hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

maximally random jammed packings of platonic solids hyperuniform long range correlations and isostaticity we generate maximally random jammed mrj packings of the four nontiling platonic solids tetrahedra octahedra dodecahedra and icosahedra using the adaptive shrinking cell method s torquato and y jiao phys rev e 80 041104 2009 such packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid the mrj packing fractions for tetrahedra octahedra dodecahedra and icosahedra are respectively 0 763 0 005 0 697 0 005 0 716 0 002 and 0 707 0 002 we find that as the number of facets of the particles increases the translational order in the packings increases while the orientational order decreases moreover we show that the mrj packings are hyperuniform i e their infinite wavelength local number density fluctuations vanish and possess quasi long range pair correlations that decay asymptotically with scaling r 4 this provides further evidence that hyperuniform quasi long range correlations are a universal feature of mrj packings of frictionless particles of general shape however unlike mrj packings of ellipsoids superballs and superellipsoids which are hypostatic mrj packings of the nontiling platonic solids are isostatic we provide a rationale for the organizing principle that the mrj packing fractions for nonspherical particles with sufficiently small asphericities exceed the corresponding value for spheres 0 64 we also discuss how the shape and symmetry of a polyhedron particle affects its mrj packing fraction

coincidence of extendible vector valued ideals with their minimal kernel abstract we provide coincidence results for vector valued ideals of multilinear operators more precisely if a is an ideal of n linear mappings we give conditions for which the equality a e 1 e n f a min e 1 e n f holds isometrically as an application we obtain in many cases that the monomials form a schauder basis of the space a e 1 e n f several structural and geometric properties are also derived using this equality we apply our results to the particular case where a is the classical ideal of extendible or pietsch integral multilinear operators similar statements are given for ideals of vector valued homogeneous polynomials

radon cosine and sine transforms on real hyperbolic space new pointwise inversion formulae are obtained for the d dimensional totally geodesic radon transform on the n dimensional real hyperbolic space 1 d n 1 in terms of polynomials of the laplace beltrami operator and intertwining fractional integrals similar results are established for hyperbolic cosine and sine transforms

robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming we have formulated the problem of generating dense packings of nonoverlapping nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive shrinking cell asc formulation s torquato and y jiao phys rev e 80 041104 2009 because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d dimensional euclidean space r d it is most suitable and natural to solve the corresponding asc optimization problem using sequential linear programming slp techniques we implement an slp solution to produce robustly a wide spectrum of jammed sphere packings in r d for d 2 3 4 5 and 6 with a diversity of disorder and densities up to the respective maximal densities a novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures locally maximally dense and mechanically stable packings besides the usual disordered ones such as the maximally random jammed state with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere for example in three dimensions we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range 0 6 0 7408 where 18 0 7408 corresponds to the density of the densest packing we also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d 2 4 5 and 6 our jammed sphere packings are characterized and compared to the corresponding packings generated by the well known lubachevsky stillinger ls molecular dynamics packing algorithm compared to the ls procedure our slp protocol is able to ensure that the final packings are truly jammed produces disordered jammed packings with anomalously low densities and is appreciably more robust and computationally faster at generating maximally dense packings especially as the space dimension increases

inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

jamming at zero temperature and zero applied stress the epitome of disorder we have studied how two and three dimensional systems made up of particles interacting with finite range repulsive potentials jam i e develop a yield stress in a disordered state at zero temperature and zero applied stress at low packing fractions phi the system is not jammed and each particle can move without impediment from its neighbors for each configuration there is a unique jamming threshold phi c at which particles can no longer avoid each other and the bulk and shear moduli simultaneously become nonzero the distribution of phi c values becomes narrower as the system size increases so that essentially all configurations jam at the same packing fraction in the thermodynamic limit this packing fraction corresponds to the previously measured value for random close packing in fact our results provide a well defined meaning for random close packing in terms of the fraction of all phase space with inherent structures that jam the jamming threshold point j occurring at zero temperature and applied stress and at the random close packing density has properties reminiscent of an ordinary critical point as point j is approached from higher packing fractions power law scaling is found for the divergence of the first peak in the pair correlation function and in the vanishing of the pressure shear modulus and excess number of overlapping neighbors moreover near point j certain quantities no longer self average suggesting the existence of a length scale that diverges at j however point j also differs from an ordinary critical point the scaling exponents do not depend on dimension but do depend on the interparticle potential finally as point j is approached from high packing fractions the density of vibrational states develops a large excess of low frequency modes indeed at point j the density of states is a constant all the way down to zero frequency all of these results suggest that point j is a point of maximal disorder and may control behavior in its vicinity perhaps even at the glass transition

even minkowski valuations a new integral representation of smooth translation invariant and rotation equivariant even minkowski valuations is established explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived moreover the action of alesker s hard lefschetz operators on these minkowski valuations is explored in detail

bases in spaces of multilinear forms over banach spaces fil dimant veronica universidad de san andres departamento de matematica y ciencias argentina

distinctive features arising in maximally random jammed packings of superballs dense random packings of hard particles are useful models of granular media and are closely related to the structure of nonequilibrium low temperature amorphous phases of matter most work has been done for random jammed packings of spheres and it is only recently that corresponding packings of nonspherical particles e g ellipsoids have received attention here we report a study of the maximally random jammed mrj packings of binary superdisks and monodispersed superballs whose shapes are defined by x1 2p xd 2p 1 with d 2 and 3 respectively where p is the deformation parameter with values in the interval 0 a sp increases from zero one can get a family of both concave 0p0 5 and convex p0 5 particles with square symmetry d 2 or octahedral and cubic symmetry d 3 in particular for p 1 the particle is a perfect sphere circular disk and for p the particle is a perfect cube square we find that the mrj densities of such packings increase dramatically and nonanalytically as one moves away from the circular disk and sphere point p 1 moreover the disordered packings are hypostatic i e the average number of contacting neighbors is less than twice the total number of degrees of freedom per particle and yet the packings are mechanically stable as a result the local arrangements of particles are necessarily nontrivially correlated to achieve jamming we term such correlated structures nongeneric the degree of nongenericity of the packings is quantitatively characterized by determining the fraction of local coordination structures in which the central particles have fewer contacting neighbors than average we also show that such seemingly special packing configurations are counterintuitively not rare as the anisotropy of the particles increases the fraction of rattlers decreases while the minimal orientational order as measured by the tetratic and cubatic order parameters increases these characteristics result from the unique manner in which superballs break their rotational symmetry which also makes the superdisk and superball packings distinctly different from other known nonspherical hard particle packings

geometrical families of mechanically stable granular packings we enumerate and classify nearly all of the possible mechanically stable ms packings of bidipserse mixtures of frictionless disks in small sheared systems we find that ms packings form continuous geometrical families where each family is defined by its particular network of particle contacts we also monitor the dynamics of ms packings along geometrical families by applying quasistatic simple shear strain at zero pressure for small numbers of particles nl16 we find that the dynamics is deterministic and highly contracting that is if the system is initialized in a ms packing at a given shear strain it will quickly lock into a periodic orbit at subsequent shear strain and therefore sample only a very small fraction of the possible ms packings in steady state in studies with ng16 we observe an increase in the period and random splittings of the trajectories caused by bifurcations in configuration space we argue that the ratio of the splitting and contraction rates in large systems will determine the distribution of ms packing geometrical families visited in steady state this work is part of our long term research program to develop a master equation formalism to describe macroscopic slowly driven granular systems in terms of collections of small subsystems

dynamic simulation of the centripetal packing of mono sized spheres this paper presents a study of the centripetal packing of mono sized spherical particles simulated by means of the granular dynamic or discrete element method a packing is formed by imposing an assumed centripetal force on particles randomly generated in a spherical space different from the conventional simulation techniques dynamic information of individual particles including transient forces and trajectory is traced in the present simulation structural properties such as packing density radial distribution function coordination number distribution and homogeneity are analyzed with particular reference to the eects of the magnitude of the centripetal force and the number of particles comparison with the literature results suggests that such a dynamic model can satisfactorily simulate the dynamics of forming a packing and produce more realistic structural information in particular it is conrmed that a centripetal packing is not homogeneous in structure becoming looser as its size or the number of particles increases the packing has a limit packing density of 0 637 0 645 an overall mean coordination number of around 6 0 and a radial distribution function of clear split second peak the centripetal force aects the rate of densication and the mean coordination number but not packing density and radial distribution function c 1999 elsevier science b v all rights reserved

range characterization of the cosine transform on higher grassmannians abstract we characterize the range of the cosine transform on real grassmannians in terms of the decomposition under the action of the special orthogonal group so n we also give a geometric interpretation of this image in terms of valuations in addition we discuss the non archimedean analogues

packing ellipsoids in an optimized cylinder abstract the paper studies packing ellipsoids of revolution in a cylindrical container of minimum volume ellipsoids can be continuously rotated and translated two nonlinear mathematical programming models are introduced exact and approximated the latter uses an optimized multi spherical approximation of ellisoids for both models the phi function technique is employed to describe analytically non overlapping and containment constraints two solution approaches are proposed to solve the packing problem computational results for up to 500 ellipsoids are provided to demonstrate efficiency of the proposed approaches

dense packings of polyhedra platonic and archimedean solids understanding the nature of dense particle packings is a subject of intense research in the physical mathematical and biological sciences the preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings we formulate the problem of generating dense packings of nonoverlapping nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem which we call the adaptive shrinking cell asc scheme this optimization problem is solved here using a variety of multiparticle initial configurations to find the dense packings of each of the platonic solids in three dimensional euclidean space mathbb r 3 except for the cube which is the only platonic solid that tiles space we find the densest known packings of tetrahedra icosahedra dodecahedra and octahedra with densities 0 823 dots 0 836 dots 0 904 dots and 0 947 dots respectively it is noteworthy that the densest tetrahedral packing possesses no long range order unlike the densest tetrahedral packing which must not be a bravais lattice packing the densest packings of the other nontiling platonic solids that we obtain are their previously known optimal bravais lattice packings we also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to platonic solids archimedean solids superballs and ellipsoids provided that what we term the asphericity ratio of the circumradius to inradius is sufficiently small the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric platonic and archimedean solids our simulation results rigorous upper bounds and other theoretical arguments lead us to the conjecture that the densest packings of platonic and archimedean solids with central symmetry are given by their corresponding densest lattice packings this can be regarded to be the analog of kepler s sphere conjecture for these solids the truncated tetrahedron is the only non centrally symmetric archimedean solid the densest known packing of which is a non lattice packing with density at least as high as 23 24 0 958 text 333 dots we discuss the validity of our conjecture to packings of superballs prisms and antiprisms as well as to high dimensional analogs of the platonic solids in addition we conjecture that the optimal packing of any convex congruent polyhedron without central symmetry generally is not a lattice packing finally we discuss the possible applications and generalizations of the asc scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra

statistical mechanics of hard ellipsoids i overlap algorithm and the contact function abstract a contact function for two arbitrary ellipsoids is derived the numerical value of the contact function is less than 1 if they overlap and greater than 1 if they do not this extends previous work by vieillard baron who derived an overlap criterion for spheroids without use of a contact function the equivalence of the two criteria has been checked by extensive numerical test with spheroids it is shown that the use of the contact function greatly facilitates the calculation of the pressure in monte carlo simulations

optimality and uniqueness of the leech lattice among lattices we prove that the leech lattice is the unique densest lattice in r24 the proof combines human reasoning with computer verification of the properties of certain explicit polynomials we furthermore prove that no sphere packing in r24 can exceed the leech lattice s density by a factor of more than 1 1 65 10 30 and we give a new proof that es is the unique densest lattice in r8

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles we apply the algorithm presented in the first part of this series of papers to systems of hard ellipses and ellipsoids the theoretical machinery needed to treat such particles including the overlap potentials is developed in full detail we describe an algorithm for predicting the time of collision for two moving ellipses or ellipsoids we present performance results for our implementation of the algorithm demonstrating that for dense systems of very aspherical ellipsoids the novel techniques of using neighbor lists and bounding sphere complexes offer as much as two orders of magnitude improvement in efficiency over direct adaptations of traditional event driven molecular dynamics algorithms the practical utility of the algorithm is demonstrated by presenting several interesting physical applications including the generation of jammed packings inside spherical containers the study of contact force chains in jammed packings and melting the densest known equilibrium crystals of prolate spheroids

polynomials on banach lattices and positive tensor products abstract this paper is the first systematic study of homogeneous polynomials on banach lattices a variety of new banach spaces and banach lattices of multilinear maps homogeneous polynomials and operators are introduced the main technique is to employ positive tensor products and quotients of positive tensor products our theorems generalize the results on orthogonally additive polynomials by benyamini lassalle and llavona 2006 in 4 the results by grecu and ryan 2005 in 14 and the results by sundaresan 1991 in 23

schauder decompositions and the fremlin projective tensor product of banach lattices abstract in this paper we first introduce a lattice decomposition and finite dimensional lattice decomposition fdld for banach lattices then we show that for a banach lattice with fdld the following are equivalent i it has the radon nikodym property ii it is a kb space iii it is a levi space and iv it is a levi space we then give a sequential representation of the fremlin projective tensor product of an atomic banach lattice with a banach lattice using this sequential representation we show that if one of the banach lattices x and y is atomic then the fremlin projective tensor product x f y has the radon nikodym property or respectively is a kb space if and only if both x and y have the radon nikodym property or respectively are kb spaces

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles i algorithmic details in this first part of a series of two papers we present in considerable detail a collision driven molecular dynamics algorithm for a system of non spherical particles within a parallelepiped simulation domain under both periodic or hard wall boundary conditions the algorithm extends previous event driven molecular dynamics algorithms for spheres and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size we present a novel partial update near neighbor list nnl algorithm that is superior to previous algorithms at high densities without compromising the correctness of the algorithm this efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes bsc these techniques will be useful in any particle based simulation including monte carlo and time driven molecular dynamics additionally we allow for a non vanishing rate of deformation of the boundary which can be used to model macroscopic strain and also alleviate boundary effects for small systems in the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation demonstrating the practical utility of the algorithm

isostaticity of constraints in amorphous jammed systems of soft frictionless platonic solids the average number of constraints per particle in mechanically stable amorphous systems of platonic solids approaches the isostatic limit at the jamming point 12 though average number of contacts are hypostatic by introducing angular alignment metrics to classify the degree of constraint imposed by each contact constraints are shown to arise as a direct result of local orientational order reflected in edge face and face face alignment angle distributions with approximately one face face contact per particle at jamming chainlike face face clusters form with finite extent a signature of amorphous jammed systems

an overlapping detection algorithm for random sequential packing of elliptical particles abstract random sequential packing of particles is a subject of intense research in many branches of physics and engineering the preponderance of preview work has focused on spherical particles and little is known about non overlapping elliptical particles in the present work a new numerical algorithm is developed to detect overlapping of ellipses by a series of sequential coordinate transformations and a novel golden section search algorithm the accuracy and efficiency of the numerical algorithm are tested and compared with algorithms developed in previous literature its stability is verified by visualizations of random sequential packing of monodispersed and polydispersed ellipses with different boundary conditions then the algorithm is applied to investigate the influence of the shape of ellipses on the random packing fraction as well as to study the influence of the shape and size of ellipses on the wall effect in the particle packing structure and some numerical results are demonstrated finally the reliability of the numerical algorithm extended to the three dimensional space is also evaluated by checking overlapping of ellipsoidal particles

random sequential adsorption of lines and ellipses infinitely many lines of zero thickness may be adsorbed onto a plane without overlap and it is shown that the number of lines adsorbed should scale as 13 where is the number of adsorption trials per unit area of surface this is confirmed by numerical simulation the adsorption of ellipses with major and minor axes 2a 2b respectively is also studied the area coverage theta approaches its maximum value theta max as alpha as to infinity where alpha is typically 1 3 as b a decreases from unity theta max first increases to a maximum value 0 58 or 0 01 when b approximately 0 5 and then decreases a feature not noted in previous work

geometrical frustration a study of four dimensional hard spheres the smallest maximum kissing number voronoi polyhedron of three dimensional 3d euclidean spheres is the icosahedron and the tetrahedron is the smallest volume that can show up in delaunay tessellation no periodic lattice is consistent with either and hence these dense packings are geometrically frustrated because icosahedra can be assembled from almost perfect tetrahedra the terms icosahedral and polytetrahedral packing are often used interchangeably which leaves the true origin of geometric frustration unclear here we report a computational study of freezing of 4d euclidean hard spheres where the densest voronoi cluster is compatible with the symmetry of the densest crystal while polytetrahedral order is not we observe that under otherwise comparable conditions crystal nucleation in four dimensions is less facile than in three dimensions which is consistent with earlier observations m skoge et al phys rev e 74 041127 2006 we conclude that it is the geometrical frustration of polytetrahedral structures that inhibits crystallization

monte carlo study of the sphere packing problem we employ the monte carlo method to study a constrained optimization problem that is packing spheres with unequal radii into a 3 d bounded region selection of the best fit solution is based on using the boltzmann factor e e t to determine the transition probability which allows us to search for the global optimal solution we determined the least numbers of packed spheres that will occupy the largest volume the optimal occupied volume found is around 44 of a bounded region volume which is obtained within a relative short computing time this suggests that our result could be able to give a good starting point for the radiosurgery treatment plan

underconstrained jammed packings of nonspherical hard particles ellipses and ellipsoids continuing on recent computational and experimental work on jammed packings of hard ellipsoids donev et al science 303 990 2004 we consider jamming in packings of smooth strictly convex nonspherical hard particles we explain why an isocounting conjecture which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle overline z 2 d f does not apply to nonspherical particles we develop first and second order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained hypoconstrained overline z l2 d f we apply an algorithm using these conditions to computer generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings even close to the sphere point we also consider packings that are nearly jammed and draw connections to packings of deformable but stiff particles finally we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

weak strong continuity of multilinear mappings and the pelczynski pitt theorem abstract in generalizing a result of pelczynski the sequential weak to norm continuity ofn linear mappings between certain banach spaces including spaces of type and cotype will be studied in particular it is shown that everyn linear continuous mapping from lp1 lpninto lqis compact if and only if i 1n 1 pi

the symmetric radon nikodým property for tensor norms abstract we introduce the symmetric radon nikod m property srn property for finitely generated s tensor norms of order n and prove a lewis type theorem for s tensor norms with this property as a consequence if is a projective s tensor norm with the srn property then for every asplund space e the canonical mapping n s e n s e is a metric surjection this can be rephrased as the isometric isomorphism q min e q e for some polynomial ideal q we also relate the srn property of an s tensor norm with the asplund or radon nikod m properties of different tensor products as an application results concerning the ideal of n homogeneous extendible polynomials are obtained as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on asplund spaces an analogous study is carried out for full tensor products

weakly continuous mappings on banach spaces abstract it is shown that every n homogeneous continuous polynomial on a banach space e which is weakly continuous on the unit ball of e is weakly uniformly continuous on the unit ball of e applications of the result to spaces of polynomials and holomorphic mappings on e are given

on the translative packing densities of tetrahedra and cuboctahedra abstract in 1900 as a part of his 18th problem hilbert asked the question to determine the density of the densest tetrahedron packings however up to now no mathematician knows the density t t of the densest translative tetrahedron packings and the density c t of the densest congruent tetrahedron packings this paper presents a local method to estimate the density of the densest translative packings of a general convex solid as an application we obtain the upper bound in 0 3673469 t t 0 3840610 where the lower bound was established by groemer in 1962 which corrected a mistake of minkowski for the density t c of the densest translative cuboctahedron packings we obtain the upper bound in 0 9183673 t c 0 9601527 in both cases we conjecture the lower bounds to be the correct answer

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

packing tiling and covering with tetrahedra it is well known that three dimensional euclidean space cannot be tiled by regular tetrahedra but how well can we do in this work we give several constructions that may answer the various senses of this question in so doing we provide some solutions to packing tiling and covering problems of tetrahedra our results suggest that the regular tetrahedron may not be able to pack as densely as the sphere which would contradict a conjecture of ulam the regular tetrahedron might even be the convex body having the smallest possible packing density

bases in the space of regular multilinear operators on banach lattices abstract for banach lattices e 1 e m and f with 1 unconditional bases we show that the monomial sequence forms a 1 unconditional basis of l r e 1 e m f the banach lattice of all regular m linear operators from e 1 e m to f if and only if each basis of e 1 e m is shrinking and every positive m linear operator from e 1 e m to f is weakly sequentially continuous as a consequence we obtain necessary and sufficient conditions for which the m fold fremlin projective tensor product e 1 e m resp the m fold positive injective tensor product e 1 e m has a shrinking basis or a boundedly complete basis

radon transforms on higher grassmannians on definit une transformee de radon r a partir des fonctions cr k n les grassmanniens des k plans projectifs dans cp n aux fonctions sur gr l n si f c gr k n et l gr l n alors rf l est l integrale de f h sur tous les k plans h qui sont dans l si r t est la transformee duale on montre sous de bonnes hypotheses sur k et l que r t r est inversible par un polynome dans les operateurs de casimir de u n 1 le groupe des isometries de cp n

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

maximum and minimum stable random packings of platonic solids motivated by the relation between particle shape and packing we measure the volume fraction ensuremath phi occupied by the platonic solids which are a class of polyhedrons with congruent sides vertices and dihedral angles tetrahedron cube octahedron dodecahedron and icosahedron shaped plastic dice were fluidized or mechanically vibrated to find stable random loose packing ensuremath phi rlp 0 51 0 54 0 52 0 51 0 50 and densest packing ensuremath phi rcp 0 64 0 67 0 64 0 63 0 59 respectively with standard deviation of ensuremath simeq ifmmode pm else textpm fi 0 01 we find that ensuremath phi obtained by all protocols peak at the cube which is the only platonic solid that can tessellate space and then monotonically decrease with number of sides this overall trend is similar but systematically lower than the maximum ensuremath phi reported for frictionless platonic solids and below ensuremath phi rlp of spheres for the loose packings experiments with ceramic tetrahedron were also conducted and higher friction was observed to lead to lower ensuremath phi

even valuations on convex bodies the notion of eveil valuation is introduced as a natural generalization of volume on cornpact convex subsets of euclidean space a recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on grassmannians this connection can be described in part using generating distributions for symmetric compact convex sets we also explore some consequences of these characterization results in convex and integral geometry recent interest in volume as a valuation on compact conlvex sets stems from hilbert s third problem which is actually an ancient problem recast in modern terms hilbert asked if two polytopes p and q can be each cut into a finite number of pieces p pn and q qm with each pi congruent to qi by a rigid euclidean motion provided that p and q have the same volume 18 this question was answered in the negative by max dehn 2 4 29 who found a functionlal on polytopes that is invariant under dissections over rigid motions while varying in value among polytopes of equal volume in other words the dehn invariant is a simple rigid motion invariant valuation on polytopes that is not equal to volume under any normalization dehn s solution left open the question of exactly what conditions oil p and q imply equidissectability over the group of rigid motions although this problem was solved by hadwiger in the case where only translations and no rotations nor reflections are permitted see 2 16 17 26 29 in the course of studying this and related problems hadwiger discovered a characterization of euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets that is a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension this result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1r as consisting of a real n 4 dimensional vector space spanned by the intrinsic volumes or quermassintegrals 16 also 20 21 31 since many standard functionals and integral operators can be interpreted as invariant valuations such as intrinsic volumes mean projections crofton and kinematic formulas what came to be known as hadwiger s chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry unfortunately hadwiger s original proof was long and difficult 161 while seeking a shorter proof of hadwiger s volume characterization the author discovered received by the editors june 24 1996 and in revised form september 29 1997 1991 mathematics subject classification primary 52a22 52a38 52a39 52b45 research supported in part by nsf grants dms 9022140 to msri and dms 9626688 to the author 1999 american mathematical society 71 this content downloaded from 207 46 13 52 on mon 24 oct 2016 04 15 57 utc all use subject to http about jstor org terms

exact constructions of a family of dense periodic packings of tetrahedra the determination of the densest packings of regular tetrahedra one of the five platonic solids is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell this analysis results in six parameter family of dense tetrahedron packings that includes as special cases recently discovered dimer packings of tetrahedra including the densest known packings with density ensuremath phi frac 4000 4671 0 856347 dots this study strongly suggests that the latter set of packings are the densest among all packings with a four particle basis whether they are the densest packings of tetrahedra among all packings is an open question but we offer remarks about this issue moreover we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings which could aid in assessing the packing efficiency of candidate dense packings

geometrical properties of rigid frictionless granular packings as a function of particle size and shape three dimensional discrete numerical simulation is used to investigate the properties of close packed frictionless granular assemblies as a function of particle polydispersity and shape unlike some experimental results simulations show that disordered packings of pinacoids eight face convex polyhedra achieve higher solid fraction values than amorphous packings of spherical or rounded particles thus fulfilling the analog of ulam s conjecture stated by jiao and co workers for random packings y jiao and s torquato phys rev e 84 041309 2011 pleee81539 375510 1103 physreve 84 041309 this seeming discrepancy between experimental and numerical results is believed to result from difficulties in overcoming inter particle friction through experimental densification processes moreover solid fraction is shown to increase further with bidispersity and peak when the volume proportion of small particles reaches 30 contrarily substituting up to 50 of flat pinacoids for isometric ones yields solid fraction decrease especially when flat particles are also elongated nevertheless particle shape seems to play a minor role in packing solid fraction compared to polydispersity additional investigations focused on the packing microstructure confirm that pinacoid packings fulfill the isostatic conjecture and that they are free of order except beyond 30 to 50 of flat or flat elongated polyhedra in the packing this order increase progressively takes the form of a nematic phase caused by the reorientation of flat or flat elongated particles to minimize the packing potential energy simultaneously this reorientation seems to increase the solid fraction value slightly above the maximum achieved by monodisperse isometric pinacoids as well as the coordination number finally partial substitution of elongated pinacoids for isometric ones has limited effect on packing solid fraction or order

free energy of singular sticky sphere clusters networks of particles connected by springs model many condensed matter systems from colloids interacting with a short range potential and complex fluids near jamming to self assembled lattices and various metamaterials under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero frequency vibrational modes yet such singular modes are at the epicenter of many interesting behaviors in the systems above we consider a system of n spherical particles and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged although the contribution to the partition function from singular clusters diverges in the limit its asymptotic value can be calculated and depends on only two parameters characterizing the depth and range of the potential the result holds for systems that are second order rigid a geometric characterization that describes all known ground state rigid sticky clusters to illustrate the applications of our theory we address the question of emergence how does crystalline order arise in large systems when it is strongly disfavored in small ones we calculate the partition functions of all known rigid clusters up to n 21 and show the cluster landscape is dominated by hyperstatic clusters those with more than 3n 6 contacts singular and isostatic clusters are far less frequent despite their extra vibrational and configurational entropies since the most hyperstatic clusters are close to fragments of a close packed lattice this underlies the emergence of order in sticky sphere systems even those as small as n 10

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

order metrics and order maps of octahedron packings abstract we apply the ideal octahedron model and the relaxation algorithm in generating octahedron packings the cubatic order parameter p 4 1 bond orientational order metric q 6 and local cubatic order parameter p 4 l o c a l of the packings are calculated and their correlations with the packing density are investigated in the order maps the border curve of packing density separates the geometrically feasible and infeasible regions in the order maps observing the transition phenomenon on the border curve we propose the concept of the maximally dense random packing mdrp as the densest packing in the random state in which the particle positions and orientations are randomly distributed and there is no nontrivial spatial correlations among particles the mdrp characterizes the onset of nontrivial spatial correlations among particles a special packing with a density about 0 7 is found in the order maps and considered to be the mdrp of octahedra the p 4 l o c a l is proposed as a new order parameter for octahedron packings which measures the average order degree in the neighborhoods of particles the p 4 1 q 6 and p 4 l o c a l evaluate the order degree of orientation bond orientation and local structures respectively and are applied simultaneously to measure the order degree of the octahedron packings their thresholds in the random state are determined by monte carlo simulations

integral geometry in minkowski spaces abstract we investigate the possibility of extending classical integral geometric results involving lower dimensional areas from euclidean space to minkowski spaces finite dimensional banach spaces of the two natural notions of area in a minkowski space due respectively to busemann and to holmes and thompson the latter turns out to be the more tractable one for the holmes thompson area we derive a translative intersection formula and in the class of hypermetric minkowski spaces full analogues of the crofton formulae for rectifiable sets and for convex bodies for the busemann k area we give a short proof of the fact that it coincides for k rectifiable sets with the k dimensional hausdorff measure induced by the minkowski metric

athermal jamming of soft frictionless platonic solids a mechanically based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless platonic solids systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape revealing that thermal motion is necessary to access high density phases we confirm that the large system jamming threshold of 0 623 0 003 for tetrahedra is consistent with experiments on tetrahedral dice also the extremely short ranged translational correlations of packed tetrahedra observed in experiments are confirmed here in contrast with those of thermally simulated glasses although highly ordered phases are observed to form for small numbers of cubes and dodecahedra the short correlation length scale suppresses ordering in large systems resulting in packings that are mechanically consistent with orientationally disordered contacts point face and edge edge contacts mild nematic ordering is observed for large systems of cubes whereas angular correlations for the remaining shapes are ultrashort ranged in particular the angular correlation function of tetrahedra agrees with that recently observed experimentally for tetrahedral dice power law scaling exponents for energy with respect to distance from the jamming threshold exhibit a clear dependence on the highest order percolating contact topology these nominal exponents are 6 4 and 2 for configurations having percolating point face or edge edge edge face and face face contacts respectively jamming contact number is approximated for small systems of tetrahedra icosahedra dodecahedra and octahedra with order and packing representative of larger systems these platonic solids exhibit hypostatic behavior with average jamming contact number between the isostatic value for spheres and that of asymmetric particles these shapes violate the isostatic conjecture displaying contact number that decreases monotonically with sphericity the common symmetry of dual polyhedra results in local translational structural similarity systems of highly spherical particles possessing icosahedral symmetry such as icosahedra or dodecahedra exhibit structural behavior similar to spheres including jamming contact number and radial distribution function these results suggest that although continuous rotational symmetry is broken by icosahedra and dodecahedra the structural features of disordered packings of these particles are well replicated by spheres octahedra and cubes which possess octahedral symmetry exhibit similar local translational ordering despite exhibiting strong differences in nematic ordering in general the structural features of systems with tetrahedra octahedra and cubes differ significantly from those of sphere packings

optimal packings of superballs dense hard particle packings are intimately related to the structure of low temperature phases of matter and are useful models of heterogeneous materials and granular media most studies of the densest packings in three dimensions have considered spherical shapes and it is only more recently that nonspherical shapes e g ellipsoids have been investigated superballs whose shapes are defined by x1 2p x2 2p x3 2p 1 provide a versatile family of convex particles p0 5 with both cubic like and octahedral like shapes as well as concave particles 0p0 5 with octahedral like shapes in this paper we provide analytical constructions for the densest known superball packings for all convex and concave cases the candidate maximally dense packings are certain families of bravais lattice packings in which each particle has 12 contacting neighbors possessing the global symmetries that are consistent with certain symmetries of a superball we also provide strong evidence that our packings for convex superballs p0 5 are most likely the optimal ones the maximal packing density as a function of p is nonanalytic at the sphere point p 1 and increases dramatically as p moves away from unity two more nontrivial nonanalytic behaviors occur at p c 1 150 9 and p o ln 3 ln 4 0 792 4 for cubic and octahedral superballs respectively where different bravais lattice packings possess the same densities the packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two dimensional superdisk counterparts y jiao et al phys rev lett 100 245504 2008 and are distinctly different from that of ellipsoid packings our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems which should deepen our understanding of the statistical thermodynamics of nonspherical particle systems

bases in spaces of homogeneous polynomials on banach spaces in this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on banach spaces

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

computer simulation of random packing of unequal particles a monte carlo simulation model for the random packing of unequal spherical particles is presented in this paper with this model the particle radii obeying a given distribution are generated and randomly placed within a cubic packing domain with high packing density and many overlaps then a relaxation iteration is applied to reduce or eliminate the overlaps while the packing space is gradually expanded the simulation is completed once the mean overlap value falls below a preset value to simulate the random close packing a vibration process is applied after the relaxation iteration for log normal distributed particles the effect of particle size standard deviation and for bidisperse particles the effects of particle size ratio and the volume fraction of large particles on packing density and on coordination number are investigated simulation results show good agreement with that obtained by experiments and by other simulations the randomness homogeneity and isotropy which have not been evaluated before for packing of distributed particles are also examined using statistical measures

densest binary sphere packings the densest binary sphere packings in the x plane of small to large sphere radius ratio and small sphere relative concentration x have historically been very difficult to determine previous research had led to the prediction that these packings were composed of a few known alloy phases including for example the alb2 hexagonal hgbr2 a nd aute2 structures and to xyn structures composed of close packed large spheres with small spheres in a number ratio of n to 1 in the interstices e g the nacl packing for n 1 however utilizing an implementation of the torquato jiao sphere packing algorithm torquato and jiao phys rev e 82 061302 2010 we have discovered that many more structures appear in the densest packings for example while all previously known densest structures were composed of spheres in small to large number ratios of one to one two to one and very recently three to one we have identified densest structures with number ratios of seven to three and five to two in a recent work hopkins et al phys rev lett 107 125501 2011 we summarized these findings in this work we present the structures of the densest known packings and provide details about their characteristics our findings demonstrate that a broad array of different densest mechanically stable structures consisting of only two types of components can form without any consideration of attractive or anisotropic interactions in addition the structures that we have identified may correspond to currently unidentified stable phases of certain binary atomic and molecular systems particularly at high temperatures and pressures

phase behavior of colloidal superballs shape interpolation from spheres to cubes the phase behavior of hard superballs is examined using molecular dynamics within a deformable periodic simulation box a superball s interior is defined by the inequality x 2q y 2q z 2q ensuremath le 1 which provides a versatile family of convex particles q ensuremath ge 0 5 with cubelike and octahedronlike shapes as well as concave particles ql0 5 with octahedronlike shapes here we consider the convex case with a deformation parameter q between the sphere point q 1 and the cube q ensuremath infty we find that the asphericity plays a significant role in the extent of cubatic ordering of both the liquid and crystal phases calculation of the first few virial coefficients shows that superballs that are visually similar to cubes can have low density equations of state closer to spheres than to cubes dense liquids of superballs display cubatic orientational order that extends over several particle lengths only for large q along the ordered high density equation of state superballs with 1lql3 exhibit clear evidence of a phase transition from a crystal state to a state with reduced long ranged orientational order upon the reduction of density for q ensuremath ge 3 long ranged orientational order persists until the melting transition the width of the apparent coexistence region between the liquid and ordered high density phase decreases with q up to q 4 0 the structures of the high density phases are examined using certain order parameters distribution functions and orientational correlation functions we also find that a fixed simulation cell induces artificial phase transitions that are out of equilibrium current fabrication techniques allow for the synthesis of colloidal superballs and thus the phase behavior of such systems can be investigated experimentally