Foreground contamination of the WMAP CMB maps from the perspective of the matched circle test

WMAP has provided CMB maps of the full sky. The raw data is subject to foreground contamination, in particular near to the Galactic plane. Foreground cleaned maps have been derived, e.g., the internal linear combination (ILC) map of Bennett et al. and the reduced foreground TOH map of Tegmark et al. Using S statistics we examine whether residual foreground contamination is left over in the foreground cleaned maps. In particular, we specify which parts of the foreground cleaned maps are sufficiently accurate for the circle-in-the-sky signature. We generalise the S statistic, called D statistic, such that the circle test can deal with CMB maps in which the contaminated regions of the sky are excluded with masks

Maass cusp forms for large eigenvalues

We investigate the numerical computation of Maass cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed r=40000. These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the 130millionth eigenvalue.Comment: 24 pages, 7 figures, 3 table

Large sets of consecutive Maass forms and fluctuations in the Weyl remainder

We explore an algorithm which systematically finds all discrete eigenvalues of an analytic eigenvalue problem. The algorithm is more simple and elementary as could be expected before. It consists of Hejhal's identity, linearisation, and Turing bounds. Using the algorithm, we compute more than one hundredsixty thousand consecutive eigenvalues of the Laplacian on the modular surface, and investigate the asymptotic and statistic properties of the fluctuations in the Weyl remainder. We summarize the findings in two conjectures. One is on the maximum size of the Weyl remainder, and the other is on the distribution of a suitably scaled version of the Weyl remainder.Comment: A version with higher resolution figures can be downloaded from http://www.maths.bris.ac.uk/~mahlt/research/T2012a.pd

Spectral resolution in hyperbolic orbifolds, quantum chaos, and cosmology

We present a few subjects from physics that have one in common: the spectral resolution of the Laplacian.Comment: 24 pages. Contribution to the TSL Expository Lecture Series V "Computational Physical Sciences 2006", Universiti Putra Malaysi

A characteristic of Bennett's acceptance ratio method

A powerful and well-established tool for free-energy estimation is Bennett's acceptance ratio method. Central properties of this estimator, which employs samples of work values of a forward and its time reversed process, are known: for given sets of measured work values, it results in the best estimate of the free-energy difference in the large sample limit. Here we state and prove a further characteristic of the acceptance ratio method: the convexity of its mean square error. As a two-sided estimator, it depends on the ratio of the numbers of forward and reverse work values used. Convexity of its mean square error immediately implies that there exists an unique optimal ratio for which the error becomes minimal. Further, it yields insight into the relation of the acceptance ratio method and estimators based on the Jarzynski equation. As an application, we study the performance of a dynamic strategy of sampling forward and reverse work values

Measuring the convergence of Monte Carlo free energy calculations

The nonequilibrium work fluctuation theorem provides the way for calculations of (equilibrium) free energy based on work measurements of nonequilibrium, finite-time processes and their reversed counterparts by applying Bennett's acceptance ratio method. A nice property of this method is that each free energy estimate readily yields an estimate of the asymptotic mean square error. Assuming convergence, it is easy to specify the uncertainty of the results. However, sample sizes have often to be balanced with respect to experimental or computational limitations and the question arises whether available samples of work values are sufficiently large in order to ensure convergence. Here, we propose a convergence measure for the two-sided free energy estimator and characterize some of its properties, explain how it works, and test its statistical behavior. In total, we derive a convergence criterion for Bennett's acceptance ratio method.Comment: 14 pages, 17 figure

Rapid computation of LL-functions attached to Maass forms

Let $L$ be a degree-$2$ $L$-function associated to a Maass cusp form. We explore an algorithm that evaluates $t$ values of $L$ on the critical line in time $O(t^{1+\varepsilon})$. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution