description of infinite orbits on multiple projective spaces abstract let g be the general linear group of the degree n 2 over an algebraically closed field k in this article we describe the orbit decomposition of a product of copies of p n 1 k under the diagonal action of g we also construct representatives of orbits if m 4 the number of orbits is infinite and we give a description of them

the production function abstract the productions function maps the inputs of a firm or a productive system onto its outputs this article expounds generalizations of the production function that include state variables organizational structures and increasing returns to scale these extensions are needed in order to explain the regularities of the empirical distributions of certain economic variables

multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

growth and fluctuations of personal income pareto s law states that the distribution of personal income obeys a power law in high income range many researchers have proposed models over a century since its discovery but the dynamical understanding of hie phenomenon necessitates empirical study on growth of individual income here we uncovered several properties in the fluctuations of individual income growth derived from income tax data including all taxpayers more than 80 000 with high income iii japan for two consecutive years 1997 and 1998 in a relatively stable period of economics in this work 8 we show i that the dynamics has approximate time reversal symmetry in the period ii that the distribution of growth rate is independent of income tax in the past yearm and iii that the pareto s law breaks down in bubble collapse of japanese economy in 1990 it is shown in 3 how these phenomenological facts are consistent with pareto law these findings provide first direct observation of dynamical process of ruuirey flow in an economic network and its relation to risky assets price fluctuations of lands and stocks

a typical copula is singular abstract we present baire category results in the class of bivariate copulas or equivalently doubly stochastic probability measures endowed with two different metrics under which the space is complete main content of the paper is that in the sense of baire categories with respect to the topology induced by the uniform metric the family of absolutely continuous copulas is of first category whereas the family of purely singular copulas is co meager and hence of second category moreover several other popular dense sub classes of copulas are considered like shuffles of min and checkerboard copulas

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

discrete decomposability of the restriction of a_q λ with respect to reductive subgroups ii micro local analysis and asymptotic k support let g c g be real reductive lie groups this paper offers a criterion on the triplet g g ir that the irreducible unitary representation ir of g splits into a discrete sum of irreducible unitary representations of a subgroup g when restricted to g each of finite multiplicity furthermore we shall give an upper estimate of the multiplicity of an irreducible unitary representation of g occurring in wrig

measuring firm size distribution with semi nonparametric densities abstract in this article we propose a new methodology based on a log semi nonparametric log snp distribution that nests the lognormal and enables better fits in the upper tail of the distribution through the introduction of new parameters we test the performance of the lognormal and log snp distributions capturing firm size measured through a sample of us firms in 2004 2015 taking different levels of aggregation by type of economic activity our study shows that the log snp provides a better fit of the firm size distribution we also formally introduce the multivariate log snp distribution which encompasses the multivariate lognormal to analyze the estimation of the joint distribution of the value of the firm s assets and sales the results suggest that sales are a better firm size measure as indicated by other studies in the literature

power laws from randomly sampled continuous time random walks abstract it has been shown by reed that random sampling a wiener process x t at times t chosen out of an exponential distribution gives rise to power laws in the distribution p x t x t we show both theoretically and numerically that this power law behaviour also follows by random sampling levy flights as continuous time random walks having fourier distribution w k e k with the exponent

generalized bernstein reznikov integrals we find a closed formula for the triple integral on spheres in mathbb r 2n times mathbb r 2n times mathbb r 2n whose kernel is given by powers of the standard symplectic form this gives a new proof to the bernstein reznikov integral formula in the n 1 case our method also applies for linear and conformal structures

symmetry breaking operators for real reductive groups of rank one abstract for a pair of real reductive groups g g we consider the space hom g g of intertwining operators between spherical principal series representations of g and of g also called symmetry breaking operators restricting to those pairs g g where dim hom g g and g and g are of real rank one we classify all symmetry breaking operators explicitly in terms of their distribution kernels this generalizes previous work by kobayashi speh for g g o 1 n 1 o 1 n to the reductive pairs g g u 1 n 1 f u 1 m 1 f f with f c h o and f u n m f in most cases all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail in addition to this family there may occur some sporadic symmetry breaking operators which we determine explicitly

company size distribution for developing countries abstract we analyze company size distribution for developing countries using the framework proposed by ramsden and kiss haypal physica a 277 2000 220 although this distribution does not fit developing countries data as good as it does to developed ones the parameters of the distribution and for developing countries are remarkably different to those for developed countries this result supports the hypothesis that parameter plays a role analogous to the temperature of the economy which could be related to the level of economic development as reported previously by saslow am j phys 67 1999 1239 also this supports the hypothesis that is related to the competitive exclusion in economics as tending to zero implies the competition free limit case where company size distribution is predicted to be a power law as reported by takayasu and okuyama fractals 6 1998 67 finally we report the goodness of fit for two functions a finite size scaling and a log normal we found that these functions fit the data better in some cases however this is not in itself sufficient evidence that those functions are an appropriate representation of the phenomenon

new results on discrete copulas and quasi copulas abstract in this paper we prove that the set of all irreducible discrete quasi copulas coincides with the set of all discrete quasi copulas with minimal range we also provide answers throughout some counterexamples to two questions posed in aguilo et al 2 regarding both discrete quasi copulas with minimal range and irreducible discrete quasi copulas some additional results concerning discrete quasi copulas are also given

branching problems in reproducing kernel spaces for a semisimple lie group g satisfying the equal rank condition the most basic family of unitary irreducible representations is the discrete series found by harish chandra in our work here we study some of the branching laws for discrete series when restricted to a subgroup h of the same type by combining classical results with recent work of kobayashi in particular we prove discrete decomposability under harish chandra s condition of cusp form on the reproducing kernel we show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

representations of reductive groups distinguished by symmetric subgroups let g be a complex connected reductive group g theta be its fixed point subgroup under a galois involution theta and h be an open subgroup of g theta we show that any h distinguished representation pi satisfies 1 pi theta simeq tilde pi where tilde pi is the contragredient representation and pi theta is the twist of pi under theta 2 mathrm dim mathbb c left pi right h le left b backslash g h right where b is a borel subgroup of g by proving the first statement we give a partial answer to a conjecture by prasad and lapid

zero sets of copulas abstract we study conditions on sets in order to be zero sets of semi copulas quasi copulas and with special attention of copulas we find necessary and sufficient conditions for characterizing the zero sets of absolutely continuous copulas and copulas whose support coincides with the closure of the complementary of the zero set moreover we study several topological properties and the lattice theoretic structure and characterize the zero sets of the class of archimedean copulas

finite multiplicity theorems for induction and restriction abstract we find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple lie group g occurring in the induced representations ind h g from irreducible representations of a closed subgroup h as corollaries we establish geometric criteria for finiteness of the dimension of hom g ind h g induction and of hom h h restriction by means of the real flag variety g p and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety

power laws in cities population financial markets and internet sites scaling in systems with a variable number of components we study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules we compare them with respect to the emergence of power laws in the size distribution of their components we show that the details specifying and enforcing the smallest size of the components are crucial as well as the rules for creating new components in particular a growing system with a xed number of components and a xed smallest component size does not converge to a power law we present a new model with variable number of components that converges to a power law for a very wide range of parameters in a very large subset of this range one obtains for the exponent the special value 1 specic for the city populations distribution we discuss the conditions in which can take dierent values in the case of the stock market the distribution of the investors wealth is related to the ratio between the new capital invested in stock and the rate of increase of the stock index c 2000 elsevier science b v all rights reserved

baire category results for exchangeable copulas considering two different metrics on the space of two dimensional copulas c we prove some baire category results for important subclasses of copulas including the families of exchangeable associative and archimedean copulas from the point of view of baire categories with respect to the uniform metric d a typical copula is not symmetric and a typical symmetric copula is not associative whereas a typical associative copula is archimedean and a typical archimedean copula is strict the results in particular answer the open question posed in 1 whether the family of associative copulas is of first category in c d

orbital symmetric spaces and finite multiplicity abstract it is proven that any algebraic symmetric space has finite multiplicity the multiplicity is then shown to be bounded provided that is the case for reductive symmetric spaces a widely believed but as yet unproven fact the method of proof involves an amalgamation of the mackey machine and the orbit method a necessary and sufficient orbital condition is also derived for the quasi regular representation of a symmetric space to be irreducible

size distribution of portuguese firms between 2006 and 2012 this study aims to describe the size distribution of portuguese firms as measured by annual sales and total assets between 2006 and 2012 giving an economic interpretation for the evolution of the distribution along the time three distributions are fitted to data the lognormal the pareto and as a particular case zipf and the simplified canonical law scl we present the main arguments found in literature to justify the use of distributions and emphasize the interpretation of scl coefficients methods of estimation include maximum likelihood modified ordinary least squares in log log scale and nonlinear least squares considering the levenberg marquardt algorithm when applying these approaches to portuguese s firms data we analyze if the evolution of estimated parameters in both lognormal power and scl is in accordance with the known existence of a recession period after 2008 this is confirmed for sales but not for assets leading to the conclusion that the first variable is a best proxy for firm size

size dependent standard deviation for growth rates empirical results and theoretical modeling we study annual logarithmic growth rates r of various economic variables such as exports imports and foreign debt for each of these variables we find that the distributions of r can be approximated by double exponential laplace distributions in the central parts and power law distributions in the tails for each of these variables we further find a power law dependence of the standard deviation sigma r on the average size of the economic variable with a scaling exponent surprisingly close to that found for the gross domestic product gdp phys rev lett 81 3275 1998 by analyzing annual logarithmic growth rates r of wages of 161 different occupations we find a power law dependence of the standard deviation sigma r on the average value of the wages with a scaling exponent beta 0 14 close to those found for the growth of exports imports debt and the growth of the gdp in contrast to these findings we observe for payroll data collected from 50 states of the usa that the standard deviation sigma r of the annual logarithmic growth rate r increases monotonically with the average value of payroll however also in this case we observe a power law dependence of sigma r on the average payroll with a scaling exponent beta 0 08 based on these observations we propose a stochastic process for multiple cross correlated variables where for each variable i the distribution of logarithmic growth rates decays exponentially in the central part ii the distribution of the logarithmic growth rate decays algebraically in the far tails and iii podobnik the standard deviation of the logarithmic growth rate depends algebraically on the average size of the stochastic variable

on the size distribution of firms additional evidence from the g7 countries abstract we analyze the average size distribution of a pool of the g7 group s firms over the period 1987 2000 in particular firm sizes are measured employing different proxies and after conditioning on business cycle phases we find that i the empirical distributions are all consistent with a power law ii point estimates suggest that only in limited cases the exponent is equal to 1 i e the resulting size distribution generally is not zipf iii regardless of the variable employed to measure firm sizes firms are distributed more equally during recessions than during expansions

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

a hitchhiker s guide to quasi copulas abstract this paper presents a review of the concept of an n quasi copula starting from the characterization of functions that can be derived from operations on random variables we recall several other characterizations and properties that have been proven in the literature highlighting the differences between the bivariate case and the higher dimensional case additionally we recall several applications of n quasi copulas such as their role in the study of bounds on sets of n copulas we also recall various classes of n quasi copulas that have been studied in the literature finally we present some questions that have not yet been answered in the literature

optimization with extremal dynamics for the traveling salesman problem abstract by mapping the optimization problems to physical systems the paper presents a general purpose stochastic optimization method with extremal dynamics it is built up with the traveling salesman problem tsp being a typical np complete problem as self organized critical processes of extremal dynamics the optimization dynamics successively updates the states of those cities with high energy consequently a near optimal solution can be quickly obtained through the optimization processes combining the two phases of greedy searching and fluctuated explorations ergodic walk near the phase transition the computational results demonstrate that the proposed optimization method may provide much better performance than other optimization techniques developed from statistical physics such as simulated annealing sa since the proposed fundamental solution is based on the principles and micromechanisms of computational systems it can provide systematic viewpoints and effective computational methods on a wide spectrum of combinatorial and physical optimization problems

on a strong metric on the space of copulas and its induced dependence measure abstract using the one to one correspondence between copulas and markov operators on l 1 0 1 and expressing the markov operators in terms of regular conditional distributions markov kernels allows to define a metric d 1 on the space of copulas c that is a metrization of the strong operator topology of the corresponding markov operators it is shown that the resulting metric space c d 1 is complete and separable and that the induced dependence measure 1 defined as a scalar times the d 1 distance to the product copula has various good properties in particular the class of copulas that have maximum d 1 distance to the product copula is exactly the class of completely dependent copulas i e copulas induced by lebesgue measure preserving transformations on 0 1 hence in contrast to the uniform distance d cannot be approximated arbitrarily well by completely dependent copulas with respect to d 1 the interrelation between d 1 and the so called convergence by mikusinski and taylor as well as the interrelation between 1 and the mutual dependence measure by siburg and stoimenov is analyzed 1 is calculated for some well known parametric families of copulas and an application to singular copulas induced by certain iterated functions systems is given

a stochastic model of firm growth abstract recently from analyses on different databases the tent shape of the distribution of firm growth rates has emerged as a robust and universal characteristic of the time evolution of corporates we add new evidence on this topic and we present a new stochastic model that under rather general assumptions provides a robust explanation for the observed regularity

volume growth temperedness and integrability of matrix coefficients on a real spherical space abstract we apply the local structure theorem from 13 and the polar decomposition of 12 to a real spherical space z g h and control the volume growth on z we define the harish chandra schwartz space on z we give a geometric criterion to ensure l p integrability of matrix coefficients on z

the power law distribution of cumulative coal production abstract the coal industry is dominated by the largest mines with 1 of coal mines in the u s being responsible for 65 of the coal cumulatively produced we show that this heavy tail can be well approximated by a power law where mine level cumulative production is inversely proportionate to distributional rank maximum likelihood and regression based procedures estimate the counter cumulative power law parameter to be less than one indicating there is no well defined mean or variance for cumulative production goodness of fit tests indicate the power law is a better fit to the data than other competing distributions including the lognormal

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

testing the pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities fat tail distributions of sizes abound in natural physical economic and social systems the lognormal and the power laws have historically competed for recognition with sometimes closely related generating processes and hard to distinguish tail properties this state of affair is illustrated with the debate between eeckhout amer econ rev 94 1429 2004 and levy amer econ rev 99 1672 2009 on the validity of zipf s law for us city sizes by using a uniformly most powerful unbiased umpu test between the lognormal and the power laws we show that conclusive results can be achieved to end this debate we advocate the umpu test as a systematic tool to address similar controversies in the literature of many disciplines involving power laws scaling fat or heavy tails in order to demonstrate that our procedure works for data sets other than the us city size distribution we also briefly present the results obtained for the power law tail of the distribution of personal identity id losses which constitute one of the major emergent risks at the interface between cyberspace and reality

the emergence of different tail exponents in the distributions of firm size variables abstract we discuss a mechanism through which inversion symmetry i e invariance of a joint probability density function under the exchange of variables and gibrat s law generate power law distributions with different tail exponents using a dataset of firm size variables that is tangible fixed assets k the number of workers l and sales y we confirm that these variables have power law tails with different exponents and that inversion symmetry and gibrat s law hold based on these findings we argue that there exists a plane in the three dimensional space log k log l log y with respect to which the joint probability density function for the three variables is invariant under the exchange of variables we provide empirical evidence suggesting that this plane fits the data well and argue that the plane can be interpreted as the cobb douglas production function which has been extensively used in various areas of economics since it was first introduced almost a century ago

from gene families and genera to incomes and internet file sizes why power laws are so common in nature we present a simple explanation for the occurrence of power law tails in statistical distributions by showing that if stochastic processes with exponential growth in expectation are killed or observed randomly the distribution of the killed or observed state exhibits power law behavior in one or both tails this simple mechanism can explain power law tails in the distributions of the sizes of incomes cities internet files biological taxa and in gene family and protein family frequencies

erratum to hausdorffness for lie algebra homology of schwartz spaces and applications to the comparison conjecture let h be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold x and a real algebraic bundle mathcal e on x let mathfrak h be the lie algebra of h let mathcal s x mathcal e be the space of schwartz sections of mathcal e we prove that mathfrak h mathcal s x mathcal e is a closed subspace of mathcal s x mathcal e of finite codimension we give an application of this result in the case when h is a real spherical subgroup of a real reductive group g we deduce an equivalence of two old conjectures due to casselman the automatic continuity and the comparison conjecture for zero homology namely let pi be a casselman wallach representation of g and v be the corresponding harish chandra module then the natural morphism of coinvariants v mathfrak h rightarrow pi mathfrak h is an isomorphism if and only if any linear mathfrak h invariant functional on v is continuous in the topology induced from pi the latter statement is known to hold in two important special cases if h includes a symmetric subgroup and if h includes the nilradical of a minimal parabolic subgroup of g

strong approximation of copulas abstract we introduce strong convergence with regard to approximation of copulas this new type of convergence is useful in dealing with the product of darsow nguyen and olsen for copulas we also provide tools for constructing strong approximations of copulas by using partitions of unity

discrete decomposability of the restriction of aq λ with respect to reductive subgroupsiii restriction of harish chandra modules and associated varieties let h g be real reductive lie groups and an irreducible unitary representation of g we introduce an algebraic formulation discretely decomposable restriction to single out the nice class of the branching problem breaking symmetry in physics in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction h this paper offers basic algebraic properties of discretely decomposable restrictions especially for a reductive symmetric pair g h and for the zuckerman vogan derived functor module and proves that the sufficient condition invent math 94 is in fact necessary a finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum an application to the restriction h of discrete series for a symmetric space g h is also given

discrete decomposability of the restriction of a q λ with respect to reductive subgroups and its applications letg g be real reductive lie groups and q a stable parabolic subalgebra of lie g this paper offers a sufficient condition on g g q that the irreducible unitary representation mathop a q limits ofg with non zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupg each of finite multiplicity as an application to purely analytic problems new results on discrete series are also obtained for some pseudo riemannian non symmetric spherical homogeneous spaces which fit nicely into this framework some explicit examples of a decomposition formula are also found in the cases wherea q is not necessarily a highest weight module

knapp stein type intertwining operators for symmetric pairs abstract for a symmetric pair g h of reductive groups we construct a family of intertwining operators between spherical principal series representations of g and h that are induced from parabolic subgroups satisfying certain compatibility conditions the operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation in particular this establishes lower bounds for multiplicities of intertwining operators we further discuss uniqueness of intertwining operators and for the rank one cases g h su 1 n f s u 1 m f u n m f with f r c h o and for the pair g h gl 4 n r gl 2 n c we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners

sklar s theorem obtained via regularization techniques abstract sklar s theorem establishes the connection between a joint d dimensional distribution function and its univariate marginals its proof is straightforward when all the marginals are continuous the hard part is the extension to the case where at least one of the marginals has a discrete component we present a new proof of this extension based on some analytical regularization techniques i e mollifiers and on the compactness with respect to the l norm of the class of copulas

real double flag varieties for the symplectic group abstract in this paper we study a key example of a hermitian symmetric space and a natural associated double flag variety namely for the real symplectic group g and the symmetric subgroup l the levi part of the siegel parabolic p s we give a detailed treatment of the case of the maximal parabolic subgroups q of l corresponding to grassmannians and the product variety of g p s and l q in particular we classify the l orbits here and find natural explicit integral transforms between degenerate principal series of l and g

on the irreconcilability of pareto and gibrat laws abstract if business firms face a multiplicative growth process in which their growth rates are laplace distributed and independent from their sizes the size cannot be distributed according to a stationary pareto distribution recent contributions using formal arguments seem to contrast with these statements we prove that the proposed formal results are wrong

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

an alternative method for modeling the size distribution of top wealth abstract the pareto distribution has been widely applied in modeling the distribution of wealth as well as top incomes cities and firms however recent evidence has shown that the pareto distribution is not consistent with many situations in which it was previously considered applicable we propose an alternative method for estimating the upper tail distribution of wealth and suggest a new lorenz curve for building models to provide such estimates applying our new models to the forbes world s billionaire lists we show that they significantly outperform the pareto lorenz curve as well as some other popular alternatives

a tale of two tails do power law and lognormal models fit firm size distributions in the mid victorian era abstract the paper explores the frequency and size distributions of firm size in a novel dataset for the mid victorian era from a recent extraction of the england and wales population censuses of 1851 1861 1871 and 1881 the paper contrasts the hypothesis of the power laws against the lognormal model for the tails of the distributions using maximum likelihood estimation log likelihood ratio clipped sample coefficient of variation umpu wilks test kolmogorov smirnov statistic among other state of the art statistical methods our results show that the power law hypothesis is accepted for the size distribution for the years 1851 and 1861 while 1871 is marginally non significant but for 1881 the test is inconclusive the paper discusses the process that generates these distributions citing recent literature that shows how after adding an i i d noise to the gibrat s multiplicative model one can recreate a power law behaviour overall the paper provides describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a pareto model and that the lognormal model is firmly rejected