singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

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

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

schrodinger operators with sparse potentials asymptotics of the fourier transform of the spectral measure we study the pointwise behavior of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with sparse potentials we find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model we also present an improved version of known results on the spectrum of such operators

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

on the inverse scattering problem for jacobi matrices with the spectrum on an interval a finite system of intervals or a cantor set of positive length when solving the inverse scattering problem for a discrete sturm liouville operator with a rapidly decreasing potential one gets reflection coefficients s and invertible operators where is the hankel operator related to the symbol s the marchenko faddeev theorem 8 in the continuous case for the discrete case see 4 6 guarantees the uniqueness of the solution of the inverse scattering problem in this article we ask the following natural question can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators are invertible can one claim that uniqueness implies invertibility or vise versa moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials so we merge here two mostly developed cases of the inverse problem for sturm liouville operators the inverse problem with almost periodic potential and the inverse problem with the fast decreasing potential

approximate max min resource sharing for structured concave optimization we present a lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an varepsilon approximate solution of the general max min resource sharing problem with m nonnegative concave constraints on a convex set b we show that this algorithm runs in o m varepsilon 2 rm ln m iterations a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy varepsilon in 0 1 in the block angular case b is the product of k convex sets blocks and each constraint function is block separable for such models an iteration of our method requires a theta varepsilon approximate solution of k independent block maximization problems which can be computed in parallel

coordination complexity of parallel price directive decomposition the general block angular convex resource sharing problem in k blocks and m nonnegative block separable coupling constraints is considered applications of this model are in combinatorial optimization network flows scheduling communication networks engineering design and finance this paper studies the coordination complexity of approximate price directive decomposition pdd for this problem i e the number of iterations required to solve the problem to a fixed relative accuracy as a function of k and m first a simple pdd method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in m in the class of all pdd methods that work with the original unrestricted blocks it is then shown that logarithmic and exponential potentials generate a polylogarithmically optimal algorithm for a wider class of pdd methods which can restrict the blocks by the coupling constraints as an application the fastest currently known deterministic approximation algorithm for minimum cost multicommodity flows is obtained

natural boundaries and spectral theory we present and exploit an analogy between lack of absolutely continuous spectrum for schrodinger operators and nat ural boundaries for power series among our new results are gen eralizations of hecke s example and natural boundary examples for random power series where independence is not assumed

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

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

fast approximation schemes for convex programs with many blocks and coupling constraints this paper presents block coordinate descent algorithms for the approximate solution of large structured convex programming problems the constraints of such problems consist of k disjoint convex compact sets b k called blocks and m nonnegative valued convex block separable inequalities called coupling or resource constraints the algorithms are based on an exponential potential function reduction technique it is shown that feasibility as well as min mix resource sharing problems for such constraints can be solved to a relative accuracy varepsilon in o k ln m varepsilon 2 ln k iterations each of which solves k block problems to a comparable accuracy either sequentially or in parallel the same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration an extension to objective and constraint functions of arbitrary sign is also presented the above results yield fast approximatio

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

pure point spectrum under 1 parameter perturbations and instability of anderson localization we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic fora we study the set of all eigenvalues of the operatora t a tp t that belong to its essential spectrum which does not depend on the parametert we prove that this set is empty for a dense set of values oft then we apply this result or its idea to questions of anderson localization for 1 dimensional schrodinger operators discrete and continuous

complex support vector machines for regression and quaternary classification the paper presents a new framework for complex support vector regression svr as well as support vector machines svm for quaternary classification the method exploits the notion of widely linear estimation to model the input out relation for complex valued data and considers two cases 1 the complex data are split into their real and imaginary parts and a typical real kernel is employed to map the complex data to a complexified feature space and 2 a pure complex kernel is used to directly map the data to the induced complex feature space the recently developed wirtinger s calculus on complex reproducing kernel hilbert spaces is employed to compute the lagrangian and derive the dual optimization problem as one of our major results we prove that any complex svm svr task is equivalent with solving two real svm svr tasks exploiting a specific real kernel which is generated by the chosen complex kernel in particular the case of pure complex kernels leads to the generation of new kernels which have not been considered before in the classification case the proposed framework inherently splits the complex space into four parts this leads naturally to solving the four class task quaternary classification instead of the typical two classes of the real svm in turn this rationale can be used in a multiclass problem as a split class scenario based on four classes as opposed to the one versus all method this can lead to significant computational savings experiments demonstrate the effectiveness of the proposed framework for regression and classification tasks that involve complex data

faster and simpler approximation algorithms for mixed packing and covering problems we propose an algorithm for approximately solving the mixed packing and covering problem given a convex compact set 0 b r n either compute x b such that f x 1 e a and g x 1 e b or decide that x b f x a g x b 0 here f g b r m are vectors whose components are m non negative convex and concave functions respectively and a b r m are constant positive vectors our algorithm requires an efficient feasibility oracle or block solver which given vectors c d r m and a r computes x b such that c tf x d tg x a or correctly decides that no such x b exists our algorithm which is based on the lagrangian or price directive decomposition method generalizes the result from k jansen approximation algorithm for the mixed fractional packing and covering problem in proceedings of 3rd ifip conference on theoretical computer science ifip tcs 2004 kluwer 2004 pp 223 236 siam journal on optimization 17 2006 331 352 and needs only o m lnm e 2ln e 1 iterations or calls to the feasibility oracle furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

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

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

approximation algorithm for the mixed fractional packing and covering problem we propose an approximation algorithm based on the lagrangian or price directive decomposition method to compute an epsilon approximate solution of the mixed fractional packing and covering problem find x in b such that f x le 1 epsilon a g x ge 1 epsilon b where f x g x are vectors with m nonnegative convex and concave functions a and b are m dimensional nonnegative vectors and b is a convex set that can be queried by an optimization oracle we propose an algorithm that needs only o m epsilon 2 ln m epsilon 1 iterations or calls to the oracle the main contribution is that the algorithm solves the general mixed fractional packing and covering problem in contrast to pure fractional packing and covering problems and to the special mixed packing and covering problem with b rz n and runs in time independent of the so called width of the problem

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

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

sparse potentials with fractional hausdorff dimension abstract we construct non random bounded discrete half line schrodinger operators which have purely singular continuous spectral measures with fractional hausdorff dimension in some interval of energies to do this we use suitable sparse potentials our results also apply to whole line operators as well as to certain random operators in the latter case we prove and compute an exact dimension of the spectral measures

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

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

spectral properties of a class of reflectionless schrodinger operators abstract we prove that one dimensional reflectionless schrodinger operators with spectrum a homogeneous set in the sense of carleson belonging to the class introduced by sodin and yuditskii have purely absolutely continuous spectra this class includes all earlier examples of reflectionless almost periodic schrodinger operators in addition we construct examples of reflectionless schrodinger operators with more general types of spectra given by the complement of a denjoy widom type domain in c which exhibit a singular component