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Courses for 2006
In the Fall of 2006, the department will offer Math-501 (Probability Theory), Math-502 (Deterministic Models), and Math-656 (Data Mining) .
In the Spring of 2007, Math-503 (Mathematical Statistics), Math-504 (Numerical Methods) and Math-605 (Intro to Financial Mathematics) will be offered.
In addition, non-mathematical electives and one credit bridge courses are available.
Math-501: Probability Theory and Applications. Fall. This is an advanced introduction to probability theory. The notions of probability measure, random variable, independence, and conditional probability will be presented, along with the frequentist interpretation of probability embodied in the Law of Large Numbers. Basic properties of discrete and continuous random variables will be developed, including means, variances, c.d.f.s, functions of random variables, generating functions, and Chebyshevs inequality. Joint distributions will be considered, along with covariances, random vectors, and changes of variable. The course will also include an introduction to stochastic processes of Gaussian and Poisson type, and a discussion of different notions of convergence, culminating in the Central Limit Theorem. Applications will be given in population studies, information theory, and Bayesian inference.
The course will use two texts: Casella/Berger, Statistical Inference, Chapters 1-5; and Brmaud, An Introduction to Probabilistic Modeling. Some previous exposure to elementary probability theory and random variables is desirable.
Instructor, room and time are to be announced.
Math-502: Deterministic Mathematical Models. Fall. This is a course on differential and difference equations with an emphasis on derivation and analysis of models of physical phenomena. The course will begin with a brief review of matrix techniques including eigenvalues. As a preliminary to deriving models, a treatment of dimensional analysis will be given, leading to the Buckingham pi-theorem. The course will then introduce first order systems of ordinary differential and difference equations with constant coefficients. Analytical techniques such as linearization, scaling, perturbation, characteristics and variational methods will be studied. Applications of this material will include age-structured populations and single-species non-linear models. The course will include an introduction to partial differential equations, with applications to conservation laws, shock formation and traffic congestion.
The primary textbook will be Applied Mathematics, 2nd Ed, David A Logan, John Wiley and Sons, 1997. Material from the books Mathematical Models, R. Haberman, Prentice-Hall, 1977, and Nonlinear Partial Differential Equations for Scientists and Engineers, L. Debnath, Birkhauser, 1997, will be used. Students will be required to purchase the Logan text.
Instructor, room and time are to be announced.
Math-503: Mathematical Statistics. Spring. This is a first course in the mathematical theory of statistical inference. The emphasis is on frequentist methods, with appropriate attention also to Bayesian methods. Statistical software (SAS) will be used in the second half of the course. Topics include principles of data reduction (sufficiency and sufficient statistics, likelihood, invariance), construction of point estimates (method of moments, maximum likelihood, Bayes estimators), criteria for point estimation (mean squared error, unbiasedness, consistency), construction of hypothesis tests (likelihood ratio, invariance, Bayesian tests), some asymptotic properties of point estimators, criteria for hypothesis tests (error probabilities and power, most powerful tests, bias), asymptotics of some large sample tests, construction of interval estimates (using a test statistic, pivotal quantities, Bayesian intervals, invariance), criteria for interval estimates (coverage probability, optimality), elements of decision theory and applications to statistical inference (Bayes rules, minimax), elements of the analysis of variance (one-way ANOVA, F-test, contrasts), elements of linear regression (least squares, tests for model parameters, pointwise and simultaneous estimation and prediction. If time permits: More on asymptotic of estimators, exact and approximate tests for contingency tables, some nonparametric tests (sign test, rank sum test), more on ANOVA.
The course will use Casella/Berger, Statistical Inference, chapters 6-12, or Bickel/Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol I (2nd Edition), starting at chapter 2.
Prerequisites: Calculus of one and several variables, some linear algebra (matrix algebra), Math-501 (Probability Theory).
Math-504: Numerical Methods. Spring. This course concerns the design and analysis of computational algorithms and techniques for solving a variety of mathematical problems. This course will provide the insight and theory behind scientific and engineering computing. The topics covered in this course includes solving nonlinear equations; numerical linear algebra and solving systems of linear equations; approximation and interpolation using Lagrange polynomials, least square polynomials and splines; numerical differentiation and integration; solving ordinary differential equations; and simulation of stochastic processes. We will also discuss issues associated with computer arithmetic, such as floating-point number representations, roundoff errors, and stability of computations, as well as error analysis and convergence of numerical schemes.
Students who take this course are expected to have background in multivariable calculus, linear algebra and differential equations. Some knowledge in one computer programming language is required.
The main text for the course will be Numerical Analysis: Mathematics of Scientific Computing and E. Ward Cheney.
Math-605: Introduction to Financial Mathematics. Spring. This is a course on mathematical finance, emphasizing mathematical models and techniques for pricing financial derivative instruments. Topics covered include financial markets of stocks, bonds, futures and options; present value analysis; Brownian motion and Ito's formula; asset price random walk; the heat equation; the Black-Scholes option pricing equation and its conversion to the heat equation; European option price as the solution of initial value problem of Black-Scholes equation; American option as a free boundary value problem; binomial trees and other numerical valuation methods; exotic options and path-dependent options; term structure and interest rate derivatives.
Textbook: Paul Wilmott, Paul Wilmott Introduces Quantitative Finance, John Wiley 2001.
Math-656: Data Exploration and Data Mining. Fall. Huge volumes of data are constantly being generated by businesses, in science, in telecommunications, and elsewhere, doubling the amount of information available in the world roughly every nine months. This course presents an introduction to computer-based methods for exploring large data sets and discovering patterns in them. It focuses on statistical aspects and computational algorithms for numerical and categorical data. After a brief review of graphical exploration methods, the course discusses linear and nonlinear methods of feature extraction and dimension reduction (variable selection, singular value decomposition, factor analysis, multidimensional scaling, artificial neural networks), data tours (grand tour, projection pursuit), clustering methods, model-based approaches such as finite mixture methods and expectation maximization, and multivariate visualization techniques. Statistical techniques such as linear regression with model selection, logistic regression and decision trees are covered in the last third of the course. The course will also discuss methods for assessing the quality of results (cross validation, bootstrap) and for combining results (bagging, boosting). Database and machine learning aspects of data mining will receive less emphasis in this course. The course will use Matlab throughout and SAS Enterprise Miner in the last third.
Prerequisites: Linear algebra (matrix methods), some previous experience with eementary statistics and probability, basic knowledge of Matlab.
Textbooks: Wendy Martinez, Exploratory Data Analysis with Matlab, Chapman & Hall 2004. George Fernandez, Data Mining Using SAS Applications.
These one-credit courses will be offered through Georgetown's School of Continuing Studies during the second summer session (July 10 - August 11, 2006), based on an as-needed basis. Actual meeting times are subject to agreement.
Bridge course credits may carry graduate credit towards other degrees, but they do not count towards the MS degree in Mathematics and Statistics.
Math-401: Matrix Methods. Matrix algebra, systems of linear equations, eigenvalues and singular values, matrix factorization.
Math-402: Methods of Analysis. Review of differential calculus of one variable, partial derivatives, constrained and unconstrained extrema, power series and Taylor series, elementary differential equations.
Math-403: Methods of Discrete Mathematics. Set theory, formal logic and methods of proof, elements of combinatorics, elements of graph theory, modular arithmetic.
Math-404: Elements of Statistics. Randomness and variability, graphical data exploration, descriptive statistics for univariate data, normal and binomial distribution, getting started with SAS.
Math-405: Computer tools. Getting started with Matlab, Mathematica, SAS, technical typesetting with TeX, technical presentations with PowerPoint, mathematical software on the Internet.