even continuity and equicontinuity in fuzzy topology we study the notion of even continuity in the category fcs of fuzzy convergence spaces and in fns the category of fuzzy neighborhood spaces we further compare this notion to equicontinuity in fus the category of fuzzy uniform spaces

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

pseudo differential operators and differential structures ii introduction let m be a smooth closed manifold the algebra of oth order pseudo differential operators on m p0 depends explicitely on the differential structure on m for its construction 1 there is a natural embedding of po as a subalgebra of l2 m the bounded operators on l2 m there is a symbol map a po co sm where sm is the tangent sphere bundle of m after taking the norm closure of p0 one obtains an extension of c algebras

fuzzy topological spaces with conical neighborhood systems abstract for a commutative integral and meet continuous quantale q a cns space is defined to be a q topological space such that the q neighborhood system of each point is a conical q filter these spaces reduce to fuzzy t locality spaces in the case that the quantale is the unit interval endowed with a left continuous t norm t and reduce to probabilistic q topological spaces in the case that q is an mv algebra for a continuous quantale q a necessary and sufficient condition is obtained for the category of cns spaces to be simultaneously reflective and coreflective in the category of stratified q topological spaces this result provides an interesting example of the interaction between properties of the quantale q and that of q topological spaces

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

the spectrum minimum for random schrodinger operators with indefinite sign potentials this paper sets out to study the spectral minimum for an operator belonging to the family of random schrodinger operators of the form h wper v where we suppose that v is of anderson type and the single site is assumed to be with an indefinite sign under some assumptions we prove that there exists 0 0 such that for any 0 0 the minimum of the spectrum of h is obtained by a given realization of the random variables

toeplitz operators and quantum mechanics abstract toeplitz operators on the segal bargmann spaces of gaussian measure square integrable entire functions on complex n space c n are studied the c algebra generated by the weyl form of the canonical commutation relations consists precisely of the uniform limits of almost periodic toeplitz operators the question of which toeplitz operators admit a symbol calculus modulo the compact operators is raised and sufficient conditions are given for such a calculus these conditions involve a notion of slow oscillation at infinity

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

a note on compactness in a fuzzy metric space we study the compactness in a fuzzy metric space we also give a generalization of niemytzki tychonoff theorem for a fuzzy metric space

on the theories of fuzzy syntopogeny for the framework fns abstract two notions of fuzzy syntopogeny have been introduced for the framework fns of fuzzy neighbourhood spaces one by katsaras and the other by mashhour morsi and el tantawy we identify a property that holds in all the special types of interest from katsara s fuzzy topogenous orders and we call it structure a we show that the katsara s fuzzy topogenous orders of structure a are completely determined by their behaviour on ordinary subsets we then establish natural bijections between them and the saturated n topogenous orders of mashhour et al those bijections are related to flushoh s ideas on fuzzy filters

a fuzzy r 1 separation axiom abstract we introduce a new r 1 separation axiom for fuzzy topological spaces we identify from the literature which lower fuzzy separation axioms are compatible with our r 1 axiom we provide alternative descriptions for this r 1 axiom in the special setting of fuzzy neighborhood spaces this special case has standard higher fuzzy separation axioms in it we establish aspects of compatibility of our r 1 axiom with those higher separation axioms as well as with fuzzy compactness

invariant probabilistic metrizability of fuzzy neighbourhood groups we introduce the notions of n local compactness and n openess in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups together with some characterizations for them in particular we prove that a fuzzy neighbourhood space is n locally compact if and only if its level spaces for all 0 1 are locally compact a similar criterion is established for n open functions we also introduce the notion of fuzzy absolute value functions on groups we show that there exists a 1 1 correspondence between invariant fuzzy probabilistic pseudo metrics on groups and fuzzy absolute value functions we establish theorems on fuzzy probabilistic pseudo metrizability of fuzzy neighbourhood groups finally fuzzy probabilistic metrizability of quotient fuzzy neighbourhood groups is also taken into account

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

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

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

fuzzy neighborhood syntopogenous structures abstract we introduce the notion of fuzzy neighbourhood syntopogenous structures as a unified theory of fuzzy neighbourhood spaces lowen fuzzy uniformities and artico moresco fuzzy proximities our primary idea is that a fuzzy neighbourhood syntopogenous structure on a set x consists of fuzzy neighbourhood of the ordinary subsets of x

the theory of global fuzzy neighborhood structures iii fuzzy uniform structures in this paper a new notion of fuzzy uniform structure is presented and investigated fuzzy uniform structures are given as special fuzzy filters of special interest are those fuzzy uniform structures which are sup fuzzy filters they can be characterized by a related notion of fuzzy syntopogenous structure as is shown fuzzy metrics gahler and gahler 1994 generate fuzzy uniform structures in our sense each fuzzy uniform structure has an associated global homogeneous fuzzy neighborhood structure and hence also an associated fuzzy topology in case of a 0 1 fuzzy uniform structure and even in a more general case a fuzzy uniform structure of the first type can be associated there are given relation to huttons s and katsaras notions of fuzzy uniform structure hutton 1977 katsaras 1979 and to the notion of fuzzy syntopogenous structure in sense of katsaras and petalas 1984

some categorical aspects of fuzzy topology research in recent years has revealed that the construct of fuzzy topological spaces behaves quite differently from that of topological spaces with respect to certain categorical properties in this paper we discuss some of these aspects since the topological construct l fts contains nontrivial both initially and finally closed full subconstructs and each such construct gives rise to a natural autonomous theory of fuzzy topology it can be said to some extent that fuzzy topology should consist of a system of closely related topology theories including the classical topology theory as a special case with each applying to one such subconstruct therefore in the first part of this paper the theory of sobriety is established for each finally and initially closed full subconstruct of l fts to illustrate this idea the second topic of this paper is the relationship between the construct of stratified l fuzzy topological spaces and several other familiar constructs in fuzzy topology for example the constructs of sostak fuzzy topological spaces and l fuzzifying topological spaces

fuzzy proximities induced by functional fuzzy separation abstract mashhour and morsi introduced an axiom of n complete regularity to characterize the lowen fuzzy uniformizable fuzzy neighbourhood spaces fns s guided by that axiom we supply a sierpinski object for the artico moresco category fp of fuzzy proximity spaces which katsaras has shown to be a topological category we define the degree of functional fuzzy separatedness of a pair of subsets of a fns we use that notion to define the cech fuzzy proximity of an n completely regular fns x we establish that it is the finest fuzzy proximity that induces the fns x we study in some detail the cech fuzzy proximities obtained from n normal fns s

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

the spectral minimum for random displacement models consider a one dimensional schrodinger operator with potential v given as follows fix a single site potential f which is supported in an interval of length less than 1 construct v by placing a translate of f into each unit interval n n 1 for an integer n where otherwise the positions of each translate are arbitrary which configuration of single sites minimizes the spectral minimum of the schrodinger operator with potential v this question is equivalent to finding the spectral minimum of the random displacement model we conjecture that the minimum is realized through pair formation of the single sites we provide a partial proof of this conjecture and additional numerical evidence for its correctness

an enriched category approach to many valued topology in this paper we discuss a basic question in fuzzy set theory what is a many valued topology on a set x an answer is presented by interpreting the axioms two logical formulas for classical topological spaces in the fuzzy setting making use of the intrinsic fuzzy partial order on the fuzzy power set of x as preparation a systematic investigation of fuzzy partial orders is presented from the viewpoint of enriched category theory

smooth pretopogenous structures abstract in this paper we present smooth extension of pretopogenous structure which is obtained by weakening the axioms of the corresponding classical structure this weakening consists in the embedding of the structure in a multiple valued logic context a natural link is established between these concepts and the concept of fuzzy pretopological spaces smooth preproximity spaces and fuzzy preuniform spaces the notation of product smooth pretopogenous structures is also introduced

the relationship between various filter notions on agl monoid abstract the notion of a generalised filter is extended to the setting of a gl monoid it is shown that there exists a one to one correspondence between the collection of generalised filters on a set x and the collection of strongly stratified l filters on x specialising to the case where l is the closed unit interval 0 c viewed as a heyting algebra we show that any strongly stratified 0 c filter on x can be uniquely identified with a saturated filter on i x with characteristic value c in this way the notion of a generalised filter unifies various filter notions in particular necessity measures and finitely additive probability measures are specific examples of generalised filters

minimizing the ground state energy of an electron in a randomly deformed lattice we provide a characterization of the spectral minimum for a random schrodinger operator of the form h delta sum i in mathbb z d q x i omega i in l 2 mathbb r d where the single site potential q is reflection symmetric compactly supported in the unit cube centered at 0 and the displacement parameters i are restricted so that adjacent single site potentials do not overlap in particular we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2 d clusters of single site potentials

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

diagonalization modulo norm ideals with lipschitz estimates abstract suppose that x d is a compact metric space and that is a regular borel measure on x let lip x denote the collection of lipschitz functions on x in this paper we investigate the problem of simultaneous diagonalization of the multiplication operators m f f lip x on l 2 x modulo norm ideals of compact operators

lowen lm fuzzy topological spaces the aim of this paper is to define and study lowen lm fuzzy topological spaces we discuss the basic properties of lowen lm fuzzy topological spaces and introduce the notions of interior lowen topology and exterior lowen topology of an lm fuzzy topology and prove that llm ftop the category of lowen lm fuzzy topological spaces is isomorphism closed and simultaneously bireflective and bicoreflective in slm ftop the category of stratified lm fuzzy topological spaces moreover we also prove that an lm fuzzy topological space is an induced lm fuzzy topological space iff it is a lowen lm fuzzy topological space and an ic lm fuzzy topological space

spectral metric spaces on extensions of c algebras we construct spectral triples on c algebraic extensions of unital c algebras by stable ideals satisfying a certain toeplitz type property using given spectral triples on the quotient and ideal our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones using our construction we find new spectral triples on the quantum 2 and 3 spheres giving a new perspective on these algebras in noncommutative geometry

fuzzy proximities compatible with lowen fuzzy uniformities abstract we introduce a notion of fuzzy proximity which behaves well with respect to the fuzzy uniformities introduced by r lowen and prove some related results we also provide some critical remarks about fuzzy topologies fuzzy proximities and fuzzy uniformities

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

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

voiculescu theorem sobolev lemma and extensions of smooth algebras we present the analytic foundation of a unified b d f extension functor ext on the category of noncommutative smooth algebras for any frechet operator ideal k combining the techniques devised by arveson and voiculescu we generalize voiculescu s theorem to smooth algebras and frechet operator ideals a key notion involved is r smoothness which is verified for the algebras of smooth functions via a noncommutative sobolev lemma the groups ext are computed for many examples

operations on fuzzy syntopogenous structures abstract initial fuzzy syntopogenous structures product and subspaces of fuzzy syntopogenous spaces and initial fuzzy uniformities are investigated continuity of functions between fuzzy syntopogenous spaces is studied

fuzzy uniform spaces in this paper we introduce the notion of fuzzy uniform spaces in section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity in sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces finally in section 5 we prove some basic results

global l neighborhood groups this paper introduces and studies the notion of global l neighborhood group which is defined as a group equipped with a global l neighborhood structure in sense of gahler et al such that both the binary operation and the unary operation of the inverse are continuous with respect to this global l neighborhood structure some examples of global l neighborhood groups are given it is shown that the l topological groups given by ahsanullah in 1984 and later by bayoumi in 2003 are special global l neighborhood groups we also show that all initial and final lifts and hence all initial and final global l neighborhood groups uniquely exist in the category l gnghgrp of global l neighborhood groups these initial and final global l neighborhood groups are defined using the initial and final global l neighborhood structures moreover we show that the l neighborhood groups defined by ahsanullah using the l neighborhood structures in sense of lowen are special global l neighborhood groups for l i is the closed unit interval

the hyperspace of fuzzy sets a natural nontopological fuzzy topological space let x be a uniform topological space then on the family ix resp d x of all nonzero functions resp nonzero uppersemicontinuous functions from x to the unit interval i a fuzzy uniform topology is constructed such that 2x resp 61 x the family of all nonvoid resp nonvoid closed subsets of x equipped with the hausdorff bourbaki structure is isomorphically injected in ix resp i x the main result of this paper is a complete description of convergence in ix by means of a notion of degree of incidence of members of ix immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications and that second it follows from its construction and properties that for each ordinary uniform topological space x there exists a natural nontopological fuzzy uniform topology on ix

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies