discrete nonlinear schrodinger equations with arbitrarily high order nonlinearities a class of discrete nonlinear schrodinger equations with arbitrarily high order nonlinearities is introduced these equations are derived from the same hamiltonian using different poisson brackets and include as particular cases the saturable discrete nonlinear schrodinger equation and the ablowitz ladik equation as a common property these equations possess three kinds of exact analytical stationary solutions for which the peierls nabarro barrier is zero several properties of these solutions including stability discrete breathers and moving solutions are investigated

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

high dimensional random apollonian networks abstract we propose a simple algorithm which produces a new category of networks high dimensional random apollonian networks with small world and scale free characteristics we derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network the values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks we estimate also analytically that the average path length of the networks increases at most logarithmically with the number of vertices

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

degree and component size distributions in the generalized uniform recursive tree we propose a generalized model for the famous uniform recursive tree urt by introducing an imperfect growth process which may generate disconnected components clusters the model undergoes an interesting phase transition from a singly connected network to a graph consisting of fully isolated nodes we investigate the distributions of degree and component sizes by both theoretical predictions and numerical simulations for the nontrivial cases we show that the network has an exponential degree distribution while its component size distribution follows a power law both of which are related to the imperfect growth process we also predict the growth dynamics of the individual components all analytical solutions are successfully contrasted with computer simulations

exact static solutions to a translationally invariant discrete phi4 model for a discrete translationally invariant 4 model introduced by barashenkov et al 2005 phys rev e 72 35602r we provide the momentum conservation law and demonstrate how the first integral of the static version of the discrete model can be constructed from a jacobi elliptic function jef solution the first integral can be written in the form of a nonlinear map from which the static solution supported by the model can be constructed a set of jef solutions including the staggered ones is derived we also report on the stability analysis for the static bounded solutions and exemplify the dynamical behaviour of the unstable solutions this work provides a road map through this illustrative example on how to fully analyse translationally invariant models in terms of their static problem its first integral their full set of static solutions and associated conservation laws

a partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres the numerical analysis of large numbers of arbitrarily distributed discrete thin fibres embedded in a continuum is a computationally demanding process in this contribution we propose an approach based on the partition of unity property of finite element shape functions that can handle discrete thin fibres in a continuum matrix without meshing them this is made possible by a special enrichment function that represents the action of each individual fibre on the matrix our approach allows to model fibre reinforced materials considering matrix fibres and interfaces between matrix and fibres individually each with its own elastic constitutive law copyright 2010 john wiley sons ltd

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

efficiency of scale free networks error and attack tolerance the concept of network efficiency recently proposed to characterize the properties of small world networks is here used to study the effects of errors and attacks on scale free networks two different kinds of scale free networks i e networks with power law p k are considered 1 scale free networks with no local clustering produced by the barabasi albert model and 2 scale free networks with high clustering properties as in the model by klemm and eguiluz and their properties are compared to the properties of random graphs exponential graphs by using as mathematical measures the global and the local efficiency we investigate the effects of errors and attacks both on the global and the local properties of the network we show that the global efficiency is a better measure than the characteristic path length to describe the response of complex networks to external factors we find that at variance with random graphs scale free networks display both on a global and on a local scale a high degree of error tolerance and an extreme vulnerability to attacks in fact the global and the local efficiency are unaffected by the failure of some randomly chosen nodes though they are extremely sensitive to the removal of the few nodes which play a crucial role in maintaining the network s connectivity

stochastically evolving networks we discuss a class of models for the evolution of networks in which new nodes are recruited into the network at random times and links between existing nodes that are not yet directly connected may also form at random times the class contains both models that produce small world networks and less tightly linked models we produce both trees appropriate in certain biological applications and networks in which closed loops can appear which model communication networks and networks of human sexual interactions one of our models is closely related to random recursive trees and some exact results known in that context can be exploited the other models are more subtle and difficult to analyze our analysis includes a number of exact results for moments correlations and distributions of coordination number and network size we report simulations and also discuss some mean field approximations if the system has evolved for a long time and the state of a random node which thus has a random age is observed power law distributions for properties of the system arise in some of these models

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

translationally invariant discrete kinks from one dimensional maps for most discretizations of the phi4 theory the stationary kink can only be centered either on a lattice site or midway between two adjacent sites we search for exceptional discretizations that allow stationary kinks to be centered anywhere between the sites we show that this translational invariance of the kink implies the existence of an underlying one dimensional map phi n 1 f phi n a simple algorithm based on this observation generates three families of exceptional discretizations

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

a 3d interface enriched generalized finite element method for weakly discontinuous problems with complex internal geometries an interface enriched generalized finite element method gfem is introduced for 3d problems with discontinuous gradient fields the proposed method differs from conventional gfem by assigning the generalized degrees of freedom to the interface nodes i e nodes generated along the interface when creating integration subdomains instead of the nodes of the original mesh a linear combination of the lagrangian shape functions in these integration subelements are then used as the enrichment functions to capture the discontinuity in the gradient field this approach provides a great flexibility for evaluating the enrichment functions including for cases where elements are intersected with multiple interfaces we show that the method achieves optimal rate of convergence with meshes which do not conform to the phase interfaces at a computational cost similar to or lower than that of conventional gfem the potential of the method is demonstrated by solving several heat transfer problems with discontinuous gradient field encountered in particulate and fiber reinforced composites and in actively cooled microvascular materials

standard nearest neighbour discretizations of klein gordon models cannot preserve both energy and linear momentum we consider nonlinear klein gordon wave equations and illustrate that standard discretizations thereof involving nearest neighbours may preserve either standardly defined linear momentum or standardly defined total energy but not both this has a variety of intriguing implications for the non potential discretizations that preserve only the linear momentum such as the self accelerating or self decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

cyclic identities for jacobi elliptic and related functions identities involving cyclic sums of terms composed from jacobi elliptic functions evaluated at p equally shifted points on the real axis were recently found these identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations we derive four master identities from which the identities discussed earlier are derivable as special cases master identities are also obtained which lead to cyclic identities with alternating signs we discuss an extension of our results to pure imaginary and complex shifts as well as to the ratio of jacobi theta functions

from regular to growing small world networks abstract we propose a growing model which interpolates between one dimensional regular lattice and small world networks the model undergoes an interesting phase transition from large to small worlds we investigate the structural properties by both theoretical predictions and numerical simulations our growing model is a complementarity for the important static watts strogatz network model

generalized finite element enrichment functions for discontinuous gradient fields a general gfem xfem formulation is presented to solve two dimensional problems characterized by c0 continuity with gradient jumps along discrete lines such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media the new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks we show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines the use of the proposed enrichments in both linear and quadratic approximations is investigated as well as their combination with interface enrichment functions available in the literature through a detailed convergence study we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems in the linear case the effectiveness of correction functions proposed in the literature is also investigated copyright 2009 john wiley sons ltd

component sizes in networks with arbitrary degree distributions we give an exact solution for the complete distribution of component sizes in random networks with arbitrary degree distributions the solution tells us the probability that a randomly chosen node belongs to a component of size s for any s we apply our results to networks with the three most commonly studied degree distributions poisson exponential and power law as well as to the calculation of cluster sizes for bond percolation on networks which correspond to the sizes of outbreaks of epidemic processes on the same networks for the particular case of the power law degree distribution we show that the component size distribution itself follows a power law everywhere below the phase transition at which a giant component forms but takes an exponential form when a giant component is present

correlations in random apollonian network abstract in this paper by both simulations and theoretical predictions we study two and three node or degree correlations in random apollonian network ran which have small world and scale free topologies using the rate equation approach under the assumption of continuous degree we first give the analytical solution for two node correlations expressed by average nearest neighbor degree annd then we revisit the degree distribution of ran using rate equation method and get the exact connection distribution based on which we derive a more accurate result for mean clustering coefficient as an average quantity of three degree correlations than the one previously reported analytical results reveal that annd has no correlations with respect to degree while clustering coefficient is dependent on degree showing a power law behavior as c k k 1 the obtained expressions are successfully contrasted with extensive numerical simulations

an interface enriched generalized fem for problems with discontinuous gradient fields summary a new generalized fem is introduced for solving problems with discontinuous gradient fields the method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges the proposed approach has several advantages over conventional generalized fem formulations such as a lower computational cost easier implementation and straightforward handling of dirichlet boundary conditions a detailed convergence study of the proposed method and a comparison with the standard fem are presented for heat transfer problems the method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard fem with conforming meshes various application problems are presented including the conjugate heat transfer problem encountered in microvascular materials copyright 2011 john wiley sons ltd

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

algorithms for refining triangular grids suitable for adaptive and multigrid techniques two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed the algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used the algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way proofs of the above properties are presented the second algorithm is a simpler improved version of the first which retains most of the properties of the latter the algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation in this sense they can be adequately combined with adaptive and or multigrid techniques for solving finite element systems examples of the application of the algorithms are given and two possible generalizations are pointed out

a hierarchical finite element method based on the partition of unity in this paper we consider the application of hierarchical functions to base approximations which are a partition of unity the particular hierarchical functions used are added to base finite element interpolations which for c0 approximations are a particular case of the partition of unity we also show how the functions may be constructed to preserve the interpolation property of the base finite element functions an application to linear elasticity is used to illustrate the properties and stability of the approximation

percolation transition in a dynamically clustered network we consider a percolationlike phenomenon on a generalization of the barab asi albert model where a modification of the growth dynamics directly allows formation of disconnected clusters the transition is located with high precision by an original numerical technique based on the comparison of the largest and second largest clusters a careful investigation focusing on finite size scaling allows us to highlight properties which would hardly be accessible by an analytical solution of cluster growth equations in the stationary limit our analysis shows that some critical features of the percolation transition are different from those observed in the case of dilution in fully grown networks at variance with other models of percolation on growing networks we also find evidence that the order parameter approaches zero as a power of the field p ensuremath p c driving the transition rather than as a stretched exponential this behavior does not agree with the berezinskii kosterlitz thouless scenario found in other similar models for describing the phase in which a giant cluster develops a key role is played by the crossover number of nodes n x ensuremath sim p ensuremath p c ensuremath ensuremath zeta with ensuremath zeta ensuremath simeq 4 this power law behavior and that of other quantities are conjectured on the basis of scaling arguments and numerical evidence

discrete inclusion models for reinforced composites comparative performance analysis and modeling challenges abstract we report the results of a comparative analysis of mesh independent discrete inclusion models and point out some shortcomings of classical approaches in the approximation of the strain field across an inclusion artificial continuity and the slip profile along an inclusion oscillatory behavior we also present novel embedded reinforcement models based on partition of unity enrichment strategies adaptive h refinement and order regularity extensions these novel models are assessed by means of mesh convergence studies and it is shown that they improve the quality of the solution by significantly decreasing local spurious oscillations in the slip profile along an inclusion

exact solutions of the saturable discrete nonlinear schrodinger equation exact solutions to a nonlinear schrodinger lattice with a saturable nonlinearity are reported for finite lattices we find two different standing wave like solutions and for an infinite lattice we find a localized soliton like solution the existence requirements and stability of these solutions are discussed and we find that our solutions are linearly stable in most cases we also show that the effective peierls nabarro barrier potential is nonzero thereby indicating that this discrete model is quite likely nonintegrable

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

discrete klein gordon models with static kinks free of the peierls nabarro potential for the nonlinear klein gordon type models we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice these discrete models are therefore free of the peierls nabarro potential previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method a relevant physical consequence of our findings is the existence of a wide class of discrete klein gordon models where slow kinks practically do not experience the action of the peierls nabarro potential such kinks are not trapped by the lattice and they can be accelerated by even weak external fields