Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Patrick Keef

  • TITLE: PROVING COMPLETENESS OF THE HAUSDORFF INDUCED METRIC SPACE

    AUTHOR: Katie Barich
    ABSTRACT: Given a metric space (X; d), we may denote a new metric space with Hausdorff metric h on the set K of the collection of all nonempty compact subsets of X. We show that if (X; d) is complete, then the Hausdorff metric space (K; h) is also complete.
    ADVISOR: Russ Gordon

  • TITLE: MATHEMATICS OF MEDICAL IMAGING: INVERTING THE RADON TRANSFORM

    AUTHOR: Kailey Bolles
    ABSTRACT: Computed Tomography (CT) and other radial imaging techniques are vital to the practice of modern medicine, allowing non-invasive examination of the inner workings of the human body. However, raw CT data must be transformed in order to become diagnostically relevant. This project examines raw CT data, modeled by the Radon transform, and methods of inversion via unfiltered backprojection, Fourier transforms, and filtered backprojection (the inverse Radon transform). We demonstrate this process through examples of "raw data" and inversion, with a focus on the influence of discrete data sets of different sizes on inversion quality.
    ADVISOR: Albert Schueller

  • TITLE: FINITE-WIDTH ELEMENTARY CELLULAR AUTOMATA

    AUTHOR: Ian Coleman
    ABSTRACT: This paper is an empirical study of eight-wide elementary cellular automata motivated by Stephen Wolfram's conjecture about widespread universality in regular elementary cellular automata. Through examples, the concepts of equivalence, reversibility, and additivity in elementary cellular automata are explored. In addition, we will view finite-width cellular automata in the context of finite-size state transition diagrams and develop foundational results about the behavior of finite-width elementary cellular automata.
    ADVISOR: Albert Schueller

  • TITLE: BUILDING LOCAL EQUATIONS OF MOTION TO MODEL CHAOTIC DATASETS

    AUTHOR: Cooper Crosby
    ABSTRACT: This paper presents some methods of dealing with the data generated by chaotic dynamical systems. By deconstructing the data into locally linear clusters and building low dimensional bases for each cluster, radial basis functions can be used to rebuild the dynamics of the system and predict future behavior.
    ADVISOR: Doug Hundley

  • TITLE: INVESTIGATION OF SOLUTIONS TO THE EQUATION x^{\ell+1} \equiv x ~ (\mathop{\rm mod} n)

    AUTHOR: Jung Hyo Koo
    ABSTRACT: This paper will study the solutions to the equation x^{\ell+1}\equiv x~(\mathop{\rm mod} n). The topic will be approached in three ways. First, we will fix \ell=1 and study the characteristics of idempotents. Secondly, we will let \ell vary within the positive integers and find characteristics of the roots to the equation. Lastly, we will use the previous results to study subsets of Zn that are cyclic groups under multiplication having powers of odd primes as orders and show exactly how many such subsets qualify as groups.
    ADVISOR: Pat Keef

  • TITLE: A CLASSIFICATION OF THE SUBGROUPS OF THE RATIONALS UNDER ADDITION

    AUTHOR: Patrick Miller
    ABSTRACT: In this paper, we examine one of the most fundamental and interesting algebraic structures, the rational numbers, from the perspective of group theory. We delve into subgroups of the rationals under addition, with the ultimate goal of completely classifying their isomorphisms, gaining insight into general torsion-free groups along the way.
    ADVISOR: Pat Keef

  • TITLE: GENERALIZATIONS OF THE RIEMANN INTEGRAL: AN INVESTIGATION OF THE HENSTOCK INTEGRAL (selected outstanding senior project)

    AUTHOR: Jonathan Wells
    ABSTRACT: The Henstock integral, a generalization of the Riemann integral that makes use of the \delta-fine tagged partition, is studied. We first demonstrate that a function is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. We find that not every derivative is Riemann integrable, and that the strong condition of uniform convergence must be applied to guarantee that the limit of Riemann integrable functions remains integrable. We investigate several properties of Henstock integral, demonstrate that every derivative is Henstock integrable, and show that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory.
    ADVISOR: Russ Gordon


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schuelaw@whitman.edu Last updated: Sat May 21 09:58:26 PDT 2011