Mathematics

The course provides an introduction to selected topics in discrete mathematics; including graph theory, combinatorics, final bodies, and code theory.

Ordinary Differential Equations (ODEs) allow us to describe many natural phenomena and are therefore essentially important in many scientific areas. This course introduces the standard and most common tools to solve differential equations, particularly Laplace transform and linear algebra method. 

An introduction to linear algebra, mainly in R^n but concluding with an introduction to abstract vector spaces. The principal topics are vectors, systems of linear equations, matrices, eigenvalues and eigenvectors, and orthogonality. The important notions of linear independence, span and bases are introduced. This course is both a preparation for the practical use of vectors, matrices, and systems of equations and also lays the groundwork for a more abstract, pure-mathematical treatment of vector spaces. Students learn how to use a computer to calculate the results of some simple matrix operations and to visualize vectors.

Asset price in discrete time, random walks, conditional expectation, elements of discrete-time martingale theory, the binomial asset pricing model, option pricing in discrete time, and -time permitting- discrete time term structure models and/or discrete time portfolio theory.

This course focuses on the basic concepts and theorems of Ring Theory and Field Theory, which are about generalized and abstracted properties of the set of integers and the set of rational numbers with respect to the four elementary arithmetic operations. Recommended Prerequisite: MATH321.

The course covers the following topics:

• Homomorphisms and Factor Rings
• Prime and Maximal Ideals
• Introduction to Extension Fields, Vector Spaces
• Algebraic Extensions
• Geometric Constructions
• Finite Fields
• Unique Factorization Domains
• Euclidean Domains
• Isomorphism Theorems, Series of Groups
• Sylow Theorems and it applications
• Automorphism of Fields
• The Isomorphism Extension Theorem
• Splitting Fields, Separable Extensions
• Galois Theory
• Cyclotomic Extensions
• Insolvability of the Quintic 

This course covers basic topology, meaning of convergence, and continuity.

This course focuses on the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include Mathematical Formulations; Finite Difference Method, Finite Volume Method, Collocation Method, Finite Element Discretization.

This course introduces and investigates fundamental concepts of linear algebra in the context of the Euclidean spaces R^n. Topics include systems of linear equations, matrices, determinants, Euclidean spaces, linear combinations and linear span, subspaces, linear independence, bases and dimension, rank of a matrix, inner products, eigenvalues and eigenvectors, diagonalization, linear transformations between Euclidean spaces, and applications.

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.