thresholds for epidemic spreading in networks we study the threshold of epidemic models in quenched networks with degree distribution given by a power law forthesusceptible infected susceptiblemodeltheactivitythresholdc vanishesinthelargesize limitonanynetworkwhosemaximumdegree kmax divergeswiththesystemsize atoddswithheterogeneous mean field hmf theory the vanishing of the threshold has nothing to do with the scale free nature of the networkbutstemsinsteadfromthelargesthubinthesystembeingactiveforanyspreadingrate 1 ffiffiffiffiffiffiffiffiffimax p and playing the role of a self sustained source that spreads the infection to the rest of the system the susceptible infected removedmodel displays insteadagreement with hmftheory and afinite threshold for scale rich networks we conjecture that on quenched scale rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state

evaluation of an analytic approximate formula for the time varying sis prevalence in different networks abstract one of the most important quantities of the exact markovian sis epidemic process is the time dependent prevalence which is the average fraction of infected nodes unfortunately the markovian sis epidemic model features an exponentially increasing computational complexity with growing network size n in this paper we evaluate a recently proposed analytic approximate prevalence function introduced in van mieghem 2016 we compare the approximate function with the n intertwined mean field approximation nimfa and with simulation of the markovian sis epidemic process the results show that the new analytic prevalence function is comparable with other approximate methods

approximate formula and bounds for the time varying susceptible infected susceptible prevalence in networks based on a recent exact differential equation the time dependence of the sis prevalence the average fraction of infected nodes in any graph is first studied and then upper and lower bounded by an explicit analytic function of time that new approximate tanh formula obeys a riccati differential equation and bears resemblance to the classical expression in epidemiology of kermack and mckendrick proc r soc london a 115 700 1927 1364 502110 1098 rspa 1927 0118 but enhanced with graph specific properties such as the algebraic connectivity the second smallest eigenvalue of the laplacian of the graph we further revisit the challenge of finding tight upper bounds for the sis and sir epidemic threshold for all graphs we propose two new upper bounds and show the importance of the variance of the number of infected nodes finally a formula for the epidemic threshold in the cycle or ring graph is presented

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

finite difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid we describe the application of the rotated staggered grid rsg finite difference technique to the wave equations for anisotropic and viscoelastic media the rsg uses rotated finite difference operators leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position this can be advantageous for modeling wave propagation in anisotropic media or complex media including high contrast discontinuities because no averaging of elastic moduli is needed the rsg can be applied both to displacement stress and to velocity stress finite difference fd schemes whereby the latter are commonly used to model viscoelastic wave propagation with a von neumann style anlysis we estimate the dispersion error of the rsg scheme in general anisotropic media in three different simulation examples all based on previously published problems we demonstrate the application and the accuracy of the proposed numerical approach

prevalence expansion in nimfa abstract the n intertwined mean field approximation nimfa is a reasonably accurate approximation of the exact sis epidemic process on a network the average fraction of infected nodes in the nimfa steady state also called the steady state prevalence in terms of the effective infection rate can be expanded into a power series around the nimfa epidemic threshold in this paper we investigate the convergence of the steady state prevalence taylor expansion we determine the radius of convergence in some special types of graphs we also show that the radius of convergence of the steady state prevalence expansion depends upon the network topology in particular the average degree of the network and the spectral gap of the adjacency matrix play a role

a discontinuous galerkin method for wave propagation in orthotropic poroelastic media with memory terms abstract poroelastic materials play an important role in biomechanical and geophysical research in this paper we investigate wave propagation in orthotropic poroelastic media by studying the time domain poroelastic wave equations both the low frequency biot s lf biot equations and the biot johnson koplik dashen biot jkd model are considered in lf biot equations the dissipation terms are proportional to the relative velocity between the fluid and the solid by a constant contrast to this the dissipation terms in the biot jkd model are in the form of time convolution memory as a result of the frequency dependence of fluid solid interaction at the underlying microscopic scale in the frequency domain the dynamic tortuosity and permeability described by darcy s law are two crucial factors in this problem and highly linked to the viscous force in the biot model the key difficulty is to handle the viscous term when the pore fluid is viscous in the biot jkd model the convolution operator involves order 1 2 shifted fractional derivatives in the time domain which is challenging to discretize in this work a new method of the multipoint pade or rational approximation for stieltjes function is applied to approximate the jkd dynamic tortuosity and then an augmented system of biot jkd model is obtained where the kernel of the memory term is replaced by the finite auxiliary variables satisfying a local system of ordinary differential equations the runge kutta discontinuous galerkin rkdg method with the un splitting method is used to compute the numerical solution and numerical examples are presented to demonstrate the high order accuracy and stability of the method compared with the existing approaches for solving the biot jkd equations the augmented system presented here require neither the storage of solution history nor the computation of the flux of the auxiliary variables

a pseudo spectral method for the simulation of poro elastic seismic wave propagation in 2d polar coordinates using domain decomposition we present a novel numerical approach for the comprehensive flexible and accurate simulation of poro elastic wave propagation in 2d polar coordinates an important application of this method and its extensions will be the modeling of complex seismic wave phenomena in fluid filled boreholes which represents a major and as of yet largely unresolved computational problem in exploration geophysics in view of this we consider a numerical mesh which can be arbitrarily heterogeneous consisting of two or more concentric rings representing the fluid in the center and the surrounding porous medium the spatial discretization is based on a chebyshev expansion in the radial direction and a fourier expansion in the azimuthal direction and a runge kutta integration scheme for the time evolution a domain decomposition method is used to match the fluid solid boundary conditions based on the method of characteristics this multi domain approach allows for significant reductions of the number of grid points in the azimuthal direction for the inner grid domain and thus for corresponding increases of the time step and enhancements of computational efficiency the viability and accuracy of the proposed method has been rigorously tested and verified through comparisons with analytical solutions as well as with the results obtained with a corresponding previously published and independently benchmarked solution for 2d cartesian coordinates finally the proposed numerical solution also satisfies the reciprocity theorem which indicates that the inherent singularity associated with the origin of the polar coordinate system is adequately handled

second order mean field susceptible infected susceptible epidemic threshold given the adjacency matrix a of a network we present a second order mean field expansion that improves on the first order n intertwined susceptible infected susceptible sis epidemic model unexpectedly we found that in contrast to first order second order mean field theory is not always possible the network size n should be large enough under the assumption of large n we show that the crucial and characterizing quantity the sis epidemic threshold ensuremath tau c obeys an eigenvalue equation more complex than the one in the first order n intertwined model however the resulting epidemic threshold is more accurate ensuremath tau c 2 ensuremath tau c 1 o frac ensuremath tau c 1 n where the first order epidemic threshold is ensuremath tau c 1 frac 1 ensuremath lambda 1 a and where ensuremath lambda 1 a is the spectral radius of the adjacency matrix a

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media it is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs however the computational cost of conventional numerical modeling methods such as finite difference method and finite element method becomes prohibitively expensive when applied to very large models we propose a generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element the application of multiscale basis functions can capture the fine scale medium property variations and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite element method for wave equation while restricting the error to low values we formulate the continuous galerkin and discontinuous galerkin formulation of the multiscale method both of which have pros and cons applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system

structure of the rational monoid algebra for boolean matrices of order 3 abstract we use the computer algebra system maple to study the 512 dimensional associative algebra q b 3 the rational monoid algebra of 3 3 boolean matrices using the lll algorithm for lattice basis reduction we obtain a basis for the radical in bijection with the 42 non regular elements of b 3 the center of the 470 dimensional semisimple quotient has dimension 14 we use a splitting algorithm to find a basis of orthogonal primitive idempotents we show that the semisimple quotient is the direct sum of simple two sided ideals isomorphic to matrix algebras m d q for d 1 1 1 2 3 3 3 3 6 6 7 9 9 12 we construct the irreducible representations of b 3 over q by calculating the representation matrices for a minimal set of generators

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

susceptible infected susceptible epidemics on the complete graph and the star graph exact analysis since mean field approximations for susceptible infected susceptible sis epidemics do not always predict the correct scaling of the epidemic threshold of the sis metastable regime we propose two novel approaches a an sis generalized model and b a modified sis model that prevents the epidemic from dying out i e without the complicating absorbing sis state both adaptations of the sis model feature a precisely defined steady state that corresponds to the sis metastable state and allow an exact analysis in the complete and star graph consisting of a central node and n leaves the n intertwined mean field approximation nimfa is shown to be nearly exact for the complete graph but less accurate to predict the correct scaling of the epidemic threshold c in the star graph which is found as c c 1 where 1 2logn 3 2loglogn and where c 1 1 n c is the first order epidemic threshold for the star in nimfa and equal to the inverse of the spectral radius of the star s adjacency matrix

primes in the semigroup of boolean matrices abstract let a b c be n n matrices of zeros and ones using boolean addition and multiplication we say that a is prime if it is not a permutation matrix and if a bc implies that b or c must be a permutation matrix conditions sufficient for a matrix to be prime are provided and a characterization of primes in terms of a nation of rank is given finally an order property of primes is used to obtain a result on prime factors

finite element methods for the simulation of waves in composite saturated poroviscoelastic media this work presents and analyzes a collection of finite element procedures for the simulation of wave propagation in a porous medium composed of two weakly coupled solids saturated by a single phase fluid the equations of motion formulated in the space frequency domain include dissipation due to viscous interaction between the fluid and solid phases with a correction factor in the high frequency range and intrinsic anelasticity of the solids modeled using linear viscoelasticity this formulation leads to the solution of a helmholtz type boundary value problem for each temporal frequency for the spatial discretization nonconforming finite element spaces are employed for the solid phases while for the fluid phase the vector part of the raviart thomas nedelec mixed finite element space is used optimal a priori error estimates for global standard and hybridized galerkin finite element procedures are derived an iterative nonoverlapping domain decomposition procedure is also presented and convergence results are derived numerical experiments showing the application of the numerical procedures to simulate wave propagation in partially frozen porous media are presented

splitting full matrix algebras over algebraic number fields abstract let k be a fixed algebraic number field and let a be an associative algebra over k given by structure constants such that a m n k holds for some positive integer n suppose that n is bounded then an isomorphism a m n k can be constructed by a polynomial time ff algorithm an ff algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields as a consequence we obtain a polynomial time ff algorithm to compute isomorphisms of central simple algebras of bounded degree over k

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

a discontinuous galerkin method for poroelastic wave propagation the two dimensional case abstract in this paper we consider a high order discontinuous galerkin dg method for modelling wave propagation in coupled poroelastic elastic media the upwind numerical flux is derived as an exact solution for the riemann problem including the poroelastic elastic interface attenuation mechanisms in both biot s low and high frequency regimes are considered the current implementation supports non uniform basis orders which can be used to control the numerical accuracy element by element in the numerical examples we study the convergence properties of the proposed dg scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions

epidemics in networks with nodal self infection and the epidemic threshold since the susceptible infected susceptible sis epidemic threshold is not precisely defined in spite of its practical importance the classical sis epidemic process has been generalized to the sis model where a node possesses a self infection rate in addition to a link infection rate and a curing rate the exact markov equations are derived from which the steady state can be computed the major advantage of the sis model is that its steady state is different from the absorbing or overall healthy state and approximates for a certain range of small 0 the in reality observed phase transition also called the metastable state that is characterized by the epidemic threshold the exact steady state analysis for the complete graph illustrates the effect of small and the quality of the first order mean field approximation the n intertwined model proposed earlier apart from duality principles often used in the mathematical literature we present an exact recursion relation for the markov infinitesimal generator

phase transition of the susceptible infected susceptible dynamics on time varying configuration model networks we present a degree based theoretical framework to study the susceptible infected susceptible sis dynamics on time varying rewired configuration model networks using this framework on a given degree distribution we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the sis process this analysis is suitable for the characterization of the phase transition and leads to three main contributions 1 we obtain a self consistent expression for the absorbing state threshold able to capture both collective and hub activation 2 we recover the predictions of a number of existing approaches as limiting cases of our analysis providing thereby a unifying point of view for the sis dynamics on random networks 3 we obtain bounds for the critical exponents of a number of quantities in the stationary state this allows us to reinterpret the concept of hub dominated phase transition within our framework it appears as a heterogeneous critical phenomenon observables for different degree classes have a different scaling with the infection rate this phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold

survival time of the susceptible infected susceptible infection process on a graph the survival time t is the longest time that a virus a meme or a failure can propagate in a network using the hitting time of the absorbing state in an uniformized embedded markov chain of the continuous time susceptible infected susceptible sis markov process we derive an exact expression for the average survival time e t of a virus in the complete graph kn and the star graph k1 n 1 by using the survival time instead of the average fraction of infected nodes we propose a new method to approximate the sis epidemic threshold c that at least for kn and k1 n 1 correctly scales with the number of nodes n and that is superior to the epidemic threshold 1 c 1 1 of the n intertwined mean field approximation where 1 is the spectral radius of the adjacency matrix of the graph g although this new approximation of the epidemic threshold offers a more intuitive understanding of the sis process it remains difficult to compare outbreaks in different graph types for example the survival in an arbitrary graph seems upper bounded by the complete graph and lower bounded by the star graph as a function of the normalized effective infection rate 1 c however when the average fraction of infected nodes is used as a basis for comparison the virus will survive in the star graph longer than in any other graph making the star graph the worst case graph instead of the complete graph finally in non markovian sis the distribution of the spreading attempts over the infectious period of a node influences the survival time even if the expected number of spreading attempts during an infectious period the non markovian equivalent of the effective infection rate is kept constant both early and late infection attempts lead to shorter survival times interestingly just as in markovian sis the survival times appear to be exponentially distributed regardless of the infection and curing time distributions

discontinuous galerkin methods for wave propagation in poroelastic media we have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three dimensional domains this method is a discontinuous galerkin method that achieves arbitrary high order accuracy on unstructured tetrahedral meshes for the low frequency range and the inviscid case by using biot s equations and darcy s dynamic laws we have built a scheme that can successfully model wave propagation in fluid saturated porous media when anisotropy of the pore structure is allowed zero inflow fluxes are used as absorbing boundary conditions a continuous arbitrary high order derivatives time integration is used for the high frequency inviscid case whereas a space time discontinuous scheme is applied for the low frequency case we conducted a numerical convergence test of the proposed methods we used a series of examples to quantify the quality of our numerical results comparing them to analytic solutions as well as numerical solutions obtained by other methodologies in partic

unified mean field framework for susceptible infected susceptible epidemics on networks based on graph partitioning and the isoperimetric inequality we propose an approximation framework that unifies and generalizes a number of existing mean field approximation methods for the susceptible infected susceptible sis epidemic model on complex networks we derive the framework which we call the unified mean field framework umff as a set of approximations of the exact markovian sis equations our main novelty is that we describe the mean field approximations from the perspective of the isoperimetric problem which results in bounds on the umff approximation error these new bounds provide insight in the accuracy of existing mean field methods such as the n intertwined mean field approximation and heterogeneous mean field method which are contained by umff additionally the isoperimetric inequality relates the umff approximation accuracy to the regularity notions of szemeredi s regularity lemma

non markovian infection spread dramatically alters the susceptible infected susceptible epidemic threshold in networks most studies on susceptible infected susceptible epidemics in networks implicitly assume markovian behavior the time to infect a direct neighbor is exponentially distributed much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible infected susceptible markovian epidemics on networks here we report the rather dramatic effect of a nonexponential infection time while still assuming an exponential curing time on the epidemic threshold by considering weibullean infection times with the same mean but different power exponent for three basic classes of graphs the erd s renyi random graph scale free graphs and lattices the average steady state fraction of infected nodes is simulated from which the epidemic threshold is deduced for all graph classes the epidemic threshold significantly increases with the power exponents hence real epidemics that violate the exponential or markovian assumption can behave seriously differently than anticipated based on markov theory

accuracy criterion for the mean field approximation in susceptible infected susceptible epidemics on networks mean field approximations mfas are frequently used in physics when a process such as an epidemic or a synchronization on a network is approximated by mfa a major hurdle is the determination of those graphs for which mfa is reasonably accurate here we present an accuracy criterion for markovian susceptible infected susceptible sis epidemics on any network based on the spectrum of the adjacency and sis covariance matrix we evaluate the mfa criterion for the complete and star graphs analytically and numerically for connected erd s renyi random graphs for small size n 14 the accuracy of mfa increases with average degree and with n precise simulations up to network sizes n 100 of the mfa accuracy criterion versus n for the complete graph star square lattice and path graphs lead us to conjecture that the worst mfa accuracy decreases for large n proportionally to the inverse of the spectral radius of the adjacency matrix of the graph

an optimal nearly analytic discrete weighted runge kutta discontinuous galerkin hybrid method for acoustic wavefield modeling abstractthe newly developed optimal nearly analytic discrete onad and the weighted runge kutta discontinuous galerkin wrkdg methods can effectively suppress the numerical dispersion caused by discretizing wave equations but it is difficult for onad to implement on flexible meshes whereas the wrkdg has high computational cost for wavefield simulations we have developed a new hybrid algorithm by combining the onad method with the wrkdg method in this hybrid algorithm the computational domain was split into several subdomains in which the subdomain for the onad method used regular cartesian grids whereas the subdomain for the wrkdg method used triangular grids the hybrid method was at least third order spatially accurate we have applied the proposed method to simulate the scalar wavefields for different models including a homogeneous model a rough topography model a fracture model and a cave model the numerical results found that the hybrid method can deal with complicated geometrical struc

susceptible infected susceptible model a comparison of n intertwined and heterogeneous mean field approximations we introduce the susceptible infected susceptible sis spreading model which is taken as a benchmark for the comparison between the n intertwined approximation and the pastor satorras and vespignani heterogeneous mean field hmf approximation of the sis model the n intertwined approximation the hmf approximation and the sis spreading model are compared for different graph types we focus on the epidemic threshold and the steady state fraction of infected nodes in networks with different degree distributions overall the n intertwined approximation is superior to the hmf approximation the n intertwined approximation is exactly the same as the hmf approximation in regular graphs however for some special graph types such as the square lattice graph and the path graph the two mean field approximations are both very different from the sis spreading model

green s functions in an infinite transversely isotropic saturated poroelastic medium the purpose of this article is to study green s functions for an infinite transversely isotropic saturated poroelastic medium in the first part equations of motions of such a medium are recalled according to biot s theory and completed by homogenization techniques for harmonic displacements the second part gives green s functions for such an infinite medium in the 3 d fourier transformed geometrical space in the third part a 1 d inverse analytical fourier transformation followed by an inverse hankel transformation leads to an analytical expression of green s functions for harmonic motions some numerical methods to compute final green s functions in time space are listed