All living mammalian cells need to consume oxygen and nutrients for cellular processes and need a way to remove waste from those cellular... Show moreAll living mammalian cells need to consume oxygen and nutrients for cellular processes and need a way to remove waste from those cellular processes. Capillary networks provide places for such exchanges to occur. The process of creating new capillaries from existing blood vessels is called angiogenesis. Understanding angiogenesis is critical to the advancement of knowledge in the life sciences, as well as in medical applications where blood vessels play an important role. Angiogenesis is a complex process composed of many subprocesses which are not yet fully understood and take place over varying temporal and spatial scales. Mathematically modeling and simulating angiogenesis, and evaluating the capillary networks that result from angiogenesis, can help further understanding of angiogenesis and improve therapeutic treatments. This thesis examines mathematical models and simulations of sprouting angiogenesis and proposes two generic models of sprouting angiogenesis based on descriptions found in educational and scientific literature. Future research opportunities for scientific study and educational study using these models as a starting place are discussed. M.S. in Applied Mathematics, May 2016 Show less
Monte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are... Show moreMonte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are not known explicitly. This thesis investigates how to reliably construct fixed width confidence intervals for μ with some prescribed absolute error tolerance, "a, relative error tolerance, "r or some generalized error criterion. To facilitate this, it is assumed that the kurtosis, , of the random variable, Y , does not exceed a user specified bound max. The key idea is to confidently estimate the variance of Y by applying Cantelli’s Inequality. A Berry-Esseen Inequality makes it possible to determine the sample size required to construct such a confidence interval. When relative error is involved, this requires an iterative process. This idea for computing μ = E(Y ) can be used to develop a numerical integration method by writing the integral as μ = E(f(x)) = RRd f(x)⇢(x)dx, where x is a d dimensional random vector with probability density function ⇢. A similar idea is used to develop an algorithm for computing p = E(Y) where Y is a Bernoulli random variable. All of the algorithms have been implemented in the Guaranteed Automatic Integration Library (GAIL). Ph.D. in Applied Mathematics, May 2016 Show less
