on the structure of stationary solutions of the navier stokes equations this paper is a supplementary section to 1 we show that without any additional hypothesis the main result in 1 theorem 1 can be considerably strengthened

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

lyapunov exponents for products of rectangular real complex and quaternionic ginibre matrices we study the joint density of eigenvalues for products of independent rectangular real complex and quaternionic ginibre matrices in the limit where the number of matrices tends to infinity it is shown that the joint probability density function for the eigenvalues forms a permanental point process for all three classes the moduli of the eigenvalues become uncorrelated and log normal distributed while the distribution for the phases of the eigenvalues depends on whether real complex or quaternionic ginibre matrices are considered in the derivation for a product of real matrices we explicitly use the fact that all eigenvalues become real when the number of matrices tends to infinity finally we compare our results with known results for the lyapunov exponents as well as numerical simulations

eigenvalue statistics of random real matrices completing ginibre s work we determine the joint probability density of eigenvalues in a gaussian ensemble of real asymmetric matrices which is invariant under orthogonal transformations the symmetry parameter r may vary from 1 antisymmetric ensemble through 0 completely asymmetric ensemble to 1 symmetric ensemble the elliptic law for the average density of eigenvalues in the limit of large dimension is recovered matrices of the type considered appear in models for neural network dynamics and dissipative quantum dynamics

on the spectra of gaussian matrices abstract we give a simple characterization of the moduli of the eigenvalues of a complex gaussian matrix in terms of x 2 distributions we also show that the spectral radius of a k k complex gaussian matrix is stochastically smaller than the norm of a k k 1 real gaussian matrix

the real linear eigenvalue problem in cn abstract in this paper we study the real linear eigenvalue problem in c n we present results concerning the location of the eigenvalues of a real linear operator and classify components of the spectrum various families of real linear operators with structure are introduced for which the eigenvalue problem can be regarded at least partially as understood we consider ways to achieve savings in computational complexity continuation techniques are implemented for locating components and subsets of the spectrum once an eigenvalue is available

real eigenvalue statistics for products of asymmetric real gaussian matrices abstract random matrices formed from i i d standard real gaussian entries have the feature that the expected number of real eigenvalues is non zero this property persists for products of such matrices independently chosen and moreover it is known that as the number of matrices in the product tends to infinity the probability that all eigenvalues are real tends to unity we quantify the distribution of the number of real eigenvalues for products of finite size real gaussian matrices by giving an explicit pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular meijer g functions we also compute the explicit form of the pfaffian correlation kernel for the correlation between real eigenvalues and the correlation between complex eigenvalues the simplest example of these the eigenvalue density of the real eigenvalues gives by integration the expected number of real eigenvalues our ability to perform these calculations relies on the construction of certain skew orthogonal polynomials in the complex plane the computation of which is carried out using their relationship to particular random matrix averages

extracting string motif bases for quorum higher than two bases of generators of motifs consisting of strings in which some positions can be occupied by a don t care provide a useful conceptual tool for their description and a way to reduce the time and space involved in the discovery process in the last few years a few algorithms have been proposed for the extraction of a basis building in large part on combinatorial properties of strings and their autocorrelations currently the most efficient techniques for binary alphabets and quorum q 2 require time quadratic in the length of the host string the present paper explores properties of motif bases for quorum q 2 both with binary and general alphabets by also showing that important results holding for quorum q 2 cannot be extended to this more general case furthermore the extraction of motifs in which a bound is set on the maximum allowed number of don t cares is addressed and suitable algorithms are proposed whose computational complexity depends on the fixed bound

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

on the distribution of a scaled condition number in this note we give the exact distribution of a scaled condition number used by demmel to model the probability that matrix inversion is difficult specifically consider a random matrix a and the scaled condition number kcd a iialif iia1 1 demmel provided bounds for the condition number distribution when a has real or complex normally distributed elements here we give the exact formula

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

spectral statistics for the difference of two wishart matrices in this work we consider the weighted difference of two independent complex wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite dimension scenario using two distinct approaches the first derivation involves the use of unitary group integral while the second one relies on applying the derivative principle the latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with monte carlo simulation results analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived additionally we provide a large dimension asymptotic result for the spectral density using the stieltjes transform approach for algebraic random matrices finally we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean

irredundant tandem motifs eliminating the possible redundancy from a set of candidate motifs occurring in an input string is fundamental in many applications the existing techniques proposed to extract irredundant motifs are not suitable when the motifs to search for are structured i e they are made of two or several subwords that co occur in a text string s of length n the main effort of this work is studying and characterizing a compact class of tandem motifs that is pairs of substrings occurring in tandem within a maximum distance of d symbols in s where d is an integer constant given in input to this aim we first introduce the concept of maximality related to four specific conditions that hold only for this class of motifs then we eliminate the remaining redundancy by defining the notion of irredundancy for tandem motifs we prove that the number of non overlapping irredundant tandem motifs is o d 2n which considering d as a constant leads to a linear number of tandems in the length of the input string this is an order of magnitude less than previously developed compact indexes for tandem extraction the notions and bounds provided for tandem motifs are generalized for the case r 2 if r is the number of subwords composing the motifs finally we also provide an algorithm to extract irredundant tandem motifs

root statistics of random polynomials with bounded mahler measure the mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coecient times the modulus of the product of its roots outside the unit circle the roots of a real degree n polynomial chosen uniformly from the set of polynomials of mahler measure at most 1 yields a pfaan point process on the complex plane when n is large with probability tending to 1 the roots tend to the unit circle and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle when this point is away from the real axis on which there is a positive probability of nding a root the scaled process degenerates to a determinantal point process with the same local statistics i e scalar kernel as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of mahler measure at most 1 three new matrix kernels appear in a neighborhood of 1 which encode information about the correlations between real roots between complex roots and between real and complex roots away from the unit circle the kernels converge to new limiting kernels which imply among other things that the expected number of roots in any open subset of c disjoint from the unit circle converges to a positive number we also give ensembles with identical statistics drawn from two dimensional electrostatics with potential theoretic weights and normal matrices chosen with regard to their topological entropy as actions on euclidean space

probability of local bifurcation type from a fixed point a random matrix perspective results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems real random matrices time delay dynamical systems companion matrices and a set of mappings known for their properties as universal approximators neural networks the eigenvalue spectrum is considered both numerically and analytically using previous work of edelman et al based upon the numerical evidence various conjectures are presented the conclusion is that in many circumstances most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues nevertheless surprising situations are presented for which the aforementioned conclusion does not hold e g real random matrices with gaussian elements with a large positive mean and finite variance

supplement on the structure of stationary solutions of the navier stokes equations this paper is a supplementary section to 1 we show that without any additional hypothesis the main result in 1 theorem 1 can be considerably strengthened note this paper cannot be read independently of 1 the numbering of equations theorems and propositions as well as cross references used here have to be understood as if this paper were an additional section to 1

pfaffian point process for the gaussian real generalised eigenvalue problem the generalised eigenvalues for a pair of n n matrices x1 x2 are defined as the solutions of the equation det x1 x2 0 or equivalently for x2 invertible as the eigenvalues of x 2 1 x 1 we consider gaussian real matrices x1 x2 for which the generalised eigenvalues have the rotational invariance of the half sphere or after a fractional linear transformation the rotational invariance of the unit disk in these latter variables we calculate the joint eigenvalue probability density function the probability pn k of finding k real eigenvalues the densities of real and complex eigenvalues the latter being related to an average over characteristic polynomials and give an explicit pfaffian formula for the higher correlation functions rho k 1 k 2 a limit theorem for pn k is proved and the scaled form of rho k 1 k 2 is shown to be identical to the analogous limit for the correlations of the eigenvalues of real gaussian matrices we show that these correlations satisfy sum rules characteristic of the underlying two component coulomb gas

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

motif patterns in 2d motif patterns consisting of sequences of intermixed solid and don t care characters have been introduced and studied in connection with pattern discovery problems of computational biology and other domains in order to alleviate the exponential growth of such motifs notions of maximal saturation and irredundancy have been formulated whereby more or less compact subsets of the set of all motifs can be extracted that are capable of expressing all others by suitable combinations in this paper we introduce the notion of maximal irredundant motifs in a two dimensional array and develop initial properties and a combinatorial argument that poses a linear bound on the total number of such motifs the remainder of the paper presents approaches to the discovery of irredundant motifs both by offline and incremental algorithms

the ginibre ensemble of real random matrices and its scaling limits we give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the pfaffian of an antisymmetric matrix formed from a 2 2 matrix kernel associated to the ensemble we apply this result to the real ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

stationary flow past a semi infinite flat plate analytical and numerical evidence for a symmetry breaking solution we consider the question of the existence of stationary solutions for the navier stokes equations describing the flow of a incompressible fluid past a semi infinite flat plate at zero incidence angle by using ideas from the theory of dynamical systems we analyze the vorticity equation for this problem and show that a symmetry breaking term fits naturally into the downstream asymptotic expansion of a solution finally in order to check that our asymptotic expressions can be completed to a symmetry breaking solution of the navier stokes equations we solve the problem numerically by using our asymptotic results to prescribe artificial boundary conditions for a sequence of truncated domains the results of these numerical computations a clearly compatible with the existence of such a solution

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

random fields and the enumerative geometry of lines on real and complex hypersurfaces we introduce a probabilistic framework for the study of real and complex enumerative geometry of lines on hypersurfaces this can be considered as a further step in the original shub smale program of studying the real zeros of random polynomial systems our technique is general and it also applies for example to the case of the enumerative geometry of flats on complete intersections we derive a formula expressing the average number e n of real lines on a random hypersurface of degree 2n 3 in mathbb r text p n in terms of the expected modulus of the determinant of a special random matrix in the case n 3 we prove that the average number of real lines on a random cubic surface in mathbb r text p 3 equals begin aligned e 3 6 sqrt 2 3 end aligned this technique can also be applied to express the number c n of complex lines on a generic hypersurface of degree 2n 3 in mathbb c text p n in terms of the expectation of the square of the modulus of the determinant of a random hermitian matrix as a special case we recover the classical statement c 3 27 we determine at the logarithmic scale the asymptotic of the quantity e n by relating it to c n whose asymptotic has been recently computed in 19 specifically we prove that begin aligned lim n rightarrow infty frac log e n log c n frac 1 2 end aligned finally we show that this approach can be used to compute the number r n 2n 3 of real lines counted with their intrinsic signs as defined in 28 on a generic real hypersurface of degree 2n 3 in mathbb r text p n

skew orthogonal polynomials and the partly symmetric real ginibre ensemble the partly symmetric real ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti symmetric gaussian random matrices such matrices typically have both real and complex eigenvalues for a fixed number of real eigenvalues an earlier work has given the explicit form of the joint eigenvalue probability density function we use this to derive a pfaffian formula for the corresponding summed up generalized partition function this pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries it can be used too to give the explicit form of the correlation functions provided certain skew orthogonal polynomials are computed this task is accomplished in terms of hermite polynomials and allows us to proceed to analyze various scaling limits of the correlations including that in which the matrices are only weakly non symmetric