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

on a dimensional reduction method i the optimal selection of basis functions abstract this paper is the first in a series of three which analyze an adaptive approximate approach for solving n 1 dimensional boundary value problems by replacing them with systems of equations in n dimensional space in this approach the unknown functions of n 1 variables are projected onto finite linear combinations of functions of just n variables this paper shows how the coefficients of these linear combinations can be chosen optimally author

reformulation of the covering and quantizer problems as ground states of interacting particles it is known that the sphere packing problem and the number variance problem closely related to an optimization problem in number theory can be posed as energy minimizations associated with an infinite number of point particles in d dimensional euclidean space r d interacting via certain repulsive pair potentials we reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in r d that generally involve single body two body three body and higher body interactions this is done by linking the covering and quantizer problems to certain optimization problems involving the void nearest neighbor functions that arise in the theory of random media and statistical mechanics these reformulations which again exemplify the deep interplay between geometry and physics allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems the covering and quantizer problems have relevance in numerous applications including wireless communication network layouts the search of high dimensional data parameter spaces stereotactic radiation therapy data compression digital communications meshing of space for numerical analysis and coding and cryptography among other examples in the first three space dimensions the best known solutions of the sphere packing and number variance problems or their dual solutions are directly related to those of the covering and quantizer problems but such relationships may or may not exist for d 4 depending on the peculiarities of the dimensions involved our reformulation sheds light on the reasons for these similarities and differences we also show that disordered saturated sphere packings provide relatively thin economical coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions in the case of the quantizer problem we derive improved upper bounds on the quantizer error using sphere packing solutions which are generally substantially sharper than an existing upper bound in low to moderately large dimensions we also demonstrate that disordered saturated sphere packings yield relatively good quantizers finally we remark on possible applications of our results for the detection of gravitational waves

exponential improvements for superball packing upper bounds abstract we prove that for all fixed p 2 the translative packing density of unit l p balls in r n is at most 2 p o 1 n with p 1 p this is the first exponential improvement in high dimensions since van der corput and schaake 1936

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

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

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

geometrical frustration in two dimensions idealizations and realizations of a hard disk fluid in negative curvature we examine a simple hard disk fluid with no long range interactions on the two dimensional space of constant negative gaussian curvature the hyperbolic plane this geometry provides a natural mechanism by which global crystalline order is frustrated allowing us to construct a tractable one parameter model of disordered monodisperse hard disks we extend free area theory and the virial expansion to this regime deriving the equation of state for the system and compare its predictions with simulations near an isostatic packing in the curved space additionally we investigate packing and dynamics on triply periodic negatively curved surfaces with an eye toward real biological and polymeric systems

memory of the unjamming transition during cyclic tiltings of a granular pile discrete numerical simulations are performed to study the evolution of the microstructure and the response of a granular packing during successive loading unloading cycles consisting of quasistatic rotations in the gravity field between opposite inclination angles we show that internal variables e g stress and fabric of the pile exhibit hysteresis during these cycles due to the exploration of different metastable configurations interestingly the hysteretic behavior of the pile strongly depends on the maximal inclination of the cycles giving evidence of the irreversible modifications of the pile state occurring close to the unjamming transition more specifically we show that for cycles with maximal inclination larger than the repose angle the weak contact network carries the memory of the unjamming transition these results demonstrate the relevance of a two phase description strong and weak contact networks for a granular system as soon as it has approached the unjamming transition

jamming graphs a local approach to global mechanical rigidity we revisit the concept of minimal rigidity as applied to frictionless repulsive soft sphere packings in two dimensions with the introduction of the jamming graph minimal rigidity is a purely combinatorial property encoded via laman s theorem in two dimensions it constrains the global average coordination number of the graph for example however minimal rigidity does not address the geometry of local mechanical stability the jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability we demonstrate how jamming graphs can be constructed using local moves via the henneberg construction such that these graphs fall under the jurisdiction of correlated percolation we then probe how jamming graphs destabilize or become unjammed by deleting a bond and computing the resulting rigid cluster distribution we also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra redundant contacts destabilizes the latter endeavor allows us to probe a disk packing in the rigid phase and uncover a potentially new diverging length scale associated with the random deletion of contacts as compared to the study of cut out or frozen in subsystems

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

approximation of skewed interfaces with tensor based model reduction procedures in this article we introduce a procedure which allows to recover the potentially very good approximation properties of tensor based model reduction procedures for the solution of partial differential equations in the presence of interfaces or strong gradients in the solution which are skewed with respect to the coordinate axes the two key ideas are the location of the interface either by solving a lower dimensional partial differential equation or by using data functions and the subsequent removal of the interface of the solution by choosing the determined interface as the lifting function of the dirichlet boundary conditions we demonstrate in numerical experiments for linear elliptic equations and the reduced basis hierarchical model reduction approach that the proposed procedure locates the interface well and yields a significantly improved convergence behavior even in the case when we only consider an approximation of the interface

the simplest model of jamming we study a well known neural network model the perceptron as a simple statistical physics model of jamming of hard objects we exhibit two regimes 1 a convex optimization regime where jamming is hypostatic and non critical 2 a non convex optimization regime where jamming is isostatic and critical we characterize the critical jamming phase through exponents describing the distribution laws of forces and gaps surprisingly we find that these exponents coincide with the corresponding ones recently computed in high dimensional hard spheres in addition modifying the perceptron to a random linear programming problem we show that isostaticity is not a sufficient condition for singular force and gap distributions for that fragmentation of the space of solutions replica symmetry breaking appears to be a crucial ingredient we hypothesize universality for a large class of non convex constrained satisfaction problems with continuous variables

granular contact force density of states and entropy in a modified edwards ensemble a method has been found to analyze edwards granular contact force probability functional for a special case as a result the granular contact force probability density functions are obtained from first principles for this case the results are in excellent agreement with the experimental and simulation data the derivation assumes edwards flat measure a density of states dos that is uniform within the metastable regions of phase space the enabling assumption supported by physical arguments and empirical evidence is that correlating information is not significantly recursive through loops in the packing maximizing a state counting entropy results in a transport equation that can be solved numerically for the present this has been done using the mean structure approximation projecting the dos across all angular coordinates to more clearly identify its predominant nonuniformities these features are 1 the grain factor psi related to grain stability and strong correlation between the contact forces on the same grain and 2 the structure factor upsilon related to newton s third law and strong correlation between neighboring grains

on a dimensional reduction method ii some approximation theoretic results abstract this paper is the second in a series of three that analyze a method of dimensional reduction it contains some results for approximation of functions on the interval 1 1 with elements from the nullspace of p to the n power n greater than or equal to 1 where p is a second order ordinary differential operator a special case of this is approximation by polynomials the one dimensional results are used as a tool to prove similar versions in several dimensions these multi dimensional results are directly related to the approximate method of dimensional reduction and they lead to statements about the convergence properties of this approach the third paper which analyzes the adaptive aspects of the method is forthcoming

k core percolation in four dimensions the k core percolation on the bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first order second order nature we investigate numerically k core percolation on the four dimensional regular lattice for k 4 the presence of a discontinuous transition is clearly established but its nature is strictly first order in particular the k core density displays no singular behavior before the jump and its correlation length remains finite for k 3 the transition is continuous

jamming transition in kinetically constrained models with reflection symmetry a class of kinetically constrained models with reflection symmetry is proposed as an extension of the fredrickson andersen model it is proved that the proposed model on the square lattice exhibits a freezing transition at a non trivial density it is conjectured by numerical experiments that the known mechanism of the singular behaviors near the freezing transition in a previously studied model spiral model is not responsible for that in the proposed model

necessary and sufficient conditions for the two point phase probability function of two phase random media constructing realizations of random media with a specified two point phase probability function s2 has attracted considerable attention in the recent literature however little is known about cond

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

inverse halftoning based on the bayesian theorem this study proposes a method which can generate high quality inverse halftone images from halftone images this method can be employed prior to any signal processing over a halftone image or the inverse halftoning used in jbig2 the proposed method utilizes the least mean square lms algorithm to establish a relationship between the current processing position and its corresponding neighboring positions in each type of halftone image including direct binary search error diffusion dot diffusion and ordered dithering after which a referenced region called a support region sr is used to extract features the sr can be obtained by relabeling the lms trained filters with the order of importance moreover the probability of black pixel occurrence is considered as a feature in this work according to this feature the probabilities of all possible grayscale values at the current processing position can be obtained by the bayesian theorem consequently the final output at this position is the grayscale value with the highest probability experimental results show that the proposed method offers better visual quality than that of mese vaidyanathan s and chang s methods in terms of human visual peak signal to noise ratio hpsnr in addition the memory consumption is also superior to mese vaidyanathan s method

realizability of point processes there are various situations in which it is natural to ask whether a given collection of k functions j r 1 r j j 1 k defined on a set x are the first k correlation functions of a point process on x here we describe some necessary and sufficient conditions on the j s for this to be true our primary examples are x d x d and x an arbitrary finite set in particular we extend a result by ambartzumian and sukiasian showing realizability at sufficiently small densities 1 r typically if any realizing process exists there will be many even an uncountable number in this case we prove when x is a finite set the existence of a realizing gibbs measure with k body potentials which maximizes the entropy among all realizing measures we also investigate in detail a simple example in which a uniform density and translation invariant 2 are specified on there is a gap between our best upper bound on possible values of and the largest for which realizability can be established

the effect of sidewall friction on dense granular flows we present novel results on flow properties using molecular dynamics simulation as well as c densitometry experiments we show that fundamentally different types of granular flow can be observed in one unique system using both frictional and frictionless confining walls we show that the differences existing between these regimes originate in the frictional properties of the walls the salient features of each regime are described in detail in particular the packing fraction and streamwise velocity profiles are shown to hinge on the frictional properties of the confining walls finally we propose a dimensionless number independent of the grain size which helps to delineate the different regimes

on a dimensional reduction method iii a posteriori error estimation and an adaptive approach abstract this paper analyzes an adaptive approximate approach for solving n 1 dimensional boundary value problems by replacing them with systems of equation in n dimensional space the discussion shows how to find reliable a posteriori estimates for the error and how these can also be used in the design of an adaptive strategy various numerical examples are presented

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

ensemble theory for force networks in hyperstatic granular matter an ensemble approach for force networks in static granular packings is developed the framework is based on the separation of packing and force scales together with an a priori flat measure in the force phase space under the constraints that the contact forces are repulsive and balance on every particle in this paper we will give a general formulation of this force network ensemble and derive the general expression for the force distribution psfd for small regular packings these probability densities are obtained in closed form while for larger packings we present a systematic numerical analysis since technically the problem can be written as a noninvertible matrix problem where the matrix is determined by the contact geometry we study what happens if we perturb the packing matrix or replace it by a random matrix the resulting psfd s differ significantly from those of normal packings which touches upon the deep question of how network statistics is related to the underlying network structure overall the ensemble formulation opens up a different perspective on force networks that is analytically accessible and which may find applications beyond granular matter

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

voronoi and void statistics for superhomogeneous point processes we study the voronoi and void statistics of superhomogeneous or hyperuniform point patterns in which the infinite wavelength density fluctuations vanish superhomogeneous or hyperuniform point patterns arise in one component plasmas primordial density fluctuations in the universe and jammed hard particle packings we specifically analyze a certain one dimensional model by studying size fluctuations and correlations of the associated voronoi cells we derive exact results for the complete joint statistics of the size of two voronoi cells we also provide a sum rule that the correlation matrix for the voronoi cells must obey in any space dimension in contrast to the conventional picture of superhomogeneous systems we show that infinitely large voronoi cells or voids can exist in superhomogeneous point processes in any dimension we also present two heuristic conditions to identify and classify any superhomogeneous point process in terms of the asymptotic behavior of the void size distribution

on the realizability of pair correlation functions abstract the pair correlation function g 2 r provides a basic geometric descriptor for many particle systems it must obey two necessary conditions i non negativity for all distances r and ii non negativity of its associated structure factor s k for all k here we utilize an improved stochastic construction algorithm for particle configurations to establish conditions in which i and ii are also sufficient i e g 2 r is in fact realizable two types of target pair correlation functions have been investigated in one two and three dimensions for hard core particles specifically a unit step function and a contact plus step pair correlation function results indicate that the former target function is realizable up to a terminal density set by necessary condition ii at which the particle core packing fraction equals 2 d in d dimensions furthermore results are consistent with the proposition that for d 1 the contact plus step function is realizable up to a terminal density due to condition ii at which the packing fraction of cores is d 2 2 d 1 torquato and stillinger j phys chem b 106 2000 8354 11406

higher order corrections to the effective potential close to the jamming transition in the perceptron model in view of the results achieved in a previously related work a altieri s franz and g parisi j stat mech 2016 093301 10 1088 1742 5468 2016 09 093301 regarding a plefka like expansion of the free energy up to the second order in the perceptron model we improve the computation here focusing on the role of third order corrections the perceptron model is a simple example of constraint satisfaction problem falling in the same universality class as hard spheres near jamming and hence allowing us to get exact results in high dimensions for more complex settings our method enables to define an effective potential or thouless anderson palmer free energy namely a coarse grained functional which depends on the generalized forces and the effective gaps between particles the analysis of the third order corrections to the effective potential reveals that albeit irrelevant in a mean field framework in the thermodynamic limit they might instead play a fundamental role in considering finite size effects we also study the typical behavior of generalized forces and we show that two kinds of corrections can occur the first contribution arises since the system is analyzed at a finite distance from jamming while the second one is due to finite size corrections we nevertheless show that third order corrections in the perturbative expansion vanish in the jamming limit both for the potential and the generalized forces in agreement with the isostaticity argument proposed by wyart and coworkers finally we analyze the relevant scaling solutions emerging close to the jamming line which define a crossover regime connecting the control parameters of the model to an effective temperature

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

1 d expansion for k core percolation the physics of k core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon examples of such a phenomenon range from orientational ordering in solid ortho para mathrm h 2 mixtures to the onset of rigidity in bar joint networks to dynamical arrest in glass forming liquids unlike ordinary k 1 and biconnected k 2 percolation the mean field k ensuremath geqslant 3 core percolation transition is both continuous and discontinuous i e there is a jump in the order parameter accompanied with a diverging length scale to determine whether or not this hybrid transition survives in finite dimensions we present a 1 d expansion for k core percolation on the d dimensional hypercubic lattice we show that to order 1 d 3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability this result suggests that the unusual hybrid nature of the mean field k core transition survives in high dimensions

high capacity data hiding for error diffused block truncation coding block truncation coding btc is an efficient compression technique with extremely low computational complexity however the blocking and false contour effects are two major deficiencies in btc which cause severe perceptual artifacts the former scheme error diffused btc edbtc can significantly improve the above issues through the visual low pass compensation on the bitmap which thus widens its possible application market yet the corresponding security issue may limit its value in this paper a method namely complementary hiding edbtc is developed to cope the above issue this paper is managed by firstly discussing when a single watermark is embedded and then multiple watermarks are employed to test the limitation of the proposed scheme herein an adaptive external bias factor is employed to control the watermark embedding and this factor also affects the image quality and robustness simultaneously experimental results demonstrate that the proposed method only requires an extremely small external bias factor to carry watermarks which enables a high capacity scenario without significantly damaging image quality

a novel classification method of halftone image via statistics matrices existing classification methods tend not to work well on various error diffusion patterns thus a novel classification method for halftone image via statistics matrices is proposed the statistics matrix descriptor of halftone image is constructed according to the characteristic of error diffusion filters on this basis an extraction algorithm is developed based on halftone image patches the feature modeling is formulated as an optimization problem and then a gradient descent method is used to seek optimum class feature matrices by minimizing the total square error a maximum likelihood method is proposed according to priori knowledge of training samples in experiments the performance evaluation method is provided and comparisons of performance between our method and seven similar methods are made then the influence of parameters performance under various attacks computational time complexity and the limitations are discussed from our experimental study it is observed that our method has lower classification error rate when compared with other similar methods in addition it is robust against usual attacks

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

tilings of space and superhomogeneous point processes we consider the construction of point processes from tilings with equal volume tiles of d dimensional euclidean space mathbb r d we show that one can generate with simple algorithms ascribing one or more points to each tile point processes which are superhomogeneous or hyperuniform i e for which the structure factor s mathbf k vanishes when the wave vector mathbf k tends to zero the exponent ensuremath gamma characterizing the leading small k behavior s k ensuremath rightarrow 0 ensuremath propto k ensuremath gamma depends in a simple manner on the nature of the correlation properties of the specific tiling and on the conservation of the mass moments of the tiles assigning one point to the center of mass of each tile gives the exponent ensuremath gamma 4 for any tiling in which the shapes and orientations of the tiles are short range correlated smaller exponents in the range 4 ensuremath dl ensuremath gamma l4 and thus always superhomogeneous for d ensuremath le 4 may be obtained in the case that the latter quantities have long range correlations assigning more than one point to each tile in an appropriate way we show that one can obtain arbitrarily higher exponents in both cases we illustrate our results with explicit constructions using known deterministic tilings as well as some simple stochastic tilings for which we can calculate s mathbf k exactly our results provide an explicit analytical construction of point processes with ensuremath gamma g4 applications to condensed matter physics and also to cosmology are briefly discussed

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

on the study of jamming percolation we investigate kinetically constrained models of glassy transitions and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales the models we investigate have constraints similar to that of the knights model introduced by toninelli biroli and fisher tbf but differing neighbor relations we find that such knights like models otherwise known as models of jamming percolation need a no parallel crossing rule for the tbf proof of a glassy transition to be valid furthermore most knights like models fail a no perpendicular crossing requirement and thus need modification to be made rigorous we also show how the no parallel crossing requirement can be used to evaluate the provable glassiness of other correlated percolation models by looking at models with more stable directions than the knights model finally we show that the tbf proof does not generalize in any straightforward fashion for three dimensional versions of the knights like models

force balance percolation we study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance i e for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions we prove rigorously that p c 1 for the two dimensional models studied numerical data indicate that the force balance percolation transition is discontinuous with a growing crossover length with perhaps the same form as the jamming percolation models suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases we find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition finally we study the dynamics of the culling procedure invoked to obtain the force balance configurations and find a dynamical exponent similar to that found in sandpile models

spatial correlations in polydisperse frictionless two dimensional packings we investigate next nearest neighbor correlations of the contact number in simulations of polydisperse frictionless packings in two dimensions we find that disks with few contacting neighbors are predominantly in contact with disks that have many neighbors and vice versa at all packing fractions this counterintuitive result can be explained by drawing a direct analogy to the aboav weaire law in cellular structures we find an empirical one parameter relation similar to the aboav weaire law that satisfies an exact sum rule constraint surprisingly there are no correlations in the radii between neighboring particles despite correlations between contact number and radius

look up table lut method for inverse halftoning in this paper we propose look up table lut based methods for inverse halftoning of images the lut for inverse halftoning is obtained from the histogram gathered from a few sample halftone images and corresponding original images the method is extremely fast no filtering is required and the psnr and visual image quality achieved is comparable to the best methods known for inverse halftoning the lut inverse halftoning method does not depend on the specific properties of the halftoning method and can be applied to any halftoning method then an algorithm for template selection for lut inverse halftoning is introduced we demonstrate the performance of the lut inverse halftoning algorithm on error diffused images and ordered dithered images we also extend lut inverse halftoning to color halftones

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

a dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction in this article we introduce a new dimensional reduction approach which is based on the application of reduced basis rb techniques in the hierarchical model reduction hmr frame work considering problems that exhibit a dominant spatial direction the idea of hmr is to perform a galerkin projection onto a reduced space which combines the full solution space in the dominant direction with a reduction space in the transverse direction the latter is spanned by modal or thonormal basis functions while so far the basis functions in the hmr approach have been chosen a priori s perotto a ern and a veneziani multiscale model simul 8 2010 pp 1102 1127 for instance as legendre or trigonometric polynomials in this work a highly nonlinear approximation is employed for the construction of the reduction space to this end we first derive a lower dimen sional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction exploiting the good ap proximation properties of rb methods we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation for an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article we introduce our method for general elliptic problems such as advection diffusion equations in two space dimensions numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach

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

numerical tests of constitutive laws for dense granular flows we numerically and theoretically study the macroscopic properties of dense sheared granular materials in this process we first consider an invariance in newton s equations explain how it leads to bagnold s scaling and discuss how it relates to the dynamics of granular temperature next we implement numerical simulations of granular materials in two different geometries simple shear and flow down an incline and show that measurements can be extrapolated from one geometry to the other then we observe nonaffine rearrangements of clusters of grains in response to shear strain and show that fundamental observations which served as a basis for the shear transformation zone stz theory of amorphous solids m l falk and j s langer phys rev e 57 7192 1998 m r s bull 25 4 02000 can be reproduced in granular materials finally we present constitutive equations for granular materials as proposed by lemaitre phys rev lett 89 064303 2002 based on the dynamics of granular temperature and stz theory and show that they match remarkably well with our numerical data from both geometries

anomalous local coordination density fluctuations and void statistics in disordered hyperuniform many particle ground states we provide numerical constructions of one dimensional hyperuniform many particle distributions that exhibit unusual clustering and asymptotic local number density fluctuations growing more slowly than the volume of an observation window but faster than the surface area hyperuniformity defined by vanishing infinite wavelength local density fluctuations provides a quantitative metric of global order within a many particle configuration and signals the onset of an inverted critical point in which the direct correlation function becomes long ranged by targeting a specified form of the structure factor at small wavenumbers s k k for 0 1 using collective density variables we are able to tailor the form of asymptotic local density fluctuations while simultaneously measuring the effect of imposing weak and strong constraints on the available degrees of freedom within the system this procedure is equivalent to finding the possibly disordered classical ground state of an interacting many particle system with up to four body interactions even in one dimension the long range effective interactions induce clustering and nontrivial phase transitions in the resulting ground state configurations we provide an analytical connection between the fraction of constrained degrees of freedom within the system and the disorder order phase transition for a class of target structure factors by examining the realizability of the constrained contribution to the pair correlation function our results explicitly demonstrate that disordered hyperuniform many particle ground states and therefore also point distributions with substantial clustering can be constructed we directly relate the local coordination structure of our point patterns to the distribution of the void space external to the particles and we provide a scaling argument for the configurational entropy analogous to spin frustated system of the disordered ground states by emphasizing the intimate connection between geometrical constraints on the particle distribution and structural regularity our work has direct implications for higher dimensional systems including an understanding of the appearance of hyperuniformity and quasi long range pair correlations in maximally random strictly jammed packings of hard spheres

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

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

hybrid lms mmse inverse halftoning technique the objective of this work is to reconstruct high quality gray level images from bilevel halftone images we develop optimal inverse halftoning methods for several commonly used halftone techniques which include dispersed dot ordered dither clustered dot ordered dither and error diffusion at first the least mean square lms adaptive filtering algorithm is applied in the training of inverse halftone filters the resultant optimal mask shapes are significantly different for various halftone techniques and these mask shapes are also quite different from the square shape that was frequently used in the literature in the next step we further reduce the computational complexity by using lookup tables designed by the minimum mean square error mmse method the optimal masks obtained from the lms method are used as the default filter masks finally we propose the hybrid lms mmse inverse halftone algorithm it normally uses the mmse table lookup method for its fast speed when an empty cell is referred the lms method is used to reconstruct the gray level value consequently the hybrid method has the advantages of both excellent reconstructed quality and fast speed in the experiments the error diffusion yields the best reconstruction quality among all three halftone techniques

k core architecture and k core percolation on complex networks abstract we analytically describe the architecture of uncorrelated complex networks with arbitrary degree distributions as a set of successively enclosed substructures k cores we find the structure of k cores their sizes and their birth points the bootstrap percolation thresholds we show that in networks with a finite mean number z 2 of the second nearest neighbours the emergence of a k core is a unique hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous phase transition in contrast if z 2 diverges the networks contain an infinite sequence of k cores which are ultra robust against random damage we explain the meaning of the order parameter for k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we demonstrate that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

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

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

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

correlated percolation and tricriticality the recent proliferation of correlated percolation models models where the addition of edges and or vertices is no longer independent of other edges and or vertices has been motivated by the quest to find discontinuous percolation transitions the leader in this proliferation is what is known as explosive percolation a recent proof demonstrates that a large class of explosive percolation type models does not in fact exhibit a discontinuous transition riordan and warnke science 333 322 2011 here we discuss two lesser known correlated percolation models the k 3 core model on random graphs and the counter balance model in two dimensions both exhibiting discontinuous transitions to search for tricriticality we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition using a powerful rate equation approach we demonstrate that a mixture of k 2 core and k 3 core vertices on the random graph exhibits a tricritical point however for a mixture of k core and counter balance vertices in two dimensions as the fraction of counter balance vertices is increased numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition i e when all vertices are counter balanced interestingly these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions in addition our results may have potential implications for glassy and jamming systems

hyperuniformity quasi long range correlations and void space constraints in maximally random jammed particle packings i polydisperse spheres hyperuniform many particle distributions possess a local number variance that grows more slowly than the volume of an observation window implying that the local density is effectively homogeneous beyond a few characteristic length scales previous work on maximally random strictly jammed sphere packings in three dimensions has shown that these systems are hyperuniform and possess unusual quasi long range pair correlations decaying as r 4 resulting in anomalous logarithmic growth in the number variance however recent work on maximally random jammed sphere packings with a size distribution has suggested that such quasi long range correlations and hyperuniformity are not universal among jammed hard particle systems in this paper we show that such systems are indeed hyperuniform with signature quasi long range correlations by characterizing the more general local volume fraction fluctuations we argue that the regularity of the void space induced by the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers to induce hyperuniformity in the medium with a linear small wave number nonanalytic behavior in the spectral density resulting in quasi long range spatial correlations scaling with r d 1 in d euclidean space dimensions a numerical and analytical analysis of the pore size distribution for a binary maximally random jammed system in addition to a local characterization of the n particle loops governing the void space surrounding the inclusions is presented in support of our argument this paper is the first part of a series of two papers considering the relationships among hyperuniformity jamming and regularity of the void space in hard particle packings

k core bootstrap percolation on complex networks critical phenomena and nonlocal effects we develop the theory of the k core bootstrap percolation on uncorrelated random networks with arbitrary degree distributions we show that the k core percolation is an unusual hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous transition we describe the properties of the k core explain the meaning of the order parameter for the k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role corona is a subset of vertices in the k core which have exactly k neighbors in the k core it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we show that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of the k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

theoretical analysis for critical fluctuations of relaxation trajectory near a saddle node bifurcation a langevin equation whose deterministic part undergoes a saddle node bifurcation is investigated theoretically it is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle node bifurcation point in the weak noise limit while the final value of the deterministic solution changes discontinuously at the point a systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method in this formulation the critical nature turns out to originate from the neutrality of exiting time from a saddle point the theoretical calculation explains results of numerical simulations

nonlinear effective medium theory of disordered spring networks disordered soft materials such as fibrous networks in biological contexts exhibit a nonlinear elastic response we study such nonlinear behavior with a minimal model for networks on lattice geometries with simple hookian elements with disordered spring constant by developing a mean field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations we provide insight into the origins of the strain stiffening and softening behavior of these systems we find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity in particular the nonlinear response is controlled by the isostatic connectivity which depends strongly on the applied strain our predictions for the strain dependence of the isostatic point as well as the strain dependent differential bulk modulus agree well with numerical results in both two and three dimensions in addition by using a mapping between the disordered network and a regular network with random forces we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results finally we discuss the limitations and implications of the developed theory 2012 american physical society

structure of finite sphere packings via exact enumeration implications for colloidal crystal nucleation we analyze the geometric structure and mechanical stability of a complete set of isostatic and hyperstatic sphere packings obtained via exact enumeration the number of nonisomorphic isostatic packings grows exponentially with the number of spheres n and their diversity of structure and symmetry increases with increasing n and decreases with increasing hyperstaticity h n c n iso where n c is the number of pair contacts and n iso 3n 6 maximally contacting packings are in general neither the densest nor the most symmetric analyses of local structure show that the fraction f of nuclei with order compatible with the bulk rhcp crystal decreases sharply with increasing n due to a high propensity for stacking faults five and near fivefold symmetric structures and other motifs that preclude rhcp order while f increases with increasing h a significant fraction of hyperstatic nuclei for n as small as 11 retain non rhcp structure classical theories of nucleation that consider only spherical nuclei or only nuclei with the same ordering as the bulk crystal cannot capture such effects our results provide an explanation for the failure of classical nucleation theory for hard sphere systems of n 10 particles we argue that in this size regime it is essential to consider nuclei of unconstrained geometry our results are also applicable to understanding kinetic arrest and jamming in systems that interact via hard core like repulsive and short ranged attractive interactions

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

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

statistical properties of determinantal point processes in high dimensional euclidean spaces the goal of this paper is to quantitatively describe some statistical properties of higher dimensional determinantal point processes with a primary focus on the nearest neighbor distribution functions toward this end we express these functions as determinants of n n matrices and then extrapolate to n this formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in euclidean spaces of dimension d we also implement an algorithm due to hough et al for generating configurations of determinantal point processes in arbitrary euclidean spaces and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the fermi sphere point process for d 1 4 this homogeneous isotropic determinantal point process discussed also in a companion paper s torquato a scardicchio and c e zachary j stat mech theory exp 2008 p11019 i s the high dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble in addition to the nearest neighbor probability distribution we are able to calculate voronoi cells and nearest neighbor extrema statistics for the fermi sphere point process and we discuss these properties as the dimension d is varied the results in this paper accompany and complement analytical properties of higher dimensional determinantal point processes developed in a prior paper

adaptive petrov galerkin methods for first order transport equations we propose stable variational formulations for certain linear unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper the central objective is to develop for such classes adaptive solution concepts with provable error reduction to adaptively resolve anisotropic solution features such as propagating singularities the presently proposed variational formulations allow in particular the employment of trial spaces spanned by directional representation systems since such systems typically given as frames are known to be stable only in l 2 special emphasis is placed on l 2 stable formulations the proposed stability concept is based on perturbations of certain ideal test spaces in petrov galerkin formulations see also l f demkowicz and j gopalakrishnan comput methods appl mech engrg 199 2010 pp 1558 1572 l demkowicz and j gopalakrishnan numer methods partial differential equations 27 2011 pp 70 105 j

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

reply to comment on jamming at zero temperature and zero applied stress the epitome of disorder we answer the questions raised by donev torquato stillinger and connelly dtsc in their preceding comment on jamming at zero temperature and zero applied stress the epitome of disorder phys rev e 70 043301 2004 we emphasize that we follow a fundamentally different approach than they have done to reinterpret random close packing in terms of the maximally random jammed framework we define the maximally random jammed packing fraction to be where the largest number of initial states chosen completely randomly have relaxed final states at the jamming threshold in the thermodynamic limit thus we focus on an ensemble of states at the jamming threshold while dtsc are interested in determining the amount of order and degree of jamming for a particular configuration we also argue that soft particle systems are as clean as those using hard spheres for studying jammed packings and point out the benefits of using soft potentials

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

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

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

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

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

optimized halftoning using dot diffusion and methods for inverse halftoning unlike the error diffusion method the dot diffusion method for digital halftoning has the advantage of pixel level parallelism however the image quality offered by error diffusion is still regarded as superior to most of the other known methods we show how the dot diffusion method can be improved by optimization of the so called class matrix by taking the human visual characteristics into account we show that such optimization consistently results in images comparable to error diffusion without sacrificing the pixel level parallelism adaptive dot diffusion is also introduced and then a mathematical description of dot diffusion is derived furthermore inverse halftoning of dot diffused images is discussed and two methods are proposed the first one uses projection onto convex sets pocs and the second one uses wavelets of these methods the wavelet method does not make use of the knowledge of the class matrix embedded multiresolution dot diffusion is also discussed which is useful for rendering at different resolutions and transmitting images progressively

modeling heterogeneous materials via two point correlation functions basic principles heterogeneous materials abound in nature and man made situations examples include porous media biological materials and composite materials diverse and interesting properties exhibited by these materials result from their complex microstructures which also make it difficult to model the materials yeong and torquato phys rev e 57 495 1998 introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions in this first part of a series of two papers we collect the known necessary conditions on the standard two point correlation function s2 r and formulate a conjecture in particular we argue that given a complete two point correlation function space s2 r of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space we provide examples of realizable two point correlation functions and suggest a set of analytical basis functions we also discuss an exact mathematical formulation of the re construction problem and prove that s2 r cannot completely specify a two phase heterogeneous material alone moreover we devise an efficient and isotropy preserving construction algorithm namely the lattice point algorithm to generate realizations of materials from their two point correlation functions based on the yeong torquato technique subsequent analysis can be performed on the generated images to obtain desired macroscopic properties these developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two point correlation functions we will mainly focus on basic principles in this paper the algorithmic details and applications of the general scheme are given in the second part of this series of two papers

contact network in nearly jammed disordered packings of hard sphere chains we present salient results of the analysis of the geometrical structure of a large fully equilibrated ensemble of nearly jammed packings of linear freely jointed chains of tangent hard spheres generated via extensive monte carlo simulations in spite the expected differences due to chain connectivity both the pair correlation function and the contact network for chain packings are found to strongly resemble those in packings of monomeric hard spheres at the maximally random jammed mrj state a remarkable finding of the present work is the tendency of chains to form closed loops at the mrj state as a consequence of chain collapse our simulations on disordered nearly jammed chain packings yield an average coordination number of 6 which fulfills the isostaticity condition and is in excellent agreement with the corresponding simulation a donev s torquato and f h stillinger phys rev e 71 011105 2005 and experimental t aste m saadatfar and t j senden phys rev e 71 061302 2005 findings for jammed packings of monatomic hard spheres an exact correspondence between the statistical mechanical ensembles of monomeric spheres and of hard sphere chains offers insights regarding the structure and topology of the contact network of hard sphere systems at the mrj state

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

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