fuzzy topological vector spaces topological generation and normability abstract increasing awareness of the utility of fuzzy systems in modeling real life situations necessitates a thorough understanding of fuzzy topological vector spaces in this paper the first of several to follow on this topic the generation of a fuzzy vector topology from an ordinary vector topology on a vector space is discussed also obtained is a condition for the fuzzy normability of a fuzzy vector topology induced by a family of fuzzy seminorms finally an example indicating that the earlier mentioned condition is only sufficient is given

fuzzy vector spaces abstract this paper defines a basis for a fuzzy vector space and its dimension a class of fuzzy vector spaces having dimension is introduced and two standard results from crisp theory are proved for this case namely d i m e1 e2 d i m e1 d i m e2 d i m e1 e2 and rmd i m i m f d i m k e r f the product of two fuzzy vector spaces under certain conditions is also characterised

the space of omega limit sets of a continuous map of the interval we first give a geometric characterization of limit sets we then use this characterization to prove that the family of limit sets of a continuous interval map is closed with respect to the hausdorff metric finally we apply this latter result to other dynamical systems

λ statistical limit points of the sequences of fuzzy numbers aytar has introduced the concepts of statistical limit and cluster points of a sequence of fuzzy numbers based on the definitions given in fridy s study for sequences of real numbers in this paper we define l statistical limit and l statistical cluster points of sequences of fuzzy numbers and discuss the relations among the sets of ordinary limit points l statistical limit points and l statistical cluster points of sequences of fuzzy numbers

fuzzy topological ordered vector spaces i abstract in an ordered vector space one can consider various fuzzy topologies determined by the order relation but it is also convenient to consider the case in which in an ordered vector space a fuzzy topology is given from the beginning in this case one requires not only that the fuzzy topology be compatible with the structure of vector space but also that it be compatible with the order relation in one sense or another

fuzzy topological vector spaces over r in this paper the concepts of fuzzy vector fuzzy vector space and fuzzy topological vector space are investigated a topology called sendograph topology which is compatible with the algebraic structure is given

statistical cluster and extreme limit points of sequences of fuzzy numbers in this paper we investigate the relations between statistical limit superior or statistical limit inferior and statistical cluster points of a statistically bounded sequence of fuzzy numbers and show that the set of statistical cluster points of a sequence of this type might be empty

on negative limit sets for one dimensional dynamics abstract in this paper we study the structure of negative limit sets of maps on the unit interval we prove that every limit set is an limit set while the converse is not true in general surprisingly it may happen that the space of all limit sets of interval maps is not closed in the hausdorff metric and thus some limit sets are never obtained as limit sets moreover we prove that the set of all recurrent points is closed if and only if the space of all limit sets is closed

on the equivalence of convergences of fuzzy sets abstract the aim of this paper is to compare d h and l convergences with the variational covergence our analysis which is carried out in is made through the introduction of the space c of compact fuzzy sets u on for which the level function 0 1 l k is continuous we prove that in this space all the above notions of convergence are equivalent

stability of asymptotically stable sets for continuous functions let k be the class of nonempty compact subsets of i 0 1 and k consist of the nonempty closed subsets of k we study the map a c i i k defined so that a f is the closure of the collection of asymptotically stable sets of f while we find that a is not continuous we do get much more positive results when studying the semicontinuity of a in particular a is lower semicontinuous at a residual set of functions in c i i so that a is continuous if we restrict our map to a residual subset of c i i

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on generalization on nguyen s theorem abstract the goal of this paper is to generalize certain results of nguyen 1 concerning the cuts of two place functions defined by the zadeh s extension principle to the case of extended two place functions defined via a sup t norm convolution

is every approximate trajectory of some process near an exact trajectory of a nearby process this paper deals with the problem can a noisy orbit be tracked by a real orbit in particular we will study the one parameter family of tent maps and the one parameter family of quadratic maps we writeg for eitherf orf withf x x forx 1 2 andf x 1 x forx 1 2 andf x x 1 x for a given we will say g permits increased parameter shadowing if for each x 0 there exists some 0 and some f 0 such that every f pseudog orbit starting in some invariant interval can be x shadowed by a realg orbit with we show thatg typically permits increased parameter shadowing

approximate and real trajectories for generic dynamical systems abstract in this paper we show that generic dynamical systems have a weak shadowing property namely that for a generic dynamical system in the space of discrete continuous dynamical systems on a compact manifold m with the usual c 0 topology given 0 there exists a 0 such that trajectories are close to real trajectories there are no dimensional restrictions on this result we also give some results on the inverse problem of tracing real trajectories by trajectories these problems are motivated by questions regarding the validity of numerical solutions of differential equations

preservation of shadowing in discrete dynamical systems abstract we look at the preservation of various notions of shadowing in discrete dynamical systems under inverse limits products factor maps and the induced maps for symmetric products and hyperspaces the shadowing properties we consider are the following shadowing h shadowing eventual shadowing orbital shadowing strong orbital shadowing the first and second weak shadowing properties limit shadowing s limit shadowing orbital limit shadowing and inverse shadowing

on λ statistical convergence of difference sequences of fuzzy numbers in this paper the concept of strongly d l p 2 cesaro summability of a sequence of fuzzy numbers is introduced also some inclusion relations between the set of strongly d l p 2 cesaro convergent and d l 2 statistically convergent sequences of fuzzy numbers are given

fuzzy topological vector spaces ii abstract some of the properties of fuzzy topological vector spaces are investigated also there are given necessary and sufficient conditions for a family of fuzzy sets in a vector space e to be the family of all neighborhoods of zero for a fuzzy linear topology

fuzzy functional analysis i basic concepts properties of continuous functions from the set of fuzzy numbers with support in a closed interval to the set of all fuzzy numbers are investigated associated to the metrics on the set of all fuzzy numbers we give two metrics on the set of all continuous functions and prove that the algebraic operations are continuous

on statistically convergent sequences of fuzzy numbers abstract in this paper we study some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically cauchy

lacunary statistical convergence of sequences of fuzzy numbers the sequence x xk of fuzzy numbers is statistically convergent to the fuzzy number x0 provided that for each 0 limln the number ofk n d xk x0 0 in this paper we study a related concept of convergence in which the set k k n is replaced by k kr 1 k kr for some lacunary sequence kr also we introduce the concept of lacunary statistically cauchy sequence and show that it is equivalent to the lacunary statistical convergence in addition the inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers are given

on statistically convergent double sequences of fuzzy numbers in this paper we continue the study the notion of statistically convergent and statistically cauchy for double sequences of fuzzy numbers

pseudo orbit shadowing in the family of tent maps on montre que les applications en tente ont la propriete d ombrage pour presque tous les parametres s mais elles n ont pas cette propriete pour un ensemble dense non denombrable des parametres on montre que pour une application tente chaque pseudo orbite peut etre approchee par une orbite actuelle d une application en tente avec une boucle peut etre legerement plus grande

on shadowing ordinary and ergodic the aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property in particular we deduce the chaotic behavior of a map with ergodic shadowing property moreover we define some kind of specification property and investigate its relation to the ergodic shadowing property

stability in the family of ω limit sets of alternating systems let f and g be elements of c i with x i 0 1 we study the limit sets x f g generated by alternating trajectories of the form x f g x f x g f x f g f x as well as the sets f g x i x f g and l f g x f g x i in particular we show that 1 if g is constant on no interval j i then there exists a residual set s c i so that the maps c i c i k and l c i c i k taking f g to f g and l f g respectively are both continuous at f g whenever f s 2 the map i c i c i k taking x f g to x f g is in the second class of baire and for any g c i there exists a residual set t i c i so that is continuous at x f g whenever x f t 3 if f is constant on no interval j i then there exists a residual set d i c i so that x f g x g f f x f g where both x g f and f x f g are adding machines of type whenever x g d

shadowing internal chain transitivity and α limit sets abstract let f x x be a continuous map on a compact metric space x and let f f and i c t f denote the set of limit sets limit sets and nonempty closed internally chain transitive sets respectively we show that if the map f has shadowing then every element of i c t f can be approximated to any prescribed accuracy by both the limit set and the limit set of a full trajectory furthermore if f is additionally expansive then every element of i c t f is equal to both the limit set and the limit set of a full trajectory in particular this means that shadowing guarantees that f f i c t f where the closures are taken with respect to the hausdorff topology on the space of compact sets whilst the addition of expansivity entails f f i c t f we progress by introducing novel variants of shadowing which we use to characterise both maps for which f i c t f and maps for which f i c t f

fuzzy vector spaces and fuzzy cosets abstract a fuzzy subspace is characterized by its level subspaces the algebraic nature of fuzzy subspaces under homomorphisms is studied certain conditions are obtained under which a given fuzzy subspace can or cannot be realized as a union of two fuzzy subspaces such that none is contained in the other the construction of a fuzzy subspace generated by a given fuzzy subset of a vector space is given the set of all the fuzzy cosets of a fuzzy subspace is shown to be a vector space and the dimension of the same is obtained a fuzzy quotient space is defined and an analogue of a consequence of the fundamental theorem of homomorphisms is obtained

universality with respect to ω limit sets abstract a discrete dynamical system on a compact metric space x is called universal with respect to limit sets if among its limit sets there is a homeomorphic copy of any limit set of any dynamical system on x by a result of pokluda and smital the unit interval admits a universal system in this paper we study the problem of the existence of universal systems on cantor spaces graphs dendrites and higher dimensional spaces

on fuzzy bases of vector spaces abstract in this paper the concept of fuzzy basis of fuzzy vector space is studied the criterion for existence of fuzzy basis is proved any countable dimensional space has a fuzzy basis there are spaces without fuzzy basis

defect controlled numerical methods and shadowing for chaotic differential equations abstract it is well known that chaotic ode s are in the classical sense unstable or ill conditioned it is not clear that variable step size codes based on local error control can solve such problems in a useful way in particular it is usually difficult to show that the apparent chaos in the resulting solution is not a numerical artifact i show here that a defect controlled method gives useful solutions for chaotic problems a pragmatically modified definition of what it means for a dynamical system to be chaotic is also presented