transient fokker planck kolmogorov equation solved with smoothed particle hydrodynamics method summary probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function pdf the time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient fokker planck kolmogorov fpk equation the fpk equation is a conservation equation of a hypothetical or abstract fluid which models the transport of probability this paper presents a generalized formalism for the resolution of the transient fpk equation by using the well known mesh free lagrangian method smoothed particle hydrodynamics numerical implementation shows notable advantages of this method in an unbounded state space 1 the conservation of total probability in the state space is explicitly written 2 no artifact is required to manage far field boundary conditions 3 the positivity of the pdf is ensured and 4 the extension to higher dimensions is straightforward furthermore thanks to the moving particles this method is adapted for a large kind of initial conditions even slightly dispersed distributions the fpk equation is solved without any a priori knowledge of the stationary distribution just a precise representation of the initial distribution is required copyright 2013 john wiley sons ltd

continuity of quantum channel capacities we prove that a broad array of capacities of a quantum channel are continuous that is two channels that are close with respect to the diamond norm have correspondingly similar communication capabilities we first show that the classical capacity quantum capacity and private classical capacity are continuous with the variation on arguments varepsilon apart bounded by a simple function of varepsilon and the channel s output dimension our main tool is an upper bound of the variation of output entropies of many copies of two nearby channels given the same initial state the bound is linear in the number of copies our second proof is concerned with the quantum capacities in the presence of free backward or two way public classical communication these capacities are proved continuous on the interior of the set of non zero capacity channels by considering mutual simulation between similar channels

squashed entanglement in infinite dimensions we analyse two possible definitions of the squashed entanglement in an infinite dimensional bipartite system direct translation of the finite dimensional definition and its universal extension it is shown that the both definitions produce the same lower semicontinuous entanglement measure possessing all basis properties of the squashed entanglement on the set of states having at least one finite marginal entropy it is also shown that the second definition gives an adequate lower semicontinuous extension of this measure to all states of the infinite dimensional bipartite system a general condition relating continuity of the squashed entanglement to continuity of the quantum mutual information is proved and its corollaries are considered continuity bound for the squashed entanglement under the energy constraint on one subsystem is obtained by using the tight continuity bound for quantum conditional mutual information proved in the appendix by using winter s technique it is shown that the same continuity bound is valid for the entanglement of formation as a result the asymptotic continuity of the both entanglement measures under the energy constraint on one subsystem is proved

a comparative study of the discontinuous galerkin and continuous supg finite element methods for computation of viscoelastic flows the performance of the discontinuous galerkin dg method and the continuous streamline upwind petrov galerkin supg method in the computation of steady viscoelastic flows have been compared in this paper for the continuous supg method it is necessary to include the elastic viscous split stress evss formulation to satisfy the compatibility requirement on the velocity stress interpolation supg evss while for the discontinuous galerkin method both the pure mixed formulation dg mix and the evss formulation dg evss have been tested we first examine a benchmark smooth problem i e the steady flow of an upper convected maxwell ucm fluid around a sphere falling along the axis of a cylindrical tube by using high order finite element approximation dg mix is found not to be a stable discretization in this problem its limiting weissenberg number decreases with increasing the interpolation order on the other hand dg evss is an accurate and stable algorithm its solution converges as the approximation order increases the so called p convergence and it produces almost identical stress boundary layers to that of the supg evss algorithm at a weissenberg number as high as 2 0 to explore benefits of the discontinuous interpolation for the stress in the dg method we further examine two benchmark problems with stress singularities by using low order interpolations on sufficiently fine meshes they are the axisymmetric steady flows of the modified chilcott and rallison mcr fluid through a four to one abrupt contraction and in a tube with the stick to slip transition in the wall velocity boundary condition in the contraction problem the dg evss method is found to be a significant improvement over the supg evss method quality of the solutions is better for dg evss and a much higher weissenberg number limit is observed in the stick slip problem both dg evss and supg evss are very robust they confront no limit of the weissenberg number and predict asymptotic stress profiles as the number becomes sufficiently large in both problems the pure mixed method dg mix produces severe stress oscillations near the singularities and fails to converge at medium weissenberg numbers

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

local existence for the fene dumbbell model of polymeric fluids we study the well posedness of a multi scale model of polymeric fluids the microscopic model is the kinetic theory of the finitely extensible nonlinear elastic fene dumbbell model the macroscopic model is the incompressible non newton fluids with polymer stress computed via the kramers expression the boundary condition of the fene type fokker planck equation is proved to be unnecessary by the singularity on the boundary other main results are the local existence uniqueness and regularity theorems for the fene model in certain parameter range

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra

a fuzzy gould type integral in this paper we introduce and study a set valued gould type integral of a real function with respect to a fuzzy multimeasure taking values in the family of all nonempty closed subsets of a banach space x some classical properties of the integral are obtained only using the hypothesis of monotonicity for the set valued fuzzy measure

unified entropy entanglement measures and monogamy of multi party entanglement we show that restricted shareability of multi qubit entanglement can be fully characterized by unified q s entropy we provide a two parameter class of bipartite entanglement measures namely unified q s entanglement with its analytic formula in two qubit systems for q 1 0 s 1 and qs 3 using unified q s entanglement we establish a broad class of the monogamy inequalities of multi qubit entanglement for q 2 0 s 1 and qs 3

some quantities represented by the choquet integral this paper shows that the choquet integral can represent some useful quantities such as supremum infimum essential supremum essential infimum mean median quantile 100 percentile and l estimators including trimmed mean

a note on convergence properties of interval valued capacity functionals and choquet integrals based on the classical theory of random sets feng and nguyen 2007 5 studied the convergence of capacity functionals of random sets in terms of choquet integrals in this paper we consider an interval valued capacity functional which is motivated by the goal to generalize a capacity functional and the choquet integral with respect to an interval valued capacity in particular we discuss some convergence theorems for interval valued capacity functionals and interval valued probability measures in the hausdorff metric and choquet weak sense

non monotonic fuzzy measures and the choquet integral abstract this paper discusses non monotonic fuzzy measures which are set functions without monotonicity and the choquet integral with respect to non monotonic fuzzy measures a concrete example shows that the choquet integral is meaningful in the case of non monotonic fuzzy measures as well as in the case of ordinary fuzzy measures every comonotonically additive positively homogeneous functional of bounded variation can be represented as a choquet integral with respect to a non monotonic fuzzy measure of bounded variation the space of such functionals is a real banach space isometrically isomorphic to the space of non monotonic fuzzy measures of bounded variation

viscoelastic flow past a cylinder drag coefficient abstract the channel flow of a viscoelastic fluid past a circular cylinder is simulated using a parallelized unstructured finite volume method fvm h s dou n phan thien j non newtonian fluid mech 77 1998 21 51 under a distributed computing environment through the parallel virtual machine pvm libraries the numerical method is based on the simpler semi implicit method for pressure linked equations revised algorithm using the simplified phan thien tanner ptt and the upper convective maxwell ucm constitutive models with the elastic viscous split stress evss formulation the parallelization of the program is implemented by a domain decomposition strategy and the pvm software it is found that the both shear thinning and normal stress contribute to the drag reduction with the ptt model the drag reduction is mainly due to shear thinning with the ucm model it is primarily due to the normal stress effect

on the frechet derivative of matrix functions abstract in 1956 rinehart 4 discussed the derivatives of matrix functions by considering differences a e a for matrices e commuting with a in that case the derivative turned out to be a in this paper the case of noncommutative a and e is treated this leads to the frechet derivative of the matrix function an explicit integral representation is obtained using an approach that is similar to the one in 5 a finite sum in polynomials of a is obtained the coefficients may be computed recursively this is useful in computing the frechet derivative which is needed for newton s method to solve nonlinear matrix equations

extension and representation of comonotonically additive functionals the domain of comonotonically additive and monotone functionals on the class of continuous functions with compact support is extended in a natural way a necessary and sufficient condition for which the extended functional is comonotonically additive and monotone is presented it is shown that the extended functional can be represented by a choquet integral when the extended functional is comonotonically additive and monotone

regular fuzzy measure and representation of comonotonically additive functional this paper discusses the functional i defined on the class of continuous functions which is comonotonically additive and monotone the notion of regular fuzzy measure is proposed and the uniqueness theorem of regular fuzzy measure is shown it is also shown that i can be represented by the difference of two choquet integrals with respect to regular fuzzy measures when the universal set x is a locally compact hausdorff space and that i can be represented by one choquet integral with respect to a regular fuzzy measure when x is a compact hausdorff space

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

on the representation of choquet integrals of set valued functions and null sets in this paper we consider the definition of set valued choquet integrals we obtain some properties and the representation of set valued choquet integrals and also we discuss some fundamental relations between null sets and set valued choquet integrals

one shot classical data compression with quantum side information and the distillation of common randomness or secret keys the task of compressing classical information in the one shot scenario is studied in the setting where the decompressor additionally has access to some given quantum side information in this hybrid classical quantum version of the famous slepian wolf problem the smooth max entropy is found to govern the number of bits into which classical information can be compressed so that it can be reliably recovered from the compressed version and quantum side information combining this result with known results on privacy amplification then yields tight bounds on the amount of common randomness and secret key that can be recovered in one shot from hybrid classical quantum systems using one way classical communication

continuity of quantum conditional information we prove continuity of quantum conditional information s 12 2 with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party this can e g be used to prove the continuity of squashed entanglement

tight uniform continuity bounds for the quantum conditional mutual information for the holevo quantity and for capacities of quantum channels we start with fannes type and winter s type tight uniform continuity bounds for the quantum conditional mutual information and their specifications for states of special types then we analyse continuity of the holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states we show that the holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying hilbert space is finite and obtain fannes type tight continuity bounds for the holevo quantity in this case in the general case conditions for local continuity of the holevo quantity for discrete and continuous ensembles are found winter s type tight continuity bound for the holevo quantity under constraint on the average energy of ensembles is obtained and applied to the system of quantum oscillators the above results are used to obtain tight and close to tight continuity bounds for basic capacities of finite dimensional chann

a sharp continuity estimate for the von neumann entropy we derive an inequality relating the entropy difference between two quantum states to their trace norm distance sharpening a well known inequality due to fannes in our inequality equality can be attained for every prescribed value of the trace norm distance

the general pettis sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure abstract in this paper we define and study the general pettis sugeno integral of a vector multifunction relative to a vector fuzzy multimeasure both taking values in a locally convex space x ordered by a closed convex pointed cone x with nonempty interior several classic properties of this integral and some estimation and comparative results with the pettis lebesgue integral are established

maximum and minimum entropy states yielding local continuity bounds given an arbitrary quantum state we obtain an explicit construction of a state e respectively e which has the maximum respectively minimum entropy among all states which lie in a specified neighborhood e ball of computing the entropy of these states leads to a local strengthening of the continuity bound of the von neumann entropy i e the audenaert fannes inequality our bound is local in the sense that it depends on the spectrum of the states e and e depend only on the geometry of the e ball and are in fact optimizers for a larger class of entropies these include the renyi entropy and the minimum and maximum entropies providing explicit formulas for certain smoothed quantities this allows us to obtain local continuity bounds for these quantities as well in obtaining this bound we first derive a more general result which may be of independent interest namely a necessary and sufficient condition under which a state maximizes a concave

extremal distributions under approximate majorization although an input distribution may not majorize a target distribution it may majorize a distribution which is close to the target here we consider a notion of approximate majorization for any distribution and given a distance we find the approximate distributions which majorize are majorized by all other distributions within the distance we call these the steepest and flattest approximation this enables one to compute how close one can get to a given target distribution under a process governed by majorization we show that the flattest and steepest approximations preserve ordering under majorization furthermore we give a notion of majorization distance this has applications ranging from thermodynamics entanglement theory and economics

a note on the interval valued generalized fuzzy integral by means of an interval representable pseudo multiplication and their convergence properties in this paper we consider the generalized fuzzy integral which is a fuzzy integral based on the idempotent pseudo addition supremum and a pseudo multiplication studied by xie and fang in 2006 the purpose of this study is to define the interval valued generalized fuzzy integral with respect to a fuzzy measure by means of an interval representable pseudo multiplication of measurable interval valued functions and to investigate some characterizations and convergence properties of them

least squares dynamic approximation method for evolution of uncertainty in initial conditions of dynamical systems we describe an approximation method to solve the probability density function transport equation i e the liouville equation which is encountered in the evolution of uncertainty of the initial values of dynamical systems a state space based method is formulated using a least squares technique that preserves the parabolic nature of the liouville equation and is flexible in terms of accuracy of representation this method is based on a global approximation in terms of analytical elementary functions with unknown parameters whose evolution equations are determined by a global least squares approximation the realizability conditions of the probability density i e the non negativity and normalization conditions are enforced at all times the method is successfully evaluated in a number of scenarios including the uncertainty evolution in a system governed by a riccati equation and a particle moving in a fluid under the influence of stokes drag force the results obtained in our examples exhibit a reasonable good agreement when compared with the solution of the probability transport equation using the method of characteristics the cost of the method is proportional to the cost of solving the deterministic system and the number of parameters used to approximate the probability density function a feature that can make the present method very advantageous in comparison with other methods in problems involving a large number of dimensions

fuzzy integrals of fuzzy valued functions abstract this paper based on the fuzzy measures and integrals given by sugeno first defines fuzzy integrals of set valued functions and then the fuzzy integral of fuzzy valued functions is obtained and discussed next it defines the fuzzy integrals of fuzzy mappings then pays much effort to the fuzzy integral of interval valued functions the properties and convergence theorems are obtained all these are generalizations of earlier results

a new spectral element method for the reliable computation of viscoelastic flow abstract a new stabilised spectral element method has been developed by the authors for the accurate integration of the mixed elliptic hyperbolic system of partial differential equations governing certain viscoelastic flows the method is illustrated by solving the benchmark problem of the flow of an oldroyd b and a ptt fluid past a cylinder in a channel results are presented to demonstrate the advantages of the proposed method over traditional galerkin type procedures in terms of accuracy and stability

maximum principle satisfying third order discontinuous galerkin schemes for fokker planck equations we design and analyze up to third order accurate discontinuous galerkin dg methods satisfying a strict maximum principle for fokker planck equations a procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution this is achievable by taking advantage of the two parameters in the numerical flux and a novel decomposition of weighted cell averages based on this result a scaling limiter for the dg method with first order euler forward time discretization is proposed to solve the one dimensional fokker planck equations strong stability preserving high order time discretizations will keep the maximum principle it is straightforward to extend the method to two and higher dimensions on rectangular meshes we also show that a modified limiter can preserve the strict maximum principle for dg schemes solving fokker planck equations as a consequence the present schemes preserve steady states numerical tests

a fast solver for fokker planck equation applied to viscoelastic flows calculations 2d fene model in this paper we introduce a new method for solving efficiently the fokker planck equation arising in the simulation of dilute polymeric solutions for a two dimensional fene finitely extensible nonlinear elastic model the structure of the fokker planck equation is used to design a fast solver the resulting equations are discretized with a spectral spectral element method application of the method to flow of a fene fluid past a confined cylinder demonstrates readily the advantages of the proposed scheme over traditional stochastic simulations

space of fuzzy measures and convergence the convergences of the net generalized sequence of fuzzy measures are discussed it is shown that three types of convergence are not equivalent in general case however they are equivalent if the universal set is finite

atoms of fuzzy measures and fuzzy integrals abstract some analytical properties of sugeno s fuzzy measures especially of atoms of fuzzy measures are investigated a fuzzy integral of riemann type is defined through atoms and it is proved that a continuous function is fuzzy integrable in the sense of riemann and has the same integral as that of sugeno

fuzzy integrals of set valued mappings and fuzzy mappings abstract the paper based on sugeno 1974 first defines the concept of fuzzy integrals of set valued mappings by the similar way of aumann 1965 then it is discussed some results similar to aumann integral are obtained these include convexity closedness fatou s lemmas and the lebesgue convergence theorem etc at last the fuzzy integral of fuzzy mappings is defined the extended results corresponding with fuzzy integrals of set valued mappings are showed

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

monotone set functions defined by choquet integral abstract given a nonnegative monotone set function and a nonnegative measurable function on a measurable space the choquet integral determines a new nonnegative monotone set function that is absolutely continuous with respect to the original one in a generalized sense for monotone set functions this new set function preserves almost all desirable structural characteristics of the original monotone set function such as continuity subadditivity superadditivity null additivity converse null additivity autocontinuity converse autocontinuity uniform autocontinuity uniform converse autocontinuity and fuzzy multiplicativity such a construction is a useful method to define sound fuzzy measures in various applications

continuity bounds on the quantum relative entropy ii the quantum relative entropy is frequently used as a distance measure between two quantum states and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory we have derived many such inequalities in previous work the present paper is a follow up on this and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned the result obtained here is more general than the corresponding one from our previous work as a corollary we obtain a sharp upper bound on the regularised relative entropy introduced by lendi farhadmotamed and van wonderen

aggregating membership values by a choquet fuzzy integral based operator a new type of aggregator the choquet fuzzy integral based operator is introduced and applied in aggregation of membership values in this paper we generalized the definition of the fuzzy integral as a tool for membership aggregation the generalization involves replacing the max or min operator in information integration with a choquet fuzzy integral based aggregator resulting in increased flexibility the characteristics of the choquet fuzzy integral are studied and a two phase aggregating process is proposed also we demonstrate a good classification result using the proposed aggregator to integrate the memberships of several clusters in a complex unconstrained handwritten digit recognition domain

on set valued fuzzy integrals abstract in the paper we extend fuzzy integrals of point valued functions which were defined by ralescu and adams to set valued functions some properties similar to auman integrals are shown and these include convexity monotonicity fatou s lemma lebesgue convergence theorem etc

choquet like integrals abstract integrals based on a pseudo addition and a pseudo multiplication with properties similar to those of the choquet and the sugeno integral respectively are introduced the structure of convenient pseudo arithmetical operations leading to a choquet like integral is studied

generalized fuzzy integrals of set valued functions in this paper based on wu s generalized fuzzy integrals of point valued function c wu et al fuzzy sets and systems 70 1995 75 a theory of generalized fuzzy integrals of fuzzy valued functions will be investigated it includes definitions properties and various kinds of convergence theorems these are extensions of results in c wu et al fuzzy sets and systems 70 1995 75 d zhang and z wang fuzzy sets and systems 54 1993 63

tight uniform continuity bounds for quantum entropies conditional entropy relative entropy distance and energy constraints we present a bouquet of continuity bounds for quantum entropies falling broadly into two classes first a tight analysis of the alicki fannes continuity bounds for the conditional von neumann entropy reaching almost the best possible form that depends only on the system dimension and the trace distance of the states almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators as applications we give new proofs with tighter bounds of the asymptotic continuity of the relative entropy of entanglement er and its regularization e r infty as well as of the entanglement of formation ef using a novel quantum coupling of density operators which may be of independent interest we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation aka entanglement cost e c e f infty second we derive analogous continuity bounds for the von neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint most importantly systems of multiple quantum harmonic oscillators while without an energy bound the entropy is discontinuous it is well known to be continuous on states of bounded energy however a quantitative statement to that effect seems not to have been known here under some regularity assumptions on the hamiltonian we find that quite intuitively the gibbs entropy at the given energy roughly takes the role of the hilbert space dimension in the finite dimensional fannes inequality

an interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure abstract in this paper the non additivity of sugeno s fuzzy measures is interpreted in terms of addition and the rationality of the choquet integral is discussed it is pointed out that a fuzzy measure on a set x expresses the interaction between the subsets of x and can be represented by an additive measure it is shown through concrete examples that the choquet integral is reasonable as an integral with respect to a fuzzy measure it is also found that the choquet integral is closely related with the representation of a fuzzy measure

fuzzy t conorm integral with respect to fuzzy measures generalization of sugeno integral and choquet integral abstract this article discusses integrals with respect to sugeno s fuzzy measures sugeno integrals and choquet integrals with respect to general fuzzy measures are generalized in the setting of a t conorm algebraic system first t conorm integrals with respect to decomposable measures based on t conorm algebraic system are defined they are a modification of the weber integral secondly they are extended to fuzzy t conorm integrals with respect to general fuzzy measures the extension is made in the same way as the extension of the lebesgue integral to the choquet integral the class of fuzzy t conorm integrals includes choquet integrals and sugeno integrals properties of the integrals especially their representation are discussed

some properties and convergence theorems of set valued choquet integrals the paper aims at discussing the set valued choquet integral which is the integral of set valued random variables with respect to capacities we mainly present representation theorems of the set valued random variable by using a sequence of choquet integrable selections and then we investigate some properties of set valued choquet integrals especially subadditive property and inequality of the metric of set valued choquet integrals we also prove fatou s lemmas lebesgue dominated convergence theorem and monotone convergence theorem of set valued choquet integrals under the weaker conditions than that in previous works

set valued lusin type theorem for null null additive set multifunctions in a previous paper we presented a set valued egoroff type theorem for monotone set multifunctions in this paper a new version of this set valued egoroff type theorem is obtained it enables a set valued lusin type theorem for null null additive monotone continuous set multifunctions to be established several applications of this lusin type theorem are given pointing out some relationships among m measurability a measurability and totally measurability of real functions we also compare our new lusin type theorem with lusin s theorem previously obtained by the second author

some general properties of unified entropies basic properties of the unified entropies are examined the consideration is mainly restricted to the finite dimensional quantum case bounds in terms of ensembles of quantum states are given both the continuity in fannes sense and stability in lesche s sense are shown for wide ranges of parameters in particular uniform estimates are obtained for the quantum renyi entropies stability properties in the thermodynamic limit are discussed as well it is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters non decreasing of all the unified entropies under projective measurements is proved

tight bound on relative entropy by entropy difference we prove a lower bound on the relative entropy between two finite dimensional states in terms of their entropy difference and the dimension of the underlying space the inequality is tight in the sense that equality can be attained for any prescribed value of the entropy difference both for quantum and classical systems we outline implications for information theory and thermodynamics such as a necessary condition for a process to be close to thermodynamic reversibility or an easily computable lower bound on the classical channel capacity furthermore we derive a tight upper bound uniform for all states of a given dimension on the variance of the surprisal whose thermodynamic meaning is that of heat capacity

on the solution of the fokker planck equation using a high order reduced basis approximation the numerical solution of the one dimensional fokker planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented the finitely extensible non linear elastic fene model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique this technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition svd the basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite dimensional subspace of the solution space this is achieved through basis enrichment and projection the reduced basis method is extended to the high dimensional fokker planck equation using the d dumbbell fene model by consideration of a high dimensional singular value decomposition some numerical results are presented to demonstrate the efficiency of the numerical scheme

non atomicity for fuzzy and non fuzzy multivalued set functions in this paper we extend the concept of set valued fuzzy measures introduced and studied by guo and zhang 2004 on set valued fuzzy measures information sciences 160 2004 13 25 to the class of multivalued set functions taking values in the family of non empty closed subsets of a real normed space also we introduce the notions of atom and non atomicity of a multivalued set function and present some of their properties we also establish an extension result by preserving non atomicity for a fuzzy multisubmeasure multimeasure

strong integral of multifunctions relative to a fuzzy measure abstract in this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of 0 with respect to a fuzzy measure unlike various ways of defining such an integral e g aumann way using different selections dunford way riemann way we use the sugeno method directly applied to multifunctions the originality of our definition is that the value of the integral is a real scalar and not usually a set some classical integral properties a convergence theorem and relationships with other integrals are presented