| What is equational definability? Jeremy Alm In undergraduate abstract algebra we learn that a group is a set with a binary operation such that ... In this talk we will show that groups can be defined by equations alone iff the "inverse" operation is included among the original (fundamental) operations. Garrett Birkhoff's varieties theorem, the most interesting theorem I encountered in graduate school, will be stated. The Spectrum of Population Projection Matrices The spectrum of population projection matrices is used to determine whether a population undergoes growth or decay. Changes in the model due to perturbations in the parameters can and often do change the eigenvalues of the matrix. This leads us to ask when is a specific eigenvalue the leading one. To answer this we give Population Pojection Matrices a mathematical definition from which we can determine whether an eigenvalue of the perturbed matrix is infact the leading eigenvalue. This allows us to place a surface into the parameter space partitioning the "good" region from the "bad" in the sence that "good" parameters give a spectral radius with at least/at most a desired magnitude. Utilizing Idle CPU Usage in an Automated Fashion for RSA Decryption Businesses and consumers continually rely on the security of RSA cryptosystems. The RSA algorithm, which was first introduced in the 1970s, has since served as a primary means of key cryptography throughout the world. As computers have increased in speed and capabilities the bit length of RSA cryptosystems has increased to lengths of 512 to 2048 bits. RSA numbers in the past have remained fairly secure because of the difficultly involved in factoring large numbers. In this presentation, we present an automated decryption technique implemented through screen savers, which attempts to break large RSA numbers. We have implemented our screen saver program on thirty plus computers, which utilize low CPU usage times to perform mathematical calculations. Our preliminary results from factoring RSA numbers up to 200 bits indicate that our technique can in fact find the factors of RSA numbers. The Time Scales Harmonic Oscillator I will be examining Chapter 5 of Moritz Simon's dissertation, entitled "The Time Scales Harmonic Oscillator." The basics of time scales and quantum mechanics will be introduced and we will see how the theory of time scales is being used to model solutions to a harmonic oscillator problem. Projective Planes and Latin Squares A latin square of order n is an n-by-n array of n symbols where each symbol appears once per row and once per column. A finite projective plane is an easily defined non-Euclidean geometry: every two points determine a unique line; every two lines meet at a unique point; there are four points, no three of which share a line. These very different objects are connected by the fact that a projective plane of order n exists if and only if there are n-1 mutually orthogonal latin squares of order n (the set of which is a so-called MAXMOLS). The definition of orthogonal latin squares and a construction of a projective plane from a MAXMOLS and vice versa will be given. Empirical Software Engineering Building software quickly and reliably remains a challenging task. Researchers, consultants and software vendors have developed many technologies that claim to help us build software faster and with fewer bugs. But, how well do these techniques really work? This talk provides an overview of the field of "empirical software engineering", which uses empirical studies to validate claims about software technologies. Numerical approximations of null controllability for structurally damped operators In this talk, we give a numerical approximation scheme to compute the null (internal) controller for a nonstandard parabolic partial differential equation (PDE). This PDE consists of a plate equation under the action of "square root" fractional damping. Because of the hinged boundary conditions which are in play, the eigenfunctions of the Dirichlet Laplacian may be used to find a sequence of null controls for corresponding finite dimensional, spectral truncations, of this system. Based on the existing theory, one can show that the sequence of (finite dimensional) null controls converges to the null controller of the (infinite dimensional) PDE system. More important, the (infinite dimensional) minimal null control obeys the asymptotics of its finite dimensional approximations. We further discuss the possibility of replacing the spectral approximations with corresponding finite difference approximations. Numerical examples will be presented. Dr. Strangeproof, or How I learned to stop worrying and love the superfluity Everyone knows that when one writes a proof of a theorem and never invokes one of the hypotheses, then it should be removed from the statement of the theorem. Even undergraduate students pick up on this bit of common knowledge. Although we encourage them in the development of this habit, the general claim, taken at face value, cannot be believed. An Analysis of Growth Rates and Population Densities of Bacterial Flagellum Salmonella typhimurium utilize long, whip-like flagella for motility in liquid. The lengths of these flagellum and the rate at which they grow are highly conserved in nature, suggesting the presence of a regulatory mechanism. At present, no one characterization of this mechanism is universally agreed upon; indeed, a battery of biological and mathematical models have been proposed. In Applied Combinatorics in Software Engineering Covering Arrays, a mathematical combinatorics structure, appear useful in limiting the number of interaction tests for a system compared to total enumeration (a combinatorial explosion). The catch is that Covering Arrays themselves are difficult to find. This talk motivates the interaction testing problem and difficulty in finding covering arrays, overviews current meta-heuristic approaches to finding CAs, and concludes with current progress of a genetic algorithm approach.
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