on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

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

categories isomorphic to fns the main result in this paper is that the category of fuzzy neighborhood spaces is isomorphic to that of fuzzifying topological spaces

on some results of analysis for fuzzy metric spaces abstract a necessary and sufficient condition for a fuzzy metric space to be complete is given we prove that a subspace of a separable fuzzy metric space is separable and every separable fuzzy metric space is second countable uniform limit theorem is generalized to fuzzy metric spaces

stratified hutton uniform spaces in this paper some categorical properties of hutton uniform spaces are investigated at first the initial and final structures for hutton uniform spaces are discussed and it is proved that the canonical functor from the construct of hutton uniform spaces to that of l topological spaces preserves initial sources secondly the notion of stratified hutton uniform spaces is introduced and it is demonstrated that every stratified hutton uniformity induces a stratified l topology and that the relationship between stratified hutton uniform spaces and hutton uniform spaces is like that between stratified l topological spaces and l topological spaces finally it is showed that the construct of uniform spaces can be embedded in the construct of stratified hutton uniform spaces as a concretely both reflective and co reflective full sub construct

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

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

extension of a class of decomposable measures using fuzzy pseudometrics in this paper we consider a topological approach to extension of t conorm based decomposable measures by introducing a fuzzy pseudometric structure on an algebra of sets we prove that every non strict continuous archimedean t conorm based decomposable measure can be extended from an algebra to the completion of this algebra under the fuzzy pseudometric and then to the sigma algebra generated by this algebra the existence of such an extension follows very simply from the well known caratheodory result however our topological proof offers an intuitive interpretation of the extension of decomposable measures

the maximal distance between imprecise point objects in this paper we propose a new mathematical model of imprecise point objects and a new algorithm for determining the maximal distance between imprecise point objects results obtained with our model can be used in various applications such as gis imprecise spatial object modelling robotics environment models and image analysis imprecise feature extraction imprecise point objects are modelled as fuzzy points in linear fuzzy space linear fuzzy space is defined over r n where the membership function is convex upper semi continuous symmetric w r t the core and linearly decreasing w r t the core distance where the core is a single point a fuzzy convex hull is defined in linear fuzzy space the distance between objects points is modelled as an imprecise point object in linear fuzzy space defined over r 1 fuzzy ordering relations are defined in distance space the maximal distance is a fuzzy set over a set of distances the maximal distance between a set of imprecise point objects is then determined as the maximal distance of fuzzy convex hull edge points an algorithm for determining the maximal distance between imprecise points belonging to one and two dimensional linear fuzzy space is given for an exact calculation of this maximal distance computational complexity of the algorithm is of polynomial growth of the fourth degree however if the maximal distance is determined approximately then the algorithms computational complexity is of power growth of the quadratic degree with respect to the cardinality of the convex edge

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

on fuzzy pseudo metric spaces this paper attempts to endow the fuzzy pseudo metric in the sense of george and veeramani with many valued topological structures it is shown that george and veeramani s fuzzy pseudo metric can be characterized by a family of compatible ordinary pseudo metrics called pseudo metric chain in this paper then two kinds of many valued topological structures fuzzifying topology and fuzzifying uniformity are generated by this fuzzy pseudo metric and their properties are also investigated

completeness in probabilistic quasi uniform spaces abstract in this paper we give a kind of cauchy completeness in probabilistic quasi uniform space by using pair filters and prove that this kind of cauchy completeness is equivalent to the completeness of induced probabilistic uniform space introduced by hohle we also give a characterization of cauchy completeness by lawvere completeness then we study the completion of probabilistic quasi uniform space and show that each t 0 separated probabilistic quasi uniform space has a t 0 cauchy completion finally in probabilistic quasi metric space we study the relationship between the completeness of induced quasi uniform space and the completeness of probabilistic quasi metric space

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

fuzzy pseudometric spaces vs fuzzifying structures in this paper we apply the representation theorem established in mardones perez and de prada vicente 2012 12 to define and study the degree in which some topological type properties of fuzzy pseudometric spaces are fulfilled fuzzifying structures which appear naturally are also investigated and the relation between these structures and fuzzy pseudometric spaces is explored

nonlocal mechanism for synchronization of time delay networks we present the interplay between synchronization of networks with heterogeneous delays and the greatest common divisor gcd of loops composing the network we distinguish between two types of networks i chaotic networks and ii population dynamic networks with periodic activity driven by external stimuli for type i in the weak chaos region the units of a chaotic network characterized by gcd 1 are in a chaotic zero lag synchronization whereas for gcd 1 the network splits into gcd clusters in which clustered units are in zero lag synchronization these results are supported by simulations of chaotic systems self consistent and mixing arguments as well as analytical solutions of bernoulli maps type ii is exemplified by simulations of hodgkin huxley population dynamic networks with unidirectional connectivity synaptic noise and distribution of delays within neurons belonging to a node and between connecting nodes for a stimulus to one node the network splits into gcd clusters in which cluster neurons are in zero lag synchronization for complex external stimuli the network splits into clusters equal to the greatest common divisor of loops composing the network spatial and the periodicity of the external stimuli temporal the results suggest that neural information processing may take place in the transient to synchronization and imply a much shorter time scale for the inference of a perceptual entity

graph operations and synchronization of complex networks the effects of graph operations on the synchronization of coupled dynamical systems are studied the operations range from addition or deletion of links to various ways of combining networks and generating larger networks from simpler ones methods from graph theory are used to calculate or estimate the eigenvalues of the laplacian operator which determine the synchronizability of continuous or discrete time dynamics evolving on the network results are applied to explain numerical observations on random scale free and small world networks an interesting feature is that when two networks are combined by adding links between them the synchronizability of the resulting network may worsen as the synchronizability of the individual networks is improved similarly adding links to a network may worsen its synchronizability although it decreases the average distance in the graph

synchronization of oscillators with random nonlocal connectivity in this paper we study the existing observation in literature about synchronization of a large number of coupled maps with random nonlocal connectivity chate and manneville chaos 2 307 1992 these connectivities which lack any spatial significance can be realized in neural nets and electrical circuits it is quite interesting and of practical importance to note that a huge number of maps can be synchronized with this connectivity we show that this synchronization stems from the fact that the connectivity matrix has a finite gap in the eigenvalue spectrum in the macroscopic limit we give a quantitative explanation for the gap we compare the analytic results with the ones quoted in the above reference we also study the departures from this highly collective behavior in the low connectivity limit and show that the behavior is almost statistical for very low connectivity of late there has been considerable attention paid to the study of coupled map lattices cml in various contexts they have been used as a computationally simple and analytically tractable model for spatiotemporal systems 1 the studies on cml s have been either in one and two dimensional lattices or with global coupling in which case there is no notion of lattice geometry 2 the higher dimensional connectivities 3 or hierarchial connectivities 4 5 are studied very little one more system that has been studied is the cml with random nonlocal couplings 6 the motivation is twofold first this is an effectively high dimensional system the phenomenology in cml in higher dimensions has not been studied much and needs further investigation second there are systems like neural nets in which the local connections do not have any spatial significance there also exist systems like electrical circuits 7 in which connectivity is at one s will and such a coupling can be easily realized thus the studies of different connectivities and their effects will be useful in designing well controlled systems in this system of random nonlocal connectivity chate and manneville have presented preliminary results 6 which show that synchronization of a large number of oscillators is easily achieved with this connectivity synchronization of oscillators in spatially extended systems such as coupled oscillators is important from various points of view by achieving synchronization one can try to build huge but more controllable and better behaved systems which are effectively low dimensional 7 sometimes synchronization may serve other purposes such as sending codes that are difficult to break 8 in various contexts this problem has been subjected to several studies in the past few years 9 we would like to show that the phenomenon of synchronization can be generally understood by investigating the eigenvalue spectrum of the connectivity matrix and can help us to understand the existing observations in this paper we will explicitly illustrate how one can separate the mode leading to spatial homogeneity from the rest we will show that there exists a finite gap between the growth rates the spatially homogeneous mode and the rest in the model studied in ref 6 we will also study the departures from this behavior for lower connectivities for the linear stability analysis of the synchronized state we will study the eigenvalue spectrum of the connectivity matrix with random nonlocal connectivity we will also study the eigenspectrum of the product of such matrices we would note that a similar model of random connectivity matrix of size n3n with k nonzero elements in each row has been investigated by cook and derrida in ref 10 in connection with the random energy model the generalized random energy model and directed polymers in random media they have obtained exact analytic results for products of such matrices in the case where the matrices are sparse and the distribution function of nonzero elements is not a d function however in the model studied by chate and manneville 6 all connections have the same weight i e the distribution function of nonzero elements is a d function

on completion of fuzzy metric spaces completions of fuzzy metric spaces in the sense of george and veeramani are discussed a complete fuzzy metric space y is said to be a fuzzy metric completion of a given fuzzy metric space x if x is isometric to a dense subspace of y we present an example of a fuzzy metric space that does not admit any fuzzy metric completion however we prove that every standard fuzzy metric space has an up to isometry unique fuzzy metric completion we also show that for each fuzzy metric space there is an up to uniform isomorphism unique complete fuzzy metric space that contains a dense subspace uniformly isomorphic to it

support vector machines for classification of input vectors with different metrics in this paper a generalization of support vector machines is explored where it is considered that input vectors have different p norms for each class it is proved that the optimization problem for binary classification by using the maximal margin principle with p and q norms only depends on the p norm if 1 p q furthermore the selection of a different bias in the classifier function is a consequence of the q norm in this approach some commentaries on the most commonly used approaches of svm are also given as particular cases

coupled maps on trees we study coupled maps on a cayley tree with local nearest neighbor interactions and with a variety of boundary conditions the homogeneous state where every lattice site has the same value and the node synchronized state where sites of a given generation have the same value are both shown to occur for particular values of the parameters and coupling constants we study the stability of these states and their domains of attraction since the number of sites that become synchronized is much higher compared to that on a regular lattice control is easier to achieve a general procedure is given to deduce the eigenvalue spectrum for these states perturbations of the synchronized state lead to different spatiotemporal structures we find that a mean field like treatment is valid on this effectively infinite dimensional lattice

some properites of fuzzy metric spaces abstract we prove that the topology induced by any complete fuzzy metric space in the sense of george and veeramani is completely metrizable we also show that every separable fuzzy metric space admits a precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete finally we generalize among other classical results the niemytzki tychonoff theorem to fuzzy metric spaces

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

hutton 0 1 quasi uniformities induced by fuzzy quasi metric spaces it is well known that given a probabilistic metric space menger space with continuous t norm t there is a hausdorff topology associated this association factorizes through strong uniformities or e l uniformities similarly any fuzzy metric space x m can be endowed with a hausdorff topology t m in the case of fuzzy quasi metric spaces a t 1 topology and again this association factorizes through quasi uniform spaces in this paper we associate to each fuzzy quasi metric space a hutton 0 1 quasi uniformity u m this allows us to give a factorization of the previous association via hutton 0 1 quasi uniformities it is also proved that the topology t m is exactly the image under lowen s functor i of the 0 1 topology induced by u m as a consequence we get a class of hutton 0 1 quasi uniformities which are probabilistic metrizable

a new approach for fuzzy topology ii abstract we point out a new approach for fuzzy topology with fuzzy logic and discuss the neighborhood structure of a point and the convergence of nets and filters in this new framework

characterizations of fuzzifying topologies by some limit structures in this paper we introduce the concepts of fuzzifying topological interiors fuzzifying topological closures and fuzzifying topological limit structures a comprehensive characterization theorem about fuzzifying topological interiors are proved characterizations of fuzzifying topologies by filter convergence structures are given

network synchronization spectral versus statistical properties we consider synchronization of weighted networks possibly with asymmetrical connections focusing on causal relations rather than the observed correlations we show that the synchronizability of networks cannot be directly inferred from their statistical properties small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization while the gross statistical network properties remain essentially unchanged consequently commonly used statistical properties including the degree distribution degree homogeneity average degree average distance degree correlation and clustering coefficient can fail to characterize the synchronizability of networks in terms of causal relations despite the observed correlations c 2006 elsevier b v all rights reserved

new support vector algorithms we propose a new class of support vector algorithms for regression and classification in these algorithms a parameter lets one effectively control the number of support vectors while this can be useful in its own right the parameterization has the additional benefit of enabling us to eliminate one of the other free parameters of the algorithm the accuracy parameter epsilon in the regression case and the regularization constant c in the classification case we describe the algorithms give some theoretical results concerning the meaning and the choice of and report experimental results

fuzzifying topology induced by a strong fuzzy metric a construction of a fuzzifying topology induced by a strong fuzzy metric is presented properties of this fuzzifying topology in particular its convergence structure are studied our special interest is in the study of the relations between products of fuzzy metrics and the products of the induced fuzzifying topologies

l metric on the space of l fuzzy numbers abstract in this article a natural l topology is constructed on the set of l fuzzy numbers which is called the standard l topology it is proved that the standard l topology can be induced by an l metric the space of trapezoidal fuzzy numbers can be regarded as a subspace of the space of l fuzzy numbers an l metric on the set of trapezoidal fuzzy numbers is presented

synchronization of chaotic networks with time delayed couplings an analytic study networks of nonlinear units with time delayed couplings can synchronize to a common chaotic trajectory although the delay time may be very large the units can synchronize completely without time shift for networks of coupled bernoulli maps analytic results are derived for the stability of the chaotic synchronization manifold for a single delay time chaos synchronization is related to the spectral gap of the coupling matrix for networks with multiple delay times analytic results are obtained from the theory of polynomials finally the analytic results are compared with networks of iterated tent maps and lang kobayashi equations which imitate the behavior of networks of semiconductor lasers

advanced sampled data synchronization control for complex dynamical networks with coupling time varying delays abstract this study addresses sampled data control in the synchronization problem of complex dynamical networks systems with a coupling time varying delay by constructing the augmented lyapunov krasovskii functionals and using some mathematical techniques a sampled data synchronization criterion is derived as the first result with the framework of linear matrix inequalities based on the first result a sampled data synchronization control method is proposed as the second result finally the effectiveness and superiority of the proposed results will be shown by comparing the maximum sampling period with the established results in two numerical examples

regularity and normality of l m fuzzy topological spaces abstract in this paper regularity and normality axioms are introduced in l m fuzzy topological spaces where l and m are completely distributive lattices with order reversing involution each l m fuzzy topological space can be regarded to be regular and normal to some degree the relations among the four separation axioms are investigated they are equivalent to each other in an l m fuzzy metric space

effects of kernel function on nu support vector machines in extreme cases how we should choose a kernel function in support vector machines svms is an important but difficult problem in this paper we discuss the properties of the solution of the spl nu svm s a variation of svm s for normalized feature vectors in two extreme cases all feature vectors are almost orthogonal and all feature vectors are almost the same in the former case the solution of the spl nu svm is nearly the center of gravity of the examples given while the solution is approximated to that of the spl nu svm with the linear kernel in the latter case although extreme kernels are not employed in practice analyzes are helpful to understand the effects of a kernel function on the generalization performance

separation axioms and the fuzzy real lines we show that for each lattice l which is completely distributive and equipped with an order reversing involution the canonical topology of the fuzzy real line l is uniformizable in the sense of hutton furthermore this uniformity induces a pseudometric which induces the canonical topology consequently l satisfies the higher order separation axioms of hutton and sarkar more generally we answer most of the open questions concerning l and its subspaces with respect to each of the separation axiom schemes of hutton and sarkar answer some of these same questions for the stratified fuzzy real line c l and its subspaces and analyze c l and its subspaces with respect to the separation axioms of rodabaugh

l fuzzy normal spaces and tietze extension theorem abstract in this paper we prove a characterization theorem for normal l fuzzy topological spaces l is an infinitely distributive lattice with an order reversing involution in the particular case l 0 1 this theorem reduces to a known result of kat tov fund math 38 1951 85 91 and tong duke math j 19 1952 289 292 as an important application we obtain a fuzzy version of tietze extension theorem this yields an affirmative answer to a question raised in a series of papers by rodabaugh fuzzy sets and systems 11 1983 163 183

pointwise pseudo metrics in l fuzzy set theory a theory of pointwise pseudo metrics is based on l fuzzy sets a pointwise pseudo metric is characterized by its remote neighborhood maps many theorems in general topology are generalized on l fuzzy sets the l fuzzy real line and the l fuzzy unit interval are pointwise pseudo metrizable a pseudo metrization theorem of pointwise uniformity is obtained relations between erceg s p metric pseudometric and pointwise p metric are discussed

on fuzzy uniformities induced by a fuzzy metric space abstract different types of fuzzy uniformities have been introduced in the literature standing out the notions due to hutton hohle and lowen the main purpose of this paper is to study several methods to endow a fuzzy metric space x m in the sense of george and veeramani with a probabilistic uniformity and with a hutton 0 1 quasi uniformity we will show the functorial behavior of these constructions as well as its relation with respect to lowen s functors and katsaras s functors which establish a relationship between the categories of probabilistic uniformities and hutton 0 1 quasi uniformities with the category of classical uniformities respectively furthermore we also study the fuzzy topologies associated with these fuzzy uniformities

extension of shi s quasi uniformities in a kubiak sostak sense abstract the aim of this paper is to study the extension of shi fu gui s quasi uniformities in a kubiak sostak sense the relationship between this extension of shi fu gui s quasi uniform molecular lattices shi s quml and the corresponding kubiak sostak extension of wang guo jun s topological molecular lattices wang s tml is discussed quml and tml denote the categories of shi s quml and wang s tml respectively cd is the category of completely distributive lattices with complete lattice morphisms as morphisms we prove that fquml the category of fuzzy quasi uniform molecular lattices the extension of shi s quml is a topological category over cd op and ftml the category of fuzzy topological molecular lattices the extension of wang s tml can be embedded in fquml we also prove that fquml is isomorphic to quml c m when m is a completely distributive lattice with multiplicative property where quml c m is the co tower extension of quml finally we study the kubiak sostak extension of hutton s quasi uniformities

an asymptotic statistical theory of polynomial kernel methods the generalization properties of learning classifiers with a polynomial kernel function are examined in kernel methods input vectors are mapped into a high dimensional feature space where the mapped vectors are linearly separated it is well known that a linear dichotomy has an average generalization error or a learning curve proportional to the dimension of the input space and inversely proportional to the number of given examples in the asymptotic limit however it does not hold in the case of kernel methods since the feature vectors lie on a submanifold in the feature space called the input surface in this letter we discuss how the asymptotic average generalization error depends on the relationship between the input surface and the true separating hyperplane in the feature space where the essential dimension of the true separating polynomial named the class is important we show its upper bounds in several cases and confirm these using computer simulations

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

synchronization of rossler oscillators on scale free topologies abstract we study the synchronization of rossler oscillators as prototypes of chaotic systems on scale free complex networks as it turns out the underlying topology crucially affects the global synchronization properties in particular we show that the existence of loops facilitates the synchronizability of the system whereas rossler oscillators do not synchronize on tree like topologies beyond a certain size moreover it is not the mere number of loops that counts for synchronization but also the type of loops by considering cayley trees modified by additional loops in different ways we find out that also the distribution of shortest path lengths between two oscillators plays an important role for the global synchronization

geometrical properties of nu support vector machines with different norms by employing the l1 or l norms in maximizing margins support vector machines svms result in a linear programming problem that requires a lower computational load compared to svms with the l2 norm however how the change of norm affects the generalization ability of svms has not been clarified so far except for numerical experiments in this letter the geometrical meaning of svms with the lp norm is investigated and the svm solutions are shown to have rather little dependency on p