block boundary value methods for solving volterra integral and integro differential equations reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind volterra integral equations and volterra integro differential equations these extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for volterra type equations numerical experiments are presented and the efficiency accuracy and stability of the schemes are confirmed

ito type stochastic fuzzy differential equations with delay abstract in the paper we give some foundations for the studies of stochastic fuzzy delayed differential equations we prove the existence and uniqueness of solutions to such the equations to obtain our result we assume that the coefficients of the equation satisfy the lipschitz condition together with linear growth condition we estimate the distance between approximate solution and exact solution also the stability of solution with respect to the initial history is shown an application of stochastic fuzzy delayed differential equations in the modeling of population growth is indicated

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

fuzzy set valued gaussian processes and brownian motions gaussian processes and brownian motion are concepts and tools in modelling important uncertain systems in many areas in view of uncertainty complexity in many real world problems we extend these tools to the case where stochastic processes can take on fuzzy sets as values in this paper we discuss fuzzy set valued gaussian processes based on the results of s li y ogura v kreinovich limit theorems and applications of set valued and fuzzy set valued random variables kluwer academic publishers dordrecht 2002 s li y ogura h t nguyen gaussian processes and martingales for fuzzy valued variables with continuous parameter inform sci 133 2001 7 21 s li y ogura f n proske m l puri central limit theorems for generalized set valued random variables j math anal appl 285 2003 250 263 n n lyashenko on limit theorems for sums of independent compact random subsets in the euclidean space j soviet math 20 1982 2187 2196 and m l puri d a ralescu the concept of normality for fuzzy random variables ann probab 13 1985 1373 1379 we also introduce the concept of fuzzy set valued brownian motion and then prove several properties of such processes

a class of compact boundary value methods applied to semi linear reaction diffusion equations abstract this paper deals with a class of compact boundary value methods cbvms for solving semi linear reaction diffusion equations slres the presented cbvms are constructed by combining a fourth order compact difference method cdm with the p order boundary value methods bvms where the former is for the spatial discretization and the latter for temporal discretization it is proven under some suitable conditions that the cbvms are locally stable and uniquely solvable and have fourth order accuracy in space and p order accuracy in time the computational effectiveness and accuracy of cbvms are further testified by applying the methods to the fisher equation besides these research we also extend the cbvms to solve the two component coupled system of slres the numerical experiment shows that the extended cbvms are effective and can arrive at the high precision

on the convergence in distribution of measurable multifunctions random sets normal integrands stochastic processes and stochastic infima the concept of the distribution function of a closed valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi convergence in distribution of normal integrands and stochastic processes in particular various compactness criteria are exhibited the connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of donsker s theorem brownian motion as a limit of normalized random walks we also suggest the potential application of the theory to the study of the convergence of stochastic infima

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

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

remarks on unboundedness of set valued ito stochastic integrals abstract the paper deals with set valued stochastic integrals which are defined as a random multifunctions integrable boundedness of a such integral is one of the most important features in potential applications unfortunately up to now there were no correct proofs of such property surprisingly also a negative answer to this problem is still not explained correctly hence the problem seems to be still undetermined we shall show that in general the answer is negative we shall provide several relatively simple examples both in convex and nonconvex valued case as a consequence we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set valued ito s integral is equivalent to its single valuedness

on set valued stochastic integrals and fuzzy stochastic equations in this paper we extend the notion of set valued and fuzzy stochastic integrals to semimartingale integrators we present their main properties and finally we establish the existence of solutions of a fuzzy integral stochastic equation driven by a brownian motion the approach is based on the existence of solutions of an appropriately formulated martingale problem for a system of stochastic inclusions and the theorem of negoita and ralescu

boundary value methods for volterra integral and integro differential equations abstract new and effective quadrature rules generated by boundary value methods are introduced we employ the introduced quadrature rules to construct quadrature methods for the second kind volterra integral equations and volterra integro differential equations these methods are shown to be effective and possess excellent convergence properties the nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme numerical simulations are presented and confirm the efficiency and accuracy of the methods

block boundary value methods for the solution of ordinary differential equations block boundary value methods are considered and applied to solve initial value problems results on their convergence and stability properties are presented and their intermediate position between the class of multistep and runge kutta methods is established

functionally fitted block methods for ordinary differential equations abstract functionally fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions in this paper we propose a new type of functionally fitted block methods for ordinary differential equations the basic theory for the proposed methods is established first we derive two sufficient conditions to ensure the existence of the functionally fitted block methods discuss their equivalence to collocation methods in a special case and independence on integration time for a set of separable basis functions we then obtain some basic characteristics of the methods by taylor series expansions and show that the order of accuracy of the r point functionally fitted block method is at least r for general ordinary differential equations experimental results are conducted to demonstrate the validity of the functionally fitted block methods

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

theoretical analysis and simulations of the generalized lotka volterra model the dynamics of generalized lotka volterra systems is studied by theoretical techniques and computer simulations these systems describe the time evolution of the wealth distribution of individuals in a society as well as of the market values of firms in the stock market the individual wealths or market values are given by a set of time dependent variables w i i 1 dots n the equations include a stochastic autocatalytic term representing investments a drift term representing social security payments and a time dependent saturation term due to the finite size of the economy the w i mathrm mathrm s turn out to exhibit a power law distribution of the form p w ensuremath sim w ensuremath 1 ensuremath ensuremath alpha it is shown analytically that the exponent ensuremath alpha can be expressed as a function of one parameter which is the ratio between the constant drift component social security and the fluctuating component investments this result provides a link between the lower and upper cutoffs of this distribution namely between the resources available to the poorest and those available to the richest in a given society the value of ensuremath alpha is found to be insensitive to variations in the saturation term which represent the expansion or contraction of the economy the results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power law distribution with 1l ensuremath alpha l2

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

fuzzy set valued stochastic lebesgue integral this paper studies lebesgue integral of a fuzzy closed set valued stochastic process with respect to the time t firstly a progressively measurable fuzzy closed set valued stochastic process is discussed and an almost everywhere problem in the former aumann type lebesgue integral of the level set process is pointed out secondly a new definition of the lebesgue integral by decomposable closure is given focusing on aumann representation theorem representation theorem and property of convexity it is proved that the fuzzy closed set valued stochastic lebesgue integral is a fuzzy closed set valued stochastic process which is widely used in the fuzzy world with randomness finally the fuzzy closed set valued stochastic lebesgue integral in l p space is studied especially on an almost everywhere problem

the origin of power law distributions in self organized criticality the origin of power law distributions in self organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process an avalanche is then regarded as a first return random walk process in a one dimensional lattice power law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions this shows that power law distributions in self organized criticality may be caused by the balance of competitive interactions at the mean time the mean spatial size for avalanches with the same lifetime is found to increase in a power law with the lifetime

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

probabilistic approach to the bak sneppen model we study here the bak sneppen model a prototype model for the study of self organized criticality in this model several species interact and undergo extinction with a power law distribution of activity bursts species are defined through their fitness whose distribution in the system is uniform above a certain threshold run time statistics is introduced for the analysis of the dynamics in order to explain the peculiar properties of the model this approach based on conditional probability theory takes into account the correlations due to memory effects in this way we may compute analytically the value of the fitness threshold with the desired precision this represents a substantial improvement with respect to the traditional mean field approach

fuzzy conditional expectation abstract the fuzzy random variables defined on a complete finite measure space are studied their fuzzy conditional expectation is defined and its properties are examined a new metric in the set of integrably bounded fuzzy variables is introduced and completeness of that metric space is proved

a characterization of banach spaces containing it is proved that a banach space contains a subspace isomorphic to 11 if and only if it has a bounded sequence with no weak cauchy subsequence the proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or boolean independent a bounded sequence of elements fn in a banach space b is said to be equivalent to the usual 11 basis provided there is a a 0 so that for all n and choices of scalars c cn

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

convergence and representation theorems for set valued random processes abstract in this paper we study set valued random processes in discrete time and with values in a separable banach space we start with set valued martingales and prove various convergence and regularity results then we turn our attention to larger classes of set valued processes so we introduce and study set valued amarts and set valued martingales in the limit finally we prove a useful property of the set valued conditional expectation

differentiation of sets in measure abstract suppose f e for each e 0 1 is a bounded borel subset of r d and f e f 0 as e 0 let a e f e f 0 be symmetric difference and p be an absolutely continuous measure on r d we introduce the notion of derivative of f e with respect to e d f e d e d a e d e such that d d e p a e e 0 q d d e a e e 0 where q is another explicitly described measure although not in r d we discuss why this sort of derivative is needed to study local point processes in neighbourhood of a set in short if sequence of point processes n n n 1 2 is given on the class of set valued mappings f f such that all f e converge to the same f f 0 then the weak limit of the local processes n n a e f e f lives on the class of derivative sets d f e d e e 0 f f we compare this notion of the derivative set valued mapping with other existing notions

a stability result for differential inclusions in banach spaces on etablit un resultat de stabilite pour les inclusions differentielles definies sur un espace de banach separable

convergence and stability of multistep methods solving nonlinear initial value problems convergence and stability of initial and boundary value multistep methods are analyzed for a class of nonlinear problems satisfying a one sided lipschitz condition the linear multistep methods are recast to handle the numerical solution globally on the time interval this allows us to use the theory of toeplitz matrices to show that under suitable assumptions global properties of the exact solution are preserved by its numerical approximation in particular a new concept of stability which avoids the unpleasant passage to one leg methods is introduced

a banach lattice approach to convergent integrably bounded set valued martingales and their positive parts denote by cf x the set of all nonempty convex closed subsets of a separable banach space x let be a complete probability space and denote by l1 cf x the complete metric space of equivalence classes of a e equal integrably bounded cf x valued functions for any preassigned filtration i we describe the space of convergent integrably bounded cf x valued martingales in terms of the closure of i 1 l1 i cf x in l1 cf x in particular we provide a formula to calculate the join of two such martingales and the positive part of such a martingale our object is achieved by considering the more general setting of a near vector lattice s d endowed with a riesz metric d by means of radstrom s embedding theorem for such spaces a link is established between the space of convergent martingales in s and the space of convergent martingales in the radstrom completion r s of s this link provides information about the former space of martingales via known properties of measure free martingales in riesz normed vector lattices applicable to r s we also apply our general results to the spaces of convergent ck x valued martingales where ck x denotes the set of all nonempty convex compact subsets of x

boundary value techniques for initial value problems in ordinary differential equations the numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered we discuss a finite difference method which was already investigated by fox in 1954 and fox and mitchell in 1957 hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown we further propose and discuss new jalerkin methods for initial value problems along the lines of the boundary value approach

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

on embedding problems of fuzzy number spaces iv in this paper we are going to follow the preceding papers to investigate the embedding problem of fuzzy number space ne 1 by means of a concrete embedding operator using the embedding operator some subclasses of ne 1 and metrices on ne 1 are also investigated

on the efficiency and optimality of random allocations abstract allocations in a random economy described by an integrably bounded closed and convex valued measurable consumption multifunction and a monotone continuous utility function are studied the notions of efficiency and optimality of allocations are introduced and compared through the use of price systems belonging to l x l x 1 a necessary and sufficient condition for optimality is obtained also for the case where the allocation belongs to lx it is shown that the efficiency prices can be chosen to be in l x 1 although l x l x 1 finally approximate optimality and efficiency are considered and also some stability results are proven when the consumption multifunction or the utility function vary in a certain sense this work is based on the theory of normal integrands of rockafellar

stochastic integral with respect to set valued square integrable martingales in this paper we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set valued square integrable martingale secondly we shall prove the representation theorem of set valued square integrable martingale thirdly we shall give the definition of stochastic integral of a stochastic process with respect to a set valued square integrable martingale and the representation theorem of this kind of integrals finally we shall prove that the stochastic integral is a set valued sub martingale

on convergence and closedness of multivalued martingales in this paper various convergence theorems and criteria of closedness of multivalued martingales submartingales and supermartingales are proved

central limit theorems for generalized set valued random variables abstract we give central limit theorems for generalized set valued random variables whose level sets are compact both in r d or in a banach space under milder conditions than those obtained recently by the latter two authors

oslo rice pile model is a quenched edwards wilkinson equation the oslo rice pile model is a sandpile like paradigmatic model of self organized criticality soc in this paper it is shown that the oslo model is in fact exactly a discrete realization of the much studied quenched edwards wilkinson equation qew nattermann et al j phys ii france 2 1483 1992 this is possible by choosing the correct dynamical variable and identifying its equation of motion it establishes for the first time an exact link between soc models and the field of interface growth with quenched disorder this connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of soc

set valued stochastic integral equations driven by martingales abstract we consider a notion of set valued stochastic lebesgue stieltjes trajectory integral and a notion of set valued stochastic trajectory integral with respect to martingale then we use these integrals in a formulation of set valued stochastic integral equations the existence and uniqueness of the solution to such the equations is proven as a generalization of set valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales

on set valued stochastic integrals in an m type 2 banach space in a separable banach space for set valued martingale several equivalent conditions based on the measurable selections are discussed and then in an m type 2 banach space at first we define single valued stochastic integral by the differential of a real valued brownian motion after that extend it to set valued case we prove that the set valued stochastic integral becomes a set valued submartingale which is different from single valued case and obtain the castaing representation theorem for the set valued stochastic integral which is applicable for set valued stochastic differential equations

some properties of set valued stochastic integrals abstract the present paper is devoted to properties of set valued stochastic integrals defined as some special type of set valued random variables in particular it is shown that if the probability base is separable or probability measure is nonatomic then defined set valued stochastic integrals can be represented by a sequence of ito s integrals of nonanticipative selectors of integrated set valued processes immediately from michael s continuous selection theorem it follows that the indefinite set valued stochastic integrals possess some continuous selections the problem of integrably boundedness of set valued stochastic integrals is considered some remarks dealing with stochastic differential inclusions are also given

numerical solution of differential algebraic equations and computation of consistent initial boundary conditions boundary value methods for the solution of differential algebraic equations are described we consider both initial and boundary value problems and derive an algorithm that does not require additional information from the user but only the initial or boundary conditions needed in theory to obtain a unique solution

radon nikody m theorem and conditional expectation of fuzzy valued measures and variables abstract the notions of fuzzy valued measure and conditional expectation of a fuzzy valued variable are defined the main result of the paper is a proof of the existence of the radon nikod m derivative of the fuzzy valued measure a convergence theorem for conditional expectations is proved as well

strong laws of large numbers for independent fuzzy set valued random variables abstract in this paper we shall present strong laws of large numbers slln s for independent not necessary identically distributed fuzzy set valued random variables whose base space is a separable banach space or an euclidean space in the sense of the extended hausdorff metric d h we apply the method to the sequence of independent identically distributed fuzzy set valued random variables to give a simple proof of slln s

on the properties of matrices defining some classes of bvms it is known that the matrices defining the discrete problem generated by a k step boundary value method bvm have a quasi toeplitz band structure 7 in particular when the boundary conditions are skipped they become toeplitz matrices in this paper by introducing a characterization of positive definiteness for such matrices we shall prove that the toeplitz matrices which arise when using the methods in the classes of bvms known as generalized bdf and top order methods have such property

stochastic set differential equations in this paper we present the existence and uniqueness of solutions to the stochastic set differential equations driven by a wiener process in the paper we also study their continuous dependence on initial conditions and a stability property

high order transverse schemes for the numerical solution of pdes many existing numerical schemes for the solution of initial boundary value problems for partial differential equations can be derived by the method of lines the pdes are converted into a system of ordinary differential equations either with initial conditions longitudinal scheme or with boundary conditions transverse scheme in particular this paper studies the performance of the transverse scheme in combination with boundary value methods moreover we do not restrict the semi discretization by the usual first or second order finite difference approximations to replace the derivative with respect to time but we use high order formulae

a convergence theorem of fuzzy valued martingales in the extended hausdorff metric h in this paper we shall give a new embedding method to prove a convergence theorem for fuzzy valued random variables in the sense of the extended hausdorff metric h without the restriction of fuzzy sets satisfying the lipschitz condition

point process model of 1 f noise vs a sum of lorentzians we present a simple point process model of 1 f noise covering different values of the exponent the signal of the model consists of pulses or events the interpulse interevent interarrival recurrence or waiting times of the signal are described by the general langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power law distribution our model is free from the requirement of a wide distribution of relaxation times and from the power law forms of the pulses it contains only one relaxation rate and yields 1 f spectra in a wide range of frequencies we obtain explicit expressions for the power spectra and present numerical illustrations of the model further we analyze the relation of the point process model of 1 f noise with the bernamont surdin mcwhorter model representing the signals as a sum of the uncorrelated components we show that the point process model is complementary to the model based on the sum of signals with a wide range distribution of the relaxation times in contrast to the gaussian distribution of the signal intensity of the sum of the uncorrelated components the point process exhibits asymptotically a power law distribution of the signal intensity the developed multiplicative point process model of 1 f noise may be used for modeling and analysis of stochastic processes in different systems with the power law distribution of the intensity of pulsing signals

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

on the conditional expectation and convergence properties of random sets in this paper we study random sets with values in a separable banach space first we establish several useful properties of the set valued conditional expectation and then prove some convergence theorems for set valued amarts and uniform amarts using the weak kuratowski mosco and hausdorff modes of set convergence

criticality in simple models of evolution we consider two apparently similar models of biological evolution that have been claimed to exhibit self organized critical behavior a careful reanalysis of these models including several analytic results for one of them suggests that they are qualitatively different we demonstrate the limitations of the mean field description of these systems we argue that a more precise definition of self organized criticality is desirable and establish several criteria in this connection

stochastic integrals of set valued processes and fuzzy processes we define the stochastic integrals of a set valued process and a fuzzy process with respect to a cylindrical brownian motion on a hilbert space we also give their properties which are useful for the study of fuzzy stochastic differential equations and stochastic differential inclusions with a set valued diffusion term

gaussian processes and martingales for fuzzy valued random variables with continuous parameter abstract the purpose of this paper is to discuss fuzzy valued gaussian processes and martingales with continuous parameter we mainly obtain a representation theorem for fuzzy valued gaussian processes after proving a new embedding theorem based on previous results about set valued and fuzzy valued martingales cf s li y ogura j fuzzy math 4 1996 905 s li y ogura fuzzy sets and systems 101 1999 453 s li y ogura ann probab 26 1998 1384 1402 s li y ogura in proc ifsa 97 vol 4 1997 pp 9 13 s li y ogura in c bertoluzza m a gil d a ralescu eds statistical modeling analysis and management of fuzzy data physica in press y ogura s li fuzzy sets and systems in press we prove optional sampling theorems for set valued and fuzzy valued martingales with continuous parameter

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

numerical determination of the avalanche exponents of the bak tang wiesenfeld model we consider the bak tang wiesenfeld sandpile model on a two dimensional square lattice of lattice sizes up to l 4096 a detailed analysis of the probability distribution of the size area duration and radius of the avalanches will be given to increase the accuracy of the determination of the avalanche exponents we introduce a new method for analyzing the data which reduces the finite size effects of the measurements the exponents of the avalanche distributions differ slightly from previous measurements and estimates obtained from a renormalization group approach

solving volterra integro differential equations by variable stepsize block bs methods properties and implementation techniques in this article block bs methods are considered for the numerical solution of volterra integro differential equations vides convergence and stability properties are analyzed a new matlab code for the solution of vides called videbs is presented numerical results using a variable stepsize implementation show the effectiveness of the proposed code

set valued assessments of solutions to stochastic differential equations with random set parameters abstract we consider stochastic differential equations depending on parameters whose uncertainty is modeled by random compact sets several approaches are discussed how to construct set valued stochastic processes from the solutions the induced lower and upper probabilities are compared to a set valued set function and a set of probability measures constructed from the distributions of the solutions and the selections of the random set

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

representation theorems set valued and fuzzy set valued ito integral in this paper we shall present representation theorems of set valued martingales and set valued processes of finite variation with continuous time we shall also obtain a representation theorem of a predictable set valued stochastic process we shall give a new definition of ito integral of a set valued stochastic process with respect to a brownian motion based on the work e j jung j h kim on set valued stochastic integrals stochastic anal appl 21 2 2003 401 418 we shall also discuss some properties of set valued ito integral especially the presentation theorem of set valued ito integral finally we extend some of above results to the fuzzy set valued case

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor

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

convergence of set valued and fuzzy valued martingales the purpose of this paper is to prove the convergence theorems of set valued and fuzzy valued martingales in kuratowski mosco sense without assuming that their values are compact or of compact level sets

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents