time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

differential equations with fuzzy parameters via differential inclusions abstract we give a definition of solutions of ordinary differential equations in r n containing parameters which are described by changing in time fuzzy sets they are defined as fuzzy subsets of the space of absolutely continuous functions we introduce a hypograph metric in the space of fuzzy sets and prove a theorem on continuous dependence of fuzzy solutions on parameters and initial conditions with respect to that metric

discrete dynamical systems in l topological spaces in this paper we investigate iteration of zadeh image operators and their discrete dynamical systems in l topological spaces some basic concepts such as limit set invariant set periodic point nonwandering point and recurrent point are discussed in weakly induced topological spaces we also study topological mixing topological transitivity and minimality for such dynamical systems while comparing with those corresponding ones in crisp spaces

a representation formula for maps on supermanifolds we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces i e following berezin leites manin etc we establish a representation formula for all pull back morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of a superspace we then derive two consequences of this result the first one is that we can integrate the data associated with a morphism in order to get a nonunique map defined on an ordinary space and uniqueness can be achieved by restriction to a scheme the second one is a simple and intuitive recipe to compute pull back images of a function on a manifold m by a map from a superspace to m

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

iterated fuzzy set systems a new approach to the inverse problem for fractals and other sets abstract images with grey or colour levels admit a natural representation in terms of fuzzy sets but without the usual probabilistic interpretation of the latter we introduce a fuzzy set approach which incorporates in part the technique of iterated function systems ifs for the construction analysis and or approximation of typically fractal sets and images the method represents a significant departure from ifs especially in the interpretation of the resulting image the introduction of grey level maps i 0 1 0 1 associated with the contractive maps w i of the ifs affords much greater flexibility in the generation of images as well as in the inverse problem

on the topological structure of spaces of fuzzy compacta we prove that the space of fuzzy compacta in a peano continuum is homeomorphic to the hilbert space 2 as a corollary we find that the spaces of fuzzy compacta in r n and 2 are also homeomorphic to hilbert space

comments with reply on optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization by m a dahleh and j b pearson the commenters point out that the sufficient condition for robust stability given in the above titled work see ibid vol 33 no 8 p 722 731 aug 1988 should be modified in their reply the authors agree with the commenters and describe the assumptions which led to the omission

choice of sampling to ensure minimum phase behaviour sufficient conditions for ensuring that a continuous stable proper minimum phase plant remains minimum phase when sampled have been derived as a corollary the critical sampling frequency the fastest sampling frequency ensuring minimum phase behaviour is shown to satisfy a simple equation an engineering method to obtain the critical sampling by use of a relay is proposed this method requires no prior knowledge of the plant

the topological structure of fuzzy sets with endograph metric for a non degenerate convex subset y of the n dimensional euclidean space r n let k y be the family of all fuzzy sets of r n which are upper semicontinuous fuzzy convex and normal with compact supports contained in y we show that the space k y with the topology of endograph metric is homeomorphic to the hilbert cube q 1 1 w iff y is compact and the space k y is homeomorphic to x n q sup x n 1 iff y is non compact and locally compact

fuzzy differential equations without fuzzy convexity classical fuzzy differential equations defined in terms of the hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets here hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions is combined with ideas of aubin on morphological equations which allow nonlocal set evolution to remove the assumption of fuzzy convexity and thus to allow fuzzy differential equations to be defined for non convex level sets this approach uses reachable sets as a more general form of set integration and in contrast to the aumann set integral does not necessarily give rise to convex sets the results presented in this paper are even more general since they concern fuzzy sets that need to be only closed without additional assumptions of convexity compactness or even normality in particular an existence and uniqueness theorem is established under the assumption that the right hand sides satisfy a one sided lipschitz condition rather than a much stronger lipschitz condition fuzzy delay differential equations are also considered from this new perspective

dynamic topologies and their applications to crisp topologies fuzzifications of crisp topologies and fuzzy topologies on the crisp real line abstract we introduce and develop topologies whose members are dynamic sets a la zeleny this development spawns the notions of phase and co phase spaces fuzzy dual dynamic dual and co fuzzy dual with the following results being obtained result a the notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of hutton gantner steinlage warren klein and lowen result b the dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic result c the fuzzy real lines re interpreted as fuzzy duals of r with the usual euclidean topology t induce on r via the notion of co fuzzy dual a class of canonical stratified fuzzy topologies co r l e g if l is a demorgan algebra and lc 1 then the compactness of a r is equivalent to the ordinary compactness of a w r t t and if l is a hutton algebra r l co r l is metrizable in the sense of hutton and erceg with erceg metric d l such that d l p r 1 p s 1 r s where pr is the crisp point or impulse with support r i e d l extends the euclidean metric on r

on the compactness of fuzzy numbers with sendograph metric in this paper by embedding the fuzzy number space with the sendograph metric into the product of two uniformly support bounded complete metric spaces we characterize compact sets of the fuzzy number space the main result is that with respect to the sendograph metric a closed set is compact if and only if it is uniformly support bounded and equi right continuous at 0

fuzzy homoclinic orbits and commuting fuzzifications fuzzy homoclinic orbits and commuting fuzzifications are defined and studied we study sufficient conditions involving commuting fuzzifications under which a crisp homoclinic orbit is preserved after fuzzification we show the usual fuzzification is commuting and therefore preserves homoclinic orbits several examples are given regarding fuzzy homoclinic orbits and commuting fuzzifications

holomorphic supercurves and supersymmetric sigma models we introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds referred to as holomorphic supercurves more precisely supercurves are morphisms from a riemann surface endowed with the structure of a supermanifold which is induced by a holomorphic line bundle to an ordinary almost complex manifold they are called holomorphic if a generalised cauchy riemann condition is satisfied we show by means of an action identity that holomorphic supercurves are special extrema of a supersymmetric action functional

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

on fuzzy multi valued functions part 2 some topological properties abstract in this paper new topological concepts connected with fuzzy multi valued functions theory as presented in the first part of this work are introduced and their properties are investigated as will be shown in future papers these concepts and properties are indispensable in connection with the analysis of fuzzy economic systems

self reference and chaos in fuzzy logic investigates a range of phenomena from dynamical systems or chaos theory which appear in a simple fuzzy logic with the introduction of self reference within that logic self referential sentences exhibit properties of fixed point attractors fixed point repellers and full chaos on the

on fuzzifications of discrete dynamical systems let x denote a locally compact metric space and f x x be a continuous map in the 1970s zadeh presented an extension principle helping us to fuzzify the dynamical system x f i e to obtain a map f for the space of fuzzy sets on x we extend an idea mentioned in p diamond a pokrovskii chaos entropy and a generalized extension principle fuzzy sets syst 61 1994 277 283 to generalize zadeh s original extension principle in this paper we study basic properties of so called g fuzzifications such as their continuity properties we also show that for any g fuzzification i a uniformly convergent sequence of uniformly continuous maps on x induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and ii a conjugacy resp a semi conjugacy between two discrete dynamical systems can be extended to a conjugacy resp a semi conjugacy between fuzzified dynamical systems throughout this paper we consider different topological structures in the space of fuzzy sets namely the sendograph endograph and levelwise topologies

the topological structure of the space of fuzzy compacta abstract a fuzzy compactum in a space x is an upper semi continuous map f a 0 1 from a nonempty compact subspace a x to the unit closed interval 0 1 such that x a f x 0 is dense in a in this paper we investigate the topological structure of the space k e x of fuzzy compacta of a separable metric space x with the hausdorff metric in particular we prove that k e x l 2 if x is an infinite locally compact locally connected connected and separable metric space

on a topological fuzzy fixed point theorem and its application to non ejective fuzzy fractals abstract a topological fuzzy fixed point theorem is given when generalizing and improving the main result by diamond kloeden and pokrovskii 1997 1 apart from the sole existence a weak local stability property called non ejectivity in the sense of browder of fuzzy fixed points is established this theorem is then applied for obtaining non ejective fuzzy fractals an alternative approach via the knaster tarski theorem is also presented

moduli of super riemann surfaces the basic properties of super riemann surfaces are presented and their supermoduli spaces are constructed in a manner suitable for the application of algebro geometric techniques to string theory

fuzzy dynamics as an alternative to statistical mechanics abstract we propose a description of continuous evolution of systems with uncertainty based on min or product t norms of fuzzy logic we show that the dynamics has a hamiltonian form in the extended state space composed from physical and information components if the min t norm is used this description is useful when available information on a system behavior is mainly ordinal and not numerical it was shown also that if the product t norm is used the dynamics on a continuous universe is similar to the stochastic processes dynamics but a probability distribution is replaced by a possibility density we show that such replacement leads to a principal extension of the stochastic process theory and correspondence differential equation of evolution for the possibility density cover most basic evolution equations in physics

identification of fuzzy relational equations by fuzzy neural networks abstract although the identification of a fuzzy system by a feedforward neural network is obviously possible whenever the system was continuous however when we use fuzzy max min neural network some problems on the learning of the network arise in this paper we present a method to identify a fuzzy relation by a fuzzy max min neural network we adapt the learning algorithm backpropagation to learning of max min neural network by using one which we will denote as smooth derivative finally we present some examples and a comparison with a similar method

dynamical systems over the space of upper semicontinuous fuzzy sets given a locally compact separable metric space e a perfect transformation defined on e induces a zadeh extension which is a transformation from the space of all upper semicontinuous fuzzy sets defined on e to itself here the latter space is equipped with the hit or miss topology in this setting each upper semicontinuous fuzzy set is identified with its hypograph a closed subset in the product space of e and 0 1 this approach does not require the relevant fuzzy sets to hold a compact support and it also overcomes the drawback of the traditional level set method further it is proved that the zadeh extension with this setting is continuous in the hit or miss topology and dynamical properties of zadeh extension regarding iteration have been explored

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

existence and uniqueness of a fuzzy solution for the nonlinear fuzzy neutral functional differential equation this work is concerned with a class of nonlinear fuzzy neutral functional differential equations specifically existence and uniqueness of fuzzy solutions for the nonlinear fuzzy neutral functional differential equation where a is a fuzzy coefficient and f and g are continuous functions are established via banach fixed point analysis approach and using the fuzzy number whose values are normal convex upper semicontinuous and compactly supported interval in en

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

regularity theorems and energy identities for dirac harmonic maps we study dirac harmonic maps from a riemann surface to a sphere open image in new window we show that a weakly dirac harmonic map is in fact smooth and prove that the energy identity holds during the blow up process

dirac harmonic maps we introduce a functional that couples the nonlinear sigma model with a spinor field open image in new window in two dimensions it is conformally invariant the critical points of this functional are called dirac harmonic maps we study some geometric and analytic aspects of such maps in particular a removable singularity theorem

evolution of systems under fuzzy dynamic laws abstract the global behavior of a system with information on fuzzy described short term evolution laws is proposed and investigated in the paper for deterministic or stochastic models such problems are solved by use of differential equations describing the short term laws but if the short term laws of evolution are known only in a fuzzy form and the information on the initial state of the system is also fuzzy or if we do not have a reasonable statistical model describing the systems environment the fuzzy approach is more appropriate we will show that the introduction of the fuzzy state fuzzy velocity and use of the fuzzy logic rules for obtaining the equation of evolution of the system leads to a consistent description of the fuzzy dynamics our approach is based on the fuzzy master equation an analog of the equation describing the markov process in stochastic dynamics for continuous evolution we obtain differential equations describing the flows of the fuzzy state two cases of weak and strong fuzziness are considered a comparison with other approaches to fuzzy dynamics is performed