overview and comparison of localic and fixed basis topological products this paper studies localic products traditional topological products and l topological products and gives a complete outline of the localic product comparisons of localic and l topological products are generally absent in the literature and this paper answers longstanding open questions in that area as well as provides a complete proof of the classical comparison theorem for localic and traditional topological products this paper contributes several l valued comparison theorems one of which states the localic and l topological products of l topologies are order isomorphic if and only if the localic product is l spatial providing l is itself spatial and the family of l topological spaces is prime separated these last two conditions always hold in the traditional setting capturing the traditional comparison theorem as a special case and the prime separation condition is satisfied by important lattice valued examples that include the fuzzy real line and the fuzzy unit interval for l any complete boolean algebra and the alternative fuzzy real line and fuzzy unit interval for l any semi frame separation conditions help control the sloppy behavior of the l topological product when l 2 and several separation conditions are studied in this context and it should be noted that localic products have a point free version of the product separation condition considered in this paper the traditional comparison theorem is carefully proved both to fill gaps in the extant literature and to motivate the l valued comparison theorem quoted above and reveal the special role played by cross sums of prime l open subsets en route characterizations are given of prime l open subsets of certain l products which in turn yield characterizations of prime open and irreducible closed subsets of traditional product spaces

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

on the category of lattice valued bornological vector spaces motivated by the theory of l bornological spaces of m abel and a sostak over a complete lattice l and the concept of topological system of s vickers this paper introduces the categories of l bornological vector spaces and systems and shows that the former is isomorphic to a full reflective subcategory of the latter

an equivalence classes model of fuzzy relational databases abstract the technique of employing sets of values for tuple components to express imprecision in relational databases was proposed by buckles and petry in their classic works on fuzzy relational databases in addition to providing an intuitively appealing scheme for representing fuzzy information the model of buckles and petry possesses several key properties of the classical relational model by employing finite scalar domains with similarity relations and special fuzzy number domains buckles and petry have demonstrated that the classical properties of uniqueness of tuple interpretations and well definedness of the relational algebra can be retained in the fuzzy relational database model the key to the preservation of these properties is the fact that scalar domains with similarity relations and the fuzzy number domains can be partitioned into equivalence classes however since equivalence classes can be constructed without assuming the existence of similarity relations or special fuzzy number domains it is desirable to generalize the fuzzy relational database model to one based only on equivalence classes in this work we show that the important properties of classical relational databases and of fuzzy relational databases are preserved in a generalized model built on equivalence relations on finite database domains further we generalize the notion of a functional dependency to the fuzzy relational model

categorical foundations of variety based bornology following the concept of topological theory of s e rodabaugh this paper introduces a new approach to lattice valued bornology which is based in bornological theories and which is called variety based bornology in particular motivated by the notion of topological system of s vickers we introduce the concept of variety based bornological system and show that the category of variety based bornological spaces is isomorphic to a full reflective subcategory of the category of variety based bornological systems

a note on the hypergraph functor the purpose of this note is to show that in the case of spatial locales l the hypergraph functor has a right adjoint functor

category theoretic aspects of chain valued frames part ii applications to lattice valued topology this paper is part ii of a two part series dealing with category theoretic aspects of chain valued frames using the categorical properties established for l frm in part i for l a complete chain this paper constructs upper free functor l and lower free functor r the functor l is used to create a class of non generated l frames factor the lpt spectrum functor through the s spectrum functor resolve the relationship between l sobriety and i l sobriety and give insight into the construction of universal l topological spaces and the functor r is used to create a class of non generated l frames factor the w l functor through a new spectrum functor s and give insight into the construction of co universal l topological spaces these facts universal spaces are anti stratified co universal spaces are stratified l frm produces both kinds of spaces via l and r l frm via its dual l loc is adjunctive with l top comprise a coherent argument that l top must accommodate both kinds of spaces resolving the philosophical debate over the place of the constant maps condition with respect to the axioms of l topologies

path integration on hermitian hyperbolic space in this paper the path integral technique is applied to the quantum motion on the hermitian hyperbolic space hh 2 the schrodinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two dimensional hyperboloid for six coordinate systems out of the twelve it is possible to find a path integral solution

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

note on the performance of direct and indirect runge kutta nystro m methods abstract this paper deals with predictor corrector iteration of runge kutta nystrom rkn methods for integrating initial value problems for special second order ordinary differential equations we consider rkn correctors based on both direct and indirect collocation techniques the paper focuses on the convergence factors and stability regions of the iterated rkn correctors it turns out that the methods based on direct collocation rkn correctors possess smaller convergence factors than those based on indirect collocation rkn correctors both families of methods have sufficiently large stability boundaries for nonstiff problems

sobriety and spatiality in varieties of algebras the paper considers a generalization of the classical papert papert isbell adjunction between the categories of topological spaces and locales to an arbitrary variety of algebras and illustrates the obtained results by the category of algebras over a given unital commutative quantale

fuzzy algebras as a framework for fuzzy topology the paper introduces a variety based version of the notion of the l m fuzzy topological space of kubiak and sostak and embeds the respective category into a suitable modification of the category of topological systems of vickers the new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature paving the way for studying the problem of interweaving algebra and topology in mathematics which was raised by denniston melton and rodabaugh in their recent research on variable basis topological systems over the category of locales

exponentially fitted two step hybrid methods for y f x y it is the purpose of this paper to derive two step hybrid methods for y f x y with oscillatory or periodic solutions specially tuned to the behaviour of the solution through the usage of the exponential fitting technique the construction of two step exponentially fitted hybrid methods is shown and their properties are discussed some numerical experiments confirming the theoretical expectations are provided

lattice valued preordered sets as lattice valued topological systems this paper provides variable basis lattice valued analogues of the well known results that the construct prost of preordered sets firstly is concretely isomorphic to a full concretely coreflective subcategory of the category top of topological spaces which employs the concept of the dual of the specialization preorder and secondly is non concretely isomorphic to a full coreflective subcategory of the category topsys of topological systems of s vickers which employs the spatialization procedure for topological systems dualizing these results one arrives at the similar properties of quasi pseudo metric spaces built over locales

errata complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines abstract we construct a fuzzy multiplication for the case l a chain in the l fuzzy real line r l which is jointly fuzzy continuous and which is the unique piecewise consistent extension of ordinary multiplication to r l distributing over the fuzzy addition constructed by the author in 47 we also specify the fuzzy integers and give an extensive analysis of fuzzy contractions and expansions these results the constructions of this paper and the results of 47 are used to prove r l and the stratified r c l are nontopologically isomorphic complete fuzzy topological hyperfields if l is a chain with l 3 we can now completely answer for l a chain the following question of 27 in the affirmative can a set of fuzzy numbers its fuzzy topology and its fuzzy addition and multiplication be defined such that these operations are jointly fuzzy continuous extensions of the usual addition and multiplication of r and can we specify the fuzzy integers these results in combination with recent papers of several authors would seem to answer in the affirmative the following philosophical question cf 34 does fuzzy topology have deep specific canonical examples

numerical search for algebraically stable two step almost collocation methods we investigate algebraic stability of the new class of two step almost collocation methods for ordinary differential equations these continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration we describe the search for algebraically stable methods using the criterion based on the nyquist stability function proposed recently by hill this criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from matlab examples of algebraically stable methods in this class are also presented

on lattice valued frames the completely distributive case we provide an extension of the notion of chain valued frame introduced by pultr and rodabaugh in category theoretic aspects of chain valued frames parts i and ii fuzzy sets and systems 159 2008 501 528 and 529 558 by relaxing the assumption that l be a complete chain as a result of this investigation we formulate the category l frm of l frames under the weaker assumption that l is a completely distributive lattice in particular l frm is complete and cocomplete finally we prove that in a certain sense the assumption of l being a completely distributive lattice cannot be weakened

category theoretic aspects of chain valued frames part i categorical and presheaf theoretic foundations this paper is part i of a two part series dealing with category theoretic aspects of chain valued frames a new categorical motivation for lattice valued frames is given from presheaves and then under the assumption that l be a complete chain it is established that standard category theoretic properties completeness cocompleteness factorization structures hold for l frm the category of l frames and l frame morphisms properties which are a platform for the upper free functor l and lower free functor r used in part ii to give a broad range of applications of chain valued frames to lattice valued topology

symmetry scattering on the hyperboloid so 2 1 so 2 in different coordinate systems the symmetry scattering theory based on the harish chandra helgason theory of spherical functions on noncompact riemannian symmetric spaces is extended to treat all spherical harmonics on the hyperboloid so 2 1 so 2 the required conditions for an extension of the symmetry scattering theory to treat all spherical harmonics on arbitrary noncompact riemannian symmetric spaces are elaborated

categorical foundations of variety based topology and topological systems the paper considers a new approach to fuzzy topology based on the concept of variety and developed in the framework of topological theories resembling those of rodabaugh as a result a categorical generalization of the notion of topological system of vickers is obtained and its theory unfolded which clarifies the relations between algebra and topology we also justify the use of semi quantales as the basic underlying structure for doing lattice valued topology as well as provide a categorical framework incorporating the theory of bitopological spaces

on fuzzification of topological categories this paper shows that l m fuzzy topology of u hohle t kubiak and a sostak is an instance of a general fuzzification procedure for topological categories which amounts to the construction of a new topological category from a given one this fuzzification procedure motivates a partial dualization of the machinery of tower extension of topological constructs of d zhang thereby providing the procedure of tower extension of topological categories with the help of this dualization we arrive at the meta mathematical result that the concept of l m fuzzy topology and the notion of approach space of r lowen are dual to each other

on scattering systems related to the group scattering systems related to the noncompact groups g in the sense that the hamiltonian of the system can be written as a function of the casimir operator of g are considered the s matrix for such systems are defined in terms of an intertwining operator of underling symmetry group g the s matrices for one dimensional scattering systems with symmetry group are classified

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

on the category set jcpos category set jcpos of lattice valued subsets of sets is introduced and studied we prove that it is topological over setxjcpos and show its natural coalgebraic subcategory

groups of dualities for arbitrary categories si and 38 the set of isomorphism classes of dualities between si and 3s carries a natural group structure in case si and 3e admit faithful representable functors to set this structure can often be described quite concretely in terms of schizophrenic objects in the sense of johnstone s book on stone spaces the general theory provided here allows for a concrete computation of that group in case si i is the category of all compact and all discrete abelian groups it is the uncountable group of algebraic automorphisms of the circle r z modulo its subgroup i of continuous automorphisms

sierpinski object for affine systems abstract motivated by the concept of sierpinski object for topological systems of s vickers presented recently by r noor and a k srivastava this paper introduces the sierpinski object for many valued topological systems and shows that it has three important properties of the crisp sierpinski space of general topology

fuzzy points and attachment in this paper a binary relation between l sets called attachment is defined by means of a suitable family of completely coprime filters in l examples are given using several kinds of lattice ordered algebras and a class of attachments is determined whose elements generalize pu liu s quasi coincidence relation mapping any l set on x to the set of the attached l points one gets a frame map from the l powerset l x to the powerset p s x of the set s x of l points on x this allows one to define functors from the category l top of l topological spaces either to the category top of topological spaces or to the category topsys of topological systems a comparison with the hypergraph functor is also included

integrable systems based on su p q homogeneous manifolds the general theory of the separation of variables in hamilton jacobi and laplace beltrami equations on the su p q hyperboloid is used to introduce completely integrable hamiltonian systems on o p q hyperboloids each of the q 1 different cartan subalgebras of su p q leads to a different integrable o p q potential different complete sets of integrals of motion are obtained for each of the integrable systems

categorically algebraic topology versus universal topology this paper continues to develop the theory of categorically algebraic catalg topology introduced as a common framework for the majority of the existing many valued topological settings to provide convenient means of interaction between different approaches motivated by the results of universal topology of h herrlich we show that a concrete category is fibre small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures which is definable by topological co axioms

interweaving algebra and topology lattice valued topological systems this paper is primarily dedicated to understanding the natural role that topological systems and lattice valued topological systems play in understanding the relationship between algebra and topology a relationship expressed by an interweaving of embeddings of essentially algebraic and topological categories this natural role is revealed by how various categories for systems relate to locales topological spaces and various categories for lattice valued topology

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

on limits and colimits of variety based topological systems the paper provides variety based extensions of the concepts of lattice valued interchange system and space introduced by denniston melton and rodabaugh and shows that variety based interchange systems incorporate topological systems of vickers state property systems of aerts chu spaces over the category of sets in the sense of pratt of p h chu and contexts of formal concept analysis of wille the paper also provides an explicit description of co limits in the category of variety based topological systems and applies the obtained results to extend the claim of denniston et al that the category of topological systems of vickers is small initially topological over the category of sets

sobriety and spatiality in categories of lattice valued algebras the paper provides an analogue of the famous equivalence between the categories of sober topological spaces and spatial locales for the framework of l m fuzzy topology of kubiak and sostak and partly to that of guido to be more general we replace locales with localic lattice valued algebras in the sense of di nola and gerla and use the respective generalized topological setting as a result it appears that the shift from crisp algebras to lattice valued algebras weakens resp strengthens considerably the classical including the point set lattice theoretic setting of rodabaugh notion of sobriety resp spatiality

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism