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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.
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.
Metric spaces, completeness, contractions, compactness, the Arzela-Ascoli theorem, Picard's theorem, Weierstrass's theorem. Banach spaces, dual spaces, weak v's strong convergence. Hilbert spaces, orthogonal sequences, weak sequential compactness, compact self-adjoint operators and their spectra, application to Sturm-Liouville theory.
Pre-req: MATH.4030 Real Analysis.
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 computers and calculators in instruction, examines some of the available software, and considers their use in a variety of areas of school mathematics, such as algebra, geometry (Euclidean and analytic) probability and statistics, and introductory calculus. Mathematics Masters degree credit for Teacher Option Only.
A first course in theory of analytic functions of one complex variable: complex differentiability and the Cauchy-Riemann equations, Cauchy Integral Theorem and Cauchy Integral Formula, Taylor and Laurent series, zeroes of analytic functions and uniqueness, the maximum modulus principle, isolated singularities and residues. Applications.
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.
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.
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.
Metric spaces, topological spaces, connectedness, compactness, the fundamental group, classifications of surfaces, Brouwer's fixed point theorem.
Pre-req: MATH 4030 Mathematical Analysis or MATH 5010 Real Analysis
This course is designed for current and prospective geometry teachers. In addition to the development of Euclidean geometry, students will become familiar with geometry applications in Geometer's Sketchpad software, and to a lesser degree with other geometry software applications including Geogebra, and Cabri. There will be an introduction to spherical and hyperbolic geometry and triangle measurements will be computed for each. Calculus based derivations of area and volume for surfaces and solids will be generated and related to Euclidean geometry topics.
Differential geometry involving curves and surfaces in 3-space. Curvature, torsion, Frenet equations, intrinsic equations, involutes and evolutes.
Ordinary and partial differential equations; Fourier series and Fourier integrals; Laplace transform; matrix theory.
Vector analysis and vector calculus; Gauss, Green, and Stokes theorems; complex analysis; calculus of variations; special functions; orthogonal functions.
Historical perspectives: Euclid's synthetic geometry, Descartes' analytic geometry, attempts to prove parallel postulate,emergence of non-Euclidean geometry's, axiomatic development of geometry, Klein's Erlanger Program; projective,affine,and metric geometries;non-Euclidean geometry's; foundations of geometry; algebraic geometry; finite geometry. Requires knowledge of linear algebra,abstract algebra for groups and fields including Galois fields, some familiarity with propositions and set-theoretic topology as covered in a course on Discrete Mathematics.
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.
Existence, uniqueness, and smoothness of solutions. The Poincaré-Bendixson theory. The neighborhoods of critical points and closed orbits. Liapunov stability. Linear and perturbed linear systems.
Linear and quasilinear first order PDE. The method of characteristics. Conservation laws and propagation of shocks. Basic theory for three classical equations of mathematical physics (in all spatial dimensions): the wave equation, the heat/diffusion equation, the Laplace/Poisson equation. Initial value problems - solution formulas. Fundamental solutions. Green's functions. Eigenfunction expansion method for initial-boundary and boundary value problems.
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.
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.
Pre-Req: MATH 2340 Differential Equations or MATH 2360 Eng Differential Equations.
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 series. 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.
Use of iterative algorithms to find exact or approximate constrained solutions to large, and often spares, systems of linear equations, and on applications, such as medical imaging, in which such problems arise. Maximization of likelihood and entropy. Emphasis on exploiting sparseness, accelerating convergence, and stabilizing calculations in the presence of noise. Block-iterative methods and bounds for singular values will be included. Basic results in matrix theory presented as needed.
Introduction to functions beyond those studied in calculus and which arise in applied mathematics, including gamma, beta, elliptic, Bessel, orthogonal polynomials ... Asymptotic approximation will be introduced.
There is currently no description available for this course.
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.
Optimization 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.
This is a methods course focusing on the applications of statistics using R programming language. Topics include: Study designs, review of inference and regression, categorical data, logistic regression, rates and proportions, and nonparametric methods. Additional topics may be considered if time permits. Only on of 92.575(R) and 92.576(SAS) may be applied toward a Masters degree in Mathematics.
Pre-req: MATH 3850 Applied Statistics and MATH 3860 Probability and Statistics I, or equivalent.
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. Cannot be used as a Math Elective.
Topics in nonasymptotic direct computational methods for statistical inference in data mining. Background in probability and statistics required.
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 course presents the mathematical foundations of Probability Theory, including the concepts of Probability Space and random variable. Various types of convergence of sequences and measurable functions will be introduced, and precise statements and proofs of the probability limit theorems (Law of Large Numbers, Central Limit Theorems, etc.) will be given. Theory of measure and Lebesgue integration will be introduced. If time permits, conditional probabilities will be discussed.
Random variables, densities, joint and conditional distributions, expectations, variance, estimation, sufficiency and completeness, hypothesis testing, limiting distributions.
Overview of quality and managing quality, Define Measure Analyze Improve Control (DMAIC), the six sigma approach to quality, visual representation of data, Pareto charts, histograms, process capability vs specification (process) limits, t-tests, ANOVA, and other statistical hypothesis testing in quality, normal probability plots, control charts, measurement system analysis, application of regression analysis to manufacturing and/or design, Minitab.
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.
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.