Inspired by the paper of Klagsbrun, Mazur and Rubin [5], this thesis investigates the disparity of 2-Selmer ranks of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. In the first part, we calculate the density of quadratic twists of E with even 2-Selmer ranks under two different counting methods. First we count twists by elements inside a large convex body of the Euclidean space that contains the integer lattice of K. The second counting method is counting quadratic twists E^L by the norms of the finite part of conductors of quadratic extensions L/K. Under both counting methods we give an explicit formula for the densities, which are finite products of local factors. In the second part of the paper we give a method that uses Tate’s algorithm to calculate the size of the cokernel of the local norm maps of E at places over 2, assuming that E has good reduction. With this method we can extend Kramer’s early work on the cokernel of the local norm maps, and compute the local factors mentioned above in some additional cases.

Chemical sensors have a deep influence in a wide variety of disciplines ranging from biomedical measurement, electrochemical analysis, detection of environmental toxins, pollution monitoring, and warfare threats, to industrial control. The pressure of legislation concerning environmental issues and increasing environmental regulations makes chemical sensing of pollutants extremely important.

The overarching goal of this proposed work is to develop cost effective routes to synthesize integrated 1-D materials onto prefabricated micro-electrodes for the detection of airborne toxins at ambient temperature as well as real-time and simultaneously measuring O3, CO, SO2, and NO2 to quantify Air Quality Index. The sensor arrays will be made out of novel conductometric/chemiresistive sensors based on functionalized single-walled carbon nanotubes (SWNTs). The array will employ sensors with varying sensing materials (organic/inorganic) and sensing mechanisms operated at optimum conditions which will enable more selective and accurate monitoring of AQI. Optimization of both the individual sensor and the sensor array will produce distinctive sensing response patterns for detecting the individual agents in the gas mixtures. Ink-jet printing will be utilized to deposit hybrid SWNTs inks to precise locations on a substrate, while electrodeposition techniques will be utilized to synthesize metal and metal oxide decorated SWNTs with controlled crystallinity, morphology, and electrical properties. The structure-property relationships of these materials, such as electrical transport, optical and gas sensing performance will be investigated, evaluated, and optimized.