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

Electrocardiogram (ECG/EKG) using FPGA

FPGAs (Field Programmable Gate Arrays) are finding wide acceptance in medical systems for their ability for rapid prototyping of a concept that requires hardware/software co-design, for performing custom processing in parallel at high data rates and be programmed in the field after manufacturing. Based on the market demand, the FPGA design can be changed and no new hardware needs to be purchased as was the case with ASICs (Application Specific Integrated Circuit) and CPLDs (Complex Programmable Logic Device). Medical companies can now move over to FPGAs saving cost and delivering highly-efficient upgradable systems. ECG (Electrocardiogram) is considered to be a must have feature for a medical diagnostic imaging system. This project attempts at implementing ECG heart-rate computation in an FPGA. This project gave me exposure to hardware engineering, learning about the low level chips like Atmel UC3A3256 micro-controller on an Atmel EVK1105 board which is used as a simulator for generating the ECG signal, the operational amplifiers for amplifying and level-shifting the ECG signal, the A/D converter chip for analog to digital conversion of the ECG signal, the internal workings of FPGA, how different hardware components communicate with each other on the system and finally some signal processing to calculate the heart rate value from the ECG signal

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