dual fuzzy neighbourhood spaces i abstract a fuzzy neighbourhood space fns is determined by means of a system of fuzzy neighbourhoods of the ordinary points of that space we provide a general method for constructing bases for all fuzzy neighbourhood systems in a natural way from the fuzzy topologies of a special type of fuzzy topological spaces fts s which we call the fuzzy neighbourhood base spaces this construction is shown to be functorial of those new fts s the full subcategory of dual fuzzy neighbourhood spaces is shown to be isomorphic to the category of fns s and a fns will coincide with its dual fts iff it is topologically generated

on regularity axioms in fuzzy neighbourhood spaces abstract two notions of regularity in fuzzy neighbourhood spaces fns s have been introduced one by t m g ahsanullah and the other by the authors we prove that those two notions coincide we then characterize a regular fns as a fns whose level topologies for all 0

pointwise estimates for heat kernels of convolution type operators the work was supported by sfb1283 of german research council the third and the forth authors were partially supported by the russian science foundation project 14 50 00150

spectral analysis of non local schrodinger operators abstract we study spectral properties of convolution operators l and their perturbations h l v x by compactly supported potentials results are applied to determine the front propagation of a population density governed by operator h with a compactly supported initial density provided that h has positive eigenvalues if there is no positive spectrum then the stabilization of the population density is proved

fuzzy closure spaces abstract fuzzy topological spaces do not constitute a natural boundary for the validity of theorems but many results can be extended to what are called fuzzy closure spaces or fcs s for short the notions of a subspace a sum and a product are extended to fcs s the hereditary additivity and productivity behaviour of compactness in fcs s is investigated and some weak forms of compactness and fuzzy continuous functions in fcs s are introduced the interaction between fuzzy proximity spaces and fcs s is investigated a necessary background is included for completeness

on fuzzy pseudo normed vector spaces abstract we provide a method for introducing fuzzy pseudo metric topologies on sets and fuzzy pseudo normed topologies on vector spaces over r or c which will be fuzzy linear topologies we define fuzzy pseudo metrics for pairs of crisp points and fuzzy pseudo norms for crisp points as fuzzy real numbers 0 as defined by hutton we define the associated fuzzy open balls with crisp points for their centres and fuzzy real numbers 0 for their radii these form a basis for the associated fuzzy pseudo metric topology the axioms governing fuzzy pseudo metric and fuzzy pseudo norm are straightforward extensions for the corresponding axioms in the crisp case the formulation conforms with zadeh s extension principle we show that katsaras concept of fuzzy seminorm fuzzy sets and systems 12 1984 143 154 is equivalent to ours in the sense that both concepts will result in same fuzzy linear topologies

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

on fuzzy neighbourhood spaces abstract we study the behaviour of the fuzzy closure operator of a fuzzy neighbourhood space fns for short with regard to the two operations of sum and conjunction on fuzzy subsets we supply sufficient conditions on a cech fuzzy closure operator to be the fuzzy closure operator of a fns finally we study quotient fuzzy neighbourhood systems with the help of the notion of dual fuzzy neighbourhood spaces of 11

topological modification of a fuzzy closure space abstract fuzzy topological spaces do not constitute a natural boundary for the validity of theorems and many results can be extended to what are called fuzzy closure spaces fuzzy closure spaces have been introduced in 6 as a generalization of closure spaces 1 in the present paper we introduce the concept of topological modification of a fuzzy closure operator we prove that the topological modification of a fuzzy closure operator u on x is the finest topological fuzzy closure operator on x coarser than u the interaction between the concepts of a subspace and the topological modification of a fuzzy closure operator is also examined

trigonometric series with general monotone coefficients we study trigonometric series with general monotone coefficients convergence results in the different metrics are obtained also we prove a hardy type result on the behavior of the series near the origin

convergence of fourier series with complex quasimonotone coefficients and coefficients of bounded variation of order m abstract l1 convergence theorems for fourier series with complex quasimonotone coefficients are generalized the class of null sequences of bounded variation of order m is extended and the point wise theorem for fourier series with such coefficients is proved

pseudo closure operators in fuzzy topological spaces abstract the concept of a pseudo closure operator s of an fts x has been introduced in 6 in this paper we give an axiomatic characterization of pseudo closure operators also we give a necessary and sufficient condition on an fts x so that is the unique fuzzy topology which induces s at the end of the paper we give a characterization of fts s of structure a 6 in terms of pseudo closure operators

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

fuzzy cores and their extensions abstract we introduce core theory simultaneously for general topology and for fuzzy topology a fuzzy core on a set x is an operator from a subfamily of i x into i x subject to certain conditions we identify all the extension of a fuzzy core fuzzy cech closure operators and to fuzzy kuratowski operators the main purpose of this theory is to characterize as we do here some properties of functions as being equivalent to their continuity with respect to certain fuzzy topologies on their domains and codomains we nominate three existing theories as special instances of core theory one is from general topology and is strongly related to applications the theory of the scott topology on a partially set the other two theories are from fuzzy topology klein s theory of closure operators and the mashhour et al theory of pseudo closure operators

semi induced fuzzy topologies abstract induced fuzzy topologies were first introduced by wong 15 and weiss 14 they were studied by lowen who called them topologically generated fuzzy topologies 7 an induced fuzzy topology includes its modification topology we generalize induced fuzzy topologies to semi induced fuzzy topologies viz those which include their modification topologies the yet more general fuzzy topologies of structure c are also introduced the fully stratified fuzzy topological spaces of structure c are precisely the fuzzy neighbourhood spaces of lowen 4 in subsequent papers this relationship will be established and benefited from the two axioms of semi induction and structure c hold trivially for crisp fuzzy topologies we call such axioms fuzzy structural axioms in fuzzy topology in this paper we analyze the axioms of structure c semi induction and induction in this context we show how these axioms are linked to the three structural asioms of structures a b and d introduced by us in 11 the basic notions and results pertaining to structures a b and d are compiled in the first section the behaviour of warren s fuzzy boundary in semi induced fuzzy topological spaces fts s for short is also explained

on the principal eigenvalue of some nonlocal diffusion problems abstract in this paper we analyze some properties of the principal eigenvalue 1 of the nonlocal dirichlet problem j u x u x u x in with u x 0 in r n here is a smooth bounded domain of r n and the kernel j is assumed to be a c 1 compactly supported even nonnegative function with unit integral among other properties we show that 1 is continuous or even differentiable with respect to continuous differentiable perturbations of the domain we also provide an explicit formula for the derivative finally we analyze the asymptotic behavior of the decreasing function 1 when the dilatation parameter 0 tends to zero or to infinity

on the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level topologies abstract in this paper we discuss the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level topologies necessary and sufficient conditions for the existence of a solution are given and it is proved that in the particular case of fuzzy neighbourhood spaces this solution is always unique

nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces abstract the nearness concept giving rise to a fuzzy topological space is a fuzzy relation of closeness between fuzzy subsets and crisp points of the space lowen 8 introduced the category of fuzzy neighbourhood spaces which we denote by fns his heuristic idea was that the fuzzy topologies in fns should be characterized by means of fuzzy neighbourhoods of crisp points it may be argued that the fuzziness of some concepts related to fns is rather restricted but besides the benefits gained from this restriction in that fns is a category of well behaved fuzzy topological spaces cf 1 5 8 9 14 17 18 we argue in this paper that this restriction acts to reduce the above mentioned concept of nearness to the following one the nearness concept giving rise to a fuzzy neighbourhood space is a fuzzy relation of closeness between the crisp subsets and the crisp points of the space such fuzzy relation may naturally arise when fuzziness is engrained in the definition of the points of the space for instance in the fuzzy unit interval of hutton 4 artico and moresco 2 introduced a concept of fuzzy proximities we show that this concept is particular to fuzzy neighbourhood spaces only we then argue that the nearness concept giving rise to a fuzzy proximity space is a fuzzy relation of proximity defined on all pairs of crisp subsets of the space

fuzzy t locality spaces abstract a lowen fuzzy topological space is said to be localizable at points if it is completely determined in some prescribed manner by a system of fuzzy neighbourhoods called here fuzzy localities of its ordinary points we introduce for each lower semicontinuous triangular norm t a new type of those spaces which we call fuzzy t locality spaces the particularity of a fuzzy t locality space is that its fuzzy interior operator is obtained from its t locality system through a definition of fuzzy containment of fuzzy sets that employs the t residuated implication operator the origins of this new theory lie in the theory of probabilistic metric spaces indeed fuzzy t locality spaces provide a fuzzy topological framework for a naturally defined notion of metric t continuity of functions between probabilistic t metric spaces in a manner described in the last two sections of this article before that we establish characterizations and aspects of good behaviour for fuzzy t locality spaces including that every t locality system has an open basis we briefly study their level topologies and we show that all topologically generated spaces are fuzzy t locality spaces we prove that the category t fls of fuzzy t locality spaces and continuous functions between them is a topological category we show that it is closed in fts under the formation of optimal lifts of sources we provide a condition on t under which t fls becomes closed in fts also under the formation of co optimal lifts of sinks

linear fuzzy neighborhood spaces some of the properties of linear fuzzy neighborhood spaces are studied a special class of these spaces is the family of all fuzzy linear spaces whose topology is given by probabilistic seminorms as in the classical case it is shown that a linear fuzzy neighborhood space is probabilistic normable iff it is hausdorff and it has an n bounded convex fuzzy neighborhood of zero

continuous model for homopolymers we consider the model for the distribution of a long homopolymer in a potential field the typical shape of the polymer depends on the temperature parameter we show that at a critical value of the temperature the transition occurs from a globular to an extended phase for various values of the temperature including those at or near the critical value we consider the limiting behavior of the polymer when its size tends to infinity

estimates of heat kernels for non local regular dirichlet forms in this paper we present new heat kernel upper bounds for a certain class of non local regular dirichlet forms on metric measure spaces including fractal spaces we use a new purely analytic method where one of the main tools is the parabolic maximum principle we deduce off diagonal upper bound of the heat kernel from the on diagonal one under the volume regularity hypothesis restriction of the jump kernel and the survival hypothesis as an application we obtain two sided estimates of heat kernels for non local regular dirichlet forms with finite effective resistance including settings with the walk dimension greater than 2

fuzzy topologies with specified level properties abstract recently the author and p wuyts have studied the problem of defining fuzzy topologies when one is given the family of closure operators or the family of level topologies this paper is concerned with the relationship between these two problems when the family of topologies is a chain with the lt property there is a natural family of closure operators that generates the wuyts maximum i e the largest fuzzy topology with the given level topologies as a corollary it is shown that the fuzzy neighborhood spaces are precisely the pseudo regular spaces in general the relationship between the two problems is weak for example given an lt family of topologies and an l ftp family of semi closure operators there is a fuzzy topology associated with both if and only if the 0 level closure generates the 0 level topology and the closure of a set is always closed in the level topology lastly it is shown that when a function s domain has a wuyts maximum as its topology fuzzy continuity is equivalent to level continuity

l 1 approximation of fourier series of complex valued functions v b stanojevic suggested in her recent paper that it would be of interest to prove a corresponding l 1 convergence theorem for fourier series with complex o regularly varying quasimonotonc coefficients the present paper will discuss this question and establish l 1 convergence and furthermore l 1 approximation theorems for complex valued integrable functions

nonlocal equations with regular varying decay solutions abstract we present a new approach for studying the large time asymptotic behavior to the nonlocal equation t u j u 0 u in the whole r n where j u r n j y u y t d y we obtain a condition so that the solutions decay in time at the rate of a regular varying function r even for non symmetric j for a wide class of r this condition involves a sharp bound for j k u 0 u 0 u t 0 or the kernels of j k and the decay rate is proved by analyzing an exponential series with regular varying coefficients classical cases including the fractional laplacian the compactly support regular kernels and algebraic decaying kernels are shown to satisfy our condition by employing the sharp young s inequality and the fourier transform technique in particular we have generalized the pioneer work by chasseigne et al 2006 11 finally we exhibit nonlocal equation that has solutions with a prescribed decay rate for a wide class of regular varying function

the productivity classes of fuzzy topologies in a previous article fuzzy sets and systems 30 1989 175 191 we introduced classes of fuzzy topological spaces fts s and other special classes of fts s those are tntimatley related to the productivity properties of fuzzy closure and fuzzy interior operators we here provide a number of necessary and sufficient conditions on a fts to be a member of one of those classes we use those characterizations to establish that each one of our classes of fts s is closed under products and quotients in the category of fuzzy topological spaces

remarks on convergence of trigonometric series with special varying coefficients abstract we give an essential generalization of telyakovskii s results on the convergence of trigonometric series with rarely changing coefficients in the norm of l 1

on product fuzzy topological spaces abstract we provide complete answers to the following three questions in fuzzy topology when is the fuzzy closure operator finitely productive on a given family of fuzzy topological spaces which need not be fully stratified when is the fuzzy interior operator finitely productive on such a family when are the coordinate projections open maps from arbitrary products of members of such a family the answer to the first question is the harder one to establish the second and the third question turn out to have same answer

generalized possibility measures we define a generalized possibility measure gpm on a set x to be a function from the power set of x to the unit interval i 0 1 whereby the measure of a finite union of subsets of x equals the maximum of their measures we characterize gpms as the infima of downward directed families of possibility measures this helps to confer an intuitive meaning on gpms we discuss all directed families of possibility measures or equivalently of possibility distributions which generate a given gpm and we identify the largest such family we introduce extensions of gpms to real functions on ix obtained through a continuous triangular norm we provide characterizations and alternative descriptions for those extensions

phase transitions for controlled markov chains on infinite graphs this paper is concerned with some long run average cost problems for controlled markov chains with a denumerable state space the criterion to be optimized contains both reward and penalty functions as a trade off between reward and penalty we observe certain phase transition phenomena our results also provide a stochastic optimal control interpretation for phase transitions of discrete homopolymers with finite attracting potentials

a level topologies criterion for lowen fuzzy uniformizability throughout x will be a nonempty set we consider a subset m of x as a crisp fuzzy set and denote this again by m we denote the strong cut of a fuzzy set e ix by and its weak cut by e i

strong asymptotic equivalence and inversion of functions in the class kc abstract in this paper we consider the class of functions kc introduced by w matuszewska 1964 studia math 24 271 279 and w matuszewska and w orlicz 1965 studia math 26 11 24 as a main result we describe in terms of the class kc when two strictly increasing functions as well as their inverse functions are asymptoticaly equivalent in the strong sense this result also gives a proper characterization of the class kc

fuzzy t neighbourhood spaces part 1 t proximities this is the first one of three articles that are intended to be published consecutively in this journal together they constitute a three part work on the unified subject of fuzzy t neighbourhood spaces and their fuzzy t proximities where t stands for any continuous triangular norm these new notions generalize to arbitrary t corresponding ones due to artico hohle lowen and moresco in this part we define and study notions of fuzzy t proximity one for each t their definition subsumes that of fuzzy proximity due to artico and moresco fuzzy sets and systems 21 1987 85 as our min proximity in particular we study proximity maps the fuzzy topological space associated with a t proximity which in the second part will be proved a fuzzy t neighbourhood space t proximities as fuzzy relations in ordinary power sets t proximal neighbourhood systems and relationships to the hohle lowen fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 the second part of this three part work will be concerned with the categories of fuzzy t neighbourhood spaces one for each t whereas we devote the third part to our t separation axioms in those categories

generating fuzzy topologies with semi closure operators let x be an l fuzzy topological space for a e l 1 and a x x ec a a if and only if g fuzzy open and g x a imply there is a e a with g a 0 under certain restrictions on a which are always satisfied if l is linearly ordered c a is a semi closure operator this paper contains necessary and sufficient conditions under which a collection of semi closure operators generates a fuzzy topology in general such a collection does not generate a unique topology the class of topologies generated is shown to be closed under suprema and its largest member is characterized the topology for the fuzzy unit interval is characterized among those generated by its a closure operators

influence of spatial correlation for directed polymers in this paper we study a model of a brownian polymer in x d introduced by rovira and tindel j funct anal 222 2005 178 201 our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy fluctuation exponent and volume exponent in particular we prove that under some assumptions very strong disorder and superdiffusivity hold at all temperatures when d 3 and provide a novel approach to petermann s superdiffusivity result in dimension one superdiffusivity of directed polymers in random environment 2000 ph d thesis we also derive results for a brownian model of pinning in a nonrandom potential with power law decay at infinity

pseudo closure operators in fuzzy neighbourhood base spaces abstract fuzzy neighbourhood base spaces and dual fuzzy neighbourhood spaces are two special types of fuzzy topological spaces they were introduced by morsi as useful tools for the theory explanation and application of lowen s fuzzy neighbourhood systems the notion of the pseudoclosure operator of a fuzzy topological space was introduced by mashhour et al as a tool in establishing a necessary and sufficient condition for the productivity of fuzzy kuratowski operators pseudo closure operators carry partial information about the fuzzy closure operators of their respective spaces we establish relationships between the pseudo closure operators of fuzzy neighbourhood base spaces their fuzzy closure operators and their level topologies we provide counterexamples to disprove other nonvalid relationships we also study some categorical aspects of the operation of taking the pseudo closure operators of fuzzy topological spaces

a nonlocal convection diffusion equation in this paper we study a nonlocal equation that takes into account convective and diffusive effects ut j u u g f u f u in rd with j radially symmetric and g not necessarily symmetric first we prove existence uniqueness and continuous dependence with respect to the initial condition of solutions this problem is the nonlocal analogous to the usual local convection diffusion equation ut u b f u in fact we prove that solutions of the nonlocal equation converge to the solution of the usual convection diffusion equation when we rescale the convolution kernels j and g appropriately finally we study the asymptotic behaviour of solutions as t when f u u q 1u with q 1 we find the decay rate and the first order term in the asymptotic regime

various fuzzy topological notions and fuzzy almost continuity abstract we extend the notion of almost continuity in the sense of husain 1966 to the fuzzy setting we obtain a pointwise characterization of the notion by dual point and fuzzy net also we formulate conditions under which the two notions of fuzzy almost continuity the one defined in this paper and the one defined by ajmal and azad 1990 turn out to be equivalent the main idea behind this work is to show that using the localized approach developed in earlier papers the study of fuzzy almost continuity can be carried out as successfully as that of its counterpart in general topology in the process we exhibit the compatibility of various existing fuzzy topological notions such as fuzzy regularity c i connectedness i 3 4 c i local connectedness i 3 4 fuzzy t 1 axiom fuzzy t 2 axiom etc

on co optimal lifts in the category of fuzzy neighbourhood spaces the category fns of fuzzy neighbourhood spaces and their continuous functions is closed under the formation of optimal and co optimal lifts in the larger topological category fts of fuzzy topological spaces but while the fuzzy neighbourhood system and the level topologies of an optimal lift of a source in fns have simple descriptions in terms of the fuzzy neighbourhood systems and the level topologies respectively of the source spaces analogous descriptions are yet to be given for both constructs for a co optimal lift of a sink in fns we here establish such descriptions with the help of a functor introduced earlier by one of us with codomain fns and domain another full subcategory of fts the category fnbs of fuzzy neighbourhood base spaces

a class of l 1 convergence it is proved that if the fourier coefficients an of e l 0 it satisfy n 2 k aa p o l for some 1 p 2 then j f o l if and only if a lg n o for cosine trigonometric series with coefficients of bounded variation and satisfying it is proved that a necessary and sufficient condition for the series to be a fourier series is an e c where q is the garrett stanojevic 4 class

a note on fuzzy neighbourhood base spaces abstract fuzzy neighbourhood base spaces fnbs introduced by morsi are a special type of stratified i topological spaces si top previously denoted as fuzzy topological spaces fts we introduce further characterizations of fnbs in terms of level sets also we construct stable subconstructs of si top in the sense of lowen and wuyts stable subconstructs of fts i j fuzzy math 1 3 1993 475 489 whose corresponding total stratified i topologies are generated by elements of i i which are greater than id i the maximal objects of such stable subconstructs have weaker properties than that of the ascending chain property and then these objects give rise to fuzzy neighbourhood base spaces therefore special types of fuzzy neighbourhood base spaces are constructed

representations of fourier coefficients in tauberian l1 convergence classes abstract it is shown that the fourier coefficients satisfying stanojevic s tauberian condition can be represented in terms of o regularly varying sequences

asymptotic behavior for a nonlocal diffusion equation in domains with holes the paper deals with the asymptotic behavior of solutions to a non local diffusion equation ut j u u lu in an exterior domain which excludes one or several holes and with zero dirichlet data on mathbb r n setminus omega when the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes on the other hand if the solution is scaled according to its decay factor close to the holes it behaves like a function that is l harmonic lu 0 in the exterior domain and vanishes in its complement the height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior the inner and the outer behaviors can be presented in a unified way through a suitable global approximation

fuzzy t neighbourhood spaces part 2 t neighbourhood systems we explore a notion of fuzzy t neighbourhood spaces for any continuous triangular norm t and we present on this notion a unified treatment our theory on one hand generalizes the theory of lowen fuzzy sets and systems 7 1982 65 from t min to arbitrary t which has been the progenitor of this work and on the other hand it is strongly related to the theory of l neighbourhoods of hohle mathematics of fuzzy sets logic topology and measure theory kluwer academic publishers dordrecht 1999 when l is restricted to the lattice 0 1 we study the t neighbourhood bases and systems of our spaces their level topologies as well as their relationship to the lowen hohle fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 and to our fuzzy t proximities introduced in part 1 we show that the nearness concept underlying a fuzzy t neighbourhood space is a fuzzy relation of closeness between its ordinary subsets and ordinary points in particular our spaces are in canonical one to one correspondence with the t probabilistic topological spaces of frank j math anal appl 34 1971 67 we demonstrate that each full subcategory t fns of fts of fuzzy t neighbourhood spaces is a topological category to do that we characterize t fns within fts and we characterize continuity of functions within t fns in terms of t neighbourhood bases

on l convergence of trigonometric series abstract in the present paper we consider the trigonometric series with r general monotone and r rest bounded variation coefficients necessary and sufficient conditions of l convergence for such series are obtained in terms of the coefficients moreover we generalize and extend the tikhonov results j math anal appl 347 2008 416 427 to the class gm r or the class rbvs r

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

some more results on completely induced fuzzy topological spaces abstract the object of the paper is to present some new results on completely induced fuzzy topological spaces and completely semi induced spaces earlier introduced by the authors in this journal when a fuzzy topological space becomes a completely induced space is discussed examples of completely induced space are also cited

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

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

a short note on fuzzy neighbourhood spaces abstract lowen introduced the concept of fuzzy neighbourhood spaces fns s and proved that the family of level topologies of a fns is an antichain wuyts characterized fns s as being those fully stratified fuzzy topological spaces fts s whose level topologies form antichains and whose fuzzy topologies are the finest possible for thier level topologies mashhour et al introduced the axiom of structure a on fts s and derived for it several characterizations and fuzzy topological consequences we use those characterizations of structure a and wuyts characterization of fns s to prove that a fts is a fns iff it is of structure a and fully stratified

completely induced fuzzy topological spaces abstract the object of this paper is to introduce and to study the concepts of completely induced fuzzy topological spaces and completely lower semi continuous functions completely lower semi continuous functions turn out to be the natural tool for studying the completely induced fuzzy topological spaces

strong and ultra separation axioms on l fuzzy topological spaces given an l fuzzy topological space x d we introduce new notions of l fuzzy separation axioms using the family of its level topology i a d a pr l we check that these are well defined l fuzzy topological concepts and we compare them with the analogous l fuzzy ones introduced in the literature we verify that these notations are not equivalent and we give many examples which illustrate this fact furthermore some results concerning the new concepts have been established

on l1 convergence of fourier series abstract in this paper we consider the trigonometric fourier series with the general monotone coefficients necessary and sufficient conditions of l 1 convergence for such a series that is f s n o 1 are obtained in terms of coefficients

tauberian conditions for l 1 convergence of fourier series it is proved that fourier series with asymptotically even coefficients and satisfying lima 1lim supn exn jpi i ajf j p 0 for some i p 2 converge in l norm if and only if i1 f n en f n e n ii o l where en t k e k recent results of stanojevic 1 bojanic and stanojevic 2 and goldberg and stanojevic 3 are special cases of some corollaries to the main theorem