions interacting through their ranks consider a time varying collection of n points on the positive real axis modeled as exponentials of n brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time if at each time point we divide the points by their sum under suitable assumptions the rescaled point process converges to a stationary distribution depending on n and the vector of drifts as time goes to infinity this stationary distribution can be exactly computed using a recent result of pal and pitman the model and the rescaled point process are both central objects of study in models of equity markets introduced by banner fernholz and karatzas in this paper we look at the behavior of this point process under the stationary measure as n tends to infinity under a certain continuity at the edge condition on the drifts we show that one of the following must happen either i all points converge to 0 or ii the maximum goes to 1 and the rest go to 0 or iii the processes converge in law to a non trivial poisson dirichlet distribution the underlying idea of the proof is inspired by talagrand s analysis of the low temperature phase of derrida s random energy model of spin glasses the main result establishes a universality property for the bfk models and aids in explicit asymptotic computations using known results about the poisson dirichlet law

fuzzy modeling with tolerances abstract we consider the problem of the resolution of fuzzy equation systems a certain number of fuzzy inputs and outputs is given how can we determine a fuzzy relation which describes the behaviour of the object under consideration when attributing this problem to fuzzy equations in the classical sense it may happen that a solution does not exist therefore we introduce fuzzy tolerances as an extension of the solvability conception for this problem we get the whole solution set by means of posets moreover we consider the inverse problem how to determine tolerances to get consistency i e at least one solution for the arising equation system

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

fuzmar an approach to aggregating market research data based on fuzzy reasoning abstract we introduce an approach for validating models which contain variables taking linguistic values drawn from a scale which is a linear ordering this approach uses operations from fuzzy logic to construct the models and the principal of minimal entropy to choose between competing models we suggest the usefulness of this approach for market research surveys we implement this approach in a test case using economic attitudinal variables to predict the purchase of major consumer products

convergence to equilibrium in wasserstein distance for fokker planck equations we describe conditions on non gradient drift diffusion fokker planck equations for its solutions to converge to equilibrium with a uniform exponential rate in wasserstein distance this asymptotic behaviour is related to a functional inequality which links the distance with its dissipation and ensures a spectral gap in wasserstein distance we give practical criteria for this inequality and compare it to classical ones the key point is to quantify the contribution of the diffusion term to the rate of convergence which to our knowledge is a novelty

quantitative concentration inequalities for empirical measures on non compact spaces we establish quantitative concentration estimates for the empirical measure of many independent variables in transportation distances as an application we provide some error bounds for particle simulations in a model mean field problem the tools include coupling arguments as well as regularity and moment estimates for solutions of certain diffusive partial differential equations

modified log sobolev inequality for a compact pure jump markov process with degenerate jumps we study the modified log sobolev inequality for a class of pure jump markov processes that describe the interactions between brain neurons as a result we obtain concentration properties for empirical approximations of the process in particular we focus on a finite and compact process with degenerate jumps inspired by the model introduced by galves and locherbach j stat phys 151 896 921 2013

resolution of composite interval valued fuzzy relation equations abstract in this study we investigate the properties and the solution procedure of composite interval valued fuzzy relation equations with max min operator this is a generalization of existing constant valued fuzzy relation problems the study includes both theoretical development and numerical analyses it is concluded that our proposed methodology not only solves the existing fuzzy relation equations but also reveals the necessary and sufficient conditions for more general problems therefore this study facilitates the understanding of solution structure and wider applications

a similarity based bidirectional approximate reasoning method for decision making systems this paper presents a similarity based bidirectional approximate reasoning method which can express the decision maker s disposition for this we shall propose a new type of similarity measure based on the ordered weighted aggregation owa operator the proposed similarity measure has a structure to involve the disposition of a decision maker by choosing a suitable weighting vector of the owa operator from this property we derive a simple and flexible type of similarity based bidirectional approximate reasoning method which can support the making of a decision under more complex situations such as industrial inspection systems for example

anomalous diffusion in correlated continuous time random walks we demonstrate that continuous time random walks in which successive waiting times are correlated by gaussian statistics lead to anomalous diffusion with the mean squared displacement r2 t t2 3 long ranged correlations of the waiting times with a power law exponent 0 2 give rise to subdiffusion of the form r2 t t 1 in contrast correlations in the jump lengths are shown to produce superdiffusion we show that in both cases weak ergodicity breaking occurs our results are in excellent agreement with simulations

on the existence of minimal solutions for fuzzy equations with tolerances abstract we investigate the existence of minimal elements of solution sets for fuzzy equation problems with tolerances where the cardinality of the basic spaces of fuzzy inputs and outputs of an object need not be finite it turns out the semi continuity of applied membership functions plays a central role crisp models with fuzzy parameters are considered as well

inverse approximate reasoning abstract we consider the problem given the conclusion of a fuzzy rule find in some sense the best data input that will produce the given conclusion we investigate the use of the generalized modus tollens to solve this problem

a parametric model for fusing heterogeneous fuzzy data presented is a model that integrates three data types numbers intervals and linguistic assessments data of these three types come from a variety of sensors one objective of sensor fusion models is to provide a common framework for data integration processing and interpretation that is what our model does we use a small set of artificial data to illustrate how problems as diverse as feature analysis clustering cluster validity and prototype classifier design can all be formulated and attacked with standard methods once the data are converted to the generalized coordinates of our model the effects of reparameterization on computational outputs are discussed numerical examples illustrate that the proposed model affords a natural way to approach problems which involve mixed data types

convergence to the equilibria for self stabilizing processes in double well landscape we investigate the convergence of mckean vlasov diffusions in a nonconvex landscape these processes are linked to nonlinear partial differential equations according to our previous results there are at least three stationary measures under simple assumptions hence the convergence problem is not classical like in the convex case by using the method in benedetto et al j statist phys 91 1998 1261 1271 about the monotonicity of the free energy and combining this with a complete description of the set of the stationary measures we prove the global convergence of the self stabilizing processes

measuring fuzzy uncertainty first this paper reviews several well known measures of fuzziness for discrete fuzzy sets then new multiplicative and additive classes are defined we show that each class satisfies five well known axioms for fuzziness measures and demonstrate that several existing measures are relatives of these classes the multiplicative class is based on nonnegative monotone increasing concave functions the additive class requires only nonnegative concave functions some relationships between several existing and the new measures are established and some new properties are derived the relative merits and drawbacks of different measures for applications are discussed a weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced finally we comment on the construction of measures that may assess all of the uncertainties associated with a physical system

continuous time random walk with correlated waiting times based on the langevin description of the continuous time random walk ctrw we consider a generalization of ctrw in which the waiting times between the subsequent jumps are correlated we discuss the cases of exponential and slowly decaying persistent power law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time in the case of exponential type correlations the sub diffusion at short times is slower than in the absence of correlations at long times the behavior of the mean squared displacement is the same as in uncorrelated ctrw for power law correlations we find subdiffusion characterized by the same exponent at all times which appears to be smaller than the one in uncorrelated ctrw interestingly in the limiting case of an extremely long power law correlations the sub diffusion exponent does not tend to zero but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations

empirical processes indexed by lipschitz functions on obtient des conditions necessaires et suffisantes sur p pour que les boules unites de bl r et lip r soient des classes de p donsker fonctionnelles

properties of measures of information in evidence and possibility theories abstract an overview of information measures recently introduced by several authors in the settting of shafer s theory of evidence is proposed new results pertaining to additivity and monotonicity properties of these measures of information are presented the interpretation of each measure of information as opposed to others is discussed the potential usefulness of measures of specificity or imprecision is suggested and a principle of minimal specificity is stated for the purpose of reconstructing a body of evidence from incomplete knowledge

distances between transition probabilities of diffusions and applications to nonlinear fokker planck kolmogorov equations abstract we estimate the total variation and kantorovich distances between transition probabilities of two diffusions with different diffusion matrices and drifts via a natural quadratic distance between the drifts and diffusion matrices applications to nonlinear fokker planck kolmogorov equations optimal control and mean field games are given

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

inverse approximate reasoning ii maximize entropy abstract we consider the problem given the conclusion of a fuzzy rule find in some sense the best data input that will produce the given conclusion in this paper we investigate the use of the principle of maximum entropy to solve this problem

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

separation and measures of fuzziness abstract the concept of separation as a specifical notion in the setting of fuzzy sets was first introduced in the case of a finite referential see dujet 1 main results are summarized here this concept is then generalized to the case of any referential finite or not by means of fuzzy integrals this study although still leaving open questions is aimed to show the interest of this separation notion and derived ones which particularly leads to measures of fuzziness not related to some previously defined negation and moreover open the way to define non pointwise negation

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

free transport entropy inequalities for non convex potentials and application to concentration for random matrices talagrand s inequalities provide a link between two fundamentals concepts of probability transportation and entropy the study of the counterpart of these inequalities in the context of free probability has been initiated by biane and voiculescu and later extended by hiai petz and ueda for convex potentials in this work we prove a free analogue of a result of bobkov and gotze in the classical setting thus providing free transport entropy inequalities for a very natural class of measures appearing in random matrix theory these inequalities are weaker than the ones of hiai petz and ueda but still hold beyond the convex case we then use this result to get a concentration estimate for beta ensembles under mild assumptions on the potential

the influence of the finite velocity on spatial distribution of particles in the frame of levy walk model abstract levy walk at the finite velocity is considered to analyze the spatial and temporal characteristics of this process the method of moments has been used the asymptotic distributions of the moments at t have been obtained for n dimensional case where the free path of particles demonstrates the power law distribution p x x 0 x 1 x 0 2 the three regimes of distribution have been distinguished ballistic diffusion and asymptotic introduction of the finite velocity requires considering of two problems propagation with distribution at the finite mathematical expectation of the free path 1 2 and propagation with distribution at the infinite mathematical expectation of the free path of the particle 0 1 in the case 1 2 the asymptotic distribution is described by the levy stable law and the effect of the finite velocity is reduced to a decrease of diffusivity at 0 1 the situation is quite different here the asymptotic distribution exhibits a u or w shape and is described as the ballistic regime of distribution the obtained moments allow to reconstruct the distribution densities of particles in one dimensional and three dimensional cases

space time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model abstract in this paper we consider a type of continuous time random walk model where the jump length is correlated with the waiting time the asymptotic behaviors of the coupled jump probability density function in the fourier laplace domain are discussed the corresponding fractional diffusion equations are derived from the given asymptotic behaviors corresponding to the asymptotic behaviors of the joint probability density function in the fourier laplace space the asymptotic behaviors of the waiting time probability density and the conditional probability density for jump length are also discussed

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

incorporating statistical model error into the calculation of acceptability prices of contingent claims the determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics given this model optimal bid and ask prices can be found by stochastic optimization however the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure we define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model this neighborhood serves as ambiguity set for a multistage stochastic optimization problem under model uncertainty we obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation moreover we prove a general large deviations result for the nested distance which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data

convergence rates for empirical barycenters in metric spaces curvature convexity and extendable geodesics this paper provides rates of convergence for empirical generalised barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry we study the validity of variance inequalities in spaces of non positive and non negative aleksandrov curvature in this last scenario we show that variance inequalities hold provided geodesics emanating from a barycenter can be extended by a constant factor we also relate variance inequalities to strong geodesic convexity while not restricted to this setting our results are largely discussed in the context of the 2 wasserstein space

higher order fuzzy entropy and hybrid entropy of a set new entropy measures such as higher order fuzzy entropy and hybrid entropy of a set have been introduced various properties along with their proofs have been included applicability of these new measures to various problems has been highlighted

the wowa operator and the interpolation function w chen and otto s interpolation method revisited abstract chen and otto introduced in 1995 an interpolation method for defining membership functions from a set of points in this paper we report some errors found in that algorithm and introduce some new conditions on the boundaries that are more general than the ones in the original work we construct in this way a new interpolation method and prove that it has a smooth shape on the boundaries this study is related to the wowa operator an aggregation function that generalizes both the weighted mean and the owa operator wowa uses an interpolation method to calculate a vector of weights as a combination of other two vectors the new method is introduced to implement this combination

similarity based inverse approximate reasoning this paper investigates the inverse problem of ap proximate reasoning to conclude a from b and a b from different angles we develop a method to and a solution to the inverse problem by the use of the law of contrapositive symme try in the process generalized modus ponens generalized modus tollens and their generating functions are studied extensively an attempt is made here to establish that the concept of similarity plays an important role in inverse approximate reasoning we apply the concept of similarity to model inverse approximate reasoning and develop a new methodology that is named similarity based inverse approximate reasoning for a solution to the problem algorithms are presented and illustrated with simple yet concrete examples we then use zadeh s compositional rule of inference cri as a combi nation of cylindrical extension and projection and thereby model generalized modus tollens in a different way we thus develop a third way to find a solution to the inverse problem of approximate reasoning a comparison between the result of application of our technique and the real static characteristic of a dc shunt motor is presented

on kac s chaos and related problems this paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by m kac cite kac1956 in his study of mean field limit for systems of n undistinguishable particles as n to infty first we quantitatively liken three usual measures of it kac s chaos some involving the all n variables other involving a finite fixed number of variables the cornerstone of the proof is a new representation of the monge kantorovich wasserstein mkw distance for symmetric n particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand and a new estimate on some mkw distance on probability measures spaces endowed with a suitable hilbert norm taking advantage of the associated good algebraic structure next we define the notion of it entropy chaos and it fisher information chaos in a similar way as defined by carlen et al cite ccllv we show that it fisher information chaos is stronger than it entropy chaos which in turn is stronger than it kac s chaos more importantly with the help of the hwi inequality of otto villani we establish a quantitative estimate between these quantities which in particular asserts that it kac s chaos plus it fisher information bound implies it entropy chaos we then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the it kac s spheres revisiting cite ccllv and giving a possible answer to cite open problem 11 ccllv additionally to the above mentioned tool we use and prove an optimal rate local clt in l infty norm for distributions with finite 6 th moment and finite l p norm for some p 1 last we investigate how our techniques can be used without assuming chaos in the context of probability measures mixtures introduced by de finetti hewitt and savage in particular we define the level 3 fisher information for mixtures and prove that it is l s c and affine as that was done in cite rr for the level 3 boltzmann s entropy

phase diagram in stored energy driven levy flight phase diagram based on the mean square displacement msd and the distribution of diffusion coefficients of the time averaged msd for the stored energy driven levy flight sedlf is presented in the sedlf a random walker cannot move while storing energy and it jumps by the stored energy the sedlf shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion depending on the coupling parameter between storing time trapping time and stored energy this stochastic process can be investigated analytically with the aid of renewal theory here we consider two different renewal processes i e ordinary renewal process and equilibrium renewal process when the mean trapping time does not diverge we analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping time distribution in particular we find that distributional behavior of time averaged msd intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge

nonprobabilistic entropies and indetermination measures in the setting of fuzzy sets theory abstract our purpose in this paper is first of all to build an axiomatic generalization for the nonprobabilistic entropy of de luca and termini in the setting of fuzzy sets theory we then build from this entropy an indetermination measure which can be used like discriminant function in pattern recognition when patterns are described by means of fuzzy sets

interval valued fuzzy backward reasoning the importance and efficiency of backward reasoning in nonfuzzy reasoning has been stressed for a long time especially in the case of expert systems and decision support systems the extension of this reasoning method to fuzzy theory however has never been considered in this paper the authors propose a definition of fuzzy backward reasoning based on the generalized modus ponens and show the necessity of considering interval valued fuzzy backward reasoning then the authors propose solving methods for fuzzy backward reasoning in the case of a rule with one or several conditions as well as in the case of several rules

inverse problem in fuzzy relational equations abstract the paper deals with the problem of handling fuzzy relational equations of the form x r y in the case of an empty solution set with respect to x x r y discussing the essence of the structure of the fuzzy relation r with respect to provided y algorithms have been proposed which enable us to modify the original equation i e the fuzzy set y in a way leading to its genuine solution or at least an approximate one it is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation a grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set y which forms a cornerstone of the proposed procedures numerical examples are provided some aspects related to the area of diagnostic models in which these equations are studied are also treated

a computer algorithm for the solution of the inverse problem of fuzzy systems abstract a computer algorithm based on a method for dealing with the so called inverse problem is presented the algorithm produces the solution semi lattice formed by the fuzzy subsets a u satisfying the fuzzy relational equation a r b where denotes maxmin composition given a fuzzy relation r u v and a fuzzy subset b v

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

entropy operator for membership function of uncertain set abstract similar to fuzzy set on a possibility space uncertain set is a set valued function on an uncertainty space and attempts to model unsharp concepts entropy provides a quantitative measurement of the uncertainty associated with an uncertain set this paper presents a formula for calculating the entropy of an uncertain set via its inverse membership function based on the formula the entropy operator is shown to satisfy positive linearity property in addition this paper proposes a concept of relative entropy to describe the divergence between the membership functions of two uncertain sets

a fuzzy approach to the evaluation of image complexity the inherently multidimensional problem of evaluating the complexity of an image is of a certain relevance in both computer science and cognitive psychology computer scientists usually analyze spatial dimensions in order to deal with automatic vision problems such as feature extraction psychologists seem more interested in the temporal dimension of complexity as a means to explore attentional models is it possible to define by merging both approaches a more general index of visual complexity the aim of this paper is the definition of objective measures of image complexity that fits with the so named perceived time towards the end we have defined a fuzzy mathematical model of visual complexity based on fuzzy measures of entropy the results obtained by applying this model to a set of pictorial images present a strong correlation with the outcomes of an experiment with human subjects based on variation of subjective temporal estimations associated with changes in visual attentional load which is also described herein

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

nonergodic property of the space time coupled ctrw dependence on the long tailed property and correlation abstract the space time coupled continuous time random walk model is a stochastic framework of anomalous diffusion with many applications in physics geology and biology in this manuscript the time averaged mean squared displacement and nonergodic property of a space time coupled continuous time random walk model is studied which is a prototype of the coupled continuous time random walk presented and researched intensively with various methods the results in the present manuscript show that the time averaged mean squared displacements increase linearly with lag time which means ergodicity breaking occurs besides we find that the diffusion coefficient is intrinsically random which shows both aging and enhancement the analysis indicates that the either aging or enhancement phenomena are determined by the competition between the correlation exponent and the waiting time s long tailed index

where do we stand on measures of uncertainty ambiguity fuzziness and the like abstract it is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty each of these types requires a measure by which the degree of uncertainty of that type can be determined two main categories of uncertainty are connected with the terms vagueness or fuzziness and ambiguity in general vagueness is associated with the difficulty of making sharp or precise distinctions in the world ambiguity on the other hand is associated with one to many relations i e situations with two or more alternatives that are left unspecified while the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness the concept of a fuzzy measure is a general framework for dealing with ambiguity several classes of measures of vagueness usually referred to as measures of fuzziness have been proposed in the literature each class is based on some underlying conception of the degree of fuzziness a general set of requirements for measures of fuzziness is formulated followed by an overview of the measures proposed in the literature measures of ambiguity are discussed within the framework of plausibility and belief measures although it does not cover all fuzzy measures this framework is sufficiently broad for most practical purposes and represents a generalization of both probability theory and possibility theory it is argued that three complementary measures of ambiguity should be employed one of them is obtained by generalizing the hartley measure of uncertainty it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures the other two are obtained by generalizing the well known shannon measure of uncertainty they measure the degree of dissonance and the degree of confusion in evidence respectively basic mathematical properties of these measures are overviewed it is also argued that each of the four types of uncertainty measures which are fundamentally different from each other can be used for measuring structural syntactic information in the same sense as the hartley and shannon measures have been used in this respect as such these measures are potentially powerful tools for dealing with systems problems such as systems modelling analysis simplification or design

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

uncertainty measures for general fuzzy relations abstract relation is one of the important concepts in rough set models some uncertainty measures for relations were proposed in recent years by generalizing information entropy however the existing theories considered only special relations and do not address mixed type data well they always transformed each attribute into a binary relation with the same structure regardless of attribute types as a result the structure information can be lost for other types of attributes in this paper a new theory about uncertainty measures for general fuzzy relations is proposed the proposed theory can overcome the weakness of the existing theories we first introduce a novel entropy to compute the uncertainty information of fuzzy binary relations and then present the concepts of joint fuzzy entropy conditional fuzzy entropy and fuzzy mutual information the basic properties of these uncertainty measures are studied finally we apply the proposed theory to perform attribute reduction of heterogeneous data sets the experimental results show that our proposed method is feasible and effective in some cases the proposed theory can provide a fundamental framework that integrates all kinds of methods for the uncertainty information of special relations

probabilistic approach for granular media equations in the non uniformly convex case we use here a particle system to prove both a convergence result with convergence rate and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex the proof of convergence is simpler than the one in carrillo mccann villani rev mat iberoamericana 19 971 1018 2003 arch rat mech anal 179 217 263 2006 all the results complete former results of malrieu ann appl probab 13 540 560 2003 in the uniformly convex case the main tool is an uniform propagation of chaos property and a direct control in wasserstein distance of solutions starting with different initial measures the deviation inequality is obtained via a t 1 transportation cost inequality replacing the logarithmic sobolev inequality which is no more clearly dimension free

random walks with infinite spatial and temporal moments the continuous time random walk of montroll and weiss has been modified by scher and lax to include a coupled spatial temporal memory we treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial and temporal moments of the memory are infinite the asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated the resulting behavior includes localized diffusive wavelike and levy s stable laws for the appropriate scaled variable we show that an infinite mean waiting time can lead to long time diffusive behavior while a finite mean waiting time is not sufficient to ensure the same

on direct and inverse problems for fuzzy equation systems with tolerances abstract we investigate the resolution of fuzzy relational equation systems with tolerances which are a certain extension of fuzzy equations considered f i in 3 5 the extension of the concept of higashi and klir 3 enables us to describe the set of solutions to our problem for given tolerances by means of posets in a second part we investigate an inverse problem given upper lower tolerances how to determine lower upper tolerances such that the arising problem becomes consistent numerical examples are given

probability distance inequalities on riemannian manifolds and path spaces abstract we construct otto villani s coupling for general reversible diffusion processes on a riemannian manifold as an application some new estimates are obtained for wasserstein distances by using a sobolev poincare type inequality introduced by latala and oleszkiewicz the corresponding concentration estimates of the measure are presented finally our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the l2 distance and the intrinsic distance in particular talagrand s inequality holds on the path space over a compact manifold

nonlinear stochastic programming with a case study in continuous switching the optimal solution as well as the objective of stochastic programming problems vary with the underlying probability measure this paper addresses stability with respect to the underlying probability measure and stability of the objective the techniques presented are employed to make problems numerically tractable which are formulated by involving numerous scenarios or even by involving a continuous probability measure the results justify clustering techniques which significantly reduce computation times while guaranteeing a desired approximation quality the second part of the paper highlights newton s method to solve the reduced stochastic recourse problems the techniques presented exploit the particular structure of the recourse function of the stochastic optimization problem the tools are finally demonstrated on a benchmark problem which is taken from electrical power flows

solvability of fuzzy relational equations and manipulation of fuzzy data abstract the problem of characterization of fuzzy relational equations with respect to their solvability property is studied some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered two stages of manipulation process are indicated combining pieces of evidence and inferring their mutual correspondence both of them are formulated and solved by the use of fuzzy relational equations the role of a solvability index is extensively studied numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers

fuzzy relation equations and inequalities abstract this paper generalizes a method of solution of fuzzy relation equations introduced by sanchez 1976 and developed by pedrycz 1982 we find the greatest or the least solution of a composite l fuzzy relation equation with arbitrary monotone composition

some new information measures for fuzzy sets abstract after reviewing some existing measures for fuzzy sets we introduce a new informative measure for discrimination between two fuzzy sets this discriminating measure reduces to the nonprobabilistic entropy of deluca and termini 7 under a special condition the divergence measure between two sets has been defined along with a large set of properties it has also been used to define an ambiguity fuzziness measure renyi s 17 probabilistic entropy of order has been extended to define nonprobabilistic entropy of a fuzzy set various properties of this definition have also been proved applications of these measures to clustering image processing vision etc are highlighted

a large deviation approach to some transportation cost inequalities new transportation cost inequalities are derived by means of elementary large deviation reasonings their dual characterization is proved this provides an extension of a well known result of s bobkov and f gotze their tensorization properties are investigated sufficient conditions and necessary conditions too for these inequalities are stated in terms of the integrability of the reference measure applying these results leads to new deviation results concentration of measure and deviations of empirical processes

further relation equations with equality and difference composition operators in this ch we will introduce other composition operators which have a plausible logical interpretation they will be called equality and difference operator respectively we first define a notion of equality and difference of any two grades of membership of a fuzzy set and of a fuzzy relation in the sequel this concept will be utilized to form the respective composition operators and the related fuzzy equations afterwards we provide a method of resolution of these equations characterizing completely the set of the solutions all the fuzzy sets involved are defined on finite sets and with membership values in 0 1

continuous time random walk with jump length correlated with waiting time abstract a coupled continuous time random walk ctrw model is proposed in which the jump length of a walker is correlated with waiting time the power law distribution is chosen as the probability density function of waiting time and the gaussian like distribution as the probability density function of jump length normal diffusion subdiffusion and superdiffusion can be realized within the present model it is shown that the competition between long tailed distribution and correlation of jump length and waiting time will lead to different diffusive behavior

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

type 1 owa operators for aggregating uncertain information with uncertain weights induced by type 2 linguistic quantifiers the owa operator proposed by yager has been widely used to aggregate experts opinions or preferences in human decision making yager s traditional owa operator focuses exclusively on the aggregation of crisp numbers however experts usually tend to express their opinions or preferences in a very natural way via linguistic terms these linguistic terms can be modelled or expressed by type 1 fuzzy sets in this paper we define a new type of owa operator the type 1 owa operator that works as an uncertain owa operator to aggregate type 1 fuzzy sets with type 1 fuzzy weights which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights the procedure for performing type 1 owa operations is analysed in order to identify the linguistic weights associated to the type 1 owa operator type 2 linguistic quantifiers are proposed the problem of how to derive linguistic weights used in type 1 owa aggregation given such type of quantifier is solved examples are provided to illustrate the proposed concepts

a rule based method to calculate exactly the widest solution sets of a max min fuzzy relational inequality abstract in this paper we propose a rule based method to calculate exactly all the widest solution sets of a fuzzy relational inequality viewed as an extension of fuzzy relational equations fuzzy relational inequalities can be applied to many fields of fuzzy logic yet they provide with more flexibility as they enable to get rid of the sensitivity of fuzzy relational equations to numerical values up to now however fuzzy relational inequalities have received few attention from researchers and few methods have been proposed to solve these inequalities the proposed analytical methods furthermore do not calculate exactly all the widest solutions sets of a fuzzy relational inequality and often lead to redundant solution sets the rule based method we propose in this paper determines easily exactly all the widest solution sets of a fuzzy relational inequality