the godbillon vey invariant of a transversely homogeneous foliation a real projective foliation is a foliation 3 with a system of local coordinates transverse to 5 modelled on rp1 so that the coordinate changes are real linear fractional transformations given a closed manifold m there is but a finite set of values in f 3 m r which the godbillon vey invariant of such foliations may assume a bound on the possible values in terms of the fundamental group is computed for m an oriented circle bundle over a surface this finite set is explicitly computed in 5 c godbillon and j vey introduced a characteristic class associated to a codimension one foliation which lies in the third real cohomology group of the ambient manifold the question was then raised whether one could assign some ge ometric significance to this invariant the problem of understanding the godbillon vey invariant in such a manner was underscored by a result of thurston 12 who showed that for any closed 3 manifold there are families of codimension one folia tions whose godbillon vey invariant varies continuously in a parameter and which take on arbitrarily large values it is the main goal of this paper to show that this situation changes when one restricts to foliations which are transversely homogeneous see 1 to explain the geometric picture which emerges it is worthwhile reviewing briefly gromov s notion

perturbation bounds for least squares problems with equality constraints we present some new results on the perturbation analysis for least squares problems with equality constraints based on an equivalent formation of the problem and the optimal perturbation result for general least squares problems

distribution of bipartite entanglement for random pure states we calculate analytic expressions for the distribution of bipartite entanglement for pure random quantum states all moments of the purity distribution are derived and an asymptotic expansion for the distribution itself is deduced an approximate expression for moments and distribution of meyer wallach entanglement for random pure states is then obtained

large deviations of the maximum eigenvalue in wishart random matrices we analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the wishart laguerre ensemble of positive definite random matrices we show that the probability that all the eigenvalues of a n n wishart matrix w xtx where x is a rectangular m n matrix with independent gaussian entries are smaller than the mean value n c decreases for large n as where 1 2 corresponds respectively to real and complex wishart matrices c n m 1 and x c is a rate sometimes also called large deviation function that we compute explicitly the result for the anti wishart case m n simply follows by exchanging m and n we also analytically determine the average spectral density of an ensemble of wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier numerical simulations are in excellent agreement with the analytical predictions

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

outlier robust pca the high dimensional case principal component analysis plays a central role in statistics engineering and science because of the prevalence of corrupted data in real world applications much research has focused on developing robust algorithms perhaps surprisingly these algorithms are unequipped indeed unable to deal with outliers in the high dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation and the dataset contains some arbitrarily corrupted observations we propose a high dimensional robust principal component analysis algorithm that is efficient robust to contaminated points and easily kernelizable in particular our algorithm achieves maximal robustness it has a breakdown point of 50 the best possible while all existing algorithms have a breakdown point of zero moreover our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sublinearly in the dimension

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

index distribution of cauchy random matrices using a coulomb gas technique we compute analytically the probability that a large n n cauchy random matrix has n positive eigenvalues where n is called the index of the ensemble we show that this probability scales for large n as where is the dyson index of the ensemble the rate function c is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis we find that the rate function around its minimum at 1 2 has a quadratic behavior modulated by a logarithmic singularity as a consequence the variance of the index scales for large n as var n cln n where c 2 2 is twice as large as the corresponding prefactor in the gaussian and wishart cases the analytical results are checked by numerical simulations and against an exact finite n formula which for 2 can be derived using orthogonal polynomials

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

exact minimum eigenvalue distribution of an entangled random pure state a recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved in fact the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems they also provide a rare exactly solvable case for the distribution of the minimum of a set of nstrongly correlated random variables for all values of n and not just for large n

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

algebraic methods for least squares problem in split quaternionic mechanics a split quaternionic least squares problem is one method of solving overdetermined sets of split quaternion linear equations ax b that is appropriate when there is error in the matrix b in this paper by means of complex representation and real representation of a split quaternion matrix we study the split quaternionic least squares problem derive two algebraic methods for finding solutions of the problems in split quaternionic mechanics

on closed anti de sitter spacetimes this article deals with the structure of the fundamental group of compact anti de sitter spacetimes i e lorentz manifolds with constant negative curvature algebraically such a manifold is the quotient of the universal cover of the homogeneous space so 2 n so 1 n by a discrete group gamma acting properly and co compactly on it this exists if and only if n is even indeed as this was observed by kulkarni u 1 d is contained in so 2 2d and acts properly transitively on so 2 2d so 1 2d it then suffices to take gamma as a co compact lattice in u 1 d the results of the present article give evidence to the question in dimension 3 are all compact anti de sitter spacetimes constructed in this way

nonstandard lorentz space forms in their recent paper 8 kulharni and raymond show that a closed 3 manifold which admits a complete lorentz metric of constant curvature 1 henceforth called a complete lorentz structure must be seifert fibered over a hyperbolic base furthermore on every such seifert fibered 3 manifold with nonzero euler class they construct such a lorentz metric moreover the lorentz structure they construct has a rather strong additional property which they call standard a lorentz structure is standard if its causal double cover possesses a timelike killing vector field equivalently it possesses a riemannian metric locally isometric to a left invariant metric on sl 2 r kulkarni and raymond asked if every closed 3 dimensional lorentz structure is standard this paper provides a negative answer to this question theorem 1 and a positive answer to the implicit question raised in 8 7 1 1 theorem 3 theorem 1 let m3 be a closed 3 manifold which admits a homogeneous lorentz structure and satisfies h m r 0 then there exists a nonstandard complete lorentz structure on m in 8 it is shown that the unit tangent bundle of a closed surface admits a homogeneous lorentz structure therefore we obtain corollary 2 there exists a complete lorentz structure on the unit tangent bundle of any closed surface f of genus greater than one which is not standard the homogeneous lorentz structures are all classified in 8 a circle bundle of euler number j over a closed surface f f 0 has a homogeneous structure if and only if j f an analogous statement holds when m has singular fibers i e when f is an orbifold we also show theorem 3 let m3 be a 3 manifold which admits a complete lorentz structure then m3 is not covered by a product f x s fa closed surface theorem 3 implies that the euler class of the seifert fiber structure of m3 is nonzero

nonintersecting brownian interfaces and wishart random matrices we study a system of n nonintersecting 1 1 dimensional fluctuating elastic interfaces vicious bridges at thermal equilibrium each subject to periodic boundary condition in the longitudinal direction and in presence of a substrate that induces an external confining potential for each interface we show that for a large sytem and with an appropriate choice of the external confining potential the joint distribution of the heights of the n nonintersecting interfaces at a fixed point on the substrate can be mapped to the joint distribution of the eigenvalues of a wishart matrix of size n with complex entries dyson index ensuremath beta 2 thus providing a physical realization of the wishart matrix exploiting this analogy to random matrix we calculate analytically i the average density of states of the interfaces ii the height distribution of the uppermost and lowermost interfaces extrema and iii the asymptotic large n distribution of the center of mass of the interfaces in the last case we show that the probability density of the center of mass has an essential singularity around its peak which is shown to be a direct consequence of a phase transition in an associated coulomb gas problem

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

perturbation theory for the rank deficient equality constrained least squares problem this paper presents perturbation theory for the equality constrained least squares problem min f in s kf g 2 with s f lf h 2 min y in c n ly h 2 lse in which l and begin array 20 c l k end array may not be of full rank the analysis generalizes the results of elden 4 perturbation theorems for the lse problem are proved numerical experiments are also given to verify the error bounds

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

large deviations of extreme eigenvalues of random matrices we calculate analytically the probability of large deviations from its mean of the largest smallest eigenvalue of random matrices belonging to the gaussian orthogonal unitary and symplectic ensembles in particular we show that the probability that all the eigenvalues of an n ifmmode times else texttimes fi n random matrix are positive negative decreases for large n as ensuremath sim mathrm exp ensuremath ensuremath beta ensuremath theta 0 n 2 where the parameter ensuremath beta characterizes the ensemble and the exponent ensuremath theta 0 mathrm ln 3 4 0 274 text 653 dots is universal we also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number ensuremath zeta thus generalizing the celebrated wigner semicircle law the density of states generically exhibits an inverse square root singularity at ensuremath zeta

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

a new technique of quaternion equality constrained least squares problem by means of complex representation of a quaternion matrix we study the relationship between the solutions of the quaternion equality constrained least squares problem and that of complex equality constrained least squares problem and obtain a new technique of finding a solution of the quaternion equality constrained least squares problem

statistical distribution of quantum entanglement for a random bipartite state we compute analytically the statistics of the renyi and von neumann entropies standard measures of entanglement for a random pure state in a large bipartite quantum system the full probability distribution is computed by first mapping the problem to a random matrix model and then using a coulomb gas method we identify three different regimes in the entropy distribution which correspond to two phase transitions in the associated coulomb gas the two critical points correspond to sudden changes in the shape of the coulomb charge density the appearance of an integrable singularity at the origin for the first critical point and the detachment of the rightmost charge largest eigenvalue from the sea of the other charges at the second critical point analytical results are verified by monte carlo numerical simulations a short account of part of these results appeared recently in nadal et al phys rev lett 104 110501 2010

special least squares solutions of the quaternion matrix equation a x b c x d e in this paper by using the real representations of quaternion matrices the particular structure of the real representations of quaternion matrices the kronecker product of matrices and the moore penrose generalized inverse we obtain the expressions of the minimal norm least squares solution the pure imaginary least squares solution and the real least squares solution for the quaternion matrix equation a x b c x d e respectively our resulting formulas only involve real matrices and therefore are simpler than those reported in yuan 2014 the corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices and therefore are very efficient and portable numerical examples are provided to illustrate the efficiency of our algorithms

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

evolution of spacelike surfaces in mathrm ads _3 by their lagrangian angle we study spacelike hypersurfaces m in an anti de sitter spacetime n of constant sectional curvature kappa kappa 0 that evolve by the lagrangian angle of their gaus maps in the two dimensional case we prove a convergence result to a maximal spacelike surface if the gaus curvature k of the initial surface m subset n and the sectional curvature of n satisfy k kappa

deformation of proper actions on reductive homogeneous spaces let g be a real reductive lie group and h a closed reductive subgroup of g we investigate the deformation of standard compact quotients of g h that is of quotients of g h by discrete groups that are uniform lattices in some closed reductive subgroup l of g acting properly and cocompactly on g h for l of real rank 1 we prove that after a small deformation in g such a group keeps acting properly discontinuously and cocompactly on g h more generally we prove that the properness of the action of any convex cocompact subgroup of l on g h is preserved under small deformations and we extend this result to reductive homogeneous spaces g h over any local field as an application we obtain compact quotients of so 2n 2 u n 1 by zariski dense discrete subgroups of so 2n 2 acting properly discontinuously

extreme statistics of complex random and quantum chaotic states complex random states have the statistical properties of the gaussian and circular unitary ensemble eigenstates of random matrix theory even though their components are correlated by the normalization constraint it is nevertheless possible to derive compact analytic formulas for their extreme values statistical properties for all dimensionalities the maximum intensity result slowly approaches the gumbel distribution even though the variables are bounded whereas the minimum intensity result rapidly approaches the weibull distribution since random matrix theory is conjectured to be applicable to chaotic quantum systems we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic the statistical behaviors are consistent with the finite n formulas

average coherence and its typicality for random mixed quantum states the wishart ensemble is a useful and important random matrix model used in diverse fields by realizing induced random mixed quantum states as a wishart ensemble with fixed unit trace using matrix integral technique we give a fast track to the average coherence for random mixed quantum states induced via partial tracing of the haar distributed bipartite pure states as a direct consequence of this result we get a compact formula for the average subentropy of random mixed states these compact formulae extend our previous work

perturbation bounds for constrained and weighted least squares problems we derive perturbation bounds for the constrained and weighted linear least squares ls problems both the full rank and rank deficient cases are considered the analysis generalizes some results of earlier works

extreme value statistics of eigenvalues of gaussian random matrices we compute exact asymptotic results for the probability of the occurrence of large deviations of the largest smallest eigenvalue of random matrices belonging to the gaussian orthogonal unitary and symplectic ensembles in particular we show that the probability that all the eigenvalues of an n ifmmode times else texttimes fi n random matrix are positive negative decreases for large n as ensuremath sim text exp text ensuremath ensuremath beta ensuremath theta 0 n 2 where the dyson index ensuremath beta characterizes the ensemble and the exponent ensuremath theta 0 text ln text 3 4 0 274653 dots is universal we compute the probability that the eigenvalues lie in the interval ensuremath zeta 1 ensuremath zeta 2 which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue as a by product we also obtain exactly the average density of states in gaussian ensembles whose eigenvalues are restricted to lie in the interval ensuremath zeta 1 ensuremath zeta 2 thus generalizing the celebrated wigner semi circle law to these restricted ensembles it is found that the density of states generically exhibits an inverse square root singularity at the location of the barriers these results are confirmed by numerical simulations some of the results presented in detail here were announced in a previous paper d s dean and s n majumdar phys rev lett 97 160201 2006

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

quaternionic quantum field theory we show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions as well as the bosons obey a second order wave equation the theory takes the form of either a functional integral with quaternion imaginary lagrangian or a schrodinger equation and transformation theory for quaternion valued wave functions with a quaternion imaginary hamiltonian the connection between the two formulations is developed in detail and many related issues including the breakdown of the correspondence principle and the hilbert space structure are discussed

lsqr algorithm with structured preconditioner for the least squares problem in quaternionic quantum theory abstract the solution of a linear quaternionic least squares qls problem can be transformed into that of a linear least squares ls problem with jrs symmetric real coefficient matrix which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse the main aim of this work is to construct a structured preconditioner to accelerate the lsqr convergence the preconditioner is based on structure preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation and the incomplete inverse upper lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four so it is reliable and has low storage requirements the performances of the lsqr algorithm with structured preconditioner are demonstrated by numerical experiments

proof of vivo pato oshanin s conjecture on the fluctuation of von neumann entropy it was recently conjectured by vivo pato and oshanin phys rev e 93 052106 2016 2470 004510 1103 physreve 93 052106 that for a quantum system of hilbert dimension mn in a pure state the variance of the von neumann entropy of a subsystem of dimension m n is given by 1 mn 1 m n mn 1 1 n m 1 m 2n 1 4n 2 mn 1 where 1 is the trigamma function we give a proof of this formula

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

number statistics for β ensembles of random matrices applications to trapped fermions at zero temperature let mathcal p ensuremath beta v n mathcal i be the probability that a n ifmmode times else texttimes fi n phantom rule 4pt 0ex ensuremath beta ensemble of random matrices with confining potential v x has n mathcal i eigenvalues inside an interval mathcal i a b on the real line we introduce a general formalism based on the coulomb gas technique and the resolvent method to compute analytically mathcal p ensuremath beta v n mathcal i for large n we show that this probability scales for large n as mathcal p ensuremath beta v n mathcal i ensuremath approx exp left ensuremath ensuremath beta n 2 ensuremath psi v n mathcal i n right where ensuremath beta is the dyson index of the ensemble the rate function ensuremath psi v k mathcal i independent of ensuremath beta is computed in terms of single integrals that can be easily evaluated numerically the general formalism is then applied to the classical ensuremath beta gaussian mathcal i ensuremath l l ensuremath beta wishart mathcal i 1 l and ensuremath beta cauchy mathcal i ensuremath l l ensembles expanding the rate function around its minimum we find that generically the number variance var n mathcal i exhibits a nonmonotonic behavior as a function of the size of the interval with a maximum that can be precisely characterized these analytical results corroborated by numerical simulations provide the full counting statistics of many systems where random matrix models apply in particular we present results for the full counting statistics of zero temperature one dimensional spinless fermions in a harmonic trap

characteristic classes and representations of discrete subgroups of lie groups a volume invariant is used to characterize those representations of a countable group into a connected semisimple lie group g which are injective and whose image is a discrete cocompact subgroup of g let it be a discrete cocompact subgroup of g and consider the analytic variety hom 7r g consisting of homomorphisms 0 tt g denote by k a maximal compact subgroup of g and x k g the associated symmetric space let m be the orbit space x tt for convenience we shall henceforth assume that n is torsionfree by selberg s lemma 12 this may be accomplished by replacing tt by a subgroup of finite index this insures that m is a compact smooth manifold having tt as its fundamental group the case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here to every representation 0 g hom 7r g we associate a foliated bundle e over m with fibre x and structure group g see e g 6 if co is a closed ginvariant differential fc form on x then we may spread co over the fibres of e copies of x to obtain a closed fc form co on e we define co 0 fj f where is any section of e moreover co 0 is independent of the choice of section for example taking co to be the g invariant volume form on x we obtain a real number u 0 which depends on 0 when x is even dimensional the chern gauss bonnet theorem implies that u 0 may be described as an euler number i e the self intersection number of any section which is a topological invariant of e when x is odd dimensional this volume invariant is related to a more recent kind of topological invariant based on bounded cohomology and due to gromov 3 and is constant on the connected components of hom 7r g the volume invariant satisfies an inequality

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

asymptotic behavior of support vector machine for spiked population model for spiked population model we investigate the large dimension n and large sample size m asymptotic behavior of the support vector machine svm classification method in the limit of n m at fixed m n we focus on the generalization performance by analytically evaluating the angle between the normal direction vectors of svm separating hyperplane and corresponding bayes optimal separating hyperplane this is an analogous result to the one shown in paul 2007 and nadler 2008 for the angle between the sample eigenvector and the population eigenvector in random matrix theorem we provide not just bound but sharp prediction of the asymptotic behavior of svm that can be determined by a set of nonlinear equations based on the analytical results we propose a new method of selecting tuning parameter which significantly reduces the computational cost a surprising finding is that svm achieves its best performance at small value of the tuning parameter under spiked population model these results are confirmed to be correct by comparing with those of numerical simulations on finite size systems we also apply our formulas to an actual dataset of breast cancer and find agreement between analytical derivations and numerical computations based on cross validation

deformation of compact clifford klein forms of indefinite riemannian homogeneous manifolds 1 1 let g be a lie group and h a closed subgroup of g if a discrete subgroup of g acts properly discontinuously and freely on g h then the double coset space g h carries naturally a manifold structure such that the quotient map g h g h is locally diffeomorphic the manifold g h is said to be a clifford klein form of g h if it is compact then is said to be a uniform lattice for g h a typical example is a compact riemann surface mg with genus g 2 which is biholomorphic to a compact clifford klein form of the poincare disk g h sl 2 r so 2 by the uniformization theorem it is important from geometric view point that a clifford klein form g h inherits any g invariant local geometric structure on g h such as indefinite riemannian metric complex structure symplectic structure causal structure and so on

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

random density matrices versus random evolution of open system we present and compare two families of ensembles of random density matrices the first static ensemble is obtained foliating an unbiased ensemble of density matrices as criterion we use fixed purity as the simplest example of a useful convex function the second dynamic ensemble is inspired in random matrix models for decoherence where one evolves a separable pure state with a random hamiltonian until a given value of purity in the central system is achieved several families of hamiltonians adequate for different physical situations are studied we focus on a two qubit central system and obtain exact expressions for the static case the ensemble displays a peak around werner like states modulated by nodes on the degeneracies of the density matrices for moderate and strong interactions good agreement between the static and the dynamic ensembles is found even in a model where one qubit does not interact with the environment excellent agreement is found but only if there is maximal entanglement with the interacting one the discussion is started recalling similar considerations for scattering theory at the end we comment on the reach of the results for other convex functions of the density matrix and exemplify the situation with the von neumann entropy

algebraic properties of the rank deficient equality constrained and weighted least squares problems abstract we study the algebraic properties of the solutions of the equality constrained least squares problem min f s v kf g v 2 with s f v h lf v 2 min z c n v v h lz v 2 lse and corresponding weighted least squares problem wls in which l and l k may not be of full rank general formulas for the solutions are given and the algebraic relations between the lse and the wls solutions are obtained

entanglement of random vectors we analytically calculate the average value of the ith largest schmidt coefficient for random pure quantum states schmidt coefficients i e eigenvalues of the reduced density matrix are expressed in the limit of large hilbert space size and for arbitrary bipartite splitting as an implicit function of index i

how many eigenvalues of a gaussian random matrix are positive we study the probability distribution of the index n sub i e the number of positive eigenvalues of an nxn gaussian random matrix we show analytically that for large n and large n sub with the fraction 0 c n sub n 1 of positive eigenvalues fixed the index distribution p n sub cn n approx exp beta n sup 2 phi c where beta is the dyson index characterizing the gaussian ensemble the associated large deviation rate function phi c is computed explicitly for all 0 c 1 it is independent of beta and displays a quadratic form modulated by a logarithmic singularity around c 1 2 as a consequence the distribution of the index has a gaussian form near the peak but with a variance delta n of index fluctuations growing as delta n approx lnn beta pi sup 2 for large n for beta 2 this result is independently confirmed against an exact finite n formula yielding delta n lnn 2 pi sup 2 c o n sup 1 for large n where the constant c for even n has the nontrivial value c gamma 1 3ln2 2 pi sup 2 approx equal 0 185 248 and gamma 0 5772 is the euler constant we also determine for large n the probability that the interval zeta sub 1 zeta sub 2 is free of eigenvalues some of these results have been announced in a recent letter phys rev lett 103 220603 2009

the radiance obstruction and parallel forms on affine manifolds a manifold m is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine when they are locally linear m is called radiant the obstruction to radiance is a one dimensional class cm with coefficients in the flat tangent bundle of m exterior powers of cm give information on the existence of parallel forms on m especially parallel volume forms as applications various kinds of restrictions are found on the holonomy and topology of compact affine manifolds introduction an affine manifold m is a manifold with a distinguished maximal atlas of charts all of whose coordinate changes are locally affine on such a manifold there is an intrinsic notion of a parallel tensor one whose components in any affine chart are constants more generally there is the notion of a polynomial tensor field of given degree in 1962 l markus conjectured in mk that a compact orientable affine n dimensional manifold has parallel volume form if and only if it is complete meaning that its universal covering is affinely isomorphic to euclidean n space rn the problem of constructing a parallel volume form determines an n dimensional twisted real cohomology class originally studied by j smillie sm2 in this paper we express this class as the nth exterior power of a twisted one dimensional real class which we call the radiance obstruction cm by computing cm in various cohomology theories cech singular de rham and others we are able to exploit cm in several ways to yield more information on the structure of affine manifolds some of these results will appear in a subsequent paper gh3 the basic tool used is a formula proved in 2 6 which expresses the cohomology class of a parallel exterior k form in terms of the k th exterior power of the radiance obstruction akcvf min 2 7 the special case of a parallel volume form on a compact n dimensional manifold m is examined the existence of such a form implies that ancm 0 in 2 8 we show that the affine holonomy group f of such an m cannot preserve a proper affine subspace in gh3 this result will be improved by showing that r preserves no proper semialgebraic subset of rn in 2 10 we show that there tend to be plenty of parallel forms on a compact affine manifold with nilpotent affine holonomy received by the editors may 26 1983 1980 mathematics subject classification primary 57r99 53c05 secondary 53c10 55r25 4c1984 amencan mathematical society 0002 9947 84 1 00 25 per page

truncated linear statistics associated with the eigenvalues of random matrices ii partial sums over proper time delays for chaotic quantum dots invariant ensembles of random matrices are characterized by the distribution of their eigenvalues lambda 1 ldots lambda n we study the distribution of truncated linear statistics of the form tilde l sum i 1 p f lambda i with p n this problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones or the smallest in this second paper we consider the same problem without this restriction which leads to a rather different analysis we introduce a new ensemble which is related but not equivalent to the thinned ensembles introduced by bohigas and pato this question is motivated by the study of partial sums of proper time delays in chaotic quantum dots which are characteristic times of the scattering process using the coulomb gas technique we derive the large deviation function for tilde l large deviations of linear statistics l sum i 1 n f lambda i are usually dominated by the energy of the coulomb gas which scales as sim n 2 implying that the relative fluctuations are of order 1 n for the truncated linear statistics considered here there is a whole region including the typical fluctuations region where the energy of the coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect because the entropy scales as sim n the relative fluctuations are of order 1 sqrt n our analysis relies on the mapping on a problem of p fictitious non interacting fermions in n energy levels which can exhibit both positive and negative effective absolute temperatures we determine the large deviation function characterizing the distribution of the truncated linear statistics and show that for the case considered here f lambda 1 lambda the corresponding phase diagram is separated in three different phases

exact variance of von neumann entanglement entropy over the bures hall measure the bures hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing in this work we study the statistical behavior of quantum entanglement over the bures hall ensemble as measured by von neumann entropy the average von neumann entropy over such an ensemble has been recently obtained whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average the starting point of the calculations is the connection between correlation functions of the bures hall ensemble and those of the cauchy laguerre ensemble the derived variance formula together with the known mean formula leads to a simple but accurate gaussian approximation of the distribution of von neumann entropy of finite size systems this gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

special least squares solutions of the quaternion matrix equation a x b with applications in this paper by applying particular structure of the real representations of quaternion matrices and the moore penrose generalized inverse we derive the expressions of the minimal norm least squares solution the pure imaginary least squares solution and the real least squares solution for the quaternion matrix equation a x b the resulting formulas only involve real matrices which are simpler than those reported in yuan et al 2013 the corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices therefore are very efficient and easily understood numerical examples are provided to illustrate the efficiency of our algorithms

large deviations of the maximum eigenvalue for wishart and gaussian random matrices we present a coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value the large deviation function that characterizes this probability is computed explicitly for wishart and gaussian ensembles the method is general and applies to other related problems e g the joint large deviation function for large fluctuations of top eigenvalues our results are relevant to widely employed data compression techniques namely the principal components analysis analytical predictions are verified by extensive numerical simulations

deterministic parallel analysis an improved method for selecting factors and principal components factor analysis and principal component analysis are used in many application areas the first step choosing the number of components remains a serious challenge our work proposes improved methods for this important problem one of the most popular state of the art methods is parallel analysis pa which compares the observed factor strengths with simulated strengths under a noise only model the paper proposes improvements to pa we first derandomize it proposing deterministic pa which is faster and more reproducible than pa both pa and deterministic pa are prone to a shadowing phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors we propose deflation to counter shadowing we also propose to raise the decision threshold to improve estimation accuracy we prove several consistency results for our methods and test them in simulations we also illustrate our methods on data from the human genome diversity project where they significantly improve the accuracy

compact support probability distributions in random matrix theory we consider a generalization of the fixed and bounded trace ensembles introduced by bronk and rosenzweig up to an arbitrary polynomial potential in the large n limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the canonical ensemble with measure exp ntr v m the mapping of the corresponding phase boundaries is illuminated in an explicit example in the case of a gaussian potential we are able to derive exact expressions for the one and two point correlator for finite n having finite support

nonuniversality of compact support probability distributions in random matrix theory the two point resolvent is calculated in the large n limit for thegeneralized fixed and bounded trace ensembles it is shown to disagree with theone of the canonical gaussian ensemble by a non universal part which is givenexplicitly for all monomial potentials v m m 2p moreover we prove thatfor the generalized fixed and bounded trace ensemble all k point resolventsagree in the large n limit despite their non universality