the effective conductivity of an infinitely interchangeable mixture we are concerned with the estimation of the effective electrical conductivity of random heterogenous materials the purpose of this paper is to discuss a property of statistical symmetry verified by the symmetric cell materials of miller this property will be referred to as infinite interchangeability the usual way to approach cell materials is through n point correlation functions the property of infinite interchangebility permits us to approach cell materials from a completely different point of view our main result is a simple algorithm based on this symmetry property for computing any coefficient of the perturbation expansion in terms of information from the dilute limit specifically knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order 2r 1 this result which was previously known for r 2 in the isotropic case and for r 1 in the anisotropic case can also be obtained from the standard correlation function approach as pointed out by milton

first passage percolation network flows and electrical resistances we show that the first passage times of first passage percolation on 2 are such that p 0n n decay geometrically as n where may represent any of the four usual first passage time processes the former estimate requires no moment condition on the time coordinates but there exists a geometrically decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin here is the time constant and 0 we study the line to line first passage times and describe applications to the maximum network flow through a randomly capacitated subsection of 2 and to the asymptotic behaviour of the electrical resistance of a subsection of 2 when the edges of the subsection are wires in an electrical network with random resistances in the latter case we show for example that if each edge resistance equals 1 or ohms with probabilities p and 1 p respectively then the effective resistance rn across opposite faces of an n by n box satisfies the following a if p 1 2 then p rn 1 as n b if p 1 2 then there exists p such that p p 1 underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim inf r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim sup r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant v p 1

inverse bounds and bulk properties of complex valued two component composites the bulk properties of composites are known to depend strongly on the microstructure this dependence can be quantified in terms of a representation introduced by d bergman which factorizes the geometry dependence from the contrast based on this analytic representation of the effective permittivity we present a general scheme to estimate the microstructural parameters such as the volume fraction and the anisotropy of two component composites the estimates are given as bounds that is the largest parameter region which is compatible with the available information thus more information produces better estimates on the microstructural parameters the method which uses complex valued measurements of bulk properties of the composite is illustrated by numerical examples

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

percolation threshold for bruggeman composites using a simple phenomenological approach we calculate the percolation threshold for bruggeman composites having microgeometry of two kinds both kinds of composites consist of spheroids whose shape follows the beta distribution at the same time the first one is a mixture of spheroids equally oriented along their revolution axis in this case the percolation threshold is shown to be the same as for an assembly of equally oriented identical spheroids whose shape corresponds to the most probable shape of the distribution for such composites the percolation threshold can vary between 0 and 1 the second one is a random mixture of the spheroids in this case the percolation threshold is expressed in terms of the gauss hypergeometric function it is shown to vary between 0 and 1 3 the derived analytical results are supplemented with numerical calculations carried out for different values of the beta distribution parameters

the ising model and critical behavior of transport in binary composite media we present a general theory for critical behavior of transport in binary composite media the theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi static case frequency dependent with complex parameters through a direct analytic correspondence between the magnetization of the ising model and the effective parameter problem of two phase random media we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and at 1 pc this is analogous to the lee yang ruelle characterization of the ising model phase transition and identifies these transport transitions with the collapse of spectral gaps in these measures

sub wavelength plasmonic crystals dispersion relations and effective properties we obtain a convergent power series expansion for the first branch of the dispersion relation for sub wavelength plasmonic crystals consisting of plasmonic rods with frequency dependent dielectric permittivity embedded in a host medium with unit permittivity the expansion parameter is h kd 2pd l where k is the norm of a fixed wavevector d is the period of the crystal and l is the wavelength and the plasma frequency scales inversely to d making the dielectric permittivity in the rods large and negative the expressions for the series coefficients also called dynamic correctors and the radius of convergence in h are explicitly related to the solutions of higher order cell problems and the geometry of the rods within the radius of convergence we are able to compute the dispersion relation and the fields and define dynamic effective properties in a mathematically rigorous manner explicit error estimates show that a good approximation to the true dispersion relation is obtained using only a few terms of the expansion the convergence proof requires the use of properties of the catalan numbers to show that the series coefficients are exponentially bounded in the h 1 sobolev norm

matroids are immune to braess paradox the famous braess paradox describes the counterintuitive phenomenon in which in certain settings an increase of resources such as a new road built within a congested network may in fact lead to larger costs for the players in an equilibrium in this paper we consider general nonatomic congestion games and give a characterization of the combinatorial property of strategy spaces for which the braess paradox does not occur in short matroid bases are precisely the required structure we prove this characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest

mixed nash equilibria in selfish routing problems with dynamic constraints we study the problem of routing traffic through a congested network consisting of m parallel links each having a certain speed moreover we are given n selfish non cooperative agents each of them willing to route her own piece of traffic on exactly one link agents are selfish in that they only pick a link which minimize the delay of their own piece of traffic in this context much effort has been lavished in the framework of mixed nash equilibria where the agent s routing choices are regulated by probability distributions one for each agent which let the system thus enter a steady state from which no agent is willing to unilaterally deviate in this work we consider situations in which some agents have constraints on the routing choice in a sense they are forbidden to route their traffic on some links we show that at most one nash equilibrium may exist and in some cases with equal speed links and where each agent is forbidden to route on at most one link we give necessary and sufficient conditions on its existence these conditions correlate the traffic load of the agents we consider also a dynamic behaviour of the network when the constraints may vary in particular when a constraint is removed we establish under which conditions the network is still in equilibrium these conditions are all effective in the sense that given a set of yes no routing constraints on each link for each agent we provide the probability distributions corresponding to the unique nash equilibrium associated to the constraints if it exists moreover these conditions and the possible nash equilibrium are computed in time o mn

spectral functions for two dimensional random conductors abstract we collect together a number of results for the spectral function for two dimensional random conductors exact solutions are available for the conductivity of sheets containing a few elliptical inclusions we show that exact results can also be obtained for polygonal holes using conformal mapping techniques for polygonal inclusions with arbitrary conductivity the conductivity can be expressed in terms of a spectral function which we have obtained numerically using a boundary element method in this method the singularities associated with the sharp corners are explicitly taken care of spectral functions can also be used to obtain an exact solution for the conductivity when the effect of pairs of circular inclusions is also included throughout this paper we express the conductivity of the composite in terms of the total induced dipole moment and exploit the simplifications due to the reciprocity theorem in two dimensions

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

effective elastic properties of random two dimensional composites abstract consider 2d two phase random composites with circular inclusions of concentration f new analytical formulae for the effective constants are deduced up to o f 4 for macroscopically isotropic composites it is shown that the second order terms o f 2 do not depend on the location of inclusions whilst the third order terms do this implies that the previous analytical methods effective medium approximation differential scheme mori tanaka approach and so forth can be valid at most up to o f 3 for macroscopically isotropic composites first the local elastic field for a finite number n of inclusions arbitrarily located on the plane are found by a method of functional equations further the limit n yields conditionally convergent series defined by the eisenstein summation method one of the series for periodic composites is the famous lattice sum s 2 deduced by rayleigh for a conductivity problem

the uniqueness property for networks with several origin destination pairs abstract we consider congestion games on networks with nonatomic users and user specific costs we are interested in the uniqueness property defined by milchtaich 2005 as the uniqueness of equilibrium flows for all assignments of strictly increasing cost functions he settled the case with two terminal networks as a corollary of his result it is possible to prove that some other networks have the uniqueness property as well by adding common fictitious origin and destination in the present work we find a necessary condition for networks with several origin destination pairs to have the uniqueness property in terms of excluded minors or subgraphs as a key result we characterize completely bidirectional rings for which the uniqueness property holds it holds precisely for nine networks and those obtained from them by elementary operations for other bidirectional rings we exhibit affine cost functions yielding to two distinct equilibrium flows related results are also proven for instance we characterize networks having the uniqueness property for any choice of origin destination pairs

multicomponent composites electrical networks and new types of continued fraction i the development of bounds on the complex effective conductivity tensor that relates the average current to the average electric field in a multicomponent composite has been hindered by lack of a suitable continued fraction representation for here a new field equation recursion method is developed which gives an expression for as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry a hierarchy of field equations is set up such that the solutions of the j 1 th order equation generate the solutions of thejth order equation consequently the effective tensor j associated with thejth order field equation is expressible as a fractional linear matrix transformation of j 1 these transformations combine to form the continued fraction expansion for 0 which is exploited in the following paper part ii to obtain bounds crude bounds on j forj 1 give narrow bounds on the continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable stieltjes functions that proved important in the development of the theory of pade approximants asymptotic analysis and the theory of orthogonal polynomials in the last century the results extend to other transport problems including conduction in polycrystalline media the viscoelasticity of composites and the response of multicomponent multiterminal linear electrical networks

topological uniqueness of the nash equilibrium for selfish routing with atomic users we consider the problem of selfish routing in a congested network shared by several users where each user wishes to minimize the cost of its own flow users are atomic in the sense that each has a nonnegligible amount of flow demand and flows may be split over different routes the total cost for each user is the sum of its link costs which in turn may depend on the user s own flow as well as the total flow on that link our main interest here is network topologies that ensure uniqueness of the nash equilibrium for any set of users and link cost functions that satisfy some mild convexity conditions we characterize the class of two terminal network topologies for which this uniqueness property holds and show that it coincides with the class of nearly parallel networks that was recently shown by milchtaich milchtaich i 2005 topological conditions for uniqueness of equilibrium in networks math oper res 30 225 244 to ensure uniqueness in nonatomic or wardrop routing games we further show that uniqueness of the link flows holds under somewhat weaker convexity conditions which apply to the mixed nash wardrop equilibrium problem we finally propose a generalized continuum game formulation of the routing problem that allows for a unified treatment of atomic and nonatomic users

statistical mechanics of conducting phase transitions the critical behavior of the effective conductivity of the random resistor network in zd near its percolation threshold is considered the network has bonds assigned the conductivities 1 and e 0 in the volume fractions p and 1 p motivated by the statistical mechanics of an ising ferromagnet at temperature t in a field h we introduce a partition function and free energy for the resistor network which establishes a direct correspondence between the two problems in particular we show that the free energies for the resistor network and the ising model both have the same type of integral representation which has the interpretation of the complex potential due to a charge distribution on 0 1 in the s 1 1 e plane for the resistor network and on the unit circle in the z exp 2 h plane for the ferromagnet based on this correspondence we develop a yang lee picture of the onset of nonanalytic behavior of the effective conductivity so that the percolation threshold p pc is characterized as a

reinforced mindlin plates with extremal stiffness for a prescribed area fraction of stiffeners we characterize the set of stiffener reinforced mindlin plates with extremal overall stiffness the method rests upon the derivation of optimal bounds of the hashin shtrikman type our method is distinct from the usual hashin shtrikman approach we make use of the underlying variational structure behind the hashin shtrikman method to show that the use of a comparison material is redundant we do this by proceeding directly and express the equilibrium equations in terms of positive definite integral operators the positivity of the operators is used to obtain a new hilbert space variational principle for the effective stiffness the associated bounds are shown to be realized by effective rigidities associated with hierarchical laminar arrangements of stiffeners

constrained traffic equilibrium in routing we study noncooperative routing in which each user is faced with a multicriterion optimization problem formulated as the minimization of one criterion subject to constraints on others we address the questions of existence and uniqueness of equilibrium we show that equilibria indeed exist but uniqueness may be destroyed due to the multicriteria nature of the problem we obtain uniqueness in some weaker sense under appropriate conditions we show that the link utilizations are uniquely determined at equilibrium we further study the normalized constrained equilibrium and apply it to pricing

the kirwan map equivariant kirwan maps and their kernels consider a hamiltonian action of a compact lie group k on a compact symplectic manifold we find descriptions of the kernel of the kirwan map corresponding to a regular value of the moment map k we start with the case when k is a torus t we determine the kernel of the equivariant kirwan map defined by goldin in go corresponding to a generic circle s t and show how to recover from this the kernel of t as described by tolman and weitsman in to we in the situation when the fixed point set of the torus action is finite similar results have been obtained in our previous papers je je ma for a compact nonabelian lie group k we will use the non abelian localization formula of je ki1 and je ki2 to establish relationships some of them obtained by tolman and weitsman in to we between ker k and ker t where t k is a maximal torus in the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of t

bounds on herglotz functions and fundamental limits of broadband passive quasistatic cloaking using a sum rule we derive new bounds on herglotz functions that generalize those given in bernland et al j phys a math theor 44 14 145205 2011 and gustafsson and sjoberg new j phys 12 4 043046 2010 these bounds apply to a wide class of linear passive systems such as electromagnetic passive materials among these bounds we describe the optimal ones and also discuss their meaning in various physical situations like in the case of a transparency window where we exhibit sharp bounds then we apply these bounds in the context of broadband passive cloaking in the quasistatic regime to refute the following challenging question is it possible to construct a passive cloaking device that cloaks an object over a whole frequency band our rigorous approach although limited to quasistatics gives quantitative limitations on the cloaking effect over a finite frequency range by providing inequalities on the polarizability tensor associated with the cloaking device we emphasize that our resul

interchangeability and bounds on the effective conductivity of the square lattice the effective conductivity of an infinitely interchangeable two component random medium is considered this class of media includes cell materials in the continuum and the bond lattice on d where the cells or bonds are randomly assigned the conductivities 1 and 2 1 2ne0 with probabilitiesp1 andp2 1 p1 a rigorous basis for the very old and widely used low volume fraction expansion of is established by proving that is an analytic function ofp2 in a suitable domain containing 0 1 in the case of the bond lattice ind 2 rigorous fourth order upper and lower bounds on valid for allp2 1 and 2 are derived the four perturbation coefficients entering into the bounds are obtained from the first order volume fraction coefficient using the method of infinite interchangeability

modelling the effective conductivity function of an arbitrary two dimensional polycrystal using sequential laminates the effective conductivity tensor of a two dimensional polycrystalline material depends on the conductivity tensor 0 of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal represented by a rotation field r x giving the orientation of the crystal at each point x here it is established that the dependence of on 0 in any polycrystal with r x held fixed can be mimicked exactly by a polycrystal constructed by sequential lamination it is first shown that the effective conductivity function is perturbed only slightly if we truncate the hilbert space of fields in the polycrystal to a finitedimensional space then the structure of this finite dimensional space of fields is shown to be isomorphic to the structure of the finite dimensional space of fields associated with the sequential laminate in particular there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields

topological conditions for uniqueness of equilibrium in networks equilibrium flow in a physical network with a large number of users e g transportation communication and computer networks need not be unique if the costs of the network elements are not the same for all users such differences among users may arise if they are not equally affected by congestion or have different intrinsic preferences whether or not forall assignments of strictly increasing cost functions each user s equilibrium cost is the same in all nash equilibria can be determined from the network topology specifically this paper shows that in a two terminal network the equilibrium costs are always unique if and only if the network is one of several simple networks or consists of several such networks connected in series the complementary class of all two terminal networks with multiple equilibrium costs forsome assignment of user specific strictly increasing cost functions is similarly characterized by an embedded network of a particular simple type

transport properties of touching cylinder pairs and of the square array of touching cylinders we consider the solution of the scalar transport problem for a pair of touching cylinders of high conductivity we obtain a closed form expression for the set of multipole coefficients characterizing the potential function in this problem we show that this expression permits the estimation of truncation errors occurring in the matrix solution of the transport problem for the square array of touching cylinders consequently we are able to study the array conductivity for arbitrarily high cylinder conductivities and confirm an asymptotic expression due to o brien we derive a perturbation formula giving the effect of small cylinder separations on array conductivity

a new route to finding bounds on the generalized spectrum of many physical operators here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depending linearly on a vector of parameters z z1 z2 zn we obtain constraints on z that ensure that green s function exists and hence which provide bounds on the generalized spectrum here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depend

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

on the existence of pure nash equilibria in weighted congestion games we study the existence of pure nash equilibria in weighted congestion games let c denote a set of cost functions we say that c is consistent if every weighted congestion game with cost functions in c possesses a pure nash equilibrium our main contribution is a complete characterization of consistency of continuous cost functions we prove that a set c of continuous functions is consistent for two player games if and only if c contains only monotonic functions and for all nonconstant functions c1 c2 c there are constants a b r such that c1 x a c2 x b for all x r 0 for games with at least three players we prove that c is consistent if and only if exactly one of the following cases holds a c contains only affine functions b c contains only exponential functions such that c x ac e x bc for some ac bc r where ac and bc may depend on c while must be equal for every c c the latter characterization is even valid for three player games finally we derive several characterizations of consistency of cost functions for games with restricted strategy spaces such as weighted network congestion games or weighted congestion games with singleton strategies

harmonization and homogenization on fractals this paper suggests a direct approach to define the laplacian the spectral dimension of nested fractals and the pre sierpinski carpet conductivity we find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy the paper is mostly aimed at the homogenization on the pre sierpinski gasket whose horizontal and nonhorizontal bonds have different conductivities a andb respectively we prove the convergence of the rescaled energies on the pre sierpinski gasket to a b e where e is the standard energy on the gasket with uniform conductivities we also find an explicit expression for the effective conductivity a b and deduce that its set of singularities turns out to be the julia set of a certain rational function a special section is devoted to the problem of the pre sierpinski carpet conductivity asymptotic behavior for this problem a new proof of barlow bass inequalities with sharper constants is given

inverse bounds of two component composites a method is presented for estimating microstructural parameters from permittivity measurements of two component composites this structural information is described by a particular positive measure in the stieltjes integral representation of the effective permittivity the dependence on the geometrical structure can be reduced to the problem of calculating the moments of the measure we present a method that uses measurement data at a set of distinct frequencies or temperatures to calculate bounds on several moments these inverse bounds are improved when the volume fraction is known or the material is isotropic composites with known geometrical structure illustrate the method

a general representation for the effective dielectric constant of a composite in its dependence on the dielectric constants ek of its homogeneous components the effective dielectric constant e of a composite is a function analytic in a domain some relevant results about the effective dielectric constant are collected including the form of and then a general representation of e as an analytic function is given in this representation the dependence on the geometry is separated from the dependence on the e k s

decentralized bandwidth control subject to two layer constraints abstract this paper is concerned with the new bandwidth allocation model that considers the structural allocation constraints suppose that a system is composed of finite groups of users and the bandwidth assigned to each user and to each group of users has pre specified upper and lower bounds if each user is granted with the utility function satisfying the standard continuity and concavity conditions the existence uniqueness fairness and optimality properties of the nash equilibrium point in the allocation game are studied to identify the equilibrium point an algorithm proved to converge globally is proposed and illustrated with a numerical example

resistivity of an infinite three dimensional stationary random electric conductor the aim of this paper is to prove that the resistivity of an infinite stationary random medium can be computed from one realization this is accomplished through the application of the pointwise degree 2 ergodic theorem for divergence free stationary random fields see 4

application of volterra wiener series for bounding the overall conductivity of hetergeneous media i general procedure a general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed it utilizes truncated volterra wiener functional series generated by the random conductivity field k x as classes of trial functions for the classical variational principles the method simplifies unifies and or generalizes the earlier proposed variational techniques of beran 3 dederichs and zeller 10 hori 15 kr6ner 17 and prager 25 the general procedure is displayed in detail in the simplest case of interest the construction of the optimal third order bounds that requires knowledge of the two and three point correlation functions for k x the evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions a perturbation solution to this equation is given for miller s cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of beran miller

stieltjes representation of the 3d bruggeman effective medium and pade approximation the paper deals with bruggeman effective medium approximation ema which is often used to model effective complex permittivity of a two phase composite we derive the stieltjes integral representation of the 3d bruggeman effective medium and use constrained pade approximation method introduced in 39 to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies the problem of reconstruction of the stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite is used to characterize its geometric structure we present two different proofs of the stieltjes analytical representation for the effective complex permittivity in the 3d bruggeman effective medium model one proof is based on direct calculation the other one is the derivation of the representation using stieltjes inversion formula we show that the continuous spectral density in the integral representation for the bruggeman ema model can be efficiently approximated by a rational function a rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition we show results of numerical rational approximation of bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium the volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values

the effective conductivity of strongly heterogeneous composites we deal with the effective conductivity m m z of two phase ordered or disordered mixtures consisting of particles of material of conductivity z inserted in a matrix of conductivity 1 we focus on finding bounds on the set of values of z for which the function m is singular or vanishes and we apply our results to the estimation of the effective conductivity of high contrast mixtures z 0 or z we find that the zeroes and singularities of the function m lie on an interval of the negative real axis which depends on the shape of the particles and the interparticle distances our results agree with previous numerical calculations for periodic arrays of spheres in some cases we show that our estimates are optimal we apply our results about the zeroes and singularities together with the complex variable method and find bounds on the effective conductivity of matrix particle random composites these bounds give good estimations even in cases of high contrast and in many cases they improve substantially over the bounds obtained by other methods for the same types of high contrast mixtures

reconstruction of spectral function from effective permittivity of a composite material using rational function approximations the paper deals with the problem of reconstruction of microstructural information from known effective complex permittivity of a composite material a numerical method for recovering geometric information from measurements of frequency dependent effective complex permittivity is developed based on stieltjes analytic representation of the effective permittivity tensor of a two component mixture we derive the stieltjes representation for the effective permittivity of the medium using the eigenfunction expansion of the solution of a boundary value problem the spectral function in this representation contains all information about the microgeometry of the mixture a discrete approximation of the spectral measure is derived from a rational pade approximation followed by its partial fractions decomposition the approach is based on the least squares minimization with regularization constraints provided by the spectral properties of the operator the method is applied to calculation of volume fractions of the components in a mixture of two materials in a bruggeman effective medium analytic model which has a continuous spectral density and to analytical models of two phase composites with coated cylindrical and ellipsoidal inclusions the numerical results of reconstruction of spectral measure for a mixture of silver and silicon dioxide and a composite of magnesium and magnesium fluoride show good agreement between theoretical and predicted values the approach is applicable to geological materials biocomposites porous media etc

the effective conductivity of a periodic array of spheres we consider the problem of determining the effective conductivity k of a composite material consisting of equal sized spheres of conductivity alpha arranged in a cubic array within a homogeneous matrix of unit conductivity we modify zuzovski brenner9s 1977 method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by mckenzie et al 1978 using a method originally devised by rayleigh 1892 on solving these equations we derive expressions for k ast to o c 9 c being the volume fraction of the spheres for simple body centred and face centred cubic arrays and also obtain numerical values for k over the whole range of alpha and c we show that these results for cubic arrays can be used to estimate ast for random arrays of identical spheres for arrays of highly conducting and nearly touching spheres batchelor o9brien 1977 showed that begin equation k ast sim begin cases k 1 ln 1 chi k 2 alpha infty chi c c max frac 1 3 rightarrow 1 2k 1 ln alpha k9 2 quad chi 1 alpha rightarrow infty end cases end equation where c max corresponds to the volume fraction when the spheres are actually touching each other and determined k 1 for the three cubic arrays our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants k 2 and k9 2 thereby obtained do not quite satisfy the relation k9 2 k 1 3 9 ln 2 k 2 given by batchelor o9brien we have been unable to find the reason for this slight discrepancy

diffusion in lattices with anisotropic scatterers we study diffusion in lattices with periodic and random arrangements of anisotropic scatterers we show using both analytical techniques based upon our previous work on asymptotic properties of multistate random walks and computer calculation that the diffusion constant for the random arrangement of scatterers is bounded above and below at an arbitrary density by the diffusion constant for an appropriately chosen periodic arrangement of scatterers at the same density we also investigate the accuracy of the low density expansion for the diffusion constant up to second order in the density for a lattice with randomly distributed anisotropic scatterers comparison of the analytical results with numerical calculations shows that the accuracy of the density expansion depends crucially on the degree of anisotropy of the scatterers finally we discuss a monotonicity law for the diffusion constant with respect to variation of the transition rates in analogy with the rayleigh monotonicity law for the effective resistance of electric networks as an immediate corollary we obtain that the diffusion constant averaged over all realizations of the random arrangement of anisotropic scatterers at density is a monotone function of the density

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

effective conductivity and skew brownian motion we consider heat conduction in a periodic body which is composed of finitely many different components the effective conductivity is represented in terms of skew brownian motion the representation formula is a fluctuation dissipation relation the dissipation term in this formula is related to the transmission of heat through the surface separating the different components of the body it is described by the skew reflections of brownian motion at these surfaces the problems caused by the discontinuity of the microscopic conductivity are handled in the framework of dirichlet forms

sample to sample fluctuations in the conductivity of a disordered medium we investigate the sample to sample fluctuations in the conductivity of a random resistor network equivalently in the diffusivity of a disordered medium with symmetric hopping rates we argue that whenever the effective conductivity is strictly positive then the fluctuations are normal i e proportional to volume 1 2 if the local conductivities are allowed to be zero then vanishes when approaching the percolation thresholdpc close topc the fluctuations are anomalous from the renormalization group on hierarchical lattices we find that atpc fluctuations and mean scale in the same fashion i e there is no independent scaling exponent for the fluctuations

spectral analysis and computation for homogenization of advection diffusion processes in steady flows advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of d involving the molecular diffusivity and a spectral measure of a self adjoint operator that depends on the characteristics of a randomly perturbed periodic flow in discrete formulations of each approach we express explicitly in terms of standard or generalized eigenvalues and eigenvectors of hermitian matrices we develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach we use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorou

hierarchies of stieltjes functions and their application to the calculation of bounds for the dielectric constant of a two component composite medium the bulk effective dielectric constant varepsilon e varepsilon 1 varepsilon 2 of a two component composite medium with a fixed microstructure considered as function of the ratio varepsilon 1 varepsilon 2 is a stieltjes function with special analytical properties it is shown how other related stieltjes functions can be constructed by exploiting partial information that is available about the values of varepsilon e varepsilon 1 varepsilon 2 and about its series expansion these functions are used to construct continued fraction representations for varepsilon e varepsilon 1 varepsilon 2 that can provide exact bounds for varepsilon e that incorporate that information explicit algorithmic constructions are developed for obtaining these bounds in the real as well as in the complex domain

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

optimal lower bounds on local stress inside random media a methodology is presented for bounding the higher l p norms 2 leq p leq infty of the local stress inside random media we present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites these bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads these results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media

bounds which incorporate morphological information for a nonlinear composite dielectric the nonlinear hashin shtrikman variational structure relies on the introduction of a comparison material in recent work a nonlinear comparison composite has been employed this has enabled both upper and lower bounds to be found that depend on the three point statistics of the microstructure through a parameter introduced by g w milton at least one of the bounds requires an estimate of the size of the trial field to be known and even if the best estimate available is used and the milton parameter is known for a particular microstructure the bounds can still be widely separated o p bruno has obtained bounds for a linear composite consisting of a matrix containing spherical inclusions where the inclusions are coated with the matrix phase the bounds remain non trivial even when the properties of the inclusions tend to either zero or infinity these bounds are used to improve the bounds for a nonlinear composite comprising either a nonlinear matrix containing linear inclusions or a linear matrix containing nonlinear inclusions for the microgeometry considered here it is possible to construct a bounded trial field and a simple example involving this approach is also described

exponent inequalities for the bulk conductivity of a hierarchical model the bulk conductivity p of the bond lattice in d is considered where the bonds have conductivity 1 with probabilityp or e 0 with probability 1 p various representations of the derivatives of p are developed these representations are used to analyze the behavior of p for e 0 near the percolation thresholdp c when the conducting backbone is assumed to have a hierarchical node link blob nlb structure this model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds exact asymptotics of frac d 2 sigma dp 2 for the nlb model are proven under some technical assumptions the proof employs a novel technique whereby frac d 2 sigma dp 2 for the nlb model with e 0 andp nearp c is computed using perturbation theory for p for two and three component resistor lattices aroundp 1 with a sequence of e s converging to 1 as one goes deeper in the hierarchy these asymptotics establish convexity of p for the nlb model nearp c and that its critical exponentt obeys the inequalities 1 t 2 ford 2 3 while 2 t 3 ford 4 the upper boundt 2 ind 3 which is realizable in the nlb class virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model

optimal bounds and microgeometries for elastic two phase composites consider two isotropic linearly elastic materials with elasticity tensors c 1 c 2 and assume c 1 c 2 we prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite rank laminates this implies that for any imposed uniform strain or stress field optimal bounds on the effective strain or stress energy per unit volume are attained by finite rank laminates explicit bounds on the strain energy with no symmetry assumptions are given in dimensions 2 and 3 these bounds are more stringent than the classical voigt reuss bounds finally explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry

oblique propagation of electromagnetic and elastic waves for an array of cylindrical fibres this paper presents analysis of electromagnetic and elastodynamic waves propagating through a doubly periodic array of cylindrical channels in oblique incidence a new method based on a multipole scattering approach has been proposed to reduce these spectral problems for partial differential equations to certain algebraic problems of the rayleigh type we obtain a formulation in terms of an eigenvalue problem that enables us to construct the high order dispersion curves and to study both photonic and phononic band gap structures in oblique incidence we also address the question of a singular perturbation induced by the conical incidence and discuss some effective properties for ferromagnetic photonic crystal fibres in the long wavelength limit

the coherent potential approximation is a realizable effective medium scheme the effective electrical conductivity of an aggregate composed of grains of various conductivities is frequently estimated by the coherent potential approximation which embodies a local effective medium concept it is proved rigorously that this approximation is exact for a wide class of hierarchical model composites made of spherical grains the starting material 0 in the hierarchy is chosen arbitrarily otherwise materialj 1 2 consists of equisized spheres sayj spheres of arbitrary conductivities embedded in materialj 1 the spatial distribution of thej spheres must satisfy a mild homogeneity condition and their radiusr j must asymptotically increase faster than exponentially withj furthermore the minimum spacing 2s j between thej spheres is such that the ratios j r j diverges on the basis of these and some further ancillary conditions it is established that the coherent potential approximation becomes asymptotically exact for the effective conductivity of materialj the results extend to other effective parameters of the composites including the thermal conductivity dielectric constant and magnetic permeability in addition the model composites and the proof of realizability may be generalized to allow non spherical grains

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

a continued fraction representation for the effective conductivity of a two dimensional polycrystal a continued fraction representation for the effective conductivity tensor of a two dimensional polycrystal is derived this representation is in terms of a sequence of positive definite symmetric matrices which characterize the underlying geometric structure of the material the proof is accomplished by considering a particular basis for the hilbert space of fields in the composite in which the linear operators relevant to determining the effective conductivity take simple forms as infinite matrices these infinite matrices are then used in the variational definition of the effective conductivity to formulate the continued fraction this continued fraction is used to derive upper and lower bounds on

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media

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

network approximation in the limit of small interparticle distance of the effective properties of a high contrast random dispersed composite we consider a high contrast two phase composite such as a ceramic polymer composite or a fiberglass composite our objective is to determine the dependence of the effective conductivity or the effective dielectric constant or the effective shear modulus of the composite on the random locations of the inclusions ceramic particles or fibers when the concentration of the inclusions is high we consider a two dimensional model and show that the continuum problem can be approximated by a discrete random network graph we use variational techniques to provide rigorous mathematical justification for this approximation in particular we have shown asymptotic equivalence of the effective constant for the discrete and continuum models in the limit when the relative interparticle distance goes to zero we introduce the geometrical interparticle distance parameter using voronoi tessellation and emphasize the relevance of this parameter due to the fact that for irregular non periodic geometries it is not uniquely determined by the volume fraction of the inclusions we use the discrete network to compute numerically for this purpose we employ a computer program which generates a random distribution of disks on the plane using this distribution we obtain the corresponding discrete network furthermore the computer program provides the distribution of fluxes in the network which is based on keller s formula for two closely spaced disks we compute the dependence of on the volume fraction of the inclusions v for monodispersed composites and obtaine results which are consistent with the percolation theory predictions for polydispersed composites random inclusions of two different sizes the dependence is not simple and is determined by the relative volume fraction v r of large and small particles we found some special values of v r for which the effective coefficient is significantly decreased the computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations

competitive routing in networks with polynomial costs we study a class of noncooperative general topology networks shared by n users each user has a given flow which it has to ship from a source to a destination we consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling and show that these costs have appealing properties that lead to predictable and efficient network flows in particular we show that the nash equilibrium is unique and is moreover efficient these properties make the polynomial cost structure attractive for traffic regulation and link pricing in telecommunication networks we finally discuss the computation of the equilibrium in the special case of the affine cost structure for a topology of parallel links

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

n point pade approximants to real valued stieltjes series with nonzero radii of convergence abstract by employing special continued fractions to two stieltjes series with nonzero radii of convergence we extend the inequalities for one point pade approximants reported by baker 1975 corollory 17 1 to the case of two point pade approximants we prove that some convergents of the continued fractions form a monotone sequences of upper and lower bounds converging uniformly to stieltjes function x1 x on compact subsets of r where r is a radius of convergence of an expansion of x f1 x at x 0 for an illustration of theoretical results we provide nontrivial numerical examples as an application to real physical problems second order pade approximants bounds on the effective conductivity of a square array of cylinders are evaluated

homogeneity in integral representations for effective properties of composite materials the effective complex conductivity of an n component composite material is considered recently an integral representation that treats the component conductivities 1 n symmetrically was developed this representation has the advantage that the moments of the positive measure in the integral are directly related to the coefficients in a perturbation expansion of around a homogeneous medium however the admissible class of measures has been difficult to characterize due to the condition that is a homogeneous function of the component conductivities here this admissible class is characterized in terms of linear relations among the moments associated with tetrahedra in fourier space zn the homogeneity is used to derive a new formula for in the two component case in which for general media is expressed in terms of the conductivities of laminates of second rank

duality relation for the maxwell system this paper is intended to establish a link between the vector maxwell system for three dimensional 3d and 2d finite photonic crystals in the low frequency limit for this we generalize the classical results of keller and dykhne chessboard problem to periodic media described by piecewise continuous permittivity profiles our theorem enlights the result of mendelson polycrystalline and multiphase media in the framework of homogenization theory of elliptic operators in fine we give illustrative examples by using both integral equation and variational approaches via the so called method of fictitious charges and finite element method

on the uniqueness of equilibrium in atomic splittable routing games in routing games with infinitesimal players it follows from well known convexity arguments that equilibria exist and are unique in routing games with atomic players with splittable flow equilibria exist but uniqueness of equilibria has been demonstrated only in limited cases in two terminal nearly parallel graphs when all players control the same amount of flow and when latency functions are polynomials of degree at most three there are no known examples of multiple equilibria in these games in this work we show that in contrast to routing games with infinitesimal players atomic splittable routing games admit multiple equilibria we demonstrate this multiplicity via two specific examples in addition we show that our examples are topologically minimal by giving a complete characterization of the class of network topologies for which multiple equilibria exist our proofs and examples are based on a novel characterization of these topologies in terms of sets of circulations

recovery of inclusion separations in strongly heterogeneous composites from effective property measurements an effective property of a composite material consisting of inclusions within a host matrix depends on the geometry and connectedness of the inclusions this dependence may be quite strong if the constituents have highly contrasting properties here we consider the inverse problem of using effective property data to obtain information on the geometry of the microstructure while previous work has been devoted to recovering the volume fractions of the constituents our focus is on their connectedness a key feature in critical behaviour and phase transitions we solve exactly a reduced inverse spectral problem by bounding the volume fraction of the constituents an inclusion separation parameter and the spectral gap of a self adjoint operator that depends on the geometry of the composite we present a new algorithm based on the mobius transformation structure of the forward bounds whose output is a set of algebraic curves in parameter space bounding regions of admissible parameter values these results advance the development of techniques for characterizing the microstructure of composite materials as an example we obtain inverse bounds on the volume fraction and separation of the brine inclusions in sea ice from measurements of its effective complex permittivity

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