level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

closed loop model set validation under a stochastic framework this paper deals with probabilistic model set validation it is assumed that the dynamics of a multi input multi output mimo plant is described by a model set with unstructured uncertainties and identification experiments are performed in closed loop a necessary and sufficient condition has been derived for the consistency of the model set with both the stabilizing controller and closed loop frequency domain experimental data fded in this condition only the euclidean norm of a complex vector is involved and this complex vector depends linearly on both the disturbances and the measurement errors based on this condition an analytic formula has been derived for the sample unfalsified probability sup of the model set some of the asymptotic statistical properties of the sup have also been briefly discussed a numerical example is included to illustrate the efficiency of the suggested method in model set quality evaluation

large deviations for the hbox sine _ beta and hbox sch _ tau processes the hbox sine beta process is the bulk point process limit of the gaussian beta ensemble for beta 1 2 and 4 this process gives the limit of the goe gue and gse random matrix models the hbox sch tau process is obtained similarly as the bulk scaling limit of the spectrum of certain discrete one dimensional random schrodinger operators both point processes have asymptotically constant average density in our chosen normalization one expects close to tfrac 1 2 pi lambda points in a large interval of length lambda we prove large deviation principles for the average densities of the processes identifying the rate function in both cases our approach is based on the representation of the counting functions using coupled systems of stochastic differential equations our techniques work for the full range of parameter values the results are novel even in the classical beta 1 2 and 4 cases for the hbox sine beta process they are consistent with the existing rigorous results on large gap probabilities and confirm the physical predictions made using log gas arguments

fast construction of robustness degradation function we develop a fast algorithm to construct the robustness degradation function which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set this function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system

asymptotics for the fredholm determinant of the sine kernel on a union of intervals in the bulk scaling limit of the gaussian unitary ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the fredholm determinant of the sine kernel frac sin x y pi x y over this interval a formal asymptotic expansion for the determinant ass tends to infinity was obtained by dyson in this paper we replace a single interval of lengths bysj wherej is a union ofm intervals and present a proof of the asymptotics up to second order the logarithmic derivative with respect tos of the determinant equals a constant expressible in terms of hyperelliptic integrals timess plus a bounded oscillatory function ofs zero ifm 1 periodic ifm 2 and in general expressible in terms of the solution of a jacobi inversion problem pluso 1 also determined are the asymptotics of the trace of the resolvent operator which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none the proofs use ideas from orthogonal polynomial theory

randomized algorithms for probabilistic robustness with real and complex structured uncertainty there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems unlike classical worst case methods these algorithms provide probabilistic estimates assessing for instance if a certain design specification is met with a given probability one of the advantages of this approach is that the robustness margins can be often increased by a considerable amount at the expense of a small risk in this sense randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information the applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres sampling in more general uncertainty sets is generally performed through overbounding at the expense of an exponential rejection rate in the paper randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general m spl delta configuration are studied in particular efficient polynomial time algorithms for uncertainty structures spl delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed

largest eigenvalue distribution in the double scaling limit of matrix models a coulomb fluid approach using thermodynamic arguments we find that the probability that there are no eigenvalues in the interval s infinity in the double scaling limit of hermitian matrix models is o exp s2 gamma 2 as s to infinity here gamma m 1 2 m 1 2 determines the mth multi critical point of the level density sigma x alpha b 1 x b 2 gamma x in b b and b2 alpha n furthermore the size of the transition zone where the eigenvalue density becomes vanishingly small at the tail of the spectrum is approximately n gamma 1 2 gamma 1 in agreement with earlier work based upon the string equation

on red black sor smoothing in multigrid optimal relaxation parameters are obtained for red black gauss seidel relaxation in multigrid solvers of a family of elliptic equations the resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator yielding simple inexpensive and fully parallelizable smoothers in many situations where less cost effective block and alternating direction schemes are commonly used

technical communique on the binomial confidence interval and probabilistic robust control the clopper pearson confidence interval has ever been documented as an exact approach in some statistics literature more recently such approach of interval estimation has been introduced to probabilistic control theory and has been referred as non conservative in control community in this note we clarify the fact that the so called exact approach is actually conservative in particular we derive analytic results demonstrating the extent of conservatism in the context of probabilistic robustness analysis this investigation encourages seeking better methods of confidence interval construction for robust control purpose

a parallel gauss seidel method for block tridiagonal linear systems a parallel variant of the block gauss seidel iteration is presented for the solution of block tridiagonal linear systems in this method parallel computations derive from a block reordering of the

a new parallel gauss seidel method based on alternating group explicit method and domain decomposition method a new parallel gauss seidel method is presented for solution of system of linear equations related to finite difference discretization of partial differential equations this method is based on domain decomposition method and local coupling between interfaces of neighbor sub domains same as alternating group explicit method this method is convergent and number of iterations for achieving convergence criteria is near the original gauss seidel method sometimes better and sometimes worse but difference is very small the convergence theory is discussed in details numerical results are given to justify the convergence and performance of the proposed iterative method

dyson s constant in the asymptotics of the fredholm determinant of the sine kernel we prove that the asymptotics of the fredholm determinant of i k where k is the integral operator with the sine kernel open image in new window on the interval 0 are given by

a probabilistic framework for problems with real structured uncertainty in systems and control the objective of this paper is twofold first the problem of generation of real random matrix samples with uniform distribution in structured spectral norm bounded sets is studied this includes an analysis of the distribution of the singular values of uniformly distributed real matrices and an efficient i e polynomial time algorithm for their generation second it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

on the distribution of the total energy of a system of non interacting fermions random matrix and semiclassical estimates abstract we consider a single particle spectrum as given by the eigenvalues of the wigner dyson ensembles of random matrices and fill consecutive single particle levels with n fermions assuming that the fermions are non interacting we show that the distribution of the total energy is gaussian and its variance grows as n 2 log n in the large n limit next to leading order corrections are also computed some related quantities are discussed in particular the nearest neighbor spacing autocorrelation function canonical and grand canonical approaches are considered and compared in detail a semiclassical formula describing as a function of n a non universal behavior of the variance of the total energy starting at a critical number of particles is also obtained it is illustrated with the particular case of single particle energies given by the imaginary part of the zeros of the riemann zeta function on the critical line

asymptotics of level spacing distributions for random matrices asymptotic formulas for the probability of finding exactly n eigenvalues in an interval of length s for large s and fixed n are given for random matrices taken from the gaussian ensembles 1 2 4 these exact results are compared with the predictions of a continuum coulomb gas model due to dyson

new stable group explicit finite difference method for solution of diffusion equation a new group explicit method for solution of diffusion equation is presented this method is based on domain decomposition concept and using asymmetric saul yev schemes for internal nodes of each sub domain and alternating group explicit method for sub domain s interfacial nodes this new method has several advantages such as good parallelism unconditional stability fully explicit nature and better accuracy than original saul yev schemes the details of implementation and proving stability are briefly discussed numerical experiments on stability and accuracy are also presented

level spacing functions and connection formulas for painleve v transcendent abstract the connection formulas for painleve v transcendents are applied for calculations of the fredholm determinant of the kernel sin x y x y on the finite interval t t the level spacing functions arising in the theory of random matrices as well as for the correlators of one dimensional xy model are expressed via this fredholm determinant its asymptotics for large t come from nonlinear second order ode with known initial conditions at t 0 the backlund transformations reduce this ode to the painleve v equation while the isomonodromic deformation method idm provides the explicit connection formulas for the necessary pv transcendent the result generalizes the well known dyson asymptotic expansion for the one level spacing function to a case of arbitrary many levels

a parallel gauss seidel method using nr data flow ordering a parallel variant of the block gauss seidel method is presented to solve the poisson equation with dirichlet boundary condition this method uses two dimensional logically connected parallel processors furthermore natural rowwise nr data flow block ordering is used so that its convergence rate is the same as that of the standard block gauss seidel method spectral radius is determined by the formula for general k x l block iterative methods numerical computations on a parallel computer are included

dyson s constants in the asymptotics of the determinants of wiener hopf hankel operators with the sine kernel let k alpha pm stand for the integral operators with the sine kernels frac sin x y pi x y pm frac sin x y pi x y acting on l2 0 dyson conjectured that the asymptotics of the fredholm determinants of i k alpha pm are given by log det i k alpha pm frac alpha 2 4 mp frac alpha 2 frac log alpha 8 frac log 2 24 pm frac log 2 4 frac 3 2 zeta 1 o 1 as in this paper we are going to give a proof of these two asymptotic formulas

about the spacing functions of the three matrix ensembles three ensembles of random matrices have been extensively studied orthogonal unitary and symplectic characterized by a parameter taking values 1 2 and 4 the probability that a randomly chosen interval of length 2t contains exactly r eigenvalues of such a matrix can be expressed in terms of the rth partial derivative of a fredholm determinant using this fact we give a new proof of some known relations between and as well as a relation between odd and even spheroidal functions

random spherical uncertainty in estimation and robustness a theorem is formulated that gives an exact probability distribution for a linear function of a random vector uniformly distributed over a ball in n dimensional space this mathematical result is illustrated via applications to a number of important problems of estimation and robustness under spherical uncertainty these include parameter estimation characterization of attainability sets of dynamical systems and robust stability of affine polynomial families

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

probabilistic robust design with linear quadratic regulators in this paper we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators lqr we consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism the approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters in particular we propose an iterative algorithm for designing a controller which is based upon subgradient iterations at each step of the sequence we first generate a random sample and then we perform a subgradient step for a convex constraint defined by the lqr problem the main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps in addition at a fixed step we compute a lower bound of the probability that a quadratically stabilizing controller is found