bi hamiltonian structure of the kermack mckendrick model for epidemics the dynamical system proposed by kermack and mckendrick 1933 to model the spread of epidemics is shown to admit bi hamiltonian structure without any restrictions on the rate constants these two inequivalent hamiltonian structures are compatible

time independent invariants of motion for the quadratic system a hamiltonian method is developed to obtain first integrals of the form p qmu and p qmu rnu for a general system of two dimensional autonomous ordinary differential equations with quadratic terms where p q and r are linear or quadratic polynomials and mu nu are two real numbers it is found that there is an intimate relationship between the polynomials and the equilibrium points of the system which is useful for determining the existence of periodic orbits and asymptotic behaviour

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

darboux method and search of invariants for the lotka volterra and complex quadratic systems the darboux method introduces algebraic solutions quite useful to obtain invariant and first integrals of polynomial differential systems here we study the 2d lotka volterra lvs and the complex quadratic system qs using straight lines for both and conics for the lvs the conditions needed to obtain these invariants are given and a study of the phase space portrait is done

multiscale aggregating discontinuities a method for circumventing loss of material stability new methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described these methods called multiscale aggregating discontinuities are based on the concept of perforated unit cells which exclude subdomains that are unstable i e exhibit loss of material stability using this concept it is possible to compute an equivalent discontinuity at the coarser scale including both the direction of the discontinuity and the magnitude of the jump these variables are then passed to the coarse scale model along with the stress in the unit cell the discontinuity is injected at the coarser scale by the extended finite element method analysis of the procedure shows that the method is consistent in power and yields a bulk stress strain response that is stable applications of this procedure to crack growth in heterogeneous materials are given copyright 2007 john wiley sons ltd

on the hamiltonian structure of 2d ode possessing an invariant the authors discuss the hamiltonian structure with respect to the canonical bracket for a 2d dynamical system and prove that this structure can be obtained simply with a rescaling of the independent variable the rescaling will provide a hamiltonian when a time independent invariant is known the hamiltonian coinciding with the invariant applications to different systems of ode in two dimensions are presented

integrability of the 2d lotka volterra system via polynomial first integrals and polynomial inverse integrating factors we present new first integrals of the two dimensional lotka volterra systems which have a polynomial inverse integrating factor moreover we characterize all the polynomial first integrals of the two dimensional lotka volterra systems

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

multigrid method for periodic heterogeneous media part 1 convergence studies for one dimensional case abstract a multi grid method for a periodic heterogeneous medium in 1 d is presented based on the homogenization theory special integrid transfer operators have been developed to simulate a low frequency response of the differential equations with oscillatory coefficients the proposed multi grid method has been proved to have a fast rate of convergence governed by the ratio q 4 q where 0

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

families of invariants of the motion for the lotka volterra equations the linear polynomials family the modified carleman embedding method already introduced by the authors to find first integrals invariants of the motion of polynomial form to the lotka volterra system is described in detail and its efficiency to treat the n dimensional system proved using this method an extensive investigation is performed for polynomials of the first degree which allow a classification of the integrals in three families for some systems possessing one invariant it is possible to find a second invariant using rescaling methods they represent very restrictive solutions implying that there exists a great number of conditions among the equation s coefficients to satisfy a proof is given that the volterra invariants can be deduced as a limit finally the interesting properties of the solutions of these systems are studied in detail

a finite element variational multiscale method for incompressible flows based on two local gauss integrations in this article we present a finite element variational multiscale vms method for incompressible flows based on two local gauss integrations and compare it with common vms method which is defined by a low order finite element space l h on the same grid as x h for the velocity deformation tensor and a stabilization parameter a the best algorithmic feature of our method is using two local gauss integrations to replace projection operator we theoretically discuss the relationship between our method and common vms method for the taylor hood elements and show that the nonlinear system derived from our method by finite element discretization is much smaller than that of common vms method computationally additionally we present numerical simulations to demonstrate the effectiveness storage computational complexity of our method finally we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method

generalized hamiltonian structures for systems in three dimensions with a rescalable constant of motion the generalized hamiltonian structures of several three dimensional dynamical systems of interest in physical applications are considered in general hamiltonians exist only for systems that possess at least one time independent constant of motion systems with only time dependent constants of motion may sometimes be rescaled and their constant of motion made time independent when this is possible the transformed system may be cast in a generalized hamiltonian formalism with non canonical structure functions

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

a two level variational multiscale method for convection dominated convection diffusion equations this paper studies the error in the efficient implementation of and time stepping methods for a variational multiscale method vms for solving convection dominated problems the vms studied uses a fine mesh c0 finite element space xh to approximate the concentration and a coarse mesh discontinuous vector finite element space lh for the large scales of the flux in the two scale discretization our tests show that these choices lead to an efficient vms whose complexity is further reduced if a locally l2 orthogonal basis for lh is used a fully implicit and a semi implicit treatment of the terms which link effects across scales are tested and compared the semi implicit vms was much more efficient the observed global accuracy of the most straightforward vms implementation was much better than the artificial diffusion stabilization and comparable to a streamline diffusion finite element method in our tests

multi grid method for periodic heterogeneous media part 2 multiscale modeling and quality control in multidimensional case abstract a multi grid method for a periodic heterogeneous medium in multidimensions is developed based on the homogenization theory special intergrid transfer operators have been developed to stimulate a low frequency response of the boundary value problem with oscillatory coefficients an adaptive strategy is developed to form a nearly optimal two scale computational model consisting of the finite element mesh entirely constructed on the microscale in the regions identified by the idealization error indicators while elsewhere the modeling level is only sufficient to capture the response of homogenized medium numerical experiments show the usefulness of the proposed adaptive multi level procedure for predicting a detailed response of composite specimens

a mathematical homogenization perspective of virial stress a continuum stress measure is derived from molecular dynamics equations using a generalized mathematical homogenization gmh theory gmh consists of solving a coupled fine scale atomistic unit cell problem and a coarse scale continuum problem the fine scale problem derived can be interpreted as a molecular statics at 0 k problem where the coarse scale problem derived is a constitutive law free continuum equation which calculates the cauchy stress directly from atomistics the continuum stress derived is compared to various versions of the virial stress formula copyright 2005 john wiley sons ltd

new 25 point stencils with optimal accuracy for 2 d heat transfer problems comparison with the quadratic isogeometric elements abstract a new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent poisson equation has been suggested on uniform square and rectangular meshes it is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error by a simple modification of the coefficients of 25 point stencils the new approach exceeds the accuracy of the quadratic isogeometric elements by four orders for the heat equation and by twelve orders for the poisson equation despite the significant increase in accuracy the computational costs of the new technique are the same as those for the conventional quadratic isogeometric elements on a given mesh the numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new approach is much more accurate than the conventional isogeometric elements at the same number of degrees of freedom hybrid methods that combine the new stencils with the conventional isogeometric and finite elements and can be applied to irregular domains are also presented

comments on hamiltonian structures for the n dimensional lotka volterra equations j math phys 36 3520 3534 1995 in a recent paper plank presents results on first integrals for the lotka volterra system lvs and their use as possible hamiltonians since he was not acquainted with already published results we feel that an embedding of his results in those of previous papers can clarify some of his statements we concentrate our comments on the results presented in his paragraphs iii and iv

new four dimensional solutions of the jacobi equations for poisson structures a four dimensional family of skew symmetric solutions of the jacobi equations for poisson structures is characterized as a consequence previously known types of poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of new family of solutions additionally it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions such as their casimir invariants symplectic structure and the algorithm for the reduction to the darboux canonical form which have been reported only for a limited sample of known finite dimensional poisson structures moreover the results developed are valid globally in phase space thus ameliorating the usual scope of darboux theorem which is of local nature

a generalized space time mathematical homogenization theory for bridging atomistic and continuum scales a generalized space time mathematical homogenization theory which constructs an equivalent continuum description directly from molecular dynamics md equations is developed the noteworthy theoretical findings of this work are i the coarse scale continuum equations pdes obtained from the homogenization of md equations are identical to those obtained from the classical homogenization of the fine scale continuum ii the lower order effective continuum properties such as the fourth order elasticity tensor have similar characteristics to those resulting from the homogenization of fine scale continuum iii the higher order effective properties such as polarization and dispersion tensors substantially differ from those of continuum due to the discreteness effect both the heterogeneous continuum and discrete media possess the size effect but only atomistic medium has the discreteness effect some demonstrative numerical examples are presented to verify the formulation copyright 2005 john wiley sons ltd

connection between the existence of first integrals and the painleve property in two dimensional lotka volterra and quadratic systems taking advantage of the considerable amount of work done in the search for first integrals invariants for the two dimensional lotka volterra system and the quadratic system lvs and qs we compare the relations needed to exhibit invariants one for the lvs at least three for the qs to the two conditions of the painleve test index and compatibility we find that eventually restricting the invariants to those which are analytic all exponents integers and thereby adding new constraints these constraints always coalesce with the two painleve conditions we conclude that straightforward application of the painleve test picks up only these simple analytic invariants and that possession of the painleve property is too strong a condition for the existence of the invariants

cluster mean field approximations with the coherent anomaly method analysis for the driven pair contact process with diffusion the cluster mean field approximations are performed up to 13 cluster sizes to study the critical behavior of the driven pair contact process with diffusion dpcpd and its precedent the pcpd in one dimension critical points are estimated by extrapolating our data to the infinite cluster size limit which are in good accordance with recent simulation results within the cluster mean field approximation scheme the pcpd and the dpcpd share the same mean field critical behavior the application of the coherent anomaly method however shows that the two models develop different coherent anomalies which lead to different true critical scaling the values of the critical exponents for the particle density the pair density the correlation length and the relaxation time are fairly well estimated for the dpcpd these results support and complement our recent simulation results for the dpcpd

generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions we derive thermo mechanical continuum equations from molecular dynamics md equations using the generalized mathematical homogenization gmh theory developed by the authors for 0 k applications gmh constructs an array of atomistic unit cell problems coupled with a thermo mechanical continuum problem the unit cell problem derived is a molecular dynamics problem defined for the perturbation from the average atomistic displacements subjected to the deformation gradient and temperature extracted from the continuum problem the coarse scale problem derived is a constitutive law free continuum thermo mechanical equation attention is restricted to heat transfer by lattice vibration phonons the method is verified on several model problems against the reference molecular dynamics solution 2006 elsevier b v all rights reserved

high order fem for multigrid homogenization the high order fem was applied to the multigrid based homogenization and proved to be an efficient method for upscaled solution approximation of problems with rapidly varying possibly noncontinuous coefficients as well as non coherent porous domains that may or may not depict a periodic micro structure and do not have to satisfy the separation condition our main contribution is an improved appropriate for bubble functions inter grid mapping consequently a fast exponential convergence of both displacements and stresses was observed in the conducted numerical experiments with respect to the number of degrees of freedom as well as the time of computation particularly the high accuracy of solution derivatives is an important advantage of this homogenization technique that is equivalent to the multiscale fem

an energetically consistent concurrent multiscale method for heterogeneous heat transfer and phase transition applications abstract a concurrent multiscale method is developed to model time dependent heat transfer and phase transitions in heterogeneous media and is formulated in a way such that the energy being exchanged between scales is conserved ensuring this energetic consistency among scales enables the implementation of high fidelity physics based models at critical locations within the coarse scale to temporally and spatially resolve highly complex and localized phenomena to achieve this only neumann boundary conditions are applied over the fine scale domain ensuring a conservative formulation the coarse scale solution is used to reconstruct these neumann boundary conditions on the fine scale which are then used to evolve a separate system of governing equations the results on the fine scale are then sent back to the coarse scale through an energy based homogenization scheme transient simulations for the heat equation are implemented with the proposed method to demonstrate its accuracy in energy conservation and effectiveness including the coupling of a phase field model at the fine scale to a coarse scale heat equation

an integrating factor matrix method to find first integrals in this paper we develop an integrating factor matrix method to derive conditions for the existence of first integrals we use this novel method to obtain first integrals along with the conditions for their existence for two and three dimensional lotka volterra systems with constant terms the results are compared to previous results obtained by other methods

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

inverse jacobi multiplier as a link between conservative systems and poisson structures some aspects of the relationship between conservativeness of a dynamical system namely the preservation of a finite measure and the existence of a poisson structure for that system are analyzed from the local point of view due to the flow box theorem we restrict ourselves to neighborhoods of singularities in this sense we characterize poisson structures around the typical zero hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non vanishing first jet by connecting with the classical poincar e center problem from the global point of view we connect the property of being strictly conservative the invariant measure must be positive with the existence of a poisson structure depending on the phase space dimension finally weak conservativeness in dimension two is introduced by the extension of inverse jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed examples including lotka volterra systems quadratic isochronous centers and non smooth oscillators are provided

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

hamiltonian method and invariant search for 2d quadratic systems the hamiltonian formalism for a two dimensional system of ode possessing an invariant of the motion not containing the time explicitly has suggested a method for the search of first integrals applied to the quadratic system it leads to the finding of two phase space configurations

diffusion limited reactions of hard core particles in one dimension we investigate three different methods to tackle the problem of diffusion limited reactions annihilation of hard core classical particles in one dimension we first extend an approach devised by lushnikov sov phys jetp 64 811 1986 and calculate for a single species the asymptotic long time and or large distance behavior of the two point correlation function based on a work by grynberg and stinchcombe phys rev e 50 957 1994 phys rev lett 74 1242 1995 76 851 1996 which was developed to treat stochastic adsorption desorption models we provide in a second step the exact two point one and two time correlation functions of lushnikov s model we then propose a formulation of the problem in terms of path integrals for pseudo fermions this formalism can be used to advantage in the multispecies case especially when applying perturbative renormalization group techniques

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

a general time dependent invariant for and integrability of the quadratic system we derive a general time dependent invariant first integral for the quadratic system qs that requires only one condition on the coefficients of the qs the general invariant could yield asymptotic behaviour of phase space trajectories with more conditions imposed on the coefficients the general invariant reduces to polynomial form and is equivalent to polynomial invariants found using a direct method for the special case of a linear polynomial invariant where one of the variables is analytically invertible the solution of the qs is reduced to a quadrature

heterogeneous multiscale methods for stiff ordinary differential equations the heterogeneous multiscale methods hmm is a general framework for the numerical approximation of multiscale problems it is here developed for ordinary differential equations containing differe

on certain new integrable second order nonlinear differential equations and their connection with two dimensional lotka volterra system in this paper we consider a second order nonlinear ordinary differential equation of the form x k1 x2 x k2 k3x x k4x3 k5x2 k6x 0 where ki s i 1 2 6 are arbitrary parameters by using the modified prelle singer procedure we identify five new integrable cases in this equation besides two known integrable cases namely i k2 0 k3 0 and ii k1 0 k2 0 k5 0 among these five four equations admit time dependent first integrals and the remaining one admits time independent first integral from the time independent first integral nonstandard hamiltonian structure is deduced thereby proving the liouville sense of integrability in the case of time dependent integrals we either explicitly integrate the system or transform to a time independent case and deduce the underlying hamiltonian structure we also demonstrate that the above second order ordinary differential equation is intimately related to the two dimensional lotka volterra system from the integrable parametric choices of the above non

a multilevel finite element method fe2 to describe the response of highly non linear structures using generalized continua abstract a general method called fe2 has been introduced which consists in describing the behavior of heterogeneous structures using a multiscale finite element model instead of trying to build differential systems to establish a stress strain relation at the macroscale a finite element computation of the representative volume element is carried out simultaneously doing so does not require any constitutive equations to be written at the macroscopic scale all non linearities come directly from the microscale in this paper we describe how this method can be used in the context of generalized continua for such continua constitutive equations are very difficult to write and a new set of material is difficult to fit to experimental data the use of fe2 models bypasses this problem because no analytical equation is needed at the macroscale an academic application is presented to show that generalized continua are necessary when the size of the heterogeneities increases and that fe2 models behave well compared to a reference solution