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

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

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

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

Hyperbolic Universes with a Horned Topology and the CMB Anisotropy

We analyse the anisotropy of the cosmic microwave background (CMB) in hyperbolic universes possessing a non-trivial topology with a fundamental cell having an infinitely long horn. The aim of this paper is twofold. On the one hand, we show that the horned topology does not lead to a flat spot in the CMB sky maps in the direction of the horn as stated in the literature. On the other hand, we demonstrate that a horned topology having a finite volume could explain the suppression of the lower multipoles in the CMB anisotropy as observed by COBE and WMAP