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

evolving loop structure in gradually tilted two dimensional granular packings granular packings especially near the jamming transition form fragile networks where small perturbations can lead to destabilization and large scale rearrangements a key stabilizing element in two dimensions is the contact loop yet surprisingly little is known about contact loop statistics in realistic granular networks in this paper we use particle dynamics to study the evolution of contact loop structure in a gradually tilted two dimensional granular bed we find that the resulting contact loop distributions 1 are sensitive to material properties 2 deviate from the expected structure of a randomly wired lattice and 3 are uniquely dependent on tilting angle also we introduce a quantitative measure of loop stability ensuremath xi and show that increased tilting results in a gradual destabilization of individual loops we briefly discuss the considerations for extending our approach to three dimensions

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

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

hua operators poisson transform and relative discrete series on line bundles over bounded symmetric domains let d g k be a bounded symmetric domain and s k l be its shilov boundary we consider the action of g with weight nu in mathbb z on functions on d viewed as sections the line bundle and the corresponding eigenspace of g invariant di erential operators the poisson transform maps hyperfunctions on the s to the eigenspaces we characterize the image in terms of hua operators on the sections of the line bundle for some special parameter the poisson transform is of szego type mapping into the relative discrete series we compute the corresponding elements in the discrete series

algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory by means of complex representation and real representation of a quaternion matrix this paper studies the problem of diagonalization of a quaternion matrix gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory

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

geometrical 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

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

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

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

hilbert series of nearly holomorphic sections on generalized flag manifolds abstract let x g p be a complex flag manifold and e x be a g homogeneous holomorphic vector bundle fix a u invariant kahler metric on x with u g maximal compact we study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in e coincides with the space of u finite smooth sections in e the degree of nearly holomorphic sections defines a u invariant filtration on this space using sheaf cohomology we determine in suitable cases the corresponding hilbert series it turns out that this is given in terms of lusztig s q analog of kostant s weight multiplicity formula and hence gives a new representation theoretic interpretation of these formulas

higher laplacians on pseudo hermitian symmetric spaces abstract one of the most fundamental operators studied in geometric analysis is the classical laplace beltrami operator on pseudo hermitian manifolds higher laplacians l m are defined for each positive integer m where l 1 coincides with the laplace beltrami operator despite their natural definition these higher laplacians have not yet been studied in detail in this paper we consider the setting of simple pseudo hermitian symmetric spaces i e let x g h be a symmetric space for a real simple lie group g equipped with a g invariant complex structure we show that the higher laplacians l 1 l 3 l 2 r 1 form a set of algebraically independent generators for the algebra d g x of g invariant differential operators on x where r denotes the rank of x for higher rank this is the first instance of a set of generators for d g x defined explicitly in purely geometric terms and confirms a conjecture of englis and peetre originally stated in 1996 for the class of hermitian symmetric spaces

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

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

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

on a class of nearly holomorphic automorphic forms with a b 1 e c and 0 n e z the point that attracts our attention is its nonholomorphic nature it is noteworthy that f is holomorphic if k 2 and also such nonholomorphy doesn t appear in the hilbert modular case therefore one may naturally ask whether this is an isolated oddity in our previous papers 10 and 13 we showed that it was not so in fact a similar eisenstein series on sp n q or on su n n of a particular weight can have the form

jacobi method for quaternion matrix singular value decomposition the study of quaternion matrices has gained interest in many areas in recent years and the problem of diagonalizing such matrices has also attracted attention in this article we present an algorithm for computing the svd of a matrix with quaternion coefficients directly in quaternion arithmetic using a generalization of classical complex jacobi methods the extension of the jacobi transformation to the quaternion case is introduced for the diagonalization of a hermitian quaternion valued matrix based on this an implicit jacobi algorithm is proposed for computing the svd of a quaternion matrix the performance of the proposed algorithm is presented and compared with an already known algorithm using a complex equivalent of the quaternion matrix and shown to be superior in execution time and accuracy

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

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

nearly holomorphic sections on compact hermitian symmetric spaces abstract let x be a kahler manifold and e be a hermitian vector bundle on x we investigate the space n x e of nearly holomorphic sections in e which generalizes the notion of nearly holomorphic functions introduced by shimura if x u k is a compact hermitian symmetric space and e is u homogeneous it turns out that n x e coincides with the space of u finite vectors in c x e and we obtain new results on the u type decomposition of the hilbert space of square integrable sections as an application we determine this decomposition for the holomorphic tangent bundle of x

topological evolution in dense granular materials a complex networks perspective abstract one of the great challenges in the science of complex materials materials capable of emergent behavior such as self organized pattern formation is deciphering their inherent structural design principles as they deform in response to external loads we have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to i uncover such design principles and ii identify suitable metrics that quantify the evolution of structure during deformation herein we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level i e a particle and its immediate neighbors as well as the macroscopic level we explore network motifs and other topological properties at these multiple length scales in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation spatially and with respect to strain key processes or rearrangement events that cause loss of connectivity within the material domain e g microbanding and force chain buckling are investigated network statistics of these processes previously shown to be major sources of energy dissipation and nonaffine deformation are then tied to corresponding trends observed in the evolving macroscopic network it is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity

anomalous force diffusion in nearly ordered packings of frictionless discs we derive analytic expressions for force propagation in packings of frictionless discs with a narrow distribution of disc sizes by expanding to first order about the known ordered solution the distribution of contact forces p f is found to be narrow at the upper surface and broadens at a rate that varies with depth being superdiffusive near the surface until crossing over to a subdiffusive regime near the fixed base furthermore the response to an isolated load propagates along the edge of a cone as in the ordered case but fluctuates under ensemble averaging by an amount that depends purely on height not on the lateral position finally we comment on ways in which the analytical framework presented here could be extended to a wider range of granular packings

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

geometric origin of mechanical properties of granular materials model granular assemblies in which grains are assumed rigid and frictionless at equilibrium under some prescribed external load are shown to possess under generic conditions several remarkable mechanical properties related to isostaticity and potential energy minimization isostaticity the uniqueness of the contact forces once the list of contacts is known is established in a quite general context and the important distinction between isostatic problems under given external loads and isostatic rigid structures is presented complete rigidity is only guaranteed on stability grounds in the case of spherical cohesionless grains otherwise the network of contacts might deform elastically in response to small load increments even though grains are perfectly rigid in general one gets an upper bound on the contact coordination number the approximation of small displacements that is introduced and discussed allows analogies to be drawn with other model systems studied in statistical mechanics such as minimum paths on a lattice it also entails the uniqueness of the equilibrium state the list of contacts itself is geometrically determined for cohesionless grains and thus the absence of plastic dissipation in rearrangements of the network of contacts plasticity and hysteresis are related to the lack of such uniqueness which can be traced back apart from intergranular friction to nonreversible rearrangements of small but finite extent in which the system jumps between two distinct potential energy minima in configuration space or to bounded tensile forces deriving from a nonconvex potential in the contacts properties of response functions to load increments are discussed on the basis of past numerical studies it is argued that provided the approximation of small displacements is valid displacements due to the rearrangements of the rigid grains in response to small load increments once averaged on the macroscopic scale are solutions to elliptic boundary value problems similar to the stokes problem for viscous incompressible flow

a structure preserving jacobi algorithm for quaternion hermitian eigenvalue problems abstract a new real structure preserving jacobi algorithm is proposed for solving the eigenvalue problem of quaternion hermitian matrix by employing the generalized jrs symplectic jacobi rotations the new quaternion jacobi algorithm can preserve the symmetry and jrs symmetry of the real counterpart of quaternion hermitian matrix moreover the proposed algorithm only includes real operations without dimension expanding and is generally superior to the state of the art algorithm numerical experiments are reported to indicate its efficiency and accuracy

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

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

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

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

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

invariant differential operators on hermitian symmetric spaces our object of study is the ring of invariant differential operators on a hermitian symmetric space g k of classical and noncompact type here as usual g is a connected noncompact semisimple lie group with finite center and k is a maximal compact subgroup of g it is well known that the ring denoted by g k is isomorphic to a polynomial ring of 1 variables 1 being the rank of g k this is true even in the nonhermitian case if 1 1 it is generated by the laplace beltrami operator z which is essentially self adjoint moreover iz2 is nonnegative in the general case an easily posable problem is to find an explicitly defined set of generators we can go one step further by focusing our attention on the nonnegativity which necessarily limits the range of eigenvalues under a suitable integrability condition on functions it is thus natural to ask whether there exist some canonically defined operators which generate 9 g k and have the property of nonnegativity the main purpose of the present paper is to give an affirmative answer to this question in fact our answer applies to a somewhat more general type of rings of differential operators which includes 9 g k as a special case to be explicit we take an irreducible representation p k gl v with a complex vector space v of finite dimension and consider the set c p of all v valued cx functions f on g such that f xk 1 p k f x for every k e k we then denote by p the ring of left invariant difrrential operators on g which map c p into itself now the complexification g of the lie algebra of g has abelian subalgebras p and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on g k at the origin for any complex vector space w let sr w denote the ring of all complex valued homogeneous polynomial functions on w of degree r and let s w er 0sr w through the adjoint representation of g on g k acts naturally on sr it is a known fact due to hua and schmid that each irreducible constituent of this representation of k has multiplicity one now our principal

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

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

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

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

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

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

statistics of the elastic behaviour of granular materials the elastic behaviour of isotropic assemblies of granular materials consisting of two dimensional bonded and non rotating particles is studied from the micromechanical viewpoint discrete element simulations have been performed of assemblies of 50 000 particles with various coordination numbers average number of contacts per particle and various ratios of tangential to normal stiffness statistics such as mean standard deviation and probability distribution function have been computed for geometrical quantities and contact displacements under compressive and shearing loading a linear relation is observed between the average number of contacts of a particle and its circumference average displacements do not conform to generalised uniform strain as is often assumed the probability density functions for normal and tangential displacements are nearly gaussian none of the considered energies satisfy an equipartition of energy between normal and tangential modes

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

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

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

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

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

on the power method for quaternion right eigenvalue problem abstract in this paper we study the power method of the right eigenvalue problem of a quaternion matrix a if a is hermitian we propose the power method that is a direct generalization of that of complex hermitian matrix when a is non hermitian by applying the properties of quaternion right eigenvalues we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector we also briefly discuss the inverse power method and shift inverse power method for the both cases the real structure preserving algorithm of the power method in the two cases are also proposed and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method