groups without faithful transitive permutation representations of small degree abstract a subgroup h of a group g is core free if h contains no non trivial normal subgroup of g or equivalently the transitive permutation representation of g on the cosets of h is faithful we study the obstacles to a group having large core free subgroups we call a subgroup d a dedekind subgroup of g if all subgroups of d are normal in g our main result is the following if a finite group g has no core free subgroups of order greater than k then g has two dedekind subgroups d 1 and d 2 such that every subgroup in g of order greater than f k has non trivial intersection with either d 1 or d 2 where f is a fixed function independent of g examples show that the dedekind subgroups need not have index bounded by a function of k and the result would not be true with one dedekind subgroup instead of two we exhibit various related properties of p groups and infinite locally finite groups without large core free subgroups including the following if g is a locally finite group with no infinite core free subgroup then every infinite subgroup of g contains a non trivial cyclic normal subgroup of g we also exhibit asymptotic bounds for some related problems including the following if a group g has a solvable subgroup of index n then g has a solvable normal subgroup of index at most n c for some absolute constant c if g is a transitive permutation group of degree n with cyclic point stabilizer subgroup then g n 7

modelling of income distribution in the european union with the fokker planck equation abstract herein we applied statistical physics to study incomes of three low medium and high income society classes instead of the two low and medium income classes studied so far in the frame of the threshold nonlinear langevin dynamics and its threshold fokker planck counterpart we derived a unified formula for description of income of all society classes by way of example of those of the european union in years 2006 and 2008 hence the formula is more general than the well known formula of yakovenko et al that is our formula well describes not only two regions but simultaneously the third region in the plot of the complementary cumulative distribution function vs an annual household income furthermore the known stylised facts concerning this income are well described by our formula namely the formula provides the boltzmann gibbs income distribution function for the low income society class and the weak pareto law for the medium income society class as expected importantly it predicts to satisfactory approximation the zipf law for the high income society class moreover the region of medium income society class is now distinctly reduced because the bottom of high income society class is distinctly lowered this reduction made in fact the medium income society class an intermediate income society class

on a conjecture of gluck let mathbf f g and b g respectively denote the fitting subgroup and the largest degree of an irreducible complex character of a finite group g a well known conjecture of d gluck claims that if g is solvable then g mathbf f g le b g 2 we confirm this conjecture in the case where mathbf f g is coprime to 6 we also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

the κ generalized distribution a new descriptive model for the size distribution of incomes this paper proposes the generalized distribution as a model for describing the distribution and dispersion of income within a population formulas for the shape moments and standard tools for inequality measurement such as the lorenz curve and the gini coefficient are given a method for parameter estimation is also discussed the model is shown to fit extremely well the data on personal income distribution in australia and in the united states c 2008 elsevier b v all rights reserved

on the orders of primitive groups with restricted nonabelian composition factors we prove that given c0 there exists c such that the following holds if g is a primitive permutation group of degree n no composition factor of which is an alternating group of degree greater than c0 or a classical group of dimension greater than c0 then g nc in particular if the nonabelian composition factors of g have bounded order then g is polynomially bounded

transitive projective planes and insoluble groups suppose that a group g acts transitively on the points of p a finite non desarguesian projective plane we prove first that the sylow 2subgroups of g are cyclic or generalized quaternion we then prove that if g is insoluble then g o g is isomorphic to sl2 5 or sl2 5 2 msc 2000 20b25 51a35

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

un renforcement de la propriete t nous proposons un renforcement de la propriete t de kazhdan en remplacant les representations unitaires par les representations dans des espaces de hilbert qui ne sont pas necessairement unitaires mais a croissance exponentielle suffisamment petite on sait qu un groupe localement compact a la propriete t si et seulement si il existe un idempotent p dans c max g tel que pour toute representation unitaire continue h de g p est le projecteur orthogonal sur le sous espace ferme de h forme des vecteurs ginvariants pour la propriete t renforcee nous posons une definition dans ce style on appelle longueur sur un groupe localement compact g une fonction continue g r verifiant g 1 g et g1g2 g1 g2 pour g g1 g2 g dans la suite pour tout groupe localement compact g on supposera choisie une mesure de haar a gauche qui munit cc g d une structure d algebre par convolution

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

criteria for the solubility and non simplicity of finite groups abstract some criteria of the non simplicity of a finite group by graph theoretical terms are derived this is then used to establish conditions under which a finite group is soluble

nonpositive curvature is not coarsely universal we prove that not every metric space embeds coarsely into an alexandrov space of nonpositive curvature this answers a question of gromov geometric group theory cambridge university press cambridge 1993 and is in contrast to the fact that any metric space embeds coarsely into an alexandrov space of nonnegative curvature as shown by andoni et al ann sci ec norm super 4 51 3 657 700 2018 we establish this statement by proving that a metric space which is q barycentric for some q in 1 infty has metric cotype q with sharp scaling parameter our proof utilizes nonlinear metric space valued martingale inequalities and yields sharp bounds even for some classical banach spaces this allows us to evaluate the bi lipschitz distortion of the ell infty grid m infty n 1 ldots m n vert cdot vert infty into ell q for all q in 2 infty from which we deduce the following discrete converse to the fact that ell infty n embeds with distortion o 1 into ell q for q o log n a rigidity theorem of ribe ark mat 14 2 237 244 1976 implies that for every n in mathbb n there exists m in mathbb n such that if m infty n embeds into ell q with distortion o 1 then q is necessarily at least a universal constant multiple of log n ribe s theorem does not give an explicit upper bound on this m but by the work of bourgain geometrical aspects of functional analysis 1985 86 springer berlin 1987 it suffices to take m n and this was the previously best known estimate for m we show that the above discretization statement actually holds when m is a universal constant

a study of the personal income distribution in australia we analyze the data on personal income distribution from the australian bureau of statistics we compare fits of the data to the exponential log normal and gamma distributions the exponential function gives a good albeit not perfect description of 98 of the population in the lower part of the distribution the log normal and gamma functions do not improve the fit significantly despite having more parameters and mimic the exponential function we find that the probability density at zero income is not zero which contradicts the log normal and gamma distributions but is consistent with the exponential one the high resolution histogram of the probability density shows a very sharp and narrow peak at low incomes which we interpret as the result of a government policy on income redistribution r 2006 elsevier b v all rights reserved

from microscopic taxation and redistribution models to macroscopic income distributions abstract we present here a general framework expressed by a system of nonlinear differential equations suitable for the modeling of taxation and redistribution in a closed society this framework allows one to describe the evolution of income distribution over the population and to explain the emergence of collective features based on knowledge of the individual interactions by making different choices of the framework parameters we construct different models whose long time behavior is then investigated asymptotic stationary distributions are found which enjoy similar properties as those observed in empirical distributions in particular they exhibit power law tails of pareto type and their lorenz curves and gini indices are consistent with some real world ones

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

the limiting distribution of the product replacement algorithm for finitely generated prosoluble groups abstract babai and pak proved that the product replacement algorithm a widely used heuristic algorithm intended to rapidly generate nearly uniformly distributed random elements in a finite group g has a flaw indeed the projection of the uniform distribution on generating n tuples onto the first component may not give the uniform distribution on elements of g in this paper we examine the difference between this distribution and the uniform one in the particular case when g is a finitely generated prosoluble group under some additional hypotheses for example if the derived subgroup is pronilpotent this bias is bounded away from 1 however even for soluble groups the problem pointed out by babai and pak is unavoidable we construct a sequence of finite soluble groups of derived length 4 for which the bias is close to 1

do wealth distributions follow power laws evidence from rich lists abstract we use data on the wealth of the richest persons taken from the rich lists provided by business magazines like forbes to verify if the upper tails of wealth distributions follow as often claimed a power law behaviour the data sets used cover the world s richest persons over 1996 2012 the richest americans over 1988 2012 the richest chinese over 2006 2012 and the richest russians over 2004 2011 using a recently introduced comprehensive empirical methodology for detecting power laws which allows for testing the goodness of fit as well as for comparing the power law model with rival distributions we find that a power law model is consistent with data only in 35 of the analysed data sets moreover even if wealth data are consistent with the power law model they are usually also consistent with some rivals like the log normal or stretched exponential distributions

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

modelling income data using two extensions of the exponential distribution abstract in this paper we propose two extensions of the exponential model to describe income distributions the exponential arctan eat and the composite eat lognormal models discussed in this paper preserve key properties of the exponential model including its capacity to model distributions with zero incomes this is an important feature as the presence of zeros conditions the modelling of income distributions as it rules out the possibility of using many parametric models commonly used in the literature many researchers opt for excluding the zeros from the analysis however this may not be a sensible approach especially when the number of zeros is large or if one is interested in accurately describing the lower part of the distribution we apply the eat and the eat lognormal models to study the distribution of incomes in australia for the period 2001 2012 we find that these models in general outperform the gamma and exponential models while preserving the capacity of the latter to model zeros

the power law distribution for the income of poor households abstract this study proposes a reverse pareto model to describe the power law behavior for the lower tail data of income distribution and illustrates an application on malaysian and italian household income data a robust method based on probability integral transform statistic is used for estimating the shape parameter of the reverse pareto model to allow for the existence of outlying observations in the lower tail data besides that the optimal threshold of reverse pareto is determined by using kolmogorov smirnov statistic it is found that the fitted reverse pareto adequately describes the lower tail data of both datasets suggesting that the power law behavior is obeyed in addition the estimated optimal threshold of reverse pareto model can be utilized as an alternative measure for the relative poverty line based on the reverse pareto model the lorenz curve gini and theil coefficients are determined and it is found that low income inequality is observed for the period of the study the fitted lorenz curve shows that nearly 80 of the total household income is owned by the bottom 80 whereas the remaining total household income is owned by the top 20 finally comparison of the reverse pareto model with some alternative distributions such as shifted reverse exponential shifted reverse stretched exponential and shifted reverse lognormal in terms of model fitting for the lower tail data is also conducted the results show that the reverse pareto model outperform all the other models

a polynomial bound for the orders of primitive solvable groups the structure of primitive solvable permutation groups has bee r thoroughly investigated since galois relying on the known structure theorems we determine the largest order a n of a primitive solvable group of degree ii theorem 3 in the course of the proof we obtain the largest order a n of a transitive solvable group of degree n as well theorem 2 although the formula describing the exact value of i is quite complicated we are able to give an asymptotic evaluation it turns out thar the order of a primitive solvable group of degree n is bounded by 24 12 for a constant c 3 24399 theorem i a similar asymptotic estimate obtained recentiy by babai cameron and pilfy 111 shows that the orders of those primitive permutation groups of degree iz in which all nonabelian simple sections have orders less than a iixed integer k are bounded by n where c c k depends on k only related estimates under the assumption of the classification of finite simple groups are given by cameron 2 strong motivation for such asymptotic estimates comes from recent results of luks 151 on the complexity of graph isomorphism testing the results of the present paper have immediate application to graphs of vaience k p we mean that the field multiplication can be defined on the k dimensional vector space over gf p so that this equ ality holds 127 002 8693 82 g g i27 i so2 gg g

harmonic maps from a simplicial complex and geometric rigidity we study harmonic maps from an admissible flat simplicial complex to a non positively curved riemannian manifold our main regularity theorem is that these maps are c 1 at the interfaces of the top dimensional simplices in addition to satisfying a balancing condition if we assume that the domain is a 2 complex then these maps are c 1 as an application we show that the regularity the balancing condition and a bochner formula lead to rigidity and vanishing theorems for harmonic maps furthermore we give an explicit relationship between our techniques and those obtained via combinatorial methods

stochastic volatility of financial markets as the fluctuating rate of trading an empirical study we present an empirical study of the subordination hypothesis for a stochastic time series of a stock price the fluctuating rate of trading is identified with the stochastic variance of the stock price as in the continuous time random walk ctrw framework the probability distribution of the stock price changes log returns for a given number of trades n is found to be approximately gaussian the probability distribution of n for a given time interval dt is nonpoissonian and has an exponential tail for large n and a sharp cutoff for small n combining these two distributions produces a non trivial distribution of log returns for a given time interval dt which has exponential tails and a gaussian central part in agreement with empirical observations r 2007 elsevier b v all rights reserved

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

random groups random graphs and eigenvalues of p laplacians abstract we prove that a random group in the triangular density model has for density larger than 1 3 fixed point properties for actions on l p spaces affine isometric and more generally 2 2 1 2 p uniformly lipschitz with p varying in an interval increasing with the set of generators in the same model we establish a double inequality between the maximal p for which l p fixed point properties hold and the conformal dimension of the boundary in the gromov density model we prove that for every p 0 2 for a sufficiently large number of generators and for any density larger than 1 3 a random group satisfies the fixed point property for affine actions on l p spaces that are 2 2 1 2 p uniformly lipschitz and this for every p 2 p 0 to accomplish these goals we find new bounds on the first eigenvalue of the p laplacian on random graphs using methods adapted from kahn and szemeredi s approach to the 2 laplacian these in turn lead to fixed point properties using arguments of bourdon and gromov which extend to l p spaces previous results for kazhdan s property t established by uk and ballmann wiatkowski

geodesic loops and periodic geodesics on a riemannian manifold diffeomorphic to s 3 let m n be a closed 2 connected riemannian manifold such that 3 m n 0 in this paper we prove that either there exists a periodic geodesic on m n of length 6d where d is the diameter of m n or at each point p m n there exists a geodesic loop of length 2d

a fixed point theorem of discrete group actions on riemannian manifolds we prove a xed point theorem for a class of discrete group acting on man ifolds of nonpositive curvature by isometry these discrete groups include cocompact lattices in simply connected semisimple p adic groups of rank at least two and large p hence it gives a geometric generalization of margulis superrigidity theorem for the archimedean representation of these groups introduction let n be a complete simply connected riemannian manifold of non positive sectional curvature n has a natural compacti cation n n n by the sphere at in nity which is de ned as the equivalent classes of geodesic rays any group action on n by isometry extends to an action on n de nition a group is said to have property f if any isometric action of on any complete simply connected manifold of nonpositive sectional curvature n has a xed point in n in term of representations of property f has the following inter pretation let h be a simple noncompact lie group with trivial center if has property f then any homomorphism h with zariski dense image is precompact inh this is because the symmetric space as sociated with h has nonpositive curvature and the image being zariski received october and in revised form july

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

improved bounds in the metric cotype inequality for banach spaces abstract it is shown that if x x is a banach space with rademacher cotype q then for every integer n there exists an even integer m n 1 1 q such that for every f z m n x we have 1 j 1 n e x f x m 2 e j f x x q m q e e x f x e f x x q where the expectations are with respect to uniformly chosen x z m n and e 1 0 1 n and all the implied constants may depend only on q and the rademacher cotype q constant of x this improves the bound of m n 2 1 q from mendel and naor 2008 13 the proof of 1 is based on a smoothing and approximation procedure which simplifies the proof of the metric characterization of rademacher cotype of mendel and naor 2008 13 we also show that any such smoothing and approximation approach to metric cotype inequalities must require m n 1 2 1 q

the coarse geometric novikov conjecture and uniform convexity abstract the coarse geometric novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero the purpose of this paper is to prove the coarse geometric novikov conjecture for spaces which admit a coarse uniform embedding into a uniformly convex banach space

enumerating finite groups of given order let f n denote the number of isomorphism classes of groups of order n let n hw p9ig be the prime factorization of n and let a a n ek and u ii n maxkl1 gi the function f n has been subject to considerable study g higman 15 and c c sims 26 showed that if n is a prime power say n p then f n n 2 27 o l a2 for a x oc p m neumann 22 proved that if the number of isomorphism classes of simple groups of order n is bounded by nca2 for some constant c then f n n 1 c a2 holds in general combining the methods of 22 with some consequences of the classification theorem of finite simple groups a mciver and neumann 21 obtained that in fact the estimates f n na2 2 and f n n1 2 1 2 hold recently d f holt 16 found a rather elegant proof for the bound

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

cat 0 spaces and expanders we construct a cat 0 space y with izeki nayatani invariant y 1 by a similar construction we also prove that there exists a cat 0 space which does not have markov type p for every p 1

systoles des surfaces plates singulieres de genre deux the extremal metrics for the isosystolic problem on surfaces of genus two are studied in this paper we show that contrary to e calabi s conjecture in genus three no flat metric with conical singularities is extremal for this problem in genus two

rigidity of warped cones and coarse geometry of expanders abstract we study the geometry of warped cones over free minimal isometric group actions and related constructions of expander graphs we prove a rigidity theorem for the coarse geometry of such warped cones namely if a group has no abelian factors then two such warped cones are quasi isometric if and only if the actions are finite covers of conjugate actions as a consequence we produce continuous families of non quasi isometric expanders and superexpanders the proof relies on the use of coarse topology for warped cones such as a computation of their coarse fundamental groups

nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature we develop a nonlinear martingale theory for time discrete processes y n n n 0 these processes are defined on any filtered probability space f f n p n n 0 and have values in a metric space n d of nonpositive curvature in the sense of a d alexandrov the defining martingale property for such processes is e y n 1 f n y n p a s where the conditional expectation on the left hand side is defined as the minimizer of the functional z ed 2 z y n 1 within the space of f n measurable maps z n we give equivalent characterization of n valued martingales using merely the usual linear conditional expectations and derive fundamental properties of these martingales for example a martingale convergence theorem finally we exploit the relation with harmonic maps it turns out that a map f m n is harmonic w r t a given markov kernel p on m if and only if it maps markov chains x n n n with transition kernel p on m onto martingales f x n n n with values in n the nonlinear heat flow f n 0 x m n of a given initial map f 0 m n at time n is obtained as the filtered expectation f n x e x f x n f k k 0 of the random map f x n similarly the unique solution to the dirichlet problem for a given map g m n and a subset d m is obtained as f x e x g x d f k k 0 in both cases a crucial role is played by the notion of filtered expectation e x f k k 0 which will be discussed in detail moreover we prove jensen s inequality for expectations and filtered expectations and we prove weak and strong laws of large numbers for sequences of i i d random variables with values in n our theory is an extension of the classical linear martingale theory and of the nonlinear theory of martingales with values in manifolds as developed for example in emery 1989 and kendall 1990 the goal is to extend the previous framework towards processes with values in metric spaces this will lead to a stochastic approach to the theory of generalized harmonic maps with values in such singular spaces as developed by jost 1994 and korevaar and schoen 1993

distribution of individual incomes in china between 1992 and 2009 abstract this paper presents comprehensive analysis of the evolution of the distribution of individual annual incomes across the majority of the population in china from 1992 2009 the cumulative distribution functions cdfs and probability density functions pdfs are presented overall the cdfs follow the gaussian function c x a e x 2 2 2 for the majority of individuals in the population while the pdfs obey the function p x b x e x 2 2 2 the width of the pdf has widened from 1992 to 2009 suggesting the factor x has been progressively skewing the curve to the right this long tail representing the high income range is reminiscent of an exponential distribution curve this indicates that a few individuals obtain extremely high incomes leading to increasing levels of financial inequality in china

upper bounds for the number of conjugacy classes of a finite group abstract for a finite group g let k g denote the number of conjugacy classes of g we prove that a simple group of lie type of untwisted rank l over the field of q elements has at most 6 q l conjugacy classes using this estimate we show that for completely reducible subgroups g of gl n q we have k g q 10 n confirming a conjecture of kovacs and robinson for finite groups g with f g a p group we prove that k g cp a where p a is the order of a sylow p subgroup of g and c is a constant for groups with o p g 1 we obtain that k g g p this latter result confirms a conjecture of iranzo navarro and monasor we also improve various earlier results concerning conjugacy classes of permutation groups and linear groups as a by product we show that any finite group g has a soluble subgroup s and a nilpotent subgroup n such that k g s and k g n 3

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

nonembeddability theorems via fourier analysis various new nonembeddability results mainly into l 1 are proved via fourier analysis in particular it is shown that the edit distance on 0 1 d has l 1 distortion we also give new lower bounds on the l 1 distortion of flat tori quotients of the discrete hypercube under group actions and the transportation cost earthmover metric

the coarse lipschitz geometry of ell_p oplus ell_q we show that for 1 p with p 2 the space l p 0 1 is not uniformly homeomorphic to ell p oplus ell 2 we also show that if 1 p 2 q the space ell p oplus ell q has unique uniform structure answering a question of johnson lindenstrauss and schechtman geom funct anal 6 430 470 1996

asymptotic results for primitive permutation groups abstract we prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostnc n where n denotes the maximal exponent occurring in the prime factorization ofn this result is applied to investigating maximal subgroup growth of infinite groups we then proceed by showing that if the point stabilizerg of a primitive groupgof degreendoes not have the alternating group alt d as a section then the order ofgis bounded by a polynomial inn this result extends a well known theorem of babai cameron and palfy it is used to prove for example that ifhis a subgroup of indexnin a groupg andhis a product ofbcyclic groups then g hg ncwherecdepends onb

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

normalizers of primitive permutation groups abstract let g be a transitive normal subgroup of a permutation group a of finite degree n the factor group a g can be considered as a certain galois group and one would like to bound its size one of the results of the paper is that a g n if g is primitive unless n 3 4 54 38 58 or 316 this bound is sharp when n is prime in fact when g is primitive out g n unless g is a member of a given infinite sequence of primitive groups and n is different from the previously listed integers many other results of this flavor are established not only for permutation groups but also for linear groups and galois groups

the statistical distribution of money and the rate of money transference abstract the distribution of money is analysed in connection with the boltzmann distribution of energy in the degenerate states of molecules plots of the population density of income distribution for various countries are well reproduced by a gamma function confirming the validity of the statistical distribution at equilibrium the equilibrium state is reached through pair wise money transference processes independently of the shape of the initial distribution and also of the detailed nature of the money transactions between the economic agents