### Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)

Completeness and compactness theorems for propositional and predicate calculi. A continuation of recursion theory, set theory, proof theory, model theory. Topics will vary from year to year in areas of mathematics and their development. Topics may include the evolution of mathematics from the Babylonian period to the eighteenth century using original sources, a history of the foundations of mathematics and the development of modern mathematics. Topics to be chosen in areas of applied mathematics and mathematical aspects of computer science. May be taken for credit two times with different topics.

Analysis of numerical methods for linear algebraic systems and least squares problems. Orthogonalization methods. Ill conditioned problems. Eigenvalue and singular value computations. Knowledge of programming recommended. MATH B. Rounding and discretization errors. Calculation of roots of polynomials and nonlinear equations. Approximation of functions.

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MATH C. Numerical differentiation and integration. Ordinary differential equations and their numerical solution. Basic existence and stability theory. Difference equations. Boundary value problems.

Functions of Matrices, Lecture 1

MATH A. Introduction to Numerical Optimization: Linear Programming 4. Linear optimization and applications. Linear programming, the simplex method, duality.

### Course Catalogue

Selected topics from integer programming, network flows, transportation problems, inventory problems, and other applications. Three lectures, one recitation. Introduction to Numerical Optimization: Nonlinear Programming 4. Convergence of sequences in Rn, multivariate Taylor series. Bisection and related methods for nonlinear equations in one variable. Equality-constrained optimization, Kuhn-Tucker theorem. Inequality-constrained optimization.

### Department of Mathematics

Introduction to convexity: convex sets, convex functions; geometry of hyperplanes; support functions for convex sets; hyperplanes and support vector machines. Linear and quadratic programming: optimality conditions; duality; primal and dual forms of linear support vector machines; active-set methods; interior methods.

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Convex constrained optimization: optimality conditions; convex programming; Lagrangian relaxation; the method of multipliers; the alternating direction method of multipliers; minimizing combinations of norms. Conjoined with MATH Floating point arithmetic, direct and iterative solution of linear equations, iterative solution of nonlinear equations, optimization, approximation theory, interpolation, quadrature, numerical methods for initial and boundary value problems in ordinary differential equations.

## Nick Higham - Functions of Matrices: Theory and Computation

Mathematical background for working with partial differential equations. Survey of finite difference, finite element, and other numerical methods for the solution of elliptic, parabolic, and hyperbolic partial differential equations. Formerly MATH Graduate students do an extra paper, project, or presentation, per instructor. Mathematical models of physical systems arising in science and engineering, good models and well-posedness, numerical and other approximation techniques, solution algorithms for linear and nonlinear approximation problems, scientific visualizations, scientific software design and engineering, project-oriented.

Graduate students will do an extra paper, project, or presentation per instructor. Probability spaces, random variables, independence, conditional probability, distribution, expectation, variance, joint distributions, central limit theorem. Prior or concurrent enrollment in MATH is highly recommended.

Random vectors, multivariate densities, covariance matrix, multivariate normal distribution. Random walk, Poisson process. Other topics if time permits.

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4. Markov chains in discrete and continuous time, random walk, recurrent events. If time permits, topics chosen from stationary normal processes, branching processes, queuing theory.

Multivariate distribution, functions of random variables, distributions related to normal. Parameter estimation, method of moments, maximum likelihood. Estimator accuracy and confidence intervals. Hypothesis testing, type I and type II errors, power, one-sample t-test.

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## Math, Statistics, and Computational Science

Hypothesis testing. Linear models, regression, and analysis of variance.

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5. Goodness of fit tests. Nonparametric statistics. Prior enrollment in MATH is highly recommended. Topics covered may include the following: classical rank test, rank correlations, permutation tests, distribution free testing, efficiency, confidence intervals, nonparametric regression and density estimation, resampling techniques bootstrap, jackknife, etc. Statistical learning refers to a set of tools for modeling and understanding complex data sets.

It uses developments in optimization, computer science, and in particular machine learning. This encompasses many methods such as dimensionality reduction, sparse representations, variable selection, classification, boosting, bagging, support vector machines, and machine learning.

Analysis of trends and seasonal effects, autoregressive and moving averages models, forecasting, informal introduction to spectral analysis. Design of sampling surveys: simple, stratified, systematic, cluster, network surveys. Sources of bias in surveys. Estimators and confidence intervals based on unequal probability sampling. Design and analysis of experiments: block, factorial, crossover, matched-pairs designs. Analysis of variance, re-randomization, and multiple comparisons. Introduction to probability.

Discrete and continuous random variables—binomial, Poisson and Gaussian distributions. Central limit theorem.