quasi coincidence and quasi fuzzy hausdorff the concept of quasi coincidence of a fuzzy point to a fuzzy set is investigated the notion of a quasi fuzzy hausdorff topological space is introduced and its close relationship with the properties of fuzzy hausdorff and hausdorf is shown

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

a nonstandard construction of levy brownian motion a nonstandard construction of levy brownian motion on d is presented which extends r m anderson s nonstandard representation of brownian motion it involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of levy brownian motion moreover a new invariance principle can be deduced in a similar way as donsker s invariance principles follows from anderson s construction

how porous is the graph of brownian motion we prove that the graph of brownian motion is almost surely porous and determine the hausdorff dimension of sets with a given porosity index in particular we show that the porosity index of the graph is yo 0 6948

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

on compact hausdorff l fuzzy spaces abstract good definitions of compactness and hausdorffness are presented for l fuzzy spaces where l is a completely distributive lattice it is then shown that compact hausdorff l fuzzy spaces are topological

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

fuzzy proximity ordered spaces in 1976 singal and lal extended the study of nearness between sets to the nachbin s 1965 topological and order structures also in 1978 dimitrijevic developed such an idea in the same manner with more study of its properties the aim of this paper is to introduce the idea of order in a fuzzy proximity space katsaras 1979 1980 1980 and construct new spaces in the area of fuzzy topological and order structures the so called fuzzy proximity ordered spaces also we study some properties of these spaces

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

semiflexible polymers in straining flows we introduce a model for semiflexible polymer chains based on the integral of an appropriate gaussian process the stiffness is characterized physically by adding a bending energy the degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity the limiting case reduces to the brownian model of completely flexible chains studied in earlier work the calculation of the partition function for the configuration statistical mechanics i e the distribution of shapes of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated gaussian process an exact formula for the partition function is presented however in practice this formula is too complicated for most computations we therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first order term which gives the first order deviation from the completely flexible case in addition to the partition function the method presented here can also deal with other quadratic functionals such as the stochastic area associated with two polymer chains

one dimensional stochastic partial differential equations and the branching measure diffusion an existence theorem in the spirit of keisler ke is proved for the simple one dimensional diffusion equation driven by white noise modulated by a non linear function of the solution this is used to obtain a density and a stochastic partial differential equation in one dimension for the critical branching diffusion studied by dawson et al

on fuzzy syntopogenous structures and preorder abstract in this paper we combine fuzzy topological structure on x with algebraic structure on x and investigate their corresponding structures and their properties first we study the preorder relation generated by an l fuzzy syntopogenous structure conversely we research the l fuzzy syntopogenous structure defined by a preorder relation second we define increasing decreasing l fuzzy syntopogenous spaces with preorder and study some properties of them finally we give an important example of increasing l fuzzy syntopogenous space i e r s r

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

fuzzy proximities and totally bounded fuzzy uniformities after the classical papers of zadeh 181 and chang 3 many concepts of general topology have been extended to fuzzy set theory in this paper we are interested to develop the study of fuzzy uniformities fuzzy proximities and of the connexions between such structures fuzzy uniformities have been introduced by lowen in 9 and with slight modifications in lo and by hutton in 5 the two approaches are quite different however the one proposed by hutton seems to suit much better to fuzzy set theory also because it allows to generalize a lot of results in the most proper way therefore in this paper we shall deal with hutton fuzzy uniformities the concept of fuzzy proximity used up to now see e g 7 8 15 1 is quite unsatisfactory indeed its fuzzyness is rather poor since these fuzzy proximities are in a canonical l 1 correspondence with the usual proximities see remark 2 6 moreover the open sets of the induced topologies are crisp and though every lowen fuzzy uniformity induces a fuzzy proximity this correspondence cannot work well since the two structures do not give the same fuzzy topology for these reasons we propose a new definition of fuzzy proximity it differs from the old one only in axiom p5 in short we could say that we privilege complementation with respect to intersection an agreement between the two operations being problematic in fuzzy set theory we point out that as a consequence a fuzzy set may be far from itself however this fact ought not to amaze anyone since it may happen just for pale sets which in a certain sense resemble the empty set nevertheless this modification enables us to associate a topology in a completely different way moreover every fuzzy uniformity induces a fuzzy proximity and vice versa we succeed to construct a fuzzy uniformity starting with a given fuzzy proximity and the induced topologies do not change at any step this implies that the topologies which admit a fuzzy proximity are exactly the completely regular ones moreover we prove that there exists a l l correspondence between fuzzy proximity spaces and a

cyclically stationary brownian local time processes local time processes parameterized by a circle defined by the occupation density up to time t of brownian motion with constant drift on the circle are studied for various random times t while such processes are typically non markovian their laplace functionals are expressed by series formulae related to similar formulae for the markovian local time processes subject to the ray knight theorems for bm on the line and for squares of bessel processes and their bridges for t the time that bm on the circle first returns to its starting point after a complete loop around the circle the local time process is cyclically stationary with same two dimensional distributions but not the same three dimensional distributions as the sum of squares of two i i d cyclically stationary gaussian processes this local time process is the infinitely divisible sum of a poisson point process of local time processes derived from brownian excursions the corresponding intensity measure on path space and similar levy measures derived from squares of bessel processes are described in terms of a 4 dimensional bessel bridge by williams decomposition of ito s law of brownian excursions

on fuzzy t1 topological spaces on etudie diverses relations entre les differentes definitions des espaces topologiques t 1 flous

forty years of hutton fuzzy unit interval this note is a very short history of a specific fuzzy topological space the hutton s fuzzy unit interval 0 1 l which occupies one of the central positions in fuzzy topology on that occasion the superiority of l valued topology over 0 1 valued topology is being discussed we then show how the hutton s concept evolved after obvious identifications we can today say that 0 1 l is the tensor product 0 1 l of the lattices 0 1 and l an account of the consequences of this new view point is hoped to appear thereafter

on fuzzy t continuous mappings abstract the concepts of fuzzy almost continuous fuzzy continuous and fuzzy weakly continuous mappings have been introduced by yalvac and by mukherjee and sinha the aim of this paper is mainly to study and to find the mutual interrelations among these concepts

mappings in fuzzy topological spaces abstract in this paper some characterizations of fuzzy strongly semicontinuous and fuzzy precontinuous mappings are given by using the notions of quasi coincident and quasi neighbourhood due to pu and liu j math anal appl 76 1980 571 599 the definitions of open and preopen mappings which are defined by mashhour et al acta math hung 41 1983 213 218 are extended to fuzzy sets

central limit theorem for the edwards model the edwards model in one dimension is a transformed path measure for standard brownian motion discouraging self intersections we prove a central limit theorem for the endpoint of the path extending a law of large numbers proved by westwater the scaled variance is characterized in terms of the largest eigenvalue of a one parameter family of differential operators introduced and analyzed by van der hofstad and den hollander interestingly the scaled variance turns out to be independent of the strength of self repellence and to be strictly smaller than one the value for free brownian motion

a critical case for brownian slow points letxt be a brownian motion and lets c be the set of realsr 0 such that uxr t xru c t 0 t h for someh h r 0 it is known thats c is empty ifc 1 a s in this paper we prove thats 1 is empty a s

brownian motion at a slow point on rappelle des resultats de greenwood et perkins 1983 84 et on etudie les proprietes probabilistes et de trajectoire du processus x t b t t b t on montre en particulier que x est un processus de markov non homogene satisfaisant une certaine equation differentielle stochastique

semi interior and semi closure of a fuzzy set on definit le semi interieur et la semi fermeture d un ensemble flou d une facon semblable a celle utilisee dans les espaces topologiques ordinaires

separation axioms in ordered fuzzy topological spaces abstract ti i 1 2 3 ordered separation axioms are defined in ordered fuzzy topological spaces and the validity of some results analogous to those in the crisp case are examined

net convergence in fuzzy neighborhood spaces our aim is to introduce a concept of net convergence in the well behaved subcategory of lowen fuzzy neighborhood spaces of the category of fuzzy topological spaces in doing so we notice that our present notion of net convergence in addition to katsaras and papageorgeou net convergence is generalized sequential convergence we also apply this notion of net convergence in fuzzy neighborhood groups and fuzzy neighborhood linear spaces

on separation axioms in fuzzy topological spaces fuzzy neighborhood spaces and fuzzy uniform spaces submitted by l zadeh a certain number of separation axioms for fuzzy topological spaces are provided all of which are good extensions of the topological t t or t2 all valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones in this paper we introduce and study a number of separation properties of fuzzy topological fuzzy neighborhood and fuzzy uniform spaces all of which are extensions of t t and t in topological spaces we started from the viewpoint that separation axioms at least t originated in the need to guarantee in some sense uniqueness of limits and we tried to generalize the topological t axiom in that direction it appears that a whole range of axioms from a fairly weak condition wt up to a rather strong one tj all give rise to acceptable extensions of the topological property moreover it seems likely that it will not be possible to single out one axiom in fuzzy topology as the best extension of topological t2 we showed in a preceding paper 181 on completions of fuzzy uniform spaces that tj as well as wt collapsing with wt in uniform spaces have a specific role to play in the same sense wt has already shown its usefulness in the study of hyperspaces 131 and t2 in its relation with compactness 191 as to our other axioms from t to wt they all seem to be natural generalizations of the corresponding properties in topological spaces and of the different t notions in fuzzy topological spaces they are moreover hereditary and t and ty excepted also productive as to the relations with other work in this field we first point out that our wt axiom is in fact the 1 hausforff separation property of hiihle 2 31 hutton and reilly in 4 and rodabaugh in 161 also studied separation current address is rijksuniversitair centrum 2020 antwerpen belgium

fuzzy hausdorff topological spaces abstract we introduce the notion of a fuzzy hausdorff topological space and make a few observations to establish the appropriateness of this notion

fuzzy separation axioms and fuzzy continuity in fuzzy bitopological spaces abstract let x 1 2 be a bitopological space and i i 1 2 be the set of all lower semi continuous functions defined from x into the closed unit interval 0 1 in this paper we have succeeded in showing goodness of extension of separation axioms for fuzzy bitopological spaces namely we prove the space x 1 2 is p t i p t iw p regular and p normal iff the fuzzy bitopological space x 1 2 is p t i p t iw p regular and p normal

some brownian functionals and their laws we develop some topics about brownian motion with a particular emphasis on the study of principal values of brownian local times we show some links between principal values and doob s h transforms of brownian motion for nonpositive harmonic functions h we also give a survey and complement some martingale approaches to ray knight theorems for local times

on internal characterizations of completelyl regular spaces abstract complete l regularity is internally characterized in terms of separating chains of open l sets a possible characterization in terms of normal and separating families of closed l sets is discussed and it is shown that spaces admitting such families are completely l regular the question of whether the converse holds true remains open some partial solutions are however given e g in the class of countably compact spaces

self intersections of 1 dimensional random walks consider a random walk snon the integers where the steps i have mean 0 and variance 2 let t be the time of first self intersection of the random walk it is shown that as t grows at rate 2 3 more precisely t 2 3 has a non degenerate limit distribution which can be described in terms of brownian motion local time

on brownian slow points it is shown that for a wiener process x t both the quantities mathop inf limits t mathop overline lim limits h text to text 0 x t text text h x t sqrt h text and mathop text sup limits t mathop underline lim limits h text to text 0 text x t text text h x t sqrt h are almost surely equal to 1

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

strong separation and strong countability in fuzzy topological spaces given a fuzzy topological space x we introduce new notions of fuzzy separation and fuzzy countability using the family of its level topologies t t 0 1 we check that these are well defined fuzzy topological concepts and we compare them with the analogous fuzzy ones introduced in the literature we verify that these notions are not equivalent and we give a large number of examples which illustrate this fact

a comparison of some hausdorff notions in fuzzy topological spaces abstract several different notions of the hausdorff separation property for fuzzy topological spaces have appeared in the literature this paper is a study of the relationships between 12 of these concepts

on convexity of fuzzy syntopogenous spaces in this paper we combine fuzzy syntopogenous structures with algebraic on x and look for their corresponding structures and typical properties first the increasing and decreasing spaces have been studied and an important example r sr has been given theorem 3 10 second two general types of convexity on l fuzzy syntopogenous structures x s have been defined the examples of them have been given and their relationship has been investigated third the equivalent description of convexity have been given finally a sequence of important properties of convexity have been obtained

weak invariance principles for local time we establish sufficient conditions to ensure that the suitably defined local times of a sequence of random walks converge weakly to brownian local time

algebraic characterization of some fuzzy topological spaces abstract let x a be a fuzzy topological space and c x a the semigroup of fuzzy continuous mappings f x a x a we prove for some fuzzy topological spaces x a and and y b that they are fuzzy homeomorphic iff c x a and c y b are isomorphic semigroups

a note on k k azad s fuzzy hausdorffness concept abstract in this paper we investigate the fuzzy hausdorffness concept of azad 1 and note that it neither generalizes the usual hausdorffness nor is a good extension of usual hausdorffness a new concept of fuzzy hausdorffness called hausdorffness related to azad s concept is then proposed and we observe that it in addition to being free from the drawbacks in azad s concept also possesses several pleasing properties

fuzzy semi preopen sets and fuzzy semi precontinuous mappings abstract the aim of this paper is to introduce and discuss the concepts of fuzzy semi preopen sets and fuzzy semi precontinuous mappings

a stochastic flow arising in the study of local times a stochastic flow of homeomorphisms of open image in new window previously studied by bass and burdzy 2 and hu and warren 4 is shown to arise in the study of the local times of brownian motion this leads to a new proof of the ray knight theorems for the flow via the classical ray knight theorems for brownian motion

on fuzzy hausdorffness concepts we continue our earlier work on fuzzy hausdorffness 11 by relating our fuzzy hausdorffness notion with those due to other authors

a note on some ft 2 concepts abstract in this note we study in particular the fuzzy hausdorffness concepts introduced by adnadjevic 1 ganguly and saha 4 and sarkar 17

trees under attack a ray knight representation of feller s branching diffusion with logistic growth we obtain a representation of feller s branching diffusion with logistic growth in terms of the local times of a reflected brownian motion h with a drift that is affine linear in the local time accumulated by h at its current level as in the classical ray knight representation the excursions of h are the exploration paths of the trees of descendants of the ancestors at time t 0 and the local time of h at height t measures the population size at time t we cope with the dependence in the reproduction by introducing a pecking order of individuals an individual explored at time s and living at time t h s is prone to be killed by any of its contemporaneans that have been explored so far the proof of our main result relies on approximating h with a sequence of harris paths h n which figure in a ray knight representation of the total mass of a branching particle system we obtain a suitable joint convergence of h n together with its local times and with the girsanov densities that introduce the dependence in the reproduction

on connections between the countability properties of x ω t and x t abstract the connections between some countability properties of a topological space x t and its generated fuzzy topological space x t are investigated

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

fuzzy syntopogenous structures and order the study of the relationship between fuzzy topological and order structures was initiated by katsaras in j math anal appl 84 1981 44 58 in m y bakier et al proc of assiut first intern conf part viii 1990 41 61 some properties of these spaces are studied in m y bakier and k el saady fuzzy sets and systems 54 1993 213 220 is by similar manner introduced the idea of order in fuzzy topological vector spaces in another work of a s mashhour et al the study of the relationship between order and lowen fuzzy uniformities is introduced and many results are studied in this paper by similar manner we will introduce the idea of order in both the fuzzy proximity space and the fuzzy syntopogenous space and we study some properties of these spaces

on separation axioms in a newly defined fuzzy topology abstract in fuzzy sets and systems 45 1992 79 82 hazra et al defined a fuzzy topology on a non empty set in a new fashion here we introduce separation axioms and some allied notions in this new set up and prove several results related to these notions

forward brownian motion we consider processes which have the distribution of standard brownian motion in the forward direction of time starting from random points on the trajectory which accumulate at infty we show that these processes do not have to have the distribution of standard brownian motion in the backward direction of time no matter which random time we take as the origin we study the maximum and minimum rates of growth for these processes in the backward direction we also address the question of which extra assumptions make one of these processes a two sided brownian motion

a comparison of some ft 2 concepts abstract recently ghanim kerre and mashhour 2 have introduced and examined a fuzzy hausdorffness concept in this paper we compare it with some of the other existing concepts of fuzzy hausdorffness and note thereby that though it is a good extension of hausdorffness it is independent of several other fuzzy hausdorffness concepts currently in use

la filtration de b l let btbe a one dimensional brownian motion and let ltbe its local time at zero we answer a question of j pitman by showing that the canonical filtrations of b and b cl are equal for all real constants c it is also shown that the filtrations of b and b cl are equal if and only if c 1

a conditioned limit theorem for random walk and brownian local time on square root boundaries we count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a local time whose inverse is a stable subordinator of a known index and a brownian motion the study of this limit process leads to some precise sample path properties of brownian motion these properties improve earlier results of dvoretsky and kahane on the existence of small oscillations in the brownian path

a pure jump markov process with a random singularity spectrum we construct a nondecreasing pure jump markov process whose jump measure heavily depends on the values taken by the process we determine the singularity spectrum of this process which turns out to be random and to depend locally on the values taken by the process the result relies on fine properties of the distribution of poisson point processes and on ubiquity theorems

l fuzzy normal spaces and tietze extension theorem abstract in this paper we prove a characterization theorem for normal l fuzzy topological spaces l is an infinitely distributive lattice with an order reversing involution in the particular case l 0 1 this theorem reduces to a known result of kat tov fund math 38 1951 85 91 and tong duke math j 19 1952 289 292 as an important application we obtain a fuzzy version of tietze extension theorem this yields an affirmative answer to a question raised in a series of papers by rodabaugh fuzzy sets and systems 11 1983 163 183

local times on curves and uniform invariance principles sufficient conditions are given for a family of local times l t ofd dimensional brownian motion to be jointly continuous as a function oft and then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of brownian motion uniformly over a large family of measures applications included some new results for intersection local times for brownian motions on 2 and 2

separation axioms in fuzzy topological ordered spaces abstract since the fuzzy topological space x may be considered as a fuzzy topological ordered space when it is realised that the non empty set x is partially ordered by agreeing that x y in x if and only if x y then the study of the fuzzy topological ordered spaces not only includes the study of the abstract fuzzy topological spaces but also reveals many generalizations of well known results concerning the abstract fuzzy topological spaces this paper provides a certain number of separation axioms for fuzzy topological ordered spaces which we label ft i order separation axioms for i 1 2 3 4 the relationships between some of the ft i order separation axioms are studied

on polymer conformations in elongational flows we consider various models of polymer conformations using paths of gaussian processes such as brownian motion in each case the calculation of the law of the moment of inertia of a random polymer structure which is equivalent to the calculation of the partition function is reduced to the problem of finding the law of a certain quadratic functional of a gaussian process we present a new method for computing the laplace transforms of these quadratic functionals which exploit their special form via the ray knight theorem and which does not involve the classical method of eigenvalue expansions we apply the method to several simple examples then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory

martingale transforms and lp norm estimates of riesz transforms on complete riemannian manifolds under the condition that the bakry emery ricci curvature is bounded from below we prove a probabilistic representation formula of the riesz transforms associated with a symmetric diffusion operator on a complete riemannian manifold using the burkholder sharp l p inequality for martingale transforms we obtain an explicit and dimension free upper bound of the l p norm of the riesz transforms on such complete riemannian manifolds for all 1 p in the euclidean and the gaussian cases our upper bound is asymptotically sharp when p 1 and when p

fuzzy t neighbourhood spaces part 3 t separation axioms we address the problem of identifying a useful set of mutually compatible separation axioms for each category t fns of t neighbourhood spaces it should contain one of t complete regularity that characterizes t uniformizability as indeed we achieve here this seems to necessitate the parameterization of most separation axioms by the triangular norm t with the exception of the two axioms t0 and r0 which have categorical definitions valid in any topological category we introduce t separation axioms in terms of t and fuzzy closure operators only making them meaningful for the whole of fts we then characterize them within t fns in terms of t fuzzy closures of crisp fuzzy subsets and t neighbourhoods we study our t axioms and establish some entailments between them our t complete regularity commits us to investigate a t real line which is a t neighbourhood space induced by a certain probabilistic t metric due to hohle fuzzy sets and systems 1 1978 311

concerning the constants in fuzzy topology on etudie les cas mathematiquement importants qui justifient l etude des topologies quasi floues

nonstandard construction of the stochastic integral and applications to stochastic differential equations ii h j keisler has recently used a nonstandard theory of lto integration due to r m anderson to construct solutions of ito integral equations by solving an associated internal difference equation in this paper we use the same general approach to find solutions y t of semimartingale integral equations of the form y t c h t c f s y w dz s where z is a given semimartingale h is a right continuous process and f s w is continuous on the space of right continuous functions with left limits with the topology of uniform convergence on compacts in addition we generalize keisler s continuity theorem and give necessary and sufficient conditions for an internal martingale to be s continuous introduction this paper is a continuation of hoover and perkins 5 notation and definitions presented in 5 are used without further introduction and references to that paper are made by simply citing the number of the theorem or lemma due to the consecutive numbering of the two papers no ambiguity will arise finally the reader is referred to 0 of 5 for a brief description of the contents of this work 8 continuous local martingales we assume throughout this section that all processes are r or r valued since the main results theorems 8 1 8 5 and 8 6 then follow immediately for higher dimensions we will show theorem 8 1 that a continuous local martingale has a q1 local martingale lifting for any internal filtration t recall that in general theorem 5 6 asserted only that such a lifting exists for some internal filtration for this reason unless otherwise stated the internal filtration under consideration will be it i t e t which was used in the original definition of t note that if x t x q m is an s continuous process i e x w is s continuous a s then st x has continuous paths a s and x is a uniform lifting of st x in the sense of keisler 6 the following lifting theorem was first proved in a more specialized setting in panetta 9 the proof given here is different received by the editors january 22 1982 the results of this paper were presented at the second victoria symposium on nonstandard analysis victoria canada june 16 20 1980 a ms mos subject cllassifications 1970 primary 60h20 02h25 secondary 60g45

basic notions for fuzzy topology 1 abstract a new notion of fuzzy real line is presented applications are obtained when solving the problem of generating fuzzy topologies by fuzzy metrics the results provide a fundamental example as a basis for a convergence based approach to fuzzy topology

on the hausdorff dimension of the brownian slow points for each c1 c2 consider the set of one sided slow points of the brownian path b defined by s c 1 c 2 omega t exists delta 0 text b t h b t in c 1 h 1 2 c 2 h 1 2 forall 0 underset raise0 3em hbox smash scriptscriptstyle leqslant h underset raise0 3em hbox smash scriptscriptstyle leqslant delta we find the hausdorff dimension of s c1 c2 as well as s c1 c2 a where a a progressively measurable set satisfying certain hypotheses in particular the hausdorff dimension of the two sided slow points s c1 c2 and the one sided and two sided slow points in the zero set of b are obtained by making an appropriate choice of a these results show for example that s c c o or o according as c is greater than or less than the smallest positive zero of m 1 2 1 2 x2 2 1 3069 where m is the confluent hypergeometric function this confirms a conjecture of burgess davis whose methods play a major role in the proofs the results are also extended to higher dimensions

on n convergence of fuzzy nets abstract in this paper we define the n convergence of fuzzy nets in a fuzzy topological space and we use it to give a characterization of some fuzzy topological concepts we give necessary conditions for the n convergence of fuzzy nets in a fuzzy topological space and we characterize this n convergence in a topologically generated fuzzy topological space we also study some characterizations of fuzzy ultranets and we introduce the concept of subordination between fuzzy nets we associate to each fuzzy net in x a prefilter on x and conversely and we give an exhaustive description of the connections between these two theories we give a theory of fuzzy convergence classes and we characterize fuzzy topologies via some special types of fuzzy convergence classes

multifractal spectra of fragmentation processes let s t t 0 be a homogeneous fragmentation of 0 1 with no loss of mass for x 0 1 we say that the fragmentation speed of x is v if and only if as time passes the size of the fragment that contains x decays exponentially with rate v we show that there is v typ 0 such that almost every point x 0 1 has speed v typ nonetheless for v in a certain range the random set g v of points of speed v is dense in 0 1 and we compute explicitly the spectrum v dim g v where dim is the hausdorff dimension

regularly open sets and a good extension on fuzzy topological spaces singal and rajvanshi 1992 have introduced the concepts of fuzzy almost separation axioms and they investigated some results on these concepts and the concepts of fuzzy almost continuity and fuzzy almost open mappings in this paper we have succeeded in showing goodness of extension of fuzzy almost separation axioms and we show by producing counterexamples that some results which appeared in the work of singal and rajvanshi are incorrect also we discuss under which conditions some of these results are true and we give an improvement of some results in singal and rajvanshi 1992

orbital stability of peakons for a generalized ch equation abstract in this letter two different kinds of peaked traveling wave solutions for a generalized camassa holm equation are considered the orbital stability of these peaked traveling wave solutions are proved

a note on α and α hausdorffness in this paper we have introduced a email protected hausdorffness in an i fuzzy topological space in short i fts in the sense of sostak and kubiak and characterize them in terms of the corresponding level topologies it is shown that these notions satisfy the hereditary productive and projective properties the degree to which an i fts is a email protected hausdorff is also discussed

on the character of convergence to brownian local time ii in this paper we consider the sequences of stochastic processes which converge weakly asn to brownian local time these processes are generated by a recurrent random walk with finite variance the main result is the following it is possible to redefine a random walk in such a way that for a wide class of processes the normalized differences between them and brownian local time converge in distribution to some stochastic process we also prove that such differences with probability one have the logarithmic upper bound it is so called strong invariance principles for local times