on strongly norm attaining lipschitz maps abstract we study the set sna m y of those lipschitz maps from a complete pointed metric space m to a banach space y which strongly attain their lipschitz norm i e the supremum defining the lipschitz norm is a maximum extending previous results we prove that this set is not norm dense when m is a length space or local or when m is a closed subset of r with positive lebesgue measure providing new examples which have very different topological properties than the previously known ones on the other hand we study the linear properties which are sufficient to get lindenstrauss property a for the lipschitz free space f m over m and show that all of them actually provide the norm density of sna m y in the space of all lipschitz maps from m to any banach space y next we prove that sna m r is weakly sequentially dense in the space of all lipschitz functions for all metric spaces m finally we show that the norm of the bidual space of f m is octahedral provided the metric space m is discrete but not uniformly discrete or m is infinite

tl subrings and tl ideals part 4 radicals of tl ideals abstract in this paper the notion of the radicals of l subsets and tl ideals of a ring is introduced and studied where t is an infinitely distributive t norm on a given complete brouwerian lattice l our main objective is to present basic properties about the radicals of tl ideals in preparation for the forthcoming two papers

operations on fuzzy ideals in this paper we give suitable definitions of the operations of l fuzzy ideals in a ring as intersection addition multiplication and residual division and we prove some fundamental properties of the operations analogous to those that hold for the operations of ordinary ideals see 3

inference of fuzzy relational tableaux from fuzzy exemplifications the paper describes a method for deriving fuzzy relational tableaux from fuzzy exemplifications in particular the work by thole zimmerman and zysno 4 is cited as they derive an operator for and using data derived from human subjects the method results in an operator or operators which is a maximal restriction of the operators implied by the data results showing the process are described and a value of and is derived using data presented in 4

fuzzy radicals and prime fuzzy ideals of ordered semigroups let s be an ordered semigroup a fuzzy subset of s is by definition an arbitrary mapping f s 0 1 where 0 1 is the usual interval of real numbers motivated by studying prime fuzzy ideals in rings semigroups and ordered semigroups and as a continuation of kehayopulu and tsingelis s works in ordered semigroups in terms of fuzzy subsets in this paper we introduce the notion of ordered fuzzy points of an ordered semigroup s and give a characterization of prime fuzzy ideals of an ordered semigroup s we also introduce the concepts of weakly prime fuzzy ideals completely prime fuzzy ideals completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup s and establish the relationship between the five classes of ideals furthermore we characterize weakly prime fuzzy ideals completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of s by their level ideals finally we introduce and investigate the fuzzy radicals of ordered semigroups by means of ordered fuzzy points and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an ordered semigroup s can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it as an application of the results of this paper the corresponding results of semigroups without order are also obtained

ehrenfest s argument extended to a formalism of nonequilibrium thermodynamics a general method of constructing dissipative equations is developed following ehrenfest s idea of coarse graining the approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations proof of the h theorem for macroscopic equations is given several examples supporting the construction are presented and generalizations are suggested

existence and uniqueness of fuzzy ideals let r be a commutative ring the idea of identifying a fuzzy ideal of r with its corresponding chain of level ideals and its range of values although widely used in the literature for the study of fuzzy ideals has never been investigated in a general setting before we investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established in particular we obtain that uniqueness always holds true for artinian rings moreover examples are presented to further support our results and shed more light on the above mentioned problem of identification we finish by raising a natural question that is left as an open problem for further investigation

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

the homomorphic image of a fuzzy subgroup is always a fuzzy subgroup let l be a completely distributive complete lattice with universal bound 0 and 1 and x any group we will prove that the image of a fuzzy group x l under a homomorphism f x y is always a fuzzy group without any restriction on and f

fuzzy ideals and fuzzy semiprime ideals some ring theoretic analogues abstract an attempt is made to examine the algebraic nature of fuzzy semiprime ideals under homomorphisms also the fuzzy nil radical of the nth power of a fuzzy prime ideal is studied and a sufficient condition for a fuzzy ideal to be fuzzy maximal is given

on t fuzzy subfield and t fuzzy vector subspaces abstract in this paper we improve the definition of t fuzzy subfield and observe some of its properties the concept of t fuzzy vector subspaces of a t fuzzy subfield of a field will also be investigated and we further introduce the concept of t fuzzy subalgebra

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

tl subrings and tl ideals part 1 basic concepts abstract in 25 27 we put forward a rudimentary frame of the theory of tl subalgebras in 26 we established a theory of tl subgroups and tl subgroups in this paper we will continue our work in concern with basic ideas provided in 25 27 introduce the notions of tl subrings and tl ideals of a ring and study some of their properties where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

t fuzzy prime ideals abstract the purpose of this paper is to define study and characterize the t fuzzy prime ideals t fuzzy strongly prime ideals and t s fuzzy weakly prime ideals proving the inclusion relationships that are satisfied among them the t fuzzy comaximal ideals and fuzzy maximal ideals are also investigated

fuzzy nil radicals and fuzzy primary ideals abstract the purpose of this paper is to extend the concept of a fuzzy set by introducing the concept of a fuzzy nil radical of a fuzzy ideal in a ring using this concept a fuzzy primary ideal is defined some fundamental results concerning these notions are proved

fuzzy subgroups fuzzy ideals and fuzzy cosets some properties abstract as an application of fuzzy subgroups a necessary and sufficient condition for a p group to be cyclic is obtained a procedure to construct the fuzzy subgroup generated by a fuzzy subset of a group is provided also the effect of a homomorphism on the sum product intersection of two fuzzy ideals is studied some elementary properties of the ring fuzzy cosets of a fuzzy ideal are also obtained

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

some comments on spectrums of prime fuzzy ideals of a ring abstract common and distinctive features of two prime spectrums one obtained by kumar and kohli using 2 valued prime fuzzy ideals and the other obtained by the author using not so restrictive definition of prime fuzzy ideals are discussed it is also shown that some results claimed by kumar and kohli regarding spectrum of 2 valued prime fuzzy ideals need some corrections

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

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

fuzzy subrings and ideals redefined abstract v q fuzzy subrings and ideals of a ring are defined the concepts of v q fuzzy semiprime prime semi primary primary and maximal ideals are introduced and characterization of such fuzzy ideals are obtained v q fuzzy cosets determined by v q fuzzy ideals and v q fuzzy radicals of v q fuzzy ideals are discussed

the lattice of fuzzy ideals of a ring the set of all fuzzy ideals of a ring r is partially ordered with respect to the natural ordering of fuzzy sets in r it has been shown that this is a lattice but not a sublattice of the lattice of all fuzzy sets in r this lattice of all fuzzy ideals of r is complete distributive and bounded and has neither atoms nor dual atoms

on fuzzy semiprimary ideals in this paper semiprimary ideals are redefined in terms of fuzzy points and it is proved that this definition is equivalent to that of kumar 1991 several results regarding semiprimary and fuzzy semiprimary ideals are derived it is shown by examples that the characterization of semiprimary ideals by prime radicals does not hold for this definition of fuzzy semiprimary ideals

fuzzy ideals and fuzzy prime ideals of a ring abstract besides obtaining certain ring theoretic analogues some peculiarities of the fuzzy setting are also discussed in particular a study of i the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals ii the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms and iii the ring of all the fuzzy cosets of a fuzzy prime fuzzy maximal ideal is carried out

fuzzy fields and fuzzy linear spaces redefined we here redefine fuzzy fields and fuzzy linear spaces which were initially defined by nanda 5 our definitions are more suitable than the earlier

irreversibility in the short memory approximation a recently introduced systematic approach to derivations of the macroscopic dynamics from the underlying microscopic equations of motions in the short memory approximation phys rev e 63 2001 066 124 is presented in detail the essence of this method is a consistent implementation of ehrenfest s idea of coarse graining realized via a matched expansion of both the microscopic and the macroscopic motions applications of this method to a derivation of the nonlinear vlasov fokker planck equation diffusion equation and hydrodynamic equations of the fluid with a long range mean field interaction are presented in full detail the advantage of the method is illustrated by the computation of the post navier stokes approximation of the hydrodynamics which is shown to be stable unlike the burnett hydrodynamics

complete metric spaces with property z are length spaces abstract a metric space m is length when every two points have an approximate midpoint while m has property z when for every x y m there exists z such that d x z d z y d x y is arbitrarily smaller than d x z and d z y answering a problem posed by garcia lirola prochazka and rueda zoca we prove that every complete metric space with property z is length as a consequence we get that for the lipschitz free space of a metric space a number of geometric properties are all equivalent like the daugavet property the diameter 2 property the absence of strongly exposed points or the absence of preserved extreme points

time structure invariance criteria for closure approximations in many areas of physics time evolution equations for moments of distributions are expressed in terms of higher order moments closure approximations are then introduced in an ad hoc fashion to reduce the higherorder moments to functions of the lower order ones herein the time structure invariance of the poisson bracket as manifested through the jacobi identity is used to derive constraint relationships on these approximations these constraints severely limit the allowable functionality of general closures and help to define the boundaries within which future investigations should concentrate s1063 651x 97 10410 x

uniqueness results in the representation of families of sets by fuzzy sets the important concept of identifying a fuzzy subset of a set r with its corresponding chain of level sets is widely used in the literature we investigate in this paper the problem of characterization of all fuzzy subsets of r that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 although our main focus is on the problem of uniqueness we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets we continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed to further support our results we present some examples which we hope will shed more light on the above mentioned problem of identification we finish by raising a few natural questions that are left as open problems for further investigation

on fuzzy ideals and fuzzy bi ideals in semigroups abstract in this paper we give some properties of fuzzy ideals and fuzzy bi ideals of semigroups and characterize semigroups that are left duo left simple and semilattices of subsemigroups in terms of fuzzy ideals and fuzzy bi ideals

fuzzy semiprimary ideals of rings abstract the concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings this class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals it is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary a study of i the level ideals of a fuzzy semiprimary ideal ii the characteristic function of a semiprimary ideal iii the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and iv the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms is carried out

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

on radicals of fuzzy submodules and primary fuzzy submodules we define in this paper the concepts of the radical of fuzzy submodules and the primary fuzzy submodules and prove some of their properties

fuzzy irreducible ideals in rings abstract in this paper the concept of a fuzzy set is extended by introducing the concept of a fuzzy irreducible ideal in a ring the characteristic function of an irreducible ideal and the level ideals of a fuzzy irreducible ideal are studied the algebraic nature of homomorphic images and preimages of fuzzy irreducible ideals is also examined some other fundamental results pertaining to this concept are discussed

on fuzzy subgroups and fuzzy ideals under triangular norms abstract in this communication we explicitly point out that the principal results of liu 1982 basically deduced from the definition of binary operation on the set f x of all fuzzy subsets of x also hold if one uses the weaker definition of product under triangular norm t fuzzy ideals with respect to the triangular norms are also defined

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

fuzzy prime spectrum of a ring ii abstract various topological properties of the fuzzy prime spectrum of a ring and certain of its subspaces as well as its functorial nature are discussed

fuzzy topological vector spaces ii abstract some of the properties of fuzzy topological vector spaces are investigated also there are given necessary and sufficient conditions for a family of fuzzy sets in a vector space e to be the family of all neighborhoods of zero for a fuzzy linear topology

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

isomorphism theorems and fuzzy k ideals of k semirings we define and study the fuzzy maximal k ideal of a k semiring also we characterize the quotient k semiring r a of a k semiring r by a fuzzy maximal k ideal a and we have some isomorphism theorems in commutative k semirings and further we have some properties of the quotient k semiring r a of all cosets of a fuzzy semiprimary semiprime primary prime k ideal a in a commutative k semiring r

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

fuzzy cosets and some fuzzy radicals abstract an attempt is made to formulate fuzzy versions of jacobson and prime radicals of a ring certain basic results pertaining to the notions of jacobson and prime radicals of a ring are extended to the fuzzy setting in particular the following is obtained the jacobson radical prime radical of the ring of fuzzy cosets of in r is zero where r is a commutative ring with identity and is the fuzzy jacobson radical fuzzy prime radical of r

fuzzy nil radicals and fuzzy semiprimary ideals of rings abstract an attempt is made to define a fuzzy semiprimary ideal by using the idea of the fuzzy nil radical of a fuzzy ideal the level ideals of a such a fuzzy ideal are examined and the ring of fuzzy cosets of such a fuzzy ideal is studied the algebraic nature of such fuzzy ideals under homomorphisms is also ascertained

fuzzy semiprime ideals in semigroups abstract in this paper we shall introduce the notion of fuzzy semiprimality in a semigroup which is an extension of semiprimality in it and characterize a semigroup that is a semilattice of simple semigroups in terms of fuzzy semiprimality

from fuzzy sets to the decompositions of non rigid sets the concept of identifying a fuzzy subset of a set r with its collection of level sets and its collection of membership values plays an important role in many real life applications such identification is possible only if the fuzzy set can be recaptured this implies that uniqueness of the fuzzy set plays an important role it has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection s of membership values of the fuzzy set be a rigid set with this in mind the authors went on to show that if s has the min max property then s is a rigid set and left the converse as an open problem we start this paper by constructing a counterexample that shows that rigid sets do not have to have the min max property we then take a closer look at non rigid sets by decomposing them into isomorphism invariant components and into s connected components in the hope of getting a characterization of non rigid sets which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set we also investigate in the case of nonuniqueness the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values

lenses and asymptotic midpoint uniform convexity abstract we introduce a new isometric geometric property namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties

radicals of fuzzy ideals abstract there are several natural ways to fuzzify the concepts of prime ideal primary ideal the radical of an ideal and the primary representation of an ideal in a commutative ring in this paper we introduce some new approaches and compare them with known ones our main goal is to unify the various possible fuzzy primary representations of a fuzzy ideal to prepare the way for the study of fuzzy nonlinear systems of equations

semidirect products of fuzzy subgroups abstract the semidirect product of two groups is an important construction in group theory in this paper we define the semidirect product of fuzzy subgroups and give conditions for it to be a fuzzy subgroup this extends the work of h sherwood on direct products of fuzzy subgroups we then give an example where a fuzzy subgroup of a semidirect product is a semidirect product of fuzzy subgroups

on fuzzy rings abstract an internal description of the fuzzy subring fuzzy ideal generated by a finite fuzzy subset of a ring is provided also a fuzzy coset of a fuzzy ideal is defined and the set of all fuzzy cosets of a fuzzy ideal is given a ring structure a fuzzy semiprime ideal is also defined and some basic properties of this fuzzy ideal are studied

prime fuzzy ideals over noncommutative rings in this paper we introduce prime fuzzy ideals over a noncommutative ring this notion of primeness is equivalent to level cuts being crisp prime ideals it also generalizes the one provided by kumbhojkar and bapat in 1993 which lacks this equivalence in a noncommutative setting semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes this allows us to introduce the fuzzy prime radical and contribute to establish the basis of a fuzzy noncommutative ring theory

certain fuzzy ideals of rings redefined abstract an attempt is made to redefine fuzzy semiprimary fuzzy primary fuzzy prime and fuzzy maximal ideals of rings these fuzzy ideals are characterized by their level ideals fuzzy analogues of various results concerning semiprimary primary prime and maximal ideals are obtained the concept of a fuzzy nil radical of a fuzzy ideal is also formulated and discussed

prime l fuzzy ideals and primary l fuzzy ideals abstract in this paper the concepts of a primary l fuzzy ideal and a prime l fuzzy ideal and p primary are introduced and some fundamental propositions proved in particular a structural theorem for a primary l fuzzy ideal belonging to a prime l fuzzy ideal is given

a characterisation of the daugavet property in spaces of lipschitz functions abstract we study the daugavet property in the space of lipschitz functions lip 0 m on a complete metric space m namely we show that lip 0 m has the daugavet property if and only if m is a length metric space this condition also characterises the daugavet property in the lipschitz free space f m moreover when m is compact we show that either f m has the daugavet property or its unit ball has a strongly exposed point if m is an infinite compact subset of a strictly convex banach space then the daugavet property of lip 0 m is equivalent to the convexity of m

on some product of fuzzy subgroups abstract a kind of product of fuzzy subsets subgroups of a group is defined analogously to the pointwise product of functions via a triangular norm t we call this product the t product some properties of the t product of fuzzy subgroups of a group are investigated in particular we are interested in the fuzzy subgroups of the internal direct product of subgroups of a group

points of differentiability of the norm in lipschitz free spaces abstract we consider convex series of molecules in lipschitz free spaces i e elements of the form n n x n y n d x n y n such that n n we characterise these elements in terms of geometric conditions on the points x n y n of the underlying metric space and determine when they are points of g teaux differentiability of the norm in particular we show that g teaux and frechet differentiability are equivalent for finitely supported elements of lipschitz free spaces over uniformly discrete and bounded metric spaces and that their tensor products with g teaux resp frechet differentiable elements of a banach space are g teaux resp frechet differentiable in the corresponding projective tensor product

supports in lipschitz free spaces and applications to extremal structure abstract we show that the class of lipschitz free spaces over closed subsets of any complete metric space m is closed under arbitrary intersections improving upon the previously known finite diameter case this allows us to formulate a general and natural definition of supports for elements in a lipschitz free space f m we then use this concept to study the extremal structure of f m we prove in particular that x y d x y is an exposed point of the unit ball of f m whenever the metric segment x y is trivial and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself we also characterize the extreme points of the positive unit ball they are precisely the normalized evaluation functionals on points of m

radicals of tl ideals in this paper we first introduce several different definitions of radicals of tl ideals in a commutative ring with identity and study their basic properties then discuss several kinds of primary tl ideals where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

banach spaces with the daugavet property a banach space x is said to have the daugavet property if every operator t x x of rank 1 satisfies 11 id tl 1 fltil we show that then every weakly compact operator satisfies this equation as well and that x contains a copy of t1 however x need not contain a copy of l1 we also study pairs of spaces x c y and operators t x y satisfying i i j t i i 1 4 1f t i i where j x y is the natural embedding this leads to the result that a banach space with the daugavet property does not embed into a space with an unconditional basis in another direction we investigate spaces where the set of operators with f id tll 1 il t l i is as small as possible and give characterisations in terms of a smoothness condition

the bishop phelps bollobas property for lipschitz maps abstract in this paper we introduce and study a lipschitz version of the bishop phelps bollobas property lip bpb property this property deals with the possibility of making a uniformly simultaneous approximation of a lipschitz map f and a pair of points at which f almost attains its norm by a lipschitz map g and a pair of points such that g strongly attains its norm at the new pair of points we first show that if m is a finite pointed metric space and y is a finite dimensional banach space then the pair m y has the lip bpb property and that both finiteness assumptions are needed next we show that if m is a uniformly gromov concave pointed metric space i e the molecules of m form a set of uniformly strongly exposed points then m y has the lip bpb property for every banach space y we further prove that this is the case for finite concave metric spaces ultrametric spaces and holder metric spaces the extension of the lip bpb property from m r to some banach spaces y and some results for compact lipschitz maps are also discussed

octahedrality in lipschitz free banach spaces the aim of this note is to study octahedrality in vector valued lipschitz free banach spaces on a metric space under topological hypotheses on it by analysing the weak star strong diameter 2 property in lipschitz function spaces also we show an example that proves that our results are optimal and that octahedrality in vector valued lipschitz free banach spaces actually relies on the underlying metric space as well as on the banach one

uniqueness in the generalized representation by fuzzy sets the concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set the problem of existence of such fuzzy sets was investigated by various authors as far as we know the problem of uniqueness of such fuzzy sets was not studied before we establish among other things necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set

preservation of thermodynamic structure in model reduction based on the availability of an invariant manifold we develop a model reduction procedure that preserves thermodynamic structure more concretely we construct the poisson and irreversible brackets of the general equation for the nonequilibrium reversible irreversible coupling of nonequilibrium thermodynamics by means of the ideas originally introduced for handling constraints the general ideas are then applied to the kramers problem that is the description of transitions between two potential wells separated by a high barrier this example reveals how a fortuitous cancellation mechanism that allows a logarithmic entropy to generate a linear diffusion equation is inherited by a master equation resulting from model reduction

octahedral norms in duals and biduals of lipschitz free spaces abstract we continue with the study of octahedral norms in the context of spaces of lipschitz functions and in their duals first we prove that the norm of f m is octahedral as soon as m is unbounded or is not uniformly discrete further we prove that a concrete sequence of uniformly discrete and bounded metric spaces k m satisfies that the norm of f k m is octahedral for every m finally we prove that if x is an arbitrary banach space and the norm of lip 0 m is octahedral then the norm of lip 0 m x is octahedral these results solve several open problems from the literature

on prime quasi prime weakly quasi prime fuzzy left ideals of semigroups in this paper we introduce and study the prime quasi prime and weakly quasi prime fuzzy left ideals of a semigroup s we give characterizations of the largest fuzzy ideal i f contained in a fuzzy left ideal f of a semigroup s and the largest fuzzy subsemigroup i f such that f is a fuzzy ideal of i f finally strongly semisimple semigroups are characterized by quasi prime fuzzy left ideals

fuzzy vector spaces under triangular norms abstract the object of this paper is to redefine fuzzy subspaces using the notion of a t norm and to obtain several results some of which are generalisations of the results of katsaras and liu 3