critical exponents of the 3d antiferromagnetic three state potts model using the coherent anomaly method abstract the antiferromagnetic three state potts model on the simple cubic lattice is studied using the coherent anomaly method cam the cam analysis provides the estimates for the critical exponents which indicate the xy universality class namely 0 011 0 351 1 309 and 4 73 this observation corroborates the results of the recent monte carlo simulations and disagrees with the proposal of a new universality class

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

conditional lie backlund symmetry second order differential constraint and direct reduction of diffusion systems abstract the equivalence relation between conditional lie backlund symmetry higher order differential constraint and direct reduction is presented in the case of evolution systems the second order nonlinear conditional lie backlund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional lie backlund symmetry of evolution systems consequently the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems

high temperature series for scalar field lattice models generation and analysis an implementation of the free embedding scheme for high temperature series generation on the body centered cubic family of lattices in arbitrary dimensiond is described series to order 21 in inverse temperature are tabulated for several scalar field models both for the magnetic susceptibility and for the second moment of the spin correlation function the critical behavior of a family of 3 dimensional double gaussian models which interpolate continuously between the spin 1 2 ising model and the gaussian model is analyzed in detail away from the gaussian model limit using confluent inhomogeneous secondorder differential approximants with our best estimate of the correction to scaling exponent 0 52 0 03 the leading exponents for the susceptibility and correlation length for this family are consistent with universality and are given by 1 237 0 002 and 0 630 0 0015 respectively and 2 0 0359 0 0007

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

nonlinear systems of the burgers type equations lie and q conditional symmetries ansatze and solutions abstract a class of nonlinear diffusion convection systems containing two burgers type equations is considered a complete description of lie symmetries is obtained for these systems new results of finding q conditional symmetries are also presented moreover a variety of lie and non lie ansatze and exact solutions of a diffusion convection system invariant under the generalized galilei algebra are constructed

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

accurate estimates of 3d ising critical exponents using the coherent anomaly method abstract an analysis of the critical behavior of the three dimensional ising model using the coherent anomaly method cam is presented various sources of errors in cam estimates of critical exponents are discussed and an improved scheme for the cam data analysis is tested using a set of mean field type approximations based on the variational series expansion approach accuracy comparable to the most precise conventional methods has been achieved our results for the critical exponents are given by 0 108 5 0 327 4 1 237 4 and 4 77 5

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

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

reduction operators of burgers equation abstract the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed a new proof of the theorem on the special no go case of regular reduction operators is presented and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described any lie reduction of the burgers equation proves to be equivalent via the hopf cole transformation to a parameterized family of lie reductions of the linear heat equation

quasi linear geostatistical theory for inversing a quasi linear theory is presented for the geostatistical solution to the inverse problem the archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations the unknown is parameterized as a realization of a random field and the estimation problem is solved in two phases structural analysis where the random field is characterized followed by estimation of the log transmissivity conditional on all observations the proposed method generalizes the linear approach of kitanidis and vomvoris 1983 the generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements i e there are enough observations that the variance of estimation error of the log transmissivity is small the methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent the applicability of the methodology is demonstrated through an example that includes structural analysis determination of best estimates and conditional simulations

on the effective viscoelastic moduli of two phase media i rigorous bounds on the complex bulk modulus the dynamic response of isotropic composites of two viscoelastic isotropic phases mixed in fixed proportions is considered in the frequency range where the acoustic wavelength is much larger than the inhomogeneities the effective bulk modulus bounds of hashin shtrikman walpole are extended to viscoelasticity in this quasi static regime where the properties of the isotropic composite can be described by complex bulk and shear moduli the effective bulk modulus is shown to be constrained to a lens shaped region of the complex plane bounded by the outermost pair of four circular arcs three circular arcs in the case of two dimensional elasticity this is proved using a new variational principle for viscoelasticity together with two established techniques for deriving bounds on effective moduli namely the translation method and the hashin shtrikman method in this application the hashin shtrikman method needs to be generalized to allow the reference tensor to have an associated quasiconvex energy microstructures are identified which have bulk moduli that correspond to various points on each of the circular arcs thus these microstructures have extremal viscoelastic behaviour when the associated arc forms one of the outermost pair the bounds and the extremal microstructures are similar to those obtained for the complex dielectric constant but the methods used here are entirely different

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

symmetries for certain coupled nonlinear schrodinger equations generalized symmetries are sought for certain coupled nonlinear schrodinger equations it is shown that all lie backlund symmetries but one are equivalent to lie point symmetries

nonclassical symmetry reductions of a porous medium equation with convection in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method the asymptotic behaviour of some of the new solutions is analysed

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

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

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

classical point symmetries of a porous medium equation the lie group formalism is applied to deduce symmetries of the porous medium equation we study those spatial forms that admit the classical symmetry group the reduction obtained from the optimal system of subalgebras are derived some new exact solutions can be obtained

the high temperature susceptibility and spin spin correlation function of the three dimensional ising model the series analysis protocol developed in a preceding paper ibid vol 20 p 1839 1987 is applied to the susceptibility and correlation function of the three dimensional ising model for the spin 1 2 ising data the author argues that the protocol needs modification to take account of trends in the data which trends arise from the nature of the series in this way the author obtains gamma 1 239 or 0 003 for all cubic lattices and for a variety of spin values the diamond lattice data lies outside this range and it is argued that this series is effectively far shorter than the others the author also estimates nu 0 632 0 002 0 003 for the bcc lattice and obtains critical point estimates for all lattice and spin models studied

a hybrid numerical method for high contrast conductivity problems we introduce and test extensively a numerical method for computing efficiently current flow in media with high contrast conductivity this method combines our analytical understanding of the form of the flow field near narrow channels with standard numerical methods elsewhere in the flow regime and so it is a hybrid numerical method

series studies of the potts model i the simple cubic ising model the finite lattice method of series expansion is generalized to the q state potts model on the simple cubic lattice it is found that the computational effort grows exponentially with the square of the number of series terms obtained unlike two dimensional lattices where the computational requirements grow exponentially with the number of terms for the ising q 2 case the authors have extended the low temperature series for the partition functions magnetization and zero field susceptibility to u26 from u20 the high temperature series for the zero field partition function is extended from nu 18 to nu 22 subsequent analysis gives critical exponents in agreement with those from field theory

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

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

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

continuous sequence of mean field approximations and critical phenomena abstract the coherent anomaly method introduced by suzuki in 1986 provides in principle a remarkably simple approach to study critical phenomena within the framework of the mean field approximation unlike the renormalization group techniques commonly employed to investigate critical phenomena suzuki s method exploits the systematic behavior of a sequence of mean field approximations in order to extract critical temperature and exponents despite its conceptual simplicity actual implementation of the cam requires the ability to treat at least three levels of mean field approximations belonging to what suzuki has termed a cardinal sequence here we present a new method based on a continuous sequence of mean field approximations from which the implementation of the cam proceeds in a straightforward manner

variational principles for complex conductivity viscoelasticity and similar problems in media with complex moduli linear processes in media with dissipation arising in conductivity optics viscoelasticity etc are considered time periodic fields in such media are described by linear differential equations for complex valued potentials the properties of the media are characterized by complex valued tensors for example by complex conductivity or complex elasticity tensors variational formulations are suggested for such problems the functionals whose euler equations coincide with the original ones are constructed four equivalent variational principles are obtained two minimax and two minimal ones the functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation the last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors

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

nonclassical symmetry solutions and the methods of bluman cole and clarkson kruskal nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of bluman and cole and clarkson and kruskal consistency conditions will be provided to show that if satisfied these two methods are equivalent in the sense that they lead to the same symmetry solutions the boussinesq equation and burgers equation are used as illustrative examples exact solutions one of which is new will be presented for burgers equation obtained from the bluman and cole method yet not obtainable by clarkson and kruskal s method

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

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

extrapolation cam theory for critical exponents we propose and test a new method for generating canonical sequences for analysis by the coherent anomaly method cam from non mean field approximations by intentionally underestimating the rate of convergence of exact diagonalization values for the mass or energy gaps of finite systems we form families of sequences of gap estimates the gap estimates cross zero with generically nonzero linear terms in their taylor expansions so that for each member of these sequences of estimates thus the cam can be used to determine our freedom in deciding exactly how to underestimate the convergence allows us to choose the sequence that displays the clearest coherent anomaly we demonstrate this approach on the two dimensional ferromagnetic ising model for which we also use it on the three dimensional ferromagnetic ising model finding in good agreement with other estimates finally we apply it to an antiferromagnetic spin 1 heisenberg chain finding at the phase transition between the haldane phase and the dimerized phase in agreement with the field theoretic prediction although the specific systems used to test the extrapolation cam procedure involve finite system sizes the method could be applied to other finite approximations such as systematic variational approximations

correlation function in ising models abstract we simulated the fourier transform of the correlation function of the ising model in two and three dimensions using a single cluster algorithm with improved estimators the simulations are in agreement with series expansion and the available exact results in d 2 which shows that the cluster algorithm can successfully be applied for correlations we show as a further result that our data do not support a hypothesis of fisher that in any d 2 lattice the fourier transform of the correlation function depends on the lattice generating function only in d 3 our simulation are again in agreement with the results from the series expansion except for the amplitudes f where we find f f 2 06 1

simultaneous analysis of several models in the three dimensional ising universality class we investigate several three dimensional lattice models believed to be in the ising universality class by means of monte carlo methods and finite size scaling these models include spin 1 models with nearestneighbor interactions on the simple cubic and on the diamond lattice for the simple cubic lattice we also include models with third neighbor interactions of varying strength and some equivalent neighbor models also included are a spin 1 model and a hard core lattice gas separate analyses of the numerical data confirm the ising like critical behavior of these systems on this basis we analyze all these data simultaneously such that the universal parameters occur only once this leads to an improved accuracy the thermal magnetic and irrelevant exponents are determined as y t51 5868 3 y h52 4816 1 and y i520 821 5 respectively the binder ratio is estimated as q5 m 2 2 m 4 50 62 341 4

diffusion in turbulence we prove long time diffusive behavior homogenization for convection diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix simple shear flow examples show that this result is sharp for flows that have stationary stream matrices

remarks on percolation and transport in networks with a wide range of bond strengths abstract in conventional percolation problems there is no simple relation between the transport exponents and the critical exponents which describe geometric properties of the percolating clusters in several problems involving bonds with a wide range of conductance however it has been found that there does exist a simple relation between the transport exponent and the geometric exponent describing the divergence of the correlation length we give a brief review of these problems

finite size scaling of the three dimensional conserved order parameter ising model via wang landau algorithm abstract the critical properties of the conserved order parameter cop version of the three dimensional ising model with zero magnetization were investigated by means of the monte carlo mc wang landau algorithm the study was carried out in appropriate restricted but dominant energy subspaces the main effort was focused on the specific heat and binder energy cumulant by using the density of states of the model for lattices with linear size l 4 60 by applying finite size scaling analysis the estimated values for the thermal critical exponent y t and critical temperature k c are in agreement with the considered as best values leading to the conclusion that this model belongs to the ising universality class the analysis revealed the necessity of corrections to scaling functions for interpreting correctly the mc data

finite element network approximation of conductivity in particle composites a new finite element method computes conductivity in some unstructured particle reinforced composite material the 2 phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed the mathematical model is a poisson type problem with discontinuous coefficients the discontinuities are huge in contrast and quantity the proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure important geometric parameters such as the volume fraction are preserved exactly the computational complexity of the method is almost proportional to the number of inclusions this is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity the discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium

solutions to nonlinear partial differential equations from symmetry enhancing and symmetry preserving constraints abstract we show how solutions to practical partial differential equations can be found by classical symmetry reductions of a larger system of equations including constraints that result in the enlarged system having a larger symmetry group or an identical symmetry group compared to the original target equation symmetry enhancing constraints provide additional similarity solutions beyond those of the standard lie algorithm for scalar equations symmetry preserving constraints enable solutions to be found more easily after reduction of variables examples are given for the cylindrical boundary layer equations and for a navier stokes formulation of schrodinger wave mechanics

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

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

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

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

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

on the determining equations for the nonclassical reductions of the heat and burgers equation abstract the determining equations for the nonclassical reductions of the heat and burgers equations are considered it is shown that both systems belong to a burgers equation hierarchy each system is written in terms of the same matrix burgers equation that is linearized via a matrix hopf cole transformation in essence it is shown that both systems can be solved simultaneously their respective solutions are then presented in a very compact form

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

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

approximate similarity reduction for singularly perturbed boussinesq equation via symmetry perturbation and direct method we investigate the singularly perturbed boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method the similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods series reduction solutions are consequently derived for the approximate symmetry perturbation method similarity reduction equations of the zero order are equivalent to the painleve iv painleve i and weierstrass elliptic equations for the approximate direct method similarity reduction equations of the zero order are equivalent to the painleve iv painleve ii painleve i or weierstrass elliptic equations the approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method

convection enhanced diffusion for random flows we analyze the effective diffusivity of a passive scalar in a two dimensional steady incompressible random flow that has mean zero and a stationary stream function we show that in the limit of small diffusivity or large peclet number with convection dominating there is substantial enhancement of the effective diffusivity our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory some of which are widely believed to be correct but have not been proved yet we show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of isichenko and kalda that the effective diffusivity behaves like 3 13 when the molecular diffusivity is small assuming some percolation theoretic facts we also analyze the effective diffusivity for a special class of convective flows random cellular flows where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

extended symmetry analysis of generalized burgers equations using enhanced classification techniques we carry out the extended symmetry analysis of the class of generalized burgers equations of the form ut uux f t x uxx 0 this enhances all the previous results on symmetries of these equations and includes the description of admissible transformations lie symmetries lie and nonclassical reductions hidden symmetries conservation laws potential admissible transformations and potential symmetries the study is based on the fact that the class is normalized and its equivalence group is finite dimensional

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

no free ends method for lattice animals and vertex models with arbitrary number of states abstract a new no free ends method for generating power series expansions for vertex models with arbitrary number of states is described it is based on first order differential equations providing a recurrent relation connecting the free ends part of the series expansion in the n 1 th order with the nth and n 1 th order of the complete expansion the number of necessary no free ends graphs can be further reduced for symmetric systems by eliminating the nodes which have two neighbors and their incident bonds are in different states

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

nonclassical potential symmetries for the burgers equation abstract in this work classes of symmetries for partial differential equations which can be written in a conserved form are found these nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation some of these symmetries are carried out for the burgers equation these symmetries are realized as the nonclassical symmetries of the associated potential system and yield many new nonlocal generators and new explicit solutions of the burgers equation

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

critical temperatures of the d 3 s 1 2 ising model the effect of confluent corrections to scaling much attention has been paid to the d 3 s 1 2 ising model on the bcc lattice in recent years because the best high temperature series expansions were available for this lattice however in order to compare series estimates of critical exponents and temperatures with independent evaluations via the monte carlo method lattices of lower coordination number must be considered since monte carlo studies are usually made on these a study of extant susceptibility series on the fcc sc diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent monte carlo analyses of the d 3 s 1 2 ising model on the sc and diamond lattices

ising universality in three dimensions a monte carlo study we investigate three ising models on the simple cubic lattice by means of monte carlo methods and finite size scaling these models are the spin 1 2 ising model with nearest neighbour interactions a spin 1 2 model with nearest neighbour and third neighbour interactions and a spin 1 model with nearest neighbour interactions the results are in accurate agreement with the hypothesis of universality analysis of the finite size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field we find that the correction to scaling amplitudes are strongly dependent on the introduction of further neighbour interactions or a third spin state in a spin 1 ising model these corrections appear to be very small this is very helpful for the determination of the universal constants of the ising model the renormalization exponents of the ising model are determined as yt 1 587 2 yh 2 4815 15 and yi 0 82 6 the universal ratio q m2 2 m4 is equal to 0 6233 4 for periodic systems with cubic symmetry the critical point of the nearest neighbour spin 1 2 model is kc 0 2216546 10

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

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

anisotropic generalization of stinchcombe s solution for the conductivity of random resistor networks on a bethe lattice our study is based on the work of stinchcombe 1974 j phys c solid state phys 7 179 and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic bethe lattice the structure of the bethe lattice is assumed to represent the normal directions of the regular lattice we calculate the anisotropic conductivity as an expansion in powers of the inverse coordination number of the bethe lattice the expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold we make a comparison of our analytical results with those of bernasconi 1974 phys rev b 9 4575 for the regular lattice

high temperature series expansions for the 2 1 dimensional ising model using efficient cluster expansion methods the known high temperature series for the vacuum energy specific heat susceptibility and mass gap of the 2 1 dimensional ising model on the square and triangular lattices have been extended by several terms estimates of the critical indices demonstrate very convincing universality with the euclidean version of the model

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

non classical symmetries and the singular manifold method revisited the connection between the singular manifold method painleve expansions truncated at the constant term and symmetry reductions of two members of a family of cahn hilliard equations is considered the conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations and is thus shown to be invalid in general given that several earlier examples demonstrate the connection between the two techniques in some cases it now becomes necessary to establish when such a relationship exists a question related to a deeper understanding of painleve analysis this issue is also briefly discussed

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

confluent singularities in 3d continuum φ4 theory resolving critical point discrepancies a new analysis of the three dimensional 4 continuum model series for critical exponents and amplitude ratios is presented by explicitly allowing for a second correction to scaling term with exponent 2 and 2 1 2 0 a revision is found of previous critical point estimates that are all in the direction of resolving the remaining discrepancies between the 3d and 4 analyses the new estimates are also consistent with those obtained from high temperature series analyses of 3d lattice ising models for example the exponents 1 238 0 630 0 0355 and correction to scaling amplitude ratio 0 77 are easily accommodated and possibly even preferred detailed verification will require further terms both in the low order perturbation theory and the high order asymptotic expansions

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