critical behavior of sandpile vortices under variable charge in the following we consider a sandpile cellular automaton model height version which also takes account of lattice cyclicity for variable charge of grains projected upon the board the size of the charge or additional grains of sand used to upset the equilibrium and to induce an avalanche is found to affect the distributional forms and in particular the convergence of the estimate for the dynamic exponent exponent estimates show slow variation with lattice dimensionl and some evidence of evolution with charge the scaling region is limited in all cases with noisy decay at all levels of charge results presented extend the work of duarte and goncalves 1990 on the triangular lattice

growth rates and explosions in sandpiles we study the abelian sandpile growth model where n particles are added at the origin on a stable background configuration in zd any site with at least 2d particles then topples by sending one particle to each neighbor we find that with constant background height h 2d 2 the diameter of the set of sites that topple has order n1 d this was previously known only for h d our proof uses a strong form of the least action principle for sandpiles and a novel method of background modification we can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d 1 on the other hand we show that if the background height 2d 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random then adding finitely many particles creates an explosion a sandpile that never stabilizes

disorder induced phase transition in a one dimensional model of rice pile abstract we propose a one dimensional rice pile model which connects the 1d btw sandpile model phys rev a 38 1988 364 and the oslo rice pile model phys rev lett 77 1997 107 in a continuous manner we found that for a sufficiently large system there is a sharp transition between the trivial critical behaviour of the 1d btw model and the self organized critical soc behaviour when there is soc the model belongs to a known universality class with the avalanche exponent 1 53

height probabilities in the abelian sandpile model on the generalized finite bethe lattice national natural science foundation of china nsfc11171134 natural science foundation of fujian province china 2011j01015

a loop reversibility and subdiffusion of the rotor router walk the rotor router model on a graph describes a discrete time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on vertices of the graph we prove the following property if at some moment of time the rotors form a closed clockwise contour on the planar graph then the clockwise rotations of rotors generate a walk which enters into the contour at some vertex v performs a number of steps inside the contour so that the contour formed by rotors becomes anti clockwise and then leaves the contour at the same vertex v this property generalizes the previously proved theorem for the case when the rotor configuration inside the contour is a cycle rooted spanning tree and all rotors inside the contour perform a full rotation we use the proven property for an analysis of the sub diffusive behavior of the rotor router walk

limiting shapes for deterministic centrally seeded growth models we study the rotor router model and two deterministic sandpile models for the rotor router model in z d levine and peres proved that the limiting shape of the growth clus ter is a sphere for the other two models only bounds in dimension 2 are known a unified approach for these models with a new parameter h the initial number of particles at each site allows to prove a number of new limiting shape results in any dimension d 1 for the rotor router model the limiting shape is a sphere for all values of h for one of the sandpile models and h 2d 2 the maximal value the limiting shape is a cube for both sandpile models the limiting shape is a sphere in the limit h finally we prove that the rotor router shape contains a diamond

root system chip firing i interval firing jim propp recently introduced a variant of chip firing on a line where the chips are given distinct integer labels hopkins mcconville and propp showed that this process is confluent from some but not all initial configurations of chips we recast their set up in terms of root systems labeled chip firing can be seen as a root firing process which allows the moves for alpha in phi whenever langle lambda alpha vee rangle 0 where phi is the set of positive roots of a root system of type a and lambda is a weight of this root system we are thus motivated to study the exact same root firing process for an arbitrary root system actually this central root firing process is the subject of a sequel to this paper in the present paper we instead study the interval root firing processes determined by for alpha in phi whenever langle lambda alpha vee rangle in k 1 k 1 or langle lambda alpha vee rangle in k k 1 for any k ge 0 we prove that these interval firing processes are always confluent from any initial weight we also show that there is a natural way to consistently label the stable points of these interval firing processes across all values of k so that the number of weights with given stabilization is a polynomial in k we conjecture that these ehrhart like polynomials have nonnegative integer coefficients

numerical study of the langevin theory for fixed energy sandpiles the recently proposed langevin equation aimed to capture the relevant critical features of stochastic sandpiles and other self organizing systems is studied numerically the equation is similar to the reggeon field theory describing generic systems with absorbing states but it is coupled linearly to a second conserved and static nondiffusive field it has been claimed to represent a different universality class including different discrete models the manna as well as other sandpiles reaction diffusion systems etc in order to integrate the equation and surpass the difficulties associated with its singular noise we follow a numerical technique introduced by dickman our results coincide remarkably well with those of discrete models claimed to belong to this universality class in one two and three dimensions this provides a strong backing for the langevin theory of stochastic sandpiles and to the very existence of this meagerly understood universality class

relations between elliptic multiple zeta values and a special derivation algebra we investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length our method is based on representing elliptic multiple zeta values as iterated integrals over eisenstein series and exploiting the connection with a special derivation algebra its commutator relations give rise to constraints on the iterated integrals over eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives conversely the above connection suggests apparently new relations in the derivation algebra under https tools aei mpg de emzv we provide relations for elliptic multiple zeta values over a wide range of weights and lengths ar x iv 1 50 7 02 25 4v 1 he pth 8 j ul 2 01 5

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

divisors on graphs binomial and monomial ideals and cellular resolutions we study various binomial and monomial ideals arising in the theory of divisors orientations and matroids on graphs we use ideas from potential theory on graphs and from the theory of delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions we show that the resolutions of all these ideals are closely related and that their mathbb z graded betti tables coincide as corollaries we give conceptual proofs of conjectures and questions posed by postnikov and shapiro by manjunath and sturmfels and by perkinson perlman and wilmes various other results related to the theory of chip firing games on graphs also follow from our general techniques and results

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

phase transition and critical behavior in a model of organized criticality we study a model of organized criticality where a single avalanche propagates through an a priori static i e organized sandpile configuration the latter is chosen according to an i i d distribution from a borel probability measure on 0 1 the avalanche dynamics is driven by a standard toppling rule however we simplify the geometry by placing the problem on a directed rooted tree as our main result we characterize which are critical in the sense that they do not admit an infinite avalanche but exhibit a power law decay of avalanche sizes our analysis reveals close connections to directed site percolation both in the characterization of criticality and in the values of the critical exponents

pattern formation in growing sandpiles with multiple sources or sinks adding sand grains at a single site in the abelian sandpile models produces beautiful but complex patterns we study the effect of sink sites on such patterns sinks change the scaling of the diameter of the pattern with the number n of sand grains added for example in two dimensions in the presence of a sink site the diameter of the pattern grows as sqrt n log n for large n whereas it grows as sqrt n if there are no sink sites in the presence of a line of sink sites this rate reduces to n 1 3 we determine the growth rates for various sink geometries along with the case when there are two lines of sink sites forming a wedge and generalizations to higher dimensions we characterize the asymptotic pattern in the large n limit for one such case the two dimensional f lattice with a single source adjacent to a line of sink sites the characterization is done in terms of the positions of different spatial features in the pattern for this lattice we also provide an exact characterization of the pattern with two sources when the line joining them is along one of the axes of the lattice

potts model partition functions on two families of fractal lattices abstract the partition function of q state potts model or equivalently the tutte polynomial is computationally intractable for regular lattices the purpose of this paper is to compute partition functions of q state potts model on two families of fractal lattices based on their self similar structures and by applying the subgraph decomposition method we divide their tutte polynomials into two summands and for each summand we obtain a recursive formula involving the other summand as a result the number of spanning trees and their asymptotic growth constants and a lower bound of the number of connected spanning subgraphs or acyclic root connected orientations for each of such two lattices are obtained

superconformal simple type and witten s conjecture abstract let x be a smooth closed connected orientable four manifold with b 1 x 0 and b x 3 and odd we show that if x has seiberg witten simple type then the so 3 monopole cobordism formula of 14 implies witten s conjecture relating the donaldson and seiberg witten invariants

mechanisms of avalanche dynamics and forms of scaling in sandpiles abstract the probability distributions of several avalanche quantities of the bak tang wiesenfeld sandpile in two dimensions display multifractal scaling this explains why this prototype model remained extremely problematic and controversial for such a long time after its introduction other models like the manna model obey instead standard finite size scaling and fall in different universality classes an analysis of time series for the waves into which avalanches can be decomposed shows that the different forms of scaling can be ascribed to the respective autocorrelation functions the intermittency of the bak tang wiesenfeld sandpile is due to this autocorrelation being long range in time unlike in the manna case a coarse graining of the wave time series elucidates these differences and the mechanism leading to multifractality

selberg s integral and linear forms in zeta values using selberg s integral we present some new euler type integral representations of certain nearly poised hypergeometric series these integrals are also shown to produce linear forms in odd and or even zeta values that generalize previous work of the author

statistics of toppling wave boundaries in deterministic and stochastic sandpile models we study numerically the statistics of curves which form the boundaries of toppling wave clusters in the deterministic bak tang and wiesenfeld sandpile model and stochastic manna model on a square lattice we consider the abelian version of each model multiple tests show that the boundary of toppling wave clusters in both deterministic and stochastic models can be described by sle curves with diffusivity 2

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

asymptotic behavior and synchronizability characteristics of a class of recurrent neural networks we propose an approach to the analysis of the influence of the topology of a neural network on its synchronizability in the sense of equal output activity rates given by a particular neural network model the model we introduce is a variation of the zhang model we investigate the time asymptotic behavior of the corresponding dynamical system in particular the conditions for the existence of an invariant compact asymptotic set and apply the results of the synchronizability analysis on a class of random scale free networks and to the classical random networks with poisson connectivity distribution

steady state and relaxation spectrum of the oslo rice pile model we show that the one dimensional oslo rice pile model is a special case of the abelian distributed processors model the exact steady state of the model is determined we show that the time evolution operator w for the system satisfies the equation w n 1 w n where n l l 1 2 for a pile with l sites this is used to prove that w has only one eigenvalue 1 corresponding to the steady state and all other eigenvalues are exactly zero also all connected time dependent correlation functions in the steady state of the pile are exactly zero for time difference greater that n generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed

renormalization group study of sandpile on the triangular lattice abstract we apply the renormalization group approach to the sandpile on the triangular lattice the only attractive fixed point is found the obtained fixed point height probabilities are compared with numerical simulations the value of critical exponent of avalanche size distribution is found to be 1 36 the probabilities of the sand transition are compared with the branching probabilities of the spanning trees on the triangular lattice which are also evaluated

height correlations in the abelian sandpile model the authors study the distribution of heights in the self organized critical state of the abelian sandpile model on a d dimensional hypercubic lattice they calculate analytically the concentration of sites having minimum allowed value in the critical state they also calculate in the critical state the probability that the heights at two sites separated by a distance r would both have minimum values and show that the lowest order r dependent term in it varies as r 2d for large r

spanning trees and recurrent configurations of a graph abstract a new explicit relationship between spanning trees and recurrent configurations of a graph is given by constructing subtree applying this relationship we determine the total number of topplings in an avalanche as applications we calculate the sizes of avalanches of the abelian sandpile model on the complete graph the wheel and the complete bipartite graph

correlations and scaling in the outflow statistics of a sandpile automaton abstract we study numerically the outflow of a two dimensional sandpile cellular automation motivated by recent experiments we calculated the correlations in the sizes of the outflow avalanches and obtained for short times exponential decay for long times a quasiperiodic behavior was observed with the characteristic time proportional to the number of sites in the system the latter feature is attributed to the abelian character of the considered model the size distribution of the outflow avalanches can be described excellently by multifractal scaling

threshold state and a conjecture of poghosyan poghosyan priezzhev and ruelle we prove a precise relationship between the threshold state of the fixed energy sandpile and the stationary state of dhar s abelian sandpile in the limit as the initial condition s0 tends to infty the former is obtained by size biasing the latter according to burst size an avalanche statistic the question of whether and how these two states are related has been a subject of some controversy since 2000

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

dynamical and spatial aspects of sandpile cellular automata the bak tang and wiesenfeld cellular automaton is simulated in 1 2 3 4 and 5 dimensions we define a new set of scaling exponents by introducing the concept of conditional expectation values scaling relations are derived and checked numerically and the critical dimension is discussed we address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2 the scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation primarily the results of our work show that the flow of sand down the slope does not have a 1 f power spectrum in any dimension although the model does show clear critical behavior with scaling exponents depending on the dimension the power spectrum behaves as 1 f 2 in all the dimensions considered

abelian cascade dynamics in bootstrap percolation abstract the culling process in bootstrap percolation is abelian since the final stable configuration does not depend on the details of the updating procedure an efficient algorithm is devised using this idea for the determination of the bootstrap percolation threshold in two dimension which takes l 2 time compared to the l 3 log l in the conventional method a generalised bootstrap percolation allowing many particles at a site is studied where continuous phase transitions are observed for all values of the threshold parameter similar results are also obtained for the continuum bootstrap percolation model

waves in the sandpile model on fractal lattices abstract the scaling properties of waves of topplings of the sandpile model on a fractal lattice are investigated the exponent describing the asymptotics of the distribution of last waves in an avalanche is found predictions for scaling exponents in the forward and backward conditional probabilities for two consecutive waves are given all predictions were examined by simulations on the sierpinski gasket and were found to be in a reasonable agreement with the numerical data

on the sandpile group of the cone of a graph abstract in this article we study the sandpile group of the cone of a graph after introducing the concept of uniform homomorphism of graphs we prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups also we establish a relationship between the sandpile group of the cone of the cartesian product of graphs and the sandpile group of the cone of their factors as an application of these results we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube

on differential equations satisfied by modular forms we use the theory of modular functions to give a new proof of a result of p f stiller which asserts that if t is a non constant meromorphic modular function of weight 0 and f is a meromorphic modular form of weight k with respect to a discrete subgroup of sl 2 commensurable with sl 2 then f as a function of t satisfies a k 1 st order linear differential equation with algebraic functions of t as coefficients furthermore we show that the schwarzian differential equation for the modular function t can be extracted from any given k 1 st order linear differential equation of this type one advantage of our approach is that every coefficient in the differential equations can be relatively easily determined

tutte polynomials of two self similar network models the tutte polynomial t g x y of a graph g or equivalently the q state potts model partition function is a two variable polynomial graph invariant of considerable importance in combinatorics and statistical physics graph operations have been extensively applied to model complex networks recently in this paper we study the tutte polynomials of the diamond hierarchical lattices and a class of self similar fractal models which can be constructed through graph operations firstly we find out the behavior of the tutte polynomial under k inflation and k subdivision which are two graph operations secondly we compute and gain the tutte polynomials of this two self similar fractal models by using their structure characteristic moreover as an application of the obtained results some evaluations of their tutte polynomials are derived such as the number of spanning trees and the number of spanning forests

generalized lambert series and arithmetic nature of odd zeta values it is pointed out that the generalized lambert series sum nolimits n 1 infty n n 2h e n nx 1 studied by kanemitsu tanigawa and yoshimoto can be found on page 332 of ramanujan s lost notebook in a slightly more general form we extend an important transformation of this series obtained by kanemitsu tanigawa and yoshimoto by removing restrictions on the parameters n and h that they impose from our extension we deduce a beautiful new generalization of ramanujan s famous formula for odd zeta values which for n odd and m 0 gives a relation between 2 m 1 and 2 nm 1 a result complementary to the aforementioned generalization is obtained for any even n and m it generalizes a transformation of wigert and can be regarded as a formula for 2 m 1 1 n applications of these transformations include a generalization of the transformation for the logarithm of dedekind eta function z zudilin and rivoal type results on transcendence of certain values and a transcendence criterion for euler s constant

pattern formation in fast growing sandpiles we study the patterns formed by adding n sand grains at a single site on an initial periodic background in the abelian sandpile models and relaxing the configuration when the heights at all sites in the initial background are low enough one gets patterns showing proportionate growth with the diameter of the pattern formed growing as n 1 d for large n in d dimensions on the other hand if sites with maximum stable height in the starting configuration form an infinite cluster we get avalanches that do not stop in this paper we describe our unexpected finding of an interesting class of backgrounds in two dimensions that show an intermediate behavior for any n the avalanches are finite but the diameter of the pattern increases as n for large n with 1 2 1 different values of can be realized on different backgrounds and the patterns still show proportionate growth the non compact nature of growth simplifies their analysis significantly we characterize the asymptotic pattern exactly for one illustrative example with 1

toppling distributions in one dimensional abelian sandpiles we consider toppling distributions for the abelian sandpile model in one dimension we study the avalanche mass and duration distributions in the thermodynamic limit we also investigate their dependence on the seeding distribution which describes how sand is dropped none of the models shows criticality

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

multiple series connected to hoffman s conjecture on multiple zeta values abstract recent results of zlobin and cresson fischler rivoal allow one to decompose any suitable p uple series of hypergeometric type into a linear combination over the rationals of multiple zeta values of depth at most p in some cases only the multiple zeta values with 2 s and 3 s are involved as in hoffman s conjecture in this text we study the depth p part of this linear combination namely the contribution of the multiple zeta values of depth exactly p we prove that it satisfies some symmetry property as soon as the p uple series does and make some conjectures on the depth p 1 part of the linear combination when p 3 our result generalizes the property that very well poised univariate hypergeometric series involve only zeta values of a given parity which is crucial in the proof by rivoal and ball rivoal that 2 n 1 is irrational for infinitely many n 1 the main feature of the proof is an algebraic approach based on representations of z 2 z p s p

apery limits and special values of l functions we describe a general method to determine the apery limits of a differential equation that has a modular function origin as a by product of our analysis we discover a family of identities involving the special values of l functions associated with modular forms the proof of these identities is independent of differential equations and apery limits

linear systems on tropical curves a tropical curve gamma is a metric graph with possibly unbounded edges and tropical rational functions are continuous piecewise linear functions with integer slopes we define the complete linear system d of a divisor d on a tropical curve gamma analogously to the classical counterpart we investigate the structure of d as a cell complex and show that linear systems are quotients of tropical modules finitely generated by vertices of the cell complex using a finite set of generators d defines a map from gamma to a tropical projective space and the image can be modified to a tropical curve of degree equal to mathrm deg d the tropical convex hull of the image realizes the linear system d as a polyhedral complex

approximations to di and tri logarithms we propose hypergeometric constructions of simultaneous approximations to polylogarithms these approximations suit for computing the values of polylogarithms and satisfy 4 term apery like polynomial recursions

patterns formed by addition of grains to only one site of an abelian sandpile abstract we study the patterns formed by adding n sand grains at a single site in the abelian sandpile model on an infinite square lattice when in the initial configuration all sites have same height when this height is 2 stable heights being 0 3 we show that the perturbed region is a square whose length increases as n for large n if all lengths are rescaled by n the pattern tends to a limiting pattern in which the locally averaged height takes piece wise constant rational values we study the structure of these patterns and also other initial configurations we introduce a toppling function that fully describes these we show that it is piecewise quadratic and determine its form as well as the relation with the height configurations of the pattern

exact results for the directed abelian sandpile models we define the directed abelian sandpile models by introducing a parameter c representing the degree of anisotropy in the avalanche processes where c 1 is for the isotropic case we calculate some quantities characterizing the self organized critical states on the one and two dimensional lattices in particular we obtain the expected number of topplings per added particle t which shows the dependence on the lattice size l as lx for large l we show that the critical exponent x does not depend on the dimensionality d at least for d 1 and 2 and that when any anisotropy is included in the system x 1 while x 2 in the isotropic system this result gives a rigorous proof of the conjecture by kadanoff et al 1989 a 39 6524 37 that the anisotropy will distinguish different universality classes we introduce a new critical exponent defined by x liml t l with as for both in d 1 and 2 we obtain 1

rational approximations to the dilogarithm the irrationality proof of the values of the dilogarithmic function l 2 z at rational points z 1 k for every integer k 5 7 is given to show this we develop the method of pade type approximations using legendre type polynomials which also derives good irrationality measures of l 2 1 k moreover the linear independence over q of the numbers 1 log 1 1 k and l 2 1 k is also obtained for each integer k 5 7

diophantine properties of numbers related to catalan s constant 1 as a linear form in 1 and even beta valuesonly the following lemma 2 is almost the same as the correspondingone in br and we only include a sketch of the proof we would like toemphasise the fact that it is essentially the well poisedness of the series thatenables us to produce this parity dependant behaviour when

avalanche prediction in abelian sandpile model abstract we study the simple two dimensional height sandpile model the action of adding one grain of sand to a randomly chosen node as the first we calculate the step by step probability of occurrence of avalanches of a given size an avalanche determination method is devised that allows to determine exactly the different avalanches that can take place and the corresponding probabilities of occurrence in each step of the evolution of the sandpile

exact solutions for a mean field abelian sandpile we introduce a model for a sandpile with n sites critical height n and each site connected to every other site it is thus a mean field model in the spin glass sense we find an exact solution for the steady state probability distribution of avalanche sizes and discuss its asymptotics for large n

scaling limit of the odometer in divisible sandpiles in a recent work levine et al ann henri poincare 17 1677 1711 2016 https doi org 10 1007 s00023 015 0433 x prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian gaussian field for the discrete torus they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field in this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus

the so 3 monopole cobordism and superconformal simple type abstract we show that the so 3 monopole cobordism formula from 15 implies that all closed oriented smooth four manifolds with b 1 0 and b 3 and odd with seiberg witten simple type satisfy the superconformal simple type condition defined by marino moore and peradze 28 27 this implies the lower bound conjectured by fintushel and stern 18 on the number of seiberg witten basic classes in terms of topological data

memory on multiple time scales in an abelian sandpile abstract we report results of a numerical analysis of the memory effects in two dimensional abelian sandpiles it is found that a sandpile forgets its instantaneous configuration in two distinct stages a fast stage and a slow stage whose durations roughly scale as n and n 2 respectively where n is the linear size of the sandpile we confirm the presence of the longer time scale by an independent diagnostic based on analysing emission probabilities of a hidden markov model applied to a time averaged sequence of avalanche sizes the application of hidden markov modelling to the output of sandpiles is novel it discriminates effectively between a sandpile time series and a shuffled control time series with the same time averaged event statistics and hence deserves further development as a pattern recognition tool for abelian sandpiles

a finite difference approximation of a two layer system for growing sandpiles we study from a numerical point of view a system of partial differential equations recently proposed in k p hadeler and c kuttler granular matter 2 1999 pp 9 18 to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded table in such a system an eikonal equation for the standing layer of the pile is coupled to an advection equation for the rolling layer we analyze a suitable explicit finite difference scheme in one dimension and discuss its properties a comparison is made with respect to the variational approach of l prigozhin european j appl math 7 1996 pp 225 235 particularly in terms of steady state solutions since the effective equilibrium configuration reached by the growing process is not completely clear for this model we show experiments in one and two dimensions which characterize the steady state solutions for different source terms

self organized criticality and thermodynamic formalism we develop a thermodynamic formalism for a dissipative version of the zhang model of self organized criticality where a parameter allows us to tune the local energy dissipation by constructing a suitable markov partition we define gibbs measures in the sense of sinai ruelle and bowen partition functions and topological pressure allowing the analysis of probability distributions of avalanches we discuss the infinite size limit in this setting in particular we show that a lee yang phenomenon occurs in the conservative case this suggests new connections to classical critical phenomena

boundary height fields in the abelian sandpile model we study the abelian sandpile model on the upper half plane and reconsider the correlations of the four height variables lying on the boundary for more convenience we carry out the analysis in the dissipative massive extension of the model and identify the boundary scaling fields corresponding to the four heights we find that they all can be accounted for by the massive pertubation of a c 2 logarithmic conformal field theory comment 19 pages 3 figures

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

algebraic aspects of abelian sandpile models the abelian sandpile models feature a finite abelian group g generated by the operators corresponding to particle addition at various sites we study the canonical decomposition of g as a product of cyclic groups g z d1 z d2 z d3 z dg where g is the least number of generators of g and di is a multiple of di 1 the structure of g is determined in terms of the toppling matrix delta we construct scalar functions linear in the height variables of the pile that are invariant under toppling at any site these invariants provide convenient coordinates to label the recurrent configurations of the sandpile for an l l square lattice we show that g l in this case we observe that the system has non trivial symmetries transcending the obvious symmetries of the square namely those coming from the action of the cyclotomic galois group gall of the 2 l 1 th roots of unity which operates on the set of eigenvalues of h these eigenvalues are algebraic integers the product of which is the order mod g mod with the help of gall we are able to group the eigenvalues into certain subsets the products of which are separately integers and thus obtain an explicit factorization of mod g mod we also use gall to define other simpler sets of toppling invariants

pu 2 monopoles ii top level seiberg witten moduli spaces and witten s conjecture in low degrees in this article a continuation of 10 we complete the proof for a broad class of four manifolds of witten s conjecture that the donaldson and seiberg witten series coincide at least through terms of degree less than or equal to c 2 where c 1 4 7 11 andandare the euler characteristic and signature of the four manifold we use our computations of chern classes for the virtual normal bundles for the seiberg witten strata from the companion article 10 a comparison of all the orientations and the pu 2 monopole cobordism to compute pairings with the links of level zero seiberg witten moduli subspaces of the moduli space of pu 2 monopoles these calculations then allow us to compute low degree donaldson invariants in terms of seiberg witten invariants and provide a partial verification of witten s conjecture

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well