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

a short proof of a kupershmidt wilson theorem this is a short elementary proof of a statement originally observed by adler then pursued by the author by kupershmidt and wilson and in a more general setting by drinfeld and sokolov

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

regularized continuum yang mills process and feynman kac functional integral giving an ultraviolet regularization and volume cut off we construct a nuclear riemannian structure on the hilbert manifold mathfrak m of gauge orbits this permits us to define a regularized laplace beltrami operator on mathfrak m and an associated global diffusion in mathfrak m governed by this enables us to define via a feynman kac integral a euclidean continuum regularized yang mills process corresponding to a suitable regularization of the kinetic term of the classical yang mills lagrangian ont mathfrak m

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

diversity of order and densities in jammed hard particle packings recently the conventional notion of random close packing has been supplanted by the more appropriate concept of the maximally random jammed mrj state this inevitably leads to the necessity of distinguishing the mrj state among the entire collection of jammed packings while the ideal method of addressing this question would be to enumerate and classify all possible jammed hard sphere configurations practical limitations prevent such a method from being employed instead we generate numerically a large number of representative jammed hard sphere configurations primarily relying on a slight modification of the lubachevsky stillinger algorithm to do so and evaluate several commonly employed order metrics for each of these packings our investigation shows that even in the large system limit jammed systems of hard spheres can be generated with a wide range of packing fractions from phi approximately 0 52 to the fcc limit phi approximately 0 74 moreover at a fixed packing fraction the variation in the order can be substantial indicating that the density alone does not uniquely characterize a packing interestingly each order metric evaluated yielded a relatively consistent estimate for the packing fraction of the maximally random jammed state phi mrj approximately 0 63 this estimate however is compromised by the weaknesses in the order metrics available and we propose several guiding principles for future efforts to define more broadly applicable metrics

mod 3 congruence and twisted signature of 24 dimensional string manifolds in this paper by combining modularity of the witten genus and the modular forms constructed by liu and wang we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds introduction let m be a 2n dimensional smooth closed oriented manifold let gtm be a riemmian metric on tm and tm the associated levi civita connection let v be a complex vector bundle over m with a hermitian metric hv and a unitary connection v let c t m be the complexified exterior algebra bundle of tm let c t m be the hermitian metric on c t m induced by gtm let dv be the riemannian volume form associated to gtm then m c t m has a hermitian metric such that for m c t m m c t m dv for x tm let c x be the clifford action on c t m defined by c x x ix where x t m corresponds to x via gtm let e1 e2 e2n be an oriented orthogonal basis of tm set 1 c e1 c e2n then one can show that is independent of the choice of the orthonormal basis and v 1 is a self adjoint element acting on c t m v such that v id c t m v let d be the exterior differentiation operator and d be the formal adjoint of d with respect to the hermitian metric the operator dsig d d 2n i 1 c ei c m ei m c t m m c t m is the signature operator and the more general twisted signature operator is defined as c f 6 dsig v 2n i 1 c ei c m v ei m c t m v m c t m v

braid matrices and structure constants for minimal conformal models using the feigin fuchs representation of minimal conformal models in a form introduced recently by one of us the braid group representation matrices describing the analytic continuation properties of conformal blocks are computed in a suitable normalization their matrix elements are shown to essentially factorize into pairs of boltzmann weights of critical rsos models in a certain limit of the spectral parameter these boltzmann weights are related to quantum groupr matrices by the vertex sos transformation we show that the crossing symmetry of the four point function in left right symmetric models follows from a quantum group relation also called crossing symmetry this observation gives a simple way to evaluate the structure constants

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

the witten genus and equivariant elliptic cohomology we construct a thom class in complex equivariant elliptic cohomology extending the equivariant witten genus this gives a new proof of the rigidity of the witten genus which exhibits a close relationship to recent work on non equivariant orientations of elliptic spectra

symmetric tensor representations quasimodular forms and weak jacobi forms abstract we establish a correspondence between vector valued modular forms with respect to a symmetric tensor representation and quasimodular forms this is carried out by first obtaining an explicit isomorphism between the space of vector valued modular forms with respect to a symmetric tensor representation and the space of finite sequences of modular forms of certain type this isomorphism uses rankin cohen brackets and extends a result of kuga and shimura who considered the case of vector valued modular forms of weight two we also obtain a correspondence between such vector valued modular forms and weak jacobi forms

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

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

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

s 1 actions and elliptic genera a proof is given of witten s conjectures for the rigidity of the index of the dirac ramond operator on the loop space of a spin manifold which admits ans1 symmetry

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

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

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

integrability of the frobenius algebra valued kadomtsev petviashvili hierarchy we introduce a frobenius algebra valued kadomtsev petviashvili kp hierarchy and show the existence of frobenius algebra valued function for this hierarchy in addition we construct its hamiltonian structures by using the adler dickey gelfand method as a byproduct of these constructions we show that the coupled kp hierarchy defined by casati and ortenzi j geom phys 56 418 449 2006 has at least n basic different local bi hamiltonian structures finally via the construction of the second hamiltonian structures we obtain some local matrix or frobenius algebra valued generalizations of classical w algebras

umbral moonshine and k3 surfaces recently 23 cases of umbral moonshine relating mock modular forms and finite groups have been discovered in the context of the 23 even unimodular niemeier lattices one of the 23 cases in fact coincides with the so called mathieu moonshine discovered in the context of k3 non linear sigma models in this paper we establish a uniform relation between all 23 cases of umbral moonshine and k3 sigma models and thereby take a first step in placing umbral moonshine into a geometric and physical context this is achieved by relating the ade root systems of the niemeier lattices to the ade du val singularities that a k3 surface can develop and the configuration of smooth rational curves in their resolutions a geometric interpretation of our results is given in terms of the marking of k3 surfaces by niemeier lattices

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

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

non local matrix generalizations of w algebras there is a standard way to define two symplectic hamiltonian structures the first and second gelfand dikii brackets on the space of ordinarymth order linear differential opeatorsl d m u1dm 1 u2dm 2 u m in this paper i consider in detail the case where theu k aren n matrix valued functions with particular emphasis on the more interesting second gelfand dikii bracket of particular interest is the reduction to the symplectic submanifoldu1 0 this reduction gives rise to matrix generalizations of the classical version of thenon linear w m algebras calledv n m algebras the non commutativity of the matrices leads tonon local terms in thesev n m algebra s i show that these algebras contain a conformal virasoro subalgebra and that combinationsw k of theu k can be formed that aren n matrices of conformally primary fields of spink in analogy with the scalar casen 1 in general however thev m n algebras have a much richer structure than thew m algebras as can be seen on the examples of thenon linear andnon local poisson brackets u2 ab u2 cd u2 ab w3 cd and w3 ab w3 cd which i work out explicitly for allm andn a matrix miura transformations is derived mapping these complicated second gelfand dikii brackets of theu k to a set of much simpler poisson brackets providing the analogoue of the free field representation of thew m algebras

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

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

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

classical w algebras we reconsider the relation between classicalw algebras and deformations of differential operators emphasizing the consistency with diffeomorphisms generators of thew algebra that arek differentials are constructed by a systematic procedure the method extends following drinfeld and sokolov tow algebras based on arbitrary simple lie algebras

string structures on loop bundles differential geometry and topology of principal loop bundles bundles of loop groups over loop spaces are investigated string structures defined as bundle extensions corresponding to the central extension of the structure group do not always exist various methods for deriving the obstruction to the existence of string structures are discussed

higher spin fields and the gelfand dickey algebra we show that in 2 dimensional field theory higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by gelfand and dickey to describe integrable nonlinear differential equations in lax form the spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined this provides a conformal field theory approach to the representation theory of gelfand dickey algebras

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

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

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

circle equivariant classifying spaces and the rational equivariant sigma genus the circle equivariant spectrum mstring mathbb c is the equivariant analogue of the cobordism spectrum mu langle 6 rangle of stably almost complex manifolds with c1 c2 0 given a rational elliptic curve c greenlees topology 44 1213 1227 2005 constructs a ring mathbb t spectrum ec representing the associated mathbb t equivariant elliptic cohomology the core of the present paper is the construction when c is a complex elliptic curve of a map of ring mathbb t spectra mstring mathbb c to ec which is the rational equivariant analogue of the sigma orientation of ando et al invent math 146 595 687 2001 we support this by a theory of characteristic classes for calculation and a conceptual description in terms of algebraic geometry in particular we prove a conjecture the first author made in ando geom topol 7 91 153 2003

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