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

three periodic solutions for perturbed second order hamiltonian systems in this paper we study the existence of three distinct solutions for the following problem u a t u f t u g t u a e in 0 t u t u 0 u t u 0 0 where r t is a real positive number a 0 t rn n is a continuous map from the interval 0 t to the set of n order symmetric matrices we propose sufficient conditions only on the potential f more precisely we assume that g satisfies only a usual growth condition which allows us to use a variational approach

homoclinic solutions for a class of subquadratic second order hamiltonian systems in this paper we study the existence of infinitely many homoclinic solutions for a class of subquadratic second order hamiltonian systems by using the variant fountain theorem we obtain a new criterion for guaranteeing that second order hamiltonian systems has infinitely many homoclinic solutions recent results from the literature are generalized and significantly improved an example is also given in this paper to illustrate our main results

infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions in this paper we consider the existence of solutions for a class of nonlinear impulsive problems with periodic boundary conditions by using critical point theory we obtain some existence theorems of infinitely many solutions for the nonlinear impulsive problem when the impulsive functions are superlinear we extend and improve some recent results

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

multiple solutions for superlinear schrodinger poisson system with sign changing potential and nonlinearity in this paper we study the following schrodinger poisson system u v x u x u f x u in r 3 u 2 in r 3 where the nonlinearity f and the potential v are allowed to be sign changing under some appropriate assumptions on v and f we obtain the existence of two different solutions of the system via the ekeland s variational principle and the mountain pass theorem

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

subharmonic solutions with prescribed minimal period for some second order impulsive differential equations abstract this paper uses critical point theory and variational methods to investigate the subharmonic solutions with prescribed minimal period for a class of second order impulsive differential equations the conditions for the existence of subharmonic solutions are established our results rectify some known results in the literature and will allow us in the future to deal with the subharmonic solutions for more extensive impulsive problems

complex network based time series analysis abstract recent works show that complex network theory may be a powerful tool in time series analysis we propose in this paper a reliable procedure for constructing complex networks from the correlation matrix of a time series an original stock time series the corresponding return series and its amplitude series are considered the degree distribution of the original series can be well fitted with a power law while that of the return series can be well fitted with a gaussian function the degree distribution of the amplitude series contains two asymmetric gaussian branches reconstruction of networks from time series is a common problem in diverse research the proposed strategy may be a reasonable solution to this problem

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

time series clustering via community detection in networks in this paper we propose a technique for time series clustering using community detection in complex networks firstly we present a method to transform a set of time series into a network using different distance functions where each time series is represented by a vertex and the most similar ones are connected then we apply community detection algorithms to identify groups of strongly connected vertices called a community and consequently identify time series clusters still in this paper we make a comprehensive analysis on the influence of various combinations of time series distance functions network generation methods and community detection techniques on clustering results experimental study shows that the proposed network based approach achieves better results than various classic or up to date clustering techniques under consideration statistical tests confirm that the proposed method outperforms some classic clustering algorithms such as k medoids diana median linkage and centroid linkage in various data sets interestingly the proposed method can effectively detect shape patterns presented in time series due to the topological structure of the underlying network constructed in the clustering process at the same time other techniques fail to identify such patterns moreover the proposed method is robust enough to group time series presenting similar pattern but with time shifts and or amplitude variations in summary the main point of the proposed method is the transformation of time series from time space domain to topological domain therefore we hope that our approach contributes not only for time series clustering but also for general time series analysis tasks

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

global renyi index of the distance matrix abstract in previous studies the heterogeneity of complex networks has been extensively studied in our study the heterogeneity of distance matrices is studied based on the renyi index of networks we define a new metric and name it global renyi index g r i and prove several properties in particular the g r i value of the distance matrix corresponding to the evenly distributed point set in the euclidean space is zero some model data were used to clarify the geometric meanings of g r i and then we studied the g r i value of financial data the results show that the g r i value in the real market changes drastically and is significantly different from the g r i value of the model generated data these results suggest that the proposed concept g r i is meaningful to the study distance matrix and provides a new perspective based on the network

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

the steady states of the kuramoto sivashinsky equation abstract the kuramoto sivashinsky equation is one of the simple nonlinear partial differential equations it represents a system in which the transport of energy through nonlinear mode coupling produces a balance between long wavelength instability and short wavelength dissipation the steady states and their stabilities give a foundation for understanding the dynamics of the system they are distinguished according to their energy levels and fall into different branches that are identified by their harmonic content various types of solution include laminar states n cell states long wave distorbed n cell states the giant states and steady traveling solutions the linear stability around the steady states gives a guide to the origin of additional steady states through bifurcation and to the identification of each steady state the index rule is also useful conjectured form the numerical observation the parameter dependence of the steady traveling waves is estimated in the limit of large values of the parameter in addition a singular perturbation calculation of traveling waves near bifurcation is carried out the dynamics of a system can be guided by the systematic study of the bifurcation and stability for example if all the stationary and periodic solutions are unstable for some parameter range chaotic behavior of the system is expected since any orbit governed by the equation has to wander around forever in the phase space the orbits will then lie on strange attractors

on ground state solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x f u in r 3 k x u 2 in r 3 and prove the existence of ground state solutions for system sp under certain assumptions on the linear and nonlinear terms some recent results from different authors are extended

new results for perturbed hamiltonian systems with impulses abstract in this paper we study the existence of three solutions for second order impulsive hamiltonian systems by using a variational method and some critical points theorems of b ricceri we obtain some new criteria for guaranteeing that the impulsive hamiltonian systems have three solutions some recent results are extended and improved some examples are presented to demonstrate the application of our main results

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

a further three critical points theorem abstract if x is a real banach space we denote by w x the class of all functionals x r possessing the following property if u n is a sequence in x converging weakly to u x and lim inf n u n u then u n has a subsequence converging strongly to u in this paper we prove the following result let x be a separable and reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional belonging to w x bounded on each bounded subset of x and whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that for each i the functional j is coercive and has a strict local not global minimum say x then for each compact interval a b i for which sup a b x j x there exists r 0 with the following property for every a b and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x has at least three solutions whose norms are less than r

solvability of second order hamiltonian systems with impulses via variational method in this paper a class of second order impulsive hamiltonian systems are considered some new existence results are obtained by using a variational method and critical point theorem due to tang and wu some recent results are extended three examples are presented to illustrate the feasibility and effectiveness of our results

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

evolution of worldwide stock markets correlation structure and correlation based graphs we investigate the daily correlation present among market indices of stock exchanges located all over the world in the time period january 1996 to july 2009 we discover that the correlation among market indices presents both a fast and a slow dynamics the slow dynamics reflects the development and consolidation of globalization the fast dynamics is associated with critical events that originate in a specific country or region of the world and rapidly affect the global system we provide evidence that the short term time scale of correlation among market indices is less than 3 trading months about 60 trading days the average values of the nondiagonal elements of the correlation matrix correlation based graphs and the spectral properties of the largest eigenvalues and eigenvectors of the correlation matrix are carrying information about the fast and slow dynamics of the correlation of market indices we introduce a measure of mutual information based on link co occurrence in networks in order to detect the fast dynamics of successive changes of correlation based graphs in a quantitative way

existence of solutions for a class of second order hamiltonian systems with impulsive effects abstract in this paper a class of second order hamiltonian systems with impulsive effects are considered we establish some sufficient conditions for the existence of solutions for the second order hamiltonian systems with impulsive effects u t f t u t a e t 0 t u 0 u t u 0 u t 0 u i t j u i t j u i t j i i j u i t j i 1 2 n j 1 2 p where t 0 0 t 1 t 2 t p t p 1 t u t u 1 t u 2 t u n t r n i i j r r i 1 2 n j 1 2 p are continuous and f 0 t r n r the solutions are sought by means of some critical point theorems finally two examples are presented to illustrate the feasibility and effectiveness of our results

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti periodic boundary conditions this paper describes the method of quasilinearization for first order nonlinear impulsive functional differential equations with anti periodic boundary conditions a monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand 2007 elsevier ltd all rights reserved

local resolution limit free potts model for community detection we report on an exceptionally accurate spin glass type potts model for community detection with a simple algorithm we find that our approach is at least as accurate as the best currently available algorithms and robust to the effects of noise it is also competitive with the best currently available algorithms in terms of speed and size of solvable systems we find that the computational demand often exhibits superlinear scaling o l 1 3 where l is the number of edges in the system and we have applied the algorithm to synthetic systems as large as 40 ifmmode times else texttimes fi 10 6 nodes and over 1 ifmmode times else texttimes fi 10 9 edges a previous stumbling block encountered by popular community detection methods is the so called resolution limit being a local measure of community structure our potts model is free from this resolution limit effect and it further remains a local measure on weighted and directed graphs we also address the mitigation of resolution limit effects for two other popular potts models

positive solutions of asymptotically linear schrodinger poisson systems with a radial potential vanishing at infinity in this paper we study the schrodinger poisson system sp u v x u x u k x f u inr3 u2 u 0 inr3 and prove the existence of positive solutions for system sp when the nonlinearity f has growth at most linear for small allowing the potential v x to vanish at infinity in addition also we obtain the nonexistence of a nontrivial positive solution for 14

multiple solutions of impulsive sturm liouville boundary value problem via lower and upper solutions and variational methods abstract in this paper we prove the existence of multiple solutions for second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is a sturm liouville operator r 1 u u 0 u 0 r 2 u u 1 u 1 the technical approach is fully based on lower and upper solutions and variational methods the interesting point is that the existence of four solutions and seven solutions is given

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

explicit construction of an inertial manifold for a reaction diffusion equation abstract an inertial manifold is constructed for the scalar reaction diffusion equation u t vu xx u with a cubic nonlinearity uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria this uniformity leads to the global parameterization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium by the introduction of local techniques near each equilibrium we succeed in constructing an inertial manifold of lowest possible dimension

multiplicity and concentration of homoclinic solutions for some second order hamiltonian systems abstract in this paper we present some new results of homoclinic solutions for second order hamiltonian systems u l t u w u t u 0 here 0 is a parameter l c r r n n and w c 1 r r n r unlike most other papers on this problem we require that l t is a positive semi definite symmetric matrix for all t r that is l t 0 is allowed to occur in some finite interval t of r under some suitable assumptions on w we prove the existence of two different homoclinic solutions u 1 u 2 which both vanish on r t as and converge to u 0 1 u 0 2 in h 1 r respectively here u 0 1 u 0 2 h 0 1 t are two nontrivial solutions of the dirichlet bvp for hamiltonian systems on the finite interval t

three solutions for a p laplacian boundary value problem with impulsive effects in this paper a p laplacian boundary value problem with impulsive effects is considered multiplicity of solutions is obtained by three critical points theorem an example is presented to illustrate main result

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

periodic solution generated by impulses for singular differential equations abstract in this paper we study the existence of a positive periodic solution for second order singular differential equations with impulsive conditions the proof is based on the mountain pass theorem we show that this positive periodic solution is generated by impulses

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

construction of inertial manifolds by elliptic regularization abstract in many cases an inertial manifold 2 m for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function from a finite dimensional hilbert space hp to another hilbert space hq the invariance property of m means that can be written as the solution of a first order partial differential equation d p g1 p p a p g2 p p 0 over hp where g1 and g2 are nonlinear functions which depend on the original dynamical system and a is a suitably stable linear operator in this paper we use a method introduced by sacker r j sacker a new approach to the perturbation theory of invariant surface comm pure appl math 18 1965 717 732 for the study of finite dimensional dynamical systems to find inertial manifolds in the infinite dimensional setting this method involves replacing the first order equation for by the regularized elliptic equation e d p g1 p p a p g2 p p with suitable boundary conditions it is shown that if a satisfies a spectral gap condition then the solutions e of the elliptic equation converge to a weak solution of 0 as e 0 furthermore m graph is an invariant manifold for the given dynamical system

constructing financial network based on pmfg and threshold method abstract based on planar maximally filtered graph p m f g and threshold method we introduced a correlation based network named p m f g based threshold network p t n we studied the community structure of p t n and applied i s o m a p algorithm to represent p t n in low dimensional euclidean space the results show that the community corresponds well to the cluster in the euclidean space further we studied the dynamics of the community structure and constructed the normalized mutual information n m i matrix based on the real data in the market we found that the volatility of the market can lead to dramatic changes in the community structure and the structure is more stable during the financial crisis

modeling multivariate time series on manifolds with skew radial basis functions we present an approach for constructing nonlinear empirical mappings from high dimensional domains to multivariate ranges we employ radial basis functions and skew radial basis functions for constructing a model using data that are potentially scattered or sparse the algorithm progresses iteratively adding a new function at each step to refine the model the placement of the functions is driven by a statistical hypothesis test that accounts for correlation in the multivariate range variables the test is applied on training and validation data and reveals nonstatistical or geometric structure when it fails at each step the added function is fit to data contained in a spatiotemporally defined local region to determine the parameters in particular the scale of the local model the scale of the function is determined by the zero crossings of the autocorrelation function of the residuals the model parameters and the number of basis functions are determined automatically from the given data and there is no need to initialize any ad hoc parameters save for the selection of the skew radial basis functions compactly supported skew radial basis functions are employed to improve model accuracy order and convergence properties the extension of the algorithm to higher dimensional ranges produces reduced order models by exploiting the existence of correlation in the range variable data structure is tested not just in a single time series but between all pairs of time series we illustrate the new methodologies using several illustrative problems including modeling data on manifolds and the prediction of chaotic time series

a class of bvps for first order impulsive integro differential equations abstract in this paper we study the impulsive integro differential equations u t f t u t tu su t t k t j 0 1 u t k i k u t k k 1 2 m u 0 au 1 bu the existence of maximal and minimal solutions to the problem is proved by using an upper and lower solutions method together with monotone iterative technique and two monotone sequences which uniformly converge to the relevant solutions are obtained an example is included to show the applicability of our results

dimension reduction of biological neuron models by artificial neural networks an artificial neural network approach to dimension reduction of dynamical systems is proposed and applied to conductance based neuron models networks with bottleneck layers of continuous time dynamical units could make a two dimensional model from the trajectories of the hodgkin huxley model and a three dimensional model from the trajectories of a six dimensional bursting neuron model nullcline analysis of these reduced models revealed the bifurcations of the dynamical system underlying firing and bursting behaviors

the multiplicity of solutions for perturbed second order hamiltonian systems with impulsive effects abstract in this paper we study the existence of multiple solutions for a class of second order impulsive hamiltonian systems we give some new criteria for guaranteeing that the impulsive hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of b ricceri we extend and improve on some recent results finally some examples are presented to illustrate our main results

an application of variational method to second order impulsive differential equation on the half line abstract in this paper we study the multiplicity of solutions for second order impulsive differential equation with a parameter on the half line by using a variational method and a three critical points theorem we give some new criteria to guarantee that the impulsive problem has at least three classical solutions also an example is given in this paper to illustrate the main results

circular nodes in neural networks in the usual construction of a neural network the individual nodes store and transmit real numbers that lie in an interval on the real line the values are often envisioned as amplitudes in this article we present a design for a circular node which is capable of storing and transmitting angular information we develop the forward and backward propagation formulas for a network containing circular nodes we show how the use of circular nodes may facilitate the characterization and parameterization of periodic phenomena in general we describe applications to constructing circular self maps periodic compression and one dimensional manifold decomposition we show that a circular node may be used to construct a homeomorphism between a trefoil knot in 3 and a unit circle we give an application with a network that encodes the dynamic system on the limit cycle of the kuramoto sivashinsky equation this is achieved by incorporating a circular node in the bottleneck layer of a three hidden layer bottleneck network architecture exploiting circular nodes systematically offers a neural network alternative to fourier series decomposition in approximating periodic or almost periodic functions

new comparison results for impulsive integro differential equations and applications we prove some new maximum principles for ordinary integro differential equations this allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first order impulsive integro differential equation

evolution of correlation structure of industrial indices of u s equity markets we investigate the dynamics of correlations present between pairs of industry indices of u s stocks traded in u s markets by studying correlation based networks and spectral properties of the correlation matrix the study is performed by using 49 industry index time series computed by k french and e fama during the time period from july 1969 to december 2011 which spans more than 40 years we show that the correlation between industry indices presents both a fast and a slow dynamics the slow dynamics has a time scale longer than 5 years showing that a different degree of diversification of the investment is possible in different periods of time moreover we also detect a fast dynamics associated with exogenous or endogenous events the fast time scale we use is a monthly time scale and the evaluation time period is a 3 month time period by investigating the correlation dynamics monthly we are able to detect two examples of fast variations in the first and second eigenvalue of the correlation matrix the first occurs during the dot com bubble from march 1999 to april 2001 and the second occurs during the period of highest impact of the subprime crisis from august 2008 to august 2009

social significance of community structure statistical view community structure analysis is a powerful tool for social networks that can simplify their topological and functional analysis considerably however since community detection methods have random factors and real social networks obtained from complex systems always contain error edges evaluating the significance of a partitioned community structure is an urgent and important question in this paper integrating the specific characteristics of real society we present a framework to analyze the significance of a social community the dynamics of social interactions are modeled by identifying social leaders and corresponding hierarchical structures instead of a direct comparison with the average outcome of a random model we compute the similarity of a given node with the leader by the number of common neighbors to determine the membership vector an efficient community detection algorithm is proposed based on the position of the nodes and their corresponding leaders then using a log likelihood score the tightness of the community can be derived based on the distribution of community tightness we establish a connection between p value theory and network analysis and then we obtain a significance measure of statistical form finally the framework is applied to both benchmark networks and real social networks experimental results show that our work can be used in many fields such as determining the optimal number of communities analyzing the social significance of a given community comparing the performance among various algorithms etc

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

multiple periodic solutions for hamiltonian systems with not coercive potential under an appropriate oscillating behavior of the nonlinear term the existence of infinitely many periodic solutions for a class of second order hamiltonian systems is established moreover the existence of two non trivial periodic solutions for hamiltonian systems with not coercive potential is obtained and the existence of three periodic solutions for hamiltonian systems with coercive potential is pointed out the approach is based on critical point theorems

impulsive periodic boundary value problems of first order differential equations in this paper by using schaeffer s theorem we prove new existence theorems for a nonlinear periodic boundary value problem of first order differential equations with impulses our results improve and generalize some known results

the existence of steady solutions of the kuramoto sivashinsky equation abstract we investigate the existence of steady solutions of the kuramoto sivashinsky equation for wave speed c 1 we show that there are two periodic solutions and two heteroclinic orbits for the resultant third order ordinary differential equation a discussion is given concerning the possibility of using these solutions to obtain the existence of more complicated patterns

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded