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

understanding complex dynamics by means of an associated riemann surface we provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated riemann surface thanks to this geometric description an explicit formula for the period of the orbits can be derived which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem for rational values of this ratio and generic values of the initial data all orbits are periodic and the system is isochronous for irrational values of the ratio there exist periodic and quasi periodic orbits for different initial data moreover the dependence of the period on the initial data shows a rich behavior and initial data can always be found with arbitrarily large periods

novel isochronous n body problems featuring n arbitrary rational coupling constants a novel class of n body problems is identified with n an arbitrary positive integer n 2 these models are characterized by newtonian accelerations equal forces equations of motion describing n equal point particles moving in the complex z plane these highly nonlinear equations feature n arbitrary coupling constants yet they can be solved by algebraic operations and if all the n coupling constants are real and rational the corresponding n body problem is isochronous its generic solutions are all completely periodic with an overall period t independent of the initial data but many solutions feature subperiods t p with p integer it is moreover shown that these models are hamiltonian

on modified versions of some solvable ordinary differential equations due to chazy we introduce modified versions featuring a deformation parameter of four explicitly solvable nonlinear autonomous ordinary differential equations odes introduced almost a century ago by chazy chazy j 1909 c r acad sci paris 148 157 9 when the deformation parameter vanishes 0 the modified odes reduce to those introduced by chazy when the deformation parameter does not vanish and is real say positive 0 then all the nonsingular solutions of these modified odes considered as functions of the real independent variable t say time are periodic with period t 2 for two of these modified odes there is also a small subset of these solutions that are also periodic with a smaller period

a nystrom algorithm for electrostatics of an anisotropic composite the construction of efficient numerical algorithms for the electrostatics of locally anisotropic composites seems to be a poorly explored area in this paper coupled first and second kind fredholm integral equations for the electrostatics of anisotropic inclusions in an anisotropic matrix are derived for a square array of circular cylinders the integral equations are solved to high accuracy through pointwise discretization in the process a recent renormalization method is extended for the evaluation of lattice sums j math phys 35 6036 1994 to encompass the calculation of sums on stretched lattices conditionally convergent stretched lattice sums are computed from physical considerations

perturbation theory for the effective diffusion constant in a medium of random scatterers we develop perturbation theory and physically motivated resummations of the perturbation theory for the problem of a tracer particle diffusing in a random medium the random medium contains point scatterers of density uniformly distributed throughout the material the tracer is a langevin particle subjected to the quenched random force generated by the scatterers via our perturbative analysis we determine when the random potential can be approximated by a gaussian random potential we also develop a self similar renormalization group approach based on thinning out the scatterers this scheme is similar to that used with success for diffusion in gaussian random potentials and agrees with known exact results to assess the accuracy of this approximation scheme its predictions are confronted with results obtained by numerical simulation

on a class of hamiltonians with classical isochronous motions and quantal equi spaced spectra hamiltonians linear in the momenta and yielding in the classical context trajectories isochronous in configuration space are considered it is shown that their motions are completely periodic in phase space as well with the same common period as the orbits in configuration space moreover it is shown that for this particular class of hamiltonians the semiclassical quantization prescription is exact so that to the isochronous character of their classical dynamics there corresponds in the quantized context an equidistant spectrum for a broad range of ordering prescriptions including non symmetrical ones examples illustrating these findings are presented

qualitative properties of systems of two complex homogeneous ode s a connection to polygonal billiards a correspondence between the orbits of a system of two complex homogeneous polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed this correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind this allows us to transfer results well known from the theory of polygonal billiards such as ergodicity the existence of periodic orbits the absence of exponential divergence the existence of additional conservation laws and the presence of discontinuities in the dynamics to the corresponding systems of ordinary differential equations it also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations

on a modified version of a solvable ode due to painleve we introduce a modified version of an ode originally introduced by painleve and we show that it possesses a lot of periodic solutions we also use a simplified version of it to verify the efficacy of an iterative approach used elsewhere in more general contexts to evince analogous properties of periodicity of systems of nonlinear odes

dynamical systems on infinitely sheeted riemann surfaces this paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi valuedness of the solutions as functions of complex time in this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral the associated riemann surface r c is known to be an infinitely sheeted covering of the complex time plane ramified at an infinite set of points whose projection in the plane is dense the main novelty of this paper is that the geometrical structure of these infinitely sheeted riemann surfaces is described in great detail which allows to study global properties of the flow such as asymptotic behaviour of the solutions periodic orbits and their stability or sensitive dependence on initial conditions the results are then compared with a numerical integration of the equations of motion following the recent approach of calogero the real time trajectories of the system are given by paths on r that are projected to a circle on the complex plane due to the branching of r the solutions may have different periods or may not be periodic at all

on the connection between hyperelliptic separability and painleve integrability we consider systems of odes which are associated with some physically significant examples shallow water equilibrium solutions travelling waves of the harry dym equation a lotka volterra system of competing species and the geodesic flow on the triaxial ellipsoid the first three are shown to share the following properties i they are hyperelliptically separable systems hss and after a suitable nonlinear time transformation become algebraically completely integrable aci and ii they are of the weak painleve type and become full painleve after the application of this transformation the geodesic flow on the other hand although it passes the usual painleve test does not possess a full set of free constants and thus one may not conclude whether it has the painleve property or not this system is also hss and becomes aci after the application of a suitable nonlinear time transformation we also combine our geometric analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches this correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available

on the non integrability of a family of duffing van der pol oscillators we investigate the non integrability of a family of duffing van der pol oscillators x alpha x x2 1 x beta x3 gamma cos omega t by studying the analytic properties of the dynamics in complex time we find that the solutions of have no worse than algebraic singularities at t with only t t 1 2 terms present in their series expansions unlike for example the alpha 0 duffing case where typically log t t terms arise still when integrating around long enough contours a remarkably intricate pattern of square root singularities emerges on different sheets which appears to prevent solutions from ever returning to the original sheet such evidence of infinitely sheeted solutions termed the iss property has also been observed in a number of hamiltonian systems and is illustrated here on a simple example of a single first order differential equation we suggest that the iss property is a necessary condition for non integrability i e non existence of a complete set of analytic single valued constants of the motion which would permit the complete integration of a dynamical system in terms of quadratures

from microstructure independent formulas for composite materials to rank one convex non quasiconvex functions examples of non quasiconvex functions that are rank one convex are rare in this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials developed to identify g closed sets of positive codimensions the algebraic methods are used to construct a set of materials of positive codimension that is closed under lamination but is not g closed the well known link between g closed sets and quasiconvex functions and sets closed under lamination and rank one convex functions is then used to construct a family of rotationally invariant nonnegative and 2 homogeneous rank one convex functions that are not quasiconvex

closed form solutions for the effective conductivity of two phase periodic composites with spherical inclusions in this paper we use approximate solutions of the nemat nasser et al to estimate the effective conductivity of two phase composites with non overlapping spherical inclusions systems with different inclusion distributions are considered cubic lattice distributions simple cubic body centered cubic and face centered cubic and random distributions the effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast fourier transform based methods for systems containing randomly distributed spherical inclusions the solutions are shown to be directly related to the static structure factor and we obtain its analytical expression in the infinite volume limit

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

a complex deformation of the classical gravitational many body problem that features many completely periodic motions a complex deformation of the newtonian equations of motion of the classical gravitational many body problem is introduced namely a many body problem that features a parameter and that reduces when this parameter vanishes to the standard equations of motion of newtonian gravitation for an arbitrary number of pointlike bodies with arbitrary masses and it is shown that when this parameter is instead positive 0 there is an open set of complex initial data such that all the complex motions originating from it are completely periodic with period t 2 and that the infinite measure of this set is a finite nonvanishing fraction of the measure of the entire set of initial data

minimum variational principles for time harmonic waves in a dissipative medium and associated variational principles of hashin shtrikman type minimization variational principles for linear elastodynamic acoustic or electromagnetic time harmonic waves in dissipative media were obtained by milton et al milton et al 2009 proc r soc a

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

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

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

duality and similarity properties of the effective permittivity of two dimensional heterogeneous medium with inclusion of fractal geometry we report a systematic finite difference time domain study on the dielectric properties of two dimensional lossless heterostructures with inclusion of deterministic fractal geometry koch s snowflake sierpinski s square and triangle constrained to sit in a circumscribed circle for a wide range of surface fractions in all the configurations investigated we observed i a strong deviation of the surface fraction dependence of the effective permittivity based on maxwell garnett analysis and ii a permittivity change with reduced perimeter according to a similarity transformation at least for the first three iterations of the fractal pattern we show that the results of our numerical analysis disagree with expectations from the duality phase interchange relation and explain this as being due to the intrinsic complexity of the morphology which requires a multipolar approach to correctly describe the microstructure features of these heterostructures

improved bounds on the conductivity of composites by interpolation different bounds on the conductivity of a composite material may improve on each other in different conductivity regimes if so the question arises of how to efficiently interpolate between the bounds in this paper i show how to do an interpolation with a two point pade approximation method for bounds on two component composites the interpolation method is shown to be in a sense the best possible the method is discussed in the context of equiaxed polycrystals where the classic hashin shtrikman bounds and the more recent null lagrangian bounds partly improve on each other denoting the principal conductivities of the crystallite sigma 1 leq sigma 2 leq sigma 3 the method gives improved lower bounds for equiaxed polycrystals which have sigma 2 0 77 sigma 1 0 23 sigma 3 geq sigma 1 sigma 3 the method also gives improved upper bounds

a further solvable three body problem in the plane the solution is provided of a three body problem in the plane which is the third of a trio recently identified as likely to display a particularly simple time evolution hence to be amenable to exact treatment this conjecture already validated by providing the solution of the first two of these three models is now completely proven by exhibiting the solution of the third this finding also demonstrates the conjectured super painleve character of certain nonlinear ordinary differential equations namely the fact that their general solution is an entire function of the independent variable

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

investigating non integrability and chaos in complex time the question of integrability of systems of ordinary differential equations odes has been studied extensively for centuries recently however owing to an upsurge of interest in chaos a lot of studies have focused on the investigation of non integrable systems where chaos often manifests itself as extremely sensitive dependence on initial conditions of a dense set of real solutions in real time here we argue that non integrability can be efficiently investigated by solving odes in complex time and review recent results which strongly suggest that for a system of odes to be non integrable it is necessary that it possess a dense set of infinitely sheeted solutions in the complex domain

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

the conductivity of a sheet containing inclusions with sharp corners we have written down an integral equation for the surface charge on the interface between an n sided regular polygonal inclusion with conductivity o 1 and a host with conductivity 0 the conductivity of a sheet with few inclusions can be determined from the induced dipole associated with this charge distribution the integral equation is solved numerically and detailed results are given for n 3 4 and 6 the circle limit is approached for large n we show that our results are consistent with the reciprocity theorem for two dimensional conducting media and we reproduce the known analytic results in the limits when the inclusion is either a hole or a superconducting region we also find the conductivity of a sheet containing a general n pointed star analytic fits to our numerical results are given which are used as input for effective medium theories the results given in this paper for the electrical conductivity apply also to the thermal conductivity and to dielectric media

a characterization of quadratic forms of type e6 e7 and e8 the goal of this article is to show that quadratic forms of type e6 e7 a nde8 are completely characterized by their even clifford algebras or equivalently by their discriminant and witt invariant 2002 elsevier science usa all rights reserved

a technique to identify solvable dynamical systems and a solvable generalization of the goldfish many body problem a simple approach is discussed which associates to solvable matrix equations solvable dynamical systems generally interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles we then focus on cases in which the auxiliary variables can be altogether eliminated reobtaining thereby via this unified approach well known solvable many body problems and moreover a solvable extension of the goldfish model

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

periodic conduction problems the fast multipole method and convergence of integral equations and lattice sums this paper presents a version of the fast multipole method fmm for integral equations describing conduction through three dimensional periodic heterogeneous media the proposed method is based on the standard rather than periodic fundamental solution and therefore it is very close to the original fmm in deriving the method particular attention is paid to convergence of arising integral equations and lattice sums it is shown that convergence can be achieved without introducing artificial compensatory sources or boundary conditions

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

a new class of isochronous dynamical systems a new class of isochronous dynamical systems is introduced and briefly discussed these systems feature in their phase space a fully dimensional region part of which can be explicitly identified where all their solutions are completely periodic periodic in all their degrees of freedom with the same period but in other regions of their phase space their evolution might be quite complicated

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

a new construction of moufang quadrangles of type and in the classification of moufang polygons by j tits and r weiss the most intricate case is by far the case of the exceptional moufang quadrangles of type e6 e7 and e8 and in fact the construction that they present is ad hoc and lacking a deeper explanation we will show how tensor products of two composition algebras can be used to construct these moufang quadrangles in characteristic different from 2 as a byproduct we will obtain a method to construct any moufang quadrangle in characteristic different from two from a module for a jordan algebra

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

order and the ubiquitous occurrence of chaos abstract for a large class of ode s which includes the van der pol equation we determine analytically the asymptotic location of the singularities in the complex t plane by integrating these ode s numerically we show that if the singularities are dense which is the generic case the solution is chaotic in the sense of sensitive dependence on initial conditions in the exceptional case that the singularities are not dense the solution exhibits order taxis chaos is ubiquitous even for first order ode s in complex t

freudenthal triple systems in arbitrary characteristic abstract let q v k be the form of degree 4 and h the alternating bilinear form that together give rise to a freudenthal triple system when the characteristic of k is different from 2 and 3 we give a form of degree 4 on a central cover of v over an arbitrary field in arbitrary characteristic and determine the group of similitudes of this group coincides with the intersection of the group of similitudes of q and the isometry group of h when the characteristic is different from 2 and 3 and has the expected structure of a suitable levi subgroup in general we extend known results about the stabilizers of elements of v and orbits of the group of similitudes and show in particular the existence of an exotic set of orbits when char k 2 and k is not perfect

new bounds on the effective thermal conductivity of n phase materials based on the perturbation solution to the effective thermal conductivity problem for an n component material we derive new third and fourth order bounds for the effective thermal conductivity of the composite the new third order bounds are accurate to third order in epsilon a epsilon b where epsilon a is the thermal conductivity of phase a and require a total of frac 1 2 n 1 2 third order correlation parameters in their evaluation for two phase composites these bounds require only one geometrical parameter and are identical to beran s bounds for n geqslant 3 the third order bounds use the same information as beran s bounds but always more restrictive than beran s bounds the new fourth order bounds are accurate to fourth order in epsilon a epsilon b and require an additional frac 1 4 n 2 n 1 2 fourth order correlation parameters for n 2 only two parameters are needed and the fourth order bounds are shown to be more restrictive than the third order beran s bounds and the second order hashin shtrikman s bounds the application of the third order bounds to a spherical cell material of miller is illustrated

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

the effective conductivity of random suspensions of spherical particles the effective conductivity of an infinite random mono disperse hard sphere suspension is reported for particle to matrix conductivity ratios of 10 and 0 01 for sphere volume fractions c up to 0 6 the conductivities are computed with a method previously described by the authors which includes both far and near field interactions and the particle configurations are generated via a monte carlo method the results are consistent with the previous theoretical work of d j jeffrey to o c 2 and the bounds computed by s torquato and f lado it is also found that the clausius mosotti equation is reasonably accurate for conductivity ratios of 10 or less all the way up to 60 by volume the calculated conductivities compare very well with those of experiments in addition percolation like numerical experiments are performed on periodically replicated cubic lattices of n nearly touching spheres with an infinite particle to matrix conductivity ratio where the conductivity is computed as spheres are removed one by one from the lattice under suitable normalization of the conductivity and volume fraction it is found that the initial volume fraction must be extremely close to maximum packing in order to observe a percolation transition indicating that the near field effects must be very large relative to far field effects these percolation transitions occur at the accepted values for simple sc bodycentred bcc and face centred fcc cubic lattices also the vulnerability of the lattices computed here are exactly those of previous investigators due to limited data above the percolation threshold we could not correlate the conductivity with a power law near the threshold however it can be correlated with a power law for large normalized volume fractions in this case the exponents are found to be 1 70 1 75 and 1 79 for sc bcc and fcc lattices respectively

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

the transition from regular to irregular motions explained as travel on riemann surfaces we introduce and discuss a simple hamiltonian dynamical system interpretable as a three body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on riemann surfaces the interest of this phenomenology illustrating the onset in a deterministic context of irregular motions is underlined by its generality suggesting its eventual relevance to understand natural phenomena and experimental investigations here only some of our main findings are reported without detailing their proofs a more complete presentation will be published elsewhere

inversion of a general hyperelliptic integral and particle motion in hořava lifshitz black hole space times the description of many dynamical problems such as the particle motion in higher dimensional spherically and axially symmetric space times is reduced to the inversion of hyperelliptic integrals of all three kinds the result of the inversion is defined locally using the algebro geometric techniques of the standard jacobi inversion problem and the foregoing restriction to the divisor for a representation of the hyperelliptic functions the klein weierstras multi variable function is introduced it is shown that all parameters needed for the calculations such as period matrices and abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and constants the cases of genus two three and four are considered in detail the method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves applications are for instance solutions to the geodesic equations in the space times of static spherically symmetric ho ava lifshitz black holes

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

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

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

general solution of a three body problem in the plane we provide and discuss the general solution of a hamiltonian three body problem in the plane characterized by newtonian equations of motion with rotation and translation invariant velocity dependent one body and two body forces the model features a nonnegative real parameter when it does not vanish all solutions are completely periodic with period t 2 when it vanishes both unbounded and confined motions are possible with a rather rich phenomenology of possible behaviour in the latter case including completely periodic motions and limit cycles

nonlinear harmonic oscillators the existence is noted of assemblies of an arbitrary number of complex oscillators or equivalently of an arbitrary even number of real oscillators characterized by newtonian equations of motion acceleration equal force with one body velocity dependent linear forces and many body velocity independent cubic forces all the nonsingular solutions of which are isochronous completely periodic with the same period as for the singular solutions as usual they emerge in the context of the initial value problem from a closed domain in phase space having lower dimensionality

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

bounds for the complex dielectric constant of a two phase composite abstract we study the effective dielectric constant of a two phase composite for the case where the dielectric constants of the two components are complex milton has derived a sequence of narrowing bounds in the complex plane limiting the possible values of the effective dielectric constant assuming that certain geometrical coefficients characterizing the medium are known we cast milton s bounds in a different form making use of auxiliary functions introduced by bergman we derive explicit expressions for the bounds up to third order and compare values for the corresponding geometrical coefficient derived for different geometries

newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2 νn 1 n z ergodicity isochrony and fractals we study the complexification of the one dimensional newtonian particle in a monomial potential we discuss two classes of motions on the associated riemann surface the rectilinear and the cyclic motions corresponding to two different classes of real and autonomous newtonian dynamics in the plane the rectilinear motion has been studied in a number of papers while the cyclic motion is much less understood for small data the cyclic time trajectories lead to isochronous dynamics for bigger data the situation is quite complicated computer experiments show that for sufficiently small degree of the monomial the motion is generically isochronous with integer period which depends in a quite sensitive way on the initial data if the degree of the monomial is sufficiently high computer experiments show essentially chaotic behavior we suggest a possible theoretical explanation of these different behaviors we also introduce a two parameter family of two dimensional mappings describing the motion of the center of the circle as a convenient representation of the cyclic dynamics we call such a mapping the center map computer experiments for the center map show a typical multifractal behavior with periodicity islands therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity

rhizospheres in spherical buildings let delta be a moufang spherical building with field of definition f let gamma be a subgroup of mathrm aut delta that acts on the set of chambers of delta with finite orbits and let m be the kernel of the natural homomorphism from mathrm aut delta to mathrm aut f we show 1 if gamma cap m 1 and there exist proper residues of delta stabilized by gamma then gamma is a descent group of delta 2 if gamma cap m has no finite quotients of order divisible by mathrm char f then gamma is completely reducible

a technique to identify solvable dynamical systems and another solvable extension of the goldfish many body problem we take advantage of the simple approach recently discussed which associates to solvable matrix equations solvable dynamical systems interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles starting from a solvable matrix evolution equation we obtain the corresponding many body model and note that in one case the auxiliary variables can be altogether eliminated obtaining thereby an also hamiltonian extension of the goldfish model the solvability of this novel model and of its isochronous variant is exhibited a related as well solvable model is also introduced as well as its isochronous variant finally the small oscillations of the isochronous models around their equilibrium configurations are investigated and from their isochronicity certain diophantine relations are evinced

integrable systems of quartic oscillators in ordinary three dimensional space a class of completely integrable and indeed solvable hamiltonian many body problems are exhibited characterized by rotation invariant newtonian equations of motion acceleration equals force with linear and cubic forces in ordinary three dimensional space the corresponding hamiltonians are of normal type with the kinetic energy quadratic in the canonical momenta and the potential energy quadratic and quartic in the canonical coordinates

teichmuller curves in moduli space eisenstein series and an application to triangular billiards there exists a teichmuller disc n containing the riemann surface ofy2 x n 1 in the genus n 1 2 teichmuller space such that the stabilizer of n in the mapping class group has a fundamental domain of finite poincare volume in n application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles n n n 2 n and to the uniform distribution of infinite billiard trajectories in the same triangles

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained

new algebraically solvable systems of two autonomous first order ordinary differential equations with purely quadratic right hand sides we identify many new solvable subcases of the general dynamical system characterized by two autonomous first order ordinary differential equations with purely quadratic right hand sides and the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations equivalently by considering the analytic continuation of these systems to complex time their algebraically solvable character corresponds to the fact that their general solution either is single valued or features only a finite number of algebraic branch points as functions of complex time the independent variable thus our results provide a major enlargement of the class of solvable systems beyond those with a single valued general solution identified by garnier about 60 years ago an interesting property of several of these new dynamical systems is the elementary character of their general solution identifiable as the roots of a polynomial with explicitly obtainable time dependent coefficients we also mention that using a time dependent change of dependent and independent variables involving the imaginary parameter i isochronous variants of each of the algebraically solvable models we identify can be explicitly exhibited by this we mean that these variants all feature the remarkable property that their generic solution is periodic with a period that is a fixed integer multiple of the basic period t 2

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