spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

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

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

eigenvalue spectra of complex networks we examine the eigenvalue spectrum of the adjacency matrix of a random scale free network with an average of p edges per vertex using the replica method we show how in the dense limit when p one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network for scale free graphs with degree distribution exponent we obtain an exact expression for the eigenvalue spectrum when 3 and show that 1 2 1 for large in the limit we recover known results for the erdos renyi random graph

hierarchical sequencing of online social graphs abstract in online communications patterns of conduct of individual actors and use of emotions in the process can lead to a complex social graph exhibiting multilayered structure and mesoscopic communities using simplicial complexes representation of graphs we investigate in depth topology of the online social network constructed from myspace dialogs which exhibits original community structure a simulation of emotion spreading in this network leads to the identification of two emotion propagating layers three topological measures are introduced referred to as the structure vectors which quantify graph s architecture at different dimension levels notably structures emerging through shared links triangles and tetrahedral faces frequently occur and range from tree like to maximal 5 cliques and their respective complexes on the other hand the structures which spread only negative or only positive emotion messages appear to have much simpler topology consisting of links and triangles the node s structure vector represents the number of simplices at each topology level in which the node resides and the total number of such simplices determines what we define as the node s topological dimension the presented results suggest that the node s topological dimension provides a suitable measure of the social capital which measures the actor s ability to act as a broker in compact communities the so called simmelian brokerage we also generalize the results to a wider class of computer generated networks investigating components of the node s vector over network layers reveals that same nodes develop different socio emotional relations and that the influential nodes build social capital by combining their connections in different layers

quantitative dunford pettis property abstract we investigate possible quantifications of the dunford pettis property we show in particular that the dunford pettis property is automatically quantitative in a sense further there are two incomparable mutually dual stronger versions of a quantitative dunford pettis property we prove that l 1 spaces and c k spaces possess both of them we also show that several natural measures of weak non compactness are equal in l 1 spaces

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

certified error bounds for uncertain elliptic equations in many applications partial differential equations depend on parameters which are only approximately known using tools from functional analysis and global optimization methods are presented for obtaining certificates for rigorous and realistic error bounds on the solution of linear elliptic partial differential equations in arbitrary domains either in an energy norm or of key functionals of the solutions given an approximate solution uncertainty in the parameters specifying the partial differential equations can be taken into account either in a worst case setting or given limited probabilistic information in terms of clouds

dealing with uncertainties in engineering problems using only available data abstract this paper is the second in the series of three addressing the problem of how to deal with uncertainties in the modelings of engineering problems using only available data the problem of the heat exchanger was selected as the example because it is sufficiently complex but still simple so that the difficulties and ideas will be apparent there are various theories on how to treat the uncertainties in the first paper in the series we use only the fuzzy set respectively the possibility theory in this paper we use the fuzzy approach for some model parameters and the probability approach for the others the third paper will use the bayesian theory

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

centrality measures for networks with community structure understanding the network structure and finding out the influential nodes is a challenging issue in large networks identifying the most influential nodes in a network can be useful in many applications like immunization of nodes in case of epidemic spreading during intentional attacks on complex networks a lot of research is being done to devise centrality measures which could efficiently identify the most influential nodes in a network there are two major approaches to this problem on one hand deterministic strategies that exploit knowledge about the overall network topology while on the other end random strategies are completely agnostic about the network structure centrality measures that can deal with a limited knowledge of the network structure are of prime importance indeed in practice information about the global structure of the overall network is rarely available or hard to acquire even if available the structure of the network might be too large that it is too much computationally expensive to calculate global centrality measures to that end a centrality measure is proposed here that requires information only at the community level indeed most of the real world networks exhibit a community structure that can be exploited efficiently to discover the influential nodes we performed a comparative evaluation of prominent global deterministic strategies together with stochastic strategies an available and the proposed deterministic community based strategy effectiveness of the proposed method is evaluated by performing experiments on synthetic and real world networks with community structure in the case of immunization of nodes for epidemic control

dynamics of directed graphs the world wide web we introduce and simulate a growth model of the world wide web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real web to update outgoing links re directing them towards constantly changing selected nodes emergent statistical correlation between the distributions of outgoing and incoming links is a key feature of the dynamics of the web the growth phase is characterized by temporal fractal structures which are manifested in the hierarchical organization of links we obtain quantitative agreement with the recent empirical data in the real web for the distributions of in and out links and for the size of the connected component in a fully grown network of n nodes we study the structure of connected clusters of nodes that are accessible along outgoing links from a randomly selected node the distributions of size and depth of the connected clusters with a giant component exhibit supercritical behavior by decreasing the control parameter average fraction of updated and added links per time step towards c n 10 the web can resume a critical structure with no giant component in it we find a different universality class when the updates of links are not allowed i e for 0 corresponding to the network of science citations

laplacian spectra as a diagnostic tool for network structure and dynamics we examine numerically the three way relationships among structure laplacian spectra and frequency synchronization dynamics on complex networks we study the effects of clustering degree distribution and a particular type of coupling asymmetry input normalization all of which are known to have effects on the synchronizability of oscillator networks we find that these topological factors produce marked signatures in the laplacian eigenvalue distribution and in the localization properties of individual eigenvectors using a set of coordinates based on the laplacian eigenvectors as a diagnostic tool for synchronization dynamics we find that the process of frequency synchronization can be visualized as a series of quasi independent transitions involving different normal modes particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes for example there are important partially synchronized states in which a set of low lying modes remain unlocked while those in the main spectral peak are locked we find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue

worst case scenario analysis for elliptic problems with uncertainty this work studies linear elliptic problems under uncertainty the major emphasis is on the deterministic treatment of such uncertainty in particular this work uses the worst scenario approach for the characterization of uncertainty on functional outputs quantities of physical interest assuming that the input data belong to a given functional set eventually infinitely dimensional this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest numerical experiments illustrate the performance of the proposed methodology

search and topology aspects in transport on scale free networks abstract using a recently proposed model physica a 332 2004 566 of information transport on complex networks we study the role of network substrates on the statistics of queuing times and correlations in traffic streams when navigation with an enlarged information horizon is applied the waiting time distribution on structured networks has a power law tail which can be parametrized by 1 1 q where the respective value of the non extensivity parameter q increases systematically with increasing graph complexity the corresponding distribution on the small world graph appears to be a stretched exponential function that cannot be classified within the same scheme we further demonstrate by looking at the anti persistence in flow time series and the transit time distributions how traffic efficiency increases with an extended information horizon a horizon of two layers approaches the critical horizon on the correlated cyclic scale free graph suggesting that the mechanisms of signaling over two to three layers ensure efficient transport processes on networks with this organizational complexity

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

a quantitative extension of a theorem of valdivia on weak compactness abstract we prove the following result let m be a subset of a banach space x with the following property there is e 0 such that f is bounded on m whenever f is a real valued and x x continuous function on x e b x then m is 2e weakly relatively compact that is the x x closure of m is contained in x 2 e b x this theorem is a quantitative extension of a theorem on weak compactness due to m valdivia that corresponds to the case e 0

spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities we study structure eigenvalue spectra and random walk dynamics in a wide class of networks with subgraphs modules at mesoscopic scale the networks are grown within the model with three parameters controlling the number of modules their internal structure as scale free and correlated subgraphs and the topology of connecting network within the exhaustive spectral analysis for both the adjacency matrix and the normalized laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees the minimally connected nodes the clustering and the average connectivity affect the central part of the spectrum the number of distinct modules leads to an extra peak at the lower part of the laplacian spectrum in cyclic graphs such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs we also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda l 2 of the laplacian further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs the distribution of first return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1 3 for trees and sigma approximately 2 3 for cyclic graphs which is independent of their mesoscopic and global structure

modeling error and adaptivity in nonlinear continuum mechanics in this paper computable global bounds on errors due to the use of various mathematical models of physical phenomena are derived the procedure involves identifying a so called fine model among a class of models of certain events and then using that model as a datum with respect to which coarser models can be compared the error inherent in a coarse model compared to the fine datum can be bounded by residual functionals unambiguously defined by solutions of the coarse model whenever there exist hierarchical classes of models in which levels of sophistication of various coarse models can be defined an adaptive modeling strategy can be implemented to control modeling error in the present work the class of models is within those embodied in nonlinear continuum mechanics

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

exact evaluation of the causal spectrum and localization properties of electronic states on a scale free network abstract a deterministic network with tree structure is considered for which the spectrum of its adjacency matrix can be exactly evaluated by a recursive renormalization approach it amounts to successively increasing number of contributions at any finite step of construction of the tree resulting in a causal chain the resulting eigenvalues can be related the full energy spectrum of a nearest neighbor tight binding model defined on this structure given this association it turns out that further properties of the eigenvectors can be evaluated like the degree of quantum localization of the tight binding eigenstates expressed by the inverse participation ratio ipr it happens that for the current model the ipr s are also suitable to be analytically expressed in terms in corresponding eigenvalue chain the resulting ipr scaling behavior is expressed by the tails of eigenvalue chains as well

quantitative grothendieck property abstract a banach space x is grothendieck if the weak and the weak convergence of sequences in the dual space x coincide the space l is a classical example of a grothendieck space due to grothendieck we introduce a quantitative version of the grothendieck property we prove a quantitative version of the above mentioned grothendieck s result and we construct a grothendieck space which is not quantitatively grothendieck we also establish the quantitative grothendieck property of l for a finite measure

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

spectra of modular and small world matrices we compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter and intra module degree distributions subject only to the constraint of finite mean connectivities we also evaluate spectra of a certain class of small world matrices generated from random graphs by introducing shortcuts via additional random connectivity components both adjacency matrices and the associated graph laplacians are investigated for the laplacians we find lifshitz type singular behaviour of the spectral density in a localized region of small values in the case of modular networks we can identify contributions of local densities of state from individual modules for small world networks we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states exhibiting only localized states in the band gaps results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances and with direct diagonalization results

on quantification of weak sequential completeness we consider several quantities related to weak sequential completeness of a banach space and prove some of their properties in general and in l embedded banach spaces improving in particular an inequality of g godefroy n kalton and d li we show some examples witnessing natural limits of our positive results in particular we construct a separable banach space x with the schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness thus providing a partial answer to a question of g godefroy

systems of fuzzy equations in structural mechanics systems of linear and nonlinear equations with fuzzy parameters are relevant to many practical problems arising in structure mechanics electrical engineering finance economics and physics in this paper three methods for solving such equations are discussed method for outer interval solution of systems of linear equations depending linearly on interval parameters fuzzy finite element method proposed by rama rao and sensitivity analysis method the performance and advantages of presented methods are described with illustrative examples extended version of the present paper can be downloaded from the web page of the utep i skalna m v rama rao a pownuk systems of fuzzy equations in structural mechanics the university of texas at el paso department of mathematical sciences research reports series texas research report no 2007 01 2007

analysis of heat kernel highlights the strongly modular and heat preserving structure of proteins abstract in this paper we study the structure and dynamical properties of protein contact networks with respect to other biological networks together with simulated archetypal models acting as probes we consider both classical topological descriptors such as modularity and statistics of the shortest paths and different interpretations in terms of diffusion provided by the discrete heat kernel which is elaborated from the normalized graph laplacians a principal component analysis shows high discrimination among the network types by considering both the topological and heat kernel based vector characterizations furthermore a canonical correlation analysis demonstrates the strong agreement among those two characterizations providing thus an important justification in terms of interpretability for the heat kernel finally and most importantly the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey notably the heat trace decay of an ensemble of varying size proteins denotes subdiffusion a peculiar property of proteins

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

greedy discrete particle swarm optimization for large scale social network clustering social computing is a new paradigm for information and communication technology social network analysis is one of the theoretical underpinnings of social computing community structure detection is believed to be an effective tool for social network analysis uncovering community structures in social networks can be regarded as clustering optimization problems because social networks are characterized by dynamics and huge volumes of data conventional nature inspired optimization algorithms encounter serious challenges when applied to solving large scale social network clustering optimization problems in this study we put forward a novel particle swarm optimization algorithm to reveal community structures in social networks the particle statuses are redefined under a discrete scenario the status updating rules are reconsidered based on the network topology a greedy strategy is designed to drive particles to a promising region to this end a greedy discrete particle swarm optimization framework for large scale social network clustering is suggested to determine the performance of the algorithm extensive experiments on both synthetic and real world social networks are carried out we also compare the proposed algorithm with several state of the art network community clustering methods the experimental results demonstrate that the proposed method is effective and promising for social network clustering

measures of weak non compactness in preduals of von neumann algebras and jbw triples abstract we prove among other results that three standard measures of weak non compactness coincide in preduals of jbw triples this result is new even for preduals of von neumann algebras we further provide a characterization of jbw triples with strongly wcg predual and describe the order of seminorms defining the strong topology as a byproduct we improve a characterization of weakly compact subsets of a jbw triple predual providing so a proof for a conjecture open for almost eighteen years on weakly compact operators from a jb triple into a complex banach space

effects of uncertainties in the domain on the solution of dirichlet boundary value problems a domain with possibly non lipschitz boundary is defined as a limit of monotonically expanding or shrinking domains with lipschitz boundary a uniquely solvable dirichlet boundary value problem dbvp is defined on each of the lipschitz domains and the limit of these solutions is investigated the limit function also solves a dbvp on the limit domain but the problem can depend on the sequences of domains if the limit domain is unstable with respect to the dbvp the core of the paper consists in estimates of the difference between the respective solutions of the dbvp on two close domains one of which is lipschitz and the other can be unstable estimates for starshaped as well as rather general domains are derived their numerical evaluation is possible and can be done in different ways

boundaries for banach spaces determine weak compactness a boundary for a real banach space is a subset of the dual unit sphere with the property that each element of the banach space attains its norm on an element of that subset trivially the pointwise convergence with respect to such a boundary is coarser than the weak topology on the banach space the boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets

theory and methodology for estimation and control of errors due to modeling approximation and uncertainty the reliability of computer predictions of physical events depends on several factors the mathematical model of the event the numerical approximation of the model and the random nature of data characterizing the model this paper addresses the mathematical theories algorithms and results aimed at estimating and controlling modeling error numerical approximation error and error due to randomness in material coefficients and loads a posteriori error estimates are derived and applications to problems in solid mechanics are presented

chain motifs the tails and handles of complex networks a great part of the interest in complex networks has been motivated by the presence of structured frequently nonuniform connectivity because diverse connectivity patterns tend to result in distinct network dynamics and also because they provide the means to identify and classify several types of complex network it becomes important to obtain meaningful measurements of the local network topology in addition to traditional features such as the node degree clustering coefficient and shortest path motifs have been introduced in the literature in order to provide complementary descriptions of the network connectivity the current work proposes a different type of motif namely chains of nodes that is sequences of connected nodes with degree 2 these chains have been subdivided into cords tails rings and handles depending on the type of their extremities e g open or connected a theoretical analysis of the density of such motifs in random and scale free networks is described and an algorithm for identifying these motifs in general networks is presented the potential of considering chains for network characterization has been illustrated with respect to five categories of real world networks including 16 cases several interesting findings were obtained including the fact that several chains were observed in real world networks especially the world wide web books and the power grid the possibility of chains resulting from incompletely sampled networks is also investigated

effects of uncertainties in the domain on the solution of neumann boundary value problems in two spatial dimensions an essential part of any boundary value problem is the domain on which the problem is defined the domain is often given by scanning or another digital image technique with limited resolution this leads to significant uncertainty in the domain definition the paper focuses on the impact of the uncertainty in the domain on the neumann boundary value problem nbvp it studies a scalar nbvp defined on a sequence of domains the sequence is supposed to converge in the set sense to a limit domain then the respective sequence of nbvp solutions is examined first it is shown that the classical variational formulation is not suitable for this type of problem as even a simple nbvp on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary a new definition of the nbvp is introduced to avoid this difficulty by means of reformulated natural boundary conditions then the convergence of solutions of the nbvp is demonstrated the uniqueness of the limit solution however depends on the stability property of the limit domain finally estimates of the difference between two nbvp solutions on two different but close domains are given

a multilevel sparse kernel based stochastic collocation finite element method for elliptic problems with random coefficients abstract a new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems bvp with random coefficients assuming that the randomness is well approximated by a finite number of random variables with given probability distributions the proposed method consists of a finite element approximation in physical space along with a stochastic collocation quadrature approach utilizing the recent multilevel sparse kernel based interpolation musik technique georgoulis et al 2013 musik is based on a multilevel sparse grid type algorithm with the basis functions consisting of directionally anisotropic gaussian radial basis functions kernels placed at directionally uniform grid points we prove that musik is interpolatory at these nodes and therefore can be naturally used to define a quadrature scheme numerical examples are also presented assessing the performance of the new algorithm in the context of high dimensional stochastic collocation finite element methods

method to find community structures based on information centrality community structures are an important feature of many social biological and technological networks here we study a variation on the method for detecting such communities proposed by girvan and newman and based on the idea of using centrality measures to define the community boundaries m girvan and m e j newman proc natl acad sci u s a 99 7821 2002 we develop an algorithm of hierarchical clustering that consists in finding and removing iteratively the edge with the highest information centrality we test the algorithm on computer generated and real world networks whose community structure is already known or has been studied by means of other methods we show that our algorithm although it runs to completion in a time o n4 is very effective especially when the communities are very mixed and hardly detectable by the other methods

tukey classification of some ideals on ω and the lattices of weakly compact sets in banach spaces abstract we study the lattice structure of the family of weakly compact subsets of the unit ball b x of a separable banach space x equipped with the inclusion relation this structure is denoted by k b x and also with the parametrized family of almost inclusion relations k l e b x where e 0 this structure is denoted by ak b x tukey equivalence between partially ordered sets and a suitable extension to deal with ak b x are used assuming the axiom of analytic determinacy we prove that separable banach spaces fall into four categories namely k b x is equivalent either to a singleton or to or to the family k q of compact subsets of the rational numbers or to the family c of all finite subsets of the continuum also under the axiom of analytic determinacy a similar classification of ak b x is obtained for separable banach spaces not containing l 1 we prove in zfc that k b x ak b x are equivalent to either 0 k q or c the lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied

modelling conflicts with cluster dynamics in networks abstract we introduce cluster dynamical models of conflicts in which only the largest cluster can be involved in an action this mimics the situations in which an attack is planned by a central body and the largest attack force is used we study the model in its annealed random graph version on a fixed network and on a network evolving through the actions the sizes of actions are distributed with a power law tail however the exponent is non universal and depends on the frequency of actions and sparseness of the available connections between units allowing the network reconstruction over time in a self organized manner e g by adding the links based on previous liaisons between units we find that the power law exponent depends on the evolution time of the network its lower limit is given by the universal value 5 2 derived analytically for the case of random fragmentation processes in the temporal patterns behind the size of actions we find long range correlations in the time series of the number of clusters and the non trivial distribution of time that a unit waits between two actions in the case of an evolving network the distribution develops a power law tail indicating that through repeated actions the system develops an internal structure with a hierarchy of units