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

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

path integration of the duffing rayleigh oscillator subject to harmonic and stochastic excitations this paper is focused on the duffing rayleigh oscillator subject to harmonic and stochastic excitations using path integration based on the gauss legendre integration scheme first applying methods of harmonic balance and multiple scales to the deterministic case the stabilities of the responses can be analyzed then the steady state periodic solution of probability density can be captured via path integration at the same time the changes of probability density induced by the intensities of harmonic and stochastic excitations are discussed in three cases

worst case to average case reductions based on gaussian measures we show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice the lattice problems we consider are the shortest vector problem the shortest independent vectors problem the covering radius problem and the guaranteed distance decoding problem a variant of the well known closest vector problem the approximation factor we obtain is n log o 1 n for all four problems this greatly improves on all previous work on the subject starting from ajtai s seminal paper generating hard instances of lattice problems in complexity of computations and proofs quad mat 13 dept math seconda univ napoli caserta italy 2004 pp 1 32 up to the strongest previously known results by micciancio siam j comput 34 2004 pp 118 169 our results also bring us closer to the limit where the problems are no longer known to be in np intersect conp our main tools are gaussian measures on lattices and the high dimensional fourier transform we start by defining a new lattice parameter which determines the amount of gaussian noise that one has to add to a lattice in order to get close to a uniform distribution in addition to yielding quantitatively much stronger results the use of this parameter allows us to simplify many of the complications in previous work our technical contributions are twofold first we show tight connections between this new parameter and existing lattice parameters one such important connection is between this parameter and the length of the shortest set of linearly independent vectors second we prove that the distribution that one obtains after adding gaussian noise to the lattice has the following interesting property the distribution of the noise vector when conditioning on the final value behaves in many respects like the original gaussian noise vector in particular its moments remain essentially unchanged

stochastic averaging of strongly non linear oscillators under combined harmonic and white noise excitations abstract a stochastic averaging procedure of strongly non linear oscillators subject to external and or parametric excitations of both harmonic and white noise forces is developed by using the so called generalized harmonic functions the procedure is applied to a duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises the averaged fokker planck kolmogrov equation is solved by using the path integration method based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration the stochastic jump of the duffing oscillator under combined harmonic and white noise excitations and its bifurcation as the system parameters frequency ratio strength of the non linearity amplitude of harmonic excitation and intensity of white noise change are examined for the first time

exact stationary solutions of averaged equations of stochastically and harmonically excited mdof quasi linear systems with internal and or external resonances abstract the exact stationary solutions of the averaged equations of stochastically and harmonically excitedndegree of freedom quasi linear systems withminternal and or external resonances are obtained as functions of bothnindependent amplitudes andmcombinations of phase angles to make the solutions more general the equivalent stochastic systems of the averaged equations are obtained by using the differential forms and exterior differentiation by considering the periodic boundary conditions with respect tomcombinations of phase angles the probability potentials of the exact stationary solutions of the equivalent stochastic systems are expanded into anm fold harmonic series ofmcombinations of phase angles and the exact stationary solutions are obtained for the case where the averaged equations belong to the class of stationary potential to examples are given to illustrate the application of the proposed procedure

mixing time and eigenvalues of the abelian sandpile markov chain the abelian sandpile model defines a markov chain whose states are integer valued functions on the vertices of a simple connected graph g by viewing this chain as a nonreversible random walk on an abelian group we give a formula for its eigenvalues and eigenvectors in terms of multiplicative harmonic functions on the vertices of g we show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual laplacian lattice while the mixing time is at most a constant times the smoothing parameter of the laplacian lattice we find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on g if the latter has a sufficiently large spectral gap then the former has a small gap in the case where g is the complete graph on n vertices we show that the sandpile chain exhibits cutoff at time 1 4 2 n logn

the criterion of the center focus for a class of planar perturbed polynomial systems with 2n 1 th degree abstract in this paper we investigate the problem of center focus for the system d x d t y k 1 n p 2 k 1 x y d y d t x k 1 n q 2 k 1 x y which is regarded as a perturbed one of a planar linear system d x d t y d y d t x where p 2 k 1 x y q 2 k 1 x y k 1 2 n are 2 k 1 th degree homogeneous polynomials in x y we shall give a simple and convenient method which can immediately distinguish that the singular point o is a center or fine focus and the stability of the singular point can be determined by the matrices consist of the coefficients of perturbed terms

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

response of nonlinear systems in probability domain using stochastic averaging determination of probability density function pdf of the response for strongly nonlinear single degree of freedom system subjected to both multiplicative and additive random excitations using stochastic averaging technique is faced with few difficulties firstly the size of excitations should be small such that the response of the system converges weakly to a markov process secondly the excitations should preferably be broad banded so that the analytical results are reasonably accurate and finally the nonlinear functions should be integrable if closed formed expressions for the result are desired the above issues are examined in the paper with a view to show a limiting values of the size parameter of excitations for which stochastic averaging technique can be applied to obtain reasonable estimates of pdf and mean square value of response b the effect of the nature of the frequency contents of excitation on the response and c the use of the stochastic averaging method employing generalized harmonic functions for nonintegrable functions representing the dynamic system it is shown that the stochastic averaging procedure provides results which compare very well with those obtained from simulation analysis not only for wide band excitations but also for narrow band excitations for a wide range of the size parameter of excitation further the procedure can be easily implemented for highly irregular functions by employing a numerical scheme with fft

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

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

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

influence of the noise spectrum on the anomalous diffusion in a stochastic system we use an effective markovian description to study the long time behavior of a nonlinear second order langevin equation with a gaussian noise when dissipation is neglected the energy of the system grows as with time a power law with an anomalous scaling exponent that depends both on the confining potential and on the high frequency distribution of the noise the asymptotic expression of the probability distribution function in phase space is calculated analytically the results are extended to the case where small dissipative effects are taken into account

stochastic stability of quasi non integrable hamiltonian systems abstract an n degree of freedom quasi non integrable hamiltonian system is first reduced to an ito equation of one dimensional averaged hamiltonian by using the stochastic averaging method developed by the first author and his coworkers the necessary and sufficient conditions for the asymptotic stability in probability of the trivial solution of the quasi non integrable hamiltonian system are then obtained approximately by examining the sample behaviors of the one dimensional diffusion process of the square root of averaged hamiltonian at the two boundaries a system of linearly and non linearly coupled two non linearly damped oscillators subject to parametric excitations of gaussian white noises is employed as an example to illustrate the procedure and the effects of non linear damping and non linear coupling on the stability are analyzed in detail

stationary response of multi degree of freedom vibro impact systems under white noise excitations a multi degree of freedom vibro impact system under white noise excitations is formulated as a stochastically excited and dissipated hamiltonian system the constraints are modelled as non linear springs according to the hertz contact law the exact stationary solution of the system is derived under certain conditions the approximate stationary solutions of the system are also obtained by using the stochastic averaging methods for quasi hamiltonian systems it is shown that the stochastic averaging method for quasi non integrable hamiltonian systems is applicable if the non linear forces according to the hertz contact law take an important role in the response of the system while the stochastic averaging method for quasi integrable hamiltonian systems is applicable if the non linear forces can be neglected an example for stochastically excited two degree of freedom vibro impact system is given to illustrate the application of the procedures in detail

stochastic averaging of strongly non linear oscillators under bounded noise excitation abstract a stochastic averaging method for strongly non linear oscillators under external and or parametric excitation of bounded noise is proposed by using the so called generalized harmonics functions the method is then applied to study the primary resonance of duffing oscillator with hardening spring under external excitation of bounded noise the stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase subsequently the method is applied to study the dynamical instability and parametric resonance of duffing oscillator with hardening spring under parametric excitation of bounded noise the primary unstable region is delineated by evaluating the lyapunov exponent of linearized system and the response and jump of non linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase

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

optimal feedback control of strongly non linear systems excited by bounded noise abstract a strategy for non linear stochastic optimal control of strongly non linear systems subject to external and or parametric excitations of bounded noise is proposed a stochastic averaging procedure for strongly non linear systems under external and or parametric excitations of bounded noise is first developed then the dynamical programming equation for non linear stochastic optimal control of the system is derived from the averaged ito equations by using the stochastic dynamical programming principle and solved to yield the optimal control law the fokker planck kolmogorov equation associated with the fully completed averaged ito equations is solved to give the response of optimally controlled system the application and effectiveness of the proposed control strategy are illustrated with the control of cable vibration in cable stayed bridges and the feedback stabilization of the cable under parametric excitation of bounded noise

averaging method using generalized harmonic functions for strongly non linear oscillators abstract an averaging method in which generalized harmonic functions are used is applied to study the approximate solutions of the strongly non linear oscillators g x e x and g x ef x t where g x is an arbitrary non linear function the method gives the approximate solutions in terms of generalized harmonic functions these functions are also periodic and are exact solutions of strongly non linear differential equations some phenomena considered include limit cycles of strongly non linear autonomous oscillators and steady state response of strongly non linear oscillators subject to weak harmonic excitation the procedure is simple and easy to apply

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

stochastic jump and bifurcation of a duffing oscillator under narrow band excitation abstract the variance of the displacement the probability densities of the displacement velocity and amplitude and the joint probability density of the displacement and velocity of the stationary response of a duffing oscillator to narrow band gaussian random excitation for a number of combinations of the parameters of the oscillator and excitation obtained from digital simulation are presented based on these results the stochastic jump and bifurcation of the system are examined it is shown that for each combination of the parameters all the statistics of the stationary response are unique and independent of initial conditions in a certain domain of the parameter space there are two more probable motions in the stationary response and jumps may occur the jump is a transition from one more probable motion to another or vice versa outside of this domain the stationary response is either nearly gaussian or like a diffused limit cycle as the parameters cross the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation it is pointed out that some previous results and explanations on this problem are incorrect

probabilistic solution of nonlinear oscillators excited by combined gaussian and poisson white noises abstract the stationary probability density function pdf solution of the stochastic response of nonlinear oscillators is investigated in this paper the external excitation is assumed to be a combination of gaussian and poisson white noises the pdf solution is governed by the generalized kolmogorov equation which is solved by the exponential polynomial closure epc method in order to evaluate the effectiveness of the epc method different nonlinear oscillators are considered in numerical analysis nonlinearity exists either in displacement or in velocity for these nonlinear oscillators the impulse arrival rate mono modal pdf and bi modal pdf are also considered in this study compared to the pdf given by monte carlo simulation the epc method presents good agreement with the simulated result which can also be observed in the tail region of the pdf solution

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

optimal bounded control of first passage failure of strongly non linear oscillators under combined harmonic and white noise excitations a procedure for designing the optimal bounded control of strongly non linear oscillators under combined harmonic and white noise excitations for minimizing their first passage failure is proposed first a stochastic averaging method for strongly non linear oscillators under combined harmonic and white noise excitations using generalized harmonic functions is introduced then the dynamical programming equations and their boundary and final time conditions for the control problems of maximizing reliability and of maximizing mean first passage time are formulated from the averaged ito equations by using the dynamical programming principle the optimal control law is derived from the dynamical programming equations and control constraint finally the conditional reliability function the conditional probability density and mean of the first passage time of the optimally controlled system are obtained from solving the backward kolmogorov equation and pontryagin equation an example is given to illustrate the proposed procedure and the results obtained are verified by using those from digital simulation

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

real solution isolation using interval arithmetic we propose a complete algorithm for real solution isolation for semialgebraic systems by using interval arithmetic the algorithm is implemented as a maple program nrealzero and its performance on several examples is reported

non probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty effects of uncertainties on the dynamic response of the nonlinear vibration systems with general form are investigated based on interval mathematics modeling the uncertain parameters as interval numbers a non probabilistic interval analysis method which estimates the range of the nonlinear dynamic response with the help of taylor series expansion is presented where the partial derivatives of the dynamic response with respect to uncertain parameters are considered to be interval numbers the sensitivity matrices of dynamic response with the uncertain parameters are derived for the presented method only the bounds on uncertain parameters are needed instead of probabilistic density distribution or statistical quantities numerical examples are used to illustrate the validity and feasibility of the presented method

nonconventional stochastic resonance it is argued on the basis of linear response theory lrt that new types of stochastic resonance sr are to be anticipated in diverse systems quite different from the one most commonly studied to date which has a static double well potential and is driven by a net force equal to the sum of periodic and stochastic terms on this basis three new nonconventional forms of sr are predicted sought found and investigated both theoretically and by analogue electronic experiment a in monostable systems b in bistable systems with periodically modulated noise and c in a system with coexisting periodic attractors in each case it is shown that lrt can provide a good quantitative description of the experimental results for sufficiently weak driving fields it is concluded that sr is a much more general phenomenon than has hitherto been appreciated

instabilities in moving bands under random tension fluctuation flexural instabilities in moving bands under random tension fluctuation are investigated the discretized equations of flexural motion represent a gyroscopic system with randomly varying stiffness explicit stability conditions based on the mean square amplitudes are established by employing a stochastic averaging procedure the effect of mean band speed on the band stability is discussed

weak centers and bifurcation of critical periods in reversible cubic systems abstract in this paper we investigate nonhomogeneous cubic differential systems with reversibility which guarantees that the systems have a center at the origin we apply the ritt wu method to process algebraic equations and inequalities using maple v 3 on a computer we give an inductive algorithm for computing the period coefficient polynomials we find the structure of solutions of systems of algebraic equations corresponding to isochronous centers and weak centers of every finite order and we derive conditions on the parameters under which the origin is an isochronous center or a weak center of finite order we show that with appropriate perturbations local bifurcation of critical periods will occur from weak centers of finite order and isochronous centers

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

numerical study of the correspondence between the dissipative and fixed energy abelian sandpile models we consider the abelian sandpile model asm on the square lattice with a single dissipative site sink particles are added one by one per unit time at random sites and the resulting density of particles is calculated as a function of time we observe different scenarios of evolution depending on the value of initial uniform density height h0 during the first stage of the evolution the density of particles increases linearly reaching a critical density c h0 the system changes its behavior and relaxes exponentially to the stationary state of the asm with density s considering initial heights 1 h0 4 we observe a dramatic decrease of the difference c h0 s when h0 is zero or negative in parallel with the asm we consider the conservative fixed energy sandpile fes the extensive monte carlo simulations show that the threshold density th h0 of the fes converges rapidly to s for h0 1

stabilizability and percolation in the infinite volume sandpile model we study the sandpile model in infinite volume on z d in particular we are interested in the question whether or not initial configurations chosen according to a stationary measure are almost surely stabilizable we prove that stabilizability does not depend on the particular procedure of stabilization we adopt in d 1 and a product measure with density 1 the known critical value for stabilizability in d 1 with a positive density of empty sites we prove that is not stabilizable furthermore we study for values of p such that is stabilizable percolation of toppled sites we find that for p 0 small enough there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail as is the case in the subcritical regime for ordinary percolation

a class of cubic differential systems with 6 tuple focus abstract this paper presents a class of cubic differential systems with the origin as a 6 tuple focus from which 6 limit cycles may be constructed for this class of differential systems the stability of the origin is given

dynamics of offshore structures part ii stochastic averaging analysis this is the second of two companion papers concerned with an investigation into the non linear dynamics of drag dominated offshore structures excited by waves the equations of motion are approximated by a linear system with a time varying damping term the excitation however is a non linear function of the wave particle kinematics to study the effect of the time varying damping the resonant response is here examined by means of a narrow band idealization a stochastic averaging of the governing equations is applied and the fokker planck equation for the response amplitude probability densities is thereby obtained from the stationary solution of this equation second order statistics are derived together with an expression for the hydrodynamic damping which is equivalent to the time varying term numerical results are given to illustrate the magnitude of this term for a typical framed structure and to provide comparisons with results obtained by the conventional technique of linearization

approach to criticality in sandpiles a popular theory of self organized criticality predicts that the stationary density of the abelian sandpile model equals the threshold density of the corresponding fixed energy sandpile we recently announced that this density conjecture is false when the underlying graph is any of z2 the complete graph kn the cayley tree the ladder graph the bracelet graph or the flower graph in this paper we substantiate this claim by rigorous proof and extensive simulations we show that driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink nevertheless we do find and prove a relationship between the two models the threshold density of the fixed energy sandpile is the point at which the driven dissipative sandpile begins to lose a macroscopic amount of sand to the sink

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

technical communique asymptotic lyapunov stability with probability one of quasi integrable hamiltonian systems with delayed feedback control the asymptotic lyapunov stability with probability one of multi degree of freedom mdof quasi integrable and nonresonant hamiltonian systems with time delayed feedback control subject to multiplicative parametric excitation of gaussian white noise is studied first the time delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi integrable and nonresonant hamiltonian system then the averaged ito stochastic differential equations are derived by using the stochastic averaging method for quasi integrable hamiltonian systems and the expression for the largest lyapunov exponent of the linearized averaged ito equations is derived finally the necessary and sufficient condition for the asymptotic lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest lyapunov exponent to be negative an example is worked out in detail to illustrate the above mentioned procedure and its validity and to show the effect of the time delay in feedback control on the largest lyapunov exponent and the stability of the system

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

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

deriving some new conditions on the existence of eight limit cycles for a cubic system abstract in james and lloyd s paper some conditions on the existence of eight limit cycles bifurcating from the origin for a cubic system were derived 1 we have currently derived some new conditions for generating eight limit cycles from the origin for this cubic system in fact deriving such conditions is reduced to finding the symbolic real solutions of systems of algebraic equations and inequalities but the involved computations are very complex we combine the ritt wu method with the budan fourier theorem to solve systems of equations and inequalities with this strategy we derive these new conditions using maplev 3 on sun sparc station 10 and sun server 1104 in addition when verifying the conditions derived in 1 by solving the system of equations and inequalities arising in 1 in a different ordering we found that one of the conditions a7 0 should be a7 0