strong laws of large numbers for adapted arrays of set valued and fuzzy valued random variables in banach space in this paper we establish some strong laws of large numbers for double adapted arrays of set valued and fuzzy valued random variables in separable banach space our results are obtained with and without conditionally compactly uniformly integrable hypothesis and they are more general than some related results

a linear code for the sawtooth and cat maps abstract the sawtooth maps are a one parameter set of piecewise linear area preserving maps on the torus for positive integer values of the parameter k they are automorphisms of the torus known as the cat maps we present a symbolic dynamics for these maps in which the symbols are integers this code is related to a practical problem of the stabilisation of a system which is perturbed by impulses the code is linear in the sense that an orbit and its code are linearly related so it is not difficult to obtain a good approximation to one from the other in practice a stationary stochastic process for generating the code is given explicitly the theory uses green function methods which are also used to study ordered periodic orbits and cantori the problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed

a strong law of large numbers for fuzzy random sets abstract an slln for fuzzy random variables in rp including other convergence theorems has been proved by klement puri and ralescu the present paper is motivated by the definitions and properties developed by puri and ralescu and concerned with an slln for banach spaced valued fuzzy random sets as an easy application of the previous results

quasiregularity and rigorous diffusion of strong hamiltonian chaos exact results are derived concerning quasiregularity and diffusion of strong chaos on resonances of the sawtooth map a chaotic ensemble of well defined quasiregularity type the sequence of resonances visited is generally a fractal set whose main characteristics the topological entropy and the hausdorff dimension are calculated exactly under some conditions using a symbolic dynamics the effect of quasiregularity on chaotic diffusion is characterized by an infinity of diffusion coefficients each associated with a fractal ensemble trapped in a periodic set of resonances in some cases these coefficients are calculated exactly and it is shown that rigorous diffusion takes place on the resonances

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

weak laws of large numbers for fuzzy random variables abstract in this paper we establish weak laws of large numbers for sums of convex compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in rp as a result weak laws of large numbers for sums of convexly tight fuzzy random variables are given

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

stability of cellular states of the kuramoto sivashinsky equation this paper is concerned with the instability property of a particular type of solution to the kuramoto sivashinsky equation namely the symmetric time independent cellular states attention is focused on the dimension of their unstable manifolds we show that as a control parameter varies the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation further change in dimension is shown to be related to hopf bifurcations from the cellular states an asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur

type specification of chaos quasiregularity of hamiltonian chaos is approached using the concept of the type of a chaotic orbit i e the sequence of resonances visited by the orbit a simple method is introduced for determining the type in the case of kicked rotor hamiltonians first applications of this method are given in the calculation and study of quantities characterizing vague tori within resonances namely survival probabilities and average dwelling times results concerning a newly defined disorder parameter characterizing quasiregularity are briefly reported

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

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

topological properties on the space of fuzzy sets abstract in this paper we introduce a new metric on the space f rp of fuzzy sets and prove that f rp is separable and complete also a characterization of compact subsets of f rp is presented and some examples are given

chaotic diffusion on periodic orbits and uniformity chaotic diffusion on periodic orbits pos is studied for the perturbed arnol d cat map exhibiting a transition from uniform to nonuniform hyperbolicity as the perturbation parameter is increased the results for the diffusion coefficient from po formulas agree very well with those obtained by standard methods using the original po formula involving a uniform nonweighted average over the pos reasonably accurate results are obtained for sufficiently small parameters corresponding also to cases of a considerably nonuniform hyperbolicity this is due to uniformity sum rules satisfied by the po lyapunov eigenvalues at fixed winding number

two periodical and quasi periodical wave solutions of the kuramoto sivashinsky equation and their stability and bifurcations abstract a nonlinear evolution equation is considered which is often encountered in modelling the behaviour of perturbations in various active dissipative media e g in problems of fluid film flow hydrodynamics wave solutions regular in space and both periodical and quasi periodical in time generating from steady state and steady state travelling waves have been found numerically stability of some solutions has been investigated and bifurcation analysis has been carried out the analysis has demonstrated that there is a sequence of bifurcations of solutions stable with respect to disturbances of the same spatial period and it has been shown that the bifurcations are related to the loss of some symmetries of the initial solution it is also shown that the regions of quiet time behaviour and those of intensive variations are separated in the temporal period of solutions with the bifurcation parameter increasing

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

a characterization of compact subsets of fuzzy number space abstract in this paper a characterization of compact subsets on the space of fuzzy numbers endowed with the skorokhod topology is presented this characterization is an analogue of the arzela ascoli theorem which characterizes compact subsets of the space c 0 1

correlations of periodic area preserving maps abstract a simple analytical decay law for correlation functions of periodic area preserving maps is obtained this law is compared with numerical experiments on the standard map the agreement between experiment and theory is good when islands are absent but poor when islands are present when islands are present the correlations have a long slowly decaying tail

on strong laws of large numbers for random upper semicontinuous functions a short proof of the strong law of large numbers for random upper semicontinuous functions fuzzy random variables with compact support is given this approach allows us to deduce many results on convergence of fuzzy random variables from the relevant results on random sets that appear as their level sets

strong convergence for weighted sums of fuzzy random sets in this paper we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper semicontinuous fuzzy sets in r p

on steady state travelling solutions of an evolution equation describing the behaviour of disturbances in active dissipative media abstract a nonlinear evolution equation is considered which is often encountered in modelling the behaviour of perturbations in various active dissipative media e g in problems of fluid film flow hydrodynamics periodic steady state travelling solutions have been found numerically for it stability of these solutions has been investigated and bifurcation analysis has been carried out the analysis has demonstrated that decrease of the wave number causes more and more new families of steady state travelling solutions a countable set of such solutions is formed in the limit when the wave number tends to zero it is also shown that time oscillating solutions can be generated from steady state ones due to bifurcation of the landau hopf type

a strong law of large numbers for fuzzy random variables abstract a limit of a sequence of fuzzy numbers is defined and its some properties are shown based on these concept and properties an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown

integrals of fuzzy number valued functions abstract using the representation theorem of fuzzy numbers we give an alternative approach for the integrals of fuzzy number valued functions and for the radon nikodym theorem of the fuzzy number valued measures

the skorokhod topology on space of fuzzy numbers in this paper we introduce the skorokhod metric on the space f r of fuzzy numbers and prove that f r is separable and complete

continuity of functions defined on the space of fuzzy sets in this paper we investigate the continuity of many useful functions defined on the space f rp of normal and upper semicontinuous fuzzy sets in rp with compact support

compactness and convexity on the space of fuzzy sets ii abstract in this paper we establish some characterizations of convex and relatively compact subsets of the space f r p of normal upper semicontinuous and compactly supported fuzzy sets in rp equipped with the skorokhod metric

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

a law of large numbers for random sets abstract we show a law of large numbers for random sets taking values in a class of subsets of a banach space which is at least larger than the class of compact subsets of the space if the banach space is of infinite dimension to see this we first define the hausdorff distance on the family of all the non empty subsets of a banach space and give some properties of it motivated by the fact that a study of fuzzy sets is to study of set valued analysis

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

convergence in distribution for level continuous fuzzy random sets in this paper we give the relationships between tightness and convergence in distribution for level continuous fuzzy random sets to this end we first obtain some characterizations of compact subsets of the space of level continuous fuzzy sets in r p and establish the tightness for level continuous fuzzy random sets

law of large numbers for uncertain random variables the law of large numbers in probability theory states that the average of random variables converges to its expected value in some sense under some conditions sometimes random factors and human uncertainty exist simultaneously in complex systems and a concept of uncertain random variable has been proposed to study this type of complex systems this paper aims to provide a law of large numbers for uncertain random variables which states that the average of uncertain random variables converges in distribution to an uncertain variable as a byproduct the convergence of a sequence of uncertain variables is also studied

hierarchy of bifurcations of space periodic structures in a nonlinear model of active dissipative media abstract this work is a detailed research of spatial periodic structures and their bifurcations in a nonlinear model of media with an active dissipation presented here is an obvious bifurcation diagram of stationary solutions attractors on a wide scale of wave numbers are studied and classified significant progress in understanding of behavior of nonstationary solutions is attained

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

periodic orbits of the sawtooth maps abstract the sawtooth maps are a one parameter family of chaotic area preserving maps of the 2 torus which become anosov systems for special values of the parameter we develop a technique for constructing explicitly all their periodic orbits as perturbations of anosov orbits combining methods of arithmetic and symbolic dynamics we find that the parameter dependence of periodic orbits is given by rational functions and compute numerically the population of periodic orbits as a function of the parameter

hamiltonian transport on unstable periodic orbits transport in hamiltonian systems is approached on the basis of the unstable periodic orbits upos embedded in the chaotic region correlation functions and diffusion rates are defined for upo ensembles of two dimensional periodic maps global diffusion in these maps is interpreted in terms of the ensembles ut containing all the upos of period t as t the relative populations of closed orbits and accelerator modes in ut approach a gaussian distribution characterized by the diffusion coefficient d evidence for the validity of this mechanism is provided for the cat and sawtooth maps prime lattice ensembles of the cat maps appear to be optimal in reproducing the global diffusion these ensembles are generically small subsets of ut for some t it is shown however that the corresponding diffusion rates assume for t odd precisely the known value of d in the entire time interval t t for t even all orbits are closed and d is reproduced only in the interval t 12 t the diffusion rates associated with ensembles ut of the sawtooth maps appear to approximate well the value of d if t is of the order of the diffusion time the case of local transport rates associated with ensembles that are localized in phase space is illustrated

kolmogorov s strong law of large numbers for fuzzy random variables in this paper kolmogorov s strong law of large numbers for sums of independent and level wise identically distributed fuzzy random variables is obtained

some properties of a new metric on the space of fuzzy numbers abstract this paper deals with some properties of a new metric here called the hausdorff skorokhod metric previously introduced by joo and kim on the space of fuzzy numbers in the euclidean space r k we first prove that convergence in the hausdorff skorokhod metric implies convergence in the sendograph metric also we show that the hausdorff skorokhod metric is not absolutely homogeneous and not translation invariant and investigate the continuity of operations and convexity on the space of fuzzy numbers equipped with the hausdorff skorokhod metric

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

measurability for fuzzy valued functions in this paper we characterize the borel field of the space of fuzzy sets endowed with the skorokhod metric as a result the measurability for a fuzzy valued function is discussed

steady solutions of the kuramoto sivashinsky equation abstract steady solutions of the kuramoto sivashinsky equation are studied these solutions are defined on the whole x line and propagate with a constant speed c 2 in time for large c 2 it is shown that the solution is unique and has a conical form for small c 2 there is a periodic solution and an infinite set of quasi periodic solutions as asserted by moser s twist map theorem numerical computations for intermediate values of c 2 suggest that below c 1 6 of every speed there is a continuum of odd quasi periodic solutions or a cantor set of chaotic solutions wrapped by infinite sequences of conic solutions

markov tree model of transport in area preserving maps abstract transport in an area preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori a model retaining a discrete set of cantori approaching a boundary circle gives the markov chain description of hanson cary and meiss the inclusion of cantori surrounding island chains and islands about islands etc gives a markov tree model with a slower decay rate the survival probability distribution is shown to decay asymptotically as a power law the decay exponent agrees reasonably well with the computations of karney and of chirikov and shepelyanski

resonances and transport in the sawtooth map we study transport in a completely chaotic hamiltonian system the hyperbolic sawtooth map analytical expressions are obtained for its cantori and resonances we show that resonances give a complete partition of phase space the flux leaking out of a resonance is given by its turnstiles whose form and areas are obtained analytically when the total flux out of a resonance becomes one third the area of an island the topology of the turnstiles changes at the same parameter value a horseshoe is formed corresponding to the orbits trapped within the resonance based on this a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained the partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles we calculate the survival probability within a resonance using the markov model the results are compared with results obtained numerically and from periodic orbit theory

supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings this paper generalizes a classical result about the space of bounded closed sets with the hausdorff metric and establishes the completeness of cb x with respect to the completeness of the metric space x where cb x is the class of fuzzy sets with nonempty bounded closed a cut sets equipped with the supremum metric d which takes the supremum on the hausdorff distances between the corresponding a cut sets in addition some common fixed point theorems for fuzzy mappings are proved and two examples are given to illustrate the validity of the main results in fixed point theory

the strong law of large numbers for fuzzy random variables abstract sequences of independent and identically distributed fuzzy random variables are considered it is shown that the strong law of large numbers holds also for fuzzy random variables this result is used to give a consistent estimator for the expectation of a fuzzy random variable

chaotic diffusion on periodic orbits the perturbed arnold cat map chaotic diffusion on periodic orbits pos is studied for the perturbed arnold cat map on a cylinder in a range of perturbation parameters corresponding to an extended structural stability regime of the system on the torus the diffusion coefficient is calculated using the following po formulas 1 the curvature expansion of the ruelle zeta function 2 the average of the po winding number squared w 2 weighted by a stability factor 3 the uniform nonweighted average of w 2 the results from formulas 1 and 2 agree very well with those obtained by standard methods for all the perturbation parameters considered formula 3 gives reasonably accurate results for sufficiently small parameters corresponding also to cases of a considerably nonuniform hyperbolicity this is due to uniformity sum rules satisfied by the po lyapunov eigenvalues at fixed w these sum rules follow from general arguments and are supported by much numerical evidence

periodic orbits and chaotic diffusion probability distributions abstract periodic orbit po formulas for chaotic diffusion probability distributions pds are examined in the case of the perturbed arnol d cat map on the cylinder this translationally invariant system exhibits a transition from uniform to nonuniform hyperbolicity as the perturbation parameter is increased two coarse grained pds describing the diffusion between unit cells of the system are studied a a pd based on po ensembles b a pd based on generic ensembles the approximate po formula for pd b gives results which fluctuate around the expected gaussian distribution for all parameters considered and thus agree qualitatively with results from standard methods the exact po formula for pd a gives similar results only for sufficiently small parameters the results for large parameters decrease monotonically relative to the gaussian distribution this deviation seems to disappear as the po period is increased

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