cone volume measure of general centered convex bodies abstract we show that the cone volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition this extends former results obtained in the discrete as well as in the symmetric case and implies among others a conjectured best possible inequality for the u functional of a convex body

volume inequalities for isotropic measures a direct approach to ball s simplex inequality is presented this approach which does not use the brascamp lieb inequality also gives barthe s characterization of the simplex for ball s inequality and extends it from discrete to arbitrary measures it also yields the dual inequality along with equality conditions and it does both for arbitrary measures

necessary subspace concentration conditions for the even dual minkowski problem abstract we prove tight subspace concentration inequalities for the dual curvature measures c q k of an n dimensional origin symmetric convex body for q n 1 this supplements former results obtained in the range q n

application of an idea of voronoĭ to john type problems abstract using an idea of vorono many john type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the 1 2 d d 1 dimensional cone of positive definite quadratic forms in this way we prove several new john type and minimum position results and give alternative versions and extensions of known results in particular we characterize minimum ellipsoidal shells of convex bodies and in the typical case show their uniqueness and determine the contact number these results are formulated also in terms of the circumradius of convex bodies next circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied then we investigate john type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body the technique used in this context finally is applied to obtain corresponding results for the mean width and the surface area

the even orlicz minkowski problem the classical minkowski problem leads to the lp minkowski problem and now to the orlicz minkowski problem existence is demonstrated for the even orlicz minkowski problem a byproduct is a new approach to the solution of the classical minkowski problem

a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

a characterization of blaschke addition a characterization of blaschke addition as a map between origin symmetric con vex bodies is established this results from a new characterization of minkowski addition as a map between origin symmetric zonoids combined with the use of l evy prokhorov metrics a full set of examples is provided that show the results are in a sense the best possible

compatible fluxes for van leer advection application of the standard second order upstream centered advection method of van leer to a coupled transport equation system of the form t a a u 0 can produce artificial extrema in the ratiot a even thoughtsatisfies a simple advection equation and is expected to preserve monotonicity hereuis velocity is mass density andais a conserved quantity such as momentum density energy density or chemical species density thustis a mass specific transport quantity such as velocity energy per unit mass temperature or species mass fraction a new flux formulation and gradient limiting procedure is presented here which eliminates these artificial extrema preserves the second order accuracy and preserves the monotone character of the van leer method even in the limit of vanishing mass density such a formulation is calledcompatible the method is noniterative i e explicit and can be employed in a general finite volume framework sample results for the transport of a square wave in one and two dimensions are provided

on displacement shallow water wave equation and symplectic solution abstract in this paper the shallow water wave problem is discussed in the lagrangian description by using the hamilton variational principle in analytical mechanics a displacement shallow water wave equation dswwe is developed for the shallow water wave problem with a sloping water bottom and wet dry interface a numerical scheme based on the discretized hamilton principle is constructed for solving the proposed displacement shallow water wave equation the proposed numerical scheme is symplectic and explicit and can preserve the total energy and mass of the shallow water system in the discrete sense the correctness of the dswwe and the effectiveness of the proposed numerical scheme are verified by using four classical numerical examples numerical examples show that the proposed method performs well with respect to the simulation of the shallow water problem with a sloping water bottom and wet dry interface

shadow systems of asymmetric lp zonotopes abstract shadow systems are used to establish new asymmetric l p volume product and asymmetric l p volume ratio inequalities along with their equality conditions these inequalities have reisner s volume product inequality for l 1 zonotopes as a special case moreover uniqueness of the extremals in the symmetric setting is obtained

subspace concentration of dual curvature measures of symmetric convex bodies we prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body 2018 international press of boston inc all rights reserved

the multidimensional positive definite advection transport algorithm nonoscillatory option abstract this paper presents a nonoscillatory option i e free of dispersive ripples of the advection algorithm described previously in j comput phys 54 1984 325 67 1986 396 the approach adopted merges the flux corrected transport methodology with the iterative formalism of the algorithm further discussion of the algorithm s accuracy is included theoretical considerations are illustrated through numerical tests and examples of applications to atmospheric fluid dynamics problems

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

on dvoretzky s theorem for subspaces of lp abstract we prove that for any 2 p and for every n dimensional subspace x of l p represented on r n whose unit ball b x is in lewis position one has the following two level gaussian concentration inequality p z e z e e z c exp c min p e 2 n e n 2 p 0 e 1 where z is the standard n dimensional gaussian vector p 0 is a constant depending only on p and c c 0 are absolute constants as a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces in particular for any 2 p and every n dimensional subspace x of l p the euclidean space l 2 k can be 1 e embedded into x with k c p min e 2 n e n 2 p where c p 0 is a constant depending only on p this improves upon the previously known estimate due to figiel lindenstrauss and milman

the orlicz brunn minkowski theory a general framework additions and inequalities the orlicz brunn minkowski theory introduced by lutwak yang and zhang is a new extension of the classical brunn minkowski theory it represents a generalization of the lp brunn minkowski theory analogous to the way that orlicz spaces generalize lp spaces for appropriate convex functions 0 1 m 0 1 a new way of combining arbitrary sets in r n is introduced this operation called orlicz addition and denoted by has several desirable properties but is not associative unless it reduces to lp addition a general framework is introduced for the orlicz brunn minkowski theory that includes both the new addition and previously introduced concepts and makes clear for the rst time the relation to orlicz spaces and norms it is also shown that orlicz addition is intimately related to a natural and fundamental generalization of minkowski addition called m addition the results obtained show roughly speaking that the orlicz brunn minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the lp brunn minkowski theory inequalities of the brunn minkowski type are obtained both for m addition and orlicz addition the new orlicz brunn minkowski inequality implies the lp brunn minkowski in equality new orlicz minkowski inequalities are obtained that generalize the lp minkowski inequality one of these has connections with the conjectured log brunn minkowski inequal ity of lutwak yang and zhang and in fact these two inequalities together are shown to split the classical brunn minkowski inequality

mean width inequalities for isotropic measures in this paper we establish barthe s mean width inequalities for continuous isotropic measures by a direct approach rather than using the brascamp lieb inequality the following results are obtained among the convex hulls of the support of isotropic mea sures on s n 1 the regular simplex inscribed in the euclidean unit ball has maximal norm in the dual situation there is a reverse result for their polar bodies moreover the case of even isotropic measures is also investigated

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

compatible energy conserving bounds preserving remap of hydrodynamic fields for an extended ale scheme abstract from the very origins of numerical hydrodynamics in the lagrangian work of von neumann and richtmyer 83 the issue of total energy conservation as well as entropy production has been problematic because of well known problems with mesh deformation lagrangian schemes have evolved into arbitrary lagrangian eulerian ale methods 39 that combine the best properties of lagrangian and eulerian methods energy issues have persisted for this class of methods we believe that fundamental issues of energy conservation and entropy production in ale require further examination the context of the paper is an ale scheme that is extended in the sense that it permits cyclic or periodic remap of data between grids of the same or differing connectivity the principal design goals for a remap method then consist of total energy conservation bounded internal energy and compatibility of kinetic energy and momentum we also have secondary objectives of limiting velocity and stress in a non directional manner keeping primitive variables monotone and providing a higher than second order reconstruction of remapped variables in particular the new contributions fall into three categories associated with energy conservation and entropy production reconstruction and bounds preservation of scalar and tensor fields and conservative remap of nonlinear fields the paper presents a derivation of the methods details of implementation and numerical results for a number of test problems the methods requires volume integration of polynomial functions in polytopal cells with planar facets and the requisite expressions are derived for arbitrary order

an edge based unstructured mesh discretisation in geospherical framework an arbitrary finite volume approach is developed for discretising partial differential equations governing fluid flows on the sphere unconventionally for unstructured mesh global models the governing equations are cast in the anholonomic geospherical framework established in computational meteorology the resulting discretisation retains proven properties of the geospherical formulation while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution the latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement a class of non oscillatory forward in time edge based solvers is developed and applied to numerical examples of three dimensional hydrostatic flows including shallow water benchmarks on a rotating sphere

how to preserve the mass fractions positivity when computing compressible multi component flows abstract we are interested in the numerical investigation of the compressible flow of a gaseous mixture considering a hyperbolic system including the euler equations for the mixture and a mass conservation equation for each species we propose a new approximation scheme for the convective term of the species equations this approximation relies on some properties of the exact solution of the riemann problem for the multi component system and applies when an upwind godunov type scheme is used for the euler equations its main interest lies in the fact that it preserves the positivity and monotonicity of the mass fractions of all species

the orlicz brunn minkowski inequality abstract the orlicz brunn minkowski theory originated with the work of lutwak yang and zhang in 2010 in this paper we first introduce the orlicz addition of convex bodies containing the origin in their interiors and then extend the l p brunn minkowski inequality to the orlicz brunn minkowski inequality furthermore we extend the l p minkowski mixed volume inequality to the orlicz mixed volume inequality by using the orlicz brunn minkowski inequality

moment entropy inequalities for a random vector the pth moment matrix is defined for a real random vector generalizing the classical covariance matrix sharp inequalities relating the pth moment and renyi entropy are established generalizing the classical inequality relating the second moment and the shannon entropy the extremal distributions for these inequalities are completely characterized

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

iterated upwind schemes for gas dynamics a class of high resolution schemes established in integration of anelastic equations is extended to fully compressible flows and documented for unsteady and steady problems through a span of mach numbers from zero to supersonic the schemes stem from iterated upwind technology of the multidimensional positive definite advection transport algorithm mpdata the derived algorithms employ standard and modified forms of the equations of gas dynamics for conservation of mass momentum and either total or internal energy as well as potential temperature numerical examples from elementary wave propagation through computational aerodynamics benchmarks to atmospheric small and large amplitude acoustics with intricate wave flow interactions verify the approach for both structured and unstructured meshes and demonstrate its flexibility and robustness

local cohomology with support in ideals of maximal minors abstract suppose that k is a field of characteristic zero x is an r s matrix of indeterminates where r s and r k x is the polynomial ring over k in the entries of x we study the local cohomology modules h i i r where i is the ideal of r generated by the maximal minors of x we identify the indices i for which these modules vanish compute h i i r at the highest nonvanishing index i r s r 1 and characterize all nonzero ones as submodules of certain indecomposable injective modules these results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings

the log brunn minkowski inequality abstract for origin symmetric convex bodies i e the unit balls of finite dimensional banach spaces it is conjectured that there exist a family of inequalities each of which is stronger than the classical brunn minkowski inequality and a family of inequalities each of which is stronger than the classical minkowski mixed volume inequality it is shown that these two families of inequalities are equivalent in that once either of these inequalities is established the other must follow as a consequence all of the conjectured inequalities are established for plane convex bodies

extremal problems for lp surface areas and john ellipsoids abstract two closely connected extremal problems in convex geometry the minimal l p surface area and the maximal l p john ellipsoid of a convex body under translation transforms are solved it is shown that there exists a unique interior point of a convex body such that they attain their extrema

orlicz john ellipsoids abstract the orlicz john ellipsoids which are in the framework of the booming orlicz brunn minkowski theory are introduced for the first time it turns out that they are generalizations of the classical john ellipsoid and the evolved l p john ellipsoids the analog of ball s volume ratio inequality is established for the new orlicz john ellipsoids the connection between the isotropy of measures and the characterization of orlicz john ellipsoids is demonstrated

lp john ellipsoids it is shown that the classical john ellipsoid the petty ellipsoid and a recently discovered dual of the legendre ellipsoid are all special cases p infty p 1 and p 2 of a family of l p ellipsoids which can be associated with a fixed convex body this insight allows for a unified view of alternative approaches to and extensions of some basic results in convex geometry

a generalized rotationally symmetric case of the centroaffine minkowski problem abstract in this paper the centroaffine minkowski problem a critical case of the l p minkowski problem in the n 1 dimensional euclidean space is studied by its variational structure and the method of blow up analyses we obtain two sufficient conditions for the existence of solutions for a generalized rotationally symmetric case of the problem

high order multi material ale hydrodynamics we present a new approach for multi material arbitrary lagrangian eulerian ale hydrodynamics simulations based on high order finite elements posed on high order curvilinear meshes the method builds on and extends our previous work in the lagrangian v a dobrev t v kolev and r n rieben siam j sci comput 34 2012 pp b606 b641 and remap r w anderson et al internat j numer methods fluids 77 2015 pp 249 273 phases of ale and depends critically on a functional perspective that enables subzonal physics and material modeling v a dobrev et al internat j numer methods fluids 82 2016 pp 689 706 curvilinear mesh relaxation is based on node movement which is determined through the solution of an elliptic equation the remap phase is posed in terms of advecting state variables between two meshes over a fictitious time interval the resulting advection equation is solved by a discontinuous galerkin dg formulation combined with a customized flux corrected transpor

optimal sobolev norms in the affine class abstract optimal sobolev norms under volume preserving affine transformations are considered it turns out that this minimal transform is equivalent to the p 2 fisher information matrix defined by lutwak lv yang and zhang furthermore some analytic inequalities regarding to the l p affine and l p sine energies for the optimal function are investigated

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

existence of solutions to the orlicz minkowski problem abstract in this paper the orlicz minkowski problem a generalization of the classical minkowski problem is studied using the variational method we obtain a new existence result of solutions to this problem for general measures

mpdata an edge based unstructured grid formulation we present an advancement in the evolution of mpdata multidimensional positive definite advection transport algorithm over the last two decades mpdata has proven successful in applications using single block structured cuboidal meshes viz cartesian meshes while employing homeomorphic mappings to accommodate time dependent curvilinear domains motivated by the strengths of the cartesian mesh mpdata we develop a new formulation in an arbitrary finite volume framework with a fully unstructured polyhedral hybrid mesh in mpdata as in any taylor series based integration method for pde the choice of data structure has a pronounced impact on the technical details of the algorithm aiming at a broad range of applications with a large number of control volume cells we select a general compact and computationally efficient edge based data structure this facilitates the use of mpdata for problems involving complex geometries and or inhomogeneous anisotropic flows where mesh adaptivity is advantageous in this paper we describe the theory and implementation of the basic finite volume mpdata and document extensions important for applications a fully monotone scheme diffusion scheme and generalization to complete flow solvers theoretical discussions are illustrated with benchmark results in two and three spatial dimensions

lyz ellipsoid and petty projection body for log concave functions abstract the aims of this paper are to develop the lyz ellipsoid and petty projection body for log concave functions which correspond to the lyz ellipsoid and petty projection body for convex bodies when restricted to the subclass of characteristic functions moreover the continuous sl n contravariant valuation on a subclass of log concave functions is classified

a semi lagrangian discontinuous galerkin method for scalar advection by incompressible flows a new conservative semi lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form the approach combines the accuracy and conservation properties of the discontinuous galerkin dg method with the computational efficiency and robustness of semi lagrangian sl techniques unconditional stability in the von neumann sense is proved for the proposed discretization in the one dimensional case a monotonization technique is then introduced based on the flux corrected transport approach this yields a multi dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard dg methods the accuracy and stability of the method are further demonstrated by two dimensional tracer advection tests in the case of incompressible flows the comparison with results obtained by standard sl and dg methods highlights several advantages of the new technique

flux corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements abstract the flux corrected transport fct algorithm is applied to the unsteady and steady state particle transport equation the proposed fct method employs the following 1 a low order positivity preserving scheme based on the application of m matrix properties 2 a high order scheme based on the entropy viscosity method introduced by guermond 1 and 3 local discrete solution bounds derived from the integral transport equation the resulting scheme is second order accurate in space enforces an entropy inequality mitigates the formation of spurious oscillations and guarantees the absence of negativities space discretization is achieved using continuous finite elements time discretizations for unsteady problems include theta schemes such as explicit and implicit euler and strong stability preserving runge kutta ssprk methods the developed fct scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady state and implicit time discretizations

lp minkowski valuations on polytopes abstract for 1 p ludwig haberl and parapatits classified l p minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity in this paper a complete classification of l p minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for p 1 including p for n 3 and p 1 there exist valuations not mentioned before

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

first order continuous and discontinuous galerkin moment models for a linear kinetic equation model derivation and realizability theory abstract we provide two new classes of moment models for linear kinetic equations in slab and three dimensional geometry they are based on classical finite elements and low order discontinuous galerkin approximations on the unit sphere we investigate their realizability conditions and other basic properties numerical tests show that these models are more efficient than classical full moment models in a space homogeneous test when the analytical solution is not smooth

cone volume measure and stability abstract we prove stability results for two central inequalities involving the cone volume measure of a centered convex body the subspace concentration conditions and the u functional volume inequality

modelling atmospheric flows with adaptive moving meshes an anelastic atmospheric flow solver has been developed that combines semi implicit non oscillatory forward in time numerics with a solution adaptive mesh capability a key feature of the solver is the unification of a mesh adaptation apparatus based on moving mesh partial differential equations pdes with the rigorous formulation of the governing anelastic pdes in generalised time dependent curvilinear coordinates the solver development includes an enhancement of the flux form multidimensional positive definite advection transport algorithm mpdata employed in the integration of the underlying anelastic pdes that ensures full compatibility with mass continuity under moving meshes in addition to satisfy the geometric conservation law gcl tensor identity under general moving meshes a diagnostic approach is proposed based on the treatment of the gcl as an elliptic problem the benefits of the solution adaptive moving mesh technique for the simulation of multiscale atmospheric flows are demonstrated the developed solver is verified for two idealised flow problems with distinct levels of complexity passive scalar advection in a prescribed deformational flow and the life cycle of a large scale atmospheric baroclinic wave instability showing fine scale phenomena of fronts and internal gravity waves

from john to gauss john positions via dual mixed volumes abstract we investigate the optimization of the dual mixed volumes w i s k 0 s k d n where k r n is a convex body d n the euclidean ball and sk runs over all positions of k when s is linear we give necessary and sufficient conditions for k to be in extremal position in terms of a decomposition of the identity we consider affine problems and we also present an approach involving parallel sections of k which can be understood as a dual fractional kubota formula

non oscillatory continuous fem for transport and shallow water flows abstract the paper presents a sign preserving continuous finite element model fem for coupled transport equations and in particular for the depth integrated shallow water equations the model incorporates an enhanced flux corrected transport method fct with a high order solution given by a generalization of the characteristic based split finite element method cbs and a predictor positive algorithm founded in a finite element upwind methodology numerical experiments explore the efficacy of the procedure with emphasis in the simulation of flows with evolutionary boundaries determined by null values of any flow property and in severe conditions of propagation such as dam breaking waves

affine moments of a random vector an affine invariant pth moment measure is defined for a random vector and used to prove sharp moment entropy inequalities that are more general and stronger than standard moment entropy inequalities

new sine ellipsoids and related volume inequalities abstract corresponding to the legendre ellipsoid and the lyz ellipsoid two new sine ellipsoids are introduced in this paper these four ellipsoids are closely related in the pythagorean relation and duality several volume inequalities and the valuation properties are obtained for two new ellipsoids