scaling and eigenstates for a class of one dimensional quasiperiodic lattices numerical and analytical calculations of the electronic properties of a one dimensional quasiperiodic lattice are presented the lattice consists of two types of bonds a and b whose distribution follows the recursion formula sl 1 sml 1snl where m n 1 2 3 and sl is a building block sequence for l or 2 an exact evaluation of the wavefunctions yields the scaling for several pairs of values of m and n numerical calculations of the wavefunction and the resistance reveal a rich self similar structure as well as localisation for one of the quasiperiodic sequences studied

on sampling controlled stochastic approximation the authors examine a novel class of stochastic approximation procedures which are based on carefully controlling the number of observations or measurements taken before each computational iteration this method referred to as sampling controlled stochastic approximation has advantages over standard stochastic approximation such as requiring less computation and the ability to handle bias in estimation the authors address the growth rate required of the number of samples and prove a general convergence theorem for the proposed stochastic approximation method in addition they present applications to optimize and also derive a sampling controlled version of the classic robbins munro algorithm

electronic properties of the tight binding fibonacci hamiltonian the analysis of the electronic properties of the tight binding fibonacci hamiltonian is carried out using dynamical systems techniques two classes of fibonacci sequences are considered corresponding to the cases when there are two types of building blocks and also when there are three types of building blocks the recursion relations for the traces of the transfer matrices are determined and studied for various extensions of the fibonacci case with the golden mean some differences are obtained between the various types of second order fibonacci sequences and are most likely due to long range order applications of this work to the transmission of light through a multilayered medium and the electrical resistance of a one dimensional quasicrystal as determined by the landauer formula are presented

symbolic dynamics for the renormalization map of a quasiperiodic schrodinger equation a rigorous analysis is given of the dynamics of the renormalization map associated to a discrete schrodinger operatorh onl 2 defined byh n n 1 n 1 vf n n wherev is a real parameter f is a certain discontinuous period 1 function and sigma left 1 sqrt 5 right mathord left vphantom left 1 sqrt 5 right 2 right kern nulldelimiterspace 2 is the golden mean the renormalization map forh is a diffeomorphism t of 3 preserving a cubic surfaces v forv 8 we prove that the non wandering set of the restriction oft tos v is a hyperbolic set on whicht is conjugate to a subshift on six symbols it follows from results in dynamical systems theory that the optimally approximating periodic operators toh have spectra which obey a global scaling law we also define a set which we call the pseudospectrum of the operatorh we prove it to be a cantor set of measure zero and obtain bounds on its hausdorff dimension it is an open question whether the pseudospectrum coincides with the spectrum ofh

sensitivity analysis from sample paths using likelihoods we modify the likelihood based method for obtaining derivatives with respect to the rate of a poisson process to that it is not necessary to know the exact value of that rate this type of modification is necessary if the method is to be used on a sample path from a real system the method is also applicable to simulation studies of certain real time control policies and may be useful in trace driven simulations the modification to the likelihood estimator is simply to use the value of the poisson rate estimated during the sample interval for regenerative systems this produces a strongly consistent asymptotically normal and asymptotically unbiased estimate of the derivative the strong law and central limit theorem are generalized to the case of estimating a derivative with respect to an unknown parameter from the exponential class of probability density functions numerical results for the m m 1 queue illustrate little difference between the estimates for the derivative of the expected delay with respect to arrival rate obtained when the arrival rate is known and unknown however both estimates are highly biased for small sample sizes this bias can be reduced by jackknifing

importance sampling for stochastic simulations importance sampling is one of the classical variance reduction techniques for increasing the efficiency of monte carlo algorithms for estimating integrals the basic idea is to replace the original random mechanism in the simulation by a new one and at the same time modify the function being integrated in this paper the idea is extended to problems arising in the simulation of stochastic systems discrete time markov chains continuous time markov chains and generalized semi markov processes are covered applications are given to a gi g 1 queueing problem and response surface estimation computation of the theoretical moments arising in importance sampling is discussed and some numerical examples given

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

beyond automated multilevel substructuring domain decomposition with rational filtering this paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems the proposed technique is purely algebraic an

block variables for deterministic aperiodic sequences we use the concept of block variables to obtain a measure of order disorder for some one dimensional deterministic aperiodic sequences for the thue morse sequence the rudin shapiro sequence and the period doubling sequence it is possible to obtain analytical expressions in the limit of infinite sequences for the fibonacci sequence we present some analytical results which can be supported by numerical arguments it turns out that the block variables show a wide range of different behaviour some of them indicating that some of the considered sequences are more random than other however the method does not give any definite answer to the question of which sequence is more disordered than the other and in this sense the results obtained are negative we compare this with some other ways of measuring the amount of order disorder in such systems and there seems to be no direct correspondence between the measures

polynomially filtered exact diagonalization approach to many body localization polynomially filtered exact diagonalization method polfed for large sparse matrices is introduced the algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix the memory requirements scale better with system size than in the state of the art shift invert approach the potential of polfed is demonstrated examining many body localization transition in 1d interacting quantum spin 1 2 chains we investigate the disorder strength and system size scaling of thouless time system size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite size effects we discuss possible scenarios regarding the many body localization transition obtaining estimates for the critical disorder strength

recursive procedures for measuring disorder in non periodic sequences abstract recursive procedures are found for calculating the kth order entropy of non periodic sequences these entropies are used as measures of disorder of the sequences and we treat the fibonacci sequence and generalizations the thue morse sequence together with generalizations and the period doubling sequence a discussion of the relative ordering of the thue morse and the fibonacci sequences is given and it is found too simplistic to use any one measure for comparison

the landauer resistance of generalized fibonacci lattices the dynamical maps approach abstract we study the electronic properties of the generalized fibonacci lattices containing finite numbers of rectangular barriers distributed quasi periodically in the framework of the kronig penney model we derive the dynamical maps which allow to calculate the landauer resistance of the considered systems the maps obtained are an extension of the existing results to 1 the more general class of distributions of rectangular barriers and 2 the case of complex unimodular matrices

convergence properties of infinitesimal perturbation analysis infinitesimal perturbation analysis ipa is a method for computing a sample path derivative with respect to an input parameter in a discrete event simulation the ipa algorithm is based on the fact that for certain parameters and any realization of a simulation the change in parameter can be made small enough so that only the times of events get shifted but their order does not change this paper considers the convergence properties of the ipa sample path derivatives in particular the question of when an ipa estimate converges to the derivative of a steady state performance measure is studied necessary and sufficient conditions for this convergence are derived for a class of regenerative processes although these conditions are not guaranteed to be satisfied in general they are satisfied for the mean stationary response time in the m g 1 queue a necessary condition for multiple ipa estimates to simultaneously converge to the derivatives of steady state throughputs in a queueing network is determined the implications of this necessary condition are that except in special cases the original ipa algorithm cannot be used to consistently estimate steady state throughput derivatives in queueing networks with multiple types of customers state dependent routing or blocking numerical studies on ipa convergence properties are also presented

a thick restart lanczos algorithm with polynomial filtering for hermitian eigenvalue problems polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric or complex hermitian matrix that are located in a given interval anywhere in the spectrum this paper describes a technique for tackling this problem by combining a thick restart version of the lanczos algorithm with deflation locking and a new type of polynomial filter obtained from a least squares technique the resulting algorithm can be utilized in a spectrum slicing approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different subintervals independently from one another

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

absence of localization in a class of schrodinger operators with quasiperiodic potential we prove that a class of discrete schrodinger operators with a quasiperiodic potential taking only a finite number of values exhibits purely continuous spectrum in particular they cannot have localized eigenvectors

evaluating non analytic functions of matrices abstract the paper revisits the classical problem of evaluating f a for a real function f and a matrix a with real spectrum the evaluation is based on expanding f in chebyshev polynomials and the focus of the paper is to study the convergence rates of these expansions in particular we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix a we present several numerical examples to illustrate our analysis

solution of the k th eigenvalue problem in large scale electronic structure calculations abstract we consider computing the k th eigenvalue and its corresponding eigenvector of a generalized hermitian eigenvalue problem of n n large sparse matrices in electronic structure calculations several properties of materials such as those of optoelectronic device materials are governed by the eigenpair with a material specific index k we present a three stage algorithm for computing the k th eigenpair with validation of its index in the first stage of the algorithm we propose an efficient way of finding an interval containing the k th eigenvalue 1 k n with a non standard application of the lanczos method in the second stage spectral bisection for large scale problems is realized using a sparse direct linear solver to narrow down the interval of the k th eigenvalue in the third stage we switch to a modified shift and invert lanczos method to reduce bisection iterations and compute the k th eigenpair with validation numerical results with problem sizes up to 1 5 million are reported and the results demonstrate the accuracy and efficiency of the three stage algorithm

measurement of disorder in non periodic sequences an information theoretic measure is introduced to compare the disorder in non periodic sequences it is shown that the measure correctly distinguishes quasiperiodic and aperiodic sequences which have been deduced from earlier studies using diffraction patterns although it is often necessary to use a set of measures depending on the order of the source used the particular sequences studied are the thue morse sequence and the generalizations of the golden mean sequence commonly studied in connection with quasicrystals

structure factor of 1d systems superlattices based on two letter substitution rules i delta bragg peaks the recent generalization to the case of arbitrary tile lengths and arbitrary scattering factors of the calculation of the structure factor of 1d substitutional systems is studied in detail this method makes a easy to find all the peaks in the diffraction spectrum of a system the well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations of two incommensurate frequencies respectively were found to be the l 0 subsets of two more general types of spectra infinite periodic or limit periodic and infinite quasiperiodic or limit quasiperiodic characterized by rational numbers of the type m nl l 0 infinity in place of the above integers substitution rules that produce quasicrystalline quasiperiodic and infinite quasiperiodic spectra give the same type of spectrum for all values of the ratio rho rho a rho b of the two tile lengths rho a and rho b this is not the case for the other rules thus the same substitution rule such as the copper mean rule can give an infinite periodic spectrum for a single rational ratio rho rho a rho b of the two tile lengths rho a and rho b a periodic like spectrum for other rational rho and a spectrum in some aspects similar to that of a random system when rho is an irrational number on the other hand a thue morse system diffracts as a periodic crystal when p not 1 but has no non trivial delta peaks when rho 1 other thue morse like systems can have infinite periodic spectra for all rho

paper a new approach to the analysis of discrete event dynamic systems we present a new time domain approach to the study of discrete event dynamical systems deds typified by queueing networks and production systems a general state space representation is developed and perturbation analysis is carried out observation of a single sample realization of such a system can be used to predict behavior over other sample realizations when some parameter is perturbed without having to make additional observations conditions under which this is always possible are investigated and explicit results for some special cases are included

sensitivity analysis for simulations via likelihood ratios we present a simple method of estimating the sensitivity of quantities obtained from simulation with respect to a class of parameters here sensitivity means the derivative of an expectation with respect to a parameter the class of parameters includes for example poisson rates discrete probabilities and the mean and variance of a normal distribution the method is extremely well suited to regenerative simulation and can be implemented on extant simulations with little effort increase in running time or memory requirements it is based on some change of measure formulas derived from likelihood ratios in addition to the theorems that underly the technique we present some numerical examples