Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials

For a non-uniform lattice in SL(2, R), we consider excursions of a random geodesic in cusp neighborhoods of the quotient finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmuller setting, we consider invariant measures for the SL(2, R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teichmuller geodesic random with respect to a SL(2,R)-invariant measure, we study its excursions in thin parts of the associated submanifold. Under a regularity hypothesis for the invariant measure, we prove similar strong laws for certain partial sums involving these excursions. The limits in these laws are related to the volume asymptotic of the thin parts. By Siegel-Veech theory, these are given by Siegel-Veech constants. As a direct consequence, we show that the word metric of mapping classes that approximate a Teichmuller geodesic ray that is random with respect to the Masur-Veech measure, grows faster than T log T

Estimation of the Sensitive Volume for Gravitational-wave Source Populations Using Weighted Monte Carlo Integration

The population analysis and estimation of merger rates of compact binaries is one of the important topics in gravitational wave (GW) astronomy. The primary ingredient in these analyses is the population-averaged sensitive volume. Typically, sensitive volume, of a given search to a given simulated source population, is estimated by drawing signals from the population model and adding them to the detector data as injections. Subsequently injections, which are simulated gravitational waveforms, are searched for by the search pipelines and their signal-to-noise ratio (SNR) is determined. Sensitive volume is estimated, by using Monte-Carlo (MC) integration, from the total number of injections added to the data, the number of injections that cross a chosen threshold on SNR and the astrophysical volume in which the injections are placed. So far, only fixed population models have been used in the estimation of the merger rates. However, as the scope of population analysis broaden in terms of the methodologies and source properties considered, due to an increase in the number of observed GW signals, the procedure will need to be repeated multiple times at a large computational cost. In this letter we address the problem by performing a weighted MC integration. We show how a single set of generic injections can be weighted to estimate the sensitive volume for multiple population models; thereby greatly reducing the computational cost. The weights in this MC integral are the ratios of the output probabilities, determined by the population model and standard cosmology, and the injection probability, determined by the distribution function of the generic injections. Unlike analytical/semi-analytical methods, which usually estimate sensitive volume using single detector sensitivity, the method is accurate within statistical errors, comes at no added cost and requires minimal computational resources.Comment: 11 pages, 1 figur

Higher Order Convergent Fast Nonlinear Fourier Transform

It is demonstrated is this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of $O{KN+C_pN\log^2N}$ such that the error vanishes as $O{N^{-p}}$ where $p\in\{1,2,3,4\}$ and $K$ is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula ($C_p=p^3$) and the implicit Adams method ($C_p=(p-1)^3$) of which the latter proves to be the most accurate family of methods for fast NFT

IPhone Securtity Analysis

The release of Apple’s iPhone was one of the most intensively publicized product releases in the history of mobile devices. While the iPhone wowed users with its exciting design and features, it also outraged many for not allowing installation of third party applications and for working exclusively with AT&T wireless services for the first two years. Software attacks have been developed to get around both limitations. The development of those attacks and further evaluation revealed several vulnerabilities in iPhone security. In this paper, we examine several of the attacks developed for the iPhone as a way of investigating the iPhone’s security structure. We also analyze the security holes that have been discovered and make suggestions for improving iPhone security