superinduction for pattern groups abstract it is well known that the representation theory of the finite group of unipotent upper triangular matrices u n over a finite field is a wild problem by instead considering approximately irreducible representations supercharacters one obtains a rich combinatorial theory analogous to that of the symmetric group where we replace partition combinatorics with set partitions this paper studies diaconis isaacs concept of superinduction in pattern groups while superinduction shares many desirable properties with usual induction it no longer takes characters to characters we begin by finding sufficient conditions guaranteeing that superinduction is in fact induction it turns out for two natural embeddings of u m in u n superinduction is induction we conclude with an explicit combinatorial algorithm for computing this induction analogous to the pieri formulas for the symmetric group

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

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

counterintuitive ground states in soft core models it is well known that statistical mechanics systems exhibit subtle behavior in high dimensions in this paper we show that certain natural soft core models such as the gaussian core model have unexpectedly complex ground states even in relatively low dimensions specifically we disprove a conjecture of torquato and stillinger who predicted that dilute ground states of the gaussian core model in dimensions 2 through 8 would be bravais lattices we show that in dimensions 5 and 7 there are in fact lower energy non bravais lattices the nearest three dimensional analog is the hexagonal close packing but it has higher energy than the face centered cubic lattice we believe these phenomena are in fact quite widespread and we relate them to decorrelation in high dimensions

comment on packing hyperspheres in high dimensional euclidean spaces it is shown that the numerical data in cond mat 0608362 are in very good agreement with the predictions of cond mat 0601573

reproducing morphologies of disorderly self assembling planar molecules with static and dynamic simulation methods by matching density abstract monte carlo and molecular dynamics simulations are the two main numerical approaches to modeling molecular self assembly and ordering conceptually however each method explores different paths through the thermodynamic landscape molecular dynamics depends on the position and momentum terms monte carlo is a static set and thus the momentum term is replaced with an energy term that is dependent on the volume and entropy until now it was unclear if a stochastic process of densifying particles would have the same internal structure as morphologies produced from classical mechanics this paper provides a systematic i e statistical analysis of the outcomes of 4032 simulations for hard core circular objects as a function of the number of molecules and the boundary conditions structural classification of the resultant ensembles averaged pair correlation function bond order parameter translational order parameter and voronoi diagrams shows that stochastic and dynamic approaches do not alter the morphology of the steric molecules we conclude that when the probability density of covering area fractions are matched the ensembles produced from the two methods will show the same level of structural disorder and positional patterns the resultant morphology from both models therefore is not a product of dynamic unrest but that of the relaxation of entropic frustration from macromolecular crowding although statistically the two methods produce similar configurations nuances arise from the static and dynamic nature of modeling as a result monte carlo is slightly better suited to modeling systems when the desired morphology is represented by a metastable state molecular dynamics on the other hand is more suited to finding defects that can arise in morphologies regardless a fixed density will result in similar morphologies from both techniques driven by similar configurational entropy

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

a linear programming algorithm to test for jamming in hard sphere packings jamming in hard particle packings has been the subject of considerable interest in recent years in a paper by torquato and stillinger j phys chem b 105 2001 a classification scheme of jammed packings into hierarchical categories of locally collectively and strictly jammed configurations has been proposed they suggest that these jamming categories can be tested using numerical algorithms that analyze an equivalent contact network of the packing under applied displacements but leave the design of such algorithms as a future task in this work we present a rigorous and practical algorithm to assess whether an ideal hard sphere packing in two or three dimensions is jammed according to the aforementioned categories the algorithm is based on linear programming and is applicable to regular as well as random packings of finite size with hard wall and periodic boundary conditions if the packing is not jammed the algorithm yields representative multi particle unjamming motions furthermore we extend the jamming categories and the testing algorithm to packings with significant interparticle gaps we describe in detail two variants of the proposed randomized linear programming approach to test for jamming in hard sphere packings the first algorithm treats ideal packings in which particles form perfect contacts another algorithm treats the case of jamming in packings with significant interparticle gaps this extended algorithm allows one to explore more fully the nature of the feasible particle displacements we have implemented the algorithms and applied them to ordered as well as random packings of circular disks and spheres with periodic boundary conditions some representative results for large disordered disk and sphere packings are given but more robust and efficient implementations as well as further applications e g non spherical particles are anticipated for the future

monte carlo simulation of hard hyperspheres in six seven and eight dimensions for low to moderate densities the pair correlation function of hard hyperspheres in six seven and eight dimensions is obtained from monte carlo simulations the value of the pair correlation function at contact is compared with the results from molecular dynamics calculations and a variety of theoretical approaches remarkably good agreement is found with the simple closed form equations of y song e a mason and r m stratt j phys chem 93 6916 6919 1989 the monte carlo results for the equation of state are compared with the theoretical expressions of m baus and j l colot phys rev a 36 3912 1987 m luban and j p j michels phys rev a 41 6796 1990 and high order virial expansions in addition in seven dimensions comparisons are made with the exact py solution provided by m robles m l de haro and a santos j chem phys 120 9113 2004 very good agreement was observed between theory and computer simulation in all dimensions

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

local density fluctuations hyperuniformity and order metrics questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges the purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d our specific objectives are to study the variance in the number of points contained within a regularly shaped window ensuremath omega of arbitrary size and to further illuminate our understanding of hyperuniform systems i e point patterns that do not possess infinite wavelength fluctuations for large windows hyperuniform systems are characterized by a local variance that grows only as the surface area rather than the volume of the window we derive two formulations for the number variance i an ensemble average formulation which is valid for statistically homogeneous systems and ii a volume average formulation applicable to a single realization of a general point pattern in the large system limit the ensemble average formulation which includes both real space and fourier representations enables us to show that a homogeneous point pattern in a hyperuniform state is at a critical point of a type with appropriate scaling laws and critical exponents but one in which the direct correlation function rather than the pair correlation function is long ranged we also prove that the non negativity of the local number variance does not add a new realizability condition on the pair correlation the volume average formulation is superior for certain computational purposes including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance we prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one dimensional hyperuniform patterns we also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one two and three dimensions for spherical windows enabling us to rank order the spatial patterns our results suggest that the local variance may serve as a useful order metric for general point patterns contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension we show that for d 3 the body centered cubic lattice has a smaller variance than the face centered cubic lattice finally for certain hyperuniform disordered point patterns we evaluate the direct correlation function structure factor and associated critical exponents exactly

functional approximation and projection of stored energy functions in computational homogenization of hyperelastic materials a probabilistic perspective abstract this work is concerned with the construction of a surrogate model for the homogenized stored energy functions defining the effective behavior of nonlinear elastic microstructures here a probabilistic standpoint is adopted and allows for the definition of a nonlinear mapping between the macroscopic deformations and the homogenized potential this functional approximation is specifically obtained by means of a polynomial chaos expansion the coefficients of which are computed through a gauss legendre quadrature rule by invoking well known results related to projections in hilbert spaces closest approximations of arbitrary potentials to standard e g ogden type models are subsequently defined and characterized by appropriate residuals numerical illustrations on various microstructures are finally provided in order to discuss the relevance of the proposed framework in particular it is shown that the surrogate model compares very well with reference results at both the macroscale and the structural scale even for moderate order expansions and that the aforementioned closest approximations constitute very accurate approximation provided that the subset onto which the homogenized response is projected is properly chosen this result readily allows for nonconcurrent coupling using standard constitutive models available in commercial codes and therefore makes the approach very attractive for engineering applications

exactly solvable disordered sphere packing model in arbitrary dimensional euclidean spaces we introduce a generalization of the well known random sequential addition rsa process for hard spheres in d dimensional euclidean space mathbb r d we show that all of the n particle correlation functions of this nonequilibrium model in a certain limit called the ghost rsa packing can be obtained analytically for all allowable densities and in any dimension this represents the first exactly solvable disordered sphere packing model in an arbitrary dimension the fact that the maximal density ensuremath phi ensuremath infty 1 2 d of the ghost rsa packing implies that there may be disordered sphere packings in sufficiently high d whose density exceeds minkowski s lower bound for bravais lattices the dominant asymptotic term of which is 1 2 d indeed we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by 2 ensuremath 0 778 65 dots d thus providing the putative exponential improvement on minkowski s 100 year old bound our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic implying the existence of disordered classical ground states for some continuous potentials

freezing transition of hard hyperspheres we investigate the system of d dimensional hard spheres in d dimensional space where d 3 for the fluid phase of these hyperspheres we generalize scaled particle theory to arbitrary d and furthermore use the virial expansion and the percus yevick integral equation for the crystalline phase we adopt cell theory based on elementary geometrical assumptions about close packed lattices regardless of the approximation applied and for dimensions as high as d550 we find a first order freezing transition which preempts the kirkwood second order instability of the fluid the relative density jump increases with d and a generalized lindemann rule of melting holds we have also used ideas from fundamental measure theory to obtain a free energy density functional for hard hyperspheres finally we have calculated the surface tension of a hypersphere fluid near a hard smooth hyper wall within scaled particle theory

universality of cluster dynamics we have studied the kinetics of cluster formation for dynamical systems of dimensions up to n 8 interacting through elastic collisions or coalescence these systems could serve as possible models for gas kinetics polymerization and self assembly in the case of elastic collisions we found that the cluster size probability distribution undergoes a phase transition at a critical time which can be predicted from the average time between collisions this enables forecasting of rare events based on limited statistical sampling of the collision dynamics over short time windows the analysis was extended to l p normed spaces p 1 to allow for some amount of interpenetration or volume exclusion the results for the elastic collisions are consistent with previously published low dimensional results in that a power law is observed for the empirical cluster size distribution at the critical time we found that the same power law also exists for all dimensions n 2 8 two dimensional l p norms and even for coalescing collisions in two dimensions this broad universality in behavior may be indicative of a more fundamental process governing the growth of clusters

an unfitted finite element method using discontinuous galerkin in this paper we present a new approach to simulations on complex shaped domains the method is based on a discontinuous galerkin dg method using trial and test functions defined on a structured grid essential boundary conditions are imposed weakly via the dg formulation this method offers a discretization where the number of unknowns is independent of the complexity of the domain we will show numerical computations for an elliptic scalar model problem in 2 and 3 convergence rates for different polynomial degrees are studied

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

structure of hard hypersphere fluids in odd dimensions the structural properties of single component fluids of hard hyperspheres in odd space dimensionalities d are studied with an analytical approximation method that generalizes the rational function approximation earlier introduced in the study of hard sphere fluids s b yuste and a santos phys rev a 43 5418 1991 the theory makes use of the exact form of the radial distribution function to first order in density and extends it to finite density by assuming a rational form for a function defined in laplace space the coefficients being determined by simple physical requirements fourier transform in terms of reverse bessel polynomials constitute the mathematical framework of this approximation from which an analytical expression for the static structure factor is obtained in its most elementary form the method recovers the solution of the percus yevick closure to the ornstein zernike equation for hyperspheres at odd dimensions the present formalism allows one to go beyond by yielding solutions with thermodynamic consistency between the virial and compressibility routes to any desired equation of state excellent agreement with available computer simulation data at d 5 and d 7 is obtained

ground states and formal duality relations in the gaussian core model we study dimensional trends in ground states for soft matter systems specifically using a high dimensional version of parrinello rahman dynamics we investigate the behavior of the gaussian core model in up to eight dimensions the results include unexpected geometric structures with surprising anisotropy as well as formal duality relations these duality relations suggest that the gaussian core model possesses unexplored symmetries and they have implications for a broad range of soft core potentials

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

spectral ewald acceleration of stokesian dynamics for polydisperse suspensions in this work we develop the spectral ewald accelerated stokesian dynamics seasd a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions seasd is based on the framework of stokesian dynamics sd with extension to compressible solvents and uses the spectral ewald se method lindbo and tornberg 2010 29 for the wave space mobility computation to meet the performance requirement of dynamic simulations we use graphic processing units gpu to evaluate the suspension mobility and achieve an order of magnitude speedup compared to a cpu implementation for further speedup we develop a novel far field block diagonal preconditioner to reduce the far field evaluations in the iterative solver and seasd nf a polydisperse extension of the mean field brownian approximation of banchio and brady 2003 39 we extensively discuss implementation and parameter selection strategies in seasd and demonstrate the spectral accuracy in the mobility evaluation and the overall o n log n computation scaling we present three computational examples to further validate seasd and seasd nf in monodisperse and bidisperse suspensions the short time transport properties the equilibrium osmotic pressure and viscoelastic moduli and the steady shear brownian rheology our validation results show that the agreement between seasd and seasd nf is satisfactory over a wide range of parameters and also provide significant insight into the dynamics of polydisperse colloidal suspensions

dimensional study of the caging order parameter at the glass transition the glass problem is notoriously hard and controversial even at the mean field level little is agreed upon regarding why a fluid becomes sluggish while exhibiting but unremarkable structural changes it is clear however that the process involves self caging which provides an order parameter for the transition it is also broadly assumed that this cage should have a gaussian shape in the mean field limit here we show that this ansatz does not hold by performing simulations as a function of spatial dimension d we find the cage to keep a nontrivial form quantitative mean field descriptions of the glass transition such as mode coupling theory density functional theory and replica theory all miss this crucial element although the mean field random first order transition scenario of the glass transition is qualitatively supported here and non mean field corrections are found to remain small on decreasing d reconsideration of its implementation is needed for it to result in a coherent description of experimental observations

algorithm for random close packing of spheres with periodic boundary conditions the isotropic algorithm is constructed for random close packing of equisized spheres with triply periodic boundary conditions all previously published packing methods with periodic boundaries were kinetics determined i e they contained a densification rate as an arbitrary parameter in contrast the present algorithm is kinetic independent and demonstrates an unambiguous convergence to the experimental results to suppress crystallization the main principles of our algorithm are 1 to form a contact network at an early stage and 2 retain contacts throughout the densification as far as possible the particles are allowed to swell by the numerical solution of the differential equations of densification the rhs of these equations is calculated efficiently from a linear system by a combination of conjugate gradient iterations and exact sparse matrix technology when an excessive contact occurs and one of the existing bonds should be broken to continue the densification an efficient criterion based on multidimensional simplex geometry is used for searching the separating bond the algorithm has a well defined termination point resulting in a perfect contact network with the average coordination number six for particle number n 1 and a system of normal reactions between the spheres maintaining the structure these forces are the counterpart of the algorithm and can be used to calculate small elastic particle deformations in a granular medium extensive calculations are presented for 50 n 5 400 and demonstrate very good agreement with the experimental packing density about 0 637 and structure

breakdown of elasticity theory for jammed hard particle packings conical nonlinear constitutive theory abstract hard particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media liquids living cells glasses and random media the nature of jammed hard particle packings is a current subject of keen interest we demonstrate that the response of jammed hard particle packings to global deformations cannot be described by linear elasticity even for small particle displacements but involves a conical nonlinear constitutive theory it is the singular nature of the hard particle potential that leads to the breakdown of linear elasticity interestingly a nonlinear theory arises because the feasible particle displacements leading to unjamming depend critically on the local spatial arrangement of the particles implying a directionality in the feasible strains that is absent in particle systems with soft potentials mathematically the set of feasible strains has a conical structure i e components of the imposed strain tensor generally obey linear inequalities the nature of the nonlinear behavior is illustrated by analyzing several specific packings finally we examine the conditions under which a packing can be considered to incompressible in the traditional sense

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

pair correlation function characteristics of nearly jammed disordered and ordered hard sphere packings we study the approach to jamming in hard sphere packings and in particular the pair correlation function g2srd around contact both theoretically and computationally our computational data unambiguously separate the narrowing d function contribution to g2 due to emerging interparticle contacts from the background contribution due to near contacts the data also show with unprecedented accuracy that disordered hard sphere packings are strictly isostatic i e the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom once rattlers are removed for such isostatic packings we derive a theoretical connection between the probability distribution of interparticle forces pfsfd which we measure computationally and the contact contribution to g2 we verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state we clearly observe a maximum in pf and a nonzero probability of zero force shedding light on long standing questions in the granular media literature we computationally observe an unusual power law divergence in the near contact contribution to g2 persistent even in the jamming limit with exponent 0 4 clearly distinguishable from previously proposed inversesquare root divergence additionally we present high quality numerical data on the two discontinuities in the split second peak of g2 and use a shared neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features finally we present the computational data on the contact contribution to g2 for vacancy diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings we find that the contact network remains isostatic even when ordering is present unlike previous studies we find that ordering has a significant impact on the shape of pf for small forces

geometrical ambiguity of pair statistics point configurations point configurations have been widely used as model systems in condensed matter physics materials science and biology statistical descriptors such as the n body distribution function g n are usually employed to characterize point configurations among which the most extensively used is the pair distribution function g 2 an intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration although it is known that the pair distance information contained in g 2 is in general insufficient to uniquely determine a point configuration this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies in this paper we present the idea of the distance space called the mathbb d space the pair distances of a specific point configuration are then represented by a single point in the mathbb d space we derive the conditions on the pair distances that can be associated with a point configuration which are equivalent to the realizability conditions of the pair distribution function g 2 moreover we derive the conditions on the pair distances that can be assembled into distinct configurations i e with structural degeneracy these conditions define a bounded region in the mathbb d space by explicitly constructing a variety of degenerate point configurations using the mathbb d space we show that pair information is indeed insufficient to uniquely determine the configuration in general we also discuss several important problems in statistical physics based on the mathbb d space including the reconstruction of atomic structures from experimentally obtained g 2 and a recently proposed decorrelation principle the degenerate configurations have relevance to open questions involving the famous traveling salesman problem

estimates of the optimal density of sphere packings in high dimensions the problem of finding the asymptotic behavior of the maximal density max of sphere packings in high euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry one century ago minkowski obtained a rigorous lower bound on max that is controlled asymptotically by 1 2d where d is the euclidean space dimension an indication of the difficulty of the problem can be garnered from the fact that exponential improvement of minkowski s bound has proved to be elusive even though existing upper bounds suggest that such improvement should be possible using a statistical mechanical procedure to optimize the density associated with a test pair correlation function and a conjecture concerning the existence of disordered sphere packings s torquato and f h stillinger exp math 15 307 2006 the putative exponential improvement on max was found with an asymptotic behavior controlled by 1 2 0 77865 d using the same methods we investigate whether this exponential im

toy model for the mean field theory of hard sphere liquids we investigate a toy model of liquid based on simplified hypernetted chain equations in very large spatial dimension d the model does not exhibit a phase transition but several regimes of behavior when d infinity can be observed in different intervals of the density

a supercharacter theory for the sylow p subgroups of the finite symplectic and orthogonal groups abstract we define the superclasses for a classical finite unipotent group u of type b n q c n q or d n q and show that together with the supercharacters defined in c a m andre a m neto supercharacters of the sylow p subgroups of the finite symplectic and orthogonal groups pacific j math 239 2 2009 201 230 they form a supercharacter theory in the sense of p diaconis i m isaacs supercharacters and superclasses for algebra groups trans amer math soc 360 5 2008 2359 2392 in particular we prove that the supercharacters take a constant value on each superclass and evaluate this value as a consequence we obtain a factorization of any superclass as a product of elementary superclasses in addition we also define the space of superclass functions and prove that it is spanned by the supercharacters as a consequence we re obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters finally we define the supercharacter table of u and prove various orthogonality relations for supercharacters similar to the well known orthogonality relations for irreducible characters

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

random sequential addition of hard spheres in high euclidean dimensions sphere packings in high dimensions have been the subject of recent theoretical interest employing numerical and theoretical methods we investigate the structural characteristics of random sequential addition rsa of congruent spheres in d dimensional euclidean space mathbb r d in the infinite time or saturation limit for the first six space dimensions 1 ensuremath le d ensuremath le 6 specifically we determine the saturation density pair correlation function cumulative coordination number and the structure factor in each of these dimensions we find that for 2 ensuremath le d ensuremath le 6 the saturation density ensuremath phi s scales with dimension as ensuremath phi s c 1 2 d c 2 d 2 d where c 1 0 202 phantom rule 0 2em 0ex 048 and c 2 0 973 phantom rule 0 2em 0ex 872 we also show analytically that the same density scaling is expected to persist in the high dimensional limit albeit with different coefficients a byproduct of this high dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by ensuremath phi s ensuremath ge d 2 1 ensuremath s 0 2 d 1 where s 0 0 1 is the structure factor at k 0 i e infinite wavelength number variance in the high dimensional limit we demonstrate that a pal asti type conjecture the saturation density in mathbb r d is equal to that of the one dimensional problem raised to the d th power cannot be true for rsa hyperspheres we show that the structure factor s k must be analytic at k 0 and that rsa packings for 1 ensuremath le d ensuremath le 6 are nearly hyperuniform consistent with the recent decorrelation principle we find that pair correlations markedly diminish as the space dimension increases up to six we also obtain kissing contact number statistics for saturated rsa configurations on the surface of a d dimensional sphere for dimensions 2 ensuremath le d ensuremath le 5 and compare to the maximal kissing numbers in these dimensions we determine the structure factor exactly for the related ghost rsa packing in mathbb r d and demonstrate that its distance from hyperuniformity increases as the space dimension increases approaching a constant asymptotic value of 1 2 our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions