Mathematics

This course covers basic mathematical logic such as propositional logic and first order (predicate) logic by studying the notions of truth, satisfaction, model, proof and Turing machine. Goedel's completeness theorem is presented and his incompleteness theorems are introduced. The course studies the completeness theorem for the first-order logic using Henkin's construction method. As a consequence compactness theorem is presented and Lowenheim-Skolem theorem as an application is studied. Turing machine, the theoretical background of the contemporary digital computer design, is introduced and compared with Goedel’s incompleteness theorems.

This is a course in single-variable calculus. It introduces precise definitions of limit, continuity, derivative, and the Riemann integral. It covers computational techniques and applications of differentiation and integration. This course concludes with an introduction to first order differential equations. Major topics include functions; limit and continuity; derivative; Intermediate Value Theorem; chain rule; implicit differentiation; higher derivatives; Mean Value Theorem; Riemann integral; Fundamental Theorem of Calculus; elementary transcendental functions and their inverses; techniques of integration; computation of area, volume and arc length using definite integrals; and first order differential equations.

Students learn how to apply various standard methods (separation of variables, integrating factors, reduction of order, undetermined coefficients) to solve certain types of differential equations (separable, first-order linear, linear with constant coefficients); give examples of differential equations for which either existence or uniqueness of solutions fails; compute the exponential of a square matrix; and use either linearization or the Lyapunov theorems to check the stability of critical points for a given autonomous system.

This course examines numerical methods and statistics essential in a wide range of engineering disciplines. Numerical methods covers computing with real numbers, numerical differentiation, integration, interpolation and curve fitting (regression analysis), solution of linear and nonlinear algebraic equations, matrix operations and applications to solution of systems of linear equations, elimination and tri-diagonal matrix algorithms, and an introduction to numerical solution of ordinary and partial differential equations. Statistics covers exploratory data analysis, probability and distribution theory including the Binomial, Poisson and Normal distributions, large sample theory including the Central Limit Theorem, elements of statistical inference including estimation, confidence intervals and hypothesis testing, one sample and two-sample t-tests and F-tests, simple and multiple linear regression and analysis of variance and statistical quality control.

This course aims at providing the basic theoretical and applied tools for a rigorous statistical analysis. Specifically, the course focuses on techniques to summarize and visualize data of different types and their possible relations, as well as on basic sampling and inferential procedures, and on the assessment of the risk associated to extrapolation and inference. In particular, students learn how to extract information from data and how to assess the reliability of such information. The course covers the following topics: collection, management, and summary of data using frequency distributions, graphical representations, and summaries; study of the relationship between two variables; statistical inference and sampling variability; theory of point estimation and confidence intervals; hypothesis testing; and simple and multiple regression models. All the descriptive and inferential tools introduced during the course are applied to data using the statistical software R - and in particular the integrated development environment (IDE) RStudio. Prerequisites: understanding of the concepts of probability theory and random variables.

This course covers the methods of reasoning; formalization and deduction in logic of propositions and first order; combinatorics; graphs and trees.

In this course, students state and prove some standard theorems in number theory, use standard theorems to solve problems in number theory including some classes of Diophantine equations, and learn to use the following: divisibility and factorization of integers: prime numbers, gcd and lcm, Euclidean algorithm, Bézout's theorem, multiplicative functions such as sums of divisors; arithmetic in the ring Z/nZ and the field Z/pZ, Euler's totient function, Chinese remainder theorem, multiplicative order and primitive roots; sums of squares, quadratic forms, discriminant, class number; and continued fractions, expansion of rationals and quadratic irrationals, Diophantine approximation, and Pell-Fermat equations.

This course introduces matrix algebra and its applications. Key topics include: systems of linear equations; linear transformations; matrix representation of linear transformations; linear operators, eigenvalues and eigenvectors; similarity invariants and canonical forms, and inner product spaces, elementary matrices, determinants and its properties, Cramer’s Rule and inverse formula, areas and volumes, vector spaces and subspaces, subspaces associated with matrices, linear independent sets and bases, coordinate systems and dimension, orthogonality and orthonormal sets, orthogonal projections and Gram-Schmidt process, least square solutions and applications. Text: David C. Lay, LINEAR ALGEBRA AND ITS APPLICATIONS. Assessment: homework (10%), midterm exam (30%), final exam (60%).

This course offers a study of the fundamentals of Python3 programming language for scientific computation (computational fluid dynamics). Topics include: basic commands for running python routines in a jupyter environment-- manipulation of files, directories, and processes, parameters of a command in POSIX format, interactive environments, and git; numerical methods for wave field models-- finite differences, finite volume method, and finite element method.