exactly solvable sandpile with fractal avalanching a simple one dimensional sandpile model is constructed which possesses exact analytical solvability while displaying both scale free behavior and fractal properties the sandpile grows by avalanching on all scales yet its shape and energy content are described by a simple continuous but nowhere differentiable analytical formula the avalanche energy distribution and the avalanche time series are both pou er lows with index 1 1 f spectra s1063 651x 99 00306 2

symmetries conservation laws and multisoliton perturbation theory on the basis of action angle variables for multisolitons new symmetries mastersymmetries are constructed for systems with known hierarchies of nonhamiltonian mastersymmetries hierarchies of hamiltonian mastersymmetries are constructed and for systems with known hamiltonian mastersymmetries a hierarchy of non hamiltonian mastersymmetries are constructed moreover with the help of the action angle variables the n soliton perturbation theory on the soliton submanifold adiabatic approximation is formulated an explicit form of the time evolution of asymptotic data under the influence of perturbation is presented

master symmetries and point particle representation of solitons hierarchies of soliton equations generated by an appropriate recursion operator are considered n non interacting galilean like point particles are connected with an n soliton solution of an arbitrary equation from the hierarchy the method of finding the particle representation of solitons is closely related to eigenstates of the recursion operator and master symmetries of the soliton equations

power of the scan statistic in detecting a changed segment in a bernoulli sequence summary fu curnow 1990 derive recursive equations to find the level of significance and power of a likelihood ratio test for a changed segment of specified length based on the scan statistic the maximum number of successes within the specified length their method is computationally feasible for segment lengths of 20 or less we present and evaluate highly accurate approximations as well as bounds for the power function of this test that are computationally feasible even for very large segment lengths we also evaluate power when the duration of the increased length used in the test statistic does not correspond to the actual length scientists dealing with data sequences modelled by independent bernoulli trials frequently seek criteria to detect changes in the underlying process certain criteria have been developed to be sensitive to situations in which the probability of success on an individual trial changes at a point in the sequence fu curnow 1990 consider a version of a two change point problem with known distance m between the change points and with the trial probability of success at the two ends being equal they give recurrence relations for the probability of type i error and the power computational difficulties arise on even large computers when the number of trials is large in the present paper we derive highly accurate approximations that allow computation of power for all segment lengths consider a sequence of n independent 0 1 random variables x1 x2 xn let 1 denote success with pr x 1 i t we wish to test the null hypothesis of a constant probability of success ho i t i t 1 n against the alternative that some contiguous sequence of m events starting at an unknown trial z has a higher probability of success fu curnow 1990 note that the generalized likelihood ratio test for this alternative rejects ho for large values of the scan statistic

distributions and size scalings for strength in a one dimensional random lattice with load redistribution to nearest and next nearest neighbors lattice and network models with elements that have random strength are useful tools in explaining various statistical features of failure in heterogeneous materials including the evolution of failure clusters and overall strength distributions and size effects models have included random fuse and spring networks where monte carlo simulation coupled to scaling analysis from percolation theory has been a common approach unfortunately severe computational demands have limited the network sizes that can be treated to gain insight at large size scales interest has returned to idealized fiber bundle models in one dimension many models can be solved exactly or asymptotically in increasing size n but at the expense of major simplification of the local stress redistribution mechanism models have typically assumed either equal load sharing among nonfailed elements or nearest neighbor local load sharing lls where a failed element redistributes its load onto its two nearest flanking survivors the present work considers a one dimensional fiber bundle model under tapered load sharing tls which assumes load redistribution to both the nearest and next nearest neighbors in a two to one ratio this rule reflects features found in a discrete mechanics model for load transfer in two dimensional fiber composites and planar lattices we assume that elements have strength 1 or 0 with probability p and q 1 ensuremath p respectively we determine the structure and probabilities for critical configurations of broken fibers which lead to bundle failure under a given load we obtain rigorous asymptotic results for the strength distribution and size effect as stackrel ensuremath rightarrow n ensuremath infty with precisely determined constants and exponents the results are a nontrivial extension of those under lls in that failure clusters are combinatorially much more complicated and contain many bridging fibers consequently certain probabilities are eigenvalues from recursive equations arising from the structure of tls next nearest neighbor effects weaken the material beyond what is predicted under lls keeping only nearest neighbor overloads our results question the validity of scaling relationships that are based largely on monte carlo simulations on networks of limited size since some failure configurations appear only in extremely large bundles the dilemma has much in common with the petersburg paradox

soliton point particles of extended evolution equations a hamiltonian formulation of the particle representation of solitons of extended hierarchies of evolution equations is presented the extension is related to the addition of new terms with x dependent coefficients

the importance of the privilege for appearance of inverse power solutions in ito equations abstract it is shown that the true cause of inverse power distributions in the ito equation is some kind of privilege which is hidden in the course of evolution of the system connections between ito equations with additive noise or and multiplicative noise with additive processes multiplicative processes multiplication of probabilities and return to the origin problem are found on the basis of two toy models the appearance of particular functions for deterministic and stochastic forces in the ito equation is explained the paper stands as the next contribution confirming the hypothesis that the adequate privilege is the cause for the origin of inverse power distributions in many phenomena

strong noise sensitivity and random graphs the noise sensitivity of a boolean function describes its likelihood to flip under small perturbations of its input introduced in the seminal work of benjamini kalai and schramm 1999 it was there shown to be governed by the first level of fourier coefficients in the central case of monotone functions at a constant critical probability here we study noise sensitivity and a natural stronger version of it addressing the effect of noise given a specific witness in the original input our main context is the erdos renyi random graph where already the property of containing a given graph is sufficiently rich to separate these notions in particular our analysis implies strong noise sensitivity in settings where the bks criterion involving the first fourier level does not apply e g when the critical value goes to 0 polynomially fast in the number of variables

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

approximate p values for local sequence alignments assume that two sequences from a finite alphabet are optimally aligned according to a scoring system that rewards similarities according to a general scoring scheme and penalizes gaps insertions and deletions under the assumption that the letters in each sequence are independent and identically distributed and the two sequences are also independent approximate p values are obtained for the optimal local alignment when either i there are at most a fixed number of gaps or ii the gap initiation cost is sufficiently large in the latter case the approximation can be written in the same form as the well known case of ungapped alignments

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

the importance of the privilege in resource redistribution models for appearance of inverse power solutions abstract a polemics with the opinion that power laws generically appear in systems with finite resources is carried on examples of simple models of redistribution of resources which lead to short tail distributions are given it is shown that the true cause of inverse power distributions is some kind of privilege which is hidden at a specific method of redistribution it is explained why some multiplicative processes lead to logexponential i e inverse power distribution functions rather than to lognormal ones the paper stands as the next contribution confirming the hypothesis that the adequate privilege is the cause of origin of inverse power distributions in many phenomena

logarithmic voltage anomalies in random resistor networks the authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold on the basis of numerical simulations on randomly diluted l l square lattice networks we find that the average value of the maximum voltage vmax p l exhibits a peak as a function of the bond concentration p which is located above the percolation threshold this peak value appears to grow logarithmically with l while the location of the peak appears to approach the percolation threshold very slowly as l increases to help understand these results they introduce the bubble model a quasi one dimensional structure in which the system length varies exponentially in the system width this model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained moreover within the bubble model the peak value of the maximum voltage increases as in l while the peak location approaches the percolation threshold as in in l in l in good agreement with our numerical results

an extreme value theory for long head runs for an infinite sequence of independent coin tosses withp heads p 0 1 the longest run of consecutive heads in the firstn tosses is a natural object of study we show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn 1 p i i d exponential random variables these results are extended to the case of the longest head run interrupted byk tails the mean length of this run is shown to be log n klog n k 1 log 1 p log k k 1 2 r1 n o 1 where log log1 p 0 577 is the euler mascheroni constant ln 1 p andr1 n is small the variance is 2 6 2 1 12 r2 n o 1 wherer2 n is again small upper and lower class results for these run lengths are also obtained and extensions discussed

algebro geometric solution of the discrete kp equation over a finite field out of a hyperelliptic curve we transfer the algebro geometric method of construction of solutions of the discrete kp equation to the finite field case we emphasize the role of the jacobian of the underlying algebraic curve in construction of the solutions we illustrate in detail the procedure on example of a hyperelliptic curve

the hausdorff dimension of level sets described by erdos renyi average abstract let 0 1 and be an integer function defined on n 0 satisfying 1 n n define the level set e r x 0 1 lim n a n n x where a n n x is the n n erdos renyi average of x 0 1 in this paper we will give descriptions for the hausdorff dimension of e r under the assumption n as n which complement simultaneously an early classic result of besicovitch and the new strong law of large number established by p erdos and a renyi moreover for the case n m ultimately where m 1 is an integer the hausdorff dimension of e r is also determined by us in the last section

on the increments of partial sum processes with multidimensional indices consider a set xn nd of iid random variables indexed by nd the positive integer d dimensional lattice points the erdos renyi 1970 law of large numbers is generalized to this case and some extensions are given the results include a theorem of chan 1976 for the d parameter brownian motion

inverse scattering variables of the kdv equation from the point of view of galilean mechanics the galilean invariance of the korteweg de vries equation is applied in order to characterize the structure of degrees of freedom displayed by the inverse scattering transform method it is found that the dynamical systems associated with the discrete scattering data variables admit a description in terms of mass position and momentum variables similar to the systems of free galilean particles on the other hand the radiation component of the kdv field associated with the continuous part of the set of scattering data turns out to be described by a new field evolving according to a linear partial differential equation

infinitely many commuting symmetries and constants of motion in involution for explicitly time dependent evolution equations the existing symmetry approach to exactly solvable time evolution equations in one spatial dimension is extended to the case that the equation depends explicitly on the time and space coordinates in this context commuting symmetries constants of motion in involution strong and hereditary symmetries squared eigenfunction operators and hamiltonian structures are discussed the cylindrical korteweg de vries equation is used as an illustrative example

poisson and compound poisson approximations in conventional and nonconventional setups it was shown in kifer israel j math 2013 that for any subshift of finite type considered with a gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically poisson distributed here we not only extend this result to all psi mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to poisson or to compound poisson distributions it turns out that for all nonperiodic points a limiting distribution is always poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is bernoulli in which case the limiting distribution always exists thus we describe essentially completely limiting distributions of multiple recurrence numbers in this setup as a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed most of the results are new also for the widely studied single recurrencies case see for instance haydn and vaienti discret contin dyn syst a 10 589 616 2004 probab theory relat fields 144 517 542 2009 abadi and saussol stoch process appl 121 314 323 2011 abadi and vergne nonlinearity 21 2871 2885 2008 as well

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

limit laws of erdos renyi shepp type on obtient la forme complete des theoremes d erdos renyi 1970 et shepp 1964 les conditions etendent celles de deheuvels devroye et lynch 1986 pour couvrir une classe de distributions plus grande

critical phenomena in sequence matching we give a generalization of the result of erdos and renyi on the length r n of the longest head run in the first n tosses of a coin consider two independent sequences x 1 x 2 cdots x m and y 1y 2 cdots y n suppose that x 1 x 2 cdots are i i d mu and y 1 y 2 cdots are i i d nu where mu and nu are possibly different distributions on a common finite alphabet s let p equiv p x 1 y 1 in 0 1 the length of the longest matching consecutive subsequence is m m n equiv max k x i r y j r for r 1 to k for some 0 leq i leq m k 0 leq j leq n k for m and n rightarrow infty with log m log mn rightarrow lambda in 0 1 our result is that there is a constant k equiv k mu nu lambda in 0 1 rbrack such that p lim m m n log 1 p mn k 1 the proof uses large deviation methods the constant k is determined from a variational formula involving the kullback liebler distance or relative entropy a simple necessary and sufficient condition for k 1 is given for the case m n lambda 1 2 and mu nu k 1 the set of mu nu lambda for which k 1 has nonempty interior the boundary of this set is the location of a phase transition the results generalize to more than two sequences and to markov chains a strong law of large numbers is given for the proportion of letters within the longest matching word the limiting proportion exhibits critical behavior similar to that of k

three level description of the domino cellular automaton motivated by the approach of kinetic theory of gases a three level description microscopic mesoscopic and macroscopic of cellular automaton is presented to provide an analytical treatment a simple domino cellular automaton with avalanches is constructed formulas concerning exact relations for density clusters avalanches and other parameters in a stationary state are derived some relations approximately valid for deviations from the stationary state are found and the adequate ito equation is constructed it provides the time evolution description of the density on the macroscopic level the results support an idea of application of the ito equation to some natural time series

scaling theory for quasibrittle structural failure this inaugural article has a twofold purpose i to present a simpler and more general justification of the fundamental scaling laws of quasibrittle fracture bridging the asymptotic behaviors of plasticity linear elastic fracture mechanics and weibull statistical theory of brittle failure and ii to give a broad but succinct overview of various applications and ramifications covering many fields many kinds of quasibrittle materials and many scales from 10 8 to 106 m the justification rests on developing a method to combine dimensional analysis of cohesive fracture with second order accurate asymptotic matching this method exploits the recently established general asymptotic properties of the cohesive crack model and nonlocal weibull statistical model the key idea is to select the dimensionless variables in such a way that in each asymptotic case all of them vanish except one the minimal nature of the hypotheses made explains the surprisingly broad applicability of the scaling laws

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

exact solution of a deterministic sandpile model in one dimension we present an exact solution of a one dimensional sandpile model for which sand is dropped along the wall and n 2 grains of sand fall over the neighboring downhill sites when the critical slope is exceeded the slopes of n consecutive sites organize into a local state the time evolution of the local states along the spatial direction shows a natural tree structure as a result various multifractals can be identified the spatial two point correlation function decreases exponentially with a correlation length of the order of the lattice spacing

exact convergence rate in the limit theorems of erdos renyi and shepp on considere le comportement asymptotique de u n sup o i n k s i k s i et on montre que le terme o k 1 2 peut etre remplace par o k 1 log k

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

the topology of probability distributions on manifolds let mathcal p be a set of n random points in mathbb r d generated from a probability measure on a m dimensional manifold mathcal m subset mathbb r d in this paper we study the homology of mathcal u mathcal p r the union of d dimensional balls of radius r around mathcal p as n rightarrow infty and r rightarrow 0 in addition we study the critical points of d mathcal p the distance function from the set mathcal p these two objects are known to be related via morse theory we present limit theorems for the betti numbers of mathcal u mathcal p r as well as for number of critical points of index k for d mathcal p depending on how fast r decays to zero as n grows these two objects exhibit different types of limiting behavior in one particular case n r m ge c log n we show that the betti numbers of mathcal u mathcal p r perfectly recover the betti numbers of the original manifold mathcal m a result which is of significant interest in topological manifold learning

strong bounds for multidimensional spacings let u1 u2 u n be independent random vectors uniformly distributed on 0 1 d we define the k th maximal spacing n k associated to u1 u1 as the k th maximal possible value of the length of a side of a d dimensional square block in 0 1 d which does not intersect the sample and which cannot be enlarged without doing so our main result is that n k n 1 log n 0 log2n 1 d almost surely as n other bounds are proposed for the limiting almost sure behavior of n k as n

entropy bounds for discrete random variables via maximal coupling this paper derives new bounds on the difference of the entropies of two discrete random variables in terms of the local and total variation distances between their probability mass functions the derivation of the bounds relies on maximal coupling and they apply to discrete random variables which are defined over finite or countably infinite alphabets loosened versions of these bounds are demonstrated to reproduce some previously reported results the use of the new bounds is exemplified for the poisson approximation where bounds on the local and total variation distances follow from stein s method

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

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

effects of pair correlation on mean field theory of btw sand pile model we extend the mean field calculation of btw sand pile model to one that includes the correlation between pairs of nearest neighbors specifically we derive dynamical equations of both one site and two site densities and solve the equations order by order starting with the mean field solution the investigation provides analytical results for both stationary and dynamic states of the sand pile near the critical point which are valid in the regime where h e2 1h e2 1 h incoming rate of sand grains e bulk dissipation rate of sand grainse bulk dissipation rate of sand grains in the stationary case we evaluate the pair correlation and the correction to the mean field single site densities due to the correlation the correction is found to be of the same order as the mean field solution in the dynamic case the initial state deviates from the stationary state by a small fluctuation which subsequently decays exponentially with the time constant being reduced from the corresponding mean field value again the correction to the time constant in this case is found comparable to the mean field value itself

dimensions of level sets related to tangential dimensions for level sets related to the tangential dimensions of bernoulli measures the hausdorff and packing dimensions are determined our results extend some classical work of besicovitch and eggleston

exact results for the one dimensional self organized critical forest fire model we present the analytic solution of the self organized critical soc forest fire model in one dimension proving soc in systems without conservation laws by analytic means under the condition that the system is in the steady state and very close to the critical point we calculate the probability that a string of ital n neighboring sites is occupied by a given configuration of trees the critical exponent describing the size distribution of forest clusters is exactly tau 2 and does not change under certain changes of the model rules computer simulations confirm the analytic results

longest runs in a sequence of m dependent random variables let x i i 1 be a sequence of random 0 s and 1 s we define the lengthr n of the longest head run as follows r n max left k leqq n mathop max limits 0 leqq i leqq n k 1 x i 1 cdots x i k 1 1 right the limit law for the distribution ofr n asn was found in 1943 the rate of convergence in that limit theorem remained unknown until recently even for bernoulli independent trials supposing that x i i 1 is a stationary sequence ofm dependent r v s we given an estimate on the rate of convergence in that limit theorem this estimate provides the unimprovable raten 1 lnn for the independent case a result of lim inf type is obtained too

burst avalanches in solvable models of fibrous materials we review limiting models for fracture in bundles of fibers with statistically distributed thresholds for breakdown of individual fibers during the breakdown process avalanches consisting of simultaneous rupture of several fibers occur and the distribution d delta of the magnitude delta of such avalanches is the central characteristics in our analysis for a bundle of parallel fibers two limiting models of load sharing are studied and contrasted the global model in which the load carried by a bursting fiber is equally distributed among the surviving members and the local model in which the nearest surviving neighbors take up the load for the global model we investigate in particular the conditions on the threshold distribution which would lead to anomalous behavior i e deviations from the asymptotics d delta sim delta 5 2 known to be the generic behavior for the local model no universal power law asymptotics exists but we show for a particular threshold distribution how the avalanche distribution can nevertheless be explicitly calculated in the large bundle limit

symmetries of the massive thirring model for a hamiltonian system every non hamiltonian symmetry gives rise to a recursion operator for symmetries using this method two recursion operators for symmetries of the massive thirring model are constructed the structure of the lie algebra of symmetries generated by these operators is given

asymptotic expansions in the poisson limit theorem asymptotic expansions for the distributions of sums of independent nonnegative integer random variables in the neighbourhood of the poisson distribution are derived together with explicit estimates of the truncation error expansions are also derived for the expectations of at most polynomially growing functions of such sums applications to the poisson binomial and poisson negative binomial approximations are considered the method used is an adaptation of the stein chen approach 1 introduction in chen 1975 stein s method of obtaining error bounds for normal approximations was introduced in the poisson context and was used to obtain rates of convergence in total variation to the poisson distribution for sums of stationary sequences of 0 1 random variables chen also observed that stein s method is in principle suited to developing asymptotic expansions as well as obtaining error estimates and used it for independent 0 1 summands to derive the second term in such an expansion together with an estimate of the remaining error kerstan 1964 had previously derived a similar result with a sharper error estimate by a quite different technique the stein chen method was refined in barbour and hall 1984 to yield the best known upper bounds for the discrepancy between the poisson distribution and that of a sum of independent 0 1 random variables as well as complementary lower bounds and new estimates for the error remaining after the second term in the asymptotic expansion were also established however although stein s method was potentially applicable also to higher order expansions it proved in practice too cumbersome to use in this paper it is shown how by means of a simple identity the stein chen method can be made to yield a full asymptotic expansion with a minimum of difficulty the error estimates obtained perform well in comparison with other estimates that are available for sums of independent 0 1 random variables they agree with those of barbour and hall when the series is truncated after one term or after two for the more general case of sums of independent nonnegative integer valued random variables the error estimates obtained after two terms in the expansion differ only slightly from those in barbour and hall comparison with kerstan s estimates for integral nonnegative summands of the error after one term of the expansion is more complicated by and large the techniques of this paper give better approximations except when the summands themselves are almost precisely poisson distributed in section 3 expansions are also obtained for the expectations of at most polynomially growing functions of the

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

a measure of dna sequence dissimilarity based on mahalanobis distance between frequencies of words summary a number of algorithms exist for searching genetic databases for biologically significant similarities in dna sequences past research has shown that word based search tools are computationally efficient and can find similarities or dissimilarities invisible to other algorithms like fasta we characterize a family of word based dissimilarity measures that define distance between two sequences by simultaneously comparing the frequencies of all subsequences of n adjacent letters i e n words in the two sequences applications to real data demonstrate that currently used word based methods that rely on euclidean distance can be significantly improved by using mahalanobis distance which accounts for both variances and covariances between frequencies of n words furthermore in those cases where mahalanobis distance may be too difficult to compute using standardized euclidean distance which only corrects for the variances of frequencies of n words still gives better performance than the euclidean distance also a simple way of combining distances obtained at different n words is considered the goal is to obtain a single measure of dissimilarity between two dna sequences the performance ranking of the preceding three distances still holds for their combined counterparts all results obtained in this paper are applicable to amino acid sequences with minor modifications

a note on the run length function for intermittent maps abstract we study the run length function for intermittent maps in particular we show that the longest consecutive zero digits resp one digits having a time window of polynomial resp logarithmic length our proof is relatively elementary in the sense that it only relies on the classical borel cantelli lemma and the polynomial decay of intermittent maps our results are compensational to the erd s renyi law obtained by denker and nicol in 8

recursion operators for infinitesimal transformations and their inverses for certain nonlinear evolution equations recursion operators generating infinitesimal transformations have been obtained systematically for certain nonlinear evolution equations their inverses have been evaluated and a double infinity of infinitesimal transformations and nonlinear equations generated

critical phenomena for sequence matching with scoring consider two independent sequences x 1 x n and y 1 y n suppose that x 1 x n are i i d x and y 1 y n are i i d y where x and y are distributions on finite alphabets x and y respectively a score f x x y r is assigned to each pair x i y j and the maximal nonaligned segment score is m n max 0 i j n 0 l 1 f x i l y j l our result is that m n log n x y a s with determined by a tractable variational formula moreover the pair empirical measure of x i l y j l during the segment where m n is achieved converges to a probability measure which is accessible by the same formula these results generalize to x i y j taking values in any polish space to intrasequence scores under shifts to long quality segments and to more than two sequences

exact convergence rates in strong approximation laws for large increments of partial sums consider partial sumss n of an i i d sequencex1x2 of centered random variables having a finite moment generating function in a neighborhood of zero the asymptotic behaviour of u n mathop max limits 0 leqq k leqq n b n s k b n s k is investigated where 1 b n n denotes an integer sequence such thatb n logn asn in particular ifb n o log p n asn for somep 1 the exact convergence rate ofu n b n n 1 0 1 is determined where n depends uponb n and the distribution ofx1 in addition a weak limit law foru n is derived finally it is shown how strong invariance takes over if mathop lim limits n to infty b n loglogn 2 log3n

improved erdos renyi and strong approximation laws for increments of partial sums let x 1 x 2 cdots be an i i d sequence with ex 1 0 ex 2 1 1 ee tx 1 infty t t 0 and partial sums s n x 1 cdots x n starting from some analogous results for the wiener process this paper studies the almost sure limiting behaviour of max 0 leq n leq n a n a 1 2 n s n a n s n as n rightarrow infty under various conditions on the integer sequence a n improvements of the erdos renyi law of large numbers for partial sums are obtained as well as strong invariance principle type versions via the komlos major tusnady approximation an appearing gap between these two results is also going to be closed

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

general large deviations of longest gaps in homogeneous poisson processes abstract let n t t 0 be a homogeneous poisson process with unit intensity the longest gap l t of this process is the maximal time subinterval between epochs of arrival times up to time t and it finds applications in the theory of reliability it has been known since the 1980s that the longest gap obeys the laws of large numbers l t ln t 1 as t almost surely however large deviation probabilities of the longest gap have not been well developed yet and in this paper we derive such large deviations with a very general speed the main tool is to establish a global estimation for the distribution function of the longest gap which is of independent interest

optimal data compression algorithm abstract a practical source coding scheme based on approximate string matching is proposed it is an approximate fixed length string matching data compression combined with a block coder based on the empirical distribution a lemma on approximate string matching which is an extension of the kac lemma is proved it is shown based on the lemma that the deterministic algorithm converts the stationary and ergodic source u into an output process v and under the assumption that v is a stationary process after the scheme has run for an infinite time the optimal compression ratio r d is achieved this reduces the problem of the universal lossy coder to the proof of stationarity of the output process v in the proposed algorithm the main advantages of the proposed method are the asymptotic sequential behavior of the encoder and the simplicity of the decoder

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

the hirota equation over finite fields algebro geometric approach and multisoliton solutions we consider the hirota equation the discrete analogue of the generalized toda system over a finite field we present the general algebro geometric method of construction of solutions of the equation as an example we construct analogues of the multisoliton solutions for which the wavefunctions and the function can be found using rational functions within the class of multisoliton solutions we isolate generalized breather type solutions which have no direct counterparts in the complex field case

an extreme value theory for sequence matching on considere des suites finies x 1 x 2 x m et y 1 y 2 y m ou les lettres x i et y i sont choisies i i d sur un alphabet denombrable avec p p x 1 y 1 0 1 on etudie la distribution de la plus longue sequence contigue de couples entre les x et les y avec au plus k mauvais couplages

approximations for distributions of scan statistics abstract certain statistical applications deal with the extremal distributions of the number of points in a moving interval or window of fixed length this article gives an approximation that is highly accurate for several of these distributions applications include the maximum cluster of points on a line or circle multiple coverage by subintervals or subarcs of fixed size the length of the longest success run in bernoulli trials and the generalized birthday problem

distribution of the length of the longest significance run on a bernoulli net and its applications we consider the length of the longest significance run in a two dimensional bernoulli net and derive its asymptotic limit distribution our results can be considered as generalizations of known theorems in significance runs we give three types of theoretical results 1 reliability style lower and upper bounds 2 erdos renyi law and 3 the asymptotic limit distribution to understand the rate of convergence to the asymptotic distributions we carry out numerical simulations the convergence rates in a variety of situations are presented to understand the relation between the length of the longest significance run s and the success probability p we propose a dynamic programming algorithm to implement simultaneous simulations insights from numerical studies are important for choosing the values of design parameters in a particular application which motivates this article the distribution of the length of the longest significance run in a bernoulli net is critical in applying a multiscale metho

an erdos renyi law with shifts abstract motivated by the comparison of dna sequences a generalization is given of the result of erdos and renyi on the length r n of the longest run of heads in the first n tosses of a coin consider two sequences x 1 x 2 x n and y 1 y 2 y n the length of the longest matching consecutive subsequence allowing shifts is m n max m x i k y j k for k 1 to m for some 0 i j n m suppose that all the letters are independent and identically distributed the length of the longest match without shifts has the same distribution as r n the length of the longest head run for a biased coin with p p x i y i described by the erdos renyi law p lim n r n log 1 p n 1 1 for matching with shifts our result is p lim n m n log 1 p n 2 1 loosely speaking allowing shifts doubles the length of the longest match the case of markov chains is also handled

poisson convergence for dissociated statistics summary a poisson limit theorem is derived for the number of large values observed among comparisons of independent but not necessarily identically distributed random variables the comparisons made need not be the same and may depend on the two variables being compared an application to the assessment of large numbers of correlation coefficients is given

a suboptimal lossy data compression based on approximate pattern matching a practical suboptimal variable source coding algorithm for lossy data compression is presented this scheme is based on approximate string matching and it naturally extends the lossless lempel ziv 1977 data compression scheme among others we consider the typical length of an approximately repeated pattern within the first n positions of a stationary mixing sequence where d percent of mismatches is allowed we prove that there exists a constant r sub 0 d such that the length of such an approximately repeated pattern converges in probability to 1 r sub 0 d log n pr but it almost surely oscillates between 1 r sub spl infin d log n and 2 r sub 1 d log n where r sub spl infin d r sub 0 d r sub 1 d 2 are some constants these constants are natural generalizations of renyi entropies to the lossy environment more importantly we show that the compression ratio of a lossy data compression scheme based on such an approximate pattern matching is asymptotically equal to r sub 0 d we also establish the asymptotic behavior of the so called approximate waiting time n sub l which is defined as the time until a pattern of length c repeats approximately for the first time we prove that log n sub l l spl rarr r sub 0 d pr as l spl rarr spl infin in general r sub 0 d r d where r d is the rate distortion function thus for stationary mixing sequences we settle in the negative the problem investigated by steinberg and gutman by showing that a lossy extension of the wyner ziv 1989 scheme cannot be optimal

an exact renormalization model for earthquakes and material failure statics and dynamics abstract earthquake events are well known to posses a variety of empirical scaling laws accordingly renormalization methods offer some hope for understanding why earthquake statistics behave in a similar way over orders of magnitude of energy we review the progress made in the use of renormalization methods in approaching the earthquake problem in particular earthquake events have been modeled by previous investigators as hierarchically organized bundles of fibers with equal load sharing we consider by computational and analytic means the failure properties of such bundles of fibers a problem that may be treated exactly by renormalization methods we show independent of the specific properties of an individual fiber that the stress and time thresholds for failure of fiber bundles obey universal albeit different scaling laws with respect to the size of the bundles the application of these results to fracture processes in earthquake events and in engineering materials helps to provide insight into some of the observed patterns and scaling in particular the apparent weakening of earthquake faults and composite materials with respect to size and the apparent emergence of relatively well defined stresses and times when failure is seemingly assured

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

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

extremes of some gaussian random interfaces in this article we give a general criterion for some dependent gaussian models to belong to maximal domain of attraction of gumbel following an application of the stein chen method studied in arratia et al ann probab 17 1 9 25 1989 we also show the convergence of the associated point process as an application we show the conditions are satisfied by some of the well known supercritical gaussian interface models namely membrane model massive and massless discrete gaussian free field fractional gaussian free field

numerical simulations of burst processes in fibre bundles we discuss a very effective numerical method for simulating fibre bundle models with equal load sharing and with local load sharing particular attention is paid to the case of the local load sharing model in which the critical load xc is defined as the average load per fibre that causes the final complete failure it is shown that xc to 0 when the size of the system n to infinity we also show analytically that the power law of the burst size distribution d delta varies as delta xi is approximately correct the exponent xi in the local load sharing case is not universal since it depends on the strength distribution as well as on the size of the system

erdős renyi laws for gibbs measures can one detect a phase transition from a single large sample of a gibbs measure what information does one get on the other gibbs distributions with the same potential we approach these questions via erd s renyi laws in particular we prove almost sure limit theorems for sets of empirical distributions of sub samples of the given one for suitable sub samples size this set converges to the set of stationary gibbs measures moreover we formulate erd s renyi laws for general families of random variables with suitable large deviation principles