community detection algorithms a comparative analysis uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents many algorithms have been proposed so far but none of them has been subjected to strict tests to evaluate their performance most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure which is very uncommon in real systems here we test several methods against a recently introduced class of benchmark graphs with heterogeneous distributions of degree and community size the methods are also tested against the benchmark by girvan and newman proc natl acad sci u s a 99 7821 2002 and on random graphs as a result of our analysis three recent algorithms introduced by rosvall and bergstrom proc natl acad sci u s a 104 7327 2007 proc natl acad sci u s a 105 1118 2008 blondel et al j stat mech theory exp 2008 p10008 and ronhovde and nussinov phys rev e 80 016109 2009 have an excellent performance with the additional advantage of low computational complexity which enables one to analyze large systems

distance dissimilarity index and network community structure we address the question of finding the community structure of a complex network in an earlier effort h zhou phys rev e 67 041908 2003 the concept of network random walking is introduced and a distance measure defined here we calculate based on this distance measure the dissimilarity index between nearest neighboring vertices of a network and design an algorithm to partition these vertices into communities that are hierarchically organized each community is characterized by an upper and a lower dissimilarity threshold the algorithm is applied to several artificial and real world networks and excellent results are obtained in the case of artificially generated random modular networks this method outperforms the algorithm based on the concept of edge betweenness centrality for yeast s protein protein interaction network we are able to identify many clusters that have well defined biological functions

from geometric quantization to moyal quantization it is shown how the moyal product of phase space functions and the weyl correspondence between symbols and operator kernels may be obtained directly using the procedures of geometric quantization applied to the symplectic groupoid constructed by doubling the phase space

moyal quantization of 2 1 dimensional galilean systems stratonovich weyl kernels are constructed for some of the coadjoint orbits of the two dimensional extended galilean group g 2 1 as an intermediate step the unitary irreducible representations associated with a given group orbit are obtained by using the kirillov mackey theory star products are defined in the sense of moyal for functions on each of these orbits the central extension of g 2 1 with parameter k is also analyzed which results from the commutator between the generators of boosts to conclude that it originates a sort of nonrelativistic remainder of the thomas precession

network landscape from a brownian particle s perspective given a complex biological or social network how many clusters should it be decomposed into we define the distance d i j from node i to node j as the average number of steps a brownian particle takes to reach j from i node j is a global attractor of i if d i j l d i k for any k of the graph it is a local attractor of i if j ensuremath in e i the set of nearest neighbors of i and d i j l d i l for any l ensuremath in e i based on the intuition that each node should have a high probability to be in the same community as its global local attractor on the global local scale we present a simple method to uncover a network s community structure this method is applied to several real networks and some discussion on its possible extensions is made

primal dual weak galerkin finite element methods for elliptic cauchy problems abstract the authors propose and analyze a well posed numerical scheme for a type of ill posed elliptic cauchy problem by using a constrained minimization approach combined with the weak galerkin finite element method the resulting euler lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable and is thus an example of the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is consistent in the sense that the system is symmetric well posed and is satisfied by the exact solution a certain stability and error estimates were derived in discrete sobolev norms including one in a weak l 2 topology some numerical results are reported to illustrate and validate the theory developed in the paper

linear canonical transformations and quantum phase a unified canonical and algebraic approach the algebra of generalized linear quantum canonical transformations is examined in the perspective of schwinger s unitary canonical operator basis formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space the representations of the wigner function in the generalized action angle unitary operator pair for certain hamiltonian systems with dynamical symmetry is examined this generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action angle wigner function

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

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

fcm based model selection algorithms for determining the number of clusters clustering is an important research topic that has practical applications in many fields it has been demonstrated that fuzzy clustering using algorithms such as the fuzzy c means fcm has clear advantages over crisp and probabilistic clustering methods like most clustering algorithms however fcm and its derivatives need the number of clusters in the given data set as one of their initializing parameters the main goal of this paper is to develop an effective fuzzy algorithm for automatically determining the number of clusters after a brief review of the relevant literature we present a new algorithm for determining the number of clusters in a given data set and a new validity index for measuring the goodness of clustering experimental results and comparisons are given to illustrate the performance of the new algorithm

recovering the initial wave amplitude for nonlinear elliptic equation with locally lipschitz source in multiple dimensional domain abstract the backward problems for the elliptic equation bpee are widely used for modelling problems in many fields such as physics geometry or engineering this paper is the first investigation of bpee in unbounded multiple dimensional domain associated with locally lipschitz source as follows 2 u x 1 2 2 u x 2 2 2 u x n 2 2 u y 2 s x 1 x 2 x n y u x 1 x 2 x n y in which x 1 x 2 x n y r n 0 l where l 0 based on the idea of the truncation approach we construct the regularized solution and obtain the h older type of its convergence rate under some assumptions on regularity of the exact solution eventually a numerical experiment for the elliptic allen cahn equation is proposed to illustrate the effectiveness and applicability of our method

quantum symmetries and the weyl wigner product of group representations in the usual formulation of quantum mechanics groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex hilbert space in accordance with wigner s theorem in the phase space formulation they have real true unitary representations in the space of square integrable functions on phase space each such phase space representation is a weyl wigner product of the corresponding hilbert space representation with its contragredient and these can be recovered by factorizing the weyl wigner product however not every real unitary representation on phase space corresponds to a group of automorphisms so not every such representation is in the form of a weyl wigner product and can be factorized the conditions under which this is possible are examined examples are presented

determining surface heat flux in the steady state for the cauchy problem for the laplace equation in this paper we consider the cauchy problem for the laplace equation in a strip where the cauchy data is given at x 0 and the flux is sought in the interval 0

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

conditional stability in an inverse problem of determining a non smooth boundary an inverse problem for determining an unknown boundary is discussed we prove conditional stability of logarithmic type for unknown lipschitz boundaries under a priori assumptions the proof is based on estimation of harmonic measures

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

the moyal lie theory of phase space quantum mechanics a lie algebraic approach to the unitary transformations in weyl quantization is discussed this approach being formally equivalent to the quantization is an extension of the classical poisson lie theory and can be used as an efficient tool in the quantum phase space transformation theory

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

optimal stability for the inverse problemof multiple cavities abstract we deal with the determination of finitely many cavities in a planar inhomogeneous conductor from one current and voltage measurement collected on the exterior boundary we prove stability estimates under essentially minimal a priori regularity assumptions we construct an explicit example showing the optimality of such stability estimates

a regularization method for a cauchy problem of the helmholtz equation we investigate a cauchy problem for the helmholtz equation a modified boundary method is used for solving this ill posed problem some holder type error estimates are obtained the numerical experiment shows that the modified boundary method works well

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

stable determination of a crack from boundary measurements we treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given we prove a theorem of continuous dependence from the data

the construction of a quantum markov partition we present a method for constructing a quantum markov partition its elements are obtained by quantizing the characteristic function of the classical rectangles the result is a set of quantum operators which behave asymptotically as projectors over the classical rectangles apart from edge and corner effects we investigate their spectral properties and different methods of construction the quantum partition is shown to induce a symbolic decomposition of the quantum evolution operator in particular an exact expression for the traces of the propagator is obtained having the same structure as the gutzwiller periodic orbit sum

bounds on integrals of the wigner function the integral of the wigner function over a subregion of the phase space of a quantum system may be less than zero or greater than one it is shown that for systems with 1 degree of freedom the problem of determining the best possible upper and lower bounds on such an integral over an possible states reduces to the problem of finding the greatest and least eigenvalues of a hermitian operator corresponding to the subregion the problem is solved exactly in the case of an arbitrary elliptical region these bounds provide checks on experimentally measured quasiprobability distributions

probabilistic framework for network partition given a large and complex network we would like to find the partition of this network into a small number of clusters this question has been addressed in many different ways in a previous paper we proposed a deterministic framework for an optimal partition of a network as well as the associated algorithms in this paper we extend this framework to a probabilistic setting in which each node has a certain probability of belonging to a certain cluster two classes of numerical algorithms for such a probabilistic network partition are presented and tested application to three representative examples is discussed

the stratonovich weyl correspondence for one dimensional kinematical groups the stratonovich weyl correspondence is a restatement of the moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively the first and most important step is to define a mapping from the manifold into the set of self adjoint operators on a hilbert space under suitable conditions this mapping is called a stratonovich weyl kernel the construction of this mapping is discussed on coadjoint orbits of the one dimensional galilei poincare and newton hooke groups as well as the two dimensional euclidean group

central difference regularization method for the cauchy problem of the laplace s equation abstract in this paper we consider the cauchy problem for the laplace s equation where the cauchy data is given at x 0 and the solution is sought in the interval 0

unresolved computation and optimal predictions we present methods for predicting the solution of time dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically but when prior statistical information can be found the sparse numerical data are viewed as constraints on the solution and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future for linear equations we offer general recipes for advancing the constraints the methods are generalized to certain classes of nonlinear problems and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed our methods are related to certain data acquisition schemes in oceanography and meteorology c 1999 john wiley sons inc

weyl wigner formulation of noncommutative quantum mechanics we address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension displaying both spatial and momentum noncommutativities by resorting to a covariant generalization of the weyl wigner transform and to the darboux map we construct an isomorphism between the operator and the phase space representations of the extended heisenberg algebra this map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space we construct the extended star product and moyal bracket and propose a general definition of noncommutative states we study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of the darboux map our approach unifies and generalizes all the previous proposals for the phase space formulation of noncommutative quantum mechanics for concreteness we rederive these proposals by restricting our formalism to some two dimensional spaces

a validity index approach for network partitions abstract the validity index has been used to evaluate the fitness of partitions produced by clustering algorithms for points in euclidean space in this paper we propose a new validity index for network partitions which can provide a measure of goodness for the community structure of networks it is defined as a product of two factors and involves the compactness and separation for each partition the simulated annealing strategy is used to minimize such a validity index function in coordination with our previous k means algorithm based on the optimal reduction of a random walker markovian dynamics on the network it is demonstrated that the algorithm can efficiently find the community structure during the cooling process the number of communities can be automatically determined without any prior knowledge of the community structure moreover the algorithm is successfully applied to three real world networks

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

wigner functions on non standard symplectic vector spaces we consider the weyl quantization on a flat non standard symplectic vector space we focus mainly on the properties of the wigner functions defined therein in particular we show that the sets of wigner functions on distinct symplectic spaces are different but have non empty intersections this extends previous results to arbitrary dimension and arbitrary constant symplectic structure as a by product we introduce and prove several concepts and results on non standard symplectic spaces which generalize those on the standard symplectic space namely the symplectic spectrum williamson s theorem and narcowich wigner spectra we also show how wigner functions on non standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation

the quantum state vector in phase space and gabor s windowed fourier transform representations of quantum state vectors by complex phase space amplitudes complementing the description of the density operator by the wigner function have been defined by applying the weyl wigner transform to dyadic operators linear in the state vector and anti linear in a fixed window state vector here aspects of this construction are explored and a connection is established with gabor s windowed fourier transform the amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations generalized bargmann representations of the state vector appear as special cases associated with gaussian windows for every choice of window amplitudes lie in a corresponding linear subspace of square integrable functions on phase space a generalized born interpretation of amplitudes is described with both the wigner function and a generalized husimi function appearing as quantities linear in an amplitude and anti linear in its complex conjugate schrodinger s time dependent and time independent equations are represented on phase space amplitudes and their solutions described in simple cases

symbolic calculus on the time frequency half plane this study concerns a special symbolic calculus of interest for signal analysis this calculus associates functions on the time frequency half plane f 0 with linear operators defined on the positive frequency signals full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way the correspondence rule is detailed and the associated wigner function is given formulas expressing the basic operation star bracket of the lie algebra of symbols which is isomorphic to that of the operators are obtained in addition it is shown that the resulting calculus is covariant under a three parameter group which contains the affine group as subgroup this observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one

validity index for crisp and fuzzy clusters abstract in this article a cluster validity index and its fuzzification is described which can provide a measure of goodness of clustering on different partitions of a data set the maximum value of this index called the pbm index across the hierarchy provides the best partitioning the index is defined as a product of three factors maximization of which ensures the formation of a small number of compact clusters with large separation between at least two clusters we have used both the k means and the expectation maximization algorithms as underlying crisp clustering techniques for fuzzy clustering we have utilized the well known fuzzy c means algorithm results demonstrating the superiority of the pbm index in appropriately determining the number of clusters as compared to three other well known measures the davies bouldin index dunn s index and the xie beni index are provided for several artificial and real life data sets

moyal quantization with compact symmetry groups and noncommutative harmonic analysis the phase space approach to quantization of systems whose symmetry group is compact and semisimple is developed from two basic principles covariance and traciality this generalizes results and methods already implemented for spin systems j c varilly and j m gracia bondia ann phys 190 107 1989 the twisted product of phase space functions is shown to be the image of group convolution in the context of a novel fourier theory on the coadjoint orbits

bounds on integrals of the wigner function the hyperbolic case wigner functions play a central role in the phase space formulation of quantum mechanics although closely related to classical liouville densities wigner functions are not positive definite and may take negative values on subregions of phase space we investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary wigner function over noncompact subregions of the phase plane with hyperbolic boundaries we show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly solvable eigenvalue problem and that the bounds differ from those on classical liouville distributions in particular we show that the total quasiprobability on such a region can be greater than 1 or less than zero c 2005 american institute of physics

a potential function method for the cauchy problem of elliptic operators abstract in this paper we deal with the cauchy problem of elliptic operators through the use of a single layer potential function we devise a numerical method for approximating the solution of the cauchy problem of elliptic operators which are well known to be highly ill posed in nature the method is based on the denseness of single layer potential functions convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method it has been observed that the use of more cauchy data will greatly improve the accuracy of the approximate solutions

least square based radial basis collocation method for solving inverse problems of laplace equation from noisy data the inverse problem of 2d laplace equation involves an estimation of unknown boundary values or the locations of boundary shape from noisy observations on over specified boundary or internal data points the application of radial basis collocation method rbcm one of meshless and non iterative numerical schemes directly induces this inverse boundary value problem ibvp to a single step solution of a system of linear algebraic equations in which the coefficients matrix is inherently ill conditioned in order to solve the unstable problem observed in the conventional rbcm an effective procedure that builds an over determined linear system and combines with least square technique is proposed to restore the stability of the solution in this paper the present work investigates three examples of ibvps using over specified boundary conditions or internal data with simulated noise and obtains stable and accurate results it underlies that least square based radial basis collocation method ls rbcm poses a significant advantage of good stability against large noise levels compared with the conventional rbcm

diffusion maps and coarse graining a unified framework for dimensionality reduction graph partitioning and data set parameterization we provide evidence that nonlinear dimensionality reduction clustering and data set parameterization can be solved within one and the same framework the main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise our construction which is based on a markov random walk on the data offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry we show that clustering in embedding spaces is equivalent to compressing operators the objective of data partitioning and clustering is to coarse grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy we show that the quantization distortion in diffusion space bounds the error of compression of the operator thus giving a rigorous justification for k means clustering in diffusion space and a precise measure of the performance of general clustering algorithms

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

conditional stability estimation for an inverse boundary problem with non smooth boundary in ℛ in this paper we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in r3 by a solution to a cauchy problem of the laplace equation assuming that the unknown part is a lipschitz continuous surface we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part the keys are the complex extension and estimates for a harmonic measure

phase space observables and isotypic spaces we give necessary and sufficient conditions for the set of neumark projections of a countable set of phase space observables to constitute a resolution of the identity and we give a criteria for a phase space observable to be informationally complete the results will be applied to the phase space observables arising from an irreducible representation of the heisenberg group

wigner measures in noncommutative quantum mechanics we study the properties of quasi distributions or wigner measures in the context of noncommutative quantum mechanics in particular we obtain necessary and sufficient conditions for a phase space function to be a noncommutative wigner measure for a gaussian to be a noncommutative wigner measure and derive certain properties of the marginal distributions which are not shared by ordinary wigner measures moreover we derive the robertson schrodinger uncertainty principle finally we show explicitly how the set of noncommutative wigner measures relates to the sets of liouville and commutative wigner measures

a mollification method for a cauchy problem for the laplace equation in this paper we consider a cauchy problem for the laplace equation in the strip region 0 x 1 y is an element of r where the cauchy data is given at x 0 and the solution is sought in the interval 0 x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data to obtain stable numerical solution we propose a mollification method using the gaussian kernel error estimate between the exact solution and its approximation is given the choices of mollification parameter for a priori and a posteriori are discussed numerical examples show that the method works well c 2011 elsevier inc all rights reserved

a new wavelet method for solving an ill posed problem abstract so far there are many papers which deal with the cauchy problem for elliptic equation however most of them are devoted to the case of constant coefficients in this paper we consider a cauchy problem for an elliptic equation with variable coefficients due to ill posedness of this problem we provide a regularization method wavelet dual least squares method to solve the problem via meyer wavelet bases with a special project method dual least squares method we can obtain error estimate between the regularized solution and exact solution

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

a new cluster validity for fuzzy clustering abstract fuzzy cluster validity criterion tends to evaluate the quality of fuzzy c partitions produced by fuzzy clustering algorithms many functions have been proposed some methods use only the properties of fuzzy membership degrees to evaluate partitions others techniques combine the properties of membership degrees and the structure of data in this paper a new heuristic method is based on the combination of two functions the search of good clustering is measured by a fuzzy compactness separation ratio the first function calculates this ratio by considering geometrical properties and membership degrees of data the second function evaluates it by using only the properties of membership degrees four numerical examples are used to illustrate its use as a validity functional its effectiveness is compared to some existing cluster validity criterion

wigner distribution functions and the representation of a non bijective canonical transformation in quantum mechanics in a previous paper with a similar title the authors showed that for bijective i e one to one onto canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of wigner distribution functions and recover the classical expectation for this kernel in the limit to 0 for non bijective canonical transformations the situation is more complex as the phase space acquires either a riemann surface structure or requires the introduction of the concept of an ambiguity group by discussing the non bijective canonical transformation taking them from an oscillator of frequency kappa 1 where kappa is an integer to one of unit frequency they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group i e the ambiguity spins they obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the riemann surface picture of their phase space while they illustrate their analysis through the representation in wigner distribution phase space of a specific non bijective canonical transformation the procedure is clearly extendable to the representations of general canonical transformations of this type