Mathematics

This course introduces students to basic concepts from abstract algebra, especially the notion of a group. The course helps prepare students for further study in abstract algebra as well as familiarize them with tools essential in many other areas of mathematics. The course is also intended to help students in the transitions from concrete to abstract mathematical thinking and from a purely descriptive view of mathematics to one of definition and deduction.

This course is the second course in a two semester sequence for the sophomore/junior level undergraduate linear algebra. It helps students understand the abstraction of linear algebra. Linear Algebra is a basic language in mathematics and has many applications in every branch of mathematics. The course covers all the topics such as vector spaces and linear transformations, matrix algebra and analysis, inner product and normed spaces in linear algebra commonly used by analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. The major goals are: to develop a systematic knowledge of the elements of linear algebra, and the ability to apply the concepts covered in classes; Fields and Vector Spaces, Linear Operators, Determinants and Eigenvalues, The Jordan Canonical Form, Orthogonality, Spectral Theory, Singular Value Decomposition, Matrix Factorization, and Infinite Dimensional Vector Spaces; to understand the elements of linear algebra with an emphasis on concepts, methods of proof, and the communication of mathematical ideas; to see how all these play a key role in many practical applications in today's technological society; Various applications of linear algebra show how linear algebra is essential not only in solving problems involving algebra, geometry, differential equations, optimization, approximation, combinatorics, but also in the fields such as biology, economics, computer graphics, electrical engineering, cryptography, political science as well as sciences; to broaden students' horizons by learning connections of one subject to other areas of linear algebra and mathematics and by mentioning results at the forefront of research.

Textbook: Mark S. Gockenbach, "FINITE-DIMENSIONAL LINEAR ALGEBRA"

Assessment: Midterm (30%), Final (50%), Attendance & Presentations (10%), Homework, Assignments, Quizzes & Class Activity (10%)  

Prerequisite:  Calculus, Linear Algebra I

This course provides a study of finite mathematical structures that are widely used in computer science. Topics include logic, number theory, methods of proof, sequences, mathematical induction, recursion, functions, probability, etc.

This course covers first- and second-order ordinary differential equations and their applications and modeling. Topics include direction fields, separable and non-homogeneous ODEs, integrating factors, Bernoulli equations, and Euler-Cauchy equations. Additional topics include power series method, Legendre Polynomials, Frobenius method, and Bessel functions. The course also provides a brief overview of linear algebra topics to assist with matrix eigenvalue problems and basics of linear systems. Other topics include Laplace transforms with related topics, such as inverse, s-shifting, derivatives, integrals, Heaviside function, t-shifting, convolution, integral equations, and solving system of ODEs.

The course covers the following: Microscopic properties of networks: adjacency matrix, vertex degree, clustering coefficient, measures of node centrality and node similarity.  Macroscopic properties of networks: degree distributions, graph modularity, and assortativity. Processes on networks: voter model, diffusion process, random walk on a graph, PageRank, and spectral distribution. Random graphs: Erdos-Renyi ensemble, graphs with a prescribed degree distribution, giant components and percolation transition.  

This is a first course on the theory and applications of numerical approximation techniques. It looks at how in practice mathematically formulated problems are solved using computers, and how computational errors are analyzed and tackled. Major topics covered include computational errors, direct method for systems of linear equations, interpolation and approximation, numerical integration, and use of MATLAB software.