In probability and statistics, Simpson’s paradox is an apparent paradox in which a trend is present in different groups, but is reversed when the groups are combined. Joel Cohen (1986) has shown that continuously distributed lifetimes can never have a Simpson’s paradox. We investigate the same question for discrete random...
Malaria is a vector-borne disease that has affected humans and other animals for a long time and which has shown high prevalence among different populations. During the beginning of the 20th century, Sir Ronald Ross and George Macdonald developed a model that represents the spread of malaria through the interaction...
We investigate the data collection problem in sensor networks. The network consists of a number of stationary sensors deployed at different sites for sensing and storing data locally. A mobile element moves from sites to sites to collect data from the sensors periodically. There are different costs associated with the...
This dissertation presents some results from various areas of probability theory, the unifying theme being the use of functional analytic intuition and techniques. We first give a result regarding the existence of certain stochastic integral representations for Banach space valued Gaussian random variables. Next we give a spectral geometric construction...
Spreadsheets are a widely used end-user programming tool. Field audits have found that 80-90% of spreadsheets created by end users contain textual and formula errors in spreadsheets. Such errors may have severe negative consequences for users in terms of productivity, credibility, or profits. To solve the problem of spreadsheet errors,...
Each chapter in this expository paper considers a mathematical model of an aspect of animal behavior, and how it affects the patterns of movement across and within a landscape. These models are all directly or indirectly related to questions in either Behavioral Ecology or Landscape Ecology, or both. I first...
In this thesis, convergence of time inhomogeneous Markov chains is studied using an adiabatic approach. The adiabatic framework considers slowly changing systems and the adiabatic time quantifies the time required for the change such that the final state of the system is close to some equilibrium state. This approach is...
Probabilistic models have been successfully applied for a wide variety of problems, such as but not limited to information retrieval, computer vision, bio-informatics and speech processing. Probabilistic models allow us to encode our assumptions about the data in an elegant fashion and enable us to perform machine learning tasks such...
This thesis deals with applications of uniaxial anisotropic crystals for microcavity resonators with partially chaotic underlying ray dynamics. We develop an implementation of the scattering matrix formalism, and relate the eigenvalues and eigenvectors of the scattering matrix to the field distribution of inside the system. Using the developed technique, we...
In this thesis, I present a new method we developed to study low frequency (< 700 cm⁻¹) vibrational dynamics: Time-resolved third-harmonic generation (TRTHG) spectroscopy. Among a variety of vibrational spectroscopy techniques, TRTHG differentiates itself with robustness, versatility, and simplicity. In TRTHG experiment, that involves only two ultrashort laser pulses, the...