how singular functions define distributions following dirac schwartz and others distributions are well understood and widely used in physics as generalized functions however a function with a nonintegrable singularity does not define a distribution automatically or unambiguously we review a variety of ways in which such distributions can be defined sometimes with inequivalent results or results containing arbitrary constants we give special attention to the function cosech2 x and its semiclassical scaling limit which have recently attracted some attention in statistical mechanics

langevin approach for the dynamics of the contact process on annealed scale free networks we study the dynamics of the contact process one of the simplest nonequilibrium stochastic processes taking place on a scale free network we consider the network topology as annealed i e all links are rewired at each microscopic time step so that no dynamical correlation can build up this is a practical implementation of the absence of correlations assumed by mean field approaches we present a detailed analysis of the contact process in terms of a langevin equation including explicitly the effects of stochastic fluctuations in the number of particles in finite networks this allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs the fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has a hard upper bound when the upper bound is soft the presence of outliers with huge connectivity perturbs the picture even more inducing an apparent shift of the critical point in light of these findings recent theoretical and numerical results in the literature are critically reviewed

on solving wave equations on fixed bounded intervals involving robin boundary conditions with time dependent coefficients abstract in this paper it is shown how characteristic coordinates or equivalently how the well known formula of d alembert can be used to solve initial boundary value problems for wave equations on fixed bounded intervals involving robin type of boundary conditions with time dependent coefficients a robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space derivative on a boundary of the interval analytical methods such as the method of separation of variables sov or the laplace transform method are not applicable to those types of problems the obtained analytical results by applying the proposed method are in complete agreement with those obtained by using the numerical finite difference method for problems with time independent coefficients in the robin boundary condition s the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables or by the finite difference method

distribution of infected mass in disease spreading in scale free networks we use scale free networks to study properties of the infected mass m of the network during a spreading process as a function of the infection probability q and the structural scaling exponent we use the standard sir model and investigate in detail the distribution of m we find that for dense networks this function is bimodal while for sparse networks it is a smoothly decreasing function with the distinction between the two being a function of q we thus recover the full crossover transition from one case to the other this has a result that on the same network a disease may die out immediately or persist for a considerable time depending on the initial point where it was originated thus we show that the disease evolution is significantly influenced by the structure of the underlying population

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

analytically solvable model of probabilistic network dynamics we present a simple model of network dynamics that can be solved analytically for fully connected networks we obtain the dynamics of response of the system to perturbations the analytical solution is an excellent approximation for random networks a comparison with the scale free network though qualitatively similar shows the effect of distinct topology

spreading and shortest paths in systems with sparse long range connections spreading according to simple rules e g of fire or diseases and shortest path distances are studied on d dimensional systems with a small density p per site of long range connections small world lattices the volume v t covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t from this the average shortest path distance l r can be calculated as a function of euclidean distance r it is found that l r approximately r for r r c the characteristic length r c which governs the behavior of shortest path lengths diverges logarithmically with l for all p 0

dynamical analysis of a iwsr rumor spreading model with considering the self growth mechanism and indiscernible degree abstract propagation of rumor as one of the most important aspects of spreading dynamics on networks which is often accompanied by the self growth of individuals and indiscernible degree of events and the analysis of this phenomenon can help us to better understand the information dissemination in the social network here with the wiseman supplemented to the classical ignorant spreader stifler i s r rumor spreading model a modified ignorant wiseman spreader stifler i w s r model is established with self growth mechanism by considering indiscernible degree of events by mean field equations of i w s r model the spreading threshold c is calculated to describe the theory of steady state dynamics of the rumor spreading numerical simulation is performed to assess the influence of self growth of individual and indiscernible degree of events on dynamic spreading process the results show that when the spreading rate is smaller than the threshold c the rumor will not spread in the population the stronger the self growth ability of the individual is the smaller the range of rumor will be the higher the indiscernible degree of events is the greater influence of the rumor these obtained findings in the present study could not only elaborate our understandings of spreading dynamics in network but also provide us with an insight to develop effective strategies of inhibiting rumor propagation

a connection between a system of random walks and rumor transmission abstract we establish a relationship between the phenomenon of rumor transmission on a population and a probabilistic model of interacting particles on the complete graph more precisely we consider variations of the maki thompson epidemic model and the frog model of random walks which were introduced in the scientific literature independently and in different contexts we analyze the markov chains which describe these models and show a coupling between them our connection shows how the propagation of a rumor in a closed homogeneously mixing population can be described by a system of random walks on the complete graph additionally we discuss further applications of the random walk model which are relevant to the modeling of different biological dynamics

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

epidemics and percolation in small world networks we study some simple models of disease transmission on small world networks in which either the probability of infection by a disease or the probability of its transmission is varied or both the resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network and we give exact solutions for the position of this threshold in a variety of cases we confirm our analytic results by numerical simulation

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

shir competitive information diffusion model for online social media abstract in online social media opinion divergences and differentiations generally exist as a result of individuals extensive participation and personalization in this paper a susceptible hesitated infected removed shir model is proposed to study the dynamics of competitive dual information diffusion the proposed model extends the classical sir model by adding hesitators as a neutralized state of dual information competition it is both hesitators and stable spreaders that facilitate information dissemination researching on the impacts of diffusion parameters it is found that the final density of stiflers increases monotonically as infection rate increases and removal rate decreases and the advantage information with larger stable transition rate takes control of whole influence of dual information the density of disadvantage information spreaders slightly grows with the increase of its stable transition rate while whole spreaders of dual information and the relaxation time remain almost unchanged moreover simulations imply that the final result of competition is closely related to the ratio of stable transition rates of dual information if the stable transition rates of dual information are nearly the same a slightly reduction of the smaller one brings out a significant disadvantage in its propagation coverage additionally the relationship of the ratio of final stiflers versus the ratio of stable transition rates presents power characteristic

critical behavior of propagation on small world networks we report numerical evidence that an epidemiclike model which can be interpreted as the propagation of a rumor exhibits critical behavior at a finite randomness of the underlying small world network the transition occurs between a regime where the rumor dies in a small neighborhood of its origin and a regime where it spreads over a finite fraction of the whole population critical exponents are evaluated through finite size scaling analysis and the dependence of the critical randomness with the network connectivity is studied the behavior of this system as a function of the small network randomness bears noticeable similarities with an epidemiological model reported recently m kuperman and g abramson phys rev lett 86 2909 2001 in spite of substantial differences in the respective dynamical rules

complex dynamic behavior of a rumor propagation model with spatial temporal diffusion terms a spatial temporal rumor propagation model with discrete and non local delays is regarded stability and hopf bifurcation are proved in a strict mathematical way sensitivity analysis is applied to search active parameters hopf bifurcation causes an oscillation phenomenon in this model numerical simulations provide a new insight into rumor spreading in online social networks rumor propagation as a typical form of social communication in online social networks has had a significant negative impact on a harmonious and stable society with the rapid development of mobile communication equipments traditional rumor propagation models which depend on ordinary differential equations ode may not be suitable for describing rumor propagation in an online social network in this paper based on reaction diffusion equations we propose a novel epidemic like model with both discrete and nonlocal delays for investigating the spatial temporal dynamics of rumor propagation by analyzing the corresponding characteristic equations of this model the local stability conditions of a boundary equilibrium point and a positive equilibrium point are established by applying the linear approximation method of nonlinear systems sufficient conditions are derived for the existence of hopf bifurcation at the above two kinds of equilibrium points moreover a sensitivity analysis method based on the density of spreading users is proposed and then in theoretical and experimental aspect we identify some sensitive parameters in the process of rumor propagation finally numerical simulations are performed to illustrate the theoretical results

the maki thompson rumor model on infinite cayley trees in this paper we study the maki thompson rumor model on infinite cayley trees the basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals ignorants spreaders and stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate one at the same rate a spreader becomes a stifler after a contact with other nearest neighbor spreaders or stiflers in this work we study this model on infinite cayley trees which is formulated as a continuous times markov chain and we extend our analysis to the generalization in which each spreader ceases to propagate the rumor right after being involved in a given number of stifling experiences we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

systematics of the relationship between vacuum energy calculations and heat kernel coefficients casimir energy is a nonlocal effect its magnitude cannot be deduced from heat kernel expansions even those including the integrated boundary terms on the other hand it is known that the divergent terms in the regularized but not yet renormalized total vacuum energy are associated with the heat kernel coefficients here a recent study of the relations among the eigenvalue density the heat kernel and the integral kernel of the operator e t h is exploited to characterize this association completely various previously isolated observations about the structure of the regularized energy emerge naturally for over 20 years controversies have persisted stemming from the fact that certain presumably physically meaningful terms in the renormalized vacuum energy density in the interior of a cavity become singular at the boundary and correlate to certain divergent terms in the regularized total energy the point of view of the present paper promises to help resolve these issues

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

on the strong unique continuation of electromagnetic schrodinger equations with random coefficients abstract stochastic partial differential equations spdes with random coefficients are useful mathematical models for the quantification of perturbations in lots of complex engineering and physical applications this paper mainly addresses the strong unique continuation property for the electromagnetic schrodinger operator with random coefficients by the method proposed by n garofalo and f h lin 10 11 appropriate multipliers with physical backgrounds have been introduced to prove a priori estimates moreover its application in an exact controllability problem has been shown in which case the boundary value determines the interior value completely

a rumor transmission model with incubation in social networks abstract in this paper we propose a rumor transmission model with incubation period and constant recruitment in social networks by carrying out an analysis of the model we study the stability of rumor free equilibrium and come to the local stable condition of the rumor equilibrium we use the geometric approach for ordinary differential equations for showing the global stability of the rumor equilibrium and when 0 1 the new model occurs a transcritical bifurcation furthermore numerical simulations are used to support the analysis at last some conclusions are presented

a rumor spreading model with variable forgetting rate abstract a rumor spreading model with the consideration of forgetting rate changing over time is examined in small world networks the mean field equations are derived to describe the dynamics of rumor spreading in small world networks further numerical solutions are conducted on livejournal an online social blogging platform to better understand the performance of the model results show that the forgetting rate has a significant impact on the final size of rumor spreading the larger the initial forgetting rate or the faster the forgetting speed the smaller the final size of the rumor spreading numerical solutions also show that the final size of rumor spreading is much larger under a variable forgetting rate compared to that under a constant forgetting rate

on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

rock scissors paper game on regular small world networks the spatial rock scissors paper game or cyclic lotka volterra system is extended to study how the spatiotemporal patterns are affected by the rewired host lattice providing uniform number of neighbours degree at each site on the square lattice this system exhibits a self organizing pattern with equal concentration of the competing strategies species if the quenched background is constructed by substituting random links for the nearest neighbour bonds of a square lattice then a limit cycle occurs when the portion of random links exceeds a threshold value this transition can also be observed if the standard link is replaced temporarily by a random one with a probability p at each step of iteration above a second threshold value of p the amplitude of global oscillation increases with time and finally the system reaches one of the homogeneous absorbing states in this case the results of monte carlo simulations are compared with the predictions of the dynamical cluster technique evaluating all the configuration probabilities on one two four and six site clusters

phase transition for the maki thompson rumour model on a small world network we consider the maki thompson model for the stochastic propagation of a rumour within a population in this model the population is made up of spreaders ignorants and stiflers any spreader attempts to pass the rumour to the other individuals via pair wise interactions and in case the other individual is an ignorant it becomes a spreader while in the other two cases the initiating spreader turns into a stifler in a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants we extend the original hypothesis of homogenously mixed population by allowing for a small world network embedding the model in such a way that interactions occur only between nearest neighbours this structure is realized starting from a k regular ring and by inserting in the average c additional links in such a way that k and c are tuneable parameters for the population architecture we prove that this system exhibits a transition between regimes of localization where the final number of stiflers is at most logarithmic in the population size and propagation where the final number of stiflers grows algebraically with the population size at a finite value of the network parameter c a quantitative estimate for the critical value of c is obtained via extensive numerical simulations

probabilistic models for stochastic elliptic partial differential equations mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively yet additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities referred to as physical requirements have not been examined systematically it is shown that current models for random coefficients constructed solely by mathematical considerations can violate physical constraints and consequently be of limited practical use we develop alternative models for the random coefficients of stochastic differential equations that satisfy both mathematical and physical constraints theoretical arguments are presented to show potential limitations of current models and establish properties of the models developed in this study numerical examples are used to illustrate the construction of the proposed models assess the performance of these models and demonstrate the sensitivity of the solutions of stochastic differential equations to probabilistic characteristics of their random coefficients

risk averse structural topology optimization under random fields using stochastic expansion methods abstract this work proposes a level set based approach for solving risk averse structural topology optimization problems considering random field loading and material uncertainty the use of random fields increases the dimensionality of the stochastic domain which poses several computational challenges related to the minimization of the excess probability as a measure of risk awareness this problem is addressed both from the theoretical and numerical viewpoints first an existence result under a typical geometrical constraint on the set of admissible shapes is proved second a level set continuous approach to find the numerical solution of the problem is proposed since the considered cost functional has a discontinuous integrand the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic polynomial chaos pc approach with a monte carlo mc sampling method for uncertainty propagation furthermore to address the increment of dimensionality induced by the random field an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the pc coefficients a key point is that the non intrusive nature of such an approach facilitates the use of high performance computing hpc to alleviate the computational burden of the problem several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal

blackmail propagation on small world networks abstract the dynamics of the blackmail propagation model based on small world networks is investigated it is found that for a given transmitting probability the dynamical behavior of blackmail propagation transits from linear growth type to logistical growth one with the network randomness p increases the transition takes place at the critical network randomness p c 1 n where n is the total number of nodes in the network for a given network randomness p the dynamical behavior of blackmail propagation transits from exponential decrease type to logistical growth one with the transmitting probability increases the transition occurs at the critical transmitting probability c 1 k where k is the average number of the nearest neighbors the present work will be useful for understanding computer virus epidemics and other spreading phenomena on communication and social networks

morphology of technological levels in an innovation propagation model we study the dynamical properties of the propagation of innovation on a two dimensional lattice random network scale free network and cayley tree in order to investigate the diversity of technological level we study the scaling property of width w n t which represents the root mean square of the technological level of agents here n is the total number of agents from the numerical simulations we find that the steady state value of w n t w sat n scales as w sat n n 1 2 when the system is in a flat ordered phase for d 2 in the flat ordered phase most of the agents have the same technological level on the other hand when the system is in a smooth disordered phase the value of w sat n does not depend on n these behaviors are completely different from those observed on a one dimensional 1d lattice by considering the effect of the underlying topology on the propagation dynamics for d 2 we also provide a mean field analysis for w sat n which agrees very well with the observed behaviors of w sat n this directly shows that the morphological properties in order disorder transition on a 1d lattice is completely different from that on higher dimensions it also provides an evidence that the upper critical dimension for the roughening transition of the propagation of innovation is d u 2

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

casimir forces for moving boundaries with robin conditions we consider a massless scalar field in 1 1 dimensions that satisfies a robin boundary condition at a non relativistic moving boundary using the perturbative approach introduced by ford and vilenkin we compute the total force on the moving boundary in contrast to what happens for the dirichlet and neumann boundary conditions in addition to a dissipative part the force acquires also a dispersive one further we also show that with an appropriate choice for the mechanical frequency of the moving boundary it is possible to turn off the vacuum dissipation almost completely

branching processes in the annni model the mean field equations of the simple cubic annni model are studied on finite lattices the results are consistent with the sequence of distinct commensurate phases 2k 13 k 1 2 3 springing from the multiphase point found using low temperature series expansions moreover evidence for new structure combination branching processes is presented which generate phases of type 2l3 m 2l 13 n or 23l m 23l 1 n where l m and n are integers

a method for solving stochastic equations by reduced order models and local approximations a method is proposed for solving equations with random entries referred to as stochastic equations ses the method is based on two recent developments the first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments the second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector referred to as stochastic reduced order model srom and uses it to construct a srom for the solution of this equation the proposed method is a direct extension of these two methods it uses sroms to select expansion points rather than selecting these points by geometrical considerations and represents the solution by linear and or higher order local approximations the implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic differential equations the method is conceptually simple non intrusive efficient relative to classical monte carlo simulation accurate and guaranteed to converge to the exact solution

the dirichlet to robin transform a simple transformation converts a function satisfying a dirichlet boundary condition to a function satisfying a robin generalized neumann condition under sufficiently restrictive conditions one can arrange that the old function is the solution of a solvable differential equation and the new one solves an interesting differential equation in the simplest cases this observation enables the exact construction of the green functions for the wave heat and schrodinger problems with a robin boundary condition the resulting physical picture is that the field can exchange energy with the boundary and a delayed reflection from the boundary results in more general situations the method allows at least approximate and local construction of the appropriate reflected solutions and hence a classical path analysis of the green functions and the associated spectral information by this method we solve the wave equation on an interval with one robin and one dirichlet endpoint and thence derive several variants of a gutzwiller type expansion for the density of eigenvalues the variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighbourhood of zero in the frequency variable

lamellar pattern formation in small world media abstract numerical and analytical techniques are used to investigate the effects of quenched disorder of small world networks on the phase ordering dynamics of lamellar patterns as modeled by the swift hohenberg equation morphologies for small and large values of the network randomness are quite different it is found that addition of shortcuts to an underlying regular lattice makes the growth of domains evolving from random initial conditions much slower at late times as the randomness increases the evolution is eventually frozen

finite dimensional models for random functions abstract truncated karhunen loeve kl representations are used to construct finite dimensional fd models for non gaussian functions with finite variances the second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation polynomial chaos and translated polynomial chaos models referred to as t pc and pct models following theoretical considerations on kl representations and t pc and pct models three numerical examples are presented to illustrate the implementation and performance of these models the pct models inherit the desirable features of both t and pc models it approximates accurately all quantities of interest considered in these examples

efficient iterative solvers for stochastic galerkin discretizations of log transformed random diffusion problems we consider the numerical solution of a steady state diffusion problem where the diffusion coefficient is the exponent of a random field the standard stochastic galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it we consider a reformulated version of this problem as a stochastic convection diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated gaussian random variables the associated galerkin matrix is nonsymmetric but sparse and allows for fast matrix vector multiplications with optimal complexity we construct and analyze two block diagonal preconditioners for this galerkin matrix for use with krylov subspace methods such as the generalized minimal residual method we test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection diffusion formulations

bayesian identification and model comparison for random property fields derived from material microstructure abstract this paper provides a bayesian framework for determining the spatial variability of the apparent material properties of two phase composites bayesian analysis is applied to learn the parameters of the random fields that characterize the mesoscale material properties using computer simulated images of the material microstructure the information from such images is used to define the likelihood function of the random field parameters given the homogenized microscale data the uncertainty in the parameter estimates is quantified through determining their full posterior distribution instead of a point estimate by application of an adaptive sampling algorithm in addition the bayesian approach allows for assessing the plausibility of different correlation models belonging to the matern class through computing their respective marginal likelihoods for various choices of the smoothness parameter the framework is applied to a composite material with large inclusion matrix stiffness ratio and useful conclusions are derived with regard to the most appropriate correlation model and its parameters

optimization as a result of the interplay between dynamics and structure abstract in this work we study the interplay between the dynamics of a model of diffusion governed by a mechanism of imitation and its underlying structure the dynamics of the model can be quantified by a macroscopic observable which permits the characterization of an optimal regime we show that dynamics and underlying network cannot be considered as separated ingredients in order to achieve an optimal behavior

stochastic collocation with kernel density estimation abstract the stochastic collocation method has recently received much attention for solving partial differential equations posed with uncertainty i e where coefficients in the differential operator boundary terms or right hand sides are random fields recent work has led to the formulation of an adaptive collocation method that is capable of accurately approximating functions with discontinuities and steep gradients these methods however usually depend on an assumption that the random variables involved in expressing the uncertainty are independent with marginal probability distributions that are known explicitly in this work we combine the adaptive collocation technique with kernel density estimation to approximate the statistics of the solution when the joint distribution of the random variables is unknown

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

distributions in spaces with thick points abstract we present a theory of distributions in a space with a thick point in dimensions n 2 generalizing the theory of thick distributions in one variable given in estrada and fulling 2007 8 the higher dimensional situation is quite different from the one dimensional case we construct topological vector spaces of thick test functions and by duality spaces of thick distributions we study several operations on these distributions both algebraic and analytic particularly partial differentiation we introduce the notion of thick delta functions at the special point not only of order 0 but of any integral order we also consider the thick distributions constructed by the hadamard finite part procedure we give formulas for the derivatives of important thick distributions including the finite part of power functions we obtain the new formula 2 p f r 1 x i x j 3 x i x j i j r 2 p f r 5 4 i j 4 n i n j for the second order thick derivatives of the finite part of r 1 in r 3 where is a thick delta of order 0

dynamics of three state excitable units on poisson vs power law random networks abstract the influence of the network topology on the dynamics of systems of coupled excitable units is studied numerically and demonstrates a lower dynamical variability for power law networks than for poisson ones this effect which reflects a robust collective excitable behavior is however lower than that observed for diffusion processes or network robustness instead the presence and number of triangles and larger loops in the networks appears as a parameter with strong influence on the considered dynamics

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

the susceptible infected recovered model on a euclidean network we consider a susceptible infected recovered sir epidemic model on a euclidean network in one dimension in which nodes at a distance l are connected with probability p l l in addition to nearest neighbors the topology of the network changes as is varied and its effect on the sir model is studied r t the recovered fraction of population up to time t and the total duration of the epidemic are calculated for different values of the infection probabilities q and a threshold behavior is observed for all up to 2 0 above the threshold value q qc the saturation value rsat attains a finite value both rsat and show scaling behavior in a finite system of size n and qc is constant for 0 qc the results are compared to static percolation phenomena and also to mean field results for finite systems discussions on the properties of the euclidean network are made in the light of the present results

exponential extinction of a stochastic predator prey model with allee effect abstract in this paper we study a stochastic predator prey model with beddington deangelis functional response and allee effect whose coefficients are dependent on expectations we prove that there is a unique global positive solution to the system with the positive initial value moreover sufficient conditions for exponential extinction are established

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

the hybrid spectral problem and robin boundary conditions the hybrid spectral problem where the field satisfies dirichlet conditions d on part of the boundary of the relevant domain and neumann n on the remainder is discussed in simple terms a conjecture for the c1 coefficient is presented and the conformal determinant on a 2 disc where the d and n regions are semi circles is derived comments on higher coefficients are made a separable second order hemisphere hybrid problem is introduced that involves robin boundary conditions and leads to logarithmic terms in the heat kernel expansion which are evaluated explicitly

mean field solution of the small world network model the small world network model is a simple model of the structure of social networks which possesses characteristics of both regular lattices and random graphs the model consists of a one dimensional lattice with a low density of shortcuts added between randomly selected pairs of points these shortcuts greatly reduce the typical path length between any two points on the lattice we present a mean field solution for the average path length and for the distribution of path lengths in the model this solution is exact in the limit of large system size and either a large or small number of shortcuts

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

casimir effect for the scalar field under robin boundary conditions a functional integral approach in this work we show how to define the action of a scalar field such that the robin boundary condition is implemented dynamically i e as a consequence of the stationary action principle we discuss the quantization of that system via functional integration using this formalism we derive an expression for the casimir energy of a massless scalar field under robin boundary conditions on a pair of parallel plates characterized by constants c1 and c2 some special cases are discussed in particular we show that for some values of c1 and c2 the casimir energy as a function of the distance between the plates presents a minimum we also discuss the renormalization at one loop order of the two point green function in the 4 theory subject to the robin boundary condition on a plate

parametric models for samples of random functions a new class of parametric models referred to as sample parametric models is developed for random elements that match sample rather than the first two moments and or other global properties of these elements the models can be used to characterize e g material properties at small scale in which case their samples represent microstructures of material specimens selected at random from a population the samples of the proposed models are elements of finite dimensional vector spaces spanned by samples eigenfunctions of karhunen loeve kl representations or modes of singular value decompositions svds the implementation of sample parametric models requires knowledge of the probability laws of target random elements numerical examples including stochastic processes and random fields are used to demonstrate the construction of sample parametric models assess their accuracy and illustrate how these models can be used to solve efficiently stochastic equations

majority vote model on a dynamic small world network abstract dynamic small world networks combine short range interactions within a fixed neighborhood with stochastic long range interactions the probability of a long range link occurring instead of a short range one is a measure of the mobility of a population here the critical properties of the majority vote model with noise on a two dimensional dynamic small world lattice are investigated via monte carlo simulation and finite size scaling analyses we construct the order disorder phase diagram and find the critical exponents associated with the continuous phase transition findings are consistent with previous results indicating that a model s transitions on static and dynamic small world networks are in the same universality class

self organized evolution in a socioeconomic environment we propose a general scenario to analyze technological changes in socio economic environments we illustrate the ideas with a model that incorporating the main trends is simple enough to extract analytical results and at the same time sufficiently complex to display a rich dynamic behavior our study shows that there exists a macroscopic observable that is maximized in a regime where the system is critical in the sense that the distribution of events follow power laws computer simulations show that in addition the system always self organizes to achieve the optimal performance in the stationary state

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

theory of rumour spreading in complex social networks we introduce a general stochastic model for the spread of rumours and derive mean field equations that describe the dynamics of the model on complex social networks in particular those mediated by the internet we use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks random graphs uncorrelated scale free networks and scale free networks with assortative degree correlations we show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system in the case of scale free networks on the other hand this threshold becomes vanishingly small in the limit of infinite system size we find that the initial rate at which a rumour spreads is much higher in scale free networks than in random graphs and that the rate at which the spreading proceeds on scale free networks is further increased when assortative degree correlations are introduced the impact of degree correlations on the final fraction of nodes that ever hears a rumour however depends on the interplay between network topology and the rumour spreading rate our results show that scale free social networks are prone to the spreading of rumours just as they are to the spreading of infections they are relevant to the spreading dynamics of chain emails viral advertising and large scale information dissemination algorithms on the internet

dynamic analysis of a stochastic rumor propagation model abstract the rapid development of the internet especially the emergence of the social networks leads rumor propagation into a new media era in this paper we are concerned with a stochastic rumor propagation model sufficient conditions for extinction and persistence in the mean of the rumor are established the threshold between persistence in the mean and extinction of the rumor is obtained compared with the corresponding deterministic model the threshold affected by the white noise is smaller than the basic reproduction number r 0 of the deterministic system

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

stability analysis of i2s2r rumor spreading model in complex networks abstract in this paper we propose i 2 s 2 r rumor spreading model with general incidence function in homogeneous networks and i 2 s 2 r rumor spreading model in heterogeneous network in the first model we establish a dynamic model for rumor propagation called i 2 s 2 r in which we consider two general incidence functions then the basic reproduction number the locally stability and the global stability of rumor free equilibrium are discussed in the second model we mainly consider another network structure i e heterogeneous network thus we propose a new dynamic model for rumor spreading by deriving the mean field equation the basic reproduction number is obtained moreover the global stability of rumor free equilibrium is established by comparison principle by using a monotone iterative algorithm the global attractivity of the positive equilibrium is proved finally we obtain some important conclusions with numerical simulations

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

semiclassical quantization of billiards with mixed boundary conditions the semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived the parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards it is also used to check in detail some recently proposed parametric spectral statistics the methods are illustrated in the analysis of the spectrum of the sinai billiard and its parametric dependence on the boundary condition on the dispersing arc

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

epidemic threshold for the sirs model on the networks abstract we study the phase transition from the persistence phase to the extinction phase for the sirs susceptible infected refractory susceptible model of diseases spreading on the networks we derive an analytical expression for the probability of the descendants to reinfect their ancestors we find that in the case of the recovery time r is larger than the infection time i the infection will flow directionally from the ancestors to descendants however the descendants will be unable to reinfect their ancestors during the time of their illness this behavior leads us to deduce that if the infection rate is high enough so that any infected node on the network infects all its neighbors during its infection time then the sirs model on the network evolves to extinction state where all the nodes on the network become susceptible moreover we assert that in order to the disease spreads frequently throughout the nodes of the networks the loops on the network are necessary which means the clustering coefficient will play an important role in this model hence unlike the other models such as sis model and sir model the sirs model has two critical thresholds which separate the persistence phase from the extinction phase when i r i e for fixed values of i and r there are two critical points for infection rate 1 and 2 where epidemic persists in between of those two points we confirm those results numerically with the simulation of a regular one dimensional sirs system and small world network