Program

On the other hand, the course introduces the theory and practice of computer programming. The emphasis is on techniques of program development using the C++ language. It aims to help students feel confident of their ability to write programs to solve problems in mathematics or engineering.

Indeed, the scope of the course is to let students become familiar with the ideas of taking integrals of scalar - valued (or vector – valued) functions over curves or surfaces, as well as the applications of the theorems of Green, Stokes and Gauss.

In addition, the course will also introduce students to the basic geometric concepts of curves and surfaces in either Euclidean spaces of dimension 2 or 3.

The prerequisites of this course are Calculus and linear algebra.

Basically speaking, the main principle of Newtonian mechanics can be realized mathematically as a second-order ordinary differential equation. To a certain extent, the past developments of mathematical tools, which are capable of addressing mathematical problems arising from this second-order O.D.E. in Newtonian mechanics, constituted the whole historical development of Calculus.

To put it in simple words, we can say that Calculus was invented for the sole purpose of solving differential equations. Indeed, being witnesses to the tremendous success of Newtonian mechanics, mathematicians and physicists since the time of Newton began to apply the main ideas of Newtonian Mechanics to all the other branches of Natural Science. In particular, in the studies of the physical phenomena of a vibrating string with fixed end-points, the first partial differential equation, which is known as the wave equation (or the d’Alembert’s equation), was derived for the very first time in the early days of the historical development of P.D.E.’s theory. A partial differential equation is, by definition, an equation satisfied by some of the partial derivatives of the unknown multi-variable function.

Partial differential equations arise naturally in the study of geometry, and in almost all physical laws of Natural Sciences or practical engineering problems. We can even say that, without the developments of the theory of partial differential equations, there could be no development of Science at all in recent human history. Indeed, some of the most famous partial differential equations which appeared in the history of Science are:Laplace equation, Wave equation, Heat equation, Maxwell equation, Schrodinger equation, Navier-Stokes equation and Euler equation, Boltzmann equation, KdV equation.

Basically, one can say that the studies of these equations encompass almost all the mathematical problems arising from Physics and different areas of engineering.

In order to solve these partial differential equations, mathematicians, physicists, and engineers had developed various kinds of mathematical methods, among these the most famous one is the Fourier analysis. As a matter of fact, the subject of Fourier analysis constitutes almost all of the modern mathematical analysis. However, as the development of the theory of partial differential equations became more mature and sophisticated, researchers began to realize the importance of acquiring a better understanding about various kinds of nonlinear phenomena. This naturally motivated the developments of the theory of dynamical systems in recent and modern times, including the recent studies of the theory of Chaos.

Besides standard tools borrowed from mathematical analysis, numerical analysis is another important methodology which is currently used in the studies of solutions to partial differential equations. Through carrying out numerical simulations with the aid of high-speed computers, the use of “Numerical Analysis” in the studies of P.D.E.’s can be regarded as carrying out mathematical experiments. These mathematical experiments through numerical simulations not only help us to gain a deeper understanding about the equation being studied, but even may provide us with clues which can lead to rigorous proofs of mathematical assertions related to the equation being studied.

Finally, but not the least, we should never forget that the sole purpose of a differential equation is to describe natural phenomena. Even priori to our efforts devoted in solving a specific differential equation, the equation itself has already revealed some of the secrets of the Nature to us. In other words, Differential equations themselves naturally speak the language of the Nature. So, it is absolutely true that knowing the basics of Physics will be extremely helpful for the studies of partial differential equations.

For example, algebraic number theory and algebraic geometry apply algebraic methods to problems in number theory and geometry, respectively. In recent years, abstract algebra also figures prominently in coding theory and cryptography, which are essential in telecommunication nowadays. In this one-year course, we will study three kinds of algebraic structures that appear most often in mathematics, namely, groups, rings, and fields, culminating in the proof of the theorem that a general quintic equation has no solutions in radicals.

The topics of lectures focus mainly on the topical programs in this department including combinatorics, differential equations, dynamic systems, scientific computations, probability and theoretical mathematics. Those interested in the selected topics are welcome to seat in the lecture, though the course is set for graduate students. No prerequisite is asked for this course.

We will organize this course into ten subjects. We first introduce the linear programming problems with modeling examples and provide a short review of the history of linear programming. The basic terminologies are defined to build the fundamental theory of linear programming and tor form a geometric interpretation of the underlying optimization process. The next topic will cover the classical simplex methods and corresponding theories including the revised simplex, duality theorem, dual simplex method, the primal-dual method. After discussing the sensitivity analysis, we will look into the concept of computational complexity and show that the simplex method, in the worst-case analysis, exhibits exponential complexity. Hence, the ellipsoid method is introduced as the first polynomial-time algorithm for linear programming. From this point onward, we focus on the nonsimplex approaches. The next subject is centered around the recent advances of Karmarkar’s algorithm and its polynomial-time solvability. We then study the affine scaling variants, containing the primal, dual, primal-dual algorithms, of Karmarkar’s methods. The concepts of central trajectory and path-following are also included. The eight topic reveals the insights of interior-point methods from both the algebraic and geometric viewpoints. It provides a platform for the comparison of different interior-point algorithms and the creation of new algorithms. We extend the results of interior-point-based linear programming techniques to quadratic and convex optimization problems with linear constrain. The important implementation issues for computer programming are addressed in the last subject.

The object of this course is the construction, analysis, and interpretation of the mathematical models to help us understand the world we live in. We shall introduce key ideas in mathematical methods and modeling, with an emphasis on the connections between mathematics and applied natural sciences. The course covers both standard and modern topics, including scaling and dimensional analysis, regular and irregular perturbation; calculus of variations and continuum mechanics, etc.