generation and monitoring of discrete stable random processes using multiple immigration population models some properties of classical population processes that comprise births deaths and multiple immigrations are investigated the rates at which the immigrants arrive can be tailored to produce a population whose steady state fluctuations are described by a pre selected distribution attention is focused on the class of distributions with a discrete stable law which have power law tails and whose moments and autocorrelation function do not exist the separate problem of monitoring and characterizing the fluctuations is studied analysing the statistics of individuals that leave the population the fluctuations in the size of the population are transferred to the times between emigrants that form an intermittent time series of events the emigrants are counted with a detector of finite dynamic range and response time this is modelled through clipping the time series or saturating it at an arbitrary but finite level whereupon its moments and correlation properties become finite distributions for the time to the first counted event and for the time between events exhibit power law regimes that are characteristic of the fluctuations in population size the processes provide analytical models with which properties of complex discrete random phenomena can be explored and in addition provide generic means by which random time series encompassing a wide range of intermittent and other discrete random behaviour may be generated

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

world wide web scaling exponent from simon s 1955 model the statistical properties of the world wide web have attracted considerable attention recently since self similar regimes were first observed in the scaling of its link structure one characteristic quantity is the number of in links k that point to a particular web page its probability distribution p k shows a pronounced power law scaling p k approximately k gamma that is not readily explained by standard random graph theory here we recall a simple and elegant model for scaling phenomena in general copy and growth processes as proposed by simon in 1955 when combined with an experimental measurement of network growth in the world wide web this classical model is able to model the in link dynamics and predicts the scaling exponent gamma 2 1 in accordance with observation

the evolution of a batch immigration death process subject to counts a bivariate batch immigration death process is developed to study the degree to which the fundamental structure of a hidden stochastic process can be inferred purely from counts of escaping individuals this question is of immense importance in fields such as quantum optics where externally based radiation elucidates the nature of the underlying electromagnetic radiation process batches of i immigrants enter the population at rate qi and each individual dies independently at rate general expressions are developed for the population size cumulants and probabilities together with those for the associated counting process the strong link between these two structures is highlighted through two specific examples involving k batch immigration for i k and schoenberg batch immigration over i 2m m 0 1 2 and shows that high quality inferences on the hidden population process can be inferred purely from externally counted observations

skew adjacency matrices of graphs the spectra of the skew adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs this leads to the following topics graphs whose skew adjacency matrices are all cospectral relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew adjacency matrices skew spectral radii and an analogue of the perron frobenius theorem and the number of skew adjacency matrices of a graph with distinct spectra

statistics of binomial number fluctuations when birth terms in the rate equation for a simple birth death population process are proportional to the difference between the number of individuals present and a larger fixed number the equilibrium population obeys a binomial distribution in this paper the statistics of such a process are investigated properties of the related counting process including interval statistics are calculated and the results used to evaluate the autocorrelation function of a binomial telegraph wave

random walks with power law fluctuations in the number of steps in this paper we derive limiting distributions for the resultant of an anisotropic random walk where the individual steps comprising the walk are drawn from distributions of arbitrary but finite variance and where the number of steps fluctuate according to a discrete power law distribution the consideration of discrete distributions with power law tails is motivated by their recently discovered relevance to complex systems and networks when this random walk is unbiased a power law decay for large values of the amplitude of the resultant occurs for a small directional bias the power law tail persists only in the direction parallel to the bias exhibiting an exponential decay in all other directions we consider the relevance of the limiting distributions to non diffusive transport

the smoluchowski coagulation equations with continuous injection we study a system of equations which models the formation of clusters by coagulation with particles of unit size being injected at a time dependent rate we observe that the criteria under which gelation occurs are the same as for the constant mass and constant monomer cases which have been studied previously we identify a variety of types of behaviour in the large time limit depending on the coagulation kernel and on the rate at which monomer is introduced into the system the results are obtained by means of exact generating function techniques matched asymptotic expansions and numerical simulations

accuracy analysis of measurements on a stable power law distributed series of events we investigate how finite measurement time limits the accuracy with which the parameters of a stably distributed random series of events can be determined the model process is generated by timing the emigration of individuals from a population that is subject to deaths and a particular choice of multiple immigration events this leads to a scale free discrete random process where customary measures such as mean value and variance do not exist however converting the number of events occurring in fixed time intervals to a 1 bit clipped process allows the construction of well behaved statistics that still retain vestiges of the original power law and fluctuation properties these statistics include the clipped mean and correlation function from measurements of which both the power law index of the distribution of events and the time constant of its fluctuations can be deduced we report here a theoretical analysis of the accuracy of measurements of the mean of the clipped process this indicates that for a fixed experiment time the error on measurements of the sample mean is minimized by an optimum choice of the number of samples it is shown furthermore that this choice is sensitive to the power law index and that the approach to poisson statistics is dominated by rare events or outliers our results are supported by numerical simulation

a multispecies birth death immigration process and its diffusion approximation abstract we consider an extended birth death immigration process defined on a lattice formed by the integers of d semiaxes joined at the origin when the process reaches the origin then it may jump toward any semiaxis with the same rate the dynamics on each ray evolves according to a one dimensional linear birth death process with immigration we investigate the transient and asymptotic behavior of the process via its probability generating function the stationary distribution when existing is a zero modified negative binomial distribution we also study a diffusive approximation of the process which involves a diffusion process with linear drift and infinitesimal variance on each ray it possesses a gamma type transient density admitting a stationary limit as a byproduct of our study we obtain a closed form of the number of permutations with a fixed number of components and a new series form of the polylogarithm function expressed in terms of the gauss hypergeometric function

intrinsic and measured statistics of discrete stochastic populations the notion that the nature of a measurement is critical to its outcome is usually associated with quantum phenomena in this paper we show that the observed statistical properties are also a function of the measurement technique in the case of simple classical populations in particular the measured and intrinsic statistics of a single population may be different while correlation and transfer of individuals between two populations may be hidden from the observer

fluctuations in a coupled population model we investigate a discrete markov process in which the immigration of individuals into one population is controlled by the fluctuations in another we examine the effect of coupling back the second population to the first through a similar mechanism and derive exact solutions for the generating functions of the population statistics we show that a stationary state exists over a certain parameter range and obtain expressions for moments and correlation functions in this regime when more than two populations are coupled cyclically transient oscillations and periodic behaviour of correlation functions are predicted we demonstrate that if the initial distribution of either population is stable or more generally has a power law tail that falls off like n 1 0 1 then for certain parameter values there exists a stationary state that is also power law but not stable this stationary state cannot be accessed from a single multiple immigrant population model but arises solely from the nonlinear interaction of the coupled system

continuous and discrete stable processes the one sided levy stable probability densities and the discrete stable distributions form a doubly stochastic poisson transform pair this relationship facilitates the formulation of a class of continuous stable stochastic processes

on tournament matrices abstract let t be a tournament of order n with adjacency matrix m we find several conditions that are equivalent to m being singular a correlation between the number of 3 cycles in t and the rank of m is established it is shown that asymptotically at least 1 2 of the tournament matrices are nonsingular we also derive bounds on the spectral radius of tournament matrices with a given row sum vector

an analytical investigation of shape control of large space structures by applied temperatures an analytical procedure for the static shape control of flexible space structures subjected to thermal distortions is developed which is based on prescribing temperatures in control elements having much higher coefficients of thermal expansion than the main structure the temperatures at the control elements are defined so as to minimize the overall thermal distortion of the structure from its ideal shape and a matrix equation is obtained which can be solved for the set of optimum control temperatures a formulation of the procedure for continuous structures governed by differential equations and a formulation for discrete finite element modeled structures governed by matrix equations are presented the equations from the continuous formulation are employed for the shape control of a simple beam distorted by nonuniform heating and the discrete formulation is applied in a general purpose finite element structural analysis computer program for the shape control of a 750 m radiometer antenna reflector dish subjected to orbital heating a reduction in thermal distortion by a factor of nearly 50 was obtained with the use of only seven control elements results for four different sets of control locations for the antenna are presented in which reductions in distortion of up to a factor of four were obtained

distinguishing population processes by external monitoring we investigate the statistical and correlation properties of two stochastic population models that give rise to identical first order probability densities we assume that the processes are monitored indirectly through measurement of the rate at which individuals emigrate from the population formulae characterizing the integrated statistics of these counting processes are derived and it is shown how they may be used to distinguish the population models

static shape control of composite plates using a curvature displacement based algorithm abstract an intuitive algorithm for the determination of voltage distribution in the application to shape control of smart structures using piezoelectric actuators is presented here this approach uses curvature as the fine tuning criteria on top of the common displacement based shape control and is an extension of the slope displacement method developed by the same authors the algorithm called the perturbation buildup voltage distribution pbvd is based on an iterative approach inspired by a previous algorithm bvd on displacement control this method aims to provide a means of targeting the desired shape of a structure by using a higher level shape attribute in this case curvature intuitive iterative parameters of the pbvd method allow the user to have better control over the degree of conformity of the structure s shape a natural consequence of this method is the smoothing of the resultant shape results show that the slopes and curvatures of the structure can be improved but at a tolerable expense of the displacement criteria another result of practical interest is the reduction of internal stresses compared to cases using pure displacement shape control

spectral radii of tournament matrices whose graphs are related by an arc reversal abstract given an irreducible tournament matrix t and a pair of distinct indices i and j let t i j be the matrix obtained from t by transposing its principal submatrix on rows and columns i and j we establish one condition on rows i and j of t under which the spectral radius of t i j is no smaller than that of t and another condition on the i th and j th entries of the left and right perron vectors of t under which the spectral radius of t i j must be strictly smaller than that of t these conditions are used to compare the spectral radii of a class of toeplitz tournament matrices and the resulting comparison sheds light on some conjectures of brualdi and li further if t yields equality in a certain lower bound on the spectral radius of a tournament matrix then for any i and j we provide simple necessary and sufficient conditions for the spectral radius of t i j to be larger than that of t to be smaller than that of t and to be equal to that of t

generation and monitoring of a discrete stable random process a discrete stochastic process with stationary power law distribution is obtained from a death multiple immigration population model emigrations from the population form a random series of events which are monitored by a counting process with finite dynamic range and response time it is shown that the power law behaviour of the population is manifested in the intermittent behaviour of the series of events

laguerre population processes we analyse a number of stochastic processes that give rise to first order number statistics governed by laguerre distributions with properties that lie between poisson and geometric random variables these distributions have hitherto been used to characterize the photon statistics of a coherent mixture of thermal and laser light here we explore a number of discrete population processes that can lead to the same first order statistics and correlation functions and highlight the distinguishing features that may be used to identify the relevant model from data

fluctuations in quantum and classical populations characterization and measurement of fluctuations in quantum and classical populations are compared a simple method for estimating the population fluctuation time is outlined and approximate quantum and classical expressions for the two time correlation function are derived the utility of the method is demonstrated through analysis of the scully lamb laser model 1967 where it is shown to give correlation times in good agreement with numerical calculations for lasers operated below at and above threshold discrepancies in the predictions of a previous similar method are shown to originate from inconsistent separation of its quantum and classical versions

on the joint statistics of stable random processes a utilitarian continuous bi variate random process whose first order probability density function is a stable random variable is constructed results paralleling some of those familiar from the theory of gaussian noise are derived in addition to the joint probability density for the process these include fractional moments and structure functions although the correlation functions for stable processes other than gaussian do not exist we show that there is coherence between values adopted by the process at different times which identifies a characteristic evolution with time the distribution of the derivative of the process and the joint density function of the value of the process and its derivative measured at the same time are evaluated these enable properties to be calculated analytically such as level crossing statistics and those related to the random telegraph wave when the stable process is fractal the proportion of time it spends at zero is finite and some properties of this quantity are evaluated an optical interpretation for which is provided

discrete scale free distributions and associated limit theorems consideration is given to the convergence properties of sums of identical independently distributed random variables drawn from a class of discrete distributions with power law tails which are relevant to scale free networks different limiting distributions and rates of convergence to these limits are identified and depend on the index of the tail for indices 2 the topology evolves to a random poisson network but the rate of convergence can be extraordinarily slow and unlikely to be yet evident for the current size of the www for example it is shown that treating discrete scale free behaviour with continuum or mean field approximations can lead to incorrect results

on the location of the spectrum of hypertournament matrices inclusion regions for the spectrum of a hypertournament matrix a are obtained based on a complex curve that relates the real and imaginary parts of the eigenvalues these results generalize and in certain cases improve the work of s kirkland linear and multilinear algebra 30 1991 261 the bounds obtained depend on the variance of the score vector their tightness is investigated using the notion of numerical range 2001 elsevier science inc all rights reserved

algebraic multiplicity of the eigenvalues of a tournament matrix let f denote the set of irreducible n x n tournament matrices here arc our main results 1 for all n 3 every matrix in k has at least three distinct eigenvalues such a matrix has exactly three distinct eigenvalues if and only if it is a hadamard tournament matrix 2 for all n 2 3 there is a matrix in y having n distinct eigenvalues 3 if cr denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z then ln 21 2 6 4 if r is the minimum perron value i e spectral radius of all matrices in r then 2 8

optimum control voltage design for constrained static shape control of piezoelectric structures design optimization of control voltage distribution for constrained static shape control of structures using piezoelectric actuators is investigated two cases are studied in the first case a scheme is developed to find the optimal control voltage distribution minimizing the square error between the desired and actuated shapes subject to a given control electric energy the optimal control voltage constrained by control energy can be found after solving an algebraic equation in terms of the lagrangian multiplier an alternative form for this algebraic equation is derived by taking advantage of an eigenvalue problem of a real symmetrical matrix which significantly reduces the computational cost for finding its roots a procedure for finding the optimal control energy is also given in the second case a process of seeking the control voltage distribution with least control energy for any given square error tolerance between the actuated and desired shapes is presented finally illustrative examples for the constrained shape control of thin plates are given to demonstrate the presented methods

shape control of vibrating simply supported rectangular plates we determine the optimum location of a given rectangular piezoceramic pzt actuator that will require the minimum voltage to annul the deflections of a simply supported rectangular linear elastic plate vibrating near one of its fundamental frequencies keeping the location of the centroid and the shape of the pzt fixed we ascertain the voltage required as a function of the length of its diagonal to nullify the deflections of the plate

the skew energy of tournaments abstract let s be the skew adjacency matrix of a digraph the skew energy of s denoted by e s is the sum of the absolute values of the all eigenvalues of s in this paper for skew adjacency matrices of tournaments of order n we will give lower and upper bounds of the skew energy we show that our lower bound is tight in the sense that it coincides with the minimum of e s of all skew adjacency matrices of tournaments of order n in addition we will introduce the skew energy which is a generalization of the skew energy

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

characterization of structural reorganization in rice piles diagnostics applied to a rice pile cellular automaton reveal different mechanisms producing power law behaviors of statistical attributes of grains which are germane to self organised critical phenomena the probability distributions for these quantities can be derived from two distinct random walk models that account for correlated clustered behavior through incorporating fluctuations in the number of steps in the walk the first model describes the distribution for a spatial quantity the resultant flight length of grains this has a power law tail caused by grains moving through a discrete power law distributed number of random steps of finite length developing this model into a random walk obtains distributions for the resultant flight length with characteristics similar to l evy distributions the second random walk model is devised to explain a temporal quantity the distribution of trapping or residence times of grains at single locations in the pile diagnostics reveal that the trapping time can be constructed as a sum of subtrapping times which are described by a l evy distribution where the number of terms in the sum is a discrete random variable accurately described by a negative binomial distribution the infinitely divisible two parameter limit distribution for the resultant of such a random walk is discussed and describes a dual scale power law behavior if the number fluctuations are strongly clustered the form for the distribution of transit times of grains results as a corollary

extremal digraphs with given clique number abstract we classify those digraphs with a given number of vertices and a given clique number that maximize the perron root of the adjacency matrix

population statistics and the counting process the authors investigate the outcome of different monitoring schemes applied to small classical populations explicit calculation for a solvable model establishes the relationship between the statistical properties of the population and those of the counting processes parallels are drawn with operator ordering in the second quantised formulation of photodetection

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

power law distributed level crossings define fractal behavior a relationship is established between the autocorrelation function of continuous gaussian and non gaussian stochastic processes and the discrete process that describes their zero or level crossings random fractals occur when the distribution for the number of crossings is described by a class of markov processes whose singlefold statistics are the discrete analog of the levy stable continuous probability densities

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

a graphical analysis of ratio scaled paired comparison data one of the building blocks of the analytic hierarchy process saaty saaty t l 1977 a scaling method for priorities in hierarchical structures math psych 15 234 281 is the ratio scaled assessment of an agent s preferences between pairs of alternatives this article shows how it is possible to visualize such data and to detect cardinal and ordinal inconsistencies in the respondent s judgments the graphical method used to display preferences is not new in itself it dates back at least to the work of gower gower j c 1977 the analysis of asymmetry and orthogonality j r barra f brodeau g romier b van cutsem eds recent developments in statistics north holland amsterdam 109 123 and many variants of his technique can be found in the literature but gower plots provide an especially valuable diagnostic tool in the context of ahp as discussed and abundantly illustrated herein

levy random walks with fluctuating step number and multiscale behavior random walks with step number fluctuations are examined in n dimensions for when step lengths comprising the walk are governed by stable distributions or by random variables having power law tails when the number of steps taken in the walk is large and uncorrelated the conditions of the levy gnedenko generalization of the central limit theorem obtain when the number of steps is correlated infinitely divisible limiting distributions result that can have levy like behavior in their tails but can exhibit a different power law at small scales for the special case of individual steps in the walk being gaussian distributed the infinitely divisible class of k distributions result the convergence to limiting distributions is investigated and shown to be ultraslow random walks formed from a finite number of steps modify the behavior and naturally produce an inner scale the single class of distributions derived have as special cases k distributions stable distributions distributions with power law tails and those characteristic of high and low frequency cascades the results are compared with cellular automata simulations that are claimed to be paradigmatic of self organized critical systems

the evolution of a single paired immigration death process the general question of whether it is possible to determine the fundamental structure of a hidden stochastic process purely from counts of escaping individuals is of immense importance in fields such as quantum optics where externally based radiation elucidates the nature of the electromagnetic radiation process although the general probability structure has been derived in an earlier paper in terms of the joint probability generating function of the hidden population size and known counts its complex nature hides some particularly intriguing features of the underlying process our current objective is therefore to examine specific immigration regimes in order to highlight the underlying saw tooth behaviour of the underlying probability and moment structures the paper first explores paired and triple immigration schemes and then introduces birth in order to show that the technique is equally successful in exposing hidden multiplicative effects these analyses uncover novel and highly illuminating features and emphasize the potential of this population counting construct for expanding into more complex multi type situations

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

eigenvalues of oriented graph matrices abstract we give bounds on the real and imaginary parts of the eigenvalues of an oriented graph matrix prove that each of the irreducible oriented graph matrices of order n 3 has at least three distinct eigenvalues and determine such irreducible matrices with three distinct eigenvalues we also consider spectral properties of the bipartite oriented graph matrices and the multiequipartite oriented graph matrices

solution of the conjecture of brualdi and li abstract we establish the conjecture of brualdi and li on the maximal perron root of a tournament matrix of even order

tournament matrices with extremal spectral properties abstract for a tournament matrix m of order n we define its walk space wm to be span m j 1 j 0 n 1 where 1 is the all ones vector we show that the dimension of wm equals the number of eigenvalues of m whose real parts are greater than 1 2 we then focus on tournament matrices whose walk space has particularly simple structure and characterize them in terms of their spectra specifically we characterize those tournament matrices such that mj1 is an eigenvector of m for some j 0 we also characterize the tournament matrices m such that jn 2m is a skew hadamard matrix throughout we illustrate our results with examples

fluctuations in the zeros of differentiable gaussian processes the stochastic point processes formed by the zero crossings or extremal points of differentiable stationary gaussian processes are studied as a function of their autocorrelation function the properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin the distribution for the number of zeros occurring in an asymptotically large interval are approximately negative binomial or binomial depending upon whether the relative variance or fano factor is greater or less than unity the correlation properties of the zeros are such that they are repelled from each other or are antibunched if the autocorrelation function of the gaussian process is characterized by a single scale size but occur in clusters if more than one characteristic scale size is present the intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves when bunching occurs the interval density becomes bimodal indicating the interval sizes within and between the clusters the interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the gaussian process at large delay times poisson distributed fluctuations of the zeros occur only exceptionally but never form a poisson process

pick s inequality and tournaments abstract let t be a tournament on n nodes and let a be its adjacency matrix a brauer and i gentry observed that an inequality due to g pick implies that im 1 2 cot 2n for all eigenvalues of a we say that t is a pick or p tournament if equality holds for some we determine when equality holds in pick s inequality for arbitrary real matrices and use this to show that the p tournaments can all be constructed from the transitive tournament m by reversing the arcs between the sets of certain node partitions or cuts u the cuts are specified by 1 n vectors u such that utw 0 where w 1 n 1 t e i n this links the cuts to cyclotomic polynomials there is at least one p tournament on n nodes if and only if n 2k k 1 up to isomorphism there is precisely one p tournament on n nodes if and only if n p or n 2p for some odd prime p for odd n up toisomorphism the only regular p tournament on n nodes has matrix zn circ 0 1 1 0 0 a composition rule is used on the matrices zp to form all p tournaments on n 2kpl nodes l 1

universality in sandpiles we perform extensive numerical simulations of different versions of the sandpile model we find that previous claims about universality classes are unfounded since the method previously employed to analyze the data suffered from a systematic bias we identify the correct scaling behavior and provide evidences suggesting that sandpiles with stochastic and deterministic toppling rules belong to the same universality class

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved