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Michael Goldstein

Publications and source records attributed to Michael Goldstein.

30 records · Page 2Linked to original sources
arXiv2014

On the Sum of the Non-Negative Lyapunov Exponents for Some Cocycles Related to the Anderson Model

We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $ W $ the lower bound is proportional to $ W^{-ε} $, for any $ ε>0 $. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $ W^{-1} $.

math-ph↗
arXiv2014

Galaxy Formation: Bayesian History Matching for the Observable Universe

Cosmologists at the Institute of Computational Cosmology, Durham University, have developed a state of the art model of galaxy formation known as Galform, intended to contribute to our understanding of the formation, growth and subsequent evolution of galaxies in the presence of dark matter. Galform requires the specification of many input parameters and takes a significant time to complete one simulation, making comparison between the model's output and real observations of the Universe extremely challenging. This paper concerns the analysis of this problem using Bayesian emulation within an iterative history matching strategy, and represents the most detailed uncertainty analysis of a galaxy formation simulation yet performed.

stat.ME↗
arXiv2014

On fluctuations and localization length for the Anderson model on a strip

We consider the Anderson model on a strip. Assuming that potentials have bounded density with considerable tails we get a lower bound for the fluctuations of the logarithm of the Green's function in a finite box. This implies an effective estimate by $ \exp(CW^2) $ for the localization length of the Anderson model on the strip of width $ W $. The results are obtained, actually, for a more general model with a non-local operator in the vertical direction.

math-ph↗
arXiv2013

Bayes linear variance structure learning for inspection of large scale physical systems

Modelling of inspection data for large scale physical systems is critical to assessment of their integrity. We present a general method for inference about system state and associated model variance structure from spatially distributed time series which are typically short, irregular, incomplete and not directly observable. Bayes linear analysis simplifies parameter estimation and avoids often-unrealistic distributional assumptions. Second-order exchangeability judgements facilitate variance learning for sparse inspection time-series. The model is applied to inspection data for minimum wall thickness from corroding pipe-work networks on a full-scale offshore platform, and shown to give materially different forecasts of remnant life compared to an equivalent model neglecting variance learning.

stat.ME↗
arXiv2008

On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger equations

We consider one-dimensional difference Schroedinger equations on the discrete line with a potential generated by evaluating a real-analytic potential function V(x) on the one-dimensional torus along an orbit of the shift x-->x+nw. If the Lyapunov exponent is positive for all energies and w, then the integrated density of states is absolutely continuous for almost every w. In this work we establish the formation of a dense set of gaps in the spectrum. Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfunctions of one scale lead to "pre-gaps" at a larger scale. To pass to actual gaps in the spectrum, we show that these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, we relate a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle using the techniques of an earlier paper by the authors. Amongst other things, we establish in this work a non-perturbative version of the co-variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of Sinai's (perturbative) description of the spectrum via his function as well as an multi-scale/finite volume approach to Anderson localization in this context. This allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase. Our infinite volume theorems hold for all Diophantine frequencies w up to a set of Hausdorff dimension zero. This is in contrast to earlier technology which only yielded exceptional sets of measure zero. Our only assumption on the potential apart from analyticity is the positivity of the Lyapunov exponent.

math.DS↗
arXiv2006

On non-perturbative Anderson localization for C^αpotentials generated by shifts and skew-shifts

In this paper we address the question of proving Anderson localization (AL) for the operator [H(x,ω)ψ](n) := - \vp(n+1) - \vp(n-1) + V\bigl(T^n_ωx\bigr)ψ(n), n\in\mathbb Z where T:\tor^2\to\tor^2 is either the shift or the skew-shift and V is only C^α(\tor^2) for some α>0. We show that under the assumption of positive Lyapunov exponents, (AL) takes place for a.e. frequency, phase, and energy.

math.DS↗
arXiv2005

Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues

We develop some non-perturbative methods for studying the IDS in almost Mathieu and related models. Assuming positive Lyapunov exponents, and assuming that the potential function is a trigonometric polynomial of degree k, we show that the Holder exponent of the IDS is almost 1/2k. We also show that this is stable under small perturbations of the potential (e.g., potentials which are close to that of almost Mathieu again give rise to almost 1/2 Holder continuous IDS). Moreover, off a set of Hausdorff dimension zero the IDS is Lipschitz. We further deduce from these properties that the IDS is absolutely continuous for almost every shift angle. The proof is based on large deviation theorems for the entries of the transfer matrices in combination with the avalanche principle for non-unimodular matrices. These two are combined to yield a multiscale approach to counting zeros of determinants which fall into small disks. In addition, we develop a new mechanism of eliminating resonant phases, and use it to obtain lower bounds on the gap between Dirichlet eigenvalues. The latter also uses the exponential localization of the eigenfunctions.

math-ph↗
arXiv2001

A Mapping Survey of the 12CO and 13CO Emission in Galaxies

We present spectra of the extended 12CO and 13CO J=1-0 emission along the major axes of 17 nearby galaxies. Spatial variations in the ratio of CO and 13CO integrated intensities, R, are found in nearly every galaxy observed. There is an overall variation in R of 20-40% from the inner 2 kpc to the disk. Roughly a third of the survey galaxies have such gradients in R detected above the 2-sigma confidence level. Though some galaxies show a lower central value of R, on average R inside 2 kpc is 10-30% higher than R outside of 2 kpc. An increase in the CO/13CO intensity ratio from disk to nucleus may imply that the conversion factor between CO intensity and H2 column density, X, is lower in galactic nuclei. Also variations in physical conditions, most notably the gas kinetic temperature, affect both R and X. Abundance variations probably do not cause the gradient in R, though we do not rule out a decrease in effective cloud column densities in galactic nuclei possibly caused by destructive starburst superwinds. A modest rise in temperature (less than a factor of 2 or 3) from outside a 2 kpc radius towards the nucleus can easily account for the observed gradient. These results support previous work implying that X is lower in the center of the Milky Way and probably most galactic nuclei. Therefore calculating H2 masses using the standard Galactic X-factor, especially within the central few kpc of galaxies, overestimates the true mass by factors of a few. The standard X-factor still appears to be appropriate for galactic disks.

astro-ph↗
arXiv2000

On nonperturbative localization with quasi-periodic potential

The two main results of the article are concerned with Anderson Localization for one-dimensional lattice Schroedinger operators with quasi-periodic potentials with d frequencies. First, in the case d = 1 or 2, it is proved that the spectrum is pure-point with exponentially decaying eigenfunctions for all potentials (defined in terms of a trigonometric polynomial on the d-dimensional torus) for which the Lyapounov exponents are strictly positive for all frequencies and all energies. Second, for every non-constant real-analytic potential and with a Diophantine set of d frequencies, a lower bound is given for the Lyapounov exponents for the same potential rescaled by a sufficiently large constant.

math-ph↗
arXiv1995

Bayes linear adjustment for variance matrices

We examine the problem of covariance belief revision using a geometric approach. We exhibit an inner-product space where covariance matrices live naturally --- a space of random real symmetric matrices. The inner-product on this space captures aspects of our beliefs about the relationship between covariance matrices of interest to us, providing a structure rich enough for us to adjust beliefs about unknown matrices in the light of data such as sample covariance matrices, exploiting second-order exchangeability specifications.

bayes-an↗
arXiv1995

Bayes linear covariance matrix adjustment for multivariate dynamic linear models

A methodology is developed for the adjustment of the covariance matrices underlying a multivariate constant time series dynamic linear model. The covariance matrices are embedded in a distribution-free inner-product space of matrix objects which facilitates such adjustment. This approach helps to make the analysis simple, tractable and robust. To illustrate the methods, a simple model is developed for a time series representing sales of certain brands of a product from a cash-and-carry depot. The covariance structure underlying the model is revised, and the benefits of this revision on first order inferences are then examined.

bayes-an↗