monte carlo simulations of the ising spin glass on lattices with finite connectivity the authors have simulated the ising spin glass model on a random lattice with a finite average coordination number and also on the bethe lattice with various different boundary conditions in particular they have calculated the overlap function p q for two independent samples for the random lattice the results are consistent with a spin glass transition above which p q converges to a dirac delta function for large n number of sites and below which p q has in addition a long tail similar to previous results obtained for the infinite range model for the bethe lattice they obtain results in the interior by discarding the two outer shells of the cayley tree when calculating the thermal averages for fixed uncorrelated boundary conditions p q seems to converge to a delta function even below the spin glass transition whereas on a closed lattice correlated boundary conditions p q has a long tail similar to its behaviour in the random lattice case

the asymmetry of stress in granular media here we show that the average stress in granular media which is defined from virtual work may be asymmetric in the absence of contact moments we specify the circumstances and amplitude of stress asymmetry and calculate the corresponding couple stress and first stress moment we also show that the average stress is always symmetric when it is alternately defined by using statics and no contact moment the stress asymmetry which results from external moments has an amplitude that decreases with the volume size the present analysis applies to two and three dimensional particles of arbitrary shapes the asymmetric stress couple stress and first stress moment are analytically calculated in a particular example with cylindrical and spherical particles

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

symmetric powers of modular representations for groups with a sylow subgroup of prime order abstract let v be a representation of a finite group g over a field of characteristic p if p does not divide the group order then molien s formula gives the hilbert series of the invariant ring in this paper we find a replacement of molien s formula which works in the case that g is divisible by p but not by p 2 we also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of v into indecomposables our methods can be applied to determine the depth of the invariant ring without computing any invariants this leads to a proof of a conjecture of the second author on certain invariants of gl 2 p

statics and kinematics of discrete cosserat type granular materials a theoretical framework is presented for the statics and kinematics of discrete cosserat type granular materials in analogy to the force and moment equilibrium equations for particles compatibility equations for closed loops are formulated in the two dimensional case for relative displacements and relative rotations at contacts by taking moments of the equilibrium equations micromechanical expressions are obtained for the static quantities average cauchy stress tensor and average couple stress tensor in analogy by taking moments of the compatibility equations micromechanical expressions are obtained for the infinitesimal kinematic quantities average rotation gradient tensor and average cosserat strain tensor in the two dimensional case alternatively these expressions for the average cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane in analogy the expressions for the average rotation gradient tensor and the average cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two dimensional case it is shown that the average particle stress tensor is always symmetrical contrary to the average stress tensor of an equivalent homogenized continuum finally discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived

constitutive relation for a particulate medium with the effect of particle rotation abstract the constitutive behavior of granular assemblies is investigated taking into account the effect of particle rotation continuum fields are assumed by tensorial polynomial expansion for the discrete variables namely particle displacements and rotations the strain measures for the packing are obtained from the particle displacement and rotation from the principle of virtual work the stress measures for the packing are expressed in terms of the contact forces the contact moments and the geometric measures for the packing structure the derived stress strain relationship is then evaluated by an example of a randomly packed assembly of circular disks loaded in two different conditions deformation of the packing calculated from the constitutive equations is compared with the results obtained from the discrete method of computer simulation to show the applicability of the method

on the definition of the stress tensor in granular media abstract this paper investigates the definition of the stress tensor within a granular assembly when inertial effects are likely to occur it is shown that the stress tensor can be expressed as a sum of two terms a first term corresponds to the standard definition of the stress according to the love weber formula this term is related to the contact forces existing within adjoining particles a second term accounts for dynamic effects related to rotation velocities and accelerations of the particles these results are checked from discrete numerical simulations in order to examine in which context the contribution of inertial effects should not be omitted with this aim the simulation of a granular specimen collapse and then a silo discharge is considered

micro mechanical analysis of deformation characteristics of three dimensional granular materials the deformation characteristics of idealized granular materials have been studied from the micro mechanical viewpoint using bagi s three dimensional micro mechanical formulation for the strain tensor bagi k 1996 mechanics of materials 22 165 177 this formulation is based on the delaunay tessellation of space into tetrahedra the set of edges of the tetrahedra can be divided into physical contacts and virtual contacts between particles bagi s formulation expresses the continuum macro scale strain as an average over all edges of their relative displacements between two successive states and the complementary area vectors this latter vector is a geometrical quantity determined from the set of edges i e from the structure of the particle packing results from discrete element method simulations of isotropic and triaxial loading of a three dimensional polydisperse packing of spheres have been used to investigate statistics of the branch vectors and complementary area vectors of edges subdivided into physical and virtual contacts and of the relative displacements of edges the investigated statistics are probability density functions and averages over groups of edges with the same orientation it is shown that these averages can be represented by second order fourier series in edge orientation edge orientations are distributed isotropically contrary to contact orientations the average lengths of the branch vectors and the normal component of the complementary area vectors are distributed isotropically with respect to the edge orientation and their average values are related to each other and to the volume fraction of the assembly the other two components of the complementary area vector are zero on average the total deformation of the assembly as given by the average of the relative displacements of the edges of the delaunay tessellation follows the uniform strain prediction however neither the deformation of the physical contact network nor of the virtual contact network has this property the average relative displacement of physical edges in the normal direction determined by the branch vector is smaller than that according to the uniform strain assumption while that of virtual contacts is larger this is caused by the high interparticle stiffness that hinders compression the reverse observation holds for the tangential component of the relative displacement vector the contribution of the deformation of the empty space between physical contacts to the continuum macro scale strain tensor is therefore very important for the understanding and the prediction of the macro scale deformation of granular materials

comparative study of different discrete element models and evaluation of equivalent micromechanical parameters abstract comparative studies of different discrete element models of a rock type material are presented the discrete element formulation employs spherical particles with the cohesive interaction model combining linear elastic behaviour with brittle failure numerical studies consisted in simulation of the uniaxial compression test two cylindrical specimens with particle size distributions yielding different degree of heterogeneity have been used macroscopic response produced by different discrete element models has been compared the main difference between the compared models consists in the evaluation of micromechanical constitutive parameters two approaches are compared in the first approach the contact stiffness and strength parameters depend on the local particle size while in the second approach global uniform contact parameters are assumed for all the contacting pairs in function of average geometric measures characterizing the particle assembly the size dependent contact parameters are calculated as functions of geometric parameters characterizing each contacting particle pair as geometric scaling parameters the arithmetic and harmonic means as well as the minimum of the radii of two contacting particles are considered two different models with size dependent contact parameters are formulated the performance of these models is compared with that of the discrete element model with global uniform contact parameters equivalence between the models with size dependent and uniform contact parameters has been checked in search of this equivalence different methods of evaluation of global uniform parameters have been studied the contact stiffness has been evaluated in terms of the average radius of the particle assembly or in terms of the averages of the arithmetic and harmonic means of the contact pair radii the geometric parameters used in the evaluation of the contact stiffness in the size dependent models the uniform contact strengths have been determined as functions of the averages of radii squares squares of arithmetic radii means or squares of minimum radii of the contacting pairs for the more homogenous specimen the models with local size dependent parameters and models with global uniform parameters give similar response the models with uniform parameters evaluated according to the averages of the geometric parameters used in the evaluation of local parameters ensure better agreement with the respective models with size dependent parameters than the models with uniform parameters evaluated according to the particle radii simulations using the more heterogenous specimen reveal differences between the considered models there are significant differences in stress strain curves as well as in the failure pattern the models with local size dependent parameters are more sensitive to the change of heterogeneity than the model with global uniform parameters

exact solution of potts random energy models with weak connectivity and of the many states potts spin glass the author generalises derrida s random energy model to the potts spin glass with p spin interactions in the limit of large p he solves the model exactly for the cases of strong and weak connectivity in the latter case he shows how to solve the model using replicas and obtains a solution for the value of the set of order parameters qr1 rsalpha 1 alpha s he then derives the large q limit of the ordinary q state potts spin glass on a lattice with finite connectivity

non cohen macaulay invariant rings of infinite groups abstract we give explicit examples of invariant rings that are not cohen macaulay for all classical groups sl n k gl n k sp 2 n k so n k and o n k where k is an algebraically closed field of positive characteristic we prove that every non trivial unipotent group over k has representations such that the invariant ring is not cohen macaulay

on the depth of invariant rings of infinite groups abstract let k be an algebraically closed field for a finitely generated graded commutative k algebra r let cmdef r dim r depth r denote the cohen macaulay defect of r let g be a linear algebraic group over k that is reductive but not linearly reductive we show that there exists a faithful rational representation v of g which we will give explicitly such that cmdef k v k g k 2 for all k n

tensorial form definitions of discrete mechanical quantities for granular assemblies this paper explains the necessity of tensorial form definitions of mechanical quantities in the discrete mechanics of granular assemblies and how to make such definitions particles are assumed to be circular in 2d or spheres in 3d first we explain the dirichlet tessellation and some important geometric tools defined from this tessellation i e the contact and dual contact cells and dual branch vectors which become necessary for the tensorial form definition comparing the fundamental quantities in discrete and continuum mechanics and regarding their mechanical properties especially of stress and strain we propose a new tensorial form definition for all of discrete mechanical quantities lastly discussions are made on the properties of these quantities especially on the internal work and the compatibility condition of strain

from particle ensembles to cosserat continua homogenization of contact forces towards stresses and couple stresses in the present contribution a transition from the dynamics of single particles to a cosserat continuum is discussed based on the definition of volume averages expressions for the macroscopic stress tensors and for the couple stress tensors are derived it is found that an ensemble of particles allows for a non symmetric macroscopic stress tensor and thus for the existence of couple stresses even if the single particles are considered as standard continua discrete element method simulations of a biaxial box are used for the validation of the proposed homogenization technique

on the depth of the invariants of the symmetric power representations of sl2 fp abstract we study the depth of the ring of invariants of sl 2 f p acting on the n th symmetric power of the natural two dimensional representation for n p these symmetric power representations are the irreducible representations of sl 2 f p over f p we prove that when the greatest common divisor of p 1 and n is less than or equal to 2 the depth of the ring of invariants is 3 we also prove that the depth is 3 for n 3 p 7 and n 4 p 5 however for n 3 p 7 the depth is 4 and for n 4 p 5 the depth is 5 in these two exceptional cases the ring of invariants is cohen macaulay

on the static and dynamical transition in the mean field potts glass we study the static as well as the glassy or dynamical transition in the mean field p state potts glass by numerical solution of the saddle point equations we investigate the static and dynamical transitions for all values of p in the non perturbative regime p 4 the static and dynamical edwards anderson parameter increase with p logarithmically this makes the glassy transition temperature lie very close to the static one we compare the main predictions of the theory with numerical simulations

stress strain relationship for granular materials based on the hypothesis of best fit abstract stress strain relationships of a granular system are derived based on the premise that the mean field of displacement is the best fit of actual particle displacements based on the best fit hypothesis two fundamental relationships are derived 1 the average strain as a function of contact displacements and 2 the mean field of contact force as a function of stress these two relationships lead to a stress strain law without the kinematic constraint of uniform strain for an assembly with isotropic packing structure closed form stress strain relationships are explicitly derived in which the modulus can be determined from the inter particle stiffness the difference between the derived stress strain relationship and that derived from voigt hypothesis is discussed the effect of fabric variation is illustrated in the theory discrete element analyses are also performed and the results are used to compare with the estimated moduli from the proposed theory in the comparison the solution bounds for granular assembly system are illustrated

analysis of three dimensional micro mechanical strain formulations for granular materials evaluation of accuracy an important objective of recent research on micro mechanics of granular materials is to develop macroscopic constitutive relations in terms of micro mechanical quantities at inter particle contacts although the micro mechanical formulation of the stress tensor is well established the corresponding formulation for the strain tensor has proven to be much more evasive still being the subject of much discussion in this paper we study various micro mechanical strain formulations for three dimensional granular assemblies following the work of bagi in two dimensions bagi 2006 all of these formulations are either based on an equivalent continuum approach or follow the best fit approach their accuracy is evaluated by comparing their results using data from discrete element method simulations on periodic assemblies to the macroscopic deformation it is found that bagi s formulation bagi 1996 which is based on the delaunay tessellation of space is the most accurate furthermore the best fit formulation based on particle displacements only did unexpectedly well in contrast to previously reported results for two dimensional assemblies

on large deformation granular strain measures for generating stress strain relations based upon three dimensional discrete element simulations abstract generating stress strain relations based upon three dimensional discrete element simulations for hierarchical multiscale constitutive modeling of granular materials at finite strain requires measures of stress and strain in the reference and current configurations the cauchy stress tensor calculated from interparticle forces and branch vectors resulting from discrete element simulations is well established but the various finite strain definitions existing in the literature are not as well studied or accepted thus the paper develops seven finite strain measures in three dimensions cartesian coordinates assumed two are calculated in the reference configuration and five are calculated in the current configuration in the process a time integrated deformation gradient is formulated with which the cauchy stress tensor can be mapped to either the kirchhoff stress first piola kirchhoff stress or second piola kirchhoff stress the difficulty in calculating strain measures for granular materials is the discrete particulate nature of the materials in the paper strain measures are motivated by the equivalent continuum method but are extended to finite strain in three dimensions simulations are conducted to test the finite strain measures and it is found that these strain measures are independent of rigid body rotation of a particle assembly granular cauchy stress is also calculated for these discrete element simulations to be able to plot stress strain curves in the current configuration finally two numerical examples with large deformation are simulated 1 cavity expansion and 2 pile penetration and their results are analyzed

the invariants of the third symmetric power representation of sl2 fp abstract for a prime p 3 we compute a finite generating set for the sl 2 f p invariants of the third symmetric power representation the proof relies on the construction of an infinite sagbi basis and uses the hilbert series calculation of hughes and kemper

a micromechanical based micropolar theory for deformation of granular solids abstract granular material perceived as a collection of particles is modelled as a micropolar continuum taking account of the discreteness and microstructure of the material the new feature of the proposed model is that the constitutive coefficients are derived in explicit terms of grain contact properties in addition the constitutive law the equilibrium equations and the compatibility equations for granular material are derived to completely define a boundary value problem using the derived constitutive law and field equations a procedure based on finite element analysis is described to obtain approximated solutions for boundary value problems an example is shown for a granular packing under boundary pressure solutions of the example are compared with that obtained from the discrete element method to show the applicability of this method based on this model the effects of particle rotation and couple stress on the deformation behavior of granular material are also studied

application of statistical mechanics to combinatorial optimization problems the chromatic number problem and q partitioning of a graph methods of statistical mechanics are applied to two important np complete combinatorial optimization problems the first is the chromatic number problem which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color the second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections both models are mapped into a frustrated potts model which is related to theq state potts spin glass for the first problem we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation the quenched model is also discussed for the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq 3 and 4 potts spin glass using parisi s replica symmetry breaking we also perform some numerical simulations to test the theoretical result and obtain very good agreement

replica symmetry breaking in finite connectivity systems a large connectivity expansion at finite and zero temperature parisi replica symmetry breaking is extended to random bond systems with finite fixed connectivity m 1 the free energy f is explicitly calculated for large m within the first stage of symmetry breaking at finite temperature t the 1 m expansion is found to diverge as t to o indeed direct evaluation at zero temperature shows that the large m expansion is in powers of 1 m introduction of symmetry breaking brings the value of f close approximately 1 for 10 m 20 to numerical estimates of banavar et al the same techniques apply to systems with average finite connectivity

homogenization and two scale simulations of granular materials for different microstructural constraints the paper presents a new method for quasi static homogenization of granular microstructures and its embedding into two scale simulations on the coarse scale a homogenized standard continuum is considered where the granular microstructures are locally attached at each point these representative volumes are defined by aggregates of discrete solid particles which may come into contact on the micromechanical side the interparticle contact mechanisms between particles is governed by a coulomb type frictional contact in particular elliptical shaped plane particles are investigated a consistent extension of classical stiff soft and periodic boundary conditions from continuous to granular microstructures induce new classes of micro to macro transitions for granular aggregates these include constraints not only for particle center displacements but also for particle rotations at a driving boundary frame the stiff and soft constraints at the driving frame of the particle aggregate induce upper and lower bounds of the particle aggregate s stiffness on the computational side we outline a unified implementation of the displacement and rotational constraints by penalty methods that proves to be convenient for straightforward integration into discrete element codes representative numerical examples with elliptical shaped particles are investigated where the macroscopic stress responses for different boundary constraints are comparatively discussed finally we embed the granular microstructures into a coarse graining discrete to finite element model where they govern the micromechanical behavior of a two scale simulation copyright 2010 john wiley sons ltd