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 | | | Department of Mathematics | | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| | Courses for 2007/2008In the Spring of 2008, the department will offer Math-503 (Mathematical Statistics), Math-504 (Numerical Methods), Math-604 (Stochastic Processes), and Math-657 (Categorical Data Analysis). Math-701 (Statistical Consulting Practicum), Math-702(Applied Mathematics Clinic). Students who wish to complete theirinternship requirement in the Spring should enroll in Math-703 (Internship). During the Summer of 2008, students can take Math-610(Combinatorics and Combinatorial Optimization). The course will beoffered during the special session, starting during the week of May 19and continuing for eight weeks, and will meet twice per week in theearly evening. The internship will also be offered during thesummer. In addition, non-mathematical electives are available to graduate students. The Biostatistics and Epidemiology graduate program will offer BIST-512 (Statistical Modeling I), BIST-513 (Statistical Modeling II), BIST-531 (Pattern recognition), and BIST-541 (Principles of Epidemiology). These courses count as electives for the Mathematics and Statistics program. BIST-541 is a non-math elective. Students who wish to take a course in the BIST program should always contact the BIST instructor first. Spring 2008 Schedule at a Glance:
In the Fall of 2007, the department is offering Math-501 (Probability Theory), Math-502 (Deterministic Models), Math-603 (Signal Processing), and Math-651 (Regression and Generalized Linear Models). In addition, Math-701 (Statistical Consulting Practicum) and Math-702 (Applied Mathematics Clinic) is being offered. Math-501: Probability Theory and Applications.Fall. This is an advanced introduction to probabilitytheory. The notions of probability measure, random variable,independence, and conditional probability will be presented, along withthe frequentist interpretation of probability embodied in the Law ofLarge Numbers. Basic properties of discrete and continuous randomvariables will be developed, including means, variances, c.d.f.s,functions of random variables, generating functions, and Chebyshev’sinequality. Joint distributions will be considered, along withcovariances, random vectors, and changes of variable. The course willalso include an introduction to stochastic processes of Gaussian andPoisson type, and a discussion of different notions of convergence,culminating in the Central Limit Theorem. Applications will be given inpopulation studies, information theory, and Bayesian inference. The course will use the book by Neil Weiss, A Course in Probability, Pearson Addison-Wesley, First Edition, 2006.Available at Amazon.com for about $80 (new) or $60 (used). Someprevious exposure to elementary probability theory and random variablesis desirable. Instructor: David Caraballo Time and Room: Tuesdays 6:15 - 8:15, St. Mary's Hall 107. Math-502: Deterministic Mathematical Models.Fall. This is a course on differential and difference equations with anemphasis on derivation and analysis of models of physical phenomena.The course will begin with a brief review of matrix techniquesincluding eigenvalues. As a preliminary to deriving models, a treatmentof dimensional analysis will be given, leading to the Buckinghampi-theorem. The course will then introduce first order systems ofordinary differential and difference equations with constantcoefficients. Analytical techniques such as linearization, scaling,perturbation, characteristics and variational methods will be studied.Applications of this material will include age-structured populationsand single-species non-linear models. The course will include anintroduction to partial differential equations, with applications toconservation laws, shock formation and traffic congestion. Textbook:David A Logan, "Applied Mathematics." Either the second or thirdedition is suitable. Either book may be purchased new or used from anysource such as Amazon.com, eBay, etc. For the second edition go toamazon.com, select Books and enter 'David A Logan, Applied Mathematics,Second Edition' in the search field to find this information. Price:$86.53. Used & new available from $40.00. For the third editionenter David A Logan, Applied Mathematics in the search field and find aprice of $110, with other new and used from about $80 Instructor: Andrew Vogt Time and Room: Thursdays 6:15 - 8:45, St. Mary's Hall 126. Math-503: Mathematical Statistics.Spring. This is a first course in the mathematical theory ofstatistical inference. The emphasis is on frequentist methods, withappropriate attention also to Bayesian methods. Topics includeprinciples of data reduction (sufficiency, completeness), constructionof point estimates (method of moments, maximum likelihood, Bayesestimators), criteria for point estimation (mean squared error, bias,consistency), construction of hypothesis tests (likelihood ratio, mostpowerful and uniformly most powerful tests, Bayesian tests), someasymptotic properties of point estimators, criteria for hypothesistests (error probabilities and power, power functions), asymptotics ofsome large sample tests, construction of interval estimates, elementsof decision theory and applications to statistical inference (Bayesrules, minimax), elements of the analysis of variance (one-way ANOVA,F-test, contrasts), elements of linear regression (least squares, tests Instructor: Kimberley Sellers. Time and Room: Tuesdays 6:15-8:45pm, St. Mary's 110 Math-504: Numerical Methods.Spring. This course concerns the design and analysis of computationalalgorithms and techniques for solving a variety of mathematicalproblems. This course will provide the insight and theory behindscientific and engineering computing. The topics covered in this courseincludes solving nonlinear equations; numerical linear algebra andsolving systems of linear equations; approximation and interpolationusing Lagrange polynomials, least square polynomials and splines;numerical differentiation and integration; solving ordinarydifferential equations; and simulation of stochastic processes. We will also discuss issues associated with computer arithmetic, suchas floating-point number representations, roundoff errors, andstability of computations, as well as error analysis and convergence ofnumerical schemes. Studentswho take this course are expected to have background in multivariablecalculus, linear algebra and differential equations. Some knowledge inone computer programming language is required. Themain text for the course will be Numerical Analysis by Tim Sauer.Optional: Charles Van Loan, Introduction to Scientific Computing. Instructor: David Gilsinn Time and Room: Thursdays 6:15-8:45pm, St. Mary's 333 Math-603: Signal Processing.Fall. This course concerns the analysis and implementation ofcomputational algorithms for the processing of digital signals.Students will learn the mathematics governing the theory and thealgorithms involved, and will learn the details of the algorithmsthrough writing programs which implement the algorithms. The topicscovered include discrete-time signals and systems, the z-transform,frequency analysis of signals and systems, sampling and reconstruction,the fast Fourier transform, digital filters, multi-rate signalprocessing, linear prediction and optimum linear filters, and powerspectrum estimation. Applications to speech processing, imageprocessing and data compression will also be discussed. Instructor: George Benke Time and Room: Mondays 6:15 - 8:45, St.Mary's 107. Math-604: Stochastic Processes.Spring. This course is an introduction to stochastic processes withoutthe use of measure theory. We will start by considering discretetime stochastic processes by covering Markov chains andmartingales. We will then consider continuous time stochasticprocesses by covering Poisson processes, general Markov processes andBrownian motion. The course will emphasize problem solving in thesense that we will introduce and study various theorems that allow usto solve problems of interest. Computer simulation of stochasticprocesses through MATLAB will be part of the course. Prerequisites: Math-501, background in ordinary differential equations. Textbook: Rick Durrett, Essentials of Stochastic Processes. Instructor: Sivan Rottenstreich Time and Room: Wednesdays 6:15-8:45pm, St. Mary's 110 Math-605: Introduction to Financial Mathematics. Spring. Not offered 2007/2008.This is a course on mathematical finance, emphasizing mathematical models and techniques for pricing financial derivativeinstruments. Topics covered include financial markets of stocks, bonds,futures and options; present value analysis; Brownian motion and Ito'sformula; asset price random walk; the heat equation; the Black-Scholesoption pricing equation and its conversion to the heat equation;European option price as the solution of initial value problem ofBlack-Scholes equation; American option as a free boundary valueproblem; binomial trees and other numerical valuation methods; exoticoptions and path-dependent options; term structure and interest ratederivatives. The textbook is "A Course in Financial Calculus" by Alison Etheridge, Math-610: Combinatorics and Combinatorial Optimization. Summer. Thecourse will cover the following topics, most of which are in thetextbook. For the others, there will be some supplementary notes.
Textbook:Combinatorial Optimization, by Cook, Cunningham, Pulleyblank, andSchrijver, Wiley-Interscience, 1997 (1st ed.). Instructor: Paul Kainen Time and Room: TBA Math-651: Regression Methods and Generalized Linear Models.Fall. Simple and multiple regression, inference and prediction, modelbuilding and diagnostics, analysis of variance (ANOVA), analysis ofcovariance (ANCOVA), generalized linear models, and other extensions astime permits (e.g. mixed models, nonlinear regression). This coursecovers both theoretical and applied aspects of regressionanalysis. Examples and illustrations will use SAS and/or R. Textbook: Kutner, Nachtsheim, Li, Applied Linear Statistical Models (5/E). Instructor: Ali Arab Time and Room: Wednesdays 6:15 - 8:45, St. Mary's G40. Math-654: Computer-intensive Statistical Methods. Not offered in Fall or Spring 2007/2008. Stochastic simulation methods are playing increasingly important rolesin science, business, and engineering. The course introduces the basictheory and some applications of stochastic simulation techniques andthen focuses on modern methods of statistical inference for problemsthat are difficult to treat with conventional parametric or asymptoticmethods. Topics: Stochastic simulation as an experiment. Random numbergeneration. Elements of Monte Carlo simulation: rejection, importancesampling, weighting. Simulation of and random sampling from data.Empirical cumulative distribution function. Nonparametric tests:Permutation, rank, and randomization tests. Jackknife procedures.Bootstrap methods for bias reduction, variance estimation, andconfidence intervals. Parametric and nonparametric estimation ofprobability density functions. Prerequisites: Math-503 (Mathematical Statistics). Textbook: James Gentle, Elements of Computational Statistics, Springer 2002, 2nd printing 2005. Math-656: Data Exploration and Data Mining. Fall. Not offered 2007/2008. Hugevolumes of data are constantly being generated by businesses, inscience, in telecommunications, and elsewhere, doubling the amount ofinformation available in the world roughly every nine months. Thiscourse presents an introduction to computer-based methods for exploringlarge data sets and discovering patterns in them. It focuses onstatistical aspects and computational algorithms for numerical andcategorical data. After a brief review of graphical explorationmethods, the course discusses linear and nonlinear methods of featureextraction and dimension reduction (variable selection, singular valuedecomposition, factor analysis, multidimensional scaling, artificialneural networks), data tours (grand tour, projection pursuit),clustering methods, model-based approaches such as finite mixturemethods and expectation maximization, and multivariate visualizationtechniques. Statistical techniques such as linear regression withmodel selection, logistic regression and decision trees are covered inthe last third of the course. The course will also discuss methods forassessing the quality of results (cross validation, bootstrap) and forcombining results (bagging, boosting). Database aspects of data miningwill receive less emphasis in this course. The course will use Matlaband WEKA 3.4. Prerequisites: Linear algebra (matrix methods), some previous experience with elementary statistics and probability, basic knowledge of Matlab. Textbooks: IanH. Witten, Eibe Frank, Data Mining: Practical MachineLearning Tools and Techniques, 2e, Morgan Kaufmann (2005), ISBN:0120884070 , and Wendy Martinez, Angel Martinez, Exploratory DataAnalysis with MATLAB, Chapman & Hall/CRC (2004), ISBN: 1584883669.The book by Witten and Frank is available for about $40, and the bookby Martinez & Martinez costs about $80. Matlab and WEKA will beinstalled in the classroom and in the departmental computer lab. Astudent version of Matlab can be purchased for $100, and WEKA is freeopen source software. Math 657: Categorical Data Analysis. Spring. This course deals with statistical models for the analysis of categorical data. Topics to be covered include inference for contingency tables, generalized linear models with emphasis on logistic regression and loglinear models, and models for clustered/repeated measures. The goal of the course is not to memorize formulae, but to understand and apply statistical concepts and techniques to real data. Most examples will be illustrated using SAS. Students are free to use the software of their preference. Textbook: Agresti, A. (2002) Categorical Data Analysis, 2nd Ed. Wiley. Optional Textbook: Stokes M.E., Davis C.S. and Koch G.G. (2000) Categorical Data Analysis using the SAS System, 2nd Ed. Wiley. Time and Room: Mondays 6:15 - 8:45, Reiss 262. Math-701: Statistical Consulting Practicum. Fall and Spring. Students will conduct statistical consulting and data analysis activities,guided by departmental faculty. Tasks will include the design ofresearch protocols, the formulation of statistical models,recommendations regarding appropriate methodologies, data analysis, andinterpretation of results. An important component will be thecommunication with clients. Thus students will be expected to co-authorwritten reports and to give oral presentations to clients. Prerequisites: Math-501 and an upper division undergraduate course inStatistics. Some familiarity with a statistical software package suchas SAS or Minitab. Textbook: We will be using statistical reference texts and online sources as needed. Instructor: Ali Arab Timeand Room: To be determined (weekday afternoons in St. Mary's). Studentswill be required to be present for a total of three hours per week fortwo credits. Math-702: Applied Mathematics Clinic. Fall and Spring. Studentswill work on science and engineering problems that will typicallyrequire a sustained effort in modeling, analysis, computation, andexposition, over the course of one or two semesters. Students will workin teams and are expected to co-author interim and final reports and togive oral presentations to clients. Prerequisites:Math-502 and an upper division undergraduate course in numericalmethods. Some familiarity with computational software such asMathematica and/or Matlab. Textbook: Reference texts and online resources. Instructor: TBA Timeand Room: To be determined (weekday late afternoon or evening in St.Mary's). Students will be required to be present for a total of threehours per week for two credits. Math-703: Internship. Fall, Spring, Summer. Industrialinternship for students in the MS Degree Program in Mathematics andStatistics. Please consult the student handbook for details, or contact the program. Instructor: STAFF Bridge CoursesThese one-credit courses will be offered through Georgetown's School of Continuing Studiesduring the summer session, typically between early June and mid Augustat times to be determined and on an as-needed basis. In 2006, Math-403 and Math-405 were offered. If you are interested in a bridge course during this summer, please contact the program. Bridgecourse credits may carry graduate credit towards other degrees, butthey do not count towards the MS degree in Mathematics and Statistics. Math-401: Matrix Methods.Matrix algebra, systems of linear equations, eigenvalues and singularvalues, 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 ofcombinatorics, elements of graph theory, modular arithmetic. Math-404: Elements of Statistics.Randomness and variability, graphical data exploration, descriptivestatistics for univariate data, normal and binomial distribution,getting started with SAS. Math-405: Computer tools.Getting started with Matlab, Mathematica, SAS, technical typesettingwith TeX, technical presentations with PowerPoint, mathematicalsoftware on the Internet. | | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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