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

field theory conjecture for loop erased random walks we give evidence that the functional renormalization group frg developed to study disordered systems may provide a field theoretic description for the loop erased random walk lerw allowing to compute its fractal dimension in a systematic expansion in e 4 d up to two loop the frg agrees with rigorous bounds correctly reproduces the leading logarithmic corrections at the upper critical dimension d 4 and compares well with numerical studies we obtain the universal subleading logarithmic correction in d 4 which can be used as a further test of the conjecture

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

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

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

size distributions of shocks and static avalanches from the functional renormalization group interfaces pinned by quenched disorder are often used to model jerky self organized critical motion we study static avalanches or shocks defined here as jumps between distinct global minima upon changing an external field we show how the full statistics of these jumps is encoded in the functional renormalizationgroup fixed point functions this allows us to obtain the size distribution p s of static avalanches in an expansion in the internal dimension d of the interface near and above d 4 this yields the mean field distribution p s s 3 2 e s 4sm where sm is a large scale cutoff in some cases calculable resumming all one loop contributions we find p s s exp c s sm 1 2 b 4 s sm where b c and are obtained to first order in 4 d our result is consistent to o with the relation a2 2 d where is the static roughness exponent often conjectured to hold at depinning our calculation applies to all static universality classes including random bond random field and random periodic disorders extended to long range elastic systems it yields a different size distribution for the case of contact line elasticity with an exponent compatible with 2 1 d to o 2 d we discuss consequences for avalanches at depinning and for sandpile models relations to burgers turbulence and the possibility that the relation be violated to higher loop order finally we show that the avalanche size distribution on a hyperplane of codimension one is in mean field valid close to and above d 4 given by p s k1 3 s s where k is the bessel k function thus hyper plane 4

closed form solutions for analysis of cylindrically conformal microstrip antennas with arbitrary radii an efficient and accurate closed form solution based on the spectral domain method of moments mom is presented to analyze the characteristics of cylindrically conformal microstrip antennas with arbitrary radii first an algorithm that combines the recursive and the uniform asymptotic expansion method in different regions is developed to calculate the spectral domain green s functions for arbitrary dielectric coated cylinders accurately next the integrands of the spectral integrals involved in mom equation are partitioned into two parts i e the fast oscillatory slowly convergent part and the well behaved part consequently all of the integrals in mom analysis can be categorized into two types through the sommerfeld identity and its extension these integrals are cast into closed forms by using the nonlinear approximation method generalized pencil of function method in this method any time consuming numerical integration procedure is eliminated which improves the computational efficiency drastically compared with the results from published references the validity and accuracy of the method are demonstrated

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

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

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

accelerating the convergence of series representing the free space periodic green s function a method for improving the convergence of the series representing the doubly periodic free space green s function is presented the method consists of successively applying three different transformations to the green s function spectral representation kummer s transformation is first applied to convert the slowly converging spectral representation into the sum of a rapidly converging series and a slowly converging series the latter series is recognized as the spectral representation of the original periodic source distribution radiating in a medium with an imaginary wavenumber application of the poisson transformation to this series renders it exponentially convergent since it effectively represents propagation of point source contributions through a medium with imaginary wavenumber finally shanks transform is plotted versus the number of terms taken in the series numerical results confirm that an improvement in the convergence rate of the series is achieved for a particular convergence criterion

stability of learning control with disturbances and uncertain initial conditions the authors investigate the effects of state disturbances output noise and errors in initial conditions on a class of learning control algorithms they present a simple learning algorithm and exhibit via a concise proof bounds on the asymptotic trajectory errors for the learned input and the corresponding state and output trajectories furthermore these bounds are continuous functions of the bounds on the initial condition errors state disturbances and output noise and the bounds are zero in the absence of these disturbances

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

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

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

efficient analysis of printed structures of arbitrary shape on coated cylinders via spatial domain mixed potential green s function we present an approach for the method of moments mom analysis of antennas of arbitrary shape printed on a cylindrical substrate we propose several key components of a code to perform this analysis with the explicit goal of adapting mom codes devised for planar stratified structures for that purpose the background medium green s function is expressed in the mixed potential mp format to allow for efficient numerical integration the spectral dyadic green s function is first recast in the spectral counterpart of the spatial mixed potential form the convergence of the spectral to spatial transform is accelerated by extraction of spectral asymptotic functions and exploitation of the azimuthal periodicity several important issues of implementation numerical stability and accuracy are addressed with a view to incorporate the spatial mp green s function into the mom analysis of printed structures of arbitrary shape using rwg basis functions the numerical results are in good agreement with those obtained by other methods and with measured data

closed form green s function representations in cylindrically stratified media for method of moments applications closed form green s function cfgf representations for cylindrically stratified media which can be used as the kernel of an electric field integral equation are developed the developed cfgf representations can safely be used in a method of moments solution procedure as they are valid for almost all possible source and field points that lie on the same radial distance from the axis of the cylinder such as the air dielectric and dielectric dielectric interfaces including the axial line rho rho and phi phi which has not been available before in the course of obtaining these expressions the conventional spectral domain green s function representations are rewritten in a different form so that i we can attack the axial line problem and ii the method can handle electrically large cylinders available acceleration techniques that exist in the literature are implemented to perform the summation over the cylindrical eigenmodes efficiently lastly the resulting expressions are transformed to the spatial domain using the discrete complex image method with the help of the generalized pencil of function method where a modified two level approach is used numerical results are presented in the form of mutual coupling between two current modes to assess the accuracy of the final spatial domain cfgf representations

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

iterative learning control for output constrained systems with both parametric and nonparametric uncertainties in this work by proposing a barrier composite energy function bcef method with a novel barrier lyapunov function blf we present a new iterative learning control ilc scheme for a class of single input single output siso high order nonlinear systems to deal with output constrained problems under alignment condition with both parametric and nonparametric system uncertainties nonparametric uncertainties such as norm bounded nonlinear uncertainties satisfying local lipschitz condition can be effectively handled backstepping design with the newly proposed blf is incorporated in analysis to ensure output constraint not violated through rigorous analysis we show that under this new ilc scheme uniform convergence of state tracking error is guaranteed in the end an illustrative example is presented to demonstrate the efficacy of the proposed ilc scheme

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

analysis of printed antennas mounted on a coated circular cylinder of arbitrary size an efficient numerical algorithm for the analysis of printed antennas on a dielectric coated circular cylinder is presented the formulation employs the spectral domain approach green s function is evaluated via olver s uniform asymptotic expansions the fourier summation is transformed into integration and the asymptotic part of the integral is evaluated in closed form the whole procedure results in a numerical algorithm that is at least as accurate and efficient as the respective one for the planar structure the present approach can be applied for any source receiver position and any cylinder radius

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

brief iterative learning control with initial rectifying action this paper addresses the initial shift problem in iterative learning control with system relative degree the tracking error caused by nonzero initial shift is detected when applying a conventional learning algorithm finite initial rectifying action is introduced in the learning algorithm and is shown effective in the improvement of tracking performance in particular robustness with respect to variable initial shifts the uniform convergence of the output trajectory to a desired one jointed smoothly with a specified transient trajectory from the starting position is ensured in the presence of fixed initial shift

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

two dimensional green s function for scattering and radiation problems in elliptically layered media a recursive solution for the computation of the two dimensional green s function for the multilayer elliptic cylinder is reported it is based on the scattering solution by this canonical scatterer through eigenfunction expansions in terms of mathieu functions the paper reports the mathematical details of the approach as well as several validation tests and comparisons

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

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

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

scaling of loop erased walks in 2 to 4 dimensions we simulate loop erased random walks on simple hyper cubic lattices of dimensions 2 3 and 4 these simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in d 4 simulations in lower dimensions were done for completeness and in order to test the algorithm in d 2 we verify with high precision the prediction d 5 4 where the number of steps n after erasure scales with the number n of steps before erasure as n nd 2 in d 3 we again find a power law but with an exponent different from the one found in the most precise previous simulations d 1 6236 0 0004 finally we see clear deviations from the naive scaling n n in d 4 while they agree only qualitatively with the leading logarithmic corrections predicted by several authors their agreement with the two loop prediction is nearly perfect

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

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

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

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

an iterative learning control approach for linear systems with randomly varying trial lengths this technical note addresses an iterative learning control ilc design problem for discrete time linear systems where the trial lengths could be randomly varying in the iteration domain an ilc scheme with an iteration average operator is introduced for tracking tasks with non uniform trial lengths which thus mitigates the requirement on classic ilc that all trial lengths must be identical in addition the identical initialization condition can be absolutely removed the learning convergence condition of ilc in mathematical expectation is derived through rigorous analysis as a result the proposed ilc scheme is applicable to more practical systems in the end two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ilc scheme for both time invariant and time varying linear systems

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

a generalized iterative learning controller against initial state error in this paper the previous results that the performance of iterative learning control ilc algorithm can be improved by adding a proportional term and or an integral term of error in d type ilc a

an average operator based pd type iterative learning control for variable initial state error this note studies the effect of variable initial state error in iterative learning control ilc systems and proposes a new ilc algorithm based on an average operator then it is shown that when the proposed algorithm is applied to linear time invariant lti systems the effect of the initial state error can be exactly estimated under a specific condition while the existing algorithms guarantee only the boundness of the error or the convergence from stochastic point of view to show the validity of the proposed algorithm a numerical example is given

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

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

an efficient asymptotic extraction approach for the green s functions of conformal antennas in multilayered cylindrical media asymptotic expressions are derived for the dyadic green s functions of antennas radiating in the presence of a multilayered cylinder where analytic representation of the asymptotic expansion coefficients eliminates the computational cost of numerical evaluation as a result the asymptotic extraction technique has been applied only once for a large summation order in addition the hankel function singularity encountered for source and evaluation points at the same radius has been eliminated using analytical integration

green s functions for probe fed arbitrary shaped cylindrical microstrip antennas the efficiency of a method of moments procedure for microstrip antennas is strongly dependent on the convergence of the inverse fourier transformation ift from the spectral to the spatial domain this is especially true when considering cylindrical microstrip antennas cmas in the cylindrical case the calculation of the two dimensional green s functions stays inherently time consuming in the planar case green s function can be reduced to a one dimensional function the problem however becomes a real bottle neck when probe feeding of a cma is considered as the traditional ift used to calculate the excitation in the spatial domain results in a huge ift time in this paper this problem is explicitly treated the asymptotical behavior of the probe related spectral green s functions gfs is rigorously derived subtracted and the corresponding reaction integrals are inverse fourier transformed without the occurrence of any cylindrical functions this results in a much improved calculation efficiency for this type of feeding evident for both thin and thick substrates numerical results for cmas with complex shaped patches are presented applicable to both smaller and larger bandwidths several structures are fabricated and measurements are in good agreement with simulations

stochastic p type d type iterative learning control algorithms this paper presents stochastic algorithms that compute optimal and sub optimal learning gains for a p type iterative learning control algorithm ilc for a class of discrete time varying linear systems the optimal algorithm is based on minimizing the trace of the input error covariance matrix the state disturbance reinitialization errors and measurement errors are considered to be zero mean white processes it is shown that if the product of the input output coupling matrices c t 1 b t is full column rank then the input error covariance matrix converges to zero in presence of uncorrelated disturbances another sub optimal p type algorithm which does not require the knowledge of the state matrix is also presented it is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub optimal p type and d type algorithms are equivalent and all converge to zero at a rate inversely proportional to the number of learning iterations a transient response pe

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

analysis and measurements of conformal patch array antennas on multilayer circular cylinder in this paper we present a method of moments mom program to analyze different configurations of arrays of cylindrical rectangular patches the patches can be located inside or on the surface of a multilayer circular cylindrical structure and can be arbitrarily rotated the antenna structure is rigorously taken into account by using proper green s functions and the array is analyzed by using an element by element approach the elements of the mom matrix are calculated in the spectral domain and special attention is given to their numerical treatment when analyzing cylindrical antennas with large radii a laboratory model is built to test different configurations of patches and to validate calculated results the agreement between measurements and calculations is very good

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

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

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

on the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths abstract in this paper a new class of stochastic impulsive differential equations involving bernoulli distribution is introduced for tracking the random discontinuous trajectory a modified tracking error associated with a piecewise continuous variable by zero order holder is defined in the sequel a new random ilc scheme by adopting global and local iteration average operators is designed too sufficient conditions to guarantee the convergence of modified tracking error are obtained by using the tools of mathematical analysis via an impulsive gronwall inequality finally two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ilc scheme to track the random discontinuous trajectory

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

further improvement for fast computation of mixed potential green s functions for cylindrically stratified media a method of fast computation of the mixed potential green s functions mpgfs in spatial domain for cylindrically stratified media has been developed based on the convergence behavior of the mpgfs in spectral domain the discrete complex image method dcim and the extrapolation technique are employed to perform the inverse fourier transform which is different from that of planarly stratified media in order to evaluate the summation of cylindrical eigenmode from order of negative infinity to order of positive infinity the green s functions in spectral domain are rearranged so that acceleration technique can be well applied to speed up the convergence of summation process for the integral along the deformed sommerfeld integral path with respect to k sub z dcim is used to speed up the computation after the quasistatic components are completely subtracted from the spectral domain green s functions the inverse transform of the remaining parts has an approximated analytical form the proposed method can be used to accurately and efficiently calculate the spatial domain mpgfs which are useful for characterizing conformal patch structures mounted on cylindrically stratified media

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

efficient analysis of input impedance and mutual coupling of microstrip antennas mounted on large coated cylinders an efficient and accurate hybrid method based on the combination of the method of moments mom with a special green s function in the space domain is presented to analyze antennas and array elements conformal to electrically large material coated circular cylinders the efficiency and accuracy of the method depend strongly on the computation of the green s function which is the kernel of the integral equation that is solved via mom for the unknown equivalent currents representing only the antenna elements three types of space domain green s function representations are used each accurate and computationally efficient in a given region of space consequently a computationally optimized analysis tool for conformal microstrip antennas is obtained input impedance of various microstrip antennas and mutual coupling between two identical antennas are calculated and compared with published results to assess the accuracy of this hybrid method

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