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

space time multiscale model for wave propagation in heterogeneous media abstract a space time multiscale model for wave propagation in heterogeneous media is developed the model builds on the authors previous work on the higher order mathematical homogenization theory with multiple spatial and temporal scales and is aimed at addressing the issues of stability and mathematical consistency starting from the weak forms of homogenized macroscopic equations of motion terms causing the solution secularity are identified and enforced to vanish this condition recovers the missing boundary conditions and gives rise to two secularity constrains imposing the uniform validity of asymptotic expansions finite element semi discretization in space along with an analytical solution in time are employed to incorporate the secularity constrains in the leading order solution and account for the slow time dependence of the leading solution pade approximation is utilized to develop the time stepping schemes on the fast time scale the formulation is verified for wave propagation problems in semi infinite and finite domains

multi scale homogenization procedure for continuum atomistic thermo mechanical problems abstract a multi scale algorithm is developed for solving continuum atomistic thermo mechanical problems the coupling of the two scales occurs through a homogenization procedure a uniform weighted residual approximation method is used to consistently model the interaction of the continuum and atomistic scales implementation of the coupling occurs through the transfer of macroscopic and microscopic variables at the macroscopic gauss points the mechanism for transfer of atomistic information to the macro level occurs through the virial stress and the heat flux vector a dynamic relaxation procedure is introduced to efficiently compute the coupled equilibrium solutions at the macroscopic and microscopic scales the algorithm has been successfully tested for thermal stress analysis cases involving constrained and unconstrained macro components and fixed temperature distributions problems involving varying temperatures and heat transfer have been solved and are shown to be in good agreement with molecular dynamics results

integrals of motion for three dimensional non hamiltonian dynamical systems the problem of finding integrals of motion of three dimensional dynamical systems is analysed the authors introduce a new type of direct method in the search of parameter values for which an integral of motion exists this method consists in proposing an ansatz for the integral that explicitly shows the dependence with respect to one of the phase space coordinates of the system they apply this procedure to the reduced three wave interaction problem and to the rabinovich system for both models new integrals of motion are found

elastic plastic analysis of arbitrary heterogeneous materials with the voronoi cell finite element method abstract in this paper a voronoi cell finite element method is developed to solve small deformation elastic plasticity problems for arbitrary heterogenous materials dirichlet tessellation of microstructural representative materials results in a network of arbitrary sided polygons called voronoi cells each voronoi cell encompasses one second phase heterogeneity at most these are natural elements for the microstructure representing the basic structural elements of the material in this paper formulations are developed for directly considering voronoi cells as elements in a finite element model without any further dissection furthermore a composite voronoi cell finite element method is developed to account for the presence of the second phase within each polygonal element various numerical elastic plastic examples are executed for validating the effectiveness of this formulation finally studies are conducted to understand the effect of size shape and distribution of second phase on the averaged and true local responses of representative material elements

modeling and simulation of piezocomposites solution procedures for large scale transient analysis of piezocomposites are developed computational efficiency of unconditionally stable and conditionally stable implicit partitioned schemes as well as explicit partitioned schemes is investigated in the context of multilevel solution methods applied to separate field equations

new solution family of the jacobi equations characterization invariants and global darboux analysis a new family of skew symmetric solutions of the jacobi partial differential equations for finite dimensional poisson systems is characterized and analyzed such family has some remarkable properties firstly it is defined for arbitrary values of the dimension and the rank secondly it is described in terms of arbitrary differentiable functions namely it is not limited to a given degree of nonlinearity additionally it is possible to determine explicitly the fundamental properties of those solutions such as their casimir invariants and the algorithm for the reduction to the darboux canonical form which have been reported only for a very limited sample of finite dimensional poisson structures moreover such analysis is carried out globally in phase space thus improving the usual local scope of the darboux theorem

generalization of solutions of the jacobi pdes associated to time reparametrizations of poisson systems the determination of solutions of the jacobi partial differential equations pdes for finite dimensional poisson systems is considered in particular a novel procedure for the construction of solution families is developed such a procedure is based on the use of time reparametrizations preserving the existence of a poisson structure as a result a method which is valid for arbitrary values of the dimension and the rank of the poisson structure under consideration is obtained in this article two main families of time reparametrizations of this kind are characterized in addition these results lead to a novel application which is also developed namely the global and constructive determination of the darboux canonical form for poisson systems of arbitrary dimension and rank two thus improving the local result provided by darboux theorem for such a case

hamiltonians and variational principles for alfven simple waves the evolution equations for the magnetic field induction b with the wave phase for alfven simple waves are expressed as variational principles and in the hamiltonian form the evolution of b with the phase which is a function of the space and time variables depends on the generalized frenet serret equations in which the wave normal n which is a function of the phase is taken to be tangent to a curve x in a 3d cartesian geometry vector space the physical variables the gas density fluid velocity gas pressure and magnetic field induction in the wave depend only on the phase three approaches are developed one approach exploits the fact that the frenet equations may be written as a 3d hamiltonian system which can be described using the nambu bracket it is shown that b as a function of the phase satisfies a modified version of the frenet equations and hence the magnetic field evolution equations can be expressed in the hamiltonian form a second approach develops an euler poincare variational formulation a third approach uses the frenet frame formulation in which the hodograph of b moves on a sphere of constant radius and uses a stereographic projection transformation due to darboux the equations for the projected field components reduce to a complex riccati equation by using a cole hopf transformation the riccati equation reduces to a linear second order differential equation for the new variable a hamiltonian formulation of the second order differential equation then allows the system to be written in the hamiltonian form alignment dynamics equations for alfven simple waves give rise to a complex riccati equation or equivalently to a quaternionic riccati equation which can be mapped onto the riccati equation obtained by stereographic projection

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

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

new global solutions of the jacobi partial differential equations abstract a new family of solutions of the jacobi partial differential equations for finite dimensional poisson systems is investigated this family is mathematically remarkable as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves this kind of poisson structure termed distinguished solutions or d solutions is defined for every nontrivial combination of values of the dimension and the rank and is also determined in terms of functions of arbitrary nonlinearity properties usually not present simultaneously in the already known solution families in addition d solutions display several properties allowing the generation of an infinity of d solutions from a given one which is an uncommon feature in the framework of the jacobi equations furthermore a special family of d solutions complying with the previous requirements is constructively characterized and analyzed examples are discussed focusing on physical implications and including an application for the global construction of the darboux canonical form

a multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks we present a multiscale model for numerical simulations of dynamics of crystalline solids the method combines the continuum nonlinear elasto dynamics model which models the stress waves and physical loading conditions and molecular dynamics model which provides the nonlinear constitutive relation and resolves the atomic structures near local defects the coupling of the two models is achieved based on a general framework for multiscale modeling the heterogeneous multiscale method hmm we derive an explicit coupling condition at the atomistic continuum interface application to the dynamics of brittle cracks under various loading conditions is presented as test examples

discrete to continuum bridging based on multigrid principles abstract the paper presents a new concurrent multiscale approach based on multigrid principles with intent of solving large molecular statics and molecular dynamics systems we show that the coarse model effective stiffness matrix obtained by variational restriction of the atomistic model effective stiffness matrix coincides with the effective stiffness matrix of the equivalent continuum scale model the equivalent continuum model is defined for both molecular statics and molecular dynamics cases and theoretical estimates of the rate of convergence for the proposed concurrent multiscale approach are obtained

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

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

on a formulation for a multiscale atomistic continuum homogenization method abstract the homogenization method is used as a framework for developing a multiscale system of equations involving atoms at zero temperature at the small scale and continuum mechanics at the very large scale the tersoff brenner type ii potential physical review letters 61 25 1988 2879 physical review b 42 15 1990 9458 is employed to model the atomic interactions while hyperelasticity governs the continuum a quasistatic assumption is used together with the cauchy born approximation to enforce the gross deformation of the continuum on the positions of the atoms the two scale homogenization method establishes coupled self consistent variational equations in which the information at the atomistic scale formulated in terms of the lagrangian stiffness tensor concurrently feeds the material information to the continuum equations analytical results for a one dimensional molecular wire and numerical experiments for a two dimensional graphene sheet demonstrate the method and its applicability

a bridging domain and strain computation method for coupled atomistic continuum modelling of solids we present a multiscale method that couples atomistic models with continuum mechanics the method is based on an overlapping domain decomposition scheme constraints are imposed by a lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap an efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material an iterative scheme is utilized to optimize the coupled configuration conditions for the regularity of the constrained matrices are determined a method for computing strain in atomistic models and handshake domains is formulated based on a moving least square approximation which includes both extensional and angle bending terms it is shown that this method exactly computes the linear strain field applications to the fracture of defected single layer atomic sheets and nanotubes are given copyright 2006 john wiley sons ltd

hierarchical model reduction at multiple scales a hierarchical model reduction approach aimed at reducing computational complexity of non linear homogenization at multiple scales is developed the method consists of the following salient features 1 formulation of non linear unit cell problems at multiple scales in terms of eigendeformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized non linear equilibrium 2 the ability to control the discretization of the eigendeformation modes at multiple scales to maintain desired accuracy and 3 hierarchical solution strategy that requires sequential solution of single scale problems a two scale formulation is verified against an one dimensional model problem for which an analytical solution can be obtained and a three scale formulation is validated against tube crash experiments copyright 2009 john wiley sons ltd

force free fields from hertz potentials we show a correspondence between magnetic fields that are eigenvectors of the curl operator and certain solutions to the source free maxwell equations when the eigenvalue of the curl is non constant then the corresponding maxwell system must be solved in a non flat spacetime we show how the generalized hertz potential method of solving maxwell s equations in curved spacetimes can be applied to determine eigenvectors of the curl in the case of a constant eigenvalue the chandrasekhar kendall eigenfunctions are recovered for non constant eigenvalue we use the formalism to produce some new force free fields

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

a class of general algorithms for multi scale analyses of heterogeneous media abstract a class of computational algorithms for multi scale analyses is developed in this paper the two scale modeling scheme for the analysis of heterogeneous media with fine periodic microstructures is generalized by using relevant variational statements instead of the method of two scale asymptotic expansion the mathematical results on the generalized convergence are utilized in the two scale variational descriptions accordingly the global local type computational schemes can be unified in association with the homogenization procedure for general nonlinear problems after formulating the problem in linear elastostatics that with local contact condition and the elastoplastic problem we present representative numerical examples along with the computational algorithm consistent with our two scale modeling strategy as well as some direct approaches

non local dispersive model for wave propagation in heterogeneous media one dimensional case non local dispersive model for wave propagation in heterogeneous media is derived from the higher order mathematical homogenization theory with multiple spatial and temporal scales in addition to the usual space time co ordinates a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long term behaviour of the homogenized solution by combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth order differential equation also known as a bad boussinesq problem regularization procedures are then introduced to construct the so called good boussinesq problem where the need for c1 continuity is eliminated numerical examples are presented to validate the present formulation copyright 2002 john wiley sons ltd

a coupled molecular dynamics and extended finite element method for dynamic crack propagation a multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain after recalling the basic equations of molecular dynamics and continuum mechanics the discretization is discussed for the continuum subdomain where the partition of unity property of finite element shape functions is used since in this fashion the crack in the wake of its tip is naturally modelled as a traction free discontinuity next the zonal coupling method between the atomistic and continuum models is recapitulated finally it is discussed how the stress has been computed in the atomic subdomain and a two dimensional computation is presented of dynamic fracture using the coupled model the result shows multiple branching which is reminiscent of recent results from simulations on dynamic fracture using cohesive zone models

congruence method for global darboux reduction of finite dimensional poisson systems a new procedure for the global construction of the casimir invariants and darboux canonical form for finite dimensional poisson systems is developed this approach is based on the concept of matrix congruence and can be applied without the previous determination of the casimir invariants recall that their prior knowledge is unavoidable for the standard reduction methods thus requiring either the integration of a system of pdes or solving some equivalent problem well the opposite in the new congruence method both the darboux coordinates and the casimir invariants arise simultaneously as the outcome of the reduction algorithm in fact the congruence algorithm proceeds only in terms of matrix algebraic transformations and direct quadratures thus avoiding the need of previously integrating a system of pdes and therefore improving previously known approaches physical examples illustrating different aspects of the theory are provided

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

an algebraic two level preconditioner for asymmetric positive definite systems a two level linear algebraic solver for asymmetric positive definite systems is developed using matrices arising from stabilized finite element formulations to motivate the approach supported by an analysis of a representative smoother the parent space is divided into oscillatory and smooth subspaces according to the eigenvectors of the associated normal system using a mesh based aggregation technique which relies only on information contained in the matrix a restriction prolongation operator is constructed various numerical examples on both structured and unstructured meshes are performed using the two level cycle as the basis for a preconditioner results demonstrate the complementarity between the smoother and the coarse level correction as well as convergence rates that are nearly independent of the problem size copyright 2001 john wiley sons ltd

multiscale enrichment based on partition of unity a new multiscale enrichment method based on the partition of unity mepu method is presented it is a synthesis of mathematical homogenization theory and the partition of unity method its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible mepu is perfectly suited for enriching the coarse scale continuum descriptions pdes with fine scale features and the quasi continuum formulations with relevant atomistic data numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi continuum formulations copyright 2005 john wiley sons ltd

hamiltonian equations in r3 the hamiltonian formulation of n 3 systems is considered in general the most general solution of the jacobi equation in r3 is proposed the form of the solution is shown to be valid also in the neighborhood of some irregular points compatible poisson structures and corresponding bi hamiltonian systems are also discussed hamiltonian structures the classification of irregular points and the corresponding reduced first order differential equations of several examples are given

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

optimal numerical solution of pdes using the variational germano identity the variational germano identity vgi is a condition that must be satisfied if the variational approximation to a partial differential equation is optimal in a user defined metric the vgi is useful as it can be utilized to dynamically determine unknown parameters in the variational approximation without any a priori knowledge of the exact solution with these values it yields solutions that are close to the optimal solution in this manuscript we examine the relation between the definition of an optimal solution and the parameters computed using the vgi we demonstrate that this relation is made precise by a so called truncated restriction operator that appears in the final expression for the model parameters and is sensitive to the criterion used for selecting the optimal solution in order to facilitate the use of the vgi for determining model parameters we derive explicit expressions for this operator when three distinct approaches are used for defining the optimal solution through numerical examples that make use of these expressions we confirm that the vgi does indeed yield model parameters and numerical solutions that are consistent with the choice of the optimal solution we also observe that the numerical solutions obtained using the vgi are close to the best numerical solution that can be obtained from a given variational approximation indicating that it yields nearly optimal values for the parameters

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

adaptive mesh computation of polycrystalline pattern formation using a renormalization group reduction of the phase field crystal model we implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes our numerical approach uses the systematic renormalization group formulation of a phase field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field a quantity that is spatially uniform except near topological defects grain boundaries and other lattice imperfections our algorithm employs a hybrid formulation of the amplitude equations combining cartesian and polar decompositions of the complex amplitude we show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions

global basis two level method for indefinite systems part 1 convergence studies a robust two level solver for high indefinite system of equations arising from the finite element discretization is developed it is shown that the optimal coarse model is spanned by the spectrum of the highest eigenmodes of the smoothing iteration matrix convergence studies conducted on a model prolongation operator reveal pathological sensitivity to any deviation from the optimal coarse model copyright 2000 john wiley sons ltd

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

quantum algebras in classical mechanics it is shown that the equations of motion do not determine their hamiltonian descriptions uniquely in particular different poisson bracket relations are admissible for a given physical problem the free symmetric top and a charged particle moving on a homogeneous and constant magnetic field for instance may be described by a classical hamiltonian theory with an su 2 q poisson bracket structure

the eighth moment of dirichlet l functions abstract we prove an asymptotic for the eighth moment of dirichlet l functions averaged over primitive characters modulo q over all moduli q q and with a short average on the critical line conditionally on grh we derive the analogous result for the fourth moment of dirichlet twists of gl 2 l functions our results match the moment conjectures in the literature in particular the constant 24 024 appears as a factor in the leading order term of the eighth moment

integrals of quadratic ordinary differential equations in r3 the lotka volterra system abstract a method already introduced by the last two authors for finding the integrable cases of three dimensional autonomous ordinary differential equations based on the frobenius integrability theorem is described in detail using this method and computer algebra the so called three dimensional lotka volterra system is studied many cases of integrability are thus found the study of this system is completed by the application of painleve analysis and the jacobi last multiplier method the methods used are of general interest and can be applied to many other systems

finite deformation plasticity for composite structures computational models and adaptive strategies abstract we develop computational models and adaptive modeling strategies for obtaining an approximate solution to a boundary value problem describing the finite deformation plasticity of heterogeneous structures a nearly optimal mathematical model consists of an averaging scheme based on approximating eigenstrains and elastic concentration factors in each micro phase by a constant in the portion of the macro domain where modeling errors are small whereas elsewhere a more detailed mathematical model based on a piecewise constant approximation of eigenstrains and elastic concentration factors is utilized the methodology is developed within the framework of statistically homogeneous composite material and local periodicity assumptions

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

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

conjugate gradient like solution algorithms for the mixed finite element approximation of the biharmonic equation applied to plate bending problems abstract discretization of the biharmonic equation with the mixed finite element method yields an indefinite linear system of equations with a special structure in this paper two variants of the conjugate gradient method are formulated that are suited for solving such systems they both require the solution of a system of linear equations in every iteration different strategies for doing this are examined an incomplete choleski decomposition is used as a preconditioner both iterative methods and the preconditioner are chosen so that optimal use can be made of the special block structure of the global system of equations

bi hamiltonian structure in frenet serret frame abstract we reduce the problem of constructing bi hamiltonian structure in three dimensions to the solutions of a riccati equation in moving coordinates of frenet serret frame all explicitly constructed examples in the literature are exhausted by constant solutions we conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi hamiltonian

breaking the 12 barrier for the twisted second moment of dirichlet l functions abstract we study the second moment of dirichlet l functions to a large prime modulus q twisted by the square of an arbitrary dirichlet polynomial we break the 1 2 barrier in this problem and obtain an asymptotic formula provided that the length of the dirichlet polynomial is less than q 51 101 q 1 2 1 202 as an application we obtain an upper bound of the correct order of magnitude for the third moment of dirichlet l functions we give further results when the coefficients of the dirichlet polynomial are more specialized

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

lower bounds for moments of l functions the moments of central values of families of l functions have recently attracted much attention and with the work of keating and snaith 2000 commun math phys 214 57 89 and 91 110 there are now precise conjectures for their limiting values we develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of l functions as an example we work out the case of the family of all dirichlet l functions to a prime modulus

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

a priori and a posteriori w 1 infty error analysis of a qc method for complex lattices in this paper we derive regularity results for equilibria of multilattices under an external force and prove a priori and a posteriori error estimates for a multiscale numerical method for computing such equilibria the estimates are derived in a w 1 infty norm in one space dimension one of the features of our analysis is that we establish an equivalent way of formulating the coarse grained problem which greatly simplifies derivation of the error bounds both a priori and a posteriori we illustrate our error estimates with numerical experiments

coarse graining molecular dynamics models using an extended galerkin projection method summary we present a new framework for coarse graining molecular dynamics models for crystalline solids the reduction method is based on a galerkin projection to a subspace whose dimension is much smaller than that of the full atomistic model to effectively reduce artificial reflections of phonons at the interface we construct extended subspaces with increasing accuracy by adding more coarse grained variables near the interface between lattice defects and the surrounding region this approach is equivalent to the generalized langevin model but it eliminates the need to precompute the memory function a well known practical difficulty further the variational formulation preserves the stability of the molecular models copyright 2014 john wiley sons ltd