the asymptotics of solutions of singularly perturbed functional differential equations distributed and concentrated delays are different abstract this paper illustrates the differences between systems with distributed delays and systems having only concentrated delays in what concerns the asymptotic rates of solutions of singularly perturbed linear retarded functional differential equations an example of a system with distributed delays shows that the introduction of a slow variable coupled with the fast variable may decrease the asymptotic rates of solutions observed when the perturbation parameter is close to zero such a situation cannot happen for ordinary differential equations or even for differential difference equations

convergence and boundary layers in singularly perturbed linear functional differential equations abstract the convergence of solutions of singularly perturbed systems of linear retarded functional differential equations to solutions of the associated degenerate problem is discussed in connection with the geometry of the flow in phase spaces defined in terms of square integrable functions

small moving rigid body into a viscous incompressible fluid we consider a single disk moving under the influence of a two dimensional viscous fluid and we study the asymptotic as the size of the solid tends to zero if the density of the solid is independent of varepsilon the energy equality is not sufficient to obtain a uniform estimate for the solid velocity this will be achieved thanks to the optimal lp lq decay estimates of the semigroup associated to the fluid rigid body system and to a fixed point argument next we will deduce the convergence to the solution of the navier stokes equations in mathbb r 2

on the motion of a rigid body with a cavity filled with a viscous liquid we study the motion of a rigid body with a cavity filled with a viscous liquid the main objective is to investigate the well posedness of the coupled system formed by the navier stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part to this end appropriate function spaces and operators are introduced and analysed by considering a completely general three dimensional bounded domain we prove the existence of weak solutions using the galerkin method in particular we show that if the initial velocity is orthogonal in a certain sense to all rigid velocities then the velocity of the system decays exponentially to zero as time goes to infinity then following a functional analytic approach inspired by kato s scheme we prove the existence and uniqueness of mild solutions finally the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data

on the global existence and convergence to steady state of navier stokes flow past an obstacle that is started from rest a new approach to the existence theory for the navier stokes equations using a technique of kato 15 further developed in combination with estimates for oseen s equation by kobayashi shibata 17 and shibata 24 has made possible the solution of a long standing open problem often referred to as finn s starting problem this paper provides the solution

homogenization of the navier stokes equations in open sets perforated with tiny holes ii non critical sizes of the holes for a volume distribution and a surface distribution of holes this paper is devoted to the homogenization of the stokes or navier stokes equations with a dirichlet boundary condition in a domain containing many tiny solid obstacles periodically distributed in each direction of the axes for obstacles of critical size it was established in part i that the limit problem is described by a law of brinkman type here we prove that for smaller obstacles the limit problem reduces to the stokes or navier stokes equations and for larger obstacles to darcy s law we also apply the abstract framework of part i to the case of a domain containing tiny obstacles periodically distributed on a surface for example in three dimensions consider obstacles of size e2 located at the nodes of a regular plane mesh of period e this provides a mathematical model for fluid flows through mixing grids based on a special form of the brinkman law in which the additional term is concentrated on the plane of the grid

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

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 weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid we consider the two dimensional motion of several non homogeneous rigid bodies immersed in an incompressible non homogeneous viscous fluid the fluid and the rigid bodies are contained in a fixed open bounded set of r 2 the motion of the fluid is governed by the navier stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies the time variation of the fluid domain due to the motion of the rigid bodies is not known a priori so we deal with a free boundary value problem the main novelty here is the demonstration of the global existence of weak solutions for this problem more precisely the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary we give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero

finding communities in networks in the strong and almost strong sense finding communities or clusters or modules in networks can be done by optimizing an objective function defined globally and or by specifying conditions which must be satisfied by all communities radicchi et al proc natl acad sci usa 101 2658 2004 define a susbset of vertices of a network to be a community in the strong sense if each vertex of that subset has a larger inner degree than its outer degree a partition in the strong sense has only strong communities in this paper we first define an enumerative algorithm to list all partitions in the strong sense of a network of moderate size the results of this algorithm are given for the zachary karate club data set which is solved by hand as well as for several well known real world problems of the literature moreover this algorithm is slightly modified in order to apply it to larger networks keeping only partitions with the largest number of communities it is shown that some of the partitions obtained are informative although they often have only a few communities while they fail to give any information in other cases having only one community it appears that degree 2 vertices play a big role in forcing large inhomogeneous communities therefore a weakening of the strong condition is proposed and explored we define a partition in the almost strong sense by substituting a nonstrict inequality to a strict one in the definition of strong community for all vertices of degree 2 results for the same set of problems as before then give partitions with a larger number of communities and are more informative

column generation algorithms for exact modularity maximization in networks finding modules or clusters in networks currently attracts much attention in several domains the most studied criterion for doing so due to newman and girvan phys rev e 69 026113 2004 is modularity maximization many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality there are few exact algorithms prominent among which is a paper by xu et al eur phys j b 60 231 2007 modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of grotschel and wakabayashi math program 45 59 1989 applied we propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming performance of the four resulting algorithms is compared on problems from the literature instances with up to 512 entities are solved exactly moreover the computing time of previously solved problems are reduced substantially

homogenization of the navier stokes equations in open sets perforated with tiny holes i abstract framework a volume distribution of holes this paper treats the homogenization of the stokes or navier stokes equations with a dirichlet boundary condition in a domain containing many tiny solid obstacles periodically distributed in each direction of the axes for example in the three dimensional case the obstacles have a size of e3 and are located at the nodes of a regular mesh of size e a suitable extension of the pressure is used to prove the convergence of the homogenization process to a brinkman type law in which a linear zero order term for the velocity is added to a stokes or navier stokes equation

two dimensional incompressible viscous flow around a small obstacle abstract in this article we study the asymptotic behavior of incompressible ideal time dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small our main purpose is to identify the equation satisfied by the limit flow we will see that the asymptotic behavior depends on the circulation around the obstacle for smooth flow around a single obstacle is a conserved quantity which is determined by the initial data we will show that if 0 the limit flow satisfies the standard incompressible euler equations in the full plane but if 0 the limit equation acquires an additional forcing term we treat this problem by first constructing a sequence of approximate solutions to the incompressible 2d euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation we use an explicit treatment of the green s

detecting complex network modularity by dynamical clustering based on cluster desynchronization properties of phase oscillators we introduce an efficient method for the detection and identification of modules in complex networks the performance of the algorithm is tested on computer generated and real world networks whose modular structure is already known or has been studied by means of other methods the algorithm attains a high level of precision especially when the modular units are very mixed and hardly detectable by the other methods with a computational effort o kn on a generic graph with n nodes and k links

edge ratio and community structure in networks a hierarchical divisive algorithm is proposed for identifying communities in complex networks to that effect the definition of community in the weak sense of radicchi et al proc natl acad sci u s a 101 2658 2004 is extended into a criterion for a bipartition to be optimal one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges a mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition this includes an exact solution of the problem of detecting indivisible communities the resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets for two problems of the former kind optimal solutions are found for five problems of the latter kind the edge ratio algorithm always appears to be competitive moreover it provides additional information in several cases notably through the use of the dendrogram summarizing the resolution finally both algorithms are compared on reduced versions of the data sets of girvan and newman proc natl acad sci u s a 99 7821 2002 and of lancichinetti et al phys rev e 78 046110 2008 results for these instances appear to be comparable

accuracy and precision of methods for community identification in weighted networks based on brief review of approaches for community identification and measurement for sensitivity characterization the accuracy and precision of several approaches for detecting communities in weighted networks are investigated in weighted networks the community structure should take both links and link weights into account and the partition of networks should be evaluated by weighted modularity q w the results reveal that link weight has important effects on communities especially in dense networks potts model and weighted extremal optimization weo algorithm work well on weighted networks then potts model and weo algorithms are used to detect communities in rhesus monkey network the results gives nice understanding for real community structure

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

detecting network communities by propagating labels under constraints we investigate the recently proposed label propagation algorithm lpa for identifying network communities we reformulate the lpa as an equivalent optimization problem giving an objective function whose maxima correspond to community solutions by considering properties of the objective function we identify conceptual and practical drawbacks of the label propagation approach most importantly the disparity between increasing the value of the objective function and improving the quality of communities found to address the drawbacks we modify the objective function in the optimization problem producing a variety of algorithms that propagate labels subject to constraints of particular interest is a variant that maximizes the modularity measure of community quality performance properties and implementation details of the proposed algorithms are discussed bipartite as well as unipartite networks are considered

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

loops and multiple edges in modularity maximization of networks the modularity maximization model proposed by newman and girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges however the applications usually correspond to simple graphs these graphs are compared to a null model where the degree distribution is maintained but edges are placed at random therefore in this null model there will be loops and possibly multiple edges sharp bounds on the expected number of loops and their impact on the modularity are derived then building upon the work of massen and doye but using algebra rather than simulation we propose modified null models associated with graphs without loops but with multiple edges graphs with loops but without multiple edges and graphs without loops nor multiple edges we validate our models by using the exact algorithm for clique partitioning of grotschel and wakabayashi

detecting the community structure in complex networks based on quantum mechanics abstract in this paper we develop a novel method to detect the community structure in complex networks this approach is based on the combination of kernel based clustering using quantum mechanics the spectral clustering technique and the concept of the bayesian information criterion we test the proposed algorithm on zachary s karate club network and the world of american college football experimental results indicate that our algorithm is efficient and effective at finding both the optimal number of clusters and the best clustering of community structures

asymptotic stability of the solutions of a linear singularly perturbed system with unbounded delay abstract sufficient conditions for asymptotic stability of the solutions of a linear singularly perturbed system of differential equations with unbounded delay have been found under the same conditions it is proved that for a locally lipschitz initial function the initial value problem for the system degenerates regularly

efficient modularity optimization by multistep greedy algorithm and vertex mover refinement identifying strongly connected substructures in large networks provides insight into their coarse grained organization several approaches based on the optimization of a quality function e g the modularity have been proposed we present here a multistep extension of the greedy algorithm msg that allows the merging of more than one pair of communities at each iteration step the essential idea is to prevent the premature condensation into few large communities upon convergence of the msg a simple refinement procedure called vertex mover vm is used for reassigning vertices to neighboring communities to improve the final modularity value with an appropriate choice of the step width the combined msg vm algorithm is able to find solutions of higher modularity than those reported previously the multistep extension does not alter the scaling of computational cost of the greedy algorithm

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

modularity and extreme edges of the internet we study the spectral properties of a diffusion process taking place on the internet network focusing on the slowest decaying modes these modes identify an underlying modular structure roughly corresponding to individual countries for instance in the slowest decaying mode the diffusion current flows from russia to u s military sites quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale free network we propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network for the 100 slowest decaying modes of the internet this fraction is 30 finally we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes

exponential estimates for singularly perturbed linear fuctional differential equations abstract exponential estimates on the fundamental matrix uniform on the perturbation parameter are obtained for singularly perturbed systems of linear retarded functional differential equations under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts the exponential rates in the estimates are computable from upper bounds on the real parts of the characteristic values of the system or of associated simpler equations differences between differential difference equations and equations with distributed delays are emphasized

existence of weak solutions for the motion of rigid bodies in a viscous fluid we study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of r d d 2 or 3 with dirichlet boundary conditions by introducing an appropriate weak formulation for the complete problem we prove existence of solutions for initial velocities in h 1 0 omega in the absence of collisions solutions exist for all time in dimension 2 whereas in dimension 3 the lifespan of solutions is infinite only for small enough data

the motion of a fluid rigid ball system at the zero limit of the rigid ball radius we study the limiting motion of a system of rigid ball moving in a navier stokes fluid in mathbb r 3 as the radius of the ball goes to zero recently dashti and robinson solved this problem in the two dimensional case in the absence of rotation of the ball dashti and robinson in arch rational mech anal 200 285 312 2011 this restriction was caused by the difficulty in obtaining appropriate uniform bounds on the second order derivatives of the fluid velocity when the rigid body can rotate in this paper we show how to obtain the required uniform bounds on the velocity fields in the three dimensional case these estimates then allow us to pass to the zero limit of the ball radius and show that the solution of the coupled system converges to the solution of the navier stokes equations describing the motion of only fluid in the whole space the trajectory of the centre of the ball converges to a fluid particle trajectory which justifies the use of rigid tracers for finding lagrangian paths of fluid flow