Elementary Number Theory: A Problem Oriented Approach

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Descriptive statistics, measures of central tendency and dispersion, correlation, probability, probability distributions and statistical inference. Use of statistical software to manage, process and analyze data. Writing of statistical programs to perform simulation experiments. Elementary Theory of Numbers. Offered every third semester as of fall Properties of integers and prime numbers, divisibility, congruence, residues and selected topics.

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Prerequisite: MATH or consent of the instructor. The Real Number System. A development of the real number system through a systematic approach to the natural numbers, integers, rationals and irrationals. Introduction to Probability and Statistics. Descriptive statistics, counting, probability axioms, discrete and continuous univariate random variables, expected values of random variables and sums of independent random variables, sampling distributions and the Central Limit Theorem, single and two sample inference for proportions and means, chi-square test of independence, simple linear regression, and correlation.

Analysis of Variance and Experimental Design. Introduction to basic concepts in statistics with applications of statistical techniques including estimation, test of hypothesis, analysis of variance and topics in experimental design. Applied Linear Regression. Introduction to basic concepts and methods in regression analysis and the application of these models to real-life situations.

Applied Nonparametric Statistics. Methods of analyzing data from non-normal populations including binomial tests, contingency tables, use of ranks, Kolmogorov-Smirnov type statistics and selected topics. Survey Sampling Methods.

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  • Theory and practice of sampling including stratified random samples, discussion of simple random samples, cluster sampling, estimating sample size, ratio estimates, subsampling, two-state sampling and analysis of sampling error. Statistical Quality Control. Uses and concepts of probability and sampling procedures. Acceptance sampling by attributes and variables, Shewhart concepts of process control, control chart process capability studies, reliability and life testing. Design of sampling plans. Categorical Data Analysis.

    Volume 42, Number 1, March 2011

    Exact inference for population proportions, comparison of population proportions for independent and dependent samples, two and three-way contingency tables, Chi-square tests of independence and homogeneity, Chi-square goodness-of-fit tests and Poisson and logistic regression. Time Series Analysis. Offered fall of even years. Elementary Differential Equations. Development of techniques for obtaining, analyzing and graphing solutions to differential equations, with emphasis on first and second order equations.

    Methods of Applied Calculus. Laplace transforms, power series and their application to differential equations. Vector differential and integral calculus; parametric curves; coordinate systems; line, surface and volume integrals; and gradient, divergence and curl including the theorems of Green, Stokes and Gauss. Mathematical Modeling I — Optimization. Linear and nonlinear optimization with an emphasis on applications in the sciences, economics and social sciences.

    Techniques studied include the simplex, Newton and Lagrange methods and Kuhn-Tucker theory. Software packages will be used to implement these methods. Nonlinear Dynamics and Chaos. Introductory study of nonlinear dynamics and chaos intended primarily for upper-level undergraduates in science and mathematics. Topics include stability, bifurcations, phase portraits, strange attractors, fractals and selected applications of nonlinear dynamics in pure and applied science. Computers may be utilized for simulations and graphics.

    Mathematical Models in Biology. Introduction to dynamical models discrete and continuous time applied to biology. Tools of mathematical analysis from linear and nonlinear dynamics will be taught, including stability analysis of equilibria, as well as appropriate use of software packages.

    Emphasis will be on model development and interpretation in the context of applications, including effective written and oral presentation. Graph Theory. Offered every third semester as of spring Graphs and their applications. Possible topics include trees, Euler paths and Hamiltonian circuits, planar graphs, digraphs, adjacency matrices, connectivity and coloring problems.

    Prerequisite: MATH or consent of instructor. Introduction to Biometrics 3, 1. This course discusses the role of statistics in biological research and interpretation of biological phenomena. The course will cover topics of sampling, correlation, regression analysis, tests of hypotheses, commonly observed distributions in natural populations, nonparametric tests, goodness-of-fit tests and ANOVA. In order to fully comprehend the statistical analysis of those publications, students will review approximately half a dozen publications from different fields of biology.

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    Complex Variables with Applications. Introduction to algebraic properties of complex numbers, analytic functions, harmonic functions, mappings of elementary functions, contour integration, series, residues, and poles and conformal mappings. Emphasis on computations and applications to fluid and heat flow. Computational Fluid Dynamics. Applications of computer models to the understanding of both compressible and incompressible fluid flows. Introduction to Acoustics.

    This course represents an introduction to sound, hearing, and vibration. Architectural, biological and environmental acoustics will also be discussed. Students will develop an ability to use mathematical models and experimental techniques to study problems in acoustics and to transfer this knowledge to analogous situations. They will also develop an ability to conduct a semester—long research or expository project and present it in written and oral form to an audience of peers. Prerequisite: MATH or permission of instructor.

    SIGACT News (ACM), Volume 42

    Mathematical Finance. An overview of the role of mathematical concepts in financial applications. Topics include continuous time finance, optimization, numerical analysis and applications in asset pricing. Securities Pricing. A quantitative treatment of the theory and method of financial securities pricing to include an examination of closed form pricing models such as the Black-Scholes and its various derivatives as well as numerical solution techniques such as binomial methods. Advanced Calculus I-II.

    Volume 42, Number 2, June 2011

    Limits, continuity, differentiation, sequences, series, integration and selected topics. History of Mathematics. Topics in the history of mathematics spanning ancient times to the present. Applied Multivariate Statistical Analysis. Multivariate statistical methods with applications.

    Number Theory

    Topics include canonical correlation, clustering, discriminant analysis, factor analysis, multivariate analysis of variance, multiple regression, multidimensional scaling and principal component analysis. Johnston, B. Jones, A. Kac, M.