the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

solvable lie algebras of dimension six six dimensional solvable lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes this completes mubarakzyanov s classification of the real six dimensional solvable lie algebras

the solution of a class of free boundary problems mellin transform and wiener hopf techniques are used to solve several free boundary problems occurring in groundwater flow this is a novel application of a technique not usually associated with free boundary problems here it is advocated as a powerful alternative to methods relying on conformal mappings in particular analytical methods can now be applied to classes of problems whose explicit solution is otherwise unobtainable relatively simple integral representations are also provided for a class of problems that otherwise require complicated analysis involving hypergeometric functions the problems are formulated in an intermediate complex plane and the mellin transform is used to solve the problems directly in this plane alternative methods require a hodograph plane to be formulated this is then conformally mapped to the intermediate plane if the conformal mapping can be done then the full solution is determined as a byproduct of this analysis the conformal mappings of some curvilinear polygons to a half plane are found to perform such mappings directly for non trivial problems involving more than three singular points requires the solution of fuchsian differential equations that involve accessory parameters and unknown points apart from degenerate cases there appears to be no constructive method of obtaining solutions to such fuchsian differential equations as a result few practical problems whose representation in the hodograph plane involves curvilinear quadrangles have been solved this paper has the aim of developing and demonstrating a simple coherent strategy for tackling the free boundary problems that lead to a class of these more complicated situations this substantially extends the range of problems that can be treated the method presented sidesteps the difficulties associated with the free point and accessory parameter and the solution for a problem where the hodograph plane is a curvilinear quadrangle with one vertex angle of 2 pi is found in the physical plane this corresponds to a point of inflection on the free surface hence it is possible to conformally map a class of curvilinear quadrangles to an upper half plane using the analytic methods developed here and this resolves an open question in conformal mapping theory two illustrative examples are also solved one involving three singular points and this provides a comparison with previous techniques the other illustrates the occurrence of a point of inflection in a relatively simple situation the alternative method advocated here is compared to the other analytical techniques available these are briefly reviewed to place the mellin transform wiener hopf approach in context

on effective resistivity and related parameters for periodic checkerboard composites an approach for finding the effective properties of two dimensional periodic rectangular checkerboard media is presented it is demonstrated that one can take a relatively simple model problem an appropriate system of joined quarter planes and that after appropriate conformal transformation one can use this to solve for either periodic or doubly periodic structures illustrative calculations involving three phase media are presented that in limiting cases recover earlier analyses useful asymptotic relations are also given

relation between effective conductivity and susceptibility of two component rhombic checkerboard the heterogeneity of composite leads to extra charge concentration at the boundaries of different phases which results in essentially nonzero effective electric susceptibility the relation between tensors of effective electric susceptibility ef and effective conductivity ef of the infinite two dimensional two component regular composite with rhombic cell structure has been established the degrees of electric field singularity at corner points of cells are found by constructing the integral equation for the effective conductivity problem the limits of weak and strong contrast of partial conductivities 1 2 are considered the results are valid for thin films and cylindrical samples

generalized non local surface susceptibility model and fresnel coefficients for the characterization of periodic metafilms with bianisotropic scatterers formulas for e m fields radiated by continuous surface polarization distributions non local effective surface susceptibility model for periodic metafilms generalized reflection and transmission coefficients for an arbitrary metafilm successful treatment of non planar scatterer arrays and spatial dispersion effects a non local surface susceptibility model for the consistent description of periodic metafilms formed by arbitrarily shaped electrically small bianisotropic scatterers is developed in this paper the rigorous scheme is based on the point dipole approximation technique and is valid for any polarization and propagation direction of an electromagnetic wave impinging upon the metafilm unlike existing approaches whose applicability is practically confined to very specific cases of incidence next the universal form of the resulting surface susceptibility matrix is employed for the derivation of the generalized fresnel coefficients for such surfaces which enable the comprehensive interpretation of several significant yet relatively unexamined physical interactions essentially these coefficients include eight distinct terms corresponding to the co polarized and cross polarized reflection and transmission coefficients for the two orthogonal eigenpolarizations of a linearly polarized incident plane wave the above formulas are then utilized for the prediction of the scattering properties of metafilms with different planar and non planar resonators which are characterized via the featured model and two previously reported local ones their comparison with numerical simulation outcomes substantiates the merits of the proposed method reveals important aspects of the underlying physics and highlights the differences between the various modeling procedures

regular hexagonal three phase checkerboard abstract a two dimensional doubly periodic three phase hexagonal structure is considered the flow in the structure is generated by three sets of vortexes sinks sources which are the same in each phase and are located in the centers of the hexagons complex analysis methods are utilized to reduce the doubly periodic r linear conjugation problem to the simpler one riemann hilbert rh problem on a three sheeted riemann surface in turn the latter problem is reduced to a rh problem involving three joined sectors on the plane which was previously investigated in 3 the limiting cases with one non conducting phase and two phases of the same conductivities are investigated all solutions derived are verified both numerically and analytically examples of relevant flow networks streamlines and equipotentials are plotted in the whole structure and separately in each phase

vorticity and symplecticity in multi symplectic lagrangian gas dynamics the lagrangian multi dimensional ideal compressible gas dynamic equations are written in a multi symplectic form in which the lagrangian fluid labels m i the lagrangian mass coordinates and time t are the independent variables and in which the eulerian position of the fluid element and the entropy are the dependent variables constraints in the variational principle are incorporated by means of lagrange multipliers the constraints are the entropy advection equation s t 0 the lagrangian map equation where is the fluid velocity and the mass continuity equation which has the form where is the jacobian of the lagrangian map in which and is the specific volume of the gas the internal energy per unit volume of the gas corresponds to a non barotropic gas the lagrangian is used to define multi momenta and to develop de donder weyl hamiltonian equations the de donder weyl equations are cast in a multi symplectic form the pullback conservation laws and the symplecticity conservation laws are obtained one class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws and another class of symplecticity laws are related to derivatives of the lagrangian energy conservation law with respect to the lagrangian mass coordinates m i we show that the vorticity symplecticity laws can be derived by a lie dragging method and also by using noether s second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action we obtain the cartan poincare form describing the equations and we discuss a set of differential forms representing the equation system

multisymplectic formulation of fluid dynamics using the inverse map we construct multisymplectic formulations of fluid dynamics using the inverse of the lagrangian path map this inverse map the back to labels map gives the initial lagrangian label of the fluid particle that currently occupies each eulerian position explicitly enforcing the condition that the fluid particles carry their labels with the flow in hamilton s principle leads to our multisymplectic formulation we use the multisymplectic one form to obtain conservation laws for energy momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to kelvin s circulation theorem we discuss how multisymplectic numerical integrators naturally arise in this approach

four phase checkerboard composites two dimensional periodic rectangular checkerboard media are considered in the situation where mean fluxes are prescribed across the structure the closed form solution is obtained in the general case where the checkerboard is constructed using four rectangular cells each having a different constant resistivity this four cell structure then repeats doubly periodically to cover the whole plane this general solution is then used to calculate the effective properties thus this four phase checkerboard encapsulates many limiting and special cases as a starting point we develop a concise closed form solution to a basic problem involving four joined quarter planes each of a different resistivity subsequent manipulations yield solutions to problems posed in increasingly convoluted domains while retaining the essentially simple structure found for joined quarter planes

efficient discretization of laplace boundary integral equations on polygonal domains we describe a numerical procedure for the construction of quadrature formulae suitable for the efficient discretization of boundary integral equations over very general curve segments while the procedure has applications to the solution of boundary value problems on a wide class of complicated domains we concentrate in this paper on a particularly simple case the rapid solution of boundary value problems for laplace s equation on two dimensional polygonal domains we view this work as the first step toward the efficient solution of boundary value problems on very general singular domains in both two and three dimensions the performance of the method is illustrated with several numerical examples

on the infinite dimensional symmetry group of the davey stewartson equations the lie algebra of the group of point transformations leaving the davey stewartson equations dse s invariant is obtained the general element of this algebra depends on four arbitrary functions of time the algebra is shown to have a loop structure a property shared by the symmetry algebras of all known 2 1 dimensional integrable nonlinear equations subalgebras of the symmetry algebra are classified and used to reduce the dse s to various equations involving only two independent variables

derivation of lagrangian density for the good boussinesq equation and multisymplectic disretizations abstract the lagrangian density for the good boussinesq equation is derived by introducing a new variable in this paper based on this density we consider the corresponding multisymplectic formulation and discretizations numerical experiments are also reported

representations for the conductivity functions of multicomponent composites the effective conductivity of a multicomponent composite material is considered integral representations for treated as a holomorphic function on a polydisk with values in a half plane are analyzed a representation for is introduced which is symmetric in the component conductivities and for which the moments of the positive measure in the integral are directly related to the coefficients in a perturbation expansion of around a homogeneous medium this second feature which is important for obtaining bounds on was previously available only in the two component case in addition a bound valid for any holomorphic function for the above type is proven

presymplectic current and the inverse problem of the calculus of variations the inverse problem of the calculus of variations asks whether a given system of partial differential equations pdes admits a variational formulation we show that the existence of a presymplectic form in the variational bicomplex when horizontally closed on solutions allows us to construct a variational formulation for a subsystem of the given pde no constraints on the differential order or number of dependent or independent variables are assumed the proof follows a recent observation of bridges hydon and lawson math proc cambridge philos soc 148 01 159 178 2010 and generalizes an older result of henneaux ann phys 140 1 45 64 1982 from ordinary differential equations odes to pdes uniqueness of the variational formulation is also discussed

statistical energy conservation principle for inhomogeneous turbulent dynamical systems understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact in such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other these complex features strongly impact practical prediction and uncertainty quantification a systematic energy conservation principle is developed here in a theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations this statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence this result applies to general inhomogeneous turbulent dynamical systems including the above applications the theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self contained treatment is presented below corollary 1 and corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds provided that the negative symmetric dissipation matrix is diagonal in a suitable basis implications of the energy principle for low order closure modeling and automatic estimates for the single point variance are discussed below

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

a universal form for the emergence of the korteweg de vries equation a new perspective on the emergence of the kortewegde vries kdv equation is presented the conventional view is that the kdv equation arises as a model when the dispersion relation of the lineariz

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

bounds for effective parameters of multicomponent media by analytic continuation recently d bergman introduced a method for obtaining bounds for the effective dielectric constant or conductivity of a two component medium this method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters we extend the method to multicomponent media using techniques of several complex variables

canonical multi symplectic structure on the total exterior algebra bundle exterior algebra bundle on m on the fibre which has dimension 2 n the form q can be locally decomposed into n classical symplectic structures when concatenated these nsymplectic structures define a partial differential operator jv which turns out to be a dirac operator with multi symplectic structure the operator jv generalizes the product operator j d dt in classical symplectic geometry and m is a generalization of the base manifold i e time in classical hamiltonian dynamics the structure generated by q provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators the operator jv is elliptic and the generalization of hamiltonian systems jvzzvs z for a section z of the total exterior algebra bundle is also an elliptic pde the inverse problem find s z for a given elliptic pde is shown to be related to a variant of the legendre transform on k forms the theory is developed for flat base manifolds but the constructions are coordinate free and generalize to riemannian manifolds with non trivial curvature some applications and implications of the theory are also discussed

integral equation methods for elliptic problems with boundary conditions of mixed type laplace s equation with mixed boundary conditions that is dirichlet conditions on parts of the boundary and neumann conditions on the remaining contiguous parts is solved on an interior planar domain using an integral equation method rapid execution and high accuracy is obtained by combining equations which are of fredholm s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results this opens up for the rapid and accurate solution of several elliptic problems of mixed type

proof of a conjecture on the conductivity of checkerboards in 1985 mortola and steffe conjectured a formula for the effective conductivity tensor of a checkerboard structure where the unit cell of periodicity is square and subdivided into four equal squares each having a different conductivity in this article their conjecture is proven the key idea is to superimpose suitably reflected potentials to obtain the solution to the dual problem this is then related back to the original problem using a well known theorem of keller thereby proving the conjecture the analysis also yields formulas relating the potentials in the four squares independently craster and obnosov have obtained a completely different proof of the conjecture

fast finite element calculation of effective conductivity of random continuum microstructures the recursive poincare steklov operator method abstract the effective conductivity of a block of composite can be extracted from the dirichlet to neumann poincare steklov operator pso for that block in this paper a domain decomposition method for computing the pso over a finite element mesh is discussed a new numerical strategy is introduced to accelerate the computation of this operator using the schur complement to calculate the pso for the smallest subdomains then recursively merging subdomain psos up to the full domain at each step of the algorithm information extraneous to the pso is discarded the effective conductivity values computed by this method are identical to those obtained from a basic finite element method an order of magnitude faster and with much less computer memory consumed as proof of concept effective conductivity measurements are presented for a percolating random fractal like microstructure across a range of phase fractions

multi symplectic lagrangian one dimensional gas dynamics the equations of lagrangian ideal one dimensional compressible gas dynamics are written in a multi symplectic form using the lagrangian mass coordinate m and time t as independent variables and in which the eulerian position of the fluid element x x m t is one of the dependent variables this approach differs from the eulerian multi symplectic approach using clebsch variables lagrangian constraints are used to specify equations for xm xt and st consistent with the lagrangian map where s is the entropy of the gas we require st 0 corresponding to advection of the entropy s with the flow we show that the lagrangian hamiltonian equations are related to the de donder weyl multi momentum formulation the pullback conservation laws and the symplecticity conservation laws are discussed the pullback conservation laws correspond to invariance of the action with respect to translations in time energy conservation and translations in m in noether s theorem the conservation law due to m translatio

periodic heterogeneous structures new explicit solutions and effective characteristics of refraction of an imposed field complex variable methods are applied to study three double periodic two phase planar heterogeneous structures explicit expressions of functionals of energy dissipation and of effective resistivities conductivities are found on the basis of strict analytical solutions of a boundary value problem for an arbitrary direction of an imposed external field comparisons with the results of bakhvalov and panasenko homogenization of processes in periodic media nauka moscow 1984 dykhne zh eksp teor fiz 59 1970 pp 110 113 soviet phys jetp 32 1971 pp 63 65 emets zh eksp teor fiz 96 1989 pp 701 711 soviet phys jetp 96 1989 keller j appl phys 34 1963 pp 991 993 and mendelson j appl phys 46 1975 pp 4740 4741 are made in particular the dissipation in the general case is shown essentially to depend on the direction of the external field for a rectangular checkerboard dissipation within the two phases does not coincide it is rigorously pr

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

multisymplectic conservation laws for differential and differential difference equations many well known partial differential equations can be written as multisymplectic systems such systems have a structural conservation law from which scalar conservation laws can be derived these c

vorticity and symplecticity in lagrangian fluid dynamics the relationship between potential vorticity pv and the symplectic form is explored for the shallow water equations governing lagrangian particle paths starting with the symplectic form the pv is found by the pull back operation to the reference space at first sight the encoding of pv in the symplectic form appears to be independent of the particle relabelling symmetry the analysis is carried a step further in two ways using the conservation of symplecticity as a starting point the fluxes of symplecticity arise as differential forms and a complete pull back of the flux forms leads to a geometric description of pv conservation secondly symmetry methods are used to give a rigorous connection between particle relabelling symplecticity and pv conservation generalizations of these issues to semi geostrophic flow and three dimensional lagrangian fluid flows and the connection with ertel s theorem are also discussed

a high order accurate algorithm for electrostatics of overlapping disks a fast and accurate algorithm for the computation of effective electric and mathematically equivalent properties of composites with nonsmooth interfaces is reported the algorithm is based on an integral equation reformulation of the electrostatic partial differential equation and a fast hierarchical technique for potential field evaluation in a numerical example 200 large and strongly inhomogeneous aggregates of randomly overlapping disks are solved with a relative error of 0 0005

upper and lower bounds for the overall properties of a nonlinear composite dielectric ii periodic microgeometry the study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix use of a nonlinear comparison material in the nonlinear hashin shtrikman variational principles permits the generation of both upper and lower bounds for any composite it is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the hashin shtrikman variational principle is finite everywhere the results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the hashin shtrikman principles

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

fourier based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields a modified green operator is proposed as an improvement of fou rier based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media contrary to other methods the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite furtherm ore it is shown that the method produces much more accurate local field s inside highly conducting and quasi insulating phases as well as in the vicinity of the phases interfaces these good propert ies stem from the discretization of green s function which is consistent with the pixel grid while retaining the local nature of the op erator that acts on the polarization field finally a fast im plementation of the direct scheme of moulinec et al 1994 that allows for parcimonious memory use is proposed

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

the symplectic evans matrix and the instability of solitary waves and fronts hamiltonian evolution equations which are equivariant with respect to the action of a lie group are models for physical phenomena such as oceanographic flows optical fibres and atmospheric flows and such systems often have a wide variety of solitary wave or front solutions in this paper we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts at the heart of the analysis is a multi symplectic formulation of hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions with a third symplectic structure with two form denoted by omega associated with a coordinate frame moving at the speed of the wave this leads to a geometric decomposition and symplectification of the evans function formulation for the linearization about solitary waves and fronts we introduce the concept of the symplectic evans matrix a matrix consisting of restricted omega symplectic forms by applying hodge duality to the exterior algebra formulation of the evans function we find that the zeros of the evans function correspond to zeros of the determinant of the symplectic evans matrix based on this formulation we prove several new properties of the evans function restricting the spectral parameter lambda to the real axis we obtain rigorous results on the derivatives of the evans function near the origin based solely on the abstract geometry of the equations and results for the large lambda behaviour which use primarily the symplectic structure but also extend to the non symplectic case the lie group symmetry affects the evans function by generating zero eigenvalues of large multiplicity in the so called systems at infinity we present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation by combining all these results a new rigorous sufficient condition for instability of solitary waves and fronts is obtained the theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as x tends to infinity to illustrate the theory it is applied to three examples a boussinesq model from oceanography a class of nonlinear schrodinger equations from optics and a nonlinear klein gordon equation from atmospheric dynamics

geometric variational discretization of continuum theories this study derives geometric variational discretization of continuum theories arising in fluid dynamics magnetohydrodynamics mhd and the dynamics of complex fluids a central role in these discretizations is played by the geometric formulation of fluid dynamics which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume preserving diffeomorphisms of the fluid domain inspired by this framework we construct a finite dimensional approximation to the diffeomorphism group and its lie algebra thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group the extension to mhd and complex fluid flow is then made through an appeal to the theory of euler poincare systems with advection which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters upon deriving a family of structured integrators for these systems we test their performance via a numerical implementation of the update schemes on a cartesian grid among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries automatic satisfaction of solenoidal constraints on vector fields good long term energy behavior robustness with respect to the spatial and temporal resolution of the discretization and applicability to irregular meshes

a statistically accurate modified quasilinear gaussian closure for uncertainty quantification in turbulent dynamical systems abstract we develop a novel second order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase space that possess a high dimensional chaotic attractor we focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent linearly unstable directions and a nonlinear energy transfer mechanism we first illustrate how existing uq schemes that rely on the gaussian assumption will fail to perform reliable uq in the presence of unstable dynamics to overcome these difficulties a modified quasilinear gaussian mqg closure is developed in two stages first we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes while preserving statistical symmetries in the second stage we develop a nonlinear mqg dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics our analysis uq schemes and conclusions are illustrated through a specific toy model the forty modes lorenz 96 system which despite its simple formulation presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes a suitable version of mqg successfully captures the mean and variance in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing very cheaply at the cost of roughly two ensemble members in a monte carlo simulation

low dimensional reduced order models for statistical response and uncertainty quantification barotropic turbulence with topography abstract a low dimensional reduced order statistical closure model is developed for quantifying the uncertainty to changes in forcing in a barotropic turbulent system with topography involving interactions between small scale motions and a large scale mean flow imperfect model sensitivity is improved through a recent mathematical strategy for calibrating model errors in a training phase where information theory and linear statistical response theory are combined in a systematic fashion to achieve the optimal model parameters statistical theories about a gaussian invariant measure and the exact statistical energy equations are also developed for the truncated barotropic equations that can be used to improve the imperfect model prediction skill a stringent paradigm model of 57 degrees of freedom is used to display the feasibility of the reduced order methods this simple model creates large scale zonal mean flow shifting directions from westward to eastward jets with an abrupt change in amplitude when perturbations are applied and prototype blocked and unblocked patterns can be generated in this simple model similar to the real natural system principal statistical responses in mean and variance can be captured by the reduced order models with desirable accuracy and efficiency with only 3 resolved modes an even more challenging regime with non gaussian equilibrium statistics using the fluctuation equations is also tested in the reduced order models with accurate prediction using the first 5 resolved modes these reduced order models also show potential for uncertainty quantification and prediction in more complex realistic geophysical turbulent dynamical systems

asymptotic analysis of quasi static transport in high constrast conductive media we show that transport in high contrast conductive media has a discrete behavior in the asymptotic limit of infinitely high contrast the effective impedance and the magnetic field in such media are given by discrete min max variational principles furthermore we show that the transport problem has an asymptotic resistor inductor capacitor network approximation we use new variational formulations of the effective impedance of the media and we assess the accuracy of the asymptotic approximation by numerical computations

the infinite dimensional evans function abstract we introduce generalized operator valued jost solutions of first order ill posed differential equations on hilbert spaces we then construct an infinite dimensional evans function for abstract differential equations as a 2 modified fredholm determinant of the operator obtained by adding the values at zero of the generalized operator valued jost solutions next we prove a formula that connects the 2 modified evans determinant and the 2 modified determinant related to the birman schwinger type operator associated to the ill posed equation using this formula we construct a holomorphic infinite dimensional evans function for second order differential operators on infinite cylinders whose zeros are the eigenvalues of the differential operators

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

isotropic conductivity of two dimensional three component symmetric composites the effective dc conductivity problem of isotropic two dimensional 2d three component symmetric regular composites is considered a simple cubic equation with one free parameter for e 1 2 3 is suggested whose solutions automatically have all the exactly known properties of that function numerical calculations on four different symmetric isotropic 2d three component regular structures show a non universal behaviour of e 1 2 3 with an essential dependence on micro structural details in contrast with the analogous two component problem the applicability of the cubic equation to these structures is discussed an extension of that equation to the description of other types of 2d three component structures is suggested including the case of random structures

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

the effective conductivity of random checkerboards an algorithm is presented for the fast and accurate solution of the electrostatic equation on multi component random checkerboards it relies on a particular choice of integral equation extended as to separate ill conditioning due to singular fields in corners from ill conditioning due to interaction of clusters of well conducting squares at large distances two separate preconditioners take care of the two separate phenomena in a series of numerical examples effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106 the achievable relative precision in these examples is on the order of 10 11

on multisymplecticity of partitioned runge kutta methods previously it has been shown that discretizing a multi hamiltonian pde in space and time with partitioned runge kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law however these previous studies use the same partitioning of the variables into two parts in both space and time this gives rise to a large number of cases to be considered each with its own set of conditions to be satisfied we present here a much simpler set of conditions covering all of these cases where the variables are partitioned independently in space and time into an arbitrary number of parts in general it is not known when such a discretization of a multi hamiltonian pde will give rise to a well defined numerical integrator however a numerical integrator that is explicit will typically be well defined in this paper we give sufficient conditions on a multi hamiltonian pde for a lobatto iiia iiib discretization in space to give rise to explicit odes and an algorithm for constructing these odes

finite analytic numerical method for two dimensional fluid flow in heterogeneous porous media starting from the investigation on the detailed flow pattern finite analytic numerical method is developed to solve the two dimensional fluid flows in heterogeneous porous media it is shown that only for some specific permeability distributions the pressure has the piecewise linear distribution where harmonic average scheme works very well in general case the pressure will have the power law behavior and its gradient will diverge as approaching the node joining the different permeability areas the nodal flow effects cause the flow fingering into the high permeability region it is a challenge problem to numerically describe the nodal fingering effects with the help of the specific properties of pressure and its gradient around the node a local analytical nodal solution is derived and then applied to construct a finite analytic numerical scheme numerical examples show that the detailed flow pattern can be reconstructed with the proposed numerical scheme under few grid refinements only with 2x2 or 3x3 subdivisions the proposed numerical scheme can provide rather accurate solutions the convergent speed of the numerical scheme is independent of the permeability heterogeneity in contrast the refinement ratio for the grid cell needs to be increased dramatically to get an accurate result when the traditional numerical method is used for strong heterogeneous cases

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

using statistical functionals for effective control of inhomogeneous complex turbulent dynamical systems abstract efficient statistical control strategies are developed for general complex turbulent systems with energy conserving nonlinearity instead of direct control on the high dimensional turbulent equations concerning a large number of instabilities a statistical functional that characterizes the total statistical structure of the complex system is adopted here as the control object first the statistical energy equation reduces the control of the complex nonlinear system to a linear statistical control problem then the explicit form of the forcing control is recovered through nonlocal inversion of the optimal control functional using approximate statistical linear response theory for attribution of the feedback through this control strategy with statistical energy conservation the explicit form of the control forcing is determined offline only requiring the initial configuration of total statistical energy change and the autocorrelation functions in the most sensitive modes of the target statistical equilibrium with no need of knowing the explicit forcing history and running the complex system the general framework of the statistical control method can be applied directly on various scenarios with both homogeneous and inhomogeneous perturbations the effectiveness of the statistical control strategy is demonstrated using the lorenz 96 system and a turbulent barotropic system with topography and a large number of instabilities

a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

expansion formulae for the homogenized determinant of anisotropic checkerboards in this paper some effective properties of anisotropic four phase periodic checkerboards are studied in two dimensional electrostatics an explicit low contrast second order expansion for the determinant of the effective conductivity is given in the case of a twophase checkerboard with commuting conductivities the expansion reduces to an explicit formula for the effective determinant valid for any contrast as soon as the second order term vanishes such an explicit formula cannot be extended to four phase checkerboards a counter example with high contrast conductivities is provided the construction of the counter example is based on a factorization principle due to astala nesi which allows us to pass from an anisotropic four phase square checkerboard to an isotropic one with the same effective determinant

multisymplectic preissman scheme for the time domain maxwell s equations from the bridges multisymplectic form of maxwell s equations we derive a multisymplectic preissman scheme which couples two time levels for 2 1 dimensional maxwell s equations the scheme is proven to preserve the discrete local energy exactly numerical results are reported to illustrate that the scheme is effective and it can get more precise numerical solutions than yee s scheme our numerical results can also indicate that the scheme keeps the discrete local energy and the global energy very well

multi symplectic formulation of near local hamiltonian balanced models we transform near local hamiltonian balanced models hbms describing nearly geostrophic fluid motion with constant coriolis parameter into multi symplectic ms systems this allows us to determine conservation of lagrangian momentum energy and potential vorticity for salmon s l1 dynamics a similar approach works for other near local balanced models such as the model the ms approach also enables us to determine a class of systems that have a contact structure similar to that of the semigeostrophic model the contact structure yields a contact transformation that makes the problem of front formation tractable the new class includes the first local model with a variable coriolis parameter that preserves all of the most useful geometric features of the semigeostrophic model

a fast direct solver for the integral equations of scattering theory on planar curves with corners we describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners it is rather comprehensive in that it applies to a wide variety of boundary value problems here we treat the neumann and dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids it achieves high accuracy is applicable to large scale problems and perhaps most importantly does not require asymptotic estimates for solutions instead the singularities of solutions are resolved numerically the approach is efficient however only in the low and mid frequency regimes once the scatterer becomes more than several hundred wavelengths in size the performance of the algorithm of this paper deteriorates significantly we illustrate our method with several numerical experiments including the solution of a neumann problem for the helmholtz equation given on a domain with nearly 10000 corner points

evans functions jost functions and fredholm determinants the principal results of this paper consist of an intrinsic definition of the evans function in terms of newly introduced generalized matrix valued jost solutions for general first order matrix valued differential equations on the real line and a proof of the fact that the evans function a finite dimensional determinant by construction coincides with a modified fredholm determinant associated with a birman schwinger type integral operator up to an explicitly computable nonvanishing factor

on the relationship between the eulerian and lagrangian descriptions of finite rigid plasticity the motivation for the present paper is to clarify certain unresolved issues pertaining to the relation between the lagrangian or referential and eulerian or spatial strain space formulations of finite plasticity for conceptual simplicity attention is confined to rigid plastic materials it is shown first that for constitutive equations in which the hardening parameter is a scalar the lagrangian and eulerian descriptions are equivalent and that additionally the choice of objective stress rate is immaterial in the light of these developments the role of objective rates is further explored in connection with more general anisotropic hardening laws which contain a shift tensor a form of the constitutive equation for the rate of the shift tensor is motivated in which the choice of objective rate is arbitrary it is then demonstrated that the structure of the constitutive equations of the theory in both the eulerian and lagrangian descriptions is form invariant under arbitrary transformations of objective rate the approach taken here contrasts with that adopted in a number of recent papers in which preference is given to one particular objective rate or another

computation of conservation laws for nonlinear lattices an algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented the algorithm is implemented in mathematica and can be used as an automated integrability test with the code diffdens m conserved densities are obtained for several well known lattice equations for systems with parameters the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist

on the lagrangian description of vorticity this paper exploits a lagrangian form of the vorticity field that is based on a formula of beltrami the merits of which do not seem to have been fully appreciated a number of new results are readily derived by using the lagrangian description and the classical vorticity theorems are also included the results being purely kinematical apply to all deformable media including viscous fluids and may also be of value in computations

an ensemble kalman filter for statistical estimation of physics constrained nonlinear regression models a central issue in contemporary science is the development of nonlinear data driven statistical dynamical models for time series of noisy partial observations from nature or a complex model it has been established recently that ad hoc quadratic multi level regression models can have finite time blow up of statistical solutions and or pathological behavior of their invariant measure recently a new class of physics constrained nonlinear regression models were developed to ameliorate this pathological behavior here a new finite ensemble kalman filtering algorithm is developed for estimating the state the linear and nonlinear model coefficients the model and the observation noise covariances from available partial noisy observations of the state several stringent tests and applications of the method are developed here in the most complex application the perfect model has 57 degrees of freedom involving a zonal east west jet two topographic rossby waves and 54 nonlinearly interacting rossby waves the perfect model has significant non gaussian statistics in the zonal jet with blocked and unblocked regimes and a non gaussian skewed distribution due to interaction with the other 56 modes we only observe the zonal jet contaminated by noise and apply the ensemble filter algorithm for estimation numerically we find that a three dimensional nonlinear stochastic model with one level of memory mimics the statistical effect of the other 56 modes on the zonal jet in an accurate fashion including the skew non gaussian distribution and autocorrelation decay on the other hand a similar stochastic model with zero memory levels fails to capture the crucial non gaussian behavior of the zonal jet from the perfect 57 mode model

effective medium approximation and exact formulae for electrokinetic phenomena in porous media electrokinetic phenomena in porous media are studied by application of the effective medium theory and the theory of duality transformation we deduce new exact relations and analytical formulae for the effective constants of the macroscopic tensor we also prove that the effective tensors obtained by these approaches coincide for 2d problems the obtained results for the electrokinetic processes are closely related to similar results derived for piezoelectric composites because of a common mathematical background

a new reiterated structure with optimal macroscopic behavior we consider homogenization of sequences of integral functionals of the type mathcal f h u int omega f h x du x dx omega subset mathbf r n the functions fh are defined in the first variable via a new reiterated structure with m microlevels in particular our results imply that the corresponding homogenized functional becomes optimal as m turns to infinity both the scalar case the conductivity problem and the vector valued case the elasticity problem are considered

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

scaling symmetries conservation laws and action principles in one dimensional gas dynamics scaling symmetries of the planar one dimensional gas dynamic equations with adiabatic index are used to obtain lagrangian and eulerian conservation laws associated with the symmetries the known eulerian symmetry operators for the scaling symmetries are converted to the lagrangian form in which the eulerian spatial position of the fluid element is given in terms of the lagrangian fluid labels conditions for a linear combination of the three scaling symmetries to be a divergence or variational symmetry of the action are established the corresponding lagrangian and eulerian form of the conservation laws are determined by application of noether s theorem a nonlocal conservation law associated with the scaling symmetries is obtained by applying a nonlocal symmetry operator to the scaling symmetry conserved vector an action principle incorporating known conservation laws using lagrangian constraints is developed noether s theorem for the constrained action principle gives the same formulas for the conserved vector as the classical noether theorem except that the lie symmetry vector field now includes the effects of nonlocal potentials noether s theorem for the constrained action principle is used to obtain nonlocal conservation laws the scaling symmetry conservation laws only apply for special forms of the entropy of the gas

analytical and numerical analysis of lensing effect for linear surface water waves through a square array of nearly touching rigid square cylinders this paper describes transport properties of linear water waves propagating within a square array of fixed square cylinders the main focus is on achieving the conditions for all angle negative refraction aanr thanks to anomalous dispersion in fluid filled periodic structures of particular interest are two limit cases when either the edges or the vertices of the cylinders come close to touching in the former case the array can be approximated by a lattice of thin water channels for which dispersion curves are given in closed form and thus frequencies at which aanr occurs whereas in the latter case the array behaves as a checkerboard with cells consisting either of water tanks or rigid cylinders for which standing modes are given in closed form the tools of choice for the present analysis are on the one hand the finite element method which solves numerically spectral problems in periodic media and on the other hand a two scale asymptotic method which provides estimates of dispersion curves and associated eigenfields through a lattice approximation namely thin water channels between rigid cylinders simple duality correspondences are found based on fourfold symmetry of square water checkerboards that allow us to get some insight into their spectra last some numerical evidence is provided for water waves focusing with no astigmatism through such arrays when they are of finite extent

checkerboard composites with separated phases recently the authors have developed a method siam j appl math 61 1839 1556 2001 capable of solving in closed form boundary value problems for four phase doubly periodic checkerboard composites with continuity between the different phases the method is based upon a novel conformal mapping that preserves the doubly periodic nature of the physical problem the aim of the current article is to explore generalizations of that approach where we now replace continuity between some phases by nonconducting or perfectly conducting strips thereby modeling debonding or electrodes in electrochemical devices the specific objective is to determine effective resistivities and related parameters for these four phase objects in a concise and explicit form

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

exact solution of a boundary value problem for a rectangular checkerboard field a class of two phase composite materials with a biperiodic structure is investigated by the methods of complex analysis two interface conditions continuity of normal component of a desired vector w and tangential component of w at the contact boundary as well as the double periodicity condition are involved in rigorous form the exact analytic solution of the corresponding generalized riemann boundary value problem is obtained the explicit values of the effective parameters namely effective resistivity and dissipation of energy of an elementary cell and resistivities along the symmetry axes are calculated in closed analytic form the coincidence of our formulae with the well known effective resistivity conductivity formula of keller 1964 dykhne 1970 and mendelson 1975 and the dissipation formula of dykhne 1970 is shown in the case of square checkerboard field the keller 1963 identity is generalized for the heterogeneous structure studied

a three phase tessellation solution and effective properties two dimensional doubly periodic three phase structures are considered in the situation where mean fluxes are applied across the structure the approach is to utilize complex variables and to use a mapping that reduces the doubly periodic problem to a m uch simpler one involving joined sectors this is a model composite structure in electrostatics and mathematically analogous areas such as porous media anti plane elasticit y heat conduction and we find various eectiv e parameters and investigate limiting cases the structure is also amenable to asymptotic methods in the case of highly varying composition and we provide these solutions partly as a check upon our analysis and partly as they are useful in their own right

conduction in a two phase plane with diamond shaped tiling the effective conductivity of a two phase two dimensional composite with diamond shaped tiling is considered this analysis based on a projection of the boundary conditions on linear combinations of solutions to the electrostatic equation that are orthonormal on the boundary generalizes results by keller j math phys 28 2516 1987 and others numerical results are given for several conductivity ratios of the two phases and for varying obtuse angles of the tiles special emphasis is given to very large and very small conductivity differences and very elongated tiles

the effective conductivity of arrays of squares large random unit cells and extreme contrast ratios an integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two component checkerboard like composites with complicated unit cells at very high contrast ratios the scheme extends recent work on multi component checkerboards at medium contrast ratios general improvement include the simplification of a long range preconditioner the use of a banded solver and a more efficient placement of quadrature points this together with a reduction in the number of unknowns allows for a substantial increase in achievable accuracy as well as in tractable system size results accurate to at least nine digits are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106 furthermore the scheme is flexible enough to handle complex valued conductivities and using a homotopy method purely negative contrast ratios examples of the accurate computation of resonant spectra are given

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

convergence of iterative methods based on neumann series for composite materials theory and practice iterative fast fourier transform methods are useful for calculating the fields in composite materials and their macroscopic response by iterating back and forth until convergence the differential constraints are satisfied in fourier space and the constitutive law in real space the methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli it is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative fast fourier transform methods we look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity obnosov theoretically some of the methods converge when the inclusions have zero or even negative conductivity however the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations there is little point in iterating beyond this accuracy improves when the grid size is reduced showing that the discrepancy is linked to the discretization finally it is shown that none of the three iterative schemes investigated over performs the others for all possible microstructures and all contrasts

on effective conductivity on z d lattice we study the effective conductivity e for a random wire problem on the d dimensional cubic lattice d d 2 in the case when random conductivities on bonds are independent identically distributed random variables we give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order in the 2d case using the duality symmetry we also derive the 6th order expansion we compare our results with the bruggeman approximation and show that in the 2d case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms

extensions of noether s second theorem from continuous to discrete systems a simple local proof of noether s second theorem is given this proof immediately leads to a generalization of the theorem yielding conservation laws and or explicit relationships between the euler lagrange equations of any variational problem whose symmetries depend on a set of free or partly constrained functions our approach extends further to deal with finite difference systems the results are easy to apply several well known continuous and discrete systems are used as illustrations

network approximation for transport properties of high contrast materials we show that the effective complex impedance of materials with conductivity and dielectric permittivity that have high contrast can be calculated approximately by solving a suitable resistor capacitor network we use variational principles extensively for the analysis and we assess the accuracy of the network approximation by numerical computations

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations