Mathematics

The course examines the foundation of infinite sets, the pigeon cage principle, repeatable permutations and combinations, graph theory concepts, Euler and Hamiltonian graphs, plane graphs , character coloring, tree, group, ring, domain, etc., and related literature for these knowledge points.

The course examines the basic theory of matrix and linear equations, including: concept, property and computation of determinants; operation of matrices; special matrices; inverse matrix; rank and elementary transformation; judgment and structure of solutions; linear relations of vectors; concept and property of eigenvalues and eigenvectors; conditions of a matrix similar to a diagonal matrix; methods to decide the sign of quadratic form; and an application on optimization.

Weekly talks by invited distinguished scholars. Topics across pure math, applied math, and statistics are allowed. The course is a requirement for masters and PHD students. 

This course introduces the use of mathematics as an effective tool in solving real-world problems through mathematical modeling and analytical and/or numerical computations. By using examples in physical, engineering, biological, and social sciences, real-world problems are converted into mathematical equations through proper assumptions and physical laws. The course provides qualitative analysis and analytical solutions for some models to interpret and explain qualitative and quantitative phenomena of the real-world problems. Major topics: introduction of modeling; dynamic (or ODE) models: population models, pendulum motion; electrical networks, chemical reaction; optimization and discrete models: profit of company and annuity; probability models: president election poll and random walk; model analysis: dimensional analysis, equilibrium and stability, bifurcation; and some typical applications. The course requires students to take prerequisites.

This course provides a bridge between mathematics and physics, mechanics, etc., and serves as a foundation for solving practical problems using mathematical tools. Through this course, students master the basic theory of common differential equations, and the application of differential equations in practical problems, including mathematics itself and physics, mechanics, economics, biology, and other fields. Topics include the source of the problem of normal differential equations, the primary solution of simple normal differential equations, the basic theory of differential equations, the structure (and solution) of the linear equation solution of constant coefficients, the theory and solution of linear differential equations, and the qualitative theory of differential equations.