the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

using a curvilinear grid to construct symmetry preserving discretizations for lagrangian gas dynamics the goal of this paper is to construct discretizations for the equations of lagrangian gas dynamics that preserve plane cylindrical and spherical symmetry in the solution of the original differential equations the new method uses a curvilinear grid that is reconstructed from a given logically rectangular distribution of nodes the sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles our procedure is exact for straight lines and circles that is it reproduces rectangular and polar grids exactly we use the method of support operators to construct a conservative finite difference method that we demonstrate will preserve spatial symmetries for certain choices of the initial grid we also introduce a curvilinear version of artificial edge viscosity that also preserves symmetry we present numerical examples to demonstrate our theoretical considerations and the robustness of the new method

on the rate of convergence in the entropic central limit theorem we study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition

analysis of material interface discontinuities and superconvergent fluxes in finite difference theory abstract an analysis of material interface discontinuities is developed and applied in finite difference theory to determine mathematically rigorous averaging techniques for material properties this result is compared with other averaging techniques particularly harmonic averaging which is often applied in practice we also develop a class of formulas of high accuracy for post processing the difference formula to compute derivatives fluxes stresses and conduct supporting numerical studies

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

the convergence of mimetic discretization for rough grids we prove that the mimetic finite difference discretizations of laplace s equation converges on rough logically rectangular grids with convex cells mimetic discretizations for the invariant operators divergence gradient and curl satisfy exact discrete analogs of many of the important theorems of vector calculus the mimetic discretization of the laplacian is given by the composition of the discrete divergence and gradient we first construct a mimetic discretization on a single cell by geometrically constructing inner products for discrete scalar and vector fields then constructing a finite volume discrete divergence and then constructing a discrete gradient that is consistent with the discrete divergence theorem this construction is then extended to the global grid we demonstrate the convergence for the two dimensional laplace equation with dirichlet boundary conditions on grids with a lower bound on the angles in the cell corners and an upper bound on the cell aspect ratios the best convergence rate to be expected is first order which is what we prove the techniques developed apply to far more general initial boundary value problems

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

support operator finite difference algorithms for general elliptic problems abstract an algorithm is developed for discretizing boundary value problems given by a general linear elliptic second order partial differential equation with general mixed or robin boundary conditions in general logically rectangular grids the continuum problem can be written as an operator equation where the operator is self adjoint and positive definite the discrete approximations have the same property consequently the matrices for the discrete problem are symmetric and positive definite also the scheme has a nearest neighbor stencil consequently the most powerful linear solvers can be applied in smooth grids the algorithm produces second order accurate solutions it is the generality of the problem general matrix coefficients general boundary conditions general logically rectangular grids that makes finding such an algorithm difficult the algorithm which is a combination of the method of support operators and the mapping method overcomes certain difficulties of the individual methods producing a high quality algorithm for solving general elliptic problems

the santalo regions of a convex body motivated by the blaschke santalo inequality we define for a convex body k in r and for t r the santalo regions s k t of k we investigate properties of these sets and relate them to a concept of affine differential geometry the affine surface area of k let k be a convex body in r for x int k the interior of k let k be the polar body of k with respect to x it is well known that there exists a unique x0 int k such that the product of the volumes k k0 is minimal see for instance sch this unique x0 is called the santalo point of k moreover the blaschke santalo inequality says that k k0 v n where vn denotes the volume of the n dimensional euclidean unit ball b 0 1 with equality if and only if k is an ellipsoid for t r we consider here the sets s k t x k k k v2 n t following e lutwak we call s k t a santalo region of k observe that it follows from the blaschke santalo inequality that the santalopoint x0 s k 1 and that s k 1 x0 if and only if k is an ellipsoid thus s k t has non empty interior for some t 1 if and only if k is not an ellipsoid in the first part of this paper we show some properties of s k t and give estimates on the size of s k t this question was asked by e lutwak the paper was written while both authors stayed at msri supported by a grant from the national science foundation msc classification 52

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

polymers as self avoiding walks a brief overview is presented of the relation of the properties of real polymers to the problem of self avoiding random walks the self consistent field method is discussed wherein the non markovian continuous self avoiding polymer is replaced by a self consistent markovian approximation an outline is presented of the method of solution of the resultant nonlinear integrodifferential equations a description is also presented of the scaling theories which provide a means for deducing some exponents in the asymptotic dependence of walk properties on the length of the walk in the limit of infinite length walks

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

the effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods abstract the support operators method of discretizing partial differential equations produces discrete analogs of continuum initial boundary value problems that exactly satisfy discrete conservation laws analogous to those satisfied by the continuum system thus the stability of the method is assured but currently there is no theory that predicts the accuracy of the method on nonuniform grids in this paper we numerically investigate how the accuracy particularly the accuracy of the fluxes depends on the definition of the inner product for discrete vector fields we introduce two different discrete inner products the standard inner product that we have used previously and a new more accurate inner product the definitions of these inner products are based on interpolation of the fluxes of vector fields the derivation of the new inner product is closely related to the use of the piola transform in mixed finite elements computing the formulas for the new accurate inner product requires a nontrivial use of computer algebra from the results of our numerical experiments we can conclude that using more accurate inner product produces a method with the same order of convergence as the standard inner product but the constant in error estimate is about three times less however the method based on the standard inner product is easier to compute with and less sensitive to grid irregularities so we recommend its use for rough grids

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

the surface measure and cone measure on the sphere of ell_p n we prove a concentration inequality for the l n p norm on the l n p sphere for p q 0 this inequality which generalizes results of schechtman and zinn 2000 is used to study the distance between the cone measure and surface measure on the sphere of l n p in particular we obtain a significant strengthening of the inequality derived by naor and romik 2003 and calculate the precise dependence of the constants that appeared there on p

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

the affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions