random matrix theory and the zeros of ζ s we study the density of the roots of the derivative of the characteristic polynomial z u z of an n n random unitary matrix with distribution given by haar measure on the unitary group based on previous random matrix theory models of the riemann zeta function s this is expected to be an accurate description for the horizontal distribution of the zeros of s to the right of the critical line we show that as n the fraction of the roots of z u z that lie in the region 1 x n 1 z 1 tends to a limit function we derive asymptotic expressions for this function in the limits x and x 0 and compare them with numerical experiments

recursive estimation methods for discrete systems in system identification the error evolution is composed of two decoupled parts one is the identifying information on the current estimation residual while the other is past arithmetic errors previous recursive algorithms only considered how to update current prediction errors up to now research has mostly been based on recursive least squares rls methods in this note a general recursive identification method is proposed for discrete systems using this new algorithm a recursive empirical frequency domain optimal parameter refop estimate is established the refop method has the advantage of resisting disturbance noise some simulations are included to illustrate the new method s reliability

mixed quasi variational inequalities in this paper we use the resolvent operator and auxiliary principle techniques to suggest and analyze several iterative algorithms for solving mixed quasi variational inequalities and related problems we study the convergence criteria of these algorithms under mild conditions we also study the global stability and existence of a unique solution of these quasi variational inequalities by using the dynamical systems approach our results represent refinement and improvement of the previously known results for variational inequalities

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

random matrix theory the exceptional lie groups and l functions there has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number theoretical l functions one example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of l functions these connections are extended here to non classical groups we focus on an explicit example the exceptional lie group g2 the value distributions for characteristic polynomials associated with the 7 and 14 dimensional representations of g2 defined with respect to the uniform invariant haar measure are calculated using two of the macdonald constant term identities a one parameter family of l functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the seven dimensional representation of g2 the random matrix calculations extend to all exceptional lie groups

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

model identification of a noncausal 2 d ar process using a causal 2 d ar model on the nonsymmetric half plane if a noncausal two dimensional 2 d autoregressive ar process is bi causal there exists a causal 2 d ar process on the nonsymmetric half plane having the same autocorrelations as the noncausal 2 d ar process a formula is presented to relate the ar coefficients of the noncausal 2 d ar process with those of the causal 2 d ar process on the nonsymmetric half plane the 2 d yule walker equations are derived for causal 2 d ar models on the nonsymmetric half plane a computationally efficient order recursive algorithm is proposed to solve the 2 d yule walker equations using the autocorrelation equivalence relation and the order recursive algorithm we can easily identify a noncausal 2 d ar process from its autocorrelations

a rank test based approach to order estimation i 2 d ar models application in system identification and parametric spectral estimation by two dimensional 2 d autoregressive ar and 2 d autoregressive moving average arma models the order selection problem is often required in this correspondence we show that the information of 2 d ar model order is implicitly contained in a correlation matrix an algorithm for 2 d quarter plane ar model order determination is proposed numerical simulations are presented to show the efficiency of the proposed singular value decomposition svd based algorithm

energy level statistics of a critical random matrix ensemble we study energy level statistics of a critical random matrix ensemble of power law banded complex hermitian matrices we compute the level compressibility via the level number variance and compare it with the analytical formula for the exactly solvable model of moshe neuberger and shapiro

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

asymptotic generalized eigenvalue distribution of block multilevel toeplitz matrices in many detection and estimation problems associated with processing of second order stationary random processes the observation data are the sum of two zero mean second order stationary processes the process of interest and the noise process in particular the main performance criterion is the signal to noise ratio snr after linear filtering the optimal snr corresponds to the maximal value of a rayleigh quotient which can be interpreted as the largest generalized eigenvalue of the covariance matrices associated with the signal and noise processes which are block multilevel toeplitz structured for m dimensional vector valued second order stationary p dimensional random processes xi1 i2 ip isin bbr m in this paper an extension of szego s theorem to the generalized eigenvalues of hermitian block multilevel toeplitz matrices is given providing information about the asymptotic distribution of those generalized eigenvalues and in particular of the optimal snr after linear filtering a simple proof of this theorem under the hypothesis of absolutely summable elements is given the proof is based on the notion of multilevel asymptotic equivalence between block multilevel matrix sequences derived from the celebrated gray approach finally a short example in wideband space time beamforming is given to illustrate this theorem

a new approach for subset 2 d ar model identification for describing textures this paper addresses the problem of identification of appropriate autoregressive ar components to describe textural regions of digital images by a general class of two dimensional 2 d ar models in analogy with univariate time series the proposed technique first selects a neighborhood set of 2 d lag variables corresponding to the significant multiple partial auto correlation coefficients a matrix is then suitably formed from these 2 d lag variables using singular value decomposition svd and orthonormal with column pivoting factorization qrcp techniques the prime information of this matrix corresponding to different pseudoranks is obtained schwarz s 1978 information criterion sig is then used to obtain the optimum set of 2 d lag variables which are the appropriate autoregressive components of the model for a given textural image a four class texture classification scheme is illustrated with such models and a comparison of the technique with the work of chellappa and chatterjee 1985 is provided

multidimensional multiple order complex parametric model identification this paper presents a way to access both the multiple order and parameters of a multidimensional complex number autoregressive ar model through matrix factorization the principle of this technique consists of the transformation of the multidimensional model to a pseudo simple input simple output ar model then performing factorization of the covariance matrix of the data this factorization then leads to a recursive form of the parameter and order estimation this paper makes two principal contributions the first is a generalization of one dimensional factored form algorithm and the second is that it makes it possible to access all the possible different orders and parameters of a multidimensional complex number ar model of any dimension whereas classical approaches are limited to at most four dimensional models computer simulation results are provided to illustrate the behavior of this method

partition functions for matrix models and isomonodromic tau functions for one matrix models with polynomial potentials the explicit relationship between the partition function and the isomonodromic tau function for the 2 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived

3 d ar model order selection via rank test procedure this paper deals with the problem of three dimensional autoregressive 3 d ar model order estimation we show that the information for the 3 d ar model order is implicitly contained in an appropriate matrix rank built from the autocorrelation function acf of the underlying 3 d gaussian process exploiting this property we develop an algorithm to estimate the order p1 p2 p3 corresponding to the quarter space qs region of support the proposed method is based upon a rank test procedure rtp using singular value decomposition svd and solving nonlinear system equations numerical simulations are presented to illustrate the performances of the proposed algorithm

two chi square statistics for determining the orders p and q of an arma p q process the theta lambda and eta functions have been previously proposed for use in choosing the autoregressive and moving average orders of an arma q p process visually two chi square statistics associated with these three functions are presented and used here to determine the orders of an arma process statistically it is shown that the two statistics are asymptotically equivalent to the quenouille walker s goodness of fit test statistic which is a lagrange multiplier test statistic some numerical examples are presented to illustrate the usefulness of the two chi square statistics as well as the three functions in arma modeling

anderson localization problems in gapless superconducting phases abstract the interplay of anderson localization and different kinds of superconducting order is most interesting in gapless cases i e for nonvanishing electron density of states at ef i present a new renormalization group result for anderson localization in the gapless type ii limit of an ising superconducting sc glass from this calculation a guess is also made for the xy superconducting glass in both cases and in contrast to localization in normal systems two renormalization constants one for field and one for coupling constant renormalization are necessary and sufficient the density of states at ef is singular with exponent for the ising sc glass i obtain vi 1 d 2 i 1 2 and i 0 while the xy sc glass has vxy 1 d 2 xy 1 and xy d 2 all in leading order of the d 2 expansion and for e ef for e ef a symmetry argument and also the calculation given here predict usual localization behaviour with v 1 d 2 0 and 2 d in both cases the effects of cooper pairs on localization in the pure superconducting glasses is compared with earlier results showing perfect coexistence of anderson localization with dirty superconductivity approximately described by a nonrandom order parameter these limiting cases are embedded in a more general field theory given here which contains three superconducting order parameters and a conventional pair breaking mechanism

modeling 2 d ar processes with various regions of support we show that there exists a causal 2 d linear process in the nonsymmetric half plane having the same autocorrelations as a noncausal 2 d linear process in the whole plane this property is called the autocorrelation equivalence relation and can be used for practical fitting and modeling of 2 d processes some causal 2 d autoregressive ar models with various regions of support are considered such as half cross half diamond quarter plane square half square half hexagon half octagon and half circle considerations of parsimony in 2 d model fitting are then focused not only on the number of parameters in our model but also most importantly on the optimal shape of the region of support their 2 d yule walker equations are derived and a computationally efficient order recursive algorithm is proposed to solve them the autocorrelation equivalence relation and the order recursive algorithm are utilized to specify a noncausal 2 d ar process as well as its spectrum from a given realization of a random field

double scaling limit in random matrix models and a nonlinear hierarchy of differential equations we study the critical behaviour of a random hermitian one matrix model with nonsymmetric interaction at a critical point in which the eigenvalue density function has a zero of degree 2m m 1 inside a cut we prove that in the generic case m 1 the model exhibits a third order phase transition in temperature we formulate an ansatz for the double scaling limit of recurrence coefficients which is consistent with the quasiperiodic asymptotics of recurrence coefficients in the low temperature region and from this ansatz we derive the painleve ii hierarchy of ordinary differential equations for the recurrence coefficients in addition we derive an integral kernel which governs the double scaling limit of correlation functions

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

generalizations of some integrals over the unitary group using the character expansion method we generalize several well known integrals over the unitary group to the case where general complex matrices appear in the integrand these integrals are of interest in the theory of random matrices and may also find applications in lattice gauge theory

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

optimum quarter plane autoregressive modeling of 2 d fields using four field lattice approach a new orthogonal four field two dimensional 2 d quarter plane lattice structure with a complete set of reflection coefficients is developed by employing appropriately defined auxiliary prediction errors this work is the generalization of the three parameter lattice filter proposed by parker and kayran 1984 after the first stage four auxiliary forward and four auxiliary backward prediction errors are generated in order to obtain a growing number of 2 d reflection coefficients at successive stages the theory has been proven by using a geometrical formulation based on the mathematical concepts of vector space orthogonal projection and subspace decomposition it is shown that all four quarter plane filters are orthogonal and thus optimum for all stages in addition to developing the basic theory a set of orthogonal backward prediction error fields for successive lattice parameter model stages is presented

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

an order recursive algorithm to solve the 3 d yule walker equations of causal 3 d ar models an order recursive algorithm is proposed to solve the 3 d yule walker equations of causal 3 d ar models it is computationally efficient and can be easily transformed into a computer program moreover it can be utilized to determine the orders of a causal 3 d ar process

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

three dimensional textured image blocks model based on wold decomposition in this paper we present a new parametric modeling of three dimensional homogeneous volumes of textured images 3d texture based on 3d wold decomposition for discrete processes this decomposition additively separates any 3d discrete homogeneous process into three mutually orthogonal components a purely indeterministic a remote past type deterministic and an evanescent component the evanescent component can be further decomposed into two orthogonal components called type 1 evanescent and type 2 evanescent assuming that the 3d image volume is a finite realization of a discrete homogeneous stochastic process the 3d texture field can be uniquely represented as a sum of four mutually orthogonal components with different spectral properties the aim of this paper is to present new parametric models able to describe both the spectral support and spatial characteristics of each component a spectral decomposition algorithm to separate a 3d texture into structured and unstructured parts is also proposed