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

translation and related bounds for the response of a nonlinear composite conductor lower bounds for the overall energy functions of a class of nonlinear composite conductors are established by two different methods the translation method and the use of a comparison linear composite when the properties of the comparison linear composite are themselves estimated by the translation method the resulting comparison bound is demonstrated never to be superior to the bound obtained by direct application of the translation method to the nonlinear composite examples show however that the bounds obtained by these two methods in fact often coincide when they do the use of the comparison medium permits the development of more refined bounds by exploiting better bounds for the linear composite particular emphasis is placed on a limiting case of material behaviour for which explicit results can be obtained a composite dielectric that displays breakdown when the local electric field exceeds a critical value the formalism delivers an upper bound for the value of the mean field in the composite which induces breakdown in the composite bounds are also established for a composite with a power law relation between current and electric field

group actions on central simple algebras abstract let g be a group f a field and a a finite dimensional central simple algebra over f on which g acts by f algebra automorphisms we study the subalgebras and ideals of a which are preserved by the group action we prove a structure theorem and two classification theorems for invariant subalgebras under suitable hypotheses on a we illustrate these results in the case of compact connected lie groups and give some other applications we also classify invariant ideals

the cavity of the optimal shape under the shear stresses the problem of optimal shape of a single cavity in an infinite 2 d elastic domain is analyzed an elastic plane is subjected to a uniform load at infinity the cavity of the fixed area is said to be optimal if it provides the minimal energy change between the homogeneous plane and the plane with the cavity we show that for the case of shear loading the contour of the optimal cavity is not smooth but is shaped as a curved quadrilateral the shape is specified in terms of conformal mapping coefficients and explicit analytical representations for components of the dipole tensor associated with the cavity are employed we also find the exact values of angles at the corners of the optimal contour the applications include the problems of optimal design for dilute composites

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

sets of conductivity and elasticity tensors stable under lamination a complete characterization is obtained of any set of constitutive conductivity or elasticity tensors that is stable under lamination between any two of the elements of the set the conditions are local and are expressed in terms of the curvature and tangent plane at points on the boundary of the set they are checked to hold for several well known examples of sets stable under a more general process of homogenization a companion paper investigates the link between these conditions and quasiconvexity 1994 john wiley sons inc

towards characterization of all 3 3 extremal quasiconvex quadratic forms given a d d quasiconvex quadratic form d 3 we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero then the form itself is an extremal quasiconvex quadratic form i e it loses its quasiconvexity whenever a convex quadratic form is subtracted from it in the special case d 3 we slightly weaken the condition namely we prove that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square then the form itself is an extremal quadratic form in the case d 3 we also prove that if the determinant of the acoustic tensor of the form is identically zero then the form is either extremal or polyconvex also if the determinant of the acoustic tensor of the form is a perfect square then the form is either extremal or polycovex or is a sum of a rank 1 form and an extremal whose acoustic tensor determinant is identically zero here we use the notion of extremality introduced by milton in 1990 2017 wiley periodicals inc

free martingale polynomials abstract in this paper we investigate the properties of free sheffer systems which are certain families of martingale polynomials with respect to the free levy processes first we classify such families that consist of orthogonal polynomials these are the free analogs of the meixner systems next we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials finally we indicate how rota s finite operator calculus can be modified for the free context

optimal bounds on the effective bulk modulus of polycrystals the classical problem of bounding the effective bulk modulus of isotropic polycrystalline aggregates built from a single anisotropic phase is studied for three dimensional composites we find that the voigt and reuss bounds are each attained by multi layered laminate geometries and hence are optimal for two dimensional composites the reuss lower bound is proved to be optimal while a new optimal upper bound is derived

minimizers for a double well problem with affine boundary conditions this paper is concerned with the existence of minimizers for functionals having a double well integrand with affine boundary conditions such functionals are related to the so called kohn strang functional which arises in optimal shape design problems in electrostatics or elasticity they are known to be not quasi convex and therefore existence of minimizers is in general guaranteed only for their quasi convex envelopes we generalize previous results in 1 and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory we also prove by a general argument that their rank one convexifications coincide with their quasi convexifications

exact relations for effective tensors of polycrystals ii applications to elasticity and piezoelectricity an important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation under lamination this requirement is so restrictive that it is possible if not always feasible to find all such relations explicitly in order to do this one needs to combine the results developed in part i of this paper and the representation theory of the rotation groups so 2 and so 3 more precisely one needs to know all rotationally invariant subspaces of the space of material moduli this paper presents an algorithm for finding all such subspaces we illustrate the workings of the algorithm on the examples of 3 dimensional elasticity where we get all the exact relations and the examples of 2 dimensional and 3 dimensional piezoelectricity where we get some possibly all of them

subrepresentation semirings and an analog of 6j symbols let v be a complex representation of the compact group g the subrepresentation semiring associated to v is the set of subrepresentations of the algebra of linear endomorphisms of v with operations induced by the matrix operations the study of these semirings has been motivated by recent advances in materials science in which the search for microstructure independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group so 3 in this case the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j symbols familiar from the quantum theory of angular momentum in this paper we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by mackey multiplicity free representations of finite groups pac j math 8 503 1958 we introduce a new class of symbols called twisted 6j symbols for these groups and we explicitly calculate the structure constan

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

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

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

rank one plus a null lagrangian is an inherited property of two dimensional compliance tensors under homogenisation assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank one tensor plus a null lagrangian there is only one symmetric one in 2d the purpose of this paper is to prove that the effective compliance tensor has the same representation rank one plus the null lagrangian this statement generalizes the well known result of hill 14 15 that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus it also generalizes the surprising discovery in 1 that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined the present paper sheds light on this effect by placing it in a more general framework and using some elliptic pde theory rather than the translation method our results allow us to calculate the polycrystalline g closures of the special class of crystals under consideration our analysis is contrasted with a two dimensional model problem for shape memory polycrystals we show that the two problems can be thought of as elastic percolation problems one elliptic one hyperbolic

optimal use of the translation method and relaxations of variational problems the translation method has been used with great success in bounding the effective moduli of composite materials we consider here the analogous method for bounding the relaxations of variational problems we optimize the bound over the set of all available translations our method is to cast this in the form of a minmax problem using techniques of nonsmooth analysis we are able to identify the optimal translation bound meanwhile proving the existence of at least one optimal combination rank one convex quadratic and null lagrangian translation the optimal translation bound proves to be a better general lower bound on relaxations of variational problems than is the polyconvexification in three dimensions in two dimensions we discuss the negative result that the optimal translation bound is exactly the polyconvexification several examples of optimal applications of translation bounds to non convex nonlinear variational problems are given

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

meixner class of non commutative generalized stochastic processes with freely independent values ii the generating function let t be an underlying space with a non atomic measure on it in comm math phys 292 99 129 2009 the meixner class of non commutative generalized stochastic processes with freely independent values omega omega t t in t was characterized through the continuity of the corresponding orthogonal polynomials in this paper we derive a generating function for these orthogonal polynomials the first question we have to answer is what should serve as a generating function for a system of polynomials of infinitely many non commuting variables we construct a class of operator valued functions z z t t in t such that z t commutes with s for any s t in t then a generating function can be understood as g z omega sum n 0 infty int t n p n omega t 1 dots omega t n z t 1 dots z t n sigma dt 1 dots sigma dt n where p n omega t 1 dots omega t n is the kernel of the nth orthogonal polynomial we derive an explicit form of g z which has a resolvent form and resembles the generating function in the classical case albeit it involves integrals of non commuting operators we finally discuss a related problem of the action of the annihilation operators partial t t in t in contrast to the classical case we prove that the operators t related to the free gaussian and poisson processes have a property of globality this result is genuinely infinite dimensional since in one dimension one loses the notion of globality

orthogonal polynomials with a resolvent type generating function the subject of this paper are polynomials in multiple non commuting variables for polynomials of this type orthogonal with respect to a state we prove a favard type recursion relation on the other hand free sheffer polynomials are a polynomial family in non commuting variables with a resolvent type generating function among such families we describe the ones that are orthogonal their recursion relations have a more special form the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality which turns out to satisfy a type of second order difference equation if the difference equation is in fact first order and the state is tracial we show that the state is necessarily a rotation of a free product state we also describe interesting examples of non tracial infinitely divisible states with orthogonal free sheffer polynomials

the relaxation of two well energies with possibly unequal moduli the elastic energy of a multiphase solid is a function of its microstructure determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect microstructure evolution and optimal design mathematically the problem is to determine the relaxation under fixed phase fraction of a multiwell energy this paper addresses two such problems in the geometrically linear setting first in two dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy with arbitrary elastic moduli and transformation strains and provide a characterization of the optimal microstructures and the associated strain second in three dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy when either 1 both elastic moduli are isotropic or 2 the elastic moduli are well ordered and the smaller elastic modulus is isotropic in both cases we impose no restrictions on the transformation strains we provide a characterization of the optimal microstructures and the associated strain we also compute a lower bound that is optimal except possibly in one regime when either 1 both elastic moduli are cubic or 2 the elastic moduli are well ordered and the smaller elastic modulus is cubic for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal in all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain in both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction we also characterize optimal microstructures under applied stress

on the relation between extremal elasticity tensors with orthotropic symmetry and extremal polynomials we prove that an elasticity tensor with orthotropic symmetry is extremal if the determinant of its acoustic tensor is an extremal polynomial that is not a perfect square

stress energy minimization as a tool in the material layout design of shallow shells abstract the present research deals with the compliance minimization problem of an elastic thin shallow shell subjected to simultaneous in plane and bending loads in this context our goal is to lay out a given amount of material in the volume of a shell assuming that the distribution in the direction transversal to its middle surface s is homogeneous the discussion hence reduces to the question of finding the optimal material arrangement on s similar problems were solved in the framework of two dimensional elasticity or kirchhoff plate theory and the present research attempts to generalize these results following the pattern emerging from the above mentioned considerations our research starts from the minimum compliance problem of a structure made of two elastic materials whose volumetric fractions are fixed the existence of a solution to thus posed optimization task is guaranteed if the fine scale microstructural composites are admitted in the analysis their constitutive tensors can be obtained by certain averaging ensuing from the theory of homogenization for periodic media additionally by the castigliano theorem the compliance minimization problem is equivalent to the one for structural stress energy in turn the lower estimation of the energy is achieved in two steps i its modification by a certain energy like functional and ii utilizing the quasiconvexity property of thus obtained expression as a result formulae describing the effective stress energy of one material shallow shell and the material distribution function are explicitly derived

on the geometry of classifying spaces and horizontal slices we study the local properties of the moduli space of a polarized calabi yau manifold let u be a neighborhood of the moduli space then we know the universal covering space v of u is a smooth manifold suppose d is the classifying space of a polarized calabi yau manifold with the automorphism group g then we prove that the map from v to d induced by the period map is a pluriharmonic map we also give a kahler metric on v which is called the hodge metric we prove that the ricci curvature of the hodge metric is negative away from zero we also proved the nonexistence of the kahler metric on the classifying space of a calabi yau threefold which is invariant under a cocompact lattice of g 1 introduction let x be a polarized simply connected calabi yau manifold that is x is a simply connected compact kahler manifold of dimension n with zero first chern class and is akform of x such that h 2 x z in this paper we study the local properties of the moduli space of the polarized calabi yau manifold x by definition is the parameter space of the complex structures over x for the fixed polarization is a quasi projective variety by a theorem of viehweg 17 suppose x is a calabi yau manifold let n d im h 1 x tx 0

polynomials of meixner s type in infinite dimensions jacobi fields and orthogonality measures abstract the classical polynomials of meixner s type hermite charlier laguerre meixner and meixner pollaczek polynomials are distinguished through a special form of their generating function which involves the laplace transform of their orthogonality measure in this paper we study analogs of the latter three classes of polynomials in infinite dimensions we fix as an underlying space a non compact riemannian manifold x and an intensity measure on it we consider a jacobi field in the extended fock space over l 2 x whose field operator at a point x x is of the form x x x x x x x where is a real parameter here x and x are respectively the annihilation and creation operators at the point x we then realize the field operators as multiplication operators in l 2 d where d is the dual of d c 0 x and is the spectral measure of the jacobi field we show that is a gamma measure for 2 a pascal measure for 2 and a meixner measure for is a levy noise measure the isomorphism between the extended fock space and l 2 d is carried out by infinite dimensional polynomials of meixner s type we find the generating function of these polynomials and using it we study the action of the operators x and x in the functional realization

a scalar inequality which bounds the effective conductivity of composites new bounds on the eigenvalues of the effective conductivity tensor in spatial dimension d 3 are proved using the compensated compactness method the inequalities are obtained using the nine quadratic null lagrangians given by the 2 times 2 minors of a jacobian matrix partial phi i partial x j it is proved that in general the bounds depend at most on four parameters

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

exact relations for effective tensors of composites necessary conditions and sufficient conditions typically the elastic and electrical properties of composite materials are strongly microstructure dependent so it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes they can also be regarded as fundamental invariances existing in a given physical context classic examples include hill s formulae for the effective bulk modulus of a two phase mixture when the phases have equal shear moduli levin s formulae linking the effective thermal expansion coefficient and effective bulk modulus of two phase mixtures and dykhne s result for the effective conductivity of an isotropic two dimensional polycrystalline material here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones the search for exact relations is reduced to a search for matrix subspaces having a structure of special jordan algebras one of many new exact relations is for the effective shear modulus of a class of three dimensional polycrystalline materials we present complete lists of exact relations for three dimensional thermoelectricity and for three dimensional thermopiezoelectric composites that include all exact relations for elasticity thermoelasticity and piezoelectricity as particular cases 2000 john wiley sons inc

compactification of complete kahler surfaces of finite volume satisfying certain curvature conditions on etudie systematiquement la compactification des varietes de kahler completes de volume fini et de courbure sectionnelle bornee

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

homogenization of dynamic laminates abstract this paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities called dynamic materials the study was initiated by lurie in his pioneering work of 1997 homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation two typical examples of periodic homogenization are considered the wave equation and maxwell s system coefficients oscillating rapidly not only in space but also in time conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed in particular we examine a cell problem for periodic composites as well as the laminate formulae the effective tensors of rank one laminates for one dimensional wave equation and the full maxwell s system are computed explicitly we also note some dramatic differences between the hyperbolic and the elliptic cases

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

compactification of complete kahler manifolds of negative ricci curvature soit m une variete de kahler complete de dimension m et soit la forme de kahler on suppose 1 ric 0 2 m est tres fortement m 2 pseudoconcave 3 le recouvrement universel de m est de stein alors m est biholomorphe a une variete quasiprojective

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

topology design with optimized self adaptive materials significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems we study the optimal shape design of a two dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology relaxation of the optimization problem via quasiconvexification or homogenization methods is required the effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material

optimal g closure bounds via stability under lamination optimal bounds for a two dimensional g closure problem are proved they are obtained by combining together bounds derived using fine properties of quasiconformal mappings and a stability property of the g closure which is known as stability under lamination

bounds on the volume fractions of two materials in a three dimensional body from boundary measurements by the translation method using the translation method of tartar murat lurie and cherkaev bounds are derived on the volume occupied by an inclusion in a three dimensional conducting body the bounds assume that electrical impedance tomography measurements have been made for three sets of pairs of current flux and voltage measurements around the boundary additionally the conductivity of the inclusion and the conductivity of the surrounding medium are assumed to be known if the boundary data dirichlet or neumann is special i e such that the fields inside the body would be uniform were the body homogeneous then the bounds reduce to those of milton and thus when the volume fraction is small to those of capdeboscq and vogelius

asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals because of the uncertain irregular microgeometry a random polycrystalline aggregate the shape and crystalline orientations of the constituent anisotropic grains of which are uncorrelated may have elastic moduli scattered over some though small intervals while the conventional macroscopic homogeneity and isotropy hypotheses for it may be considered only as approximate with corresponding uncertainty our formal bounds are proposed to provide the estimates on those uncertainties in asymptotic sense explicit expressions for the aggregates of trigonal crystals classes 3m 32 3m are derived

on the effective elasticity of a two dimensional homogenised incompressible elastic composite the set of effective elasticity tensors for all two dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described in two dimensions the effective tensors are completely characterised by bounds on their eigenvalues

multiphase interchange inequalities phase interchange inequalities have been studied since the early work of keller j math phys 5 548 1964 they constrain the effective conductivity of composite materials which are obtained from each other for fixed configuration by interchanging the position of the phases optimal results in this direction for the case of a two phase composite are due to keller in spatial dimension d 2 and to avellaneda et al j appl phys 63 4989 1988 in dimension d 3 in this paper new inequalities in spatial dimension d 2 and d 3 which are valid when an arbitrary number of phases is present are proven when specialized to two phase composites they agree with those of keller in d 2 and of avellaneda et al in d 3 respectively

free meixner states free meixner states are a class of functionals on non commutative polynomials introduced in ans06 they are characterized by a resolvent type form for the generating function of their orthogonal polynomials by a recursion relation for those polynomials or by a second order non commutative differential equation satisfied by their free cumulant functional in this paper we construct an operator model for free meixner states by combinatorial methods we also derive an operator model for their free cumulant functionals this in turn allows us to construct a number of examples some of these examples are shown to be trivial in the sense of being free products of functionals which depend on only a single variable or rotations of such free products on the other hand the multinomial distribution is a free meixner state and is not a product neither is a large class of tracial free meixner states which are analogous to the simple quadratic exponential families in statistics

geometry of domains with the uniform squeezing property abstract we introduce the notion of domains with uniform squeezing property study various analytic and geometric properties of such domains and show that they cover many interesting examples including teichmuller spaces and hermitian symmetric spaces of non compact type the properties supported by such manifolds include pseudoconvexity hyperconvexity kahler hyperbolicity vanishing of cohomology groups and quasi isometry of various invariant metrics it also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of serre about stein properties of holomorphic fiber bundles

rigid perfectly plastic two dimensional polycrystals we consider rigid perfectly plastic polycrystals in the two dimensional anti plane shear context the yield sets of the grains are identified with rectangles in the plane centred at the origin whose sides have length 2 and 2 m the limit m corresponds to the grains being rigid in one direction and ductile in the orthogonal direction we show that for large values of m there exist polycrystals whose effective yield sets are large in all directions more precisely for each value of m we construct a polycrystal whose yield set contains the set f f f f where f m o 1 we also show that the yield set of any isotropic polycrystal is contained in the ball of radius 4 m centred at the origin this bound results as an application of the div curl lemma the new component of our analysis which allowed us to obtain sharper results is that we consider simultaneously not only two but an infinite number of admissible stress fields whose averages have different directions

a link between sets of tensors stable under lamination and quasiconvexity a link is found between quasiconvexity and the conditions for a set l of conductivity or elasticity tensors to be stable under lamination these conditions derived in the companion paper are shown here to be equivalent to the condition that for every point b on the boundary of the set l an operator tb dependent on the tangent plane and curvature of the set at b is a quasiconvex translation operator a separate class of quasiconvex translation operators is obtained which are candidates for proving that l is stable under homogenization the region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set l and has curvature at that point not greater than the curvature of l the conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around l remains unresolved it is proved that l can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their legendre transforms 1994 john wiley sons inc

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

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

on the effective viscoelastic moduli of two phase media iii rigorous bounds on the complex shear modulus in two dimensions the translation and the hashin shtrikman methods are used to provide bounds on the effective complex shear modulus of a two phase two dimensional viscoelastic composite they are both given by inequalities that depend on six parameters the best bounds are obtained by optimizing these parameters over the admissible set which is larger for the translation method than for the hashin shtrikman method consequently the translation method generally leads to tighter bounds than would be obtained via the standard hashin shtrikman approach equivalence classes of two dimensional viscoelastic composites directly analogous to similar classes for the pure elastic problem are found combination of the simplified versions of the translation or the hashin shtrikman type bounds and this equivalency results in simple algorithms for computing tight bounds for any choice of phase moduli and volume fractions the bounds constrain the effective shear modulus to lie inside a region of the complex plane bounded by arcs of circles the four points which correspond to the hashin shtrikman walpole bounds on the shear modulus of an elastic composite always lie inside or on the boundary of the bounding region in many cases the bounding region tends to hug the corresponding parallelogram in the complex compliance plane having these four points as vertices

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

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

variational estimates of the self consistent type for the effective behaviour of some model nonlinear polycrystals a new class of self consistent estimates for nonlinear polycrystals is obtained by making use of the classical self consistent estimates for the linear comparison polycrystal in the nonlinear variational procedure of debotton and ponte castaneda 1995 a distinctive feature of the new self consistent estimates is that they satisfy the recent bound of kohn and little 1997 for two dimensional ideally plastic polycrystals which is known to be significantly more restrictive than the taylor bound at large grain anisotropy in addition the new self consistent estimates are found to satisfy the taylor and hashin shtrikman bounds which are sharper than the kohn little bound at low grain anisotropy this result suggests that the new self consistent estimates may also be more accurate than other self consistent estimates like those arising from the incremental model of hill 1965 for three dimensional nonlinear polycrystals

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

optimal bounds for conduction in two dimensional multiphase polycrystalline media a complete geometrical characterization is given of the set of all possible effective conductivity tensors for two dimensional composites made of an arbitrary number of given anisotropic phases it is rigorously established that any polycrystalline composite formed from an arbitrary possibly infinite number of phases can be replaced by a composite formed from only two of the given phases without altering the effective conductivity tensor

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

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

fourth order moments of nonnegative measures on s2 and applications this paper is mainly devoted to a study of the fifteen dimensional set of all fourth order moments of a nonnegative radon borel regular measure on s 2 we seek a complete characterization of the set a closed cone with the help of the hilbert decomposition theorem for nonnegative polynomials of degree four in two variables special emphasis is placed on the boundary of this set which is shown to be generated by atomic measures made of five dirac masses or fewer which are located on the intersection of s 2 with the zero set of a quadratic form as a consequence every point of the moment set is generated by atomic measures made of six dirac masses or less this study may be viewed as a contribution to the moment problem note that the analogous two dimensional case has been analyzed in avellaneda milton am potential applications for this result are however manifold especially in the field of homogenization which is of especial interest to us in the setting of linearized elasticity effective properties of fine mixtures of two phases are investigated specifically the goal is to analyze the coefficients of the linear second order elliptic system satisfied by the weak limit u of the solution fields w to a sequence of elasticity problems of the form

compactifying complete kahler einstein manifolds of finite topological type and bounded curvature siu yau 16 studied the compactification of complete kaihler manifolds of finite volume and of riemannian sectional curvature pinched between two negative constants in 12 the first author of the present article started a systematic study of the more general problem of compactifying complete kwhler manifolds of finite volume and of bounded curvature a number of conjectures were formulated which can all be regarded as conjectural generalizations of the compactification of arithmetic varieties i e quotients of bounded symmetric domains q by torsion free arithmetic subgroups of aut q in a differentialgeometric setting here and henceforth all discrete groups of automorphisms will be assumed torsion free in the same article we treated the case of kaihler surfaces and proved

homogenisation of a class of fourth order equations with application to incompressible elasticity synopsis upon formalising an analogy between two dimensional stokes flow and two dimensional isotropic conductivity we exhibit a class of fourth order equations which behave isomorphically like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours in particular lipton s result on the g closure problem for mixtures of two incompressible elastic materials is recovered in the two dimensional case

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

exact relations for effective tensors of polycrystals i necessary conditions the set of all effective moduli of a polycrystal usually has a nonempty interior when it does not we say that there is an exact relation for effective moduli this can indeed happen as evidenced by recent results 4 10 12 on polycrystals in this paper we describe a general method for finding such relations for effective moduli of laminates the method is applicable to any physical setting that can be put into the hilbert space framework developed by milton 13 the idea is to use the w function of milton 13 that transforms a lamination formula into a convex combination the method reduces the problem of finding exact relations to a problem from representation theory of so d d 2 or 3 corresponding to a particular physical setting when this last problem is solved there is a finite amount of calculation required to be done in order to answer the question completely at present each candidate relation has to be examined separately in order to confirm the stability under homogenization we apply our general theory to the settings of conductivity and two dimensional elasticity

compactification of complete kahler surfaces with negative ricci curvature it is a classical result of satake 17 baily borel 3 and ash mumford rapoport tai 2 that quotients of bounded symmetric domain by arithmetic groups of automorphism can be compactified the methods employed are essentially algebraic in nature along another direction andreott i grauert 1 siu yau 19 and nadel tsuji 14 obtained results which cover the above manifolds using the analytic properties of pseudoconcavity on the other hand it is believed that complete k ihler manifolds satisfying purely differential geometric conditions of bounded sectional curvature and negative ricci curvature can be compactified using differential geometric methods in 12 mok started a systematic scheme of attacking the problem and results in the work of mok zhong 13 the present article can be considered as a continuation of this scheme in particular we prove the conjecture in case of dimensional two more precisely we prove