contact forces in anisotropic frictional granular materials the probability density functions pdfs of contact forces in anisotropic cohesionless and frictional granular materials are studied numerically and theoretically using discrete element simulations of biaxial deformation of a large two dimensional assembly consisting of 200 000 disks it is observed that the pdfs for the normal and tangential components of the contact forces depend significantly on contact orientation the pdfs exhibit exponential decay and the pdf for the tangential component of the contact forces is not always symmetrical with respect to zero tangential force the shape of the pdf for the normal component of the contact forces changes with shear strain a qualitative explanation for this change is given that is related to the biaxial deformation mechanism in which the disrupted contacts are predominantly oriented in the direction of the minor principle stress a maximum entropy method is employed to study these pdfs theoretically using a prescribed stress tensor as constraint it is found that the theoretical results correspond qualitatively to many of the results obtained from the discrete element simulations discrepancies between theory and simulations are attributed to the fact that the kinematics have not been taken into account in the theory

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

inferring a tree from lowest common ancestors with an application to the optimization of relational expressions we present an algorithm for constructing a tree to satisfy a set of lineage constraints on common ancestors we then apply this algorithm to synthesize a relational algebra expression from a simple tableau a problem arising in the theory of relational databases

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

finding maximal leaf agreement isomorphic descendent subtrees from phylogenetic trees with different species phylogenetic trees are an important tool to help in the understanding of relationships between objects that evolve through time in particular molecular sequences in this paper we consider two descendent subtree comparison problems on phylogenetic trees given a set of k phylogenetic trees whose leaves are drawn from 1 2 n and the leaves for two arbitrary trees are not necessary the same we first present a linear time algorithm to final all the maximal leaf agreement descendent subtrees based on this result we also present a linear time algorithm to find all the maximal leaf agreement isomorphic descendent subtrees

on finding common subtrees abstract let t and r be two arbitrary ordered trees t r whose nodes are labelled over an alphabet a we devise a simple solution for detecting all the common subtrees in o t time and space if the size of a is finite and o t log min a t time otherwise we solve the problem of finding in t and r all occurrences if any of any given tree b in either o t b or o b t b time this requires to set up a simple data structure in o t time that allows to find all maximal subtrees of b in o b time and to solve other related problems

average complexity of moore s and hopcroft s algorithms in this paper we prove that for the uniform distribution on complete deterministic automata the average time complexity of moore s state minimization algorithm is o nloglogn where n is the number of states in the input automata and the number of letters in the alphabet is fixed then an unusual family of implementations of hopcroft s algorithm is characterized for which the algorithm will be proved to be always faster than moore s algorithm finally we present experimental results on the usual implementations of hopcroft s algorithm

a hitchhiker s guide to descriptional complexity through analytic combinatorics nowadays increasing attention is being given to the study of the descriptional complexity in the average case although the underlying theory for such a study seems intimidating one can obtain interesting results in this area without too much effort in this gentle introduction we take the reader on a journey through the basic analytical tools of that theory giving some illustrative examples using regular expressions additionally new asymptotic average case results for several e nfa constructions are presented in a unified framework it turns out that asymptotically and in the average case the complexity gap between the several constructions is significantly larger than in the worst case furthermore one of the e nfa constructions approaches the corresponding e free nfa construction asymptotically and on average

on the average state and transition complexity of finite languages we investigate the average case state and transition complexity of deterministic and nondeterministic finite automata when choosing a finite language of a certain size n uniformly at random from all finite languages of that particular size here size means that all words of the language are either of length n or of length at most n it is shown that almost all deterministic finite automata accepting finite languages over a binary input alphabet have state complexity q 2 nn while nondeterministic finite automata are shown to perform better namely the nondeterministic state complexity is in q 2 n interestingly in both cases the aforementioned bounds are asymptotically like in the worst case however the nondeterministic transition complexity is shown to be again q 2 nn the case of unary finite languages is also considered moreover we develop a framework that allows us to investigate the average case complexity of operations like e g union intersection complementation and reversal on finite languages in this setup

probability density functions of contact forces for cohesionless frictional granular materials a theory is developed for the probability density functions of contact forces for cohesionless frictional granular materials in quasi static equilibrium this theory is based on a maximum information entropy principle with an expression for information entropy that is appropriate for granular materials entropy is maximized under the constraints of a prescribed stress and that the normal component of the contact force is compressive and that the tangential component of the contact force is limited by coulomb friction the theory results in a dependence of the probability density function for the tangential contact forces on the friction coefficient the theoretical predictions are compared with results from discrete element simulations on isotropic two dimensional assemblies under hydrostatic stress good qualitative agreement is found for means and standard deviations of contact forces and the shape of the probability density functions while the quantitative agreement is fairly good discrepancies between theory and simulations such as the difference in shape of the probability density function for the normal force and the observed dependence on elastic properties of the exponential decay rate of tangential forces are attributed to the fact that the method does not take into account any kinematics which are essential in relation to elastic effects

model for force fluctuations in bead packs we study theoretically the complex network of forces that is responsible for the static structure and properties of granular materials we present detailed calculations for a model in which the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead of the pile we compare our results for the force distribution function for this model including exact results for certain contact angle probability distributions with numerical simulations of force distributions in random sphere packings this model reproduces many aspects of the force distribution observed both in experiment and in numerical simulations of sphere packings our model is closely related to some that have been studied in the context of self organized criticality we present evidence that in the force distribution context critical power law force distributions occur only when a parameter hidden in other interpretations is tuned our numerical mean field and exact results all indicate that for almost all contact distributions the distribution of forces decays exponentially at large forces

sparse dynamic programming for evolutionary tree comparison constructing evolutionary trees for species sets is a fundamental problem in biology unfortunately there is no single agreed upon method for this task and many methods are in use current practice dictates that trees be constructed using different methods and that the resulting trees should be compared for consensus it has become necessary to automate this process as the number of species under consideration has grown we study one formalization of the problem the maximum agreement subtree mast problem the mast problem is as follows given a set a and two rooted trees ct 0 and ct 1 leaf labeled by the elements of a find a maximum cardinality subset b of a such that the topological restrictions of ct 0 and ct 1 to b are isomorphic in this paper we will show that this problem reduces to unary weighted bipartite matching uwbm with an o n 1 o 1 additive overhead we also show that uwbm reduces linearly to mast thus our algorithm is optimal unless uwbm can be solved in near linear time the overall running time of our algorithm is o n 1 5 log n improving on the previous best algorithm which runs in o n 2 we also derive an o n c sqrt log n time algorithm for the case of bounded degrees whereas the previously best algorithm runs in o n 2 as in the unbounded case

an o n log n algorithm for the maximum agreement subtree problem for binary trees the maximum agreement subtree problem is the following given two rooted trees whose leaves are drawn from the same set of items e g species find the largest subset of these items so that the portions of the two trees restricted to these items are isomorphic we consider the case which occurs frequently in practice i e the case when the trees are binary and give an o nlog n time algorithm for this problem

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

the state complexity of random dfas the state complexity of a deterministic finite state automaton dfa is the number of states in its minimal equivalent dfa we study the state complexity of random n state dfas over a k symbol alphabet drawn uniformly from the set n n k 2 n of all such automata we show that with high probability the latter is k n o n log n for a certain explicit constant k

uniform random generation of decomposable structures using floating point arithmetic the em recursive method formalized by nijenhuis and wilf cite niwi78 and systematized by flajolet van cutsem and zimmermann cite flziva94 is extended here to floating point arithmetic the resulting adz method enables one to generate decomposable data structures both labelled or unlabelled uniformly at random in expected o n 1 epsilon time and space after a preprocessing phase of o n 2 epsilon time which reduces to o n 1 epsilon for context free grammars

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

enumeration and generation with a string automata representation the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks in this work we present a unique string representation for complete initially connected deterministic automata icdfas with n states over an alphabet of k symbols for these strings we give a regular expression and show how they are adequate for exact and random generation allow an alternative way for enumeration and lead to an upper bound for the number of icdfas the exact generation algorithm can be used to partition the set of icdfas in order to parallelize the counting of minimal automata and thus of regular languages a uniform random generator for icdfas is presented that uses a table of pre calculated values based on the same table an optimal coding for icdfas is obtained

maximum agreement subtree in a set of evolutionary trees metrics and efficient algorithms the maximum agreement subtree approach is one method of reconciling different evolutionary trees for the same set of species an agreement subtree enables choosing a subset of the species for whom the restricted subtree is equivalent under a suitable definition in all given evolutionary trees recently dynamic programming ideas were used to provide polynomial time algorithms for finding a maximum homeomorphic agreement subtree of two trees generalizing these methods to sets of more than two trees yields algorithms that are exponential in the number of trees unfortunately it turns out that in reality one is usually presented with more than two trees sometimes as many as thousands of trees in this paper we prove that the maximum homeomorphic agreement subtree problem is cal np complete for three trees with unbounded degrees we then show an approximation algorithm of time o kn5 for choosing the species that are not in a maximum agreement subtree of a set of k trees our approximation is guaranteed to provide a set that is no more than 4 times the optimum solution while the set of evolutionary trees may be large in practice the trees usually have very small degrees typically no larger than three we develop a new method for finding a maximum agreement subtree of k trees of which one has degree bounded by d this new method enables us to find a maximum agreement subtree in time o knd 1 n2d

average stresses and force fluctuations in noncohesive granular materials a lattice model is presented for investigating the fluctuations in static granular materials under gravitationally induced stress the model is similar in spirit to the scalar q model of coppersmith et al phys rev e 53 4673 1996 but ensures balance of all components of forces and torques at each site the geometric randomness in real granular materials is modeled by choosing random variables at each site consistent with the assumption of cohesionless grains configurations of the model can be generated rapidly allowing the statistical study of relatively large systems for a two dimensional 2d system with rough walls the model generates configurations consistent with continuum theories for the average stresses unlike the q model without requiring the assumption of a constitutive relation for a 2d system with periodic boundary conditions the model generates single grain force distributions similar to those obtained from the q model with a singular distribution of q s

enumeration and random generation of accessible automata we present a bijection between the set a n of deterministic and accessible automata with n states on a k letters alphabet and some diagrams which can themselves be represented as partitions of a set of kn 1 elements into n non empty subsets this combinatorial construction shows that the asymptotic order of the cardinality of a n is related to the stirling number knn our bijective approach also yields an efficient random sampler for the uniform distribution of automata with n states its complexity is o n 3 2 using the framework of boltzmann samplers

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

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

inferring evolutionary trees with strong combinatorial evidence we consider the problem of inferring the evolutionary tree of a set of n species we propose a quartet reconstruction method which specifically produces trees whose edges have strong combinatorial evidence let q be a set of resolved quartets defined on the studied species the method computes the unique maximum subset q of q which is equivalent to a tree and outputs the corresponding tree as an estimate of the species phylogeny we use a characterization of the subset q due to bandelt and dress adv appl math 7 1986 309 343 to provide an o n4 incremental algorithm for this variant of the np hard quartet consistency problem moreover when chosing the resolution of the quartets by the four point method fpm and considering the cavender farris model of evolution we show that the convergence rate of the q method is at worst polynomial when the maximum evolutive distance between two species is bounded we complete these theoretical results by an experimental study on real and simulated data sets the results show that i as expected the strong combinatorial constraints it imposes on each edge leads the q method to propose very few incorrect edges ii more surprisingly the method infers trees with a relatively high degree of resolution

parametric random generation of deterministic tree automata uniform random generators deliver a simple empirical means to estimate the average complexity of an algorithm we present a general rejection algorithm that generates sequential letter to letter transducers up to isomorphism we also propose an original parametric random generation algorithm to produce sequential letter to letter transducers with a fixed number of transitions we tailor this general scheme to randomly generate deterministic tree walking automata and deterministic top down tree automata we apply our implementation of the generator to the estimation of the average complexity of a deterministic tree walking automata to nondeterministic top down tree automata construction we also implemented

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

sampling different kinds of acyclic automata using markov chains we propose algorithms that use markov chain techniques to generate acyclic automata uniformly at random we first consider deterministic accessible and acyclic automata then focus on the class of minimal acyclic automata in each case we explain how to define random local transformations that describe an ergodic and symmetric markov chain the distribution of the automaton obtained after t random steps in this markov chain tends to the uniform distribution as t tends to infinity

a large deviations principle for the maki thompson rumour model abstract we consider the stochastic model for the propagation of a rumour within a population which was formulated by maki and thompson 20 sudbury 22 established that as the population size tends to infinity the proportion of the population never hearing the rumour converges in probability to 0 2032 watson 23 later derived the asymptotic normality of a suitably scaled version of this proportion we prove a corresponding large deviations principle with an explicit formula for the rate function

force density function relationships in 2 d granular media an integral transform relationship is developed to convert between two important probability density functions distributions used in the study of contact forces in granular physics developing this transform has now made it possible to compare and relate various theoretical approaches with one another and with the experimental data despite the fact that one may predict the cartesian probability density and another the force magnitude probability density also the transforms identify which functional forms are relevant to describing the probability density observed in nature and so the modified bessel function of the second kind has been identified as the relevant form for the cartesian probability density corresponding to exponential forms in the force magnitude distribution furthermore it is shown that this transform pair supplies a mathematical framework sufficient for describing the evolution of the force magnitude distribution under shearing apart from the choice of several coefficients whose