the set of smooth quasi periodic schrodinger cocycles with positive lyapunov exponent is not open one knows that the set of quasi periodic schrodinger cocycles with positive lyapunov exponent is open and dense in analytic topology in this paper we construct cocycles with positive lyapunov exponent which can be arbitrarily approximated by ones with zero lyapunov exponent in the space of mathcal c l 1 le l le infty smooth quasi periodic cocycles which shows that the set of quasi periodic schrodinger cocycles with positive lyapunov exponent is not open in smooth topology

strange non chaotic attractors in quasiperiodically forced circle maps the occurrence of strange non chaotic attractors sna in quasiperiodically forced systems has attracted considerable interest over the last two decades in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos their existence was first described by millionscikov and later by vinograd and also herman for quasiperiodic rm sl 2 mathbb r cocycles and by grebogi et al and later keller for so called pinched skew products however except for these two particular classes there are still hardly any rigorous results on the topic despite a large number of numerical studies confirming the widespread existence of sna in quasiperiodically forced systems here we prove the existence of sna in quasiperiodically forced circle maps under rather general conditions which can be stated in terms of mathcal c 1 estimates as a consequence we obtain the existence of sna for parameter sets of positive measure in suitable parameter families these sna carry the unique physical measure of the system which determines the behaviour of lebesgue almost all initial conditions finally we show that the dynamics are minimal in the considered situations the results apply in particular to a forced version of the arnold circle map for this example we also describe how the first arnold tongue collapses and looses its regularity due to the presence of strange non chaotic attractors and a related unbounded mean motion property

positive hausdorff dimensional spectrum for the critical almost mathieu operator we show that there exists a dense set of frequencies with positive hausdorff dimension for which the hausdorff dimension of the spectrum of the critical almost mathieu operator is positive

almost reducibility and non perturbative reducibility of quasi periodic linear systems in this paper we prove that a quasi periodic linear differential equation in sl 2 with two frequencies 1 is almost reducible provided that the coefficients are analytic and close to a constant in the case that is diophantine we get the non perturbative reducibility we also obtain the reducibility and the rotations reducibility for an arbitrary irrational under some assumption on the rotation number and give some applications for schrodinger operators our proof is a generalized kam type iteration adapted to all irrational

the regularizing properties of anisotropic radial basis functions in the present work we consider the problem of interpolating scattered data using radial basis functions rbf in general it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution the work focuses on a metric regularization approach based on a new class of rbf called anisotropic rbf the work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric supporting it by numerical examples

brief paper strong γk γclh stabilization with a new slack variable approach this paper deals with the strong c email protected c lh stabilization problem a new bounded real lemma is provided by adding two slack variables which provides extra freedom to include more control constraints in handling robust control problems with the extended bounded real lemma a new sufficient condition is proposed for nth order strong c email protected c lh stabilization it is also proved that the design of an lxnth order l 2 strong c email protected c lh controller can be transformed into that of an nth order controller for an associated plant which tackles the possible order problem furthermore the h stable stabilization and strong stabilization problems can also be solved by relaxing certain constraints accordingly despite the bmi constraints the path following and alternative iteration methods are adopted to propose procedures for both nth and lxnth order controller designs the effectiveness is indicated by several numerical examples

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

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

cantor and band spectra for periodic quantum graphs with magnetic fields we provide an exhaustive spectral analysis of the two dimensional periodic square graph lattice with a magnetic field we show that the spectrum consists of the dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant lyapunov function of a suitable kronig penney hamiltonian in particular between any two dirichlet eigenvalues the spectrum is a cantor set for an irrational flux and is absolutely continuous and has a band structure for a rational flux the dirichlet eigenvalues can be isolated or embedded subject to the choice of parameters conditions for both possibilities are given we show that generically there are infinitely many gaps in the spectrum and the bethe sommerfeld conjecture fails in this case

positive lyapunov exponents for schrodinger operators with quasi periodic potentials we present a new simple way to estimate the rate of exponential growth lyapunov exponent of solutions of the finite difference schrodinger equation h e psi n mathop limits def psi n 1 psi n 1 lambda f alpha n theta psi n heref is a non constant real analytic function of period 1 and is irrational for large we prove that the lyapunov exponent is positive for every energye in the spectrum ofh and a e in particular the absolutely continuous spectrum ofh is empty in the continuum we study the quasi periodic operator onl2 r h frac d 2 dx 2 k 2 cos x cos alpha x theta for largek and show that for wide intervals of low energies the lyapunov exponent is positive the main idea which originated from m herman s subharmonic argument 11 is to deform the phase to the complex plane this enables us to avoid small denominator problems by moving them off the axis making estimates much easier to perform we recover the information for real using an elementary extension of jensen s formula subharmonicity

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

a lower bound on the lyapunov exponent for the generalized harper s model we obtain a lower bound for the lyapunov exponent of a family of discrete schrodinger operators hu n u n 1 u n 1 2a 1 cos 2 pi theta n alpha u n 2a 2 cos 4 pi theta n alpha u n that incorporates both a 1 and a 2 thus going beyond the herman s bound

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

some new results on strong gamma_k gamma_ cl h_ infty stabilization the strong gammak gammacl hinfin stabilization problem is addressed in this paper based on the bounded real lemma an improved sufficient condition on strong hinfin stabilization as well as its dual form is derived which extends the are constraint in the former results into an inequality constraint two path following algorithms are proposed to solve the associated optimization problems with bmi constraints which lead to full order controllers a numerical example is presented to demonstrate the effectiveness of the result

approximate interpolation with applications to selecting smoothing parameters in this paper we study the global behavior of a function that is known to be small at a given discrete data set such a function might be interpreted as the error function between an unknown function and a given approximant we will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region we will apply these results to spline smoothing and show that a specific a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant this has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels

on the absolutely continuous spectrum of one dimensional quasi periodic schrodinger operators in the adiabatic limit cet article est consacre a l etude du spectre de certaines familles d equations de schrodinger quasi periodiques sur l axe reel lorsque les varietes iso energetiques adiabatiques sont etendues dans la direction des positions nous demontrons que dans un intervalle d energie ou ceci est le cas le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres generalisees correspondantes sont des fonctions de bloch floquet

capacity of reproducing kernel spaces in learning theory the capacity of reproducing kernel hilbert spaces rkhs plays an essential role in the analysis of learning theory covering numbers and packing numbers of balls of these reproducing kernel spaces are important measurements of this capacity we first present lower bound estimates for the packing numbers by means of nodal functions then we show that if a mercer kernel is c sup s for some s 0 being not an even integer the rkhs associated with this kernel can be embedded into c sup s 2 this gives upper bound estimates for the covering number concerning sobolev smooth kernels examples and applications to v sub spl gamma dimension and tikhonov 1977 regularization are presented to illustrate the upper and lower bound estimates

brief on simultaneous h control and strong h stabilization in this note the simultaneous h control problem is addressed it is first shown that the existence of a single controller which simultaneously stabilizes two systems with a given h norm level is equivalent to that of a strong h stabilization controller of an associated system moreover the result is extended to the case involving r 2 systems then a sufficient condition for the strong h stabilization is proposed based on the bounded real lemma finally a simple numerical example is presented to show the effectiveness of the result

the lyapunov exponents of generic volume preserving and symplectic maps we show that the integrated lyapunov exponents of c1 volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding oseledets splitting is trivial all lyapunov exponents are equal to zero or else dominated uniform hyperbolicity in the projective bundle almost everywhere we deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold almost every orbit either is projectively hyperbolic or has all lyapunov exponents equal to zero similarly for a residual subset of all c1 symplectic diffeomorphisms on any compact manifold either the diffeomorphism is anosov or almost every point has zero as a lyapunov exponent with multiplicity at least 2 finally given any set s gl d satisfying an accessibility condition for a residual subset of all continuous s valued cocycles over any measure preserving homeomorphism of a compact space the oseledets splitting is either dominated or trivial the condition on s is satisfied for most common matrix groups and also for matrices that arise from discrete schr nodinger operators

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

what are lyapunov exponents and why are they interesting taking as inspiration the fields medal work of artur avila i d like to introduce you to lyapunov exponents my plan is to show how lyapunov exponents play a key role in three areas in which avila s results lie smooth ergodic theory billiards and translation surfaces and the spectral theory of 1 dimensional schrodinger operators but first what are lyapunov exponents let s begin by viewing them in one of their natural habitats the barycentric subdivision of a triangle is a collection of 6 smaller triangles obtained by joining the midpoints of the sides to opposite vertices here s what happens when you start with an equilateral triangle and repeatedly barycentrically subdivide

adaptive interpolation based on polya frequency functions translation invariant interpolation of frequency domain functions by means of fourier transforms of polya frequency functions which belong to the bochner class of interpolation kernels is presented for the implementation we use an adaptive interpolation process which is a variant of the recently introduced adaptive residual subsampling method moreover it is shown that adaptive residual subsampling with polya interpolation kernels is an excellent preprocessing interface when used in conjunction with the well known vector fitting vf rational modeling algorithm

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

second phase transition line we study the phase transion line of the almost mathieu operator that separates arithmetic regions corresponding to singular continuous and a e pure point regimes and prove that both purely singular continuous and a e pure point spectrum occur for dense sets of frequencies

a numerical scheme for solving a class of logarithmic integral equations arisen from two dimensional helmholtz equations using local thin plate splines abstract this paper presents a numerical method for solving logarithmic fredholm integral equations which occur as a reformulation of two dimensional helmholtz equations over the unit circle with the robin boundary conditions the method approximates the solution utilizing the discrete collocation method based on the locally supported thin plate splines as a type of free shape parameter radial basis functions the local thin plate splines establish an efficient and stable technique to estimate an unknown function by a small set of nodes instead of all points over the solution domain to compute logarithm like singular integrals appeared in the method we use a particular nonuniform gauss legendre quadrature rule since the scheme does not require any mesh generations on the domain it can be identified as a meshless method the error estimate of the proposed method is presented numerical results are included to show the validity and efficiency of the new technique these results also confirm that the proposed method uses much less computer memory in comparison with the method established on the globally supported thin plate splines moreover it seems that the algorithm of the presented approach is attractive and easy to implement on computers

spectral characteristics of the unitary critical almost mathieu operator we discuss spectral characteristics of a one dimensional quantum walk whose coins are distributed quasi periodically the unitary update rule of this quantum walk shares many spectral characteristics with the critical almost mathieu operator however it possesses a feature not present in the almost mathieu operator namely singularity of the associated cocycles this feature is however present in the so called extended harper s model we show that this operator has empty absolutely continuous spectrum and that the lyapunov exponent vanishes on the spectrum hence this model exhibits cantor spectrum of zero lebesgue measure for all irrational frequencies and arbitrary phase which in physics is known as hofstadter s butterfly in fact we will show something stronger namely that all spectral parameters in the spectrum are of critical type in the language of avila s global theory of analytic quasiperiodic cocycles we further prove that it has empty point spectrum for each irrational frequency and away from a frequency dependent set of phases having lebesgue measure zero the key ingredients in our proofs are an adaptation of avila s global theory to the present setting self duality via the fourier transform and a johnson type theorem for singular dynamically defined cmv matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting

examples of discontinuity of lyapunov exponent in smooth quasiperiodic cocycles we study the regularity of the lyapunov exponent for quasiperiodic cocycles t a where t is an irrational rotation x x 2 on s1 and a cl s1 sl 2 r 0 l for any fixed l 0 1 2 and any fixed of bounded type we construct dl cl s1 sl 2 r such that the lyapunov exponent is not continuous at dl in cl topology we also construct such examples in a smaller schrodinger class

on the hausdorff dimension of the spectrum of the thue morse hamiltonian we show that the hausdorff dimension of the spectrum of the thue morse hamiltonian has a common positive lower bound for all coupling

local rigidity of reducibility of analytic quasi periodic cocycles on gl n c abstract in this paper we consider the reducibility problem of an analytic d dimensional quasi periodic cocycle on g l n c we prove that for any diophantine vector and diagonal matrix c satisfying some arithmetic conditions if a cocycle a can be conjugated to the constant cocycle c by some 0 b with b l 2 t d g l n c and a is sufficiently close to some diagonal matrix it is analytically reducible moreover b is actually analytic if it is continuous

design of stable controllers attaining low h sup infinity weighted sensitivity strong stabilization and sensitivity reduction for a linear time invariant single input single output system are considered stable controllers are practically desirable in the presence of possible actuator or sensor failures it is verified precisely that the optimal controller of h sup infinity weighted sensitivity minimization is almost always unstable when the plant has more than two zeros in the right half plane an algorithm for attaining the low sensitivity property for stable controllers is presented it is based on nevanlinna pick interpolation

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

absolutely continuous spectrum of multifrequency quasiperiodic schrodinger operator abstract in this paper we prove that for any d frequency analytic quasiperiodic schrodinger operator if the frequency is weak liouvillean and the potential is small enough then the corresponding operator has absolutely continuous spectrum moreover in the case d 2 we even establish the existence of ac spectrum under small potential and some super liouvillean frequency and this result is optimal due to a recent counterexample of avila and jitomirskaya 6

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

analytic quasi perodic cocycles with singularities and the lyapunov exponent of extended harper s model we show how to extend and with what limitations avila s global theory of analytic sl 2 c cocycles to families of cocycles with singularities this allows us to develop a strategy to determine the lyapunov exponent for the extended harper s model for all values of parameters and all irrational frequencies in particular this includes the self dual regime for which even heuristic results did not previously exist in physics literature the extension of avila s global theory is also shown to imply continuous behavior of the le on the space of analytic m 2 mathbb c cocycles this includes rational approximation of the frequency which so far has not been available

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

arithmetic spectral transitions for the maryland model we give a precise description of spectra of the maryland model h u n un 1 un 1 tan n un for all values of parameters we introduce an arithmetically defined index and show that for sc h e e and pp h e e since ac h this gives a complete description of the spectral decomposition for all values of parameters and making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion the set of eigenvalues can be explicitly identified for all parameters using the quantization condition we also establish for the first time for this or any other model a quantization condition for singular continuous spectrum an arithmetically defined measure zero set that supports singular continuous measures for all parameters 2017 wiley periodicals inc

denjoy constructions for fibered homeomorphisms of the torus we construct different types of quasiperiodically forced circle homeomorphisms with transitive but non minimal dynamics concerning the recent poincare like classification by jager and stark for this class of maps we demonstrate that transitive but non minimal behaviour can occur in each of the different cases this closes one of the last gaps in the topological classification actually we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets for example we show that in some of the examples we construct the unique minimal set is a cantor set and its intersection with each vertical fibre is uncountable and nowhere dense but may contain isolated points we also prove that minimal sets of the latter kind cannot occur when the dynamics are given by the projective action of a quasiperiodic sl 2 r cocycle more precisely we show that for a quasiperiodic sl 2 cocycle any minimal proper subset of the torus either is a union of finitely many continuous curves or contains at most two points on generic fibres

singular continuous spectrum for a class of almost periodic jacobi matrices we consider operators parametrized by a o on 2 with matrix ojy i o i fl o y with an cos 2ttcw 6 if ce is a liouville number and 2 we prove that for a e 0 the operator s spectral measures are all singular continuous we consider the operator on l2 z depending upon three parameters x a 0 1 x a 0 u n u n 1 u n 1 x cos 2iran 6 u n in this note we will sketch the proof of the following result whose detailed proof will appear elsewhere 3 theorem 1 fix a an irrational number obeying

the avalanche principle and negative curvature we use the geometric structure of the hyperbolic upper half plane to provide a new proof of the avalanche principle introduced by m goldstein and w schlag in the context of mathrm sl 2 mathbb r matrices this approach allows to interpret and extend this result to arbitrary text cat 1 metric spaces through the proof we deduce a polygonal schur theorem for these spaces

singular spectrum of lebesgue measure zero for one dimensional quasicrystals the spectrum of one dimensional discrete schrodinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the lyapunov exponent if and only if the lyapunov exponent exists uniformly this is used to obtain the cantor spectrum of zero lebesgue measure for all aperiodic subshifts with uniform positive weights this covers in particular all aperiodic subshifts arising from primitive substitutions including new examples such as e g the rudin shapiro substitution our investigation is not based on trace maps instead it relies on an oseledec type theorem due to a furman and a uniform ergodic theorem due to the author

sharp phase transitions for the almost mathieu operator it is known that the spectral type of the almost mathieu operator amo depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency we study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place which solves jitomirskaya s conjecture together with a previous work by avila this gives the sharp description of phase transitions for the amo for the a e phase

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

global density of reducible quasi periodic cocycles on bf t 1 times roman su 2 we prove that given a in a set of total haar measure in t 1 r z the set of a c t 1 su 2 for which the skew product system a t 1 su 2 t 1 su 2 a y a y is reducible that is a b a 0 b 1 for some a 0 su 2 b c t 1 su 2 is dense for the c topology

on the synthesis of stable spl hscr sup spl infin controllers a sufficient condition is derived for the existence of a stable suboptimal spl hscr sup spl infin controller it is shown that if a certain algebraic riccati equation determined from a parameterization of all suboptimal spl hscr sup spl infin controllers has a positive definite stabilizing solution then a stable spl hscr sup spl infin controller of order 2n can be constructed where n is the order of the generalized plant

multidimensional continued fractions dynamical renormalization and kam theory the disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors following ideas of dani lagarias and kleinbock margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space sl d mathbb z backslash sl d mathbb r the space of lattices of covolume one that indeed yields best possible approximations to any irrational vector the algorithm is ideally suited for a number of dynamical applications that involve small divisor problems as an example we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

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

localization of multivariate interpolation and smoothing methods multivariate smoothing or interpolation methods for scattered data are frequently localized by computing the value of the required function at a point from data in some small neighborhood of the point we describe the piecewise nature of the resulting function and show how a smoothing of the usually discontinuous function can be carried out by a convolution which preserves interpolation

on condition numbers associated with radial function interpolation abstract estimates on spectral condition numbers for scattered data interpolation matrices associated with a certain class of conditionally positive definite radial functions of order 0 including the gaussians are provided in addition lp condition number estimates with 1 p are discussed the estimates on the spectral condition numbers lead to necessary and sufficient conditions for such condition numbers to be independent of the number of data sites

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

stable controller synthesis for linear time invariant systems this paper is concerned with the problem of suboptimal stable mixed h 2 h x control for linear time invariant systems the designed controller with the same order as that of the considered system is required to satisfy a prescribed h x performance bound or a prescribed degree of stability by reducing the stable controller synthesis problem to a multiobjective state feedback control problem for two different state models sufficient conditions for the solvability of the considered problem are given in terms of solutions to algebraic riccati equations and matrix inequalities an lmi based iterative algorithm is developed to solve the stable controller synthesis problem the proposed algorithm is shown to be convergent examples are given to illustrate the proposed methods

linearization of quasiperiodically forced circle flows beyond brjuno condition we prove that an analytic quasiperiodically forced circle flow with a not super liouvillean base frequency and which is close enough to some constant rotation is c rotations reducible provided its fibered rotation number is diophantine with respect to the base frequency as a corollary we obtain that among such systems the linearizable ones and those displaying mode locking are locally dense for the c topology

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

phase transition and semi global reducibility for 1d continuous schodinger operators with large analytic quasi periodic potentials of two frequencies one knows that the spectral measure is singular at the bottom of the spectrum and purely absolutely continuous in the upper part of the spectrum so there is a phase transition when energy increases in this paper we obtain the exact power law for the phase transition in energy by the semi global reducibility theory of analytic quasi periodic linear systems

full measure reducibility and localization for quasiperiodic jacobi operators a topological criterion abstract we establish a topological criterion for connection between reducibility to constant rotations and dual localization for the general family of analytic quasiperiodic jacobi operators as a corollary we obtain the sharp arithmetic phase transition for the extended harper s model in the positive lyapunov exponent region

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

robust strong stabilization via modified popov controller synthesis in this paper the authors merge the parameter dependent lyapunov function framework used to construct robust popov controllers with the a priori and a posteriori approaches for obtaining stable compensators specifically the authors derive constructive sufficient conditions that yield robust lyapunov and asymptotically stable stabilizing popov dynamic compensators

synthesis of stable h controller via the chain scattering framework abstract a sufficient condition for the existence of suboptimal stable stabilizing h controllers is given by exploiting the free parameter in the parameterization of stabilizing controllers and using the chain scattering framework we reformulate the h strong stabilization problem as an equivalent h optimization problem which can be solved via only one algebraic riccati equation a parameterization of all suboptimal stable stabilizing h controllers is also given

integrated multiquadric radial basis function approximation methods promising numerical results using once and twice integrated radial basis functions have been recently presented in this work we investigate the integrated radial basis function irbf concept in greater detail connect to the existing rbf theory and make conjectures about the properties of irbf approximation methods the irbf methods are used to solve pdes

continuity of the measure of the spectrum for quasiperiodic schrodinger operators with rough potentials we study discrete quasiperiodic schrodinger operators on ell 2 mathbb z with potentials defined by holder functions we prove a general statement that for 1 2 and under the condition of positive lyapunov exponents measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants an important ingredient in our analysis is a general result on uniformity of convergence from above in the subadditive ergodic theorem for strictly ergodic cocycles

spectral theory of extended harper s model and a question by erdős and szekeres the extended harper s model proposed by d j thouless in 1983 generalizes the famous almost mathieu operator allowing for a wider range of lattice geometries parametrized by three coupling parameters by permitting 2d electrons to hop to both nearest and next nearest neighboring nnn lattice sites while still exhibiting its characteristic symmetry aubry andre duality previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space one of which is characterized by the positivity of the lyapunov exponent in this paper we complete the picture with a description of the spectral measures over the entire remaining self dual region for all irrational values of the frequency parameter the magnetic flux in the model most notably we prove that in the entire interior of this regime the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the nnn interaction becomes symmetric in physics literature extensive numerical analysis had indicated such spectral collapse however so far not even a heuristic argument for this phenomenon could be provided on the other hand in the remaining part of the self dual region the spectral measures are singular continuous irrespective of such symmetry the analysis requires some rather delicate number theoretic estimates which ultimately depend on the solution of a problem posed by erd s and szekeres on the product prod k 1 n 1 z a k publ de l institut mathematique paris 1950

anderson transitions for a family of almost periodic schrodinger equations in the adiabatic case this work is devoted to the study of a family of almost periodic one dimensional schrodinger equations using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit we prove the existence of an asymptotically sharp anderson transition in the low energy region more explicitly we prove the existence of energy intervals containing only singular spectrum and of other energy intervals containing absolutely continuous spectrum the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions the analysis may be viewed as utilizing a complex wkb method for adiabatic perturbations of periodic schrodinger equations the transition energies are interpreted in terms of phase space tunneling

absolute continuity of the integrated density of states for the almost mathieu operator with non critical coupling we show that the integrated density of states of the almost mathieu operator is absolutely continuous if and only if the coupling is non critical we deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase settling the measure theoretical case of problem 6 of barry simon s list of schrodinger operator problems for the twenty first century

on the persistence of invariant curves for fibered holomorphic transformations we consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations we define a fibered rotation number associated to an invariant curve we show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a brjuno type arithmetical condition nevertheless an extra complex parameter is added to the problem and the persistence becomes a one complex codimension property

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

the algebraic geometry of harper operators following an approach developed by gieseker kn rrer and trubowitz for discretized schr dinger operators we study the spectral theory of harper operators in dimensions 2 and 1 as a discretized model of magnetic laplacians from the point of view of algebraic geometry we describe the geometry of an associated family of bloch varieties and compute their density of states finally we also compute some spectral functions based on the density of states we discuss the difference between the cases with rational or irrational parameters for the two dimensional harper operator the compactification of the bloch variety is an ordinary variety in the rational case and an ind pro variety in the irrational case this gives rise at the algebro geometric level of bloch varieties to a phenomenon similar to the hofstadter butterfly in the spectral theory in dimension 2 the density of states can be expressed in terms of period integrals over fermi curves where the resulting elliptic integrals are independent of the parameters in dimension 1 for the almost mathieu operator with a similar argument we find the usual dependence of the spectral density on the parameter which gives rise to the well known hofstadter butterfly picture

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

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

on topological genericity of the mode locking phenomenon we study circle homeomorphisms extensions over a strictly ergodic homeomorphism under a very mild restriction we show that the fibered rotation number is locally constant on an open and dense subset of all circle homeomorphisms extensions homotopic to the trivial extension in the complement of this set we find a dense subset in which every map is conjugate to a direct product our result provides a generalisation of avila bochi damanik s result on mathrm sl 2 mathbb r cocycles and jager wang zhou s result on quasi periodically forced maps to a broader setting

embedding of analytic quasi periodic cocycles into analytic quasi periodic linear systems and its applications in this paper we prove that any analytic quasi periodic cocycle close to constant is the poincare map of an analytic quasi periodic linear system close to constant which bridges both methods and results in quasi periodic linear systems and cocycles we also show that the almost reducibility of an analytic quasi periodic linear system is equivalent to the almost reducibility of its corresponding poincare cocycle by the local embedding theorem and the equivalence we transfer the recent local almost reducibility results of quasi periodic linear systems hou and you in invent math 190 209 260 2012 to quasi periodic cocycles and the global reducibility results of quasi periodic cocycles avila in almost reducibility and absolute continuity 2010 avila et al in geom funct anal 21 1001 1019 2011 to quasi periodic linear systems finally we give a positive answer to a question of avila et al geom funct anal 21 1001 1019 2011 and use it to study point spectrum of long range quasi periodic operator with liouvillean frequency the embedding also holds for some nonlinear systems