relative symmetric polynomials in this article we introduce the notion of a relative symmetric polynomial with respect to a permutation group and an irreducible character and we give answers for some natural questions about their structures in order to study symmetric polynomials with respect to linear characters we define the concept of relative vandermonde polynomial finally we present some interesting research problems concerning relative symmetric polynomials

global behavior of positive solutions of nonlinear three point boundary value problems we investigate the structure of the positive solution set for nonlinear three point boundary value problems of the form u h t f u 0 u 0 0 u 1 lambdau eta where eta epsilon 0 1 is given lambda epsilon 0 1 n is a parameter f epsilon c 0 infinity 0 infinity satisfies f s 0 for s 0 and h epsilon c 0 1 0 infinity is not identically zero on any subinterval of 0 1 our main results demonstrate the existence of continua of positive solutions of the above problem c 2004 elsevier ltd all rights reserved

entropy based tests of uniformity abstract the power properties of an entropy based test are investigated when used for testing uniformity on 0 1 percentage points and power against seven alternatives are reported compared with other tests of uniformity the entropy based test possesses good power properties for many alternatives some asymptotic null and alternative distributions are derived for sample sizes up to 100 the table of percentage points provides a practical guide for using this test a theory of entropy based tests of distributional hypotheses other than uniformity is outlined

equality of symmetrized tensors and the flag variety author s berget andrew abstract in this note we give a transparent proof of a result of da cruz and dias da silva on the equality of symmetrized decomposable tensors this will be done by explaining that their result follows from the fact that the coordinate ring of a flag variety is a unique factorization domain

average is over abstract the popular perception of statistical distributions is depicted by the iconic bell curve which comprises of a massive bulk of middle class values and two thin tails one of small left wing values and one of large right wing values the shape of the bell curve is unimodal and its peak represents both the mode and the mean thomas friedman the famous new york times columnist recently asserted that we have entered a human era in which average is over in this paper we present mathematical models for the phenomenon that friedman highlighted while the models are derived via different modeling approaches they share a common foundation inherent tipping points cause the models to phase shift from a normal bell shape statistical behavior to an anomalous statistical behavior the unimodal shape changes to an unbounded monotone shape the mode vanishes and the mean diverges hence i there is an explosion of small values ii large values become super large iii middle class values are wiped out leaving an infinite rift between the small and the super large values and iv average is over indeed

maximization of statistical heterogeneity from shannon s entropy to gini s index abstract different fields of science apply different quantitative gauges to measure statistical heterogeneity statistical physics and information theory commonly use shannon s entropy which measures the randomness of probability laws whereas economics and the social sciences commonly use gini s index which measures the evenness of probability laws motivated by the principle of maximal entropy we explore the maximization of statistical heterogeneity for probability laws with a given mean in the four following scenarios i shannon entropy maximization subject to a given dispersion level ii gini index maximization subject to a given dispersion level iii shannon entropy maximization subject to a given gini index iv gini index maximization subject to a given shannon entropy analysis of these four scenarios results in four different classes of heterogeneity maximizing probability laws yielding an in depth description of both the marked differences and the interplay between the physical randomness based and the economic evenness based approaches to the maximization of statistical heterogeneity

irreducible representations of the full linear group we obtain some results on young diagrams based on these results we construct bases for young symmetry classes of tensors using these bases we obtain a complete reduction of the representation a ma a gl n c and irreducible matrix representations of the full linear group

existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions abstract in this paper by using fractional power of operators and sadovskii fixed point theorem the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions is studied the results are a generalization and continuation of the recent results on this issue an example is provided to illustrate the theory

local model influence abstract this article develops a general method for assessing the influence of model assumptions in a bayesian analysis we assume that model choices are indexed by a hyperparameter with some given initial choice we use the term model to encompass both the sampling model and the prior distribution we wish to assess the effect of changing the hyperparameter away from the initial choice we are performing a sensitivity analysis with the hyperparameter defining our perturbations we use the kullback leibler divergence to measure the difference between posteriors corresponding to different choices of the hyperparameter we also measure the change in priors if small changes in the priors lead to large changes in posteriors the choice of hyperparameter is influential the second order difference in the kullback leibler divergence is expressed by fisher information matrices the relative change in posteriors compared with priors may be summarized by the relative eigenvalue of the posterior and prior fishe

the existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 α 2 this paper is mainly concerned with the existence of mild solutions for fractional differential equations with nonlocal conditions of order 1 a 2 the results are obtained by the fixed point theorem combined with solutions operator theorems

on 1 f noise and other distributions with long tails abstract it is shown following shockley shockley w 1957 proc ire 45 279 290 that when a population is engaged in tasks whose completion requires the successful conclusion of many independent subtasks the distribution function for successes in the primary task is log normal it is also shown that when the dispersion of the log normal distribution is large the distribution is mimicked by a 1 x distribution over a wide range of x this argument provides a generic set of processes that yields the much observed 1 x distribution and will also lead to a 1 f noise spectrum it is commonly found that distributions that seem to be log normal over a broad range say to the 95th percentile of a population change to an inverse fractional power pareto distribution for the last few percentile annual income distributions are examples with this structure the very wealthy generally achieve their superwealth through amplification processes that are not available to most we have introduced a simple amplification model to characterize the transition from a log normal distribution to an inverse power pareto tail

decomposable critical tensors abstract let 1 t be a partition of m and 1 1 its conjugate partition denote also by the irreducible c character of sm associated with let v be a finite dimensional vector space over c the reach of an element of the symmetry class of tensors v symmetry class of tensors associated with is defined the concept of critical element is introduced as an element whose reach has dimension equal to 1 it is observed that in mv the notions of critical element and decomposable element coincide known results for decomposable elements of mv are extended to critical elements of v in particular for a basis of mv induced by a basis of v generalized plucker polynomials are constructed in a way that the set of their common roots contains the set of the families of components of decomposable critical elements of v

a langevin approach to the log gauss pareto composite statistical structure abstract the distribution of wealth in human populations displays a log gauss pareto composite statistical structure its density is log gauss in its central body and follows power law decay in its tails this composite statistical structure is further observed in other complex systems and on a logarithmic scale it displays a gauss exponential structure its density is gauss in its central body and follows exponential decay in its tails in this paper we establish an equilibrium langevin explanation for this statistical phenomenon and show that i the stationary distributions of langevin dynamics with sigmoidal force functions display a gauss exponential composite statistical structure ii the stationary distributions of geometric langevin dynamics with sigmoidal force functions display a log gauss pareto composite statistical structure this equilibrium langevin explanation is universal as it is invariant with respect to the specific details of the sigmoidal force functions applied and as it is invariant with respect to the specific statistics of the underlying noise

nonlocal cauchy problems governed by compact operator families abstract we study the existence of mild solutions for the nonlocal cauchy problem u t au t f t u t t t 0 t 0 t u t 0 g u u 0 and give new criterions which extend some existing results in this area

information indices unification and applications the unified framework of information theoretic statistics was established by kullback 1959 since then numerous information indices have been developed in various contexts this paper represents many of these indices in a unified context the unification thread is the discrimination information function information indices are all logarithmic measures of discrepancy between two probability distributions first we present a summary of informational aspects of the basic information functions a unification of various information theoretic modeling approaches and some explication in terms of traditional measures we then tabulate a unified representation of assortments of information indices developed in the literature for maximum entropy modeling covariate information and influence diagnostics the subjects of these indices include parametric model fitting nonparametric entropy estimation categorical data analysis the linear and exponential family regression and time series the coverage however is not exhaustive the tabulation includes sampling theory and bayesian indices but the focus is on interpretation as descriptive measures and inferential properties are noted tangentially finally applications of some information indices are illustrated through modeling duration data for sprint s churned customer and choice of long distance provider

flags and equality of tensors abstract we state a necessary and sufficient condition for equality of decomposable symmetrized tensors when the symmetrizer is associated to an irreducible character of s p corresponding to the partition of p and the tensors are the images by this symmetrizer of the tensor product of a family of vectors of rank partition we make an application of this result to the representation of flags by projective points

econophysical anchoring of unimodal power law distributions the sciences are abundant with size distributions whose densities have a unimodal shape and power law tails both at zero and at infinity the quintessential examples of such unimodal and power law upl distributions are the sizes of income and wealth in human societies while the tails of upl distributions are precisely quantified by their corresponding power law exponents their bulks are only qualitatively characterized as unimodal consequently different statistical models of upl distributions exist the most popular considering lognormal bulks in this paper we present a general econophysical framework for upl distributions termed the anchoring method this method i universally approximates upl distributions via three anchors set at zero at infinity and at an intermediate point between zero and infinity e g the mode ii is highly versatile and broadly applicable iii encompasses the existing statistical models of upl distributions as special cases iv facilitates the introduction of new statistical models of upl distributions and v yields a socioeconophysical analysis of upl distributions

duality and stationary distributions of wealth distribution models we analyze a class of energy and wealth redistribution models characterizing their stationary measures and showing that they have a discrete dual process in particular we show that the wealth distribution model with non zero saving propensity can never have invariant product measures we also introduce diffusion processes associated to the wealth distribution models by instantaneous thermalization

an r squared measure of goodness of fit for some common nonlinear regression models abstract for regression models other than the linear model r squared type goodness of fit summary statistics have been constructed for particular models using a variety of methods we propose an r squared measure of goodness of fit for the class of exponential family regression models which includes logit probit poisson geometric gamma and exponential this r squared is defined as the proportionate reduction in uncertainty measured by kullback leibler divergence due to the inclusion of regressors under further conditions concerning the conditional mean function it can also be interpreted as the fraction of uncertainty explained by the fitted model

from shape to randomness a classification of langevin stochasticity abstract the langevin equation perhaps the most elemental stochastic differential equation in the physical sciences describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise the langevin equation is in effect a mechanism that maps the stochastic white noise input to a stochastic output a stationary steady state distribution in the case of potential wells and a transient extremum distribution in the case of potential gradients in this paper we explore the degree of randomness of the langevin equation s stochastic output and classify it a la mandelbrot into five states of randomness ranging from infra mild to ultra wild we establish closed form and highly implementable analytic results that determine the randomness of the langevin equation s stochastic output based on the shape of the langevin equation s potential field

measuring statistical evenness a panoramic overview motivated by the question how equal is the distribution of wealth within a given human population economics devised an impressive toolbox of quantitative measures of societal egalitarianism including the lorenz curve and the following indices gini pietra hoover amato hirschman theil and atkinson these quantitative measures considered in the broader context of general data sets with positive values are in effect general gauges of statistical evenness while the application of gini s index grew beyond economics and reached diverse fields of science the aforementioned evenness toolbox has largely remained within the confines of the social sciences the aim of this paper is to expose this evenness toolbox to the physics community by presenting a comprehensive evenness based approach to a fundamental problem in science the measurement of statistical heterogeneity

a maximum entropy approach to recovering information from multinomial response data abstract the classical maximum entropy me approach to estimating the unknown parameters of a multinomial discrete choice problem which is equivalent to the maximum likelihood multinomial logit ml estimator is generalized the generalized maximum entropy gme model includes noise terms in the multinomial information constraints each noise term is modeled as the mean of a finite set of a priori known points in the interval 1 1 with unknown probabilities where no parametric assumptions about the error distribution are made a gme model for the multinomial probabilities and for the distributions associated with the noise terms is derived by maximizing the joint entropy of multinomial and noise distributions under the assumption of independence the gme formulation reduces to the me in the limit as the sample grows large or when no noise is included in the entropy maximization further even though the gme and the logit estimators are conceptually different the dual gme model is related to a gener

on the physical interpretation of statistical data from black box systems abstract in this paper we explore the physical interpretation of statistical data collected from complex black box systems given the output statistics of a black box system and considering a class of relevant markov dynamics which are physically meaningful we reverse engineer the markov dynamics to obtain an equilibrium distribution that coincides with the output statistics observed this reverse engineering scheme provides us with a conceptual physical interpretation of the black box system investigated five specific reverse engineering methodologies are developed based on the following dynamics langevin geometric langevin diffusion growth collapse and decay surge in turn these methodologies yield physical interpretations of the black box system in terms of conceptual intrinsic forces temperatures and instabilities the application of these methodologies is exemplified in the context of the distribution of wealth and income in human societies which are outputs of the complex black box system called the economy

thermodynamics of inequalities from precariousness to economic stratification growing economic inequalities are observed in several countries throughout the world following pareto the power law structure of these inequalities has been the subject of much theoretical and empirical work but their nonequilibrium dynamics e g after a policy change remains incompletely understood here we introduce a thermodynamical theory of inequalities based on the analogy between economic stratification and statistical entropy within this framework we identify the combination of upward mobility with precariousness as a fundamental driver of inequality we formalize this statement by a second law inequality displaying upward mobility and precariousness as thermodynamic conjugate variables we estimate the time scale for the relaxation of the wealth distribution after a sudden change of the after tax return on capital our method can be generalized to gain insight into the dynamics of inequalities in any markovian model of socioeconomic interactions

controllability of semilinear boundary problems with nonlocal initial conditions the main purpose of this paper is the existence of solutions and controllability for semilinear boundary problems with nonlocal initial conditions we show that the solutions are given by a variation of constants formula which allows us to study the exact controllability for this kind of problems with control and nonlinear terms at the boundary the included application to a size structured population equation provides a motivation for abstract results

first passages in bounded domains when is the mean first passage time meaningful we study the first passage statistics to adsorbing boundaries of a brownian motion in bounded two dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries from extensive numerical analysis we obtain the probability p ensuremath omega distribution of the random variable ensuremath omega ensuremath tau 1 ensuremath tau 1 ensuremath tau 2 which is a measure for how similar the first passage times ensuremath tau 1 and ensuremath tau 2 are of two independent realizations of a brownian walk starting at the same location we construct a chart for each domain determining whether p ensuremath omega represents a unimodal bell shaped form or a bimodal m shaped behavior while in the former case the mean first passage time mfpt is a valid characteristic of the first passage behavior in the latter case it is an insufficient measure for the process strikingly we find a distinct turnover between the two modes of p ensuremath omega characteristic for the domain shape and the respective location of absorbing and reflective boundaries our results demonstrate that large fluctuations of the first passage times may occur frequently in two dimensional domains rendering quite vague the general use of the mfpt as a robust measure of the actual behavior even in bounded domains in which all moments of the first passage distribution exist

semilinear integrodifferential equations with nonlocal initial conditions abstract of concern is the cauchy problem for semilinear integrodifferential equations with nonlocal initial conditions under general and natural hypotheses we establish some new theorems about the existence and uniqueness of solutions for the cauchy problem as a consequence we unify and extend the corresponding theorems given previously for the cauchy problem for differential equations or integrodifferential equations with nonlocal initial conditions moreover we present two examples one of which comes from heat conduction in materials with memory to show that the existing results are not applicable to them in contrast with ours

equality of immanantal decomposable tensors we state a necessary and sufficient condition for equality of two nonzero decomposable symmetrized tensors when the symmetrizer is associated with an irreducible character of the symmetric group of degree m

the misconception of mean reversion the notion of random motion in a potential well is elemental in the physical sciences and beyond quantitatively this notion is described by reverting diffusions asymptotically stationary diffusion processes which are simultaneously i driven toward a reversion level by a deterministic force and ii perturbed off the reversion level by a random white noise the archetypal example of reverting diffusions is the ornstein uhlenbeck process which is mean reverting in this paper we analyze reverting diffusions and establish that i if the magnitude of the perturbing noise is constant then the diffusion s stationary density is unimodal and the diffusion is mode reverting ii if the magnitude of the perturbing noise is non constant then in general neither is the diffusion s stationary density unimodal nor is the diffusion mode reverting in the latter case we further establish a result asserting when unimodality and mode reversion do hold in particular we demonstrate that the notion of mean reversion which is fundamental in economics and finance is a misconception as mean reversion is an exception rather than the norm

existence and regularity of solutions to nonlocal retarded differential equations in this paper the existence and uniqueness of different types of solutions to a class of semilinear retarded differential equations with nonlocal history conditions are obtained by a fixed point argument also finite dimensional approximation of these solutions in a hilbert space is established

principal information theoretic approaches 2000 principal information theoretic approaches journal of the american statistical association vol 95 no 452 pp 1349 1353

beyond lognormal inequality the lorenz flow structure abstract observed from a socioeconomic perspective the intrinsic inequality of the lognormal law happens to manifest a flow generated by an underlying ordinary differential equation in this paper we extend this feature of the lognormal law to a general lorenz flow structure of lorenz curves objects that quantify socioeconomic inequality the lorenz flow structure establishes a general framework of size distributions that span continuous spectra of socioeconomic states ranging from the pure communism extreme to the absolute monarchy extreme this study introduces and explores the lorenz flow structure analyzes its statistical properties and its inequality properties unveils the unique role of the lognormal law within this general structure and presents various examples of this general structure beyond the lognormal law the examples include the inverse pareto and pareto laws which often govern the tails of composite size distributions

the pareto law of incomes an explanation and an extension abstract a stochastic model for the generation of observed income distributions is used to provide an explanation for the pareto law of incomes analysis of the model also yields a prediction of paretian power law behaviour in the lower tail of the distribution and this is shown to occur in a number of empirical distributions a tractable four parameter distribution is derived and shown to fit extremely well to a number of different empirical income distributions

existence of mild solutions for semilinear evolution equations with non local initial conditions abstract in this paper we study the existence of mild solutions for a class of semilinear evolution equations with non local initial conditions and extend some related results in this direction

existence of solutions for neutral functional differential evolution equations with nonlocal conditions abstract in this paper by using fractional power of operators and sadovskii s fixed point theorem we study the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions the results we obtained are a generalization and continuation of the recent results on this issue in the end an example is given to show the application of our results

pareto tails and lognormal body of us cities size distribution abstract we consider a distribution which consists of lower tail pareto lognormal body and upper tail pareto to estimate the size distribution of all us cities this distribution fits the data more accurately than a distribution that comprises of only lognormal and the upper tail pareto

colorings and equality of tensors we study necessary conditions for equality of two nonzero star products of vectors in symmetry classes of tensors associated to irreducible characters of the symmetric group of degree m these conditions depend upon the notion of coloring of a family of vectors introduced in linear and multilinear algebra 27 1990 25

topography of chance we present a model of multiplicative langevin dynamics that is based on two foundations the langevin equation and the notion of multiplicative evolution the model is a nonlinear mechanism transforming a white noise input to a dynamic equilibrium output using a single control an underlying convex u shaped potential function the output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories the model generates each admissible combination of the output s shape and randomness in a universal and robust fashion moreover practically all the probability distributions that are supported on the positive half line and that are commonly encountered and applied across the sciences can be reverse engineered by this model hence this model is a universal equilibrium mechanism in the context of multiplicative dynamics for the robust generation of chance the model s output in turn the properties of the produced chance the output s shape and randomness are determined with mathematical precision by the control s landscape its topography thus a topographic map of chance is established as a particular application probability distributions with power law tails are shown to be universally and robustly generated by controls on the edge of convexity convex u shaped potential functions with asymptotically linear wings

the size distributions of all indian cities abstract we apply five distributions lognormal double pareto lognormal lognormal upper tail pareto pareto tails lognormal and pareto tails lognormal with differentiability restrictions to estimate the size distribution of all indian cities since india contains numerous small cities it is important to explicitly model the lower tail behavior for studying the distribution of all indian cities our results rigorously confirm using both graphical and formal statistical tests that among these five distributions pareto tails lognormal is a better suited parametrization of the indian city size data verifying that the indian city size distribution exhibits a strong reverse pareto in the lower tail lognormal in the mid range body and pareto in the upper tail

proportionate vs disproportionate distribution of wealth of two individuals in a tempered paretian ensemble abstract we study the distribution p of the random variable x 1 x 1 x 2 where x 1 and x 2 are the wealths of two individuals selected at random from the same tempered paretian ensemble characterized by the distribution x x x 1 where 0 is the pareto index and x is the cut off function we consider two forms of x a bounded function x 1 for l x h and zero otherwise and a smooth exponential function x exp l x x h in both cases x has moments of arbitrary order we show that for 1 p always has a unimodal form and is peaked at 1 2 so that most probably x 1 x 2 for 0 1 we observe a more complicated behavior which depends on the value of l h in particular for c a certain threshold value p has a three modal for a bounded x and a bimodal m shape for an exponential x form which signifies that in such ensembles the wealths x 1 and x 2 are disproportionately different

the multilinear algebra of jose dias da silva and the portuguese school of mathematics abstract in portugal multilinear algebra has a strong group of researchers that achieved in the last years some impressive results in this area this paper surveys the results obtained by this group and the current state of knowledge on the different problems

relative entropy measures of multivariate dependence abstract there has been a lot of work on measures of dependence or association for bivariate probability distributions or bivariate data these measures usually assume that the variables are both continuous or both categorical in comparison there is very little work on multivariate or conditional measures of dependence the purpose of this article is to discuss measures of multivariate dependence and measures of conditional dependence based on relative entropies these measures are conceptually very general as they can be used for a set of variables that can be a mixture of continuous ordinal categorical and nominal categorical variables for continuous or ordinal categorical variables a certain transformation of relative entropy to the interval 0 1 leads to generalizations of the correlation multiple correlation and partial correlation coefficients if all variables are nominal categorical the relative entropies are standardized to take a maximum of 1 and then transformed so that in the bivar

existence uniqueness and regularity of solutions to semilinear nonlocal functional differential problems abstract in this paper we consider a semilinear nonlocal functional differential problem in a banach space we first establish the existence uniqueness and continuation of a mild solution we then prove some regularity results for this mild solution under different conditions finally we consider some applications of the abstract results

existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions abstract in this work we establish several results about the existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions we assume that the linear part generates a strongly continuous semigroup on a general banach space

from entropy maximization to equality maximization gauss laplace pareto and subbotin abstract the entropy maximization paradigm of statistical physics is well known to generate the omnipresent gauss law in this paper we establish an analogous socioeconomic model which maximizes social equality rather than physical disorder in the context of the distributions of income and wealth in human societies we show that on a logarithmic scale the laplace law is the socioeconomic equality maximizing counterpart of the physical entropy maximizing gauss law and that this law manifests an optimized balance between two opposing forces i the rich and powerful striving to amass ever more wealth and thus to increase social inequality and ii the masses struggling to form more egalitarian societies and thus to increase social equality our results lead from log gauss statistics to log laplace statistics yield paretian power law tails of income and wealth distributions and show how the emergence of a middle class depends on the underlying levels of socioeconomic inequality and variability also in the context of asset prices with laplace distributed returns our results imply that financial markets generate an optimized balance between risk and predictability

econophysical visualization of adam smith s invisible hand abstract consider a complex system whose macrostate is statistically observable but yet whose operating mechanism is an unknown black box in this paper we address the problem of inferring from the system s macrostate statistics the system s intrinsic force yielding the observed statistics the inference is established via two diametrically opposite approaches which result in the very same intrinsic force a top down approach based on the notion of entropy and a bottom up approach based on the notion of langevin dynamics the general results established are applied to the problem of visualizing the intrinsic socioeconomic force adam smith s invisible hand shaping the distribution of wealth in human societies our analysis yields quantitative econophysical representations of figurative socioeconomic forces quantitative definitions of poor and rich and a quantitative characterization of the poor get poorer and the rich get richer phenomena