dual fuzzy neighbourhood spaces i abstract a fuzzy neighbourhood space fns is determined by means of a system of fuzzy neighbourhoods of the ordinary points of that space we provide a general method for constructing bases for all fuzzy neighbourhood systems in a natural way from the fuzzy topologies of a special type of fuzzy topological spaces fts s which we call the fuzzy neighbourhood base spaces this construction is shown to be functorial of those new fts s the full subcategory of dual fuzzy neighbourhood spaces is shown to be isomorphic to the category of fns s and a fns will coincide with its dual fts iff it is topologically generated

probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

ergodicity conditions for two dimensional markov chains on the positive quadrant the basic problem considered in this paper is that of determining conditions for recurrence and transience for two dimensional irreducible markov chains whose state space is z 2 z xz assuming bounded jumps and a homogeneity condition malyshev 7 obtained necessary and sufficient conditions for recurrence and transience of two dimensional random walks on the positive quadrant unfortunately his hypothesis that the jumps of the markov chain be bounded rules out for example the poisson arrival process in this paper we generalise malyshev s theorem by means of a method that makes novel use of the solution to laplace s equation in the first quadrant satisfying an oblique derivative condition on the boundaries this method which allows one to replace the very restrictive boundedness condition by a moment condition and a lower boundedness condition is of independent interest

t syntopogenous structures compatible with fuzzy t uniformities and fuzzy t neighbourhoods structures in this paper we continue the discussion of t syntopogenous spaces which were introduced by the second author in 2012 this manuscript shows that the t syntopogenous spaces agree well with hashem morsi fuzzy t neighbourhood spaces 2003 fuzzy t proximity spaces 2002 and hohle fuzzy t uniform spaces 1982 also we show that each syntopogenous space in the sense of csaszar 1963 generates a t syntopogenous space finally we deduce the concept of fuzzy t neighbourhood syntopogenous structures and some related results

on regularity axioms in fuzzy neighbourhood spaces abstract two notions of regularity in fuzzy neighbourhood spaces fns s have been introduced one by t m g ahsanullah and the other by the authors we prove that those two notions coincide we then characterize a regular fns as a fns whose level topologies for all 0

on fuzzy pseudo normed vector spaces abstract we provide a method for introducing fuzzy pseudo metric topologies on sets and fuzzy pseudo normed topologies on vector spaces over r or c which will be fuzzy linear topologies we define fuzzy pseudo metrics for pairs of crisp points and fuzzy pseudo norms for crisp points as fuzzy real numbers 0 as defined by hutton we define the associated fuzzy open balls with crisp points for their centres and fuzzy real numbers 0 for their radii these form a basis for the associated fuzzy pseudo metric topology the axioms governing fuzzy pseudo metric and fuzzy pseudo norm are straightforward extensions for the corresponding axioms in the crisp case the formulation conforms with zadeh s extension principle we show that katsaras concept of fuzzy seminorm fuzzy sets and systems 12 1984 143 154 is equivalent to ours in the sense that both concepts will result in same fuzzy linear topologies

fuzzy syntopogeous structures compatible with lowen fuzzy uniformities and artico moresco fuzzy proximities abstract in this paper we give a new definition of a fuzzy syntopogenous structure it turns out that this definition agrees very well with the fuzzy neighbourhood structures the lowen fuzzy uniformities and the artico moresco fuzzy proximities more precisely it is shown that there is a one to one correspondence between the fuzzy neighbourhood structures and the so called perfect fuzzy topogenous structures also there is a one to one correspondence between the artico moresco fuzzy proximities and the symmetrical fuzzy topogenous structures finally it is proved that there exists a one to one correspondence between the lowen fuzzy uniformities and the symmetrical biperfect fuzzy syntopogenous structures

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

on fuzzy neighbourhood spaces abstract we study the behaviour of the fuzzy closure operator of a fuzzy neighbourhood space fns for short with regard to the two operations of sum and conjunction on fuzzy subsets we supply sufficient conditions on a cech fuzzy closure operator to be the fuzzy closure operator of a fns finally we study quotient fuzzy neighbourhood systems with the help of the notion of dual fuzzy neighbourhood spaces of 11

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

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

sequential decomposition of the matrix riccati equation and its application to the linear quadratic regulator problem the differential matrix riccati equation for the multi input multi output linear quadratic optimal regulator problem is considered two methods are presented successive and parallel that decompose this equation into a set of riccati equations that correspond to optimal regulator problems of possibly reduced dimensions the additivity of the solutions to the equations obtained by the successive sequential decomposition method sdm and the additivity of the inverses of the solutions to the equations obtained by the parallel sdm are established some duality relations between the successive and the parallel methods are presented via the use of the adjoint riccati equation the theory developed is extended to the algebraic matrix riccati equation as a limiting case the application of the sdms in the infinite time linear quadratic regulator problem is investigated special attention is paid to the partially cheap problem where the cost of some of the regulator controls or of their combinations tends asym

pseudo closure operators in fuzzy topological spaces abstract the concept of a pseudo closure operator s of an fts x has been introduced in 6 in this paper we give an axiomatic characterization of pseudo closure operators also we give a necessary and sufficient condition on an fts x so that is the unique fuzzy topology which induces s at the end of the paper we give a characterization of fts s of structure a 6 in terms of pseudo closure operators

semi induced fuzzy topologies abstract induced fuzzy topologies were first introduced by wong 15 and weiss 14 they were studied by lowen who called them topologically generated fuzzy topologies 7 an induced fuzzy topology includes its modification topology we generalize induced fuzzy topologies to semi induced fuzzy topologies viz those which include their modification topologies the yet more general fuzzy topologies of structure c are also introduced the fully stratified fuzzy topological spaces of structure c are precisely the fuzzy neighbourhood spaces of lowen 4 in subsequent papers this relationship will be established and benefited from the two axioms of semi induction and structure c hold trivially for crisp fuzzy topologies we call such axioms fuzzy structural axioms in fuzzy topology in this paper we analyze the axioms of structure c semi induction and induction in this context we show how these axioms are linked to the three structural asioms of structures a b and d introduced by us in 11 the basic notions and results pertaining to structures a b and d are compiled in the first section the behaviour of warren s fuzzy boundary in semi induced fuzzy topological spaces fts s for short is also explained

a new method for evaluating the log likelihood gradient the hessian and the fisher information matrix for linear dynamic systems a method is presented for evaluating the log likelihood gradient score the hessian and the fisher information matrix of the parameters of linear dynamic stochastic systems the method incorporates the optimal kalman smoothing equations and is therefore ideal for simultaneous state estimation and parameter identification the result can be used for efficient implementation of gradient based algorithms for maximum likelihood identification of the unknown system parameters and for assessing the mean square estimation accuracy

nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces abstract the nearness concept giving rise to a fuzzy topological space is a fuzzy relation of closeness between fuzzy subsets and crisp points of the space lowen 8 introduced the category of fuzzy neighbourhood spaces which we denote by fns his heuristic idea was that the fuzzy topologies in fns should be characterized by means of fuzzy neighbourhoods of crisp points it may be argued that the fuzziness of some concepts related to fns is rather restricted but besides the benefits gained from this restriction in that fns is a category of well behaved fuzzy topological spaces cf 1 5 8 9 14 17 18 we argue in this paper that this restriction acts to reduce the above mentioned concept of nearness to the following one the nearness concept giving rise to a fuzzy neighbourhood space is a fuzzy relation of closeness between the crisp subsets and the crisp points of the space such fuzzy relation may naturally arise when fuzziness is engrained in the definition of the points of the space for instance in the fuzzy unit interval of hutton 4 artico and moresco 2 introduced a concept of fuzzy proximities we show that this concept is particular to fuzzy neighbourhood spaces only we then argue that the nearness concept giving rise to a fuzzy proximity space is a fuzzy relation of proximity defined on all pairs of crisp subsets of the space

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 t locality spaces abstract a lowen fuzzy topological space is said to be localizable at points if it is completely determined in some prescribed manner by a system of fuzzy neighbourhoods called here fuzzy localities of its ordinary points we introduce for each lower semicontinuous triangular norm t a new type of those spaces which we call fuzzy t locality spaces the particularity of a fuzzy t locality space is that its fuzzy interior operator is obtained from its t locality system through a definition of fuzzy containment of fuzzy sets that employs the t residuated implication operator the origins of this new theory lie in the theory of probabilistic metric spaces indeed fuzzy t locality spaces provide a fuzzy topological framework for a naturally defined notion of metric t continuity of functions between probabilistic t metric spaces in a manner described in the last two sections of this article before that we establish characterizations and aspects of good behaviour for fuzzy t locality spaces including that every t locality system has an open basis we briefly study their level topologies and we show that all topologically generated spaces are fuzzy t locality spaces we prove that the category t fls of fuzzy t locality spaces and continuous functions between them is a topological category we show that it is closed in fts under the formation of optimal lifts of sources we provide a condition on t under which t fls becomes closed in fts also under the formation of co optimal lifts of sinks

fuzzy neighbourhood structures and fuzzy quasi uniformities abstract it is shown that every fuzzy neighborhood structure n is given by some fuzzy quasi uniformity also n is induced by some fuzzy uniformity if it is completely regular

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

generalized possibility measures we define a generalized possibility measure gpm on a set x to be a function from the power set of x to the unit interval i 0 1 whereby the measure of a finite union of subsets of x equals the maximum of their measures we characterize gpms as the infima of downward directed families of possibility measures this helps to confer an intuitive meaning on gpms we discuss all directed families of possibility measures or equivalently of possibility distributions which generate a given gpm and we identify the largest such family we introduce extensions of gpms to real functions on ix obtained through a continuous triangular norm we provide characterizations and alternative descriptions for those extensions

the productivity classes of fuzzy topologies in a previous article fuzzy sets and systems 30 1989 175 191 we introduced classes of fuzzy topological spaces fts s and other special classes of fts s those are tntimatley related to the productivity properties of fuzzy closure and fuzzy interior operators we here provide a number of necessary and sufficient conditions on a fts to be a member of one of those classes we use those characterizations to establish that each one of our classes of fts s is closed under products and quotients in the category of fuzzy topological spaces

on product fuzzy topological spaces abstract we provide complete answers to the following three questions in fuzzy topology when is the fuzzy closure operator finitely productive on a given family of fuzzy topological spaces which need not be fully stratified when is the fuzzy interior operator finitely productive on such a family when are the coordinate projections open maps from arbitrary products of members of such a family the answer to the first question is the harder one to establish the second and the third question turn out to have same answer

a level topologies criterion for lowen fuzzy uniformizability throughout x will be a nonempty set we consider a subset m of x as a crisp fuzzy set and denote this again by m we denote the strong cut of a fuzzy set e ix by and its weak cut by e i

the rate of convergence of a homogeneous markov chain arising from two queue networks abstract this paper is concerned with the type of random walks in two dimensions considered by malyshev 1 which describe a wide class of two queue systems sufficient conditions are established for the random walks to have various rate of convergence results including ergodicity geometric ergodicity and sub geometric rates for a variety of rate functions

fuzzy t neighbourhood spaces part 1 t proximities this is the first one of three articles that are intended to be published consecutively in this journal together they constitute a three part work on the unified subject of fuzzy t neighbourhood spaces and their fuzzy t proximities where t stands for any continuous triangular norm these new notions generalize to arbitrary t corresponding ones due to artico hohle lowen and moresco in this part we define and study notions of fuzzy t proximity one for each t their definition subsumes that of fuzzy proximity due to artico and moresco fuzzy sets and systems 21 1987 85 as our min proximity in particular we study proximity maps the fuzzy topological space associated with a t proximity which in the second part will be proved a fuzzy t neighbourhood space t proximities as fuzzy relations in ordinary power sets t proximal neighbourhood systems and relationships to the hohle lowen fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 the second part of this three part work will be concerned with the categories of fuzzy t neighbourhood spaces one for each t whereas we devote the third part to our t separation axioms in those categories

on fuzzy distances and their use in image processing under imprecision abstract this paper proposesa classification of fuzzy distances with respect to the requirements needed for applications in image processing under imprecision we distinguish on the one hand distances that basically compare only the membership functions representing the concerned fuzzy objects and on the other hand distances that combine spatial distance between objects and membership functions to our point of view the second class of methods finds more general applications in image processing since these methods take into account both spatial information and information related to the imprecision attached to the image objects new distances based on mathematical morphology are proposed in this second class

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

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

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

mixed mode solution to the partially singular discrete time filtering problem by sequential decomposition a sequential decomposition method is developed for solving the stationary discrete time algebraic matrix riccati equation this decomposition is applied to the partially singular stationary filtering problem where some of the system outputs are free of measurement noise explicit expressions are obtained for the minimum error covariance matrices in the case where the number of the noise free measurements is equal to the number of independent inputs the method is generalized also to the case where the singular subsystem is non square

fuzzy uniformities induced by fuzzy proximities abstract given a fuzzy proximity we construct a fuzzy uniformity in the sense of lowen which turns out to be the coarse stone among those which induce we show that some basic theorems of general topology about precompact uniformities have a nice extension to fuzzy set topology

constructions in the category of fuzzy t proximity spaces abstract this paper is devoted to study the initial fuzzy t proximity structures of the fuzzy t proximity spaces defined by hashem and morsi in 2002 where t stands for any continuous triangular norm in this paper we show that all initial and final lifts in the category t fps of these fuzzy t proximity spaces exist and hence the initial and final fuzzy t proximity structures exist we introduce a characterization for the initial fuzzy t proximity structures so as a special initial fuzzy t proximity spaces the subspaces and product spaces of these fuzzy t proximity spaces can also be characterized we also show that the fuzzy topology associated to the initial fuzzy t proximity structure of a family of fuzzy t proximity spaces coincides with the initial fuzzy topology of the family of fuzzy topologies associated to these fuzzy t proximity spaces

uniform environments as a general framework for metrics and uniformities this paper puts theories of classical uniformity probabilistic uniformity hutton uniformity classical metric probabilistic metric and hutton metric into one framework by introducing the concept of uniform environment which is a functor from the category of sets to the opposite category of involutive unital quantales and involutive closed unital maps subject to certain conditions the results show that uniform environments and their transformations provide a nice framework to study relationships between various notions of uniformity in the literature

pseudo closure operators in fuzzy neighbourhood base spaces abstract fuzzy neighbourhood base spaces and dual fuzzy neighbourhood spaces are two special types of fuzzy topological spaces they were introduced by morsi as useful tools for the theory explanation and application of lowen s fuzzy neighbourhood systems the notion of the pseudoclosure operator of a fuzzy topological space was introduced by mashhour et al as a tool in establishing a necessary and sufficient condition for the productivity of fuzzy kuratowski operators pseudo closure operators carry partial information about the fuzzy closure operators of their respective spaces we establish relationships between the pseudo closure operators of fuzzy neighbourhood base spaces their fuzzy closure operators and their level topologies we provide counterexamples to disprove other nonvalid relationships we also study some categorical aspects of the operation of taking the pseudo closure operators of fuzzy topological spaces

various fuzzy topological notions and fuzzy almost continuity abstract we extend the notion of almost continuity in the sense of husain 1966 to the fuzzy setting we obtain a pointwise characterization of the notion by dual point and fuzzy net also we formulate conditions under which the two notions of fuzzy almost continuity the one defined in this paper and the one defined by ajmal and azad 1990 turn out to be equivalent the main idea behind this work is to show that using the localized approach developed in earlier papers the study of fuzzy almost continuity can be carried out as successfully as that of its counterpart in general topology in the process we exhibit the compatibility of various existing fuzzy topological notions such as fuzzy regularity c i connectedness i 3 4 c i local connectedness i 3 4 fuzzy t 1 axiom fuzzy t 2 axiom etc

on co optimal lifts in the category of fuzzy neighbourhood spaces the category fns of fuzzy neighbourhood spaces and their continuous functions is closed under the formation of optimal and co optimal lifts in the larger topological category fts of fuzzy topological spaces but while the fuzzy neighbourhood system and the level topologies of an optimal lift of a source in fns have simple descriptions in terms of the fuzzy neighbourhood systems and the level topologies respectively of the source spaces analogous descriptions are yet to be given for both constructs for a co optimal lift of a sink in fns we here establish such descriptions with the help of a functor introduced earlier by one of us with codomain fns and domain another full subcategory of fts the category fnbs of fuzzy neighbourhood base spaces

a comparison of various uniformities in fuzzy topology it is known that the construct fts of stratified fuzzy topological spaces contains non trivial concretely both reflective and coreflective full subconstructs since each of these subconstructs is both initially and finally closed in fts it gives rise to a perfectly viable and autonomous theory of fuzzy topology in this paper this idea is applied to analyze the interrelations between the various notions of uniformity in fuzzy set theory the results show that these notions can be put into a coherent and clear picture if we interpret them as notions of uniformity for certain subconstructs of fts that are concretely both reflective and coreflective or some superconstructs of fuzzy topological spaces

fuzzy t neighbourhood spaces part 2 t neighbourhood systems we explore a notion of fuzzy t neighbourhood spaces for any continuous triangular norm t and we present on this notion a unified treatment our theory on one hand generalizes the theory of lowen fuzzy sets and systems 7 1982 65 from t min to arbitrary t which has been the progenitor of this work and on the other hand it is strongly related to the theory of l neighbourhoods of hohle mathematics of fuzzy sets logic topology and measure theory kluwer academic publishers dordrecht 1999 when l is restricted to the lattice 0 1 we study the t neighbourhood bases and systems of our spaces their level topologies as well as their relationship to the lowen hohle fuzzy t uniformities j math anal appl 82 1981 370 manuscripta math 38 1982 289 and to our fuzzy t proximities introduced in part 1 we show that the nearness concept underlying a fuzzy t neighbourhood space is a fuzzy relation of closeness between its ordinary subsets and ordinary points in particular our spaces are in canonical one to one correspondence with the t probabilistic topological spaces of frank j math anal appl 34 1971 67 we demonstrate that each full subcategory t fns of fts of fuzzy t neighbourhood spaces is a topological category to do that we characterize t fns within fts and we characterize continuity of functions within t fns in terms of t neighbourhood bases

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

discrete regulators with time scale separation a procedure for decomposition and control design of linear discrete regulators with fast and slow modes is presented the composite and reduced order controls are formulated in a standard form and upper bounds on their performance are derived an eighth order power model is worked out to demonstrate the analytical development

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

some more results on completely induced fuzzy topological spaces abstract the object of the paper is to present some new results on completely induced fuzzy topological spaces and completely semi induced spaces earlier introduced by the authors in this journal when a fuzzy topological space becomes a completely induced space is discussed examples of completely induced space are also cited

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

fuzzy t neighbourhood spaces part 3 t separation axioms we address the problem of identifying a useful set of mutually compatible separation axioms for each category t fns of t neighbourhood spaces it should contain one of t complete regularity that characterizes t uniformizability as indeed we achieve here this seems to necessitate the parameterization of most separation axioms by the triangular norm t with the exception of the two axioms t0 and r0 which have categorical definitions valid in any topological category we introduce t separation axioms in terms of t and fuzzy closure operators only making them meaningful for the whole of fts we then characterize them within t fns in terms of t fuzzy closures of crisp fuzzy subsets and t neighbourhoods we study our t axioms and establish some entailments between them our t complete regularity commits us to investigate a t real line which is a t neighbourhood space induced by a certain probabilistic t metric due to hohle fuzzy sets and systems 1 1978 311

a short note on fuzzy neighbourhood spaces abstract lowen introduced the concept of fuzzy neighbourhood spaces fns s and proved that the family of level topologies of a fns is an antichain wuyts characterized fns s as being those fully stratified fuzzy topological spaces fts s whose level topologies form antichains and whose fuzzy topologies are the finest possible for thier level topologies mashhour et al introduced the axiom of structure a on fts s and derived for it several characterizations and fuzzy topological consequences we use those characterizations of structure a and wuyts characterization of fns s to prove that a fts is a fns iff it is of structure a and fully stratified

optimal nearly singular estimation of continuous linear stationary uniform rank systems the problem of optimal estimation of linear continuous systems is considered for the case where all the noise signals are stationary white and the intensity of the measurement noise tends uniformly to zero explicit expressions for the estimation procedures and the corresponding minimum error covariance matrices of the estimate are obtained in terms of the measurement noise intensity and the zero properties of the system these expressions are derived for the case where the first non zero markov parameter of the system is of full rank for the infinite and the finite time filtering problems they provide a simple geometric interpretation to the structure of the error covariance matrices and they distinctly show the difference between the estimation of minimum and non minimum phase systems the theory is developed both for square and non square systems

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

completely induced fuzzy topological spaces abstract the object of this paper is to introduce and to study the concepts of completely induced fuzzy topological spaces and completely lower semi continuous functions completely lower semi continuous functions turn out to be the natural tool for studying the completely induced fuzzy topological spaces

strong and ultra separation axioms on l fuzzy topological spaces given an l fuzzy topological space x d we introduce new notions of l fuzzy separation axioms using the family of its level topology i a d a pr l we check that these are well defined l fuzzy topological concepts and we compare them with the analogous l fuzzy ones introduced in the literature we verify that these notations are not equivalent and we give many examples which illustrate this fact furthermore some results concerning the new concepts have been established

brief paper design of observer based controllers for a class of discrete systems design of observers and observer based controllers for linear discrete control systems with fast and slow modes are considered in this paper the time separation is expressed in terms of an inequality relating norms of system submatrices it is shown that reconstruction of fast and slow states can be accomplished by a full order observer the gains of which are computed using a two stage procedure then provided that the fast subsystem is asymptotically stable it is shown that a low order observer based controller can be designed with independent gain matrices to stabilize the original discrete system the theoretical analysis is illustrated by a ninth order boiler system model