asymptotics in asep with step initial condition in previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition particles beginning at the positive integers there it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a fredholm determinant here we use this formula to obtain three asymptotic results for the positions of these particles in one an apparently new distribution function arises and in another the distribution function f2 arises the latter extends a result of johansson on tasep to asep and hence proves kpz universality for asep with step initial condition

on subordinacy and spectral multiplicity for a class of singular differential operators the spectral multiplicity of self adjoint operators h associated with singular differential expressions of the form is investigated based on earlier work of i s kac and recent results on subordinacy complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of i the boundary behaviour of titchmarsh weyl m functions and ii the asymptotic properties of solutions of lu u at the endpoints a and b in particular it is shown that h has multiplicity two if and only if l is in the limit point case at both a and b and the set of all for which no solution of lu u is subordinate at either a or b has positive lebesgue measure the results are completely general subject only to minimal restrictions on the coefficients p r q r and w r and the assumption of separated boundary conditions when l is in the limit circle case at both endpoints

on schur flows for the finite schur dmkdv flows a non local poisson structure is introduced and shown to be linked via backlund darboux transformations to linear and quadratic poisson structures for the toda lattice two different lax representations for the schur flows are used one to construct backlund darboux transformations and the other to solve the cauchy problem via the trigonometric moment problem

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

matrix orthogonal polynomials on the unit circle the properties of matrix orthogonal polynomials on the unit circle are investigated beginning with their recurrence formulas the techniques of scattering theory and banach algebras are used in the investigation a matrix generalization of a theorem of baxter is proved

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

cmv the unitary analogue of jacobi matrices we discuss a number of properties of cmv matrices by which we mean the class of unitary matrices studied recently by cantero moral and velazquez we argue that they play an equivalent role among unitary matrices to that of jacobi matrices among all hermitian matrices in particular we describe the analogues of well known properties of jacobi matrices foliation by co adjoint orbits a natural symplectic structure algorithmic reduction to this shape lax representation for an integrable lattice system ablowitz ladik and the relation to orthogonal polynomials as offshoots of our analysis we will construct action angle variables for the finite ablowitz ladik hierarchy and describe the long time behavior of this system 2006 wiley periodicals inc

nonabelian nonlinear lattice equations on finite interval we apply the inverse spectral problem method to the class of non abelian nonlinear lattice equations on the finite interval the integrable discrete nonlinear schrodinger and discrete modified korteveg de vries equations are considered as examples in the latter case the large time asymptotics for solutions are found

a simple adaptive feedback scheme for identical synchronizing chaotic systems with uncertain parameters an adaptive feedback control scheme for identical synchronizing two linearly coupled chaotic systems and identifying uncertain parameters is proposed in comparison with the previous methods it inherited the present scheme is more flexible and accessible this is because that the prerequisites on the given chaotic systems are much looser and the adaptive feedback controllers involved are relatively independent moreover the present scheme is a thoroughly amendatory one by which the uncertain parameters can be identified three concrete chaotic systems are employed to show the effectiveness of the presented scheme in addition how the coefficient in the adaptive law affects the convergence rate and the final strength is studied numerically

chaos synchronization and parameter estimation of single degree of freedom oscillators via adaptive control this paper addresses the problem of synchronizing a class of single degree of freedom oscillators with uncertain parameters a modified adaptive control scheme is proposed to achieve globally asymptotic stable synchronization between the master and slave oscillators with arbitrary different initial conditions based on the barbalat s lemma one of the advantages of the method is that it requires only a scalar driving signal which may be easily designed in practical applications on the other hand the method can ensure that the unknown parameters in the slave oscillator would be completely estimated with an adaptive updating law numerical simulations are performed on two examples to verify the analytical results

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

a quantum dynamical approach to matrix khrushchev s formulas khrushchev s formula is the cornerstone of the so called khrushchev theory a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle this formula can be understood as a factorization of the schur function for an orthogonal polynomial modification of a measure on the unit circle no such formula is known in the case of matrix valued measures this constitutes the main obstacle to generalize khrushchev theory to the matrix valued setting which we overcome in this paper it was recently discovered that orthogonal polynomials on the unit circle and their matrix valued versions play a significant role in the study of quantum walks the quantum mechanical analogue of random walks in particular schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system a topic in which khrushchev s formula has profound and surprising implications we will show that this connection between schur functions and quantum walks is behind a simple proof of khrushchev s formula via quantum diagrammatic techniques for cmv matrices this does not merely give a quantum meaning to a known mathematical result since the diagrammatic proof also works for matrix valued measures actually this path counting approach is so fruitful that it provides different matrix generalizations of khrushchev s formula some of them new even in the case of scalar measures furthermore the path counting approach allows us to identify the properties of cmv matrices which are responsible for khrushchev s formula on the one hand this helps to formalize and unify the diagrammatic proofs using simple operator theory tools on the other hand this is the origin of our main result which extends khrushchev s formula beyond the cmv case as a factorization rule for schur functions related to general unitary operators 2015 wiley periodicals inc

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

blocks in the asymmetric simple exclusion process in earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that the mth particle from the left is at site x at time t they were expressed in general as sums of multiple integrals and for the case of step initial condition as an integral involving a fredholm determinant in the present work these results are generalized to the case where the mth particle is the left most one in a contiguous block of l particles the earlier work depended in a crucial way on two combinatorial identities and the present work begins with a generalization of these identities to general l

schur parameter pencils for the solution of the unitary eigenproblem abstract let u v be an n n pencil with unitary matrices u and v an algorithm is presented which reduces u and v simultaneously to unitary block diagonal matrices g o q h up and g e q h vp with block size at most two it is an o n 3 process using householder eliminations and it is backward stable in the special case v i the block diagonal matrices g o g h e can be normalized so that their entries are just the schur parameters of the hessenberg condensed form of u we call g o g e a schur parameter pencil it can also be derived from u v by a lanczos like process for the solution of the eigenvalue problem for g o g e a qr type algorithm can be developed based on this unitary reduction of a pencil u v to a schur parameter pencil the condensed form is preserved throughout the process each iteration step needs only o n operations this method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the qr method for the hermitian eigenvalue problem

on a theorem of kac and gilbert we prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum

anderson localization for quasi periodic cmv matrices and quantum walks abstract we consider cmv matrices both standard and extended with analytic quasi periodic verblunsky coefficients and prove anderson localization in the regime of positive lyapunov exponents this establishes the cmv analog of a result bourgain and goldstein proved for discrete one dimensional schrodinger operators we also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi periodic coins

crossed products of c algebras and spectral analysis of quantum hamiltonians we study spectral properties of a hamiltonian by analyzing the structure of certain c algebras to which it is affiliated the main tool we use for the construction of these algebras is the crossed product of abelian c algebras generated by the classical potentials by actions of groups we show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians this scheme provides a unified approach to the study of hamiltonians of anisotropic and many body systems including quantum fields

synchronizing chaotic dynamics with uncertainties based on a sliding mode control design the synchronization of two chaotic systems with uncertainties is studied in this paper a feedback controller is provided based on a sliding mode control design a kind of extended state observer is used to compensate for the systems uncertainties such as the structure difference or parameter mismatching using only the available synchronizing error then the feedback controller becomes physically realizable based on the states of the observer and can be used to synchronize two continuous chaotic systems illustrative examples of the synchronization of duffing and van der pol oscillators as well as two lorenz systems with parameter mismatching are proposed to show the effectiveness of this method

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

a fredholm determinant representation in asep in previous work tracy and widom in commun math phys 279 815 844 2008 the authors found integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice the dynamics are uniquely determined once the initial state is specified in this note we restrict our attention to the case of step initial condition with particles at the positive integers and consider the distribution function for the mth particle from the left in tracy and widom commun math phys 279 815 844 2008 an infinite series of multiple integrals was derived for the distribution in this note we show that the series can be summed to give a single integral whose integrand involves a fredholm determinant we use this determinant representation to derive non rigorously at this writing a scaling limit

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

szegő difference equations transfer matrices and orthogonal polynomials on the unit circle we develop the theory of orthogonal polynomials on the unit circle based on the szeg recurrence relations written in matrix form the orthogonality measure and c function arise in exactly the same way as weyl s function in the weyl approach to second order linear differential equations on the half line the main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one dimensional schrodinger operators discrete and continuous and the notion of subordinacy from the gilbert pearson theory we study the relations between transfer matrices and the structure of orthogonality measures the theory is illustrated by the szeg equations with reflection coefficients having bounded variation

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

trace class perturbations and the absence of absolutely continuous spectra we show that various hamiltonians and jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices

bulk edge correspondence of one dimensional quantum walks we outline a theory of symmetry protected topological phases of one dimensional quantum walks we assume spectral gaps around the symmetry distinguished points 1 and 1 in which only discrete eigenvalues are allowed the phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure however our theory requires no translation invariance whatsoever and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations even those that cannot be continuously contracted to the identity more precisely we define two indices for every walk characterizing the behavior far to the right and far to the left respectively their sum is a lower bound on the number of eigenstates at 1 and 1 for a translation invariant system the indices add up to zero so one of them already characterizes the phase by joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel so the theory predicts bound states at 1 or 1 this is a rigorous statement of bulk edge correspondence the results also apply to the hamiltonian case with a single gap at zero

cmv matrices in random matrix theory and integrable systems a survey we present a survey of recent results concerning a remarkable class of unitary matrices the cmv matrices we are particularly interested in the role they play in the theory of random matrices and integrable systems throughout the paper we also emphasize the analogies and connections to jacobi matrices

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

formulas for joint probabilities for the asymmetric simple exclusion process in earlier work the authors tracy c a and widom h integral formulas for the asymmetric simple exclusion process commun math phys 279 815 2008 obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process here formulas are obtained for joint probabilities for several particles in the case of a single particle the derivation here is simpler than the one in the earlier work for one of its main results

algebro geometric solutions of the baxter szegő difference equation we derive theta function representations of algebro geometric solutions of a discrete system governed by a transfer matrix associated with an extension of the trigonometric moment problem studied by szeg and baxter we also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro geometric solutions

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

integral formulas for the asymmetric simple exclusion process in this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice mathbb z with nearest neighbor hopping rates p to the right and q 1 p to the left for the most part we consider an n particle system but for certain of these formulas we can take the n to infty limit first we obtain for the n particle system a formula for the probability of a configuration at time t given the initial configuration for this we use bethe ansatz ideas to solve the master equation extending a result of schutz for the case n 2 the main results of the paper derived from this are integral formulas for the probability for given initial configuration that the mth left most particle is at x at time t in one of these formulas we can take the n to infty limit and it gives the probability for an infinite system where the initial configuration is bounded on one side for the special case of the totally asymmetric simple exclusion process tasep our formulas reduce to the known ones

a difference equation arising from the trigonometric moment problem having random reflection coefficients an operator theoretic approach abstract operators associated with polynomials orthogonal on the unit circle and their doubly infinite analogs are considered we assume that the elements in the operators are obtained from a stationary stochastic process and we investigate the properties of the spectrum of these operators lyapunov exponents are introduced and we examine their relationship with the absolutely continuous components of the spectrum of the above operators

inverse problem for polynomials orthogonal on the unit circle polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated it is shown that the coefficients are in fact quasi periodic the measures associated with these quasi periodic coefficients are exhibited and necessary and sufficient conditions relating quasi periodicity and spectral measures of this type are given analogs for polynomials orthogonal on subsets of the real line are also presented