uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

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

minimum probability of error for asynchronous gaussian multiple access channels consider a gaussian multiple access channel shared by k users who transmit asynchronously independent data streams by modulating a set of assigned signal waveforms the uncoded probability of error achievable by optimum multiuser detectors is investigated it is shown that the k user maximum likelihood sequence detector consists of a bank of single user matched filters followed by a viterbi algorithm whose complexity per binary decision is o 2 k the upper bound analysis of this detector follows an approach based on the decomposition of error sequences the issues of convergence and tightness of the bounds are examined and it is shown that the minimum multiuser error probability is equivalent in the iow noise region to that of a single user system with reduced power these results show that the proposed multiuser detectors afford important performance gains over conventional single user systems in which the signal constellation carries the entire burden of complexity required to achieve a given performance level

the cauchy problem in one dimensional nonlinear viscoelasticity abstract the initial value problem for a nonlinear hyperbolic volterra equation which models the motion of an unbounded viscoelastic bar is studied under physically motivated assumptions the existence of a unique globally defined classical solution is established provided the initial data are sufficiently smooth and small boundedness and asymptotic behavior are also discussed this analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels

weak solutions of a class of quasilinear hyperbolic integro differential equations describing viscoelastic materials global weak solutions of scalar second order quasilinear hyperbolic integro differential equations with singular kernels are constructed perturbations of rest states are shown to propagate with finite speed smoothing effects of the solution operator are exhibited and conditions for the asymptotic stability of rest states are given the equations arise in viscoelasticity

application of a quantum ensemble model to linguistic analysis abstract a new set of parameters to describe the behavior of word frequencies in texts is proposed an analogy between the word frequency distribution and the bose distribution is suggested and the notion of temperature is introduced for this case the calculations are made for english ukrainian and the guinean maninka languages a correlation between the language structure the level of analyticity and the defined parameters is shown to exist

indecomposable error sequences in multiuser detection in this paper we provide a graph based characterization of the set of indecomposable sequences that are useful in the computation of an upper bound to error probability of maximum likelihood ml detection in code division multiple access cdma multiuser systems we apply this characterization to a k user symmetric system and to a two user two rate system it leads to a precise calculation of the bound which is then compared to the performance of a decorrelator

global smooth solutions of the equations of motion of a nonlinear fluid with fading memory the initial value problem associated with the equation of motion of an incompressible simple fluid with exponentially fading memory is studied suitable function spaces are constructed and a local existence theorem is established by the method pf artificial viscosity several energy integrals are obtained in order to estimate the size of solutions in suitable function spaces it is shown that the solution are globally defined when the intial data are small and that they decay to equilibrium are time tends to infinity

a local existence and uniqueness theorem for a k bkz fluid we consider the three dimensional motion of a viscoelastic liquid occupying all of space the constitutive law is assumed to be of the form suggested by kaye 13 and bernstein kearsley zapas 2 an existence and uniqueness result for solutions of the initial value problem on sufficiently short time intervals is proved using kato s theory of quasilinear hyperbolic equations

a new ber upper bound for multiuser downlink systems we consider improvement of bit error rate ber upper bounds based on the notion of decomposable error vectors devs introduced by s verdu for downlink multiuser space time systems over rayleigh fading channels although omission of devs and limiting before averaging lba technique are known to improve the standard union bound ub considerably it is mathematically intractable to obtain a bound in a closed form in this paper assuming maximum likelihood ml detection for downlink multiuser space time spreading sts systems we introduce a new modification of the decomposability of error vectors and propose a new ber upper bound we prove that the proposed bound is tighter than the standard union bound and the bound based on the overall decomposability the numerical results show that the proposed bound is also tighter than the bound based on positive sum decomposability psd in the scenario of downlink multiuser transmission

hamiltonian dynamics of preferential attachment prediction and control of network dynamics are grand challenge problems in network science the lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades here we study the dynamics of networks evolving according to preferential attachment pa known to approximate well the large scale growth dynamics of a variety of real networks we show that this dynamics is hamiltonian thus casting the study of complex networks dynamics to the powerful canonical formalism in which the time evolution of a dynamical system is described by hamilton s equations we derive the explicit form of the hamiltonian that governs network growth in pa this hamiltonian turns out to be nearly identical to graph energy in the configuration model which shows that the ensemble of random graphs generated by pa is nearly identical to the ensemble of random graphs with scale free degree distributions in other words pa generates nothing but random graphs with power law degree distribution the extension of the developed canonical formalism for network analysis to richer geometric network models with non degenerate groups of symmetries may eventually lead to a system of equations describing network dynamics at small scales

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

a nonlinear functional differential equation in banach space with applications to materials with fading memory abstract the report studies a nonlinear functional differential equation in banach space this equation is an abstract form of the equations of motion for nonlinear materials with fading memory its basic structure is hyperbolic in character so that global smooth solutions should not be expected in general memory effects however may induce a dissipative mechanism which although very subtle is effective so long as the solution is small the report shows that if the memory is dissipative in an appropriate sense then the history value problem associated with our equation has a unique global smooth solution provided the initial history and forcing are suitably smooth and small the proof combines a fixed point argument to establish local existence with a chain of global a priori energy type estimates the abstract results are then applied to establish global existence of smooth solutions to certain history value problems associated with the motion of nonlinear materials with fading memory under assumptions which are realistic within the framework of continuum mechanics

weak solutions of hyperbolic parabolic volterra equations the existence of a global weak solution satisfying certain a priori l bounds of the equation ut t x ft k t s a ux x s x ds f t x is established the kernel k is locally integrable and log convex and a has only one local minimum which is positive introduction the purpose of this paper is to establish the existence of a global weak solution of the equation 1 t ut t x k k t s a ux x s x ds f t x t o x e 0 1 u o x uo x x e o 1 u t o u t 1 o t o where the subscripts denote differentiation with respect to the corresponding variables and to show that this weak solution satisfies certain a priori l bounds equation 1 is of interest because it appears in mathematical models of viscoelasticity see 18 and 31 but also because it is of a form somewhere between a parabolic and a hyperbolic equation this is due to the fact that when k 1 we get the nonlinear wave equation utt t x a ux x t x ft t x and if on the other hand k is replaced by the delta functional then we get the nonlinear diffusion equation ut t x a ux x t x f t x the nonlinear wave equation that corresponds to the completely elastic case in mechanics has a weak solution with both ux and ut bounded and initially smooth solutions develop singularities in the derivatives in finite time the received by the editors september 22 1992 1991 mathematics subject classification primary 45k05 secondary 45d05 45n05 35k99 35l99 73f99

global smooth solutions for a class of parabolic integrodifferential equations the existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved if the integral kernel behaves like t near t 0 o with a 2 3 an existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for o

clustering of random scale free networks we derive the finite size dependence of the clustering coefficient of scale free random graphs generated by the configuration model with degree distribution exponent 2l ensuremath gamma l3 degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets we also find that for values of ensuremath gamma ensuremath approx 2 clustering is virtually size independent and at the same time becomes a de facto non self averaging topological property this implies that a single instance network is not representative of the ensemble even for very large network sizes

signal detection for mimo isi channels an iterative greedy improvement approach in this paper we consider the signal detection for multiple input multiple output intersymbol interference mimo isi channels with diverse assumptions on the channel knowledge perfect blind trained etc this general problem is cast into a unifying bayesian statistics framework with this formulation the optimal detector is the one maximizing the posterior signal density marginal maximum a posteriori map since the marginal map is hard to deal with a joint map formulation is proposed as a reasonable substitute that maximizes the posterior joint signal and channel density it is also shown that for independent and identically distributed i i d signals the two formulations will lead to very close results the joint map formulation leads to an iterative projection algorithm that alternates between the optimization over channel parameters and signaling matrices the bottleneck of iterative projections is on the finite alphabet constrained quadratic minimization we show that the notion of error decomposition can be bridged with greedy optimizations to construct iterative greedy search algorithms and examine their performance a particularization called full greedy search is shown to be able to reach the global optimum maximum likelihood solutions starting with any initialization since potential constraints in computational complexity may prohibit the application of this version of greedy search we explore the performance loss for greedy search implementations with complexity constraints arriving at deterministic performance bounds and a bit error rate ber upper bound the effect of model imprecision is also theoretically characterized based on the theoretical development an iterative local optimization with interference cancellation loic algorithm is proposed to achieve low complexity and exploit the finite alphabet constraint motivated by the sylvester structure it approximates the full greedy search by focusing on local error sequences it can also be regarded as a flexible interference cancellation strategy with noncausal information and iterative computations an empirical comparison of detectors with perfect channel knowledge demonstrated that the proposed loic algorithms can offer very attractive ber complexity tradeoffs

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

lax wendroff methods for hyperbolic history value problems this paper is concerned with lax wendroff methods for a class of hyperbolic history value problems these problems have the feature that globally in time smooth solutions exist if the data are sufficiently small and that solutions develop singularities for large data we prove second order convergence of the lax wendroff method for smooth solutions and investigate numerically the dependence on the initial data we demonstrate the occurrence of shock type singularities and compare the results to the quasilinear wave equation without volterra term

on computing verdu s upper bound for a class of maximum likelihood multiuser detection and sequence detection problems the upper bound derived by verdu 1986 is often used to evaluate the bit error performance of both the maximum likelihood ml sequence detector for single user systems and the ml multiuser detector for code division multiple access cdma systems this upper bound which is based on the concept of indecomposable error vectors ievs can be expensive to compute because in general the ievs may only be obtained using an exhaustive search we consider the identification of ievs for a particular class of ml detection problems commonly encountered in communications by exploiting the properties of the ievs for this case we develop an iev generation algorithm which has a complexity substantially lower than that of the exhaustive search we also show that for specific communication systems such as duobinary signaling the expressions of verdu s upper bound can be considerably simplified

on a class of quasilinear partial integrodifferential equations with singular kernels abstract we prove local and global existence theorems for a model equation in nonlinear viscoelasticity in contrast to previous studies we allow the memory function to have a singularity we approximate the equation by equations with regular kernels and use energy estimates to prove convergence of the approximate solutions

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

on a geometric view of multiuser detection for synchronous ds cdma channels we consider a geometric view of multiuser detection and present results on the problem of efficient computation of the optimal decision regions voronoi diagram for synchronous direct sequence code division multiple access ds cdma among other results it is shown that verdu s 1986 algebraic notion of indecomposable error sequences relates directly to the geometric construction of the voronoi diagram for multiuser detection based on the presented geometric results we introduce two multiuser detectors whose performance compares favorably to some popular existing detectors

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

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

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

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

information entropy and power law distributions for chaotic systems the power law is found for density distributions for the chaotic systems of most different nature physical geophysical biological economical social etc on the basis of the maximum entropy principle for the renyi entropy its exponent q is expressed as a function q of the renyi parameter the difference between the renyi and boltzmann shannon entropies a modified lyapunov functional r for the same power law distribution is negative and as a function of has a well defined minimum at which remains within the narrow range from 1 5 to 3 when varying other characteristic parameters of any concrete systems relevant variations of the exponent q are found within the range 1 3 5 the same range of observable values of q is typical for the various applications where the power law distribution takes place it is known under the following names triangular or trapezoidal in physics and technics gutenberg richter law in geophysics zipf pareto law in economics and the humanities lotka low in science of science etc as the negative r indicates self organisation of the system the negative minimum of r corresponds to the most self organised state thus the comparison between the calculated range of variations of q and observable values of the exponent q testifies that the most self organised states are as a rule realised regardless of the nature of a chaotic system

generalized entropies and the transformation group of superstatistics superstatistics describes statistical systems that behave like superpositions of different inverse temperatures so that the probability distribution is where the kernel f is nonnegative and normalized f d 1 we discuss the relation between this distribution and the generalized entropic form the first three shannon khinchin axioms are assumed to hold it then turns out that for a given distribution there are two different ways to construct the entropy one approach uses escort probabilities and the other does not the question of which to use must be decided empirically the two approaches are related by a duality the thermodynamic properties of the system can be quite different for the two approaches in that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes the transformation group is the euclidean group in one dimension

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

multiplicative processes and power laws takayasu sato and takayasu phys rev lett 79 966 1997 revisited the question of stochastic processes with multiplicative noise which have been studied in several different contexts over the past decades we focus on the regime found for a generic set of control parameters in which stochastic processes with multiplicative noise produce intermittency of a special kind characterized by a power law probability density distribution we briefly explain the physical mechanism leading to a power law probability distribution function and provide a list of references for these results dating back from a quarter of century we explain how the formulation in terms of the characteristic function developed by takayasu sato and takayasu can be extended to exponents ensuremath mu g2 which explains the reason for the lucky coincidence the multidimensional generalization of the results of takayasu sato and takayasu and the present status of the problem are briefly summarized the discovery of stretched exponential tails in the presence of the cutoff introduced by takayasu sato and takayasu is explained theoretically we end by briefly listing applications

theory of analogous force on number sets abstract a general statistical thermodynamic theory that considers given sequences of x integers to play the role of particles of known type in an isolated elastic system is proposed by also considering some explicit discrete probability distributions px for natural numbers we claim that they lead to a better understanding of probabilistic laws associated with number theory sequences of numbers are treated as the size measure of finite sets by considering px to describe complex phenomena the theory leads to derive a distinct analogous force fx on number sets proportional to px x t at an analogous system temperature t in particular this leads to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits the theory also allows to establish recursion relations to predict sequences of fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the benford s law in fibonacci numbers a possible relevance to prime numbers is also analyzed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

linear multiuser detectors for synchronous code division multiple access channels under the assumptions of symbol synchronous transmissions and white gaussian noise the authors analyze the detection mechanism at the receiver comparing different detectors by their bit error rates in the low background noise region and by their worst case behavior in a near far environment where the received energies of the users are not necessarily similar optimum multiuser detection achieves important performance gains over conventional single user detection at the expense of computational complexity that grows exponentially with the number of users it is shown that in the synchronous case the performance achieved by linear multiuser detectors is similar to that of optimum multiuser detection attention is focused on detectors whose linear memoryless transformation is a generalized inverse of the matrix of signature waveform crosscorrelations and on the optimum linear detector it is shown that the generalized inverse detectors exhibit the same degree of near far resistance as the optimum multiuser detectors the optimum linear detector is obtained

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

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

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

stochastic multiplicative processes with reset events we study a stochastic multiplicative process with reset events it is shown that the model develops a stationary power law probability distribution for the relevant variable whose exponent depends on the model parameters two qualitatively different regimes are observed corresponding to intermittent and regular behavior in the boundary between them the mean value of the relevant variable is time independent and the exponent of the stationary distribution equals ensuremath 2 the addition of diffusion to the system modifies in a nontrivial way the profile of the stationary distribution numerical and analytical results are presented

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

asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel in this paper we consider a problem which arises in viscoelasticity we prove an exponential decay of solutions under weaker assumptions than the ones frequently used in the literature in particular the kernels we are considering are not necessarily exponentially decaying to zero as was assumed before the present results improve also a previous work of the authors

empirical likelihood as a goodness of fit measure summary the method of empirical likelihood can be viewed as one of allocating probabilities to an n cell contingency table so as to minimise a goodness of fit criterion it is shown that when the cressie read power divergence statistic is used as the criterion confidence regions enjoying the same convergence rates as those found for empirical likelihood can be obtained for the entire range of values of the cressie read parameter including 1 maximum entropy 0 empirical likelihood and 1 pearson s z2 it is noted that in the power divergence family empirical likelihood is the only member which is bartlettcorrectable however simulation results suggest that for the mean using a scaled f distribution yields more accurate coverage levels for moderate sample sizes

global existence and asymptotic behavior for one dimensional thermoelasticity of integral type with hyperbolic heat conduction abstract we consider an initial boundary value problem for thermoelasticity of integral type with hyperbolic heat conduction we show that for small initial data in h 1 norm there exist globally defined solutions which are exponentially stable in bounded spatial domains

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

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

maximum likelihood sequence detection for intersymbol interference channels a new upper bound on error probability maximum likelihood sequence detection of digital signaling subject to intersymbol interference and additive white gaussian noise is considered a new approach to the error probability analysis results in an upper bound which is tighter than the forney bound in addition in the case of infinite length intersymbol interference a locally convergent bound is shown to hold under fairly general assumptions on the signal autocorrelation

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

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

entropy distribution and condensation in random networks with a given degree distribution the entropy of network ensembles characterizes the amount of information encoded in the network structure and can be used to quantify network complexity and the relevance of given structural properties observed in real network datasets with respect to a random hypothesis in many real networks the degrees of individual nodes are not fixed but change in time while their statistical properties such as the degree distribution are preserved here we characterize the distribution of entropy of random networks with given degree sequences where each degree sequence is drawn randomly from a given degree distribution we show that the leading term of the entropy of scale free network ensembles depends only on the network size and average degree and that entropy is self averaging meaning that its relative variance vanishes in the thermodynamic limit we also characterize large fluctuations of entropy that are fully determined by the average degree in the network finally above a certain threshold large fluctuations of the average degree in the ensemble can lead to condensation meaning that a single node in a network of size n can attract o n links

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

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

on the validation of the newcomb benford law and the weibull distribution in neuromuscular transmission abstract the neuromuscular junction represents a relevant substrate for revealing important biophysical mechanisms of synaptic transmission in this context calcium ions are important in the synapse machinery providing the nerve impulse transmission to the muscle fiber in this work we carefully investigated whether intervals of spontaneous electrical activity recorded in seven different calcium concentrations conform to the newcomb benford law our analysis revealed that electrical discharge of neuromuscular junction yields the expected values for newcomb benford law for first and second digits for different calcium concentrations on the other hand first two digits archived conformity especially for concentrations much above the physiological level we next examined previous theoretical studies establishing a relation between the law and lognormal and weibull distributions we showed that the weibull distribution is more appropriate to fit the intervals as compared to the lognormal distribution altogether the present findings strongly suggest that the electrical discharge dynamics is characterized by base invariance and scale invariance being the time intervals also well described by weibull statistics

q exponential weibull and q weibull distributions an empirical analysis abstract in a comparative study the q exponential and weibull distributions are employed to investigate frequency distributions of basketball baskets cyclone victims brand name drugs by retail sales and highway length in order to analyze the intermediate cases a distribution the q weibull one which interpolates the q exponential and weibull ones is introduced it is verified that the basketball baskets distribution is well described by a q exponential whereas the cyclone victims and brand name drugs by retail sales ones are better adjusted by a weibull distribution on the other hand for highway length the q exponential and weibull distributions do not give satisfactory adjustment being necessary to employ the q weibull distribution furthermore the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law q exponential with q 1 controversy

a class of hyperbolic volterra integrodifferential equations on considere l existence des solutions pour l equation integrodifferentielle de volterra u t a t u t 0 t g t s u s ds f 0 t f 1 t u t t 0 u 0 u 0

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

random texts exhibit zipf s law like word frequency distribution it is shown that the distribution of word frequencies for randomly generated texts is very similar to zipf s law observed in natural languages such as english the facts that the frequency of occurrence of a word is almost an inverse power law function of its rank and the exponent of this inverse power law is very close to 1 are largely due to the transformation from the word s length to its rank which stretches an exponential function to a power law function

the dependence of frequency distributions on multiple meanings of words codes and signs the dependence of the frequency distributions due to multiple meanings of words in a text is investigated by deleting letters by coding the words with fewer letters the number of meanings per code

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

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

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

one dimensional dynamical systems and benford s law one dimensional projections of at least almost all orbits of many multi dimensional dynamical systems are shown to follow benford s law i e their base b mantissa distribution is asymptotically logarithmic typically for all bases b as a generalization and uniflcation of known results it is proved that under a generic non resonance condition on a 2c d d for every z 2c d real and imaginary part of each non trivial component of a n z n2n0 and e at z t 0 follow benford s law also benford behavior is found to be ubiquitous for several classes of non linear maps and difierential equations in particular emergence of the logarithmic mantissa distribution turns out to be generic for complex analytic maps t with t 0 0 jt0 0 j 1 the results signiflcantly extend known facts obtained by other e g number theoretical methods and also generalize recent flndings for one dimensional systems

randomness versus specifics for word frequency distributions the text length dependence of real word frequency distributions can be connected to the general properties of a random book it is pointed out that this finding has strong implications when decidi

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

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

information content of turbulence we treat a turbulent velocity field as a message in the same way as a book or a picture all messages can be described by their entropy per symbol h defined as in shannon s theory of communication in a turbulent flow as the reynolds number re increases more correlated degrees of freedom are excited and participate in the turbulent cascade experiments in a turbulent soap film suggest that the spatial entropy density h is a decreasing function of re namely h proportionality logre const in the logistic map also analyzed here increasing the control parameter r increases h a modified logistic map with additional coupling to past iterations suggests the significance of correlations

statistical mechanical approach to human language abstract we use the formulation of equilibrium statistical mechanics in order to study some important characteristics of language using a simple expression for the hamiltonian of a language system which is directly implied by the zipf law we are able to explain several characteristic features of human language that seem completely unrelated such as the universality of the zipf exponent the vocabulary size of children the reduced communication abilities of people suffering from schizophrenia etc while several explanations are necessarily only qualitative at this stage we have nevertheless been able to derive a formula for the vocabulary size of children as a function of age which agrees rather well with experimental data

complex networks created by aggregation we study aggregation as a mechanism for the creation of complex networks in this evolution process vertices merge together which increases a number of highly connected hubs we study a range of complex network architectures produced by the aggregation fat tailed in particular scale free distributions of connections are obtained for both networks with a finite number of vertices and growing networks we observe a strong variation of a network structure with growing density of connections and find the phase transition of the condensation of edges finally we demonstrate the importance of structural correlations in these networks

a tsallis statistics based neural network model for novel word learning abstract we invoke the tsallis entropy formalism a nonextensive entropy measure to include some degree of non locality in a neural network that is used for simulation of novel word learning in adults a generalization of the gradient descent dynamics realized via nonextensive cost functions is used as a learning rule in a simple perceptron the model is first investigated for general properties and then tested against the empirical data gathered from simple memorization experiments involving two populations of linguistically different subjects numerical solutions of the model equations corresponded to the measured performance states of human learners in particular we found that the memorization tasks were executed with rather small but population specific amounts of nonextensivity quantified by the entropic index q our findings raise the possibility of using entropic nonextensivity as a means of characterizing the degree of complexity of learning in both natural and artificial systems