on relational possibilistic clustering this paper initially describes the relational counterpart of possibilistic c means pcm algorithm called relational pcm or rpcm rpcm is then improved to better handle arbitrary dissimilarity data first a re scaling of the pcm membership function is proposed in order to obtain zero membership values when the distance to prototype equals the maximum value allowed in bounded dissimilarity measures second a heuristic method of reference distance initialisation is provided which diminishes the known pcm tendency of producing coincident clusters finally rpcm improved with our initialisation strategy is tested on both synthetic and real data sets with satisfactory results

robust fuzzy clustering of relational data popular relational data clustering algorithms relational dual of fuzzy c means rfcm non euclidean rfcm nerfcm both by hathaway et al and fanny by kaufman and rousseeuw are examined a new algorithm which is a generalization of fanny called the fuzzy relational data clustering frc algorithm is introduced having an identical objective functional as rfcm however the frc does not have the restriction of rfcm which is that the relational data is derived from euclidean distance as the measure of dissimilarity between the objects and it also does not have limitations of fanny including the use of a fixed membership exponent or a fuzzifier exponent m the frc algorithm is further improved by incorporating the concept of dave s object data noise clustering nc algorithm done by proposing a concept of noise dissimilarity next based on the constrained minimization which includes an inequality constraint for the memberships and corresponding kuhn tucker conditions a noise resistant frc algorithm is derived which works well for all types of non euclidean dissimilarity data thus it is shown that the extra computations for data expansion spl beta spread transformation required by the nerfcm algorithm are not necessary this new algorithm is called robust non euclidean fuzzy relational data clustering robust ne frc and its robustness is demonstrated through several numerical examples advantages of this new algorithm are faster convergence robustness against outliers and ability to handle all kinds of relational data including non euclidean the paper also presents a new and better interpretation of the noise class

anti modularity and anti community detecting in complex networks abstract many networks of interest in sciences and social research can be divided naturally into anti communities the problem of detecting and characterizing such anti community structure has attracted recent attention in this paper we first define the anti modularity as a quantitative measure of anti community partitioning on a network we also theoretically and empirically show the reliability of anti modularity as a measurement of the quality of an anti community partitioning a label propagation algorithm lpad for anti community detection is proposed experimental results on synthetic and real world networks show that our algorithm lpad can obtain higher quality anti community partitioning than other methods

additive spectral method for fuzzy cluster analysis of similarity data including community structure and affinity matrices an additive spectral method for fuzzy clustering is proposed the method operates on a clustering model which is an extension of the spectral decomposition of a square matrix the computation proceeds by extracting clusters one by one which makes the spectral approach quite natural the iterative extraction of clusters also allows us to draw several stopping rules to the procedure this applies to several relational data types differently normalized network structure data the first eigenvector subtracted affinity between multidimensional vectors the pseudo inverse laplacian transformation and conventional relational data including in house data of similarity between research topics according to working of a research center the method is experimentally compared with several classic and recent techniques and shown to be competitive

a method of relational fuzzy clustering based on producing feature vectors using fastmap the first stage of organizing objects is to partition them into groups or clusters the clustering is generally done on individual object data representing the entities such as feature vectors or on object relational data incorporated in a proximity matrix this paper describes another method for finding a fuzzy membership matrix that provides cluster membership values for all the objects based strictly on the proximity matrix this is generally referred to as relational data clustering the fuzzy membership matrix is found by first finding a set of vectors that approximately have the same inter vector euclidian distances as the proximities that are provided these vectors can be of very low dimension such as 5 or less fuzzy c means fcm is then applied to these vectors to obtain a fuzzy membership matrix in addition two dimensional vectors are also created to provide a visual representation of the proximity matrix this allows comparison of the result of automatic clustering to visual clustering the method proposed here is compared to other relational clustering methods including nerfcm rouben s method and windhams a p method various clustering quality indices are also calculated for doing the comparison using various proximity matrices as input simulations show the method to be very effective and no more computationally expensive than other relational data clustering methods the membership matrices that are produced by the proposed method are less crisp than those produced by nerfcm and more representative of the proximity matrix that is used as input to the clustering process

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

proper fredholm submanifolds of hilbert space on considere des varietes et des sous varietes de hilbert riemanniennes on demontre qu une sous variete de fredholm propre a une structure de fredholm naturelle et on montre que les operateurs de forme sont compacts et que chaque fonction distance euclidienne fa satisfait la condition c de palais et smale on etudie la geometrie des sous varietes de fredholm propre a fibre normal plat

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

community detection by fuzzy clustering abstract how to measure the similarity between nodes is of great importance for fuzzy clustering when we use the approach to uncover communities in complex networks in this paper we first measure the similarity between nodes in a network based on edge centralities and model the network as a fuzzy relation then two fuzzy transitive rules rule i and rule ii are applied on the relation respectively by which the similarity information can be transferred from one node to another in the network until the relation reaches a stable state by choosing different thresholds our method finally can partition the network into several non overlapping subgroups we compare our method with some state of the art methods on the lfr benchmark and real world networks we find that our method based on rule i can correctly identify communities when the similarity between nodes of same groups is greater than that of different groups while it is just opposite to rule ii our method achieves better results than the state of the art methods when the pre planted communities of the random networks are vaguer

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

numerical comparison of the rfcm and ap algorithms for clustering relational data we describe and compare two fuzzy clustering algorithms based on optimizing penalty functions of relational data more specifically the ap algorithm of windham is contrasted with the relational fuzzy c means approach through a numerical example on a small artificial data set defined by windham our results indicate that while the two algorithms seem to cluster objects using very different mathematical criteria the numerical results can be quite similar however a second numerical example using a slightly distorted version of windham s data seems to exhibit real differences between clusterings obtained by the two approaches

establishing performance evaluation structures by fuzzy relation based cluster analysis the purpose of this study is to establish hierarchical structures for the performance evaluation of vague humanistic complicated systems to overcome the difficulties due to partial information and the vagueness of human knowledge and recognition a fuzzy relation based clustering method is proposed to model this evaluation first the effects of different max t i compositions on the formation of clusters are discussed then an improved clustering algorithm is developed to produce several partition trees with different levels and clusters according to different t i norm compositions to demonstrate the usefulness of the proposed algorithm the academic departments of higher education were considered using actual engineering school data in taiwan three performance evaluation structures are established by using max t 1 max t 2 and max t 3 compositions the results show that the proposed fuzzy hierarchical approach is useful and practical for performance evaluations of complicated humanistic systems

topological nonrigidity of nonuniform lattices we say that an arbitrary manifold m m is topologically rigid relative to its ends if it satisfies the following condition if n n is any other manifold with a compact subset c n for which a proper homotopy equivalence h n n m m is a homeomorphism on n n c then there is a compact subset k n and a proper homotopy ht n n m m from h to a homeomorphism such that ht and h agree on n n k for all t 0 1 we say that a manifold m without boundary is properly rigid or absolutely topologically rigid if we eliminate the requirements that h be a homeomorphism on n n c and agree with ht on n n k for all t 0 1 along the lines of the classical borel conjecture that all closed aspherical manifolds are topologically rigid farrell and jones 8 provide the following important theorem

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

a relative higher index theorem diffeomorphisms and positive scalar curvature abstract we prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity we apply this theorem to study riemannian metrics of positive scalar curvature on manifolds for every two metrics of positive scalar curvature on a closed manifold and a galois cover of the manifold we define a secondary higher index class non vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature in the special case where one metric is induced from the other by a diffeomorphism of the manifold we obtain a formula for computing this higher index class in particular it follows that the higher index class lies in the image of the baum connes assembly map

unitary structure in representations of infinite dimensional groups and a convexity theorem in this paper we show that a kac moody algebra fl a associated to a symmetrizable generalized cartan matrix a carries a contravariant hermitian form which is positive definite on all root spaces we deduce that every integrable highest weight g a module l a carries a contravariant positivedefinite hermitian form this allows us to define the moment map and prove a generalization of the schur horn kosta nt heckman atiya h pressley convexity theorem the proofs are based on an identity which also gives estimates for the action of 9 a on fl a and l a we hope that the main idea behind the paper is apparent it is to use the interplay between the coadjoint and the highest weight representations we are grateful to v guillemin for an introduction to the moment map

the energy function and homogeneous harmonic maps for equivariant maps from a compact homogeneous space into an adjoint orbit of a compact lie group it is shown that the energy function is the restriction of a quadratic function on the lie algebra providing the orbit has the metric induced from the lie algebra one obtains a simple version of the harmonic map equation and an identity relating the energy and the square of the norm of the moment map several applications are given including an example which illustrates how a change of metric in a flag manifold affects the harmonicity of equivariant maps from the two sphere to the flag manifold

weighting links based on edge centrality for community detection abstract link weights have the equally important position as links in complex networks and they are closely associated with each other for the emergence of communities how to assign link weights to make a clear distinction between internal links of communities and external links connecting communities is of vital importance for community detection edge centralities provide a powerful approach for distinguishing internal links from external ones here we first use edge centralities such as betweenness information centrality and edge clustering coefficient to weight links of networks respectively to transform unweighted networks into weighted ones and then a weighted function that both considers links and link weights is adopted on the weighted networks for community detection we evaluate the performance of our approach on random networks as well as real world networks better results are achieved on weighted networks with stronger weights of internal links of communities and the results on unweighted networks outperform that of weighted networks with weaker weights of internal links of communities the availability of our findings is also well supported by the study of granovetter that the weak links maintain the global integrity of the network while the strong links maintain the communities especially in the karate club network all the nodes are correctly classified when we weight links by edge betweenness the results also give us a more comprehensive understanding on the correlation between links and link weights for community detection

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

topology of homology manifolds anr homology n manifolds are nite dimensional absolute neighborhood retracts x with the property that for every x 2 x hi x x fxg is 0 for i 6 n and z for i n topological manifolds are natural examples of such spaces to obtain nonmanifold examples we can take a manifold whose boundary consists of a union of integral homology spheres and glue on the cone on each one of the boundary components the resulting space is not a manifold if the fundamental group of any boundary component is a nontrivial perfect group it is a consequence of the double suspension theorem of cannon and edwards that as in the examples above the singularities of polyhedral anr homology manifolds are isolated there are however many examples of anr homology manifolds which have no manifold points whatever see 12 for a good exposition of the relevant theory the purpose of this paper is to begin a surgical classi cation of anr homology manifolds sometimes referred to in the sequel simply as homology manifolds one way to approach this circle of ideas is via the problem of characterizing topological manifolds among anr homology manifolds in cannon s work on the double suspension problem 6 it became clear that in dimensions greater than 4 the right transversality hypothesis is the following weak form of general position

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

topological dynamics of the weil petersson geodesic flow we prove topological transitivity for the weil petersson geodesic flow for real two dimensional moduli spaces of hyperbolic structures our proof follows a new approach that combines the density of singular unit tangent vectors the geometry of cusps and convexity properties of negative curvature we also show that the weil petersson geodesic flow has horseshoes invariant sets with positive topological entropy and that there are infinitely many hyperbolic closed geodesics whose number grows exponentially in length furthermore we note that the volume entropy is infinite

identification of overlapping community structure in complex networks using fuzzy c means clustering abstract identification of overlapping communities clusters in a complex network is a general problem in data mining of network data sets in this paper we devise a novel algorithm to identify overlapping communities in complex networks by the combination of a new modularity function based on generalizing ng s q function an approximation mapping of network nodes into euclidean space and fuzzy c means clustering experimental results indicate that the new algorithm is efficient at detecting both good clusterings and the appropriate number of clusters

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

an exhaustion of locally symmetric spaces by compact submanifolds with corners letx be a riemannian symmetric space of noncompact type and rank 2 and let be a non uniform irreducible lattice on the locally symmetric quotientv x we construct an exhaustion functionh v 0 whose sublevel sets h s are compact submanifolds ofv with corners the top dimensional boundary faces of h s are parts of certain horospheres that join together at the corners it can be shown that actually h s is a submanifold with corners isomorphic to the borel serre compactification ofv

modular symbols and the topological nonrigidity of arithmetic manifolds in this paper we address the existence of smooth manifolds proper homotopy equivalent to nonuniform arithmetic manifolds m g k that are not homeomorphic to it while the manifolds m are properly rigid if rank 2 we show that the so called virtual structure group has infinite rank as a vector space if rank 4 2015 wiley periodicals inc

a possibilistic fuzzy c means clustering algorithm in 1997 we proposed the fuzzy possibilistic c means fpcm model and algorithm that generated both membership and typicality values when clustering unlabeled data fpcm constrains the typicality values so that the sum over all data points of typicalities to a cluster is one the row sum constraint produces unrealistic typicality values for large data sets in this paper we propose a new model called possibilistic fuzzy c means pfcm model pfcm produces memberships and possibilities simultaneously along with the usual point prototypes or cluster centers for each cluster pfcm is a hybridization of possibilistic c means pcm and fuzzy c means fcm that often avoids various problems of pcm fcm and fpcm pfcm solves the noise sensitivity defect of fcm overcomes the coincident clusters problem of pcm and eliminates the row sum constraints of fpcm we derive the first order necessary conditions for extrema of the pfcm objective function and use them as the basis for a standard alternating optimization approach to finding local minima of the pfcm objective functional several numerical examples are given that compare fcm and pcm to pfcm our examples show that pfcm compares favorably to both of the previous models since pfcm prototypes are less sensitive to outliers and can avoid coincident clusters pfcm is a strong candidate for fuzzy rule based system identification

maximizing modularity intensity for community partition and evolution most previous studies of community partition have often focused on the network topology the topology and link weights are in fact closely associated with each other for community formation in complex networks this paper proposes a function modularity intensity a variation of modularity density d value for evaluating the cohesiveness of a community which considers links between vertices as well as link weights the results showed that maximizing the modularity intensity not only can resolve the resolution limits problem but also achieve better performance for community partition to further evaluate the function and clarify the weight topology correlation with communities we give a simple model to simulate the topology and link weights development for community evolution and use the modularity intensity to capture communities of networks in each step of this process in this model a network is treated as a fuzzy relation and two operations of the fuzzy relation are used to make the link weights stronger and weaker respectively with the growth of the network in each evolutionary step by simulation experiments we found that in the model the modularity intensity catches communities of networks that undergo a gradual transition from faintness to clearness our model also reproduces the finding of granovetter that strong links are confined mainly in tight communities and the links between communities are predominantly weak from the results above we believe that the modularity intensity gives a more comprehensive evaluation for communities and by studying the weight topology correlation this model provides a new view for community evolution

cluster analysis based in fuzzy relations in this paper cluster analysis based on fuzzy relations is investigated tamura s max min n step procedure is extended to all types of max t compositions a max t similarity relation matrix is obtained by beginning with a proximity relation matrix based on the proposed max tn step procedure then a clustering algorithm is created for the max t similarityrelation matrix three critical max t compositions of max min max prod and max are compared the max composition is recommended as the 2rst choice among them several examples give more perspectives for di3erent choices of max t compositions finally the topic of incomplete data via max t compositions is discussed max t compositions can be e3ectively used to treat the t connected incomplete data c 2001 elsevier science b v all rights reserved

a general theory of canonical forms si g est un groupe de lie compact et m une g variete riemannienne d orbites principales de codimension k alors une section ou une forme canonique pour m est une sous variete fermee lisse a k dimensions de m qui rencontre toutes les orbites de m orthogonalement