An introduction to discrete mathematics, including combinatorics and graph theory. The necessary background tools in set theory, logic, recursion, relations, and functions are also included. Masters degree credit for Teacher Option Only.
Pre-Req: MS Teacher Option only
The class is aimed to give rigorous foundations to the basic concepts of Calculus such as limits of sequences and functions, continuity, Riemann integration. The main focus is given to rigorous proofs rather than computations. Tentative topics are: Real numbers (algebraic, order and distance structures); Archimedean property; Sequences and their limits. Bolzano-Weierstrass theorem; Cauchy sequences and completeness; Limit of a function; Continuity of a function at a point and on a set; Uniform continuity; Open and closed sets, idea of compactness, compactness of a closed interval; Sequences of functions, uniform convergence; Riemann integration.
Prerequisites: Calculus I-III or equivalent, Discrete Structures or equivalent.
The class is aimed to give a rigorous exposition of the measure theory and Lebesgue integration. These techniques are widely used in modern analysis, in particular, in mathematical probability. Tentative topics are: Families of subsets (Sigma-algebras, Rings, Semi-rings and Dynkin systems); Measure; Caratheodorys extension theorem; Measurable functions; Various types of convergence (almost everywhere, almost uniformly, in measure); Lebesgue integral; Fatou, monotone convergence and dominated convergence theorems; Product measures, Fubinis theorem; Signed measures; Absolute continuity and Radon-Nikodim theorem.
Prerequisites: 92.501 Real Analysis or equivalent
Development of number systems, including axiomatic and constructive treatment of the integers and the reals; sequences and series; functions of a real variable and their properties, including continuity, derivatives and integrals; functions of several real variables, including partial derivatives and multiple integration; differential equations and applications; metric spaces.
Masters degree credit for the Teacher Option only.
Pre-Req: MS Teacher Option only
This course provides a solid basis for further study in statistics and data analysis or in pattern recognition and operations research. It is especially appropriate for students with an undergraduate science or engineering major who have not had a rigorous calculus-based probability and statistics course. The course covers the topics in probability models, random variables, expected values, important discrete and continuous distributions, limit theorems, and basic problems of statistical inference: estimation and testing.
Explores the roles of mainframes, PCs and hand calculators in instruction, examine some of the available software, and consider their use in a variety of areas of secondary mathematics, such as algebra, geometry (Euclidean and analytic) probability and statistics, and introductory calculus.
Masters degree credit for Teacher Option Only.
Discusses complex numbers, functions of a complex variable, mappings, derivatives, analytic functions, elementary functions. Laurent series, residues and poles, contour integration.
Differentiation and integration of complex analytic functions. Cauchy's integral theorem and formula. Singularities and Laurent series. Theory of residues and applications. Harmonic functions. Conformal mapping.
Study of primes, congruences, number-theoretic functions, Diophantine approximation, quadratic forms and quadratic number fields. Additional topics as time permits.
The course combines theory with applications and covers both fundamental topics in statistical inference and their applications in data analysis. Discussions of the theoretical topics of statistical estimation and hypotheses testing will be complemented by analyzing simulated and real data sets. The course is taught at the computer lab equipped with MINITAB, SAS and other packages. Students will learn how statistical theory helps using statistical software, how to choose the right tool for the problem at hand and how to interpret the output.
Topics to be covered include point and interval estimation, hypotheses testing, maximum likelihood estimation, likelihood ratio and related tests, applications of statistical inference to commonly used statistical models, such as one-sample, two-sample and many-sample (ANOVA) models, linear regression models, goodness-of-fit tests and contingency tables, and elements of statistical quality control and experimental design. Time permitting, topics in nonparametric and robust statistics will also be covered.
Pre-requisite; 92.386, 92.509 or equivalent.
Pre-Req: 92.386 Prob & Stats I, and 92.509 Prob & Math Stats, or equivalent
Focuses on: mathematical resources, ability to use heuristics, the student's beliefs about the use of mathematics to solve problems, and the student's self-confidence as a problem solver. Effective strategies for incorporating problem solving in the curriculum will also be discussed.
Elementary group theory, groups, cosets, normal subgroups, quotient groups, isomorphisms, homomorphisms, applications.
Pre-Req: 92.221 Linear Algebra I
Discuss elementary rings an field theory, quotient rings and ideals, homorphism of rings, rings of polynomials, algebraic estensions, automorphisms of fields, separable extensions, and the Galolis Theory.
Pre-Req: 92.521 Abstract Algebra I
Sets and maps; vector spaces and linear maps, matrix of linear maps, solving systems of equations, scalar products and orthogonality, eigenvalues and applications. Masters degree credit for Teachers Option Only.
Pre-Req: MS Teacher Option only
Metric spaces, topological spaces, connectedness, compactness, the fundamental group, classifications of surfaces, Brouwer's fixed point theorem.
Provides a wide survey of topics related to secondary school geometry; axiomatic systems and Euclidean geometry; constructions in geometry; analytic geometry; and introduction to Noneuclidean geometry.
Pre-Req: MS Teacher Option only
Ordinary and partial differential equations; Fourier series and Fourier integrals; Laplace transform; matrix theory.
Pre-Req: 92.231 Calculus III and 92.234 Differential Equations or 92.236 Eng Differential Equations)
Vector analysis and vector calculus; Gauss, Green, and Stokes theorems; complex analysis; calculus of variations; special functions; orthogonal functions.
Pre-Req: 92.231 Calculus III and 92.234 Differential Equations or 92.236 Eng Differential Equations)
Examines ancient numeral systems, Babylonian and Egyptian mathematics, Pythagorean mathematics, duplication, trisection, and quadrature, Euclid's Elements and Greek mathematics after Euclid, Hindu and Arabian mathematics, European mathematics from 500 to 1600, origins of modern mathematics, analytic geometry, the history of calculus. Also covers the transition to the twentieth century and contemporary perspectives.
Pre-Req: 92.132 Calculus II
Introduction to partial differential equations in the plane and space with engineering applications. Solution of initial and boundary value problems. Complex variables and transform theory.
Representation of signals: Fourier analysis, fast Fourier transforms, orthogonal expansions. Transformation of signals: linear filters, modulation; band-limited signals; sampling; uncertainty principle; Windows and extrapolation.
Signal processing and Fourier analysis fundamentals in one and several variables, deterministic algorithms for image reconstruction with non-diffracting sources, daling wiht noisy data and (briefly) the problem of diffracting sources, physically realistic statistical models for the data, the use of statistical parameter estimation as a reconstruction paradigm, likelihood maximization, the
expectation maximization" (EM) iterative algorithm, applications of the EM algorithm to emission and transmission tomography, controlling noise through Bayesian "maximum a posteriori" estimation, and acceleration of convergence using incremental optimization or block-iterative methods.
Applications of mathematics to real life problems. Topics include dimensional analysis, population dynamics wave and heat propagation, traffic flow.
Pre-requisite: 92.132 Calculus II.
The first variational problem, necessary conditions. Euler's equation. Generalization to dependent and independent variables. Constraints and Lagrange multipliers. Application to dynamics and elasticity. Direct methods.
Introduction to time-frequency localization of signals; frames; windowed Fourier transforms; continuous and discrete wavelet transforms; time frequency sampling theorems; othonormal bases of wavelets; algebraic wavelet theory; applications to electrodynamics and optics.
The objective of this course is to give students an opportunity to learn
how to use a computer algebra system in the context of reviewing some of
the key mathematical topics that are used in the life sciences. The
first half of the course includes a review of mathematical topics
ranging from trigonometry through differential equations. A parallel
introduction to a computer algebra system is also included in the first
half. In the second half, students will study a mathematical topic such
as pattern recognition or models for growth and complete a project
using the computer algebra system. (UMassOnline)
Introduction and review of Taylor serries. Finding roots of F(x)=0. Numerical interpolation and extrapolation. Curve fitting and nonlinear best fits. Numerical differentiation and integration. Differential equations, initial and boundary value problems.
Pre-Req: 92.231 Calculus III and 92.234 Differential Equations or 92.236 Eng Differential Equations
Principles of Floating Point Arithmetic. The IEEE standard. Forward and Backward error analysis. Error estimation for basic procedures: summation, dot product, etc. Numerical Linear Algebra: Gauss-Jordan elimination, Cholesky decomposition. Elements of Fortran and LAPACK. Computing with integer arithmetic. Hermit Normal Form, Smith Normal Form, LLL-reduction. Discrete Fourier Transform and divide and conquer methods. Fast integer multiplication, fast matrix multiplication.
Overview of descriptive statistics, data analysis, probability of events, discrete random variables, continuous random variables, normal, binomial and other probability distributions, central limittheorem, survey sampling, estimation, hypothesis testing, regression, experimental design, analysis of categorical data, nonparametric statistics. Masters degree credit for Teachers Option Only.
Pre-Req: MS Teacher Option only
Opti8mization without calculus; geometric programming; convex sets and convex functions; review of linear algebra; linear programming and the simplex method; convex programming; iterative barrier-function methods; iterative penalty-function methods; iterative least-squares algorithms; iterative methods with positivity constraints; calculus of variations; applications to signal processing, medical imaging, game theory.
Pre-Req: 92.231 Calculus III and 92.234 Differential Equations or 92.236 Eng Differential Equations)
An introduction to creation and manipulation of databases and statistical analysis using SAS software. SAS is widely used in the pharmaceutical industry, medical research and other areas.
Introduces statistical methods for analyzing data obtained from lifetime testing of products. Statistical failure models, testing reliability hypotheses and accelerated life testing.
Topics: Basic Principles of Counting, Burnside's Lemma, Growth of Functions, Big-O, Big-Theta, Big-Omega notation, The Notion of Algorithm, Pseudocode, Graphs, Trees, Forests, Networks, Algorithms for Graphs and Networks, Graph Minors, Combinatorial Optimization, Software Implementations of Graphs, Software for Graph Theory and its Applications.
Pre-requisites: 92.321 Discrete I or 91.102 Computing II
Terminology, theorems, algorithms, and applications of graph theory. Trees, circuits, and connectivity. Hamiltonian and Eulerian graphs. Shortest routes, matching, network flows. Covering, coloring, Ramsey theory.
Building models for discrete time series and dynamic systems and their use in forecasting and control. Stationary and non-stationary time series models. Box-Jenkins (ARMA) and other techniques.
Markov chains and processes, random walks, stationary, independent increments, and Poisson processes. Ergodicity. Examples (e.g., diffusion, queuing theory, etc.).
This is a course in mathematical probability that gives rigorous proofs to various limit theorems in probability (zero-one laws, laws of large numbers, central limit theorems) that, in particular, constitute a basis for most of the statistical techniques. Tentative topics are: Sigma-algebras of random events, probability measures; Random variables and their distributions, moments; Independent events and sigma-algebras, independent random variables; infinite products of probability spaces; Zero-One Laws and Laws of Large Numbers; Characteristic functions and their properties; Weak convergence of measures and central Limit Theorem; Radon-Nikodim theorem and conditional expectations.
Prerequisites: 92.502 Measure and Integration or equivalent
Pre-req: 92.502 Real Analysis II or Instructor permission
Random variables, densities, joint and conditional distributions, expectations, variance, estimation, sufficiency and completeness, hypothesis testing, limiting distributions.
Simple random sampling, systematic sampling, stratified random sampling, multistage cluster sampling, regression estimation, ratio estimation, effect of costs on sample allocation and nonsampling errors.
Introduction to statistical methods useful in quality assurance. Description of control charts for attributes and variables, process-capability analysis and acceptance sampling.
Model building via linear regression models. Method of least squares, theory and practice. Checking for adequacy of a model, examination of residuals, checking outliers. Practical hand on experience in linear model building on real data sets. Prerequisite: Probability, Biostatistics, or Statistics for Engineering and Science.
Nonlinear model building via the method of least squares. Discriminant and factor analysis, principal components, profile analysis, canonical correlation, cluster analysis. Experience on real data sets.
How to design, carry out, and analyze experiments. Randomized block designs, randomization, blocking, matching, analysis of variance and covariance, control of extraneous variables.
Shannon theory including information measure and transmission rates and capacities. Elements of coding theory.
Product sigma-algebras and product measures. Fubini's Theorem; Probability spaces, random variables and their distribuions, moments; Independent events and sigma-algebras, independent random variables; Products and sums of independent random variables, Infinite products of probability spaces; Zero-One Laws and Laws of large numers; Characteristic functions and their properties, uniqueness; weak convergence of measures and central Limit Theorem; More limit theorems; Absolutely continuous and mutually singular measures, Radon-Nikodim Theeorem; Conditional expectations.
Intended to satisfy individual student needs. Topics include various fields of mathematics.
Intended to satisfy individual student needs. Topics include various fields of mathematics.
Advanced topics in various fields of mathematics and related fields. Since topical coverage varies from term to term, a student may be allowed to receive credit more than once for this course.
