inverse avalanches in the abelian sandpile model we define and study the inverse of particle addition process in the abelian sandpile model we show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches we study the probability distribution of s 1 the number of untopplings in the first inverse avalanche for a square lattice we determine prob s 1 exactly for s 1 0 1 2 and 3 for large s 1 we show that prob s 1 varies as s 1 11 8 in the direct avalanches this is related to the probability distribution of the number of sites which topple as often as the site where the particle was added these results are verified by numerical simulations

scaling and river networks a landau theory for erosion we propose a coarse grained theory for the formation of a river network in the form of a langevin equation for the erosion of the landscape coupled to a conservation law for the surface water flow we claim that this is the universal form for large scale behavior we show by simulations of a discrete model that represents the same dynamics that the slope area law the basin size distribution law and horton s laws agree with real rivers we discuss the relationship to optimal channel networks and to self organized criticality

sandpiles on a sierpinski gasket abstract we perform extensive simulations of the sandpile model on a sierpinski gasket critical exponents for waves and avalanches are determined we extend the existing theory of waves to the present case this leads to an exact value for the exponent w which describes the distribution of wave sizes w ln 9 5 ln 3 numerically it is found that the number of waves in an avalanche is proportional to the number of distinct sites toppled in the avalanche this leads to a conjecture for the exponent that determines the distribution of avalanche sizes 1 w ln 27 5 ln 3 our predictions are in good agreement with the numerical results

necessary and sufficient conditions for oscillation of second order neutral differential equations consider the second order nonlinear neutral differential equation with delays e d2 dt2 y t py t q t f y t 0 for t 0 where q t f x are continuous functions q t 0 yf y 0 if y 0 and 0 0 0 when f y satisfies either the superlinear or sublinear conditions which include the special case f y y y 1 of 1 and 0 1 respectively we give necessary and sufficient conditions for the oscillation of all continuable solutions of e when p 0 in e these results reduce to the well known theorems of atkinson and belohorec in the special case when f y y y 1 1

expansion and contraction of avalanches in the two dimensional abelian sandpile we present a detailed analysis of large scale simulations of avalanches in the 2d abelian sandpile model we compare statistical properties of two different decompositions of avalanches into clusters of topplings and waves of topplings auxiliary critical exponents are introduced and the existence of the exponent governing the contraction of avalanches claimed in our previous work priezzhev et al prl 76 2093 1996 is confirmed we also give more elaborated argumentation for the exact values of the exponents characterizing the statistics of waves

time consistent non zero sum stochastic differential reinsurance and investment game under default and volatility risks abstract this paper investigates a non zero sum stochastic differential game between two mean variance insurers these two insurers are concerned about their terminal wealth and the relative performance compared with each other we assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk free asset a stock and a defaultable bond the price process of stock is driven by the constant elasticity of variance cev model and the defaultable bond recovers a proportion of value at default so these two insurers are faced with insurance risk volatility risk and default risk the non zero sum goal of these insurers is to maximize the mean variance utility of a weighted value of their terminal and relative wealths we solve the mean variance problem in the time consistent case and establish the extended hamilton jacobi bellman systems for the post default case and the pre default case respectively furthermore we derive the closed form solutions of the nash equilibrium reinsurance and investment strategies for these two insurers in the end of this paper we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies

the infinite volume limit of dissipative abelian sandpiles we construct the thermodynamic limit of the stationary measures of the bak tang wiesenfeld sandpile model with a dissipative toppling matrix sand grains may disappear at each toppling we prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic markov process leaving it invariant we show how to extend the dhar formalism of the abelian group of toppling operators to infinite volume in order to obtain a compact abelian group with a unique haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches

critical dynamics of self organizing eulerian walkers the model of self organizing eulerian walkers is numerically investigated on the square lattice the critical exponents for the distribution of a number of steps tau l and visited sites tau s characterizing the process of transformation from one recurrent configuration to another are calculated using the finite size scaling analysis two different kinds of dynamical rules are considered the results of simulations show that both the versions of the model belong to the same class of universality with the critical exponents tau l tau s 1 75 pm 0 1

universality in sandpile models a classification of sandpile models into universality classes is presented on the basis of extensive numerical simulations in which we measure an extended set of exponents the manna two state model s s manna j phys a 24 l363 1991 is found to belong to a universality class of random neighbor models which is distinct from the universality class of the original model of bak tang and wiesenfeld p bak c tang and k wiesenfeld phys rev lett 59 381 1987 directed models are found to belong to a universality class which includes the directed model introduced and solved by dhar and ramaswamy d dhar and r ramaswamy phys rev lett 63 1659 1989

waves of topplings in an abelian sandpile abstract we study the structure of the avalanche process in the 2d abelian sandpile model an avalanche may be represented as a sequence of waves each consisting of sites that toppled only once in that wave it is shown that the waves are in one to one correspondence with the set of two rooted spanning trees and can be described in terms of green functions we argue that the probability distribution of waves of size s varies as 1 s for large s we prove also that the avalanches started at the open boundary consist of only one wave and find the asymptotic distribution of their sizes which varies as s 3 2 we establish the equivalence between waves and inverse avalanches introduced by dhar and manna and reproduce their result 11 8 for the critical exponent of the size distribution of the first inverse avalanche or in our terms the last wave

distribution of sizes of erased loops for loop erased random walks the loop erased random walk lerw is a simpler variant of the well known self avoiding walk saw problem which retains the no self intersection property of saws but is closer to the ordinary random walk problem in this paper we study the distribution of sizes of erased loops for lerws on arbitrary graphs and relate it to the distribution of sizes of loops formed when a bond is added to a random spanning tree on the same graph in the themodynamic limit this distribution has a power law tail and we express the exponent in terms of the fractal dimension of the chemical paths on spanning trees we show also how this relation is modified for deterministic fractals in spite of the fact that the lerw model is somewhat more tractable analytically than the saw problem the number of papers devoted to this problem has remained rather small the model was defined by lawler 1 who called it the loop erased self avoiding walk lesaw this terminology is somewhat inappropriate and

conformal field theory correlations in the abelian sandpile model we calculate all multipoint correlation functions of all local bond modifications in the two dimensional abelian sandpile model both at the critical point and in the model with dissipation the set of local bond modifications includes as the most physically interesting case all weakly allowed cluster variables the correlation functions show that all local bond modifications have scaling dimension 2 and can be written as linear combinations of operators in the central charge ensuremath 2 logarithmic conformal field theory in agreement with a form conjectured earlier by mahieu and ruelle in phys rev e 64 066130 2001 we find closed form expressions for the coefficients of the operators and describe methods that allow their rapid calculation we determine the fields associated with adding or removing bonds both in the bulk and along open and closed boundaries some bond defects have scaling dimension 2 while others have scaling dimension 4 we also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries

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

mean field self organized criticality the authors consider a variant of directed percolation in which the set of nodes is itself dependent on the progress of the percolation process for the mean field model studied the system is driven to its critical point from a broad range of initial conditions an advantage of the model is its explicit and easy solvability

emergence of quasiunits in the one dimensional zhang model we study the zhang model of sandpile on a one dimensional chain of length l where a random amount of energy is added at a randomly chosen site at each time step we show that in spite of this randomness in the input energy the probability distribution function of energy at a site in the steady state is sharply peaked and the width of the peak decreases as l 1 2 for large l we discuss how the energy added at one time is distributed among different sites by topplings with time we relate this distribution to the time dependent probability distribution of the position of a marked grain in the one dimensional abelian model with discrete heights we argue that in the large l limit the variance of energy at site x has a scaling form l 1 g x l where g xi varies as ln 1 xi for small xi which agrees very well with the results from numerical simulations

exact solution of a stochastic directed sandpile model we introduce and analytically solve a directed sandpile model with stochastic toppling rules the model clearly belongs to a different universality class from its counterpart with deterministic toppling rules previously solved by dhar and ramaswamy the critical exponents are d 7 4 ensuremath tau 10 7 in two dimensions and d 3 2 ensuremath tau 4 3 in one dimension the upper critical dimension of the model is three at which the exponents apart from logarithmic corrections reach their mean field values d 2 ensuremath tau 3 2

self organized critical state in a directed sandpile automaton on bethe lattices equivalence to site percolation the author considers a directed sandpile automaton on the bethe lattice and shows that the system evolves ergodically to every configuration with equal probability irrespective of the initial state the author makes it clear that the system is driven into the self organized critical state which is in equivalence to site percolation at the critical percolation threshold on the bethe lattice

time consistent investment reinsurance strategy for mean variance insurers with a defaultable security abstract this paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean variance criterion in a jump diffusion risk model the insurer can purchase proportional reinsurance or acquire new insurance business and invest in a financial market consisting of a risk free asset a stock and a defaultable bond in particular the correlation between the insurance risk model and the financial market is also considered from a game theoretic perspective the extended hamilton jacobi bellman systems of equations are established for the post default case and the pre default case respectively in both cases closed form expressions for the optimal time consistent investment reinsurance strategies and the corresponding value functions are derived moreover some properties of optimal strategies value functions and efficient frontiers are discussed either analytically or numerically

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

critical exponents of the sand pile models in two dimensions abstract we study three sand pile automaton models namely the critical height critical slope and the critical laplacian models in two dimensions in which the stability criterion of the sand columns depend on the zeroth first and the second derivatives of the sand height function we carried out simulations on system sizes up to 2048 2048 and up to 108 avalanches were generated the exponents of the critical height model were calculated by taking into account the strong corrections to scaling in order to determine the exponents of the critical laplacian model accurately we introduced a height restriction in the toppling criterion that maintains universality but accelerates convergence to the steady state by orders of magnitude we see clear scaling in the critical height and the critical laplacian models and find that they belong to different universality classes however we do not find any scaling in the critical slope model

unified view of scaling laws for river networks scaling laws that describe the structure of river networks are shown to follow from three simple assumptions these assumptions are 1 river networks are structurally self similar 2 single channels are self affine and 3 overland flow into channels occurs over a characteristic distance drainage density is uniform we obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent we further demonstrate that the two predominant descriptions of network structure tokunaga s law and horton s laws are equivalent in the case of landscapes with uniform drainage density the results are tested with data from both real landscapes and a special class of random networks

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

steady state of stochastic sandpile models we study the steady state of the abelian sandpile models with stochastic toppling rules the particle addition operators commute with each other but in general these operators need not be diagonalizable we use their abelian algebra to determine their eigenvalues and the jordan block structure these are then used to determine the probability of different configurations in the steady state we illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example for systems of size 12 and also study the density profile in the steady state

sandpile model on the sierpinski gasket fractal we investigate the sandpile model on the two dimensional sierpinski gasket fractal we find that the model displays interesting critical behavior and we analyze the distribution functions of avalanche sizes lifetimes and topplings and calculate the associated critical exponents t51 5160 04 a51 6360 04 and m51 3660 04 the avalanche size distribution shows power law behavior modulated by logarithmic oscillations which can be related to the discrete scale invariance of the underlying lattice such a distribution can be formally described by introducing a complex scaling exponent t t1id where the real part t corresponds to the power law and the imaginary part d is related to the period of the logarithmic oscillations s1063 651x 96 03907 4

self organized critical pinball machine abstract the nature of self organized criticality soc is pin pointed with a simple mechanical model a pinball machine its phase space is fully parameterized by two integer variables one describing the state of an on going game the other describing the state of the machine this is the simplest possible soc system having only two degrees of freedom and no spatial correlations yet is not solvable by analytical means

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

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

area distribution for directed random walks we study the probability distribution for the area under a directed random walk in the plane the walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed this model captures some of the qualitative features of earthquakes avalanches and other self organized critical phenomena in one dimension by finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4 3

geometry of river networks i scaling fluctuations and deviations this paper is the first in a series of three papers investigating the detailed geometry of river networks branching networks are a universal structure employed in the distribution and collection of material large scale river networks mark an important class of two dimensional branching networks being not only of intrinsic interest but also a pervasive natural phenomenon in the description of river network structure scaling laws are uniformly observed reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists to improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure here we report a theoretical and empirical study of fluctuations about and deviations from scaling we examine data for continent scale river networks such as the mississippi and the amazon and draw inspiration from a simple model of directed random networks we center our investigations on the scaling of the length of a subbasin s dominant stream with its area a characterization of basin shape known as hack s law we generalize this relationship to a joint probability density and provide observations and explanations of deviations from scaling we show that fluctuations about scaling are substantial and grow with system size we find strong deviations from scaling at small scales which can be explained by the existence of a linear network structure at intermediate scales we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate at large scales we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape the extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution

the upper critical dimension of the abelian sandpile model the existing estimation of the upper critical dimension of the abelian sandpile model is based on a qualitative consideration of avalanches as self avoiding branching processes we find an exact representation of an avalanche as a sequence of spanning subtrees of two component spanning trees using equivalence between chemical paths on the spanning tree and loop erased random walks we reduce the problem to determination of the fractal dimension of spanning subtrees then the upper critical dimension du 4 follows from lawler s theorems for intersection probabilities of random walks and loop erased random walks

distribution of sizes of erased loops of loop erased random walks in two and three dimensions we show that in the loop erased random walk problem the exponent characterizing probability distribution of areas of erased loops is superuniversal in d dimensions the probability that the erased loop has an area a varies as a 2 for large a independent of d for 2 d 4 we estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d 2 and 3 by large scale monte carlo simulations our estimate of the fractal dimension z in two dimensions is consistent with the known exact value 5 4 in three dimensions we get z 1 6183 0 0004 the exponent for the distribution of durations of avalanche in the three dimensional abelian sandpile model is determined from this by using scaling relations

optimal investment policies for a firm with a random risk process exponential utility and minimizing the probability of ruin we consider a firm that is faced with an uncontrollable stochastic cash flow or random risk process there is one investment opportunity a risky stock and we study the optimal investment decision for such firms there is a fundamental incompleteness in the market in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy since the random risk process ensures that there is always a positive probability of ruin bankruptcy we therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin in particular we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level this policy also minimizes the probability of ruin we prove that when there is no risk free interest rate this policy is equivalent to the policy that maximizes utility from terminal wealth for a fixed terminal time when the firm has an exponential utility function this validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility when there is a positive risk free interest rate the conjecture is shown to be false we also solve for the optimal policy for the related objective of minimizing the expected discounted penalty paid upon ruin

scaling of waves in the bak tang wiesenfeld sandpile model n with aen ae0 n a similar ansatzes should according to this view hold for the number of topplings s which difiers from a because sites can topple more than once in an avalanche and for the radius and duration of avalanches but recent inves tigations 10 12 show that the two dimensional btw model may be characterized by a multifractal behavior where difierent moments of a are governed by exponents aen which are not linear in n and are indeed not related to each other for difierent n difierent reasons for this have been proposed in 10 11 and in 12 notice that multi fractality of avalanches can be proven for certain variants of the 1 d sandpile model 14 deviations from pure power laws had been seen already in early simulations but were usually interpreted as fl nite size efiects due to avalanches which touch the bound ary of the lattice to illustrate that this is most likely not true and that there is a real problem with simple scaling we show in fig 1 the ratio ps x l pa x l of the integrated distributions ps x l r 1 x dx 0 ps x 0 l for d 2 and for difierent values of l in these simula tions cylindrical boundary conditions were used open at y l 2 and periodic at x 0 l and data were col lected only for avalanches starting at y 0 in this way we hope to have minimized boundary efiects also since we do not make separate flts to ps x l and pa x l we have none of the uncertainties inherent in such flts due

oscillatory behavior of higher order nonlinear neutral forced differential equations with oscillating coefficients in this paper the higher order nonlinear neutral forced differential equations of the form x t p t x t t n i 1mq i t f i x s i t s t t t 0 where n m n n 2 and m 1 are studied a new necessary and sufficient theorem for the oscillation of bounded solutions and a sufficient criterion for the existence of bounded positive solutions of the above equations are obtained for general p t and s t in particular our results improve some known results

rare events and breakdown of simple scaling in the abelian sandpile model due to intermittency and conservation the abelian sandpile in two dimensions obeys multifractal rather than finite size scaling in the thermodynamic limit a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution thus rare events shape most of the scaling pattern and preserve a meaning for effective exponents which can be determined on the basis of numerical and exact results

self organizing height arrow model numerical and analytical results the recently introduced self organizing height arrow ha model is studied numerically on the square lattice and analytically on the bethe lattice the concentration of occupied sites and critical exponents of distributions of avalanches are evaluated for two slightly different versions of the model the exponents found for distributions of avalanches by mass area duration and appropriate fractal dimensions are close to those for the original sandpile model which suggests that the ha model belongs to the same universality class for comparison the concentration of occupied sites in the ha model is calculated exactly on the bethe lattice with coordination number q 4

critical exponents for boundary avalanches in two dimensional abelian sandpile we investigate the properties of boundary avalanches in 2d abelian sandpile model asm we construct the one to one correspondence of boundary avalanches and two rooted spanning trees using the connection between the obtained graph representation and lattice green functions we calculate the exact values of critical exponents for size and lifetime distributions of avalanches starting at the open boundary that forms an angle alpha we find that the probability of a boundary avalanche of the size s varies as s 1 pi 2 alpha for large s and the probability of an avalanche of lifetime t varies as t 1 4 pi 5 alpha for large t the obtained values are verified by numerical simulations

the noise spectrum in the model of self organised criticality the explanation of 1 f noise with the help of models of self organised criticality makes use of combining scaling and the distributed time constant picture limits of applicability of this type of arguments are presented the numerical study of the two dimensional model of bak tang and wiesenfeld 1987 shows that although non trivial scaling is present the power spectrum of the total process has a approximately 1 f2 behaviour

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

robust optimal investment and reinsurance of an insurer under variance premium principle and default risk abstract this paper considers a robust optimal investment and reinsurance problem under model ambiguity and default risk for an insurer who can trade in a saving account a stock and a defaultable bond and aims to maximize the minimal expected cara utility the surplus process of the insurer is assumed to follow the cramer lundberg model in particular both the insurance and reinsurance premium are assumed to be calculated via the variance premium principle by using the dynamic programming approach we study the pre default case and post default case respectively then closed form expressions for the optimal strategies and the corresponding value function are derived finally numerical examples are given to illustrate our main results and we discuss relevant economic insights obtained from these results

self organized criticality in a sandpile model with threshold dissipation we study a nonconservative sandpile model in one dimension in which if the height at any site exceeds a threshold value the site topples by transferring one particle along each bond connecting it to its neighbors its height is then set to 1 irrespective of the initial value the model shows nontrivial critical behavior we solve this model analytically in one dimension for all driving rates we calculate all the two point correlation functions in this model and find that the average local height decreases as the inverse of the distance from the nearest boundary and the power spectrum of fluctuations of the total mass varies as 1 f

eulerian walkers as a model of self organized criticality we propose a new model of self organized criticality a particle is dropped at random on a lattice and moves along directions specified by arrows at each site as it moves it changes the direction of the arrows according to fixed rules on closed graphs these walks generate euler circuits on open graphs the particle eventually leaves the system and a new particle is then added the operators corresponding to particle addition generate an abelian group same as the group for the abelian sandpile model on the graph we determine the critical steady state and some critical exponents exactly using this equivalence

retracted nonoscillatory solutions of second order nonlinear neutral delay equations in this paper we study nonoscillatory solutions of a class of second order nonlinear neutral delay differential equations with positive and negative coefficients some sufficient conditions for existence of nonoscillatory solutions are obtained

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

oscillatory behaviour of higher order neutral type nonlinear forced differential equation with oscillating coefficients abstract oscillation criteria are given for higher order neutral type nonlinear forced differential equation with oscillating coefficients of the form y t p t y t n i 1 m q i t f i y i t s t

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

asymmetric abeiian sandpile models in the abelian sandpile models introduced by dhar long time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system dhar proposed a simple procedure theburning algorithm as a possible test of whether a configuration is recurrent and later with majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric we observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles we also obtain necessary and sufficient conditions for the validity of the original test the results are applied to a family of deterministic one dimensional sandpile models originally studied by lee liang and tzeng

analysis and computation of an optimality equation arising in an impulse control problem with discrete and costly observations abstract the framework of discrete and costly observations provides an attractive mathematical tool for stochastic control of biological and ecological systems in natural environment however this framework is still germinating and its applications in these research areas are still rare in this framework the system dynamics to be controlled is continuous time while the observations by the decision maker are discrete time the control problem of the system dynamics has a time hybridized type nature by which the optimality equation has a different form from the conventional ones the objectives as well as contributions of this paper are formulation of the optimality equation focusing on population management problems its detailed mathematical analysis numerical discretization and a demonstrative application we show that the optimality equation is solvable uniquely and its solution can be approximated recursively a numerical method to discretize the optimality equation is established with a simple finite difference scheme and the recursive formula mathematical analysis results such as monotonicity stability and convergence of the numerical method are discussed as well the present model is then applied to a population management problem finally an advanced problem subject to model ambiguity is also analyzed

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

robust optimal proportional reinsurance and investment strategy for an insurer with defaultable risks and jumps abstract in this paper we consider a robust optimal proportional reinsurance and investment problem in a model with default risks and jumps for an insurer whose objective is to maximize the expected exponential utility of terminal wealth the surplus process of the insurer is assumed to follow a brownian motion with drift which is an approximation of the classical compound poisson risk model assume that the insurer is allowed to purchase proportional reinsurance and invest in a financial market which consists of a risk free asset a defaultable bond and a risky asset whose price process is governed by a jump diffusion model using stochastic control approach we establish the robust hamilton jacobi bellman equations for the post default case and the pre default case respectively furthermore we derive the closed form expressions of the optimal strategies and the corresponding value functions in both cases finally we provide numerical examples to illustrate the effects of some model parameters on the robust optimal strategies

scaling optimality and landscape evolution a nonlinear model is studied which describes the evolution of a landscape under the effects of erosion and regeneration by geologic uplift by mean of a simple differential equation the equation already in wide use among geomorphologists and in that context obtained phenomenologically is here derived by reparametrization invariance arguments and exactly solved in dimension d 1 results of numerical simulations in d 2 show that the model is able to reproduce the critical scaling characterizing landscapes associated with natural river basins we show that configurations minimizing the rate of energy dissipation optimal channel networks are stationary solutions of the equation describing the landscape evolution numerical simulations show that a careful annealing of the equation in the presence of additive noise leads to configurations very close to the global minimum of the dissipated energy characterized by mean field exponents we further show that if one considers generalized river network configurations in which splitting of the flow i e braiding and loops are allowed the minimization of the dissipated energy results in spanning loopless configurations under the constraints imposed by the continuity equations this is stated in the form of a general theorem applicable to generic networks suggesting that other branching structures occurring in nature may possibly arise as optimal structures minimizing a cost function

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

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

some results and a conjecture for manna s stochastic sandpile model we present some analytical results for the stochastic sandpile model studied earlier by manna in this model the operators corresponding to particle addition at different sites commute the eigenvalues of operators satisfy a system of coupled polynomial equations for an l l square we construct a nontrivial toppling invariant and hence a ladder operator which acting on eigenvectors of the evolution operator gives new eigenvectors with different eigenvalues for periodic boundary conditions in one direction one more toppling invariant can be constructed we show that there are many forbidden subconfigurations and only an exponentially small fraction of all stable configurations are recurrent we obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration and conjecture a formula for its exact value for finite size rectangles

abelian sandpile model on the bethe lattice the authors study bak tang and wiesenfeld s abelian sandpile model 1987 of self organised criticality on the bethe lattice exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained they also determine the probability distribution of the number of distinct sites that topple at least once the number of toplings at the origin and the total number of toplings in an avalanche the probability that an avalanche consists of more than n toplings varies as n 1 2 for large n the probability that its duration exceeds t decreases as 1 t for large t these exponents are the same as for the critical percolation clusters in mean field theory

optimal investment and proportional reinsurance for a jump diffusion risk model with constrained control variables this paper considers the optimal control problem with constraints for an insurer the risk process is assumed to be a jump diffusion process and the risk can be reduced through a proportional reinsurance in addition the surplus can be invested in the financial market consists of one risk free asset and one risky asset the diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims the objective of the insurer is to maximize the expected exponential utility of terminal wealth this optimization problem is studied in two cases depending on the diffusion term s explanations in all cases with normal constraints on the control variables the value functions and the corresponding optimal strategies are given in a closed form numerical simulations are presented to illustrate the effects of parameters on the optimal strategies as well as the economic meaning behind

numerical determination of the avalanche exponents of the bak tang wiesenfeld model we consider the bak tang wiesenfeld sandpile model on a two dimensional square lattice of lattice sizes up to l 4096 a detailed analysis of the probability distribution of the size area duration and radius of the avalanches will be given to increase the accuracy of the determination of the avalanche exponents we introduce a new method for analyzing the data which reduces the finite size effects of the measurements the exponents of the avalanche distributions differ slightly from previous measurements and estimates obtained from a renormalization group approach

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

self organized formulation of standard percolation phenomena abstract we consider standard percolation processes such as epidemic processes with or without immunization we show that their dynamics can be formulated so that they mimic self organized critical phenomena the wetting probability p needs not to be fine tuned to its critical value p c in order to arrive at criticality but it rather emerges as a singularity in some time dependent distribution on the one hand this casts doubts on the significance of self organized as opposed to ordinary criticality on the other hand it suggests very efficient algorithms where percolation problems are studied at several values of p in a single run as an example we apply such an algorithm to directed percolation in 2 1 dimensions where it allows a very precise determination of critical behavior

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

oscillation of second order nonlinear neutral differential equations abstract in this paper some sufficient conditions for oscillation and nonoscillation are obtained for the second order nonlinear neutral differential equation x t p t x t q t f x t 0 t 0 where 0 p t 1 q t 0 0

absorbing state phase transition for driven dissipative stochastic dynamics on ℤ we study the long time behavior of conservative interacting particle systems in the activated random walk model for reaction diffusion systems and the stochastic sandpile we prove that both systems undergo an absorbing state phase transition

structure of two dimensional sandpile i height probabilities the height probabilities of the two dimensional abelian sandpile model are the fractionial numbers of lattice sites having heights 1 2 3 4 a combinatorial method for evaluation of these quantities is proposed the method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice exact analytical expressions for all probabilities are obtained

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

logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension we study numerically the dynamical properties of the btw model on a square lattice for various dimensions the aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean field exponents our results are consistent with the assumption that the scaling behavior of the four dimensional btw model is characterized by the mean field exponents with additional logarithmic corrections we benefit in our analysis from the exact solution of the directed btw model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension similar logarithmic corrections forms fit the numerical data for the four dimensional btw model strongly suggesting that the value of the upper critical dimension is four