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

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

a scalable fully implicit compressible euler solver for mesoscale nonhydrostatic simulation of atmospheric flows a fully implicit solver is developed for the mesoscale nonhydrostatic simulation of atmospheric flows governed by the compressible euler equations to spatially discretize the euler equations on a height based terrain following mesh we apply a cell centered finite volume scheme in which an advection upstream splitting ausm up method with a piecewise linear reconstruction is employed to achieve second order accuracy for the low mach flow a second order explicit first step single diagonal coefficient diagonally implicit runge kutta esdirk method with adaptive time stepping is applied to stabilize physically insignificant fast waves and accurately integrate the euler equations in time the nonlinear system arising at each time step is solved by using a jacobian free newton krylov schwarz nks algorithm to accelerate the convergence and improve the robustness we employ a class of additive schwarz preconditioners in which the subdomain jacobian matrix is constructed using a first order spatial discretization several test cases are used to validate the correctness of the scheme and examine the performance of the solver large scale results on a supercomputer with up to 18 432 processor cores are provided to show the parallel performance of the proposed method

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

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

aubry mather theory for conformally symplectic systems in this article we develop an analogue of aubry mather theory for a class of dissipative systems namely conformally symplectic systems and prove the existence of interesting invariant sets which in analogy to the conservative case will be called the aubry and the mather sets besides describing their structure and their dynamical significance we shall analyze their attracting repelling properties as well as their noteworthy role in driving the asymptotic dynamics of the 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

existence of c 1 critical subsolutions of the hamilton jacobi equation let m be a c second countable manifold without boundary we denote by tm the tangent bundle and by tm m the canonical projection a point in tm will be denoted by x v with x m and v tx m 1 x in the same way a point of the cotangent space t m will be denoted by x p with x m and p t x m a linear form on the vector space tx m we will suppose that g is a complete riemannian metric on m for v tx m the norm v is g v v 1 2 we will denote by the dual norm on t x m we will also use the notations v x for v tx m and p x for v t x m we will assume in the whole paper that h t m r is a function of class at least c2 which satisfies the following three conditions 1 uniform superlinearity for every k 0 there exists c k r such that

a fully spectral methodology for magnetohydrodynamic calculations in a whole sphere abstract we present a fully spectral methodology for magnetohydrodynamic mhd calculations in a whole sphere the use of jones worland polynomials for the radial expansion guarantees that the physical variables remain infinitely differentiable throughout the spherical volume furthermore we present a mathematically motivated and systematic strategy to relax the very stringent time step constraint that is present close to the origin when a spherical harmonic expansion is used for the angular direction the new constraint allows for significant savings even on relatively simple solutions as demonstrated on the so called full sphere benchmark a specific problem with a very accurately known solution the numerical implementation uses a 2d data decomposition which allows it to scale to thousands of cores on present day high performance computing systems in addition to validation results we also present three new whole sphere dynamo solutions that present a relatively simple structure

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

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

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

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

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

a schwarz waveform moving mesh method an r refinement moving mesh method is considered for solving time dependent partial differential equations pdes the resulting coupled system consisting of the physical pde and a moving mesh pde is solved by a schwarz waveform relaxation method in particular the computational space time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of pdes on each subdomain dirichlet boundary conditions are used to pass solution information between neighboring regions the efficacy of this approach is demonstrated for some model problems for problems where the solutions evolve on disparate time scales in different regions of the spatial domain this approach demonstrates the significant savings in computational time and effort which are possible

stage value predictors and efficient newton iterations in implicit runge kutta methods the prediction of stage values in implicit runge kutta methods is important both for overall efficiency as well as for the design of suitable control strategies for the method the purpose of this paper is to construct good stage value predictors for implicit methods and to verify their behavior in practical computations we show that for stiffly accurate methods of low stage order it is necessary to use several predictors in other words a continuous extension for the method will not yield the best results we also investigate how to gain additional efficiency in the newton iterations used to correct the prediction error this leads to new control strategies with respect to refactorization of jacobians that seek to globally minimize total work per unit time of integration

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

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

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

control strategies for the iterative solution of nonlinear equations in ode solvers in the numerical solution of odes by implicit time stepping methods a system of nonlinear equations has to be solved each step it is common practice to use fixed point iterations or in the stiff case some modified newton iteration the convergence rate of such methods depends on the stepsize similarly a stepsize change may force a refactorization of the iteration matrix in the newton solver this paper develops new strategies for handling the iterative solution of nonlinear equations in ode solvers these include automatic switching between fixed point and newton iterations investigating the optimal convergence rate with respect to total work per unit step a strategy for when to reevaluate the jacobian a strategy for when to refactorize the iteration matrix coordination with stepsize control examples will be given showing that the new overall strategy works efficiently in particular the new strategy admits a restrained stepsize variation without refactorizations thus permitting the use of a smoother stepsize sequence the strategy is of equal importance for runge kutta and multistep methods

convergence of the solutions of the discounted hamilton jacobi equation we consider a continuous coercive hamiltonian h on the cotangent bundle of the compact connected manifold m which is convex in the momentum if u lambda m rightarrow mathbb r is the viscosity solution of the discounted equation begin aligned lambda u lambda x h x mathrm d x u lambda c h end aligned where c h is the critical value we prove that u lambda converges uniformly as lambda rightarrow 0 to a specific solution u 0 m rightarrow mathbb r of the critical equation begin aligned h x mathrm d x u c h end aligned we characterize u 0 in terms of peierls barrier and projected mather measures as a corollary we infer that the ergodic approximation as introduced by lions papanicolaou and varadhan in 1987 in their seminal paper on homogenization of hamilton jacobi equations selects a specific corrector in the limit

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

the lax oleinik semi group a hamiltonian point of view the weak kam theory was developed by fathi in order to study the dynamics of convex hamiltonian systems it somehow makes a bridge between viscosity solutions of the hamilton jacobi equation and mather invariant sets of hamiltonian systems although this was fully understood only a posteriori these theories converge under the hypothesis of convexity and the richness of applications mostly comes from this remarkable convergence in the present course we provide an elementary exposition of some of the basic concepts of weak kam theory in a companion lecture albert fathi exposes the aspects of his theory which are more directly related to viscosity solutions here on the contrary we focus on dynamical applications even if we also discuss some viscosity aspects to underline the connections with fathi s lecture the fundamental reference on weak kam theory is the still unpublished book of albert fathi textit weak kam theorem in lagrangian dynamics although we do not offer new results our exposition is original in several aspects we only work with the hamiltonian and do not rely on the lagrangian even if some proofs are directly inspired from the classical lagrangian proofs this approach is made easier by the choice of a somewhat specific setting we work on rm d and make uniform hypotheses on the hamiltonian this allows us to replace some compactness arguments by explicit estimates for the most interesting dynamical applications however the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic in space hamiltonians our exposition is centered on the cauchy problem for the hamilton jacobi equation and the lax oleinik evolution operators associated to it dynamical applications are reached by considering fixed points of these evolution operators the weak kam solutions the evolution operators can also be used for their regularizing properties this opens a second way to dynamical applications

evaluating numerical ode dae methods algorithms and software until recently the testing of ode dae software has been limited to simple comparisons and benchmarking the process of developing software from a mathematically specified method is complex it entails constructing control structures and objectives selecting iterative methods and termination criteria choosing norms and many more decisions most software constructors have taken a heuristic approach to these design choices and as a consequence two different implementations of the same method may show significant differences in performance yet it is common to try to deduce from software comparisons that one method is better than another such conclusions are not warranted however unless the testing is carried out under true ceteris paribus conditions moreover testing is an empirical science and as such requires a formal test protocol without it conclusions are questionable invalid or even false we argue that ode dae software can be constructed and analyzed by proven standard scientific techniques instead of heuristics the goals are computational stability reproducibility and improved software quality we also focus on different error criteria and norms and discuss modifications to daspk and radau5 finally some basic principles of a test protocol are outlined and applied to testing these codes on a variety of problems

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

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

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

adaptive time stepping and computational stability we investigate the effects of adaptive time stepping and other algorithmic strategies on the computational stability of ode codes we show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability a series of computational experiments with the standard implementation of dassl and a modified version including stepsize control based on digital filters is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense the inherent performance and stability of dassl are therefore much greater than the standard implementation seems to suggest

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

weak kam from a pde point of view viscosity solutions of the hamilton jacobi equation and aubry set we introduce the notion of a viscosity solution for the first order hamilton jacobi equation in the more general setting of manifolds to obtain a weak kam theory using only tools from partial differential equations this work should be accessible to people with no prior knowledge of the subject

stabilized explicit runge kutta methods for multi asset american options american derivatives have become very popular instruments in financial markets however they are more complicated to price than european options since at each time level we have to determine not only the option value but also whether or not it should be exercised several procedures have been proposed to dissolve these difficulties but they usually involve the solution of nonlinear partial differential equations pdes in the case of multi dimensional problems solving these equations is a very challenging task in this paper we propose stabilized explicit runge kutta serk methods to solve this kind of problems they can easily be applied to many different classes of problems with large dimensions and they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes as for example lapack and blas packages or multigrid or iterative methods applied together with newton type algorithms and are especially well suited for the method of lines mol discretizations of parabolic pdes

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

adaptive stepsize based on control theory for stochastic differential equations the numerical solution of stochastic differential equations sdes has been focussed recently on the development of numerical methods with good stability and order properties these numerical implementations have been made with fixed stepsize but there are many situations when a fixed stepsize is not appropriate in the numerical solution of ordinary differential equations much work has been carried out on developing robust implementation techniques using variable stepsize it has been necessary in the deterministic case to consider the best choice for an initial stepsize as well as developing effective strategies for stepsize control the same of course must be carried out in the stochastic case in this paper proportional integral pi control is applied to a variable stepsize implementation of an embedded pair of stochastic runge kutta methods used to obtain numerical solutions of nonstiff sdes for stiff sdes the embedded pair of the balanced milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change the extension of these stepsize controllers from a digital filter theory point of view via pi with derivative pid control will also be implemented the implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy

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

lasry lions approximations for discounted hamilton jacobi equations abstract we study the lasry lions approximation using the kernel determined by the fundamental solution with respect to a time dependent tonelli lagrangian this approximation process is also applied to the viscosity solutions of the discounted hamilton jacobi equations

an improved approximate newton method for implicit runge kutta formulas abstract implicit runge kutta irk methods such as the s stage radau iia method with s 3 5 or 7 for solving stiff ordinary differential equation systems have excellent stability properties and high solution accuracy orders but their high computing costs in solving their nonlinear stage equations have seriously limited their applications to large scale problems to reduce such a cost several approximate newton algorithms were developed including a commonly used one called the simplified newton method in this paper a new approximate jacobian matrix and two new test rules for controlling the updating of approximate jacobian matrices are proposed yielding an improved approximate newton method theoretical and numerical analysis show that the improved approximate newton method can significantly improve the convergence and performance of the simplified newton method