fake airy functions and the asymptotics of reflectionlessness two classes of analytic refractive index profile p2 z in whose reflection coefficients r are zero for all values of a parameter in are studied as in to 0 the aim is to understand why r 0 rather than r varies as exp 1 in as for generic profiles the authors find that reflectionlessness is a consequence of the fact that transition points of p2 zeros or poles in the complex z plane form tight clusters whose size vanishes with in which can be regarded neither as coalesced nor well separated expansion near a cluster yields the local wave not as the usual airy function whose stokes phenomenon generates reflection but as bessel functions of half integer order fake airy functions which are exactly trigonometric functions with no stokes phenomenon and so no reflection

the stone cech compactification in the category of fuzzy topological spaces abstract in this paper we obtain a reflective subcategory c of the category fts of fuzzy topological spaces the associated reflection has properties similar to those of the stone cech compactification and in effect is an extension of it we study relations between and in particular subcategories of fts is completely determined in the case of fuzzy topological spaces topologically generated

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently given

compact g fuzzy topological spaces abstract considering complete boolean algebras g as sets of truth values a new concept of compactness so called probabilistic compactness is introduced to g fuzzy topological spaces the aim of this paper is to show that the most important theorems of the theory of ordinary compact spaces remain true e g probabilistic compactness is preserved under projective limits every probabilistic compact space has an unique g fuzzy uniformity being compatible with the underlying g fuzzy topology etc finally using the selection theorem due to kuratowski and ryll nardzewski a non trivial example of a probabilistic compact space is given

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

a theory of fuzzy uniformities with applications to the fuzzy real lines abstract for each completely distributive lattice l with order reversing involution the fuzzy real line r l is uniformizable by a uniformity which both generates the canonical fuzzy topology and induces a pseudometric generating the canonical topology if l is also a chain the usual addition and multiplication defined on r r 0 1 extend jointly fuzzy continuously to and on r l three fundamental questions in fuzzy sets until now are question a if l 1 l 2 is r l 1 uniformly isomorphic to r l 2 in some sense question b for each chain l is jointly uniformly continuous in a sense which guarantees its joint continuity on r l question c is r l a complete pseudometric space in some sense we construct categories qu and u using the quasi uniformities of b hutton which enable us to answer these questions in the affirmative these results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative question d does fuzzy topology have deep specific canonical examples

applications of localic separation axioms compactness axioms representations and compactifications to poslat topological spaces fuzzy topology first appeared about twenty seven years ago with chang s initial paper 1968 yet the fundamental issues of compactness and low order separation axioms remain unresolved this paper resolves these issues with respect to topological compactification categorical compactification and stone representation theorems using the theory of locales and previous work of the author in fuzzy sobriety our results give the first compactification reflectors for all traditional fuzzy topology construct classes of generalized stone representations of boolean algebras and distributive lattices and show categorically that the fuzzy analogue of classical compact hausdorff spaces is the categories of chang compact localic regular sober fuzzy topological spaces

fuzzy points and local properties of fuzzy topological spaces abstract we prove that three theorems of c k wong on local properties of fuzzy topology are wrong thereby we discuss his notions of fuzzy point c 1 separability and local compactness

category theoretic properties of fuzzy topological spaces abstract in this paper we establish some important category theoretic properties in particular smallness conditions for categories of fuzzy topological spaces we also obtain a non existence of compactifications

a characterization of fuzzy compactifications abstract it is shown that if x is a fuzzy t2 space then x has a fuzzy t2 compactification if and only if x is a weakly induced ultra completely regular space also for an arbitrary fuzzy topological space a characterization is given of the set of all ultra fuzzy compactifications

variation of the action in the classical time dependent harmonic oscillator an exact result from the exact solution of certain time dependent classical harmonic oscillators in one dimension the authors investigate the behaviour of the adiabatic invariant j t for a subclass of such potentials delta j identical to j infinity j infinity 0 whatever the regime they show that this does not necessarily imply a breakdown of the commonly accepted asymptotic exponential law for delta j

fuzzy closed and fuzzy perfect mappings abstract it is shown that a fuzzy continuous map f is fuzzy perfect iff f i z is fuzzy closed for every fuzzy topological space z

on l tychonoff spaces topics of this paper include 1 basic terminology and auxiliary observations 2 pointfree characterizations of completely l regular spaces 3 embedding spaces of prescribed weight applications to l tychonoff and l to spaces 4 remarks on liu s characterizations of complete l regularity involving points 5 extending continuous l real functions from h compact subspaces 6 conditional completeness of c x r l 7 remarks on the compactification of liu and luo 8 l creg l creg and l tych l tych 9 lower order axioms for the l line a new coherent system of separation axioms for l topological spaces 10 open questions

fuzzy stone ccech type compactifications abstract in this paper we investigate the fuzzy sets with lattice precisely fuzzy lattice values some new global properties on the weakly induced spaces and induced spaces are obtained based on these results we take the induced space of real unit interval i as new basic space and establish a reasonable theory of stone cech compactifications via fuzzy imbedding theory 5 and n compactness 10 12 for so called t 3 1 2 and weakly induced spaces its stone cech type compactification always exists we give a counterexample to show that the previous stone cech compactification 6 is not the largest t2 compactification of a given space compared with this we now have a positive results for the new stone cech type compactification moreover although the preorder among the compactifications for a given space does not generally satisfy the anti symmetric law we fortunately establish uniqueness of the largest t2 type compactification among the compactifications for a given space

on employment of semidefinite functions in stability of delayed equations abstract four new stability theorems for nonautonomous delayed equations are established by the lyapunov razumikhin approach with functions and its derivatives semidefinite to do this assume the collection of right hand side translations in time has limit points which govern limiting equations a corollary which relates global asymptotic stability for the equation to that for limiting equations and an illustrative example are given to show the applications of the established theorems

g fuzzy topologies on algebraic structures abstract considering complete boolean algebras as sets of truth values the structure of a fuzzy topological group fuzzy topological ring etc is specified the probabilistic completion of ordinary topological algebraic structures shows the applicability of these concepts to the theory of stochastic processes e g a new definition of the stochastic integral is presented in section 5

on fuzzy topological spaces abstract in this paper a few separation properties and some aspects of subspace fuzzy topology have been studied where both the crisp and the fuzzy elements have been taken into consideration since the conventional definition of compactness is not quite meaningful in hausdorff fuzzy spaces as introduced by us a new more natural definition of proper compactness is given and a few properties resulting from this are established

totally bounded fuzzy syntopogenous structures abstract some of the properties of the totally bounded fuzzy syntopogenous structures and the totally bounded fuzzy quasi uniformities are studied it is also proved that the restriction to a subset y of the fuzzy real line of the usual fuzzy uniformity is totally bounded iff y is bounded

the epireflective hull of the sierpinski object in fts abstract in the category top of topological spaces the epireflective hull of the standard two point sierpinski space is known to coincide with the subcategory of t0 topological spaces in this note we determine the epireflective hulls of the recently found sierpinski objects in the categories fts fns and top of respectively fuzzy topological spaces fuzzy neighborhood spaces and topologically generated fuzzy topological spaces we thereby identify the corresponding t0 objects in these categories and note that the categories fts fns and top are universal

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

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices