an empirical non parametric likelihood family of data based benford like distributions abstract a mathematical expression known as benford s law provides an example of an unexpected relationship among randomly selected sequences of first significant digits fsds newcomb note on the frequency of use of the different digits in natural numbers am j math 4 1881 39 40 and later benford the law of anomalous numbers proc am philos soc 78 4 1938 551 572 conjectured that fsds would exhibit a weakly monotonic decreasing distribution and proposed a frequency proportional to the logarithmic rule unfortunately the benford fsd function does not hold for a wide range of scale invariant multiplicative data to confront this problem we use information theoretic methods to develop a data based family of alternative benford like exponential distributions that provide null hypotheses for testing purposes two data sets are used to illustrate the performance of generalized benford like distributions

decoding least effort and scaling in signal frequency distributions abstract here assuming a general communication model where objects map to signals a power function for the distribution of signal frequencies is derived the model relies on the satisfaction of the receiver hearer communicative needs when the entropy of the number of objects per signal is maximized evidence of power distributions in a linguistic context some of them with exponents clearly different from the typical 2 of zipf s law is reviewed and expanded we support the view that zipf s law reflects some sort of optimization but following a novel realistic approach where signals e g words are used according to the objects e g meanings they are linked to our results strongly suggest that many systems in nature use non trivial strategies for easing the interpretation of a signal interestingly constraining just the number of interpretations of signals does not lead to scaling

statistical dynamics of religion evolutions abstract a religion affiliation can be considered as a degree of freedom of an agent on the human genre network a brief review is given on the state of the art in data analysis and modelization of religious questions in order to suggest and if possible initiate further research after using a statistical physics filter we present a discussion of the evolution of 18 so called religions as measured through their number of adherents between 1900 and 2000 some emphasis is made on a few cases presenting a minimum or a maximum in the investigated time range thereby suggesting a competitive ingredient to be considered besides the well accepted at birth attachment effect the importance of the external field is still stressed through an avrami late stage crystal growth like parameter the observed features and some intuitive interpretations point to opinion based models with vector rather than scalar like agents

multidimensional adaptive relevance vector machines for uncertainty quantification we develop a bayesian uncertainty quantification framework using a local binary tree surrogate model that is able to make use of arbitrary bayesian regression methods the tree is adaptively constructed using information about the sensitivity of the response and is biased by the underlying input probability distribution the local bayesian regressions are based on a reformulation of the relevance vector machine model that accounts for the multiple output dimensions a fast algorithm for training the local models is provided the methodology is demonstrated with examples in the solution of stochastic differential equations

universal features in the growth dynamics of religious activities we quantify and analyze the growth dynamics of a religious group in 140 countries for a 47 year period 1959 2005 we find that i the distribution of annual logarithmic growth rates exhibits the same functional form for distinct size scales and ii the standard deviation of growth rates scales with size as a power law both findings hold for distinct measures of religious activity these results are in surprising agreement with those found in the study of economic activities and scientific research suggesting that religious activities are governed by universal growth mechanisms we also compare the empirical findings on religious activities with the predictions of general models recently proposed in the context of complex organizations our findings should provide useful information for a better understanding of the mechanisms governing the growth of religion

dynamical model and nonextensive statistical mechanics of a market index on large time windows the shape and tails of partial distribution functions pdf for a financial signal i e the s and the turbulent nature of the markets are linked through a model encompassing tsallis nonextensive statistics and leading to evolution equations of the langevin and fokker planck type a model originally proposed to describe the intermittent behavior of turbulent flows describes the behavior of normalized log returns for such a financial market index for small and large time windows and both for small and large log returns these turbulent market volatility of normalized log returns distributions can be sufficiently well fitted with a ensuremath chi 2 distribution the transition between the small time scale model of nonextensive intermittent process and the large scale gaussian extensive homogeneous fluctuation picture is found to be at ca a 200 day time lag the intermittency exponent ensuremath kappa in the framework of the kolmogorov log normal model is found to be related to the scaling exponent of the pdf moments thereby giving weight to the model the large value of ensuremath kappa points to a large number of cascades in the turbulent process the first kramers moyal coefficient in the fokker planck equation is almost equal to zero indicating no restoring force a comparison is made between normalized log returns and mere price increments

relations between allometric scalings and fluctuations in complex systems the case of japanese firms abstract to elucidate allometric scaling in complex systems we investigated the underlying scaling relationships between typical three scale indicators for approximately 500 000 japanese firms namely annual sales number of employees and number of business partners first new scaling relations including the distributions of fluctuations were discovered by systematically analyzing conditional statistics second we introduced simple probabilistic models that reproduce all these scaling relations and we derived relations between scaling exponents and the magnitude of fluctuations

verhulst lotka volterra vlv model of ideological struggle abstract a model for ideological struggles is formulated the underlying set is a closed one like a country but in which the population size is variable in time the dynamics of the struggle is described by model equations of verhulst lotka volterra kind several ideologies compete to increase their number of adepts such followers can be either converted from one ideology to another or become followers of an ideology though being previously ideologically free a reverse process is also allowed two kinds of conversions are considered unitary conversion e g by means of mass communication tools or binary conversion e g by means of interactions between people it is found that the steady state when it exists depends on the number of ideologies moreover when the number of ideologies increases some tension arises between them this tension can change in the course of time we propose to measure the ideology tensions through an appropriately defined scale index finally it is shown that a slight change in the conditions of the environment can prevent the extinction of some ideology after almost collapsing the ideology can spread again and can affect a significant part of the country s population two kinds of such resurrection effects are described as phoenix effects

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

upper bound on the heat transport in a layer of fluid of infinite prandtl number rigid lower boundary and stress free upper boundary we obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite prandtl number with rigid lower boundary and stress free upper boundary because of the asymmetric boundary conditions the solutions of the euler lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on the upper and lower boundary of the fluid the obtained bound on the convective heat transport and the corresponding wave number are between the values for a fluid layer with two rigid boundaries and a fluid layer with two stress free boundaries

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

uncertainty propagation using infinite mixture of gaussian processes and variational bayesian inference uncertainty propagation in flow through porous media problems is a challenging problem this is due to the high dimensionality of the random property fields e g permeability and porosity as well as the computational complexity of the models that are involved the usual approach is to construct a surrogate response surface and then use it instead of the expensive model to carry out the uncertainty propagation task however the construction of the surrogate surface is hampered by various aspects such as the limited number of model evaluations that one can afford the curse of dimensionality multi variate responses with non trivial correlations potential localized features of the response and or discontinuities in this work we extend upon the concept of the multi output gaussian process mgp to effectively deal with all of these difficulties simultaneously this non trivial extension involves an infinite mixture of mgp s that is trained using variational bayesian inference prior to observing any data a dirichlet process is used to generate the components of the mgp mixture the bayesian nature of the model allows for the quantification of the uncertainties due to the limited number of simulations i e we can derive error bars for the statistics of interest the automatic detection of the mixture components by the variational inference algorithm is able to capture discontinuities and localized features without adhering to ad hoc constructions finally correlations between the components of multi variate responses are captured by the underlying mgp model in a natural way

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

understanding scaling through history dependent processes with collapsing sample space history dependent processes are ubiquitous in natural and social systems many such stochastic processes especially those that are associated with complex systems become more constrained as they unfold meaning that their sample space or their set of possible outcomes reduces as they age we demonstrate that these sample space reducing ssr processes necessarily lead to zipf s law in the rank distributions of their outcomes we show that by adding noise to ssr processes the corresponding rank distributions remain exact power laws p x x where the exponent directly corresponds to the mixing ratio of the ssr process and noise this allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds noisy ssr processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from 2 to we discuss several applications showing how ssr processes can be used to understand zipf s law in word frequencies and how they are related to diffusion processes in directed networks or aging processes such as in fragmentation processes ssr processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative preferential or self organized critical processes

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

a generalized multi resolution expansion for uncertainty propagation with application to cardiovascular modeling computational models are used in a variety of fields to improve our understanding of complex physical phenomena recently the realism of model predictions has been greatly enhanced by transitioning from deterministic to stochastic frameworks where the effects of the intrinsic variability in parameters loads constitutive properties model geometry and other quantities can be more naturally included a general stochastic system may be characterized by a large number of arbitrarily distributed and correlated random inputs and a limited support response with sharp gradients or event discontinuities this motivates continued research into novel adaptive algorithms for uncertainty propagation particularly those handling high dimensional arbitrarily distributed random inputs and non smooth stochastic responses in this work we generalize a previously proposed multi resolution approach to uncertainty propagation to develop a method that improves computational efficiency can handle arbitrarily distributed random inputs and non smooth stochastic responses and naturally facilitates adaptivity i e the expansion coefficients encode information on solution refinement our approach relies on partitioning the stochastic space into elements that are subdivided along a single dimension or in other words progressive refinements exhibiting a binary tree representation we also show how these binary refinements are particularly effective in avoiding the exponential increase in the multi resolution basis cardinality and significantly reduce the regression complexity for moderate to high dimensional random inputs the performance of the approach is demonstrated through previously proposed uncertainty propagation benchmarks and stochastic multi scale finite element simulations in cardiovascular flow

breather and soliton wave families for the sine gordon equation the approach of steuerwaldlamb for obtaining exact solutions of the sinegordon equation is further developed through transformations by means of which complicated solutions of the corresponding odes are obtained without any explicit integration polynomial and nonpolynomial transformations are discussed and several of them are used to obtain breather and soliton solutions of the sinegordon equation the obtained solutions are discussed from the point of view of the electrodynamics of the long josephson junction

gaussian processes with built in dimensionality reduction uncertainty quantification uq tasks such as model calibration uncertainty propagation and optimization under uncertainty typically require several thousand evaluations of the underlying computer codes to cope with the cost of simulations one replaces the real response surface with a cheap surrogate based e g on polynomial chaos expansions neural networks support vector machines or gaussian processes gp however the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases this curse of dimensionality can only be addressed if the response exhibits some special structure that can be discovered and exploited a wide range of physical responses exhibit a special structure known as an active subspace as an as is a linear manifold of the stochastic space characterized by maximal response variation the idea is that one should first identify this low dimensional manifold project the high dimensional input onto it and then link the projection to the output if the dimensionality of the as is low enough then learning the link function is a much easier problem than the original problem of learning a high dimensional function the classic approach to discovering the as requires gradient information a fact that severely limits its applicability furthermore and partly because of its reliance to gradients it is not able to handle noisy observations the latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators e g through molecular dynamics codes in this work we develop a probabilistic version of as which is gradient free and robust to observational noise our approach relies on a novel gaussian process regression with built in dimensionality reduction in particular the as is represented as an orthogonal projection matrix that serves as yet another covariance function hyper parameter to be estimated from the data to train the model we design a two step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the stiefel manifold i e the manifold of matrices with orthogonal columns the additional benefit of our probabilistic formulation is that it allows us to select the dimensionality of the as via the bayesian information criterion we validate our approach by showing that it can discover the right as in synthetic examples without gradient information using both noiseless and noisy observations we demonstrate that our method is able to discover the same as as the classical approach in a challenging one hundred dimensional problem involving an elliptic stochastic partial differential equation with random conductivity finally we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system

multi output separable gaussian process towards an efficient fully bayesian paradigm for uncertainty quantification computer codes simulating physical systems usually have responses that consist of a set of distinct outputs e g velocity and pressure that evolve also in space and time and depend on many unknown input parameters e g physical constants initial boundary conditions etc furthermore essential engineering procedures such as uncertainty quantification inverse problems or design are notoriously difficult to carry out mostly due to the limited simulations available the aim of this work is to introduce a fully bayesian approach for treating these problems which accounts for the uncertainty induced by the finite number of observations our model is built on a multi dimensional gaussian process that explicitly treats correlations between distinct output variables as well as space and or time the proper use of a separable covariance function enables us to describe the huge covariance matrix as a kronecker product of smaller matrices leading to efficient algorithms for carrying out inference and predictions the novelty of this work is the recognition that the gaussian process model defines a posterior probability measure on the function space of possible surrogates for the computer code and the derivation of an algorithmic procedure that allows us to sample it efficiently we demonstrate how the scheme can be used in uncertainty quantification tasks in order to obtain error bars for the statistics of interest that account for the finite number of observations

explaining the uneven distribution of numbers in nature the laws of benford and zipf abstract the distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of benford s law we analyze in detail this property for different data sets and give a general explanation for the origin of the benford s law in terms of multiplicative processes we show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity finally we derive a relation between the generalized benford s law and the popular zipf s law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics

scaling behavior in a proportional voting process we perform a statistical analysis on the proportional elections held in brazil in october 1998 we show that the distribution of votes among candidates for the whole country follows a power law n v ensuremath propto v ensuremath ensuremath alpha with ensuremath alpha 1 00 ifmmode pm else textpm fi 0 02 extending over two orders of magnitude the voting distributions for several states of the federation also display scale invariant behavior with ensuremath alpha ensuremath approx 1 we argue that this particular voting system can be modeled as a typical multiplicative process in which the choice of the candidate is governed by a product of probabilities

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

benford s law a poisson perspective abstract benford s law is a counterintuitive statistical law asserting that the distribution of leading digits taken from a large ensemble of positive numerical values that range over many orders of scale is logarithmic rather than uniform as intuition suggests in this paper we explore benford s law from a poisson perspective considering ensembles of positive numerical values governed by poisson process statistics we show that this poisson setting naturally accommodates benford s law and i establish a poisson characterization and a poisson multidigit extension of benford s law ii study a system invariant leading digit distribution which generalizes benford s law and establish a poisson characterization and a poisson multidigit extension of this distribution iii explore the universal emergence of the system invariant leading digit distribution couple this universal emergence to the universal emergence of the weibull and frechet extreme value distributions and distinguish the special role of benford s law in this universal emergence iv study the continued fractions counterpart of the system invariant leading digit distribution and establish a poisson characterization of this distribution and v unveil the elemental connection between the system invariant leading digit distribution and its continued fractions counterpart this paper presents a panoramic poisson approach to benford s law to its system invariant generalization and to its continued fractions counterpart

the first digit frequencies in data of turbulent flows abstract considering the first significant digits noted d in data sets of dissipation for turbulent flows the probability to find a given number d 1 or 2 or 9 would be 1 9 for a uniform distribution instead the probability closely follows newcomb benford s law namely p d log 1 1 d the discrepancies between newcomb benford s law and first digits frequencies in turbulent data are analysed through shannon s entropy the data sets are obtained with direct numerical simulations for two types of fluid flow an isotropic case initialized with a taylor green vortex and a channel flow results are in agreement with newcomb benford s law in nearly homogeneous cases and the discrepancies are related to intermittent events thus the scale invariance for the first significant digits which supports newcomb benford s law seems to be related to an equilibrium turbulent state namely with a significant inertial range a matlab octave program provided in appendix is such that part of the presented results can easily be replicated

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

the order of magnitude for moments for certain cotangent sums abstract we settle a question on the rate of growth of the moments of cotangent sums considered by the authors in their previous papers 7 8

adaptation and its impact on the dynamics of a system of three competing populations abstract we investigate how the adaptation of the competition coefficients of the competing populations for the same limited resource influences the system dynamics in the regions of the parameter space where chaotic motion of shilnikov kind exists we present results for two characteristic values of the competition coefficient adaptation factor the first value 0 05 belongs to the small interval of possible negative values of for this a transition to chaos by period doubling bifurcations occurs and a window of periodic motion exists between the two regions of chaotic motion with increasing the system becomes more dissipative and the number of the windows of periodic motion increases when 1 0 a region of transient chaos is observed after the last window of periodic motion we verify the picture of the system dynamics by power spectra histograms and autocorrelations and calculate the lyapunov exponents and kaplan yorke dimension finally we discuss the eligibility of the investigated system for a topological analysis

benford s law and first letter of words a universal first letter law fll is derived and described it predicts the percentages of first letters for words in novels the fll is akin to benford s law bl of first digits which predicts

the concept of force in population dynamics abstract the area of population dynamics has a rich history of the development and analysis of models of biological and social phenomena using ordinary differential equations this paper describes a method for understanding the influence one variable exerts on another in such models as a force with the relative effects of these forces providing a narrative explanation of the curvature in variable behaviour using the stock flow form of a model a symbolic notation is developed that identifies the forces with the causal pathways of the model s feedback loops a force is measured by its impact defined as the ratio of acceleration to rate of change computed by differentiation along its associated pathway between variables different phases of force dominance are determined to enhance the standard stability analysis of the models providing an explanation of model behaviour in newtonian mechanical terms the concepts developed are applied to well known models from mathematical biology the spruce budworm model where force dominance identifies scenarios that give clarity to intervention points and the lotka volterra predator prey model where the analysis highlights the importance of dissipative forces in achieving stability conclusions are drawn on the explanatory power of this approach with suggestions made for future work

long range anticorrelations and non gaussian behavior of a leaky faucet we find that intervals between successive drops from a leaky faucet display scale invariant long range anticorrelations characterized by the same exponents of heart beat to beat intervals of healthy subjects this behavior is also confirmed by numerical simulations on lattice and it is faucet width and flow rate independent the histogram for the drop intervals is also well described by a levy distribution with the same index for both histograms of healthy and diseased subjects this additional result corroborates the evidence for similarities between leaky faucets and healthy hearts underlying dynamics

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

the significant digit law in statistical physics abstract the occurrence of the nonzero leftmost digit i e 1 2 9 of numbers from many real world sources is not uniformly distributed as one might naively expect but instead the nature favors smaller ones according to a logarithmic distribution named benford s law we investigate three kinds of widely used physical statistics i e the boltzmann gibbs bg distribution the fermi dirac fd distribution and the bose einstein be distribution and find that the bg and fd distributions both fluctuate slightly in a periodic manner around benford s distribution with respect to the temperature of the system while the be distribution conforms to it exactly whatever the temperature is thus benford s law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature furthermore various elegant properties of benford s law especially the mantissa distribution of data sets are discussed

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

rise and fall of historic tram networks logistic approximation and discontinuous events abstract a logistic approximation was used to describe in terms of total length l and population h variables the growth and decay of historic transportation systems three successive stages separated for sharp discontinuities were detected for several european tramway and metro systems corresponding to a fast initial growth followed by an intermediate step of slow growth and a final stage of rapid decay a common generalized behaviour was obtained in the l h vs h variations relative to critical values of l and h parameters defined from the maximum in the l h ratio

applications of the first digit law to measure correlations the quasiempirical benford law predicts that the distribution of the first significant digit of random numbers obtained from mixed probability distributions is surprisingly meaningful and reveals some universal behavior we generalize this finding to examine the joint first digit probability of a pair of two random numbers and show that undetectable correlations by means of the usual covariance based measure can be identified in the statistics of the corresponding first digits we illustrate this new measure by analyzing the correlations and anticorrelations of the positions of two interacting particles in their quantum mechanical ground state this suggests that by using this measure the presence or absence of correlations can be determined even if only the first digit of noisy experimental data can be measured accurately

benford s law and the fsd distribution of economic behavioral micro data abstract in this paper we focus on the first significant digit fsd distribution of european micro income data and use information theoretic entropy based methods to investigate the degree to which benford s fsd law is consistent with the nature of these economic behavioral systems we demonstrate that benford s law is not an empirical phenomenon that occurs only in important distributions in physical statistics but that it also arises in self organizing dynamic economic behavioral systems the empirical likelihood member of the minimum divergence entropy family is used to recover country based income fsd probability density functions and to demonstrate the implications of using a benford prior reference distribution in economic behavioral system information recovery

incidence of q statistics in rank distributions we show that size rank distributions with power law decay often only over a limited extent observed in a vast number of instances in a widespread family of systems obey tsallis statistics the theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the lyapunov exponent is negligible or vanishes the relevant statistical mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints and the resulting duality of entropy indexes is seen to portray physically relevant information whereas the value of the index fixes the distribution s power law exponent that for the dual index 2 ensures the extensivity of the deformed entropy

evolution on trees a brief review about the extinctions of species and the case of genocide in a darwinistic punctuated equilibrium evolution model abstract numerical and theoretical investigations of species evolution in presence of extinction rules are recalled in the framework of punctuated equilibrium evolution models previous results on the bak sneppen model are mentioned together with extensions taking into account a variable number of species conditions for limited growth due to species interactions external fields or intrinsic population density conditions have already been considered an original genocide model is presented as a step towards describing effects due to the extinction of species critical exponents for i the avalanche distribution ii the steady state approach and iii the tree size are given a mean field approximation is shown to be quite accurate for describing the critical state the screening of species from the evolution process is not numerically studied

covid 19 flattening the curve and benford s law for many countries attempting to control the fast rising number of coronavirus cases and deaths the race is on to flatten the curve since the spread of coronavirus disease 2019 covid 19 has taken on pandemic proportions in the absence of significant control interventions the curve could be steep with the number of covid 19 cases growing exponentially in fact this level of proliferation may already be happening since the number of patients infected in italy closely follows an exponential trend thus we propose a test when the numbers are taken from an exponential distribution it has been demonstrated that they automatically follow benford s law bl as a result if the current control interventions are successful and we flatten the curve i e we slow the rate below an exponential growth rate then the number of infections or deaths will not obey bl for this reason bl may be useful for assessing the effects of the current control interventions and may be able to answer the question how flat is flat enough in this study we used an epidemic growth model in the presence of interventions to describe the potential for a flattened curve and then investigated whether the epidemic growth model followed bl for ten selected countries with a relatively high mortality rate among these countries south korea showed a particularly high degree of control intervention although all of the countries have aggressively fought the epidemic our analysis shows that all countries except for japan satisfied bl indicating the growth rates of covid 19 were close to an exponential trend based on the simulation table in this study bl test shows that the data from japan is incorrect

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

beyond the zipf mandelbrot law in quantitative linguistics abstract in this paper the zipf mandelbrot law is revisited in the context of linguistics despite its widespread popularity the zipf mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus in particular we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation finally it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by c tsallis

benford s law and complex atomic spectra we found that in transition arrays of complex atomic spectra the strengths of electric dipolar lines obey benford s law which means that their significant digits follow a logarithmic distribution favoring the smallest values this indicates that atomic processes result from the superposition of uncorrelated probability laws and that the occurrence of digits reflects the constraints induced by the selection rules furthermore benford law can be a useful test of theoretical spectroscopic models its applicability to the statistics of electric dipolar lines can be understood in the framework of random matrix theory and is consistent with the porter thomas law

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

a lower bound in an approximation problem involving the zeros of the riemann zeta function abstract we slightly improve the lower bound of baez duarte balazard landreau and saias in the nyman beurling formulation of the riemann hypothesis as an approximation problem we construct hilbert space vectors which could prove useful in the context of the so called hilbert polya idea

generalised central limit theorems for growth rate distribution of complex systems we introduce a solvable model of randomly growing systems consisting of many independent subunits scaling relations and growth rate distributions in the limit of infinite subunits are analysed theoretically various types of scaling properties and distributions reported for growth rates of complex systems in a variety of fields can be derived from this basic physical model statistical data of growth rates for about 1 million business firms are analysed as a real world example of randomly growing systems not only are the scaling relations consistent with the theoretical solution but the entire functional form of the growth rate distribution is fitted with a theoretical distribution that has a power law tail

dynamical consequences of adaptation of the growth rates in a system of three competing populations we investigate the nonlinear dynamics of a system of populations competing for the same limited resource for the case where each of the populations adapts its growth rate to the total number of individuals in all populations we consider regions of parameter space where chaotic motion of the shilnikov kind exists and present results for two characteristic values of the growth ratio adaptation factor r r 0 15 and 5 negative r can lead to vanishing of regions of chaotic motion and to a stabilization of a fixed point of the studied model system of differential equations positive r lead to changes of the shape of the bifurcation diagrams in comparison with the bifurcation diagrams for the case without adaptation for the case r 5 we observe transition to chaos by period doubling bifurcations windows of periodic motion between the regions of chaotic motion and a region of transient chaos after the last window of periodic motion the lyapunov dimension for the chaotic attractors is close to two and the lyapunov spectrum has a structure which allows a topological analysis of the attractors of the investigated system

the brjuno functions and their regularity properties we show that various possible versions of the brjuno function based on different kinds of continued fraction developments are all equivalent and we study their regularity l p bmo and holder properties through a systematic analysis of the functional equation which they fulfill

multi output local gaussian process regression applications to uncertainty quantification we develop an efficient bayesian uncertainty quantification framework using a novel treed gaussian process model the tree is adaptively constructed using information conveyed by the observed data about the length scales of the underlying process on each leaf of the tree we utilize bayesian experimental design techniques in order to learn a multi output gaussian process the constructed surrogate can provide analytical point estimates as well as error bars for the statistics of interest we numerically demonstrate the effectiveness of the suggested framework in identifying discontinuities local features and unimportant dimensions in the solution of stochastic differential equations

the law of the leading digits and the world religions abstract benford s law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger ones we investigate here numerical data on the country wise adherent distribution of seven major world religions i e christianity islam buddhism hinduism sikhism judaism and baha ism to see if the proportion of the leading digits occurring in the distribution conforms to benford s law we find that the adherent data of all the religions except christianity excellently does conform to benford s law furthermore unlike the adherent data on christianity the significant digit distribution of the three major christian denominations i e catholicism protestantism and orthodoxy obeys the law thus in spite of their complexity general laws can be established for the evolution of religious groups

simple model of evolution with variable system size a simple model of biological extinction with variable system size is presented that exhibits a power law distribution of extinction event sizes the model is a generalization of a model recently introduced by newman proc r soc lond b265 1605 1996 both analytical and numerical analysis show that the exponent of the power law distribution depends only marginally on the growth rate g at which new species enter the system and is equal to the one of the original model in the limit g to infty a critical growth rate g c can be found below which the system dies out under these model assumptions stable ecosystems can only exist if the regrowth of species is sufficiently fast

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations we discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities as simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of weierstrass and jacobi we show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions this system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation the methodology is illustrated by obtaining solitary wave solutions for the generalized korteweg devries equation and by obtaining solutions of the higher order korteweg devries equation

the first digit frequencies of prime numbers and riemann zeta zeros prime numbers seem to be distributed among the natural numbers with no law other than that of chance however their global distribution presents a quite remarkable smoothness such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes in this paper we show that a generalization of the well known first digit benford s law which addresses the rate of appearance of a given leading digit d in datasets describes with astonishing precision the statistical distribution of leading digits in the prime number sequence moreover a reciprocal version of this pattern also takes place in the sequence of the non trivial riemann zeta zeros we prove that the prime number theorem is in the final analysis responsible for these patterns

scale free networks with exponent one a majority of studied models for scale free networks have degree distributions with exponents greater than two real networks however can demonstrate essentially more heavy tailed degree distributions we explore two models of scale free equilibrium networks that have the degree distribution exponent ensuremath gamma 1 p q ensuremath sim q ensuremath ensuremath gamma such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high our models exploit a rewiring mechanism they are local in the sense that no knowledge of the network structure apart from the immediate neighborhood of the vertices is required these models generate uncorrelated networks in the infinite size limit where they are solved explicitly we investigate finite size effects by the use of simulations we find that both models exhibit disassortative degree degree correlations for finite network sizes in addition we observe a markedly degree dependent clustering in the finite networks we indicate a real world network with a similar degree distribution

revisiting the benford law when the benford like distribution of leading digits in sets of numerical data is expectable abstract the benford law states that the frequencies of decimal digits at the first place of numbers corresponding to various kinds of statistical or experimental data are not equal changing from 0 3 for 1 to 0 04 for 9 the corresponding frequencies distribution is described by the logarithmic function as is shown in the present article the benford distribution is a particular case of a more general mathematical statement namely if a function describing the dependence between two measurable quantities has a positive second derivative then the mentioned above frequencies decrease for digits from 1 to 9 the exact benford distribution is valid for the exponential function only explicit expressions for frequencies of leading digits are obtained and specified for the power logarithmic and tangent functions as examples the kinematic experiment was performed to illustrate the above results also the tabulated data on thermal conductivities of liquids confirm the proposed formula for frequencies distribution

statistical distributions connected to motion of substance in a channel of a network abstract we discuss a model of motion of substance through the nodes of a channel of a network the channel can be modeled by a chain of urns where each urn can exchange substance with the neighboring urns in addition the urns can exchange substance with the network nodes and the extension of the model with respect to the previous research is that we consider also the possibility for exchange of substance among the urns and the environment of the network we consider stationary regime of motion of substance through a finite channel stationary regime of exchange of substance along the chain of urns and obtain a class of statistical distributions for the substance in the channel our attention is concentrated on this class of distributions and we show that for the case of finite channel the class of distributions contains as particular cases truncated versions of the families of distributions of katz ord kemp etc several characteristics of the obtained class of distributions are calculated the theory for the case of chain containing infinite number of urns is presented in the appendix

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

a stochastic multiscale framework for modeling flow through random heterogeneous porous media flow through porous media is ubiquitous occurring from large geological scales down to the microscopic scales several critical engineering phenomena like contaminant spread nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system any realistic modeling of these transport phenomena has to resolve two key issues i the multi length scale variations in permeability that these systems exhibit and ii the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes a stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features a stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process recent developments in linear and non linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model an adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations spdes the framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given

explicit estimates of sums related to the nyman beurling criterion for the riemann hypothesis abstract we give an estimate for sums appearing in the nyman beurling criterion for the riemann hypothesis these sums contain the mobius function and are related to the imaginary part of the estermann zeta function the estimate is remarkably sharp in comparison to other sums containing the mobius function the bound is smaller than the trivial bound essentially the number of terms by a fixed power of that number the exponent is made explicit the methods intensively use tools from the theory of continued fractions and from the theory of fourier series

first digit law in nonextensive statistics nonextensive statistics characterized by a nonextensive parameter q is a promising and practically useful generalization of the boltzmann statistics to describe power law behaviors from physical and social observations we here explore the unevenness of the first digit distribution of nonextensive statistics analytically and numerically we find that the first digit distribution follows benford s law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature the fluctuation decreases when q increases and the result converges to benford s law exactly as q approaches 2 the relevant regularities between nonextensive statistics and benford s law are also presented and discussed

q exponential distribution in urban agglomeration usually the studies of distributions of city populations have been reduced to power laws in such analyses a common practice is to consider cities with more than one hundred thousand inhabitants here we argue that the distribution of cities for all ranges of populations can be well described by using a q exponential distribution this function which reproduces the zipf mandelbrot law is related to the generalized nonextensive statistical mechanics and satisfies an anomalous decay equation

the dynamics of coupled logistic social groups a society made up of a network of social groups is investigated each group is partitioned into two mutually exclusive subsets with the movement of members between the two subsets being modelled via a logistic like equation we consider various ways in which the groups in the network may influence each other via both group size and the utility groups place on the possible subsets scenarios where social groups act as agenda setters for the rest of the society are considered a number of analytic and numerical results are presented

on the nature of benford s law we study multiplicative and a ne sequences of real numbers de1ned by n j 1 j n j j where j and j are sequences of positive real numbers in the multiplicative case j 0 for all j we investigate the conditions under which the leading digits k of n j have the following probability distribution known as benford s law p k log10 k 1 k we present two main results first we show that contrary to the usual assumption in the literature j does not necessarily need to come from a chaotic or independent random process for benford s law to hold the multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing second we give conditions under which the distribution of the 1rst digits of an a ne process displays benford s law our proofs use techniques from ergodic theory c 2002 elsevier science b v all rights reserved

universal features in the growth dynamics of complex organizations we analyze the fluctuations in the gross domestic product gdp of 152 countries for the period 1950 1992 we find that i the distribution of annual growth rates for countries of a given gdp decays with fatter tails than for a gaussian and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with results on firm growth these results are consistent with the hypothesis that the evolution of organizations with complex structure is governed by similar growth mechanisms s0031 9007 98 07339 6 pacs numbers 02 50 ey 01 75 m 05 40 j 05 45 b in the study of physical systems the analysis of the scaling properties of the fluctuations has been shown to give important information regarding the underlying processes responsible for the observed macroscopic behavior in contrast most studies on the time evolution of economic time series have concentrated on average growth rates 1 18 here we investigate the possibility that the study of fluctuations in economics may also lead to a better understanding of the mechanisms responsible for the observed dynamics 19 23 we therefore analyze the fluctuations in the growth rate of the gross domestic product gdp of 152 countries during the period 1950 1992 24 we will show that i the distribution of annual growth rates for countries of a given gdp is consistent for a certain range with an exponential decay and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with