generalizations of chebyshev polynomials and polynomial mappings in this paper we show how polynomial mappings of degree from a union of disjoint intervals onto generate a countable number of special cases of generalizations of chebyshev polynomials we also derive a new expression for these generalized chebyshev polynomials for any genus from which the coefficients of can be found explicitly in terms of the branch points and the recurrence coefficients we find that this representation is useful for specializing to polynomial mapping cases for small where we will have explicit expressions for the recurrence coefficients in terms of the branch points we study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials we also derive an explicit expression for the discriminant for the genus 1 case of our chebyshev polynomials that is valid for any configuration of the branch point

a hybrid pseudo force laplace transform method for non linear transient response of a suspended cable a hybrid numerical scheme involving the combination of the laplace transform technique and the pseudo force method is proposed to analyze the non linear transient response of a suspended cable subjected to arbitrary dynamic loading a theoretical model of the cable with multi degree of freedom is first obtained through discretization of the partial differential equations by finite difference approximation the non linear governing equations take into account the effects of quadratic and cubic geometric non linearities the proposed method deals with the non linear effects as pseudo forces and then establishes an iterative solution scheme in the alternating laplace time domain by means of fast numerical laplace transform this method eschews a time stepping process and therefore is computationally efficient it also readily deals with the viscoelastic damping with frequency dependent model parameters and the hysteresis damping in terms of complex stiffness models numerical examples are presented to evaluate the dynamic responses of suspended cables under a concentrated sinusoidal force and a distributed random excitation and to identify the non linear response properties by comparison with the linear vibration the validity and accuracy of the proposed method is also verified by comparing the results with those obtained by using the direct time integration

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

global existence for energy critical waves in 3 d domains we prove that the defocusing quintic wave equation with dirichlet boundary conditions is globally well posed on for any smooth compact domain the main ingredient in the proof is an spectral projector estimate obtained recently by smith and sogge combined with a precise study of the boundary value problem

orthogonal and extremal polynomials on several intervals abstract let a1 1 where is positive on el are characterised by orthogonality conditions and some new results are proved then the connection with polynomials orthogonal with respect to h where has an odd number of zeros in each interval a2j a2j 1 j 1 l 1 is shown and a full description of orthogonal polynomials having periodic respectively asymptotic periodic recurrence coefficients is given finally the asymptotic behaviour of polynomials orthogonal on several intervals is discussed

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

on the defect of compactness for the strichartz estimates of the schrodinger equations abstract in this paper we prove that every sequence of solutions to the linear schrodinger equation with bounded data in h1 r d d 3 can be written up to a subsequence as an almost orthogonal sum of sequences of the type h d 2 2nv t tn h2n x xn hn where v is a solution of the linear schrodinger equation with a small remainder term in strichartz norms using this decomposition we prove a similar one for the defocusing h1 critical nonlinear schrodinger equation assuming that the initial data belong to a ball in the energy space where the equation is solvable this implies in particular the existence of an a priori estimate of the strichartz norms in terms of the energy

polynomial inverse images and polynomial inequalities valid for polynomials pn of degree at most n in this paper we are primarily interested in what form these inequalities take on several intervals we shall see that the extension to general sets involves the equilibrium measure of these sets we shall give the precise form of the bernstein inequality for arbi trary compacts and an asymptotical ly best form of the markoff inequality for sets consisting of finitely many intervals actually in this case we shall prove different markoff inequalities one one associated with each one of the endpoints of the system of intervals the proofs will heavily use sets that are obtained as the inverse images of intervals under special polynomial mappings we shall see that the original bernstein and

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings

orthogonal and l q extremal polynomials on inverse images of polynomial mappings abstract let t be a polynomial of degree n and let k be a compact set with c first it is shown if zero is a best approximation to f from p n on k with respect to the lq norm q 1 then zero is also a best approximation to f t on t 1 k with respect to the l q t norm where t arises from by the transformation t in particular t is the equilibrium measure on t 1 k if is the equilibrium measure on k for q i e the sup norm a corresponding result is presented in this way polynomials minimal on several intervals on lemniscates on equipotential lines of compact sets etc are obtained special attention is given to lq minimal polynomials on julia sets next based on asymptotic results of widom we show that the minimum deviation of polynomials orthogonal with respect to a positive measure on t 1 k behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour too some open problems are also given

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

asymptotics of chebyshev polynomials ii dct subsets of mathbb r we prove szego double acute widom asymptotics for the chebyshev polynomials of a compact subset of which is regular for potential theory and obeys the parreau widom and dct conditions

christoffel functions and tura n determinants on several intervals abstract we consider orthogonal polynomials on the real line with an underlying asymptotic periodic structure for the coefficients in the three term recurrence relation these polynomials live on a set e which consists of several intervals we give a survey of some techniques for obtaining asymptotic results of the orthogonal polynomials on the oscillatory region e from these results we obtain a useful technique for approximating the weight function on e by means of christoffel functions and shifted turan determinants

kotani last problem and hardy spaces on surfaces of widom type this is a small theory of non almost periodic ergodic families of jacobi matrices with purely however absolutely continuous spectrum the reason why this effect may happen is that under our axioms we found an analytic condition on the resolvent set that is responsible for exactly equivalent to this effect

periodic jacobi matrices on trees abstract we begin the systematic study of the spectral theory of periodic jacobi matrices on trees including a formal definition the most significant result that appears here for the first time is that these operators have no singular continuous spectrum we review important previous results of sunada and aomoto and present several illuminating examples we present many open problems and conjectures that we hope will stimulate further work

eigenproblems for tridiagonal 2 toeplitz matrices and quadratic polynomial mappings abstract given a system of monic orthogonal polynomials mops p n x n 0 we characterize all the sequences of monic orthogonal polynomials q n x n 0 such that q 1 x x b q 2 n x p n 2 x n 0 1 2 where 2 is a fixed polynomial of degree exactly 2 and b is a fixed complex number with an appropriate choice of the mops p n x n 0 our results enables us to solve the eigenproblem of a tridiagonal 2 toeplitz matrix giving an alternative proof to a recent result by m j c gover we also find the relations between the jacobi matrices corresponding to the mops p n x n 0 and q n x n 0 finally we show that if p n x n 0 is a semiclassical orthogonal polynomial sequence then so is q n x n 0 and in particular we analyze the classical case in detail

analysis of the hum control operator and exact controllability for semilinear waves in uniform time we present a regularity result for the hum optimal control associated with the interior control of linear waves we use this analysis together with strichartz inequalities to get results on the exact controllability for subcritical nonlinear waves in a bounded domain of mathbb r 3

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

dynamic analysis of large displacement cable motion with experimental verification cables used in engineering applications may undergo large displacement motion when subjected to dynamic loads particularly for cables with relatively low tension as cables are much weaker in the out of plane motion than the in plane motion three dimensional dynamic analysis is often necessary even when the excitation contains only a small out of plane component this paper presents the dynamic analysis of three dimensional cable motion accounting for axial flexural and torsional deformations as well as geometric non linearity due to large displacements and rotations the comprehensive study covers analytical formulation numerical strategy based on an iterative finite difference scheme and experimental verification by means of shaking table tests a specific problem of cable motion due to support excitation is used to illustrate the asymmetry and sensitivity of the dynamic tension response associated with geometric non linearity of large displacement cable motion the shaking table tests validate the accuracy of the numerical results obtained for both two dimensional and three dimensional cases

iterative solution of indefinite symmetric linear systems by methods using orthogonal polynomials over two disjoint intervals in the classical chebyshev algorithm for solving symmetric positive definite linear systems ax b the approximate solution is of the form x k x 0 s k a r 0 where r 0 b ax 0 is the initial residual and s k is a polynomial of degree k 1 which minimizes 1 lambda s k lambda infty where cdot infty is the uniform norm on some interval s containing the spectrum of a but not the origin if a is not positive definite the set s can be taken to be the union of two disjoint intervals a b and c d with b 0 c and a generalization of the chebyshev algorithm based on the minimax polynomial has been considered by several authors in this paper we propose a new technique based upon the least squares polynomial in the set s i e the polynomial t k which minimizes 1 lambda t k lambda w where cdot w is an l 2 norm with respect to a weight function w lambda defined on s with a suitable weight function w lambda the po

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

space time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables abstract this paper presents space time numerical simulation and validation of analytical predictions for the finite amplitude forced dynamics of suspended cables the main goal is to complement analytical and numerical solutions accomplishing overall quantitative qualitative comparisons of nonlinear response characteristics by relying on an approximate kinematically non condensed planar modeling a simply supported horizontal cable subject to a primary external resonance and a 1 1 or 1 1 vs 2 1 internal resonance is analyzed to obtain analytical solution a second order multiple scales approach is applied to a complete eigenfunction based series of nonlinear ordinary differential equations of cable damped forced motion accounting for both quadratic cubic geometric nonlinearities and multiple modal contributions local scenarios of cable uncoupled coupled responses and associated stability are predicted based on chosen reduced order models as a cross checking tool numerical simulation of the associated nonlinear partial differential equations describing the dynamics of the actual infinite dimensional system is carried out using a finite difference technique employing a hybrid explicit implicit integration scheme based on system control parameters and initial conditions cable amplitude displacement and tension responses are numerically assessed thoroughly validating the analytically predicted solutions as regards the actual existence the meaningful role and the predominating internal resonance of coexisting competing dynamics some methodological aspects are noticed along with a discussion on the kinematically approximate versus exact as well as planar versus non planar cable modeling

an electrostatic interpretation of the zeros of sieved ultraspherical polynomials in an earlier work castillo et al j math anal appl 455 1801 1821 2017 it was proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations in this work we use this fact to derive in a unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind in particular we derive ordinary differential equations for these polynomials as an application we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of these polynomials mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field

orthogonality properties of iterated polynomial mappings we consider a measure defined on a complex contour and its associated orthogonal polynomials the action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given iterating the transformation provides an invariant measure whose support is the julia set corresponding to the polynomial transformation it appears that up to a constant the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials which fulfill a three term recursion relation an algorithm is given to compute the coefficients of this recursion relation which can be interpreted as a linear extension of the iterative procedure

asymptotics of chebyshev polynomials i subsets of r we consider chebyshev polynomials t n z for infinite compact sets e that is the monic polynomials minimizing the sup norm t n e on e we resolve a 45 year old conjecture of widom that for finite gap subsets of r his conjectured asymptotics which we call szeg widom asymptotics holds we also prove the first upper bounds of the form t n e qc e n where c e is the logarithmic capacity of e for a class of e s with an infinite number of components explicitly for those e that obey a parreau widom condition

on stabilization and control for the critical klein gordon equation on a 3 d compact manifold abstract in this article we study the internal stabilization and control of the critical nonlinear klein gordon equation on 3 d compact manifolds under a geometric assumption slightly stronger than the classical geometric control condition we prove exponential decay for some solutions bounded in the energy space but small in a lower norm the proof combines profile decomposition and microlocal arguments this profile decomposition analogous to the one of bahouri and gerard 1999 2 on r 3 is performed by taking care of possible geometric effects it uses some results of s ibrahim 2004 21 on the behavior of concentrating waves on manifolds

orthogonal polynomials of equilibrium measures supported on cantor sets the equilibrium measure of a compact set is a fundamental object in logarithmic potential theory we compute numerically this measure and its orthogonal polynomials when the compact set is a cantor set defined by an iterated function system we first construct sequences of discrete measures via the solution of large systems of non linear equations that converge weakly to the equilibrium measure successively we compute their jacobi matrices via standard procedures enhanced for the scope numerical estimates of the convergence rate to the limit jacobi matrix are provided which show stability and efficiency of the whole procedure as a companion result we also compute jacobi matrices in two other cases equilibrium measures on finite sets of intervals and balanced measures of iterated function systems these algorithms can reach large polynomial orders therefore we study the asymptotic behavior of the orthogonal polynomials and by a natural extension of the concept of regular root asymptotics we derive an efficient scheme for the computation of complex green s functions and of related conformal mappings

non linear responses of suspended cables to primary resonance excitations we investigate the non linear forced responses of shallow suspended cables we consider the following cases 1 primary resonance of a single in plane mode and 2 primary resonance of a single out of plane mode in both cases we assume that the excited mode is not involved in an autoparametric resonance with any other mode we analyze the system by following two approaches in the first we discretize the equations of motion using the galerkin procedure and then apply the method of multiple scales to the resulting system of non linear ordinary differential equations to obtain approximate solutions discretization approach in the second we apply the method of multiple scales directly to the non linear integral partial differential equations of motion and associated boundary conditions to determine approximate solutions direct approach we then compare results obtained with both approaches and discuss the influence of the number of modes retained in the discretization procedure on the predicted solutions

a note on the semiglobal controllability of the semilinear wave equation we study the internal controllability of the semilinear wave equation v tt x t delta v x t f x v x t mathbbm 1 omega u x t for some nonlinearities f which can produce several nontrivial steady states one of the usual hypotheses to get semiglobal controllability is to assume that f x v v geq 0 in this case a stabilization term u gamma x v t makes any solution converging to zero the semiglobal controllability then follows from a theorem of local controllability and the time reversibility of the equation in this paper the nonlinearity f can be more general so that the solutions of the damped equation may converge to an equilibrium other than 0 to prove semiglobal controllability we study the controllability inside a compact attractor and show that it is possible to travel from one equilibrium point to another by using the heteroclinic orbits

oscillations and concentration effects in semilinear dispersive wave equations let un be a sequence of smooth solutions to a dispersive nonlinear wave equation formula inr1 3 with uniformly compactly supported cauchy data converging weakly to 0 inh1 r3 l2 r3 let vn be the sequence of solutions to the linear wave equation with the same cauchy data we show thatun vngoes strongly to 0 in the energy spacec 0 t h1 c1 0 t l2 if f is a subcritical nonlinearity in the critical casef u u5 we show that this property is equivalent tovn 0 inl 0 t l6 then we give sharp sufficient conditions on microlocal measures associated to the data the proof relies on a microlocal version of p l lions concentration compacity soit un une suite de solutions regulieres d une equation d ondes non lineaire dispersive formula dansr1 3 avec donnees de cauchy supportees dans un compact fixe et convergeant faiblement vers 0 dansh1 r3 l2 r3 soit vn la suite des solutions de l equation des ondes lineare avec les memes donnees de cauchy on montre queun vnconverge fortement vers 0 dans l espace d energiec 0 t h1 c1 0 t l2 sifest une non linearite sous critique dans le cas critiquef u u5 on montre que cette propriete equivaut a la convergence forte devnvers 0 dansl 0 t l6 on en donne alors des conditions suffisantes optimales en termes de mesures microlocales associees aux donnees la demonstration est basee sur une version microlocale de la concentration compacite de p l lions

a limit q 1 for the big q jacobi polynomials we study a new family of classical orthogonal polynomials here called big 1 jacobi polynomials which satisfy apart from a 3 term recur rence relation an eigenvalue problem with differential operators of dunkl type these polynomials can be obtained from the big q jacobi polynomials in the limit q 1 an explicit expression of these polynomials in terms of gauss hypergeometric functions is found the big 1 jacobi polynomials are orthogo nal on the union of two symmetric intervals of the real axis we show that the big 1 jacobi polynomials can be obtained from the bannai ito polynomials when the orthogonality support is extended to an infinite number of points we further indicate that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1 the novelty lies in the fact that l is a differential difference operator of special type namely l is a linear operator which is of first order in the derivative operator x and contains also the reflection operator r which acts as rf x f x roughly speaking one can say that l belongs to the class of dunkl operators 8 which contain both the operators x and r nevertheless the operator l differs from the standard dunkl operators in a fundamental way indeed l preserves the linear space of polynomials of any given maximal degree this basic property allows to construct a complete system of polynomials pn x n 0 1 2 as eigenfunctions of the operator l guided by the q 1 limit of the little q jacobi polynomials we derived in 20 an explicit expression of the polynomials pn x in terms of gauss hypergeometric functions we also found explicitly the recurrence coefficients and s that the polynomials pn x are orthogonal on the interval 1 1 with a weight function related to the weight function of the generalized jacobi polynomials 7 we also proved that they admit the dunkl classical property 5 and further demonstrated that the operator l together with the multiplication operator x form a special case of the askey wilson algebra aw 3 21 corresponding to the parameter q 1

profile decompositions for wave equations on hyperbolic space with applications the goal for this paper is twofold our first main objective is to develop bahouri gerard type profile decompositions for waves on hyperbolic space recently such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy with an eye towards further applications we develop this theory in a fairly general framework which includes the case of waves on hyperbolic space perturbed by a time independent potential our second objective is to use the profile decomposition to address a specific nonlinear problem namely the question of global well posedness and scattering for the defocusing energy critical semi linear wave equation on three dimensional hyperbolic space possibly perturbed by a repulsive time independent potential using the concentration compactness rigidity method introduced by kenig and merle we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t rightarrow pm infty this proof serves as a blueprint for the arguments in the work 45 by the authors where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

on some n 1 symmetric linear functionals abstract many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle particular aspects are treated for quadratic or cubic decompositions in this paper we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality this technique is based on the decomposition with respect to the cyclic group of order n the main theorem is applied to the orthogonality on n 1 symmetric curves some particular cases are singled out these results may be useful in studying hermite pade approximations vector continued fractions and dynamical systems

sharp lower bounds for the widom factors on the real line abstract we derive lower bounds for the l p norms of monic extremal polynomials with respect to compactly supported probability measures we obtain a sharp universal lower bound for all 0 p and all measures in the szeg class and an improved lower bound on l 2 norm for several classes of orthogonal polynomials including jacobi polynomials isospectral torus of a finite gap set and orthogonal polynomials with respect to the equilibrium measure of an arbitrary non polar compact subset of r

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

universality limits for generalized jacobi measures abstract in this paper universality limits are studied in connection with measures which exhibit power type singular behavior somewhere in their support we extend the results of lubinsky for jacobi measures supported on 1 1 to generalized jacobi measures supported on a compact subset of the real line where the singularity can be located in the interior or at an endpoint of the support the analysis is based upon the riemann hilbert method christoffel functions the polynomial inverse image method of totik and the normal family approach of lubinsky

inverse images of polynomial mappings and polynomials orthogonal on them let j be a polynomial with complex coefficients first we study the inverse images of the real and imaginary axes under a polynomial mapping j in detail then for an arbitrary polynomial and a sequence pn of orthogonal polynomials the orthogonality behaviour of the sequence of polynomials pn j n n is investigated in particular necessary and sufficient conditions are given such that pn j n n is a subsequence of polynomials orthogonal with respect to a positive measure supported on a compact subset of the real line

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

gauss tchebycheff quadrature formulas it is well known that the tchebycheff weight function 1 x 2 1 2 is the only weight function up to a linear transformation for which then point gauss quadrature formula has equal weights for alln in this paper we describe explicitly all weight functions which have the property that then k point gauss quadrature formula has equal weights for allk where n k n 1 n 2 is an arbitrary subsequence of furthermore results on the possibility of tchebycheff quadrature on several intervals are given

jordan algebras and orthogonal polynomials we illustrate how jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials the big 1 jacobi polynomials are eigenfunctions of a first order operator of dunkl type we consider an algebra that has this operator up to constants as one of its three generators and whose defining relations are given in terms of anticommutators it is a special case of the askey wilson algebra aw 3 we show how the structure and recurrence relations of the big 1 jacobi polynomials are obtained from the representations of this algebra we also present ladder operators for these polynomials and point out that the big 1 jacobi polynomials satisfy the hahn property with respect to a generalized dunkl operator

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

an upper bound for the norm of the chebyshev polynomial on two intervals abstract let e 1 1 1 1 be the union of two real intervals and consider the chebyshev polynomial of degree n on e that is that monic polynomial which is minimal with respect to the supremum norm on e for its norm called the n th chebyshev number of e an upper bound in terms of elementary functions of and is given the proof is based on results of n i achieser in the 1930s in which the norm is estimated with the help of zolotarev s transformation using jacobi s elliptic and theta functions

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

null controllability of the burgers system with distributed controls this paper is devoted to present some positive and negative controllability results for the viscous burgers equation more precisely in the context of null controllability with distributed controls we present sharp estimates of the minimal time of controllability t r of initial data of l 2 norm less or equal to r in particular we see that global null controllability does not hold in general unless the control is exerted

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

the mechanics of highly extensible cables abstract the mechanics of highly extensible cables are studied numerically the governing equations for the cable motion are reformulated using euler parameters and we employ a nonlinear stress strain relation also bending stiffness terms are included to ensure a well posed problem when tension becomes very low thus the singularity associated with euler angle formulation is removed and the model allows for shock formation while it can accommodate zero or negative tension along the cable span implicit time integration and non uniform grid along the cable are adopted for the numerical solution of the governing equations the model is employed to investigate 1 the dynamical behaviour of the breaking and post breaking of an initially taut cable and 2 the dynamic response of a tethered near surface buoy subject to wave excitation for a breaking cable we find that the speed of snapback which can have potentially catastrophic effects is proportional to the initial strain level but the principal parameter controlling the cable behaviour is the time it takes for the cable to fracture in the case of a tethered buoy in waves we find that beyond a threshold wave amplitude the system begins to exhibit first zero tension then followed by snapping response while the buoy performs chaotic motion