concentration in a thin euclidean shell for log concave measures abstract a weak version of a conjecture stated by kannan lovasz and simonovits claims that an isotropic log concave probability on r n should be concentrated in a thin euclidean shell in the following way 1 t 0 n x r n 1 t n x n 1 t n 1 c e c t where 1 2 and c and c are positive absolute constants for 1 10 02 this inequality has been established by klartag by combining different approaches introduced by klartag and by guedon paouris and the author we improve this result by showing that the inequality 1 holds with 1 8

cosine methods for second order hyperbolic equations with time dependent coefficients we analyze efficient high order accurate methods for the approximation of the solutions of linear second order hyperbolic equations with time dependent coefficients the methods are based on galerkin type discretizations in space and on a class of fourth order accurate two step cosine time stepping schemes preconditioned iterative techniques are used to solve linear systems with the same operator at each time step the schemes are supplemented by single step high order starting procedures and need no evaluations of derivatives of operators l2 optimal error estimates are proved throughout

comparison between the probability distribution of returns in the heston model and empirical data for stock indexes abstract we compare the probability distribution of returns for the three major stock market indexes nasdaq s p500 and dow jones with an analytical formula recently derived by dr gulescu and yakovenko for the heston model with stochastic variance for the period of 1982 1999 we find a very good agreement between the theory and the data for a wide range of time lags from 1 to 250 days on the other hand deviations start to appear when the data for 2000 2002 are included we interpret this as a statistical evidence of the major change in the market from a positive growth rate in 1980s and 1990s to a negative rate in 2000s

scaling of levy student processes abstract student s t distributions are widely used in financial studies as heavy tailed alternatives to normal distributions as these distributions are not closed under convolution there exist no levy processes with student s t marginals at all points in time in this article we show that a student s t approximation of these marginals is still suitable while not exact using this approximation we are able to describe the scaling behavior of such levy student processes and the parameters of its marginal distributions by a simple analytical scaling law this scaling law drastically simplifies the use of levy student processes as a general diffusion process in various interdisciplinary applications we explicitly provide an application in the context of modelling high frequency price returns

characteristic times in stock market indices abstract in this study we analyze the standard and poor s 500 index data of the new york stock exchange for more than 32 years using a simple random walk model we demonstrate that the proper variable to look at is the logarithmic return in the statistical analysis we have done fittings to the levy distribution using either the index data as such or pre processing it with arch garch or igarch methods which tend to remove the time dependent variance for short times the truncated levy distribution is found to fit the data quite well since this is not a stable distribution the scaling behavior observed for short times should brake down for longer times we demonstrate that the characteristic time where this cross over starts is of the order of one day

non commutative polynomials of independent gaussian random matrices the real and symplectic cases in ht2 haagerup and thorbjo rnsen prove the following extension of voiculescu s random matrix model cf v2 theorem 2 2 for each n let x1 n xr n be r independent complex self adjoint random matrices from the class open image in new window and let x1 xr be a semicircular system in a c probability space then for any polynomial p in r non commuting variables the convergence

the littlewood offord problem and invertibility of random matrices abstract we prove two basic conjectures on the distribution of the smallest singular value of random n n matrices with independent entries under minimal moment assumptions we show that the smallest singular value is of order n 1 2 which is optimal for gaussian matrices moreover we give a optimal estimate on the tail probability this comes as a consequence of a new and essentially sharp estimate in the littlewood offord problem for i i d random variables x k and real numbers a k determine the probability p that the sum k a k x k lies near some number v for arbitrary coefficients a k of the same order of magnitude we show that they essentially lie in an arithmetic progression of length 1 p

convergence of runge kutta approximations for parabolic problems with neumann boundary conditions we analyze a class of algebraically stable runge kutta standard galerkin methods for inhomogeneous linear parabolic equations with time dependent coefficients under neumann boundary conditions and derive an l 2 error bound of o h r k min nu p 3 over2 provided frac k h is bounded

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

fully discrete galerkin approximations of parabolic boundary value problems with nonsmooth boundary data in this paper we study the convergence properties of a fully discrete galerkin approximation with a backwark euler time discretization scheme an approach based on semigroup theory is used to deal with the nonsmooth dirichlet boundary data which cannot be handled by standard techniques this approach gives rise to optimal rates of convergence inl p o t l 2 norms for boundary conditions inl p o t l 2 1 p

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

pricing european options with a log student s t distribution a gosset formula abstract the distributions of returns for stocks are not well described by a normal probability density function pdf student s t distributions which have fat tails are known to fit the distributions of the returns we present pricing of european call or put options using a log student s t distribution which we call a gosset approach in honour of w s gosset the author behind the nom de plume student the approach that we present can be used to price european options using other distributions and yields the black scholes formula for returns described by a normal pdf

twice ramanujan sparsifiers we prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices as linear sized spectral sparsifiers of complete graphs are expanders our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs in particular we prove that for every d 1 and every undirected weighted graph g v e w on n vertices there exists a weighted graph h v f tilde w with at most lceil d n 1 rceil edges such that for every x in mathbb r v x t l g x leq x t l h x leq bigl frac d 1 2 sqrt d d 1 2 sqrt d bigr cdot x t l g x where l g and l h are the laplacian matrices of g and h respectively thus h approximates g spectrally at least as well as a ramanujan expander with dn 2 edges approximates the complete graph we give an elementary deterministic polynomial time algorithm for constructing h

the potential for parallelism in runge kutta methods part 1 rk formulas in standard form the authors examine the potential for parallelism in runge kutta rk methods based on formulas in standard one step form both negative and positive results are presented many of the negative results are based on a theorem that bounds the order of an rk formula in terms of the minimal polynomial associated with its coefficient matrix the positive results are largely examples of prototypical formulas that offer a potential for effective coarse grain parallelism on machines with a few processors

on the limit of the largest eigenvalue of the large dimensional sample covariance matrix in this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity the above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products sp matrix exist

treatment of kurtosis in financial markets abstract since mandelbrot 1963 2 highlighted the fact that data on the yield of financial assets exhibit leptokurtosis different distributions have been presented as alternatives to the normal distribution so far little consideration has been given to the capacity that these distributions have to recover the kurtosis of the sample data our work aims to present distributions which given the broad range of their kurtosis have the capacity to perform adjustment on many occasions where other distributions fail while also being capable of recovering the peakedness of the empirical data another key characteristic of these distributions is that they are defined within a bounded domain in the same way as the sample data an empirical application of these distributions is presented within the financial field by using daily returns

financial data and the skewed generalized t distribution this paper develops a skewed extension of the generalized t gt distribution introducedby mcdonald and newey 1988 in particular the paper derives the mathematical moments and other properties of the distribution and assesses its ability to fit the empirical distribution of several financial series characterized by skewness and excess kurtosis in all cases the skewed gt provides an excellent fit to the empirical distribution of data

negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous l 2 dirichlet boundary data this paper treats time and space finite element approximations of the solution to the nonhomogeneous wave equation with l 2 boundary terms and smooth right hand side for the case of l 2 boundary data the rates of convergence in negative norms are derived in the case of smooth forcing term and zero boundary data optimal rates of convergence in positive norms are provided

scalar intermittency in the variational theory of turbulence abstract the evolution of the probability density function pdf of temperature differences passive or active scalar t x r t x versus the scale r is considered on the point of view of the variational theory the pdf at scale r appears as a superposition of rescaled large scale pdfs with a log normal distribution of the scaling factors as in the velocity case the variance of this log normal distribution behaves as a power law versus r the exponent going to zero as the reynolds or grasshof numbers go to infinity

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis

universality results for the largest eigenvalues of some sample covariance matrix ensembles for sample covariance matrices with i i d entries with sub gaussian tails when both the number of samples and the number of variables become large and the ratio approaches one it is a well known result of soshnikov that the limiting distribution of the largest eigenvalue is same that of gaussian samples in this paper we extend this result to two cases the first case is when the ratio approaches an arbitrary finite value the second case is when the ratio becomes infinite or arbitrarily small

on mixed finite element methods for linear elastodynamics we construct and analyze finite element methods for approximating the equations of linear elastodynamics using mixed elements for the discretization of the spatial variables we consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth order in time fully discrete approximations l 2 optimal order error estimates are proved for the approximations of displacement and stress

error estimates for semidiscrete galerkin type approximations to semilinear evolution equations with nonsmooth initial data error estimates for the semidiserete galerkin method for abstract semilinear evolution equations with non smooth initial data are given in concrete cases almost optimal order of convergence for linear finite elements results

on the convergence of the extremal eigenvalues of empirical covariance matrices with dependence consider a sample of a centered random vector with unit covariance matrix we show that under certain regularity assumptions and up to a natural scaling the smallest and the largest eigenvalues of the empirical covariance matrix converge when the dimension and the sample size both tend to infinity to the left and right edges of the marchenko pastur distribution the assumptions are related to tails of norms of orthogonal projections they cover isotropic log concave random vectors as well as random vectors with i i d coordinates with almost optimal moment conditions the method is a refinement of the rank one update approach used by srivastava and vershynin to produce non asymptotic quantitative estimates in other words we provide a new proof of the bai and yin theorem using basic tools from probability theory and linear algebra together with a new extension of this theorem to random matrices with dependent entries

microscopic origin of non gaussian distributions of financial returns abstract in this paper we study the possible microscopic origin of heavy tailed probability density distributions for the price variation of financial instruments we extend the standard log normal process to include another random component in the so called stochastic volatility models we study these models under an assumption akin to the born oppenheimer approximation in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process in this approximation we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero drift modified log returns we verify that the dow jones industrial average index indeed follows this scaling we then focus on two popular stochastic volatility models the heston and hull white models in particular we show that in the hull white model the resulting probability distribution of log returns in this approximation corresponds to the tsallis t student distribution the tsallis parameters are given in terms of the microscopic stochastic volatility model finally we show that the log returns for 30 years dow jones index data is well fitted by a tsallis distribution obtaining the relevant parameters

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

log similarity for turbulent flows abstract we critically examine a recently proposed representation for turbulent energy spectra called here log similarity which is predicted both by the multifractal model and the variational approach of the small scale intermittency within this representation spectra are homothetic in the log log plot the homothety factor depending on the taylor scale based reynolds number r experimental results on energy spectra and velocity structure functions of order 2 3 and 6 measured in different flows at several r show that this factor is r a measurable exponent which characterises intermittency in the inertial range experimentally displays the existence of a transition in the neighbourhood of r 400 and for r tending to infinity behaves as 1 in r r

runge kutta methods for parabolic equations and convolution quadrature we study the approximation properties of runge kutta time discretizations of linear and semilinear parabolic equations including incompressible navier stokes equations we derive asymptotically sharp error bounds and relate the temporal order of convergence which is generally noninteger to spatial regularity and the type of boundary conditions the analysis relies on an interpretation of runge kutta methods as convolution quadratures in a different context these can be used as efficient computational methods for the approximation of convolution integrals and integral equations they use the laplace transform of the confolution kernal via a discrete operational calculus 23 refs 2 tabs

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

a bayesian robust detection of shift in the risk structure of stock market returns abstract in this article we provide a statistical procedure for the analysis of stock market prices that is robust toward departures from the normal distribution assumption and that can detect and evaluate a shift of parameters at an unknown time point the method is an adaptation of a bayesian inferential procedure developed by box and tiao that allows data to deviate moderately from the normal distribution model it is applied to a set of u s stock market prices for 1971 1974 in addition to the detection of shift in distribution parameters the procedure is also applied to the examination of shift of the beta coefficients that represent the degree of undiversifiable systematic risk of individual securities implications of the empirical findings for financial theories and their applications are sketched

a finite element method for equations of one dimensional nonlinear thermoelasticity abstract this paper presents a new coupled scheme for the equations of one dimensional nonlinear thermoelasticity by using the finite element method in the space variable and the difference method in the time variable the existence and the uniqueness of the approximate solutions are proved and the bounds of the error are analyzed

letter to the editor convergence of fem with interpolated coefficients for semilinear hyperbolic equation to solve spatially semidiscrete approximative solution of a class of semilinear hyperbolic equations the finite element method fem with interpolated coefficients is discussed by use of semidiscrete finite element for linear problem as comparative function the error estimate in l norm is derived by the nonlinear argument in chen structure theory of superconvergence of finite elements hunan press of science and technology changsha 2001 in chinese this indicates that convergence of fems with interpolated coefficients for a semilinear equation is similar to that of classical fems

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

continuous finite elements in space and time for the nonhomogeneous wave equation finite element approximation in space and time for the wave equation with a forcing term is considered the rates of convergence in h1 l2 and l2 h 1 topology reconstructing the best approximation properties of the subspaces are derived

stock market return distributions from past to present abstract we show that recent stock market fluctuations are characterized by the cumulative distributions whose tails on short minute time scales exhibit power scaling with the scaling index 3 and this index tends to increase quickly with decreasing sampling frequency our study is based on high frequency recordings of the s p500 dax and wig20 indices over the interval may 2004 may 2006 our findings suggest that dynamics of the contemporary market may differ from the one observed in the past this effect indicates a constantly increasing efficiency of world markets

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

nonextensive statistical features of the polish stock market fluctuations the statistics of return distributions on various time scales constitutes one of the most informative characteristics of the financial dynamics here we present a systematic study of such characteristics for the polish stock market index wig20 over the period 04 01 1999 31 10 2005 for the time lags ranging from 1min1min up to 1 h this market is commonly classified as emerging still on the shortest time scales studied we find that the tails of the return distributions are consistent with the inverse cubic power law as identified previously for majority of the mature markets within the time scales studied a quick and considerable departure from this law towards a gaussian can however be traced interestingly all the forms of the distributions observed can be comprised by the single q gaussians which provide a satisfactory and at the same time compact representation of the distribution of return fluctuations over all magnitudes of their variation the corresponding nonextensivity parameter q was found to systematically decrease when increasing the time scales the temporal correlations quantified here in terms of multifractality provide further arguments in favor of nonextensivity

cosine methods for a class of semilinear second order wave equations abstract we analyze fully discrete methods of fourth or second order temporal accuracy for the approximation of the solutions of a class of semilinear second order hyperbolic equations with a nonlinear term that depends on x t u u t u the methods are based on galerkin type discretizations in space and on a class of two step cosine time stepping schemes by suitably extrapolating from previous values in the nonlinear term we compute the approximations by solving linear systems of equations at each time step l 2 optimal error estimates are proved

the limit of the smallest singular value of random matrices with i i d entries abstract let a i j 1 i j be i i d real valued random variables with zero mean and unit variance and let an integer sequence n m m 1 satisfy m n m z for some z 0 1 for each m n denote by a m the n m m random matrix a i j 1 i n m 1 j m and let s m a m be its smallest singular value we prove that the sequence n m 1 2 s m a m m 1 converges to 1 z almost surely our result does not require boundedness of any moments of a i j s higher than the 2 nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold

quantitative estimates of the convergence of the empirical covariance matrix in log concave ensembles let k be an isotropic convex body in rn given e 0 how many independent points xi uniformly distributed on k are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability our paper answers this question from 12 more precisely let x rn be a centered random vector with a log concave distribution and with the identity as covariance matrix an example of such a vector x is a random point in an isotropic convex body we show that for any e 0 there existsc e 0 su ch that if n c e n and xi i n are i i d copies of 1 n x then n i 1 xi xi id e with probability larger than 1 exp c n

runge kutta nystrom methods for hyperbolic problems with time dependent coefficients abstract galerkin fully discrete approximations for hyperbolic equations with time dependent coefficients are analyzed the schemes are based on implicit runge kutta nystrom methods and are coupled with preconditioned iterative methods for the efficient albeit approximate solution of the resulting linear systems arbitrarily high order both conditionally and unconditionally stable methods are considered in addition it is shown that a specific class of methods exhibit parallel features that can be exploited to reduce the final execution time to that of a low order method

error estimates for the semidiscrete finite element approximation of a damped wave equation with a smoothing property error estimates are given for the semidiscrete finite element approximation of an inhomogeneous wave equation damped by a constant times the positive square root of the elliptic operator in the wave equation times the time derivative of the solution the damping term smooths the solution the smoothing property is used to obtain higher order error estimates for positive time even when the initial data are non smooth

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

finite difference discretizations of some initial and boundary value problems with interface we analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation the schrodinger equation and the wave equation by finite difference methods extending the concept of the elliptic projection well known from the analysis of galerkin finite element methods to our finite difference case we prove second order error estimates in space and time in the 12 norm

controlling the least eigenvalue of a random gram matrix abstract consider a n p random matrix x with i i d rows we show that the least eigenvalue of n 1 x x is bounded away from zero with high probability when p n y for some fixed y in 0 1 and normalized orthogonal projections of rows are not too close to zero the principal difference from the previous results is that y can be arbitrarily close to one our results cover many cases of interest in high dimensional statistics and random matrix theory

finite element approximations of nonlinear elastic waves in this paper we study finite element methods for a class of problems of nonlinear elastodynamics we discretize the equations in space using galerkin methods for the temporal discretization the construction of our schemes is based on rational approximations of cos x and ex we analyze semidiscrete as well as secondand fourth order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal order l2 error estimates for some schemes a taylor type technique is used so that only linear systems of equations need be solved at each time step in the proofs we need various estimates for a nonlinear elliptic projection the proofs of which are also established in the paper

some convergence estimates for semidiscrete type schemes for time dependent nonselfadjoint parabolic equations l2 norm error estimates are shown for semidiscrete continuous in time galerkin finite element type approximations to solutions of general time dependent nonselfadjoint second order parabolic equations under dirichlet boundary conditions the semidiscrete solutions are defined in terms of given methods for the corresponding elliptic problem such as the standard galerkin method in which the boundary conditions are satisfied exactly but also methods for which this is not necessary the results are proved by energy arguments and include estimates for the homogeneous equation with both smooth and nonsmooth initial data

convergence estimates for semidiscrete parabolic equation approximations semidiscrete methods for approximating the solutions of initial boundary value problems for parabolic equations are studied the construction of these semidiscrete methods is based upon the availability of several different galerkin type approximation methods for the associated elliptic steadystate problem the properties required of the spacial discretization methods are listed and estimates of the error made by the resulting semidiscrete approximations and of their time derivatives are given in particular estimates are given that require only weak smoothness assumptions on the initial data verifications of the required properties for various galerkin type methods are provided

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

on runge kutta methods for parabolic problems with time dependent coefficients analyse des approximations completement discretes de galerkin pour les equations paraboliques a coefficients dependant du temps utilisation des methodes de runge kutta implicites

higher order single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients error estimates are proved for finite element approximations to the solution of an initial boundary value problem for a second order hyperbolic partial differential equation with time dependent coefficients optimal order rates of convergence in l 2 are shown for single step fully discrete schemes the order nu of the single step schemes is greater than or equal to one and less than or equal to four a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained these methods avoid the repeated work of new matrix factorizations at each time level

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

numerical approximations for a miscible displacement process in porous media a reservoir simulation problem describing a miscible displacement process is stated and analyzed and a semidiscrete numerical scheme based on a galerkin method with mesh refinements is presented for its solution the underlying physical problem involves two miscible incompressible fluids of equal viscosity flowing in a porous medium injection and production wells are rigorously modelled which leads to a formulation containing nearly singular source and sink terms the singularities arising in the solution are described in some detail and it is shown how knowledge of their forms motivate the choice of an appropriate mesh refinement for the numerical scheme optimal order quadratic convergence is shown for the resulting semidiscrete approximation using piecewise linear elements

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

power law estimates for the central limit theorem for convex sets abstract we investigate the rate of convergence in the central limit theorem for convex sets established in b klartag a central limit theorem for convex sets invent math in press 8 we obtain bounds with a power law dependence on the dimension these bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets

estimating the distribution of volatility of realized stock returns and exchange rate changes abstract realized stock return volatility is modelled with a distribution based on the laplace distribution the moment properties of suggested volatility distribution are derived the properties of distribution correspond to the empirical regularities found in the finance literature ml estimator for is also provided the advantage of laplace approach lies in estimating from returns distribution f x directly instead of volatility distribution based on bias sensitive standard deviation estimates the goodness to fit tests with 5 day standard deviations of daily hex closing price returns in period 3 1 1983 4 3 2003 daily s p500 closing stock index returns in period 1 3 1950 27 3 2003 and daily usd euro exchange rate changes in period 28 12 1978 28 2 2003 support the suggested volatility distribution model

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients l2 norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time dependent coefficients optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data the initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

exponential distribution of financial returns at mesoscopic time lags a new stylized fact we study the probability distribution of stock returns at mesoscopic time lags return horizons ranging from about an hour to about a month while at shorter microscopic time lags the distribution has power law tails for mesoscopic times the bulk of the distribution more than 99 of the probability follows an exponential law the slope of the exponential function is determined by the variance of returns which increases proportionally to the time lag at longer times the exponential law continuously evolves into gaussian distribution the exponential to gaussian crossover is well described by the analytical solution of the heston model with stochastic volatility r 2004 elsevier b v all rights reserved

discrete maximal parabolic regularity for galerkin finite element methods for nonautonomous parabolic problems the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the lowest order time discontinuous galerkin solution of linear parabolic

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

covariance estimation for distributions with 2 e moments we study the minimal sample size n n n that suffices to estimate the covariance matrix of an n dimensional distribution by the sample covariance matrix in the operator norm with an arbitrary fixed accuracy we establish the optimal bound n o n for every distribution whose k dimensional marginals have uniformly bounded 2 e moments outside the sphere of radius o k in the specific case of log concave distributions this result provides an alternative approach to the kannan lovasz simonovits problem which was recently solved by adamczak et al j amer math soc 23 2010 535 561 moreover a lower estimate on the covariance matrix holds under a weaker assumption uniformly bounded 2 e moments of one dimensional marginals our argument consists of randomizing the spectral sparsifier a deterministic tool developed recently by batson spielman and srivastava siam j comput 41 2012 1704 1721 the new randomized method allows one to control the spectral edges of the sample covariance matrix via the stieltjes transform evaluated at carefully chosen random points

higher order single step fully discrete approximations for nonlinear second order hyperbolic equations abstract error estimates are proved for finite element approximations to the solution of an initial boundary value problem for nonlinear second order hyperbolic partial differential equations optimal order rates of convergence in l 2 are shown for single step fully discrete schemes which are linearized by extrapolations the order of accuracy in time is 2 3 or 4 a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained

error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions semidiscrete finite element error estimates for linear parabolic equations are derived under minimal regularity with the help of l2 projectors then analogous minimal regularity semidiscrete error estimates for semilinear parabolic equations are derived

stability of runge kutta methods for stiff ordinary differential equations this work analyzes the integration of initial value problems for stiff systems of ordinary differential equations by runge kutta methods the author uses the characterization of stiff initial value problems due to kreiss the jacobian matrix is essentially negative dominant and satisfies a relative lipschitz condition the existence and regularity of the numerical solution are established and conditions under which the runge kutta formula is stable are given

increasing subsequences and the hard to soft edge transition in matrix ensembles our interest is in the cumulative probabilities pr l t l for the maximum length of increasing subsequences in poissonized ensembles of random permutations random fixed point free involutions and reversed random fixed point free involutions it is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively the gap probabilities can be written as a sum over correlations for certain determinantal point processes from these expressions a proof can be given that the limiting form of pr l t l in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively thereby reclaiming theorems due to baik deift johansson and baik rains

error estimates for single step fully discrete approximations for nonlinear second order hyperbolic equations abstract optimal order rates of convergence are proved for fully discrete approximations for nonlinear second order hyperbolic equations which are based on a finite element approximation in the space variables and a single step method in the time variable the r h s of the differential equation can depend on u u 1 or u and the coefficients can depend on u this paper generalizes the work of bales where the r h s and coefficients were allowed to depend on u

differentiability and convergence rates of approximating semigroups for retarded functional differential equations in this paper the averaging approximations i e semidiscrete finite difference scheme for linear retarded functional differential equations are considered in the context of the semigroups on the hilbert space r n times l 2 it is shown that the approximating semigroups are all differentiable uniformly with respect to the index n determining the mesh size this fact makes it possible to prove the existence of a uniform exponential decay or growth rate for the approximating semigroups which can be made arbitrarily close to the decay respectively growth rate of the original semigroup more importantly the uniform differentiability makes it possible to establish convergence rates that improve with time and to prove the convergence of the approximating semigroups in the uniform operator topology the dependence of the convergence rate on the initial conditions and system parameters is also established

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

empirical distributions of chinese stock returns at different microscopic timescales abstract we study the distributions of event time returns and clock time returns at different microscopic timescales using ultra high frequency data extracted from the limit order books of 23 stocks traded in the chinese stock market in 2003 we find that the returns at the one trade timescale obey the inverse cubic law for larger timescales 2 32 trades and 1 5 min the returns follow the student distribution with power law tails with the decrease in timescale the tail becomes fatter which is consistent with the variational theory in turbulence