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Geometric Field Stability And Normal Field Curvature Of Solution Sets Of Ordinary Differential Equations In Two Variables, Leslie L. Kerns

Theses, Dissertations and Capstones

The classical linearization approach to stability theory determines whether or not a system is stable in the vicinity of its equilibrium points. This classical approach partly depends on the validity of the linear approximation. The definition of stability developed in this article takes a different approach and uses a curvature function to assess the relative locations of solutions within a field of solutions (the underlying solution set of the ODE). The present approach involves calculations that directly yield stability information, without having to enter into the often lengthy eigenvalue-eigenvector method. The present results both complement and are compatible with the …

The Dynamics Of Newton's Method On Cubic Polynomials, Shannon N. Miller

Theses, Dissertations and Capstones

The field of dynamics is itself a huge part of many branches of science, including the motion of the planets and galaxies, changing weather patterns, and the growth and decline of populations. Consider a function f and pick x0 in the domain of f . If we iterate this function around the point x0, then we will have the sequence x0, f (x0), f (f (x0)), f (f (f (x0))), ..., which becomes our dynamical system. We are essentially interested in the end behavior of this system. Do …

Solving Higher Order Dynamic Equations On Time Scales As First Order Systems, Elizabeth R. Duke

Theses, Dissertations and Capstones

Time scales calculus seeks to unite two disparate worlds: that of differential, Newtonian calculus and the difference calculus. As such, in place of differential and difference equations, time scales calculus uses dynamic equations. Many theoretical results have been developed concerning solutions of dynamic equations. However, little work has been done in the arena of developing numerical methods for approximating these solutions. This thesis work takes a first step in obtaining numerical solutions of dynamic equations|a protocol for writing higher-order dynamic equations as systems of first-order equations. This process proves necessary in obtaining numerical solutions of differential equations since the Runge-Kutta …

Applications Of Computational Homology, Christopher Aaron Johnson

Theses, Dissertations and Capstones

Homology is a field of topology that classifies objects based on the number of n- dimensional holes (cuts, tunnels, voids, etc.) they possess. The number of its real life ap- plications is quickly growing, which requires development of modern computational meth- ods. In my thesis, I will present methods of calculation, algorithms, and implementations of simplicial homology, alpha shapes, and persistent homology.

The Alpha Shapes method represents a point cloud as the union of balls centered at each point, and based on these balls, a complex can be built and homology computed. If the balls are allowed to grow, one …