a unifying study between modal like operators topologies and fuzzy sets the paper presents the essential connections between modal like operators topologies and fuzzy sets we show for example that each fuzzy set determines a preorder and an alexandrov topology and that similar correspondences hold also for the other direction further a category for preorder based fuzzy sets is defined and it is shown that its equivalent subcategory of representatives is isomorphic to the categories of preordered sets and alexandrov spaces moreover joins meets and complements for the objects in this category of representatives are determined this suggests how to define for fuzzy subsets of a certain universe the lattice operations in a canonical way

implication structures fuzzy subsets and enriched categories we show that there are natural ways to make a fuzzy set into an enriched category and that in doing so some basic results in fuzzy set theory are applications of enriched category theory thus we argue that the theory of enriched categories is a useful tool for fuzzy set theorists

closedness of the category of liminf complete fuzzy orders it is shown that the category of liminf complete fuzzy orders and liminf continuous maps valued in the complete residuated lattice based on a continuous t norm on the unit interval is monoidal closed and that this category is cartesian closed if and only if the t norm is the minimum

triangular norms position paper ii general constructions and parameterized families this second part out of three of a series of position papers on triangular norms for part i see triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press deals with general construction methods based on additive and multiplicative generators and on ordinal sums also included are some constructions leading to non continuous t norms and a presentation of some distinguished families of t norms

the categorical topology approach to fuzzy topology and fuzzy convergence it is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology we first look at interesting subcategories of fts and determine whether they are bireflective and or coreflective we also describe the set of all subcategories of fts which are at the same time bireflective and coreflective due to the fact that top is one of these subcategories of fts ffs itself can be neither cartesian closed nor extensional we therefore secondly construct a supercategory of bts which is a topological quasitopos the construction of this quasitopos is based on the theory of convergence in fts

interpolation of fuzzy data analytical approach and overview we propose a general framework for the interpolation problem our framework stems from the classical elaboration of the problem we introduce the notion of an interpolating fuzzy function and show how this function can be characterized we examine and analyze previously published fuzzy interpolation algorithms to choose those algorithms that can be represented analytically we also propose an analytic solution of the interpolation problem that unifies various algorithmic approaches

moment stability of nonlinear stochastic systems with time delays based on cal h representation technique in this note moment stability of nonlinear stochastic systems with time delays is investigated based on a kind of kronecker algebra and the h representation technique we obtain the second order moment equations of nonlinear stochastic systems with time delays by the comparison principle and the established moment equations we get the comparative systems of the nonlinear stochastic systems with time delays then we establish moment stability theorems for the systems with the stability properties of the comparative systems finally an example is presented to illustrate the efficiency of our results

fundamental study complete and directed complete ω categories let w be a commutative unital quantale complete and directed complete w categories are the core objects in quantitative domain theory this paper based on the theory of f completeness for enriched categories presents a systematic investigation of completeness and directed completeness of w categories

on the computation of steady state compressible flows using a discontinuous galerkin method computation of compressible steady state flows using a high order discontinuous galerkin finite element method is presented in this paper an accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities a physics based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave a recently developed fast low storage p multigrid method is used for solving the governing compressible euler equations to obtain steady state solutions the method is applied to compute a variety of compressible flow problems on unstructured grids numerical experiments for a wide range of flow conditions in both 2d and 3d configurations are presented to demonstrate the accuracy of the developed discontinuous galerkin method for computing compressible steady state flows

connecting fuzzy preorders fuzzy consequence operators and fuzzy closure and co closure systems abstract the relations between fuzzy preorders fuzzy consequence operators fuzzy closure systems and co closure systems lattice structures are studied given a continuous t norm we will prove that the complete lattice of all fuzzy preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators moreover we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices a solution to the following open problem when two families of fuzzy possible valuations produce the same fuzzy implication relation is given concept of interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through interior operator is shown

granular variable precision l fuzzy rough sets based on residuated lattices abstract firstly as for the beginning of this paper we introduce the granular variable precision l fuzzy rough sets based on residuated lattices with arbitrary l fuzzy relations and then the equivalent expressions for the lower and upper approximation operators are given with l fuzzy implications and l fuzzy coimplications on arbitrary l fuzzy relations which can calculate the lower and upper approximation operators efficiently finally some basic properties of the granular variable precision l fuzzy rough sets on residuated lattices are investigated and the conclusions on the granular variable precision fuzzy rough sets are generalized into the situations of the truth values set becoming residuated lattices

fixed points and generators of fuzzy relations abstract the fixed points of a t indistinguishability operator are characterized as its generators in the sense of the representation theorem of valverde the geometric description of the set of fixed points of a reflexive and symmetric fuzzy relation based on this characterization gives a way to explicitly calculate it

the aggregation of transitive fuzzy relations revisited abstract in this paper we consider the problem of aggregating a collection of transitive fuzzy binary relations in such a way that the aggregation process preserves transitivity specifically we focus our efforts on the characterization of those functions that aggregate a collection of fuzzy binary relations which are transitive with respect to a collection of t norms preserving the transitivity we characterize them in terms of triangular triplets further the relationship between triangular triplets the monotonicity of the aggregation function and an appropriate dominance notion is explicitly stated special attention is paid to a few classes of transitive fuzzy binary relations that are relevant in the literature concretely we describe in terms of triangular triplets those functions that aggregate a collection of fuzzy pre orders fuzzy partial orders relaxed indistinguishability relations indistinguishability relations and equalities a surprising relationship between functions that aggregate transitive fuzzy relations into a t m transitive fuzzy relation and those that aggregate relaxed indistinguishability relations is shown a few consequences of the new results are also provided for those cases in which all the t norms of the given collection are the same some celebrated results are retrieved as a particular case from the exposed theory

on the relationship between limit spaces many valued topological spaces and many valued preorders let l 1 be a residuated complete lattice with the underlying lattice l a meet continuous lattice this paper presents a systematic investigation of the interrelationship between the categories of limit spaces l topological spaces and l preorders the results exhibit a close connection between these different mathematical structures

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

on the topological properties of fuzzy rough sets this paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe the tc axiom for fuzzy topology is proposed it is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies tc axiom and conversely a fuzzy topology which obey tc axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation that is to say there exists a one to one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy tc axiom

a stone type duality for st0 stratified alexandrov l topological spaces abstract we show that for every complete lattice a both the set of completely prime elements and the set of completely coprime elements are one to one corresponding to ch a the set of complete lattice homomorphisms from a to the two element lattice 2 a complete lattice is called completely generated a cg lattice for short if it is generated by the set of completely prime elements or equivalently by the set of completely coprime elements then we restudy the duality between the category of t 0 alexandrov topological spaces and the category of cg lattices by means of ch a with these preparations for a frame l as the truth value table we introduce s t 0 separation axiom for stratified alexandrov l topological spaces and finally establish a duality between the category of s t 0 stratified alexandrov l topological spaces and the category of completely generated complete l ordered sets we also investigate some properties of the s t 0 axiom for example it is hereditary by closed subspaces and productive

a p multigrid discontinuous galerkin method for the euler equations on unstructured grids a p multigrid p polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature of this p multigrid method is to use different time integration schemes on different approximation levels resulting in an accurate fast and low memory method that can be used to accelerate the convergence of the euler equations to a steady state for discontinuous galerkin methods the developed method is used to compute the compressible flows for a variety of test problems on unstructured grids the numerical results obtained strongly indicate the order independent property of this p multigrid method an overall speed up factor more than one order of magnitude for both second and third order solutions of all test cases in comparison with the explicit method is demonstrated

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

fast p multigrid discontinuous galerkin method for compressible flows at all speeds ap multigrid wherep is the polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature ofthisp multigrid method is the application of an explicit smoother on the higher level approximations p 0 and an implicit smoother on the lowest level approximation p 0 resulting in a fast and low storage method that can be efficiently used to accelerate the convergence to a steady state solution furthermore this p multigrid method can be naturally applied to compute the flows with discontinuities in which a monotonic limiting procedure is usually required for discontinuous galerkin methods an accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries krivodonova l and berger m high order implementation of solid wall boundary conditions in curved geometries journal of computational physics vol 211 no 2 2006 pp 492 512 and luo h baum j d and lohner r on the computation of steady state compressible flows using a discontinuous galerkin method international journal for numerical methods in engineering vol 73 no 5 2008 pp 597 623 a variety of compressible flow problems for a wide range of flow conditions from low mach number to supersonic in both two dimensional and three dimensional configurations are computed to demonstrate the performance ofthisp multigrid method the numerical results obtained strongly indicate the order independent property of this p multigrid method and demonstrate that this method is orders of magnitude faster than its explicit counterpart the performance comparison with a finite volume method shows that using this p multigrid method the discontinuous galerkin method provides a viable attractive competitive and probably even superior alternative to the finite volume method for computing compressible flows at all speeds

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

on relationships among fuzzy approximation operators fuzzy topology and fuzzy automata the fuzzy approximation operator associated with an approximation space x r turns out to be a saturated kuratowski fuzzy closure operator on x precisely when the relation r on x is reflexive and transitive it is shown that all the level topologies of the fuzzy topology so obtained on x coincide these observations are then applied to fuzzy automata

semi discrete discontinuous galerkin methods and stage exceeding order strong stability preserving runge kutta time discretizations this paper investigates the use of a special class of strong stability preserving ssp runge kutta time discretization methods in conjunction with discontinuous galerkin dg finite element spatial discretizatons the class of ssp methods investigated here is defined by the property that the number of stages s is greater than the order k of the method from analysis cfl conditions for the linear l 2 stability of the methods defined using the s k ssp schemes are obtained that are less restrictive than those of the standard so called rkdg methods that use s k ssp runge kutta schemes the improvement in the cfl conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages given that the cfl conditions for linear stability are what must be respected in practice in order to maintain high order accuracy the use of the s k ssp schemes results in rkdg methods that are more efficient than those previously defined furthermore with the application of a slope limiter the nonlinear stability properties of the forward euler method and the dg spatial discretization which have been previously proven are preserved with these methods under less restrictive cfl conditions than those required for linear stability thus more efficient rkdg methods that possess the same favorable accuracy and stability properties of the standard rkdg methods are obtained numerical results verify the cfl conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new rkdg methods

finitary solvability conditions for systems of fuzzy relation equations in this paper we consider the problem of solving systems of fuzzy relation equations two types of these systems with different compositions are considered and processed simultaneously a new sufficient condition and new solvability criteria are proposed for systems of both types all these conditions are finitary and thus can be easily verified the sufficient condition and criteria of solvability characterize a relationship between a skeleton nxn matrix n is a number of equations and a vector of the right hand side

new results on single axioms for l fuzzy rough approximation operators abstract this paper further studies axiomatic characterizations of l fuzzy rough sets where l denotes a residuated lattice single axioms for upper l fuzzy rough approximation operators are investigated in two different approaches which describe upper l fuzzy approximation operators with ordinary l fuzzy operations and l fuzzy product operation respectively we characterize upper l fuzzy rough approximation operators by only one axiom with ordinary l fuzzy operations when the l fuzzy relation is euclidean as well as the composition of euclidean with serial resp reflexive we also discuss the single axioms for upper l fuzzy rough approximation operators with l fuzzy product operation on arbitrary l fuzzy relations that are not limited as either general l fuzzy relations or symmetric l fuzzy relations the single axioms for lower l fuzzy rough approximation operators are discussed with ordinary l fuzzy operations regardless of the regularity of residuated lattice l

on un suitable fuzzy relations to model approximate equality in this paper we state that fuzzy equivalence relations in general are not suitable to model approximate equality since then the notion of transitivity is counter intuitive to substantiate this we investigate some of the undesirable results caused by transitivity among other things in the case of approximate reasoning we then introduce a new framework to model approximate equality i e the concept of a pseudometric based resemblance relation we go into the properties of this new kind of fuzzy relation and illustrate it by means of some examples

some results on the degree of symmetry of fuzzy relations abstract in this paper we investigate the degree of the symmetry of fuzzy relations on a set x based on the fuzzy e equality derived from uninorms with unit element e the degree of the symmetry of fuzzy relations are defined by two different approaches the type i degree of symmetry and the type ii degree of symmetry the main work of this paper include first we discuss some basic properties of those two degrees especially for the relationship between them in particular for continuous t norms we give a necessary and sufficient condition such that the type i degree of symmetry and the type ii degree of symmetry are equal and then we obtain that the type ii degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type i degree of symmetry with respect to whether they preserve the fuzzy e equalities or not in the meantime for a special case with continuous t norms we provide a necessary and sufficient condition such that the type i degree of symmetry preserves the fuzzy equalities finally for conjunctive left continuous idempotent uninorms with neutral element e 0 1 we find out a symmetric fuzzy relation s which is close enough to the given fuzzy relation r with respect to fuzzy e equality such that the degree of fuzzy e equality of s and r is the type ii degree of symmetry of r in particular we obtain analogous results for continuous t norms and one class of left continuous t norms

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

fuzzy complete lattices in this paper based on l fuzzy posets previously introduced by the third author l fuzzy complete lattices are defined which are generalizations of usual complete lattices and coincide with wagner s complete and cocomplete w categories enriched over the frame l and are consequently a special kind of complete w lattices defined by lai and zhang however tarski fixed point theorem for the l fuzzy complete lattices is proved in a different way from that by lai and zhang furthermore some fuzzy powerset operators are suggested they are not only generalizations of ordinary powerset operators but also generalizations of l valued zadeh powerset operators and their properties are discussed

topological properties of the class of generators of an indistinguishability operator the present paper investigates the relations between topologies in the classical sense fuzzy topologies in lowen s sense and indistinguishability operators as well as constructing topologies and fuzzy topologies by means of indistinguishability operators furthermore as a particular consequence of the results in this paper a characterization theorem for pseudo metrizable topological spaces in the classical sense is also surprisingly obtained

on many valued stratified l fuzzy convergence spaces based on complete residuated lattices the relations between stratified l fuzzy convergence spaces and enriched l fuzzy topological spaces are studied it is shown that as a reflective subcategory the category of enriched l fuzzy topological spaces can embedded in the category of stratified l fuzzy convergence spaces also the specialization l preorder of stratified l fuzzy convergence spaces is investigated

fuzzy t transitive relations eigenvectors and generators abstract after some preliminaries the set of generators of a generalized equality relation t indistinguishability operator is studied this set is identified with the set of the eigenvectors of the relation the relation between the fuzzy and metric topologies derived from these equalities is stablished the concept of basis is introduced and the construction of a procedure in order to calculate explicitly a basis of a t indistinguishability operator for t archimedean is proposed

representations and constructions of similarity based fuzzy orderings this contribution advocates the dissemination of similarity based fuzzy orderings a generalized concept of fuzzy orderings which appears to be unavoidable as soon as the classical correspondences between preorderings equivalence relations and orderings are taken into account in contrast to earlier papers on the topic this paper is focused on representation and construction results which once more demonstrate the feasibility of the proposed concept

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

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

a survey of fuzzifications of frames the papert papert isbell adjunction and sobriety this paper gives a survey of fuzzifications of frames the papert papert isbell adjunction between the categories of topological spaces and locales as well as the related concept of sobriety it then establishes categorical isomorphisms between yao l frames zhang liu l frames and l algebras lastly it studies the relationships between l sobriety modified l sobriety and i l sobriety the obtained results suggest that modified l sobriety is the most fruitful notion between the above concepts of lattice valued sobriety

fuzzy relation equations and inequalities with two unknowns and their applications abstract in this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise we show that solutions to each of these systems form a complete lattice and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations we also demonstrate the application of solutions to these systems in data reduction

laplacian matrices and alexandrov topologies of digraphs abstract we explore the spectral properties of digraph laplacians and how they relate to topological properties of digraphs such as openness closure and strong connectedness under the alexandrov topology

discontinuous galerkin finite element method for euler and navier stokes equations a finite element method for the euler and navier stokes equations has been developed the spatial discretization involves the discontinuous galerkin finite element method and lax friedrichs flux method the temporal discretizations used are the explicit runge kutta time integrations the scheme is formally second order accurate in space and time a dynamic mesh algorithm is included to simulate flows over moving bodies the inviscid flows passing through a channel with circular arc bump through the naca 0012 airfoil and the laminar flows passing over a flat plate with shock interaction are investigated to confirm the accuracy and convergence of the finite element method also the unsteady flow through a pitching naca 0012 airfoil is performed to prove the capability of the present method

i fuzzy alexandrov topologies and specialization orders the aim of this paper is to investigate the relationships between certain generalized topological structures and fuzzy order structures on a universe set x in details a fuzzy preordered relation induces an i fuzzy alexandrov topology and an i fuzzy topology induces a fuzzy preordered relation which turns out to be a specialization order in i fuzzy setting in addition the representation theorem of fuzzy preorders by i fuzzy topologies is obtained these results coincide with the conclusions obtained in literature in the version of two valued logic

the relationship between l fuzzy rough set and l topology various fuzzy generalizations of rough approximations have been proposed in the literature this paper is devoted to the discussion of the relationship between l fuzzy rough sets and l topologies on an arbitrary universe finally one to one correspondence between the set of all reflexive transitive l relations and the set of all alexandrov l topologies is obtained

towards probabilistic partial metric spaces diagonals between distance distributions abstract the quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories motivated by the categorical interpretation of partial metric spaces we are led to investigate the quantaloid of diagonals between distance distributions which is expected to establish the categorical foundation of probabilistic partial metric spaces observing that the quantale of distance distributions w r t an arbitrary continuous t norm is non divisible we precisely characterize diagonals between distance distributions and prove that one step functions are the only distance distributions on which the set of diagonals coincides with the generated down set

a short note on l fuzzy approximation spaces and l fuzzy pretopological spaces in this note we investigate the relationship between l fuzzy pretopological spaces and l fuzzy approximation spaces based on the reflexive l fuzzy relations from a categorical viewpoint