Curriculum

Course

Introduce

Core courses

The Course of Functional Analysis

1. nonlinear operators in infinity dimensional normed linear spaces

2. the basic theory of topology linear spaces

3. Frechet-derivative and Gateaux-derivative of nonlinear operators.

4. topology degree theory and its applications

5. Variational methods and its applications

Stability theory on ordinary differential equations

General theory of the ordinary differential equations; Prepareknowledge, basic tools; Basic theorems of Lyapunov’s direct method; The expansion of Lyapunov’s direct method; The stability theories of the linear systems; The applications of several dynamic state systems of typical model.

Advanced combinatorics

Advanced combinatorics is, loosely, the science of counting. This is the area of mathematics in which we study problems of existence, enumeration and structuration of discrete objects (usually finite) with certain characteristic arrangements of their elements or subsets. It includes basic enumeration methods, sieve formulas, identities, theorem of Polya, Ramsey theory and so on.

Complex Network Theory

Advanced combinatorics is, loosely, the science of counting. This is the area of mathematics in which we study problems of existence, enumeration and structuration of discrete objects (usually finite) with certain characteristic arrangements of their elements or subsets. It includes basic enumeration methods, sieve formulas, identities, theorem of Polya, Ramsey theory and so on.

Linear and integer programming

As an important part of Operations Research and Optimization Theory, linear and Integer Programming has found its applications in many areas, such as, industry, military affairs, economics and management science. This course covers the theoretical and computational aspects of linear and integer programming with some meaningful applications. The part of linear programming includes polyhedron theory, duality theory, simplex method, dual simplex method, prime-dual algorithm and some recent topic. The part of integer programming includes cutting plane algorithms, branch and bound approach, dynamic programming and some theoretical issues. The application part includes transportation and assignment problems, the shortest path and maximum flow problem in networks, and application of linear programming in Game Theory

Elements of Algebra

The content of the course should cover the following chapters:

Chapter 1 Groups.

It includes the related definitions and properties of groups, cyclic groups, normal subgroups, quotient groups, homomorphism theorems, direct sums, and decompositions, etc.

Chapter 2 Rings.

It includes the related definitions and properties of rings, ideas, quotient rings, and homomorphism theorems, etc.

Chapter 3 Modules.

It includes the related definitions and properties of modules, free modules, and projective modules, etc.

Specialized courses

Foundations of Probability Theory

Stochastic process is the quantitative description of the dynamic relationship between a series of random events. It is an important tool in various fields of natural science, engineering and social sciences. The theory of stochastic processes has been widely used in weather forecast, statistical physics, astrophysics, mathematical economics, population theory, reliability and computer science etc. The main contents include some typical point process, stochastic differential equation and martingale theory. Through the course of learning to enable students to masterthe basic theory and research method of stochastic process.

Control theory

The concepts and methods of the stability decomposition for the large-scale dynamic systems; The stability of the linear large-scale dynamic systems; The stability of the linear time-invarying large-scale dynamic systems with time-delays; The stability of the linear time-varying large-scale dynamic systems with time-delays; The stability of the discrete large-scale dynamic systems; The applications of the stability for large-scale systems in actually; The prepare principle and the un-conditions stability; the optimal control and sub-optimal control of the systems with time-delays; the stability, stabilization and robust stability of the large-scale dynamic system.

Combinatorial Optimization

Combinatorial Optimization is one of the youngest and most active areas of discrete mathematics, and it probably its driving force today.A main motivation is that thousands of real-life problems can be formulated as abstract combinatorial optimization problems. This course includes the most important ideas, theoretical results, and algorithms in combinatorial optimization. It covers some algorithms of classical topics in combinatorial optimization, and the applications of graph theory, linear and integer programming, matroid theory and complexity theory in the design and analysis of algorithms.

Theory of Functional Differential Equations

FDE mainly include the following equations:

Delay type, neutral type, mixed type, complex deviating argument type. The subjects studied in FDE mainly contains: well posedness, stability and boundary, periodic solutions, oscillations and asymptotic, almost periodic solutions, attractors ect.

Biomathmatics

Mathematical biology or biomathematics is an interdisciplinary field of academic study which aims at modelling natural, biological processes using mathematical techniques and tools. It has both practical and theoretical applications in biological research. It is concerned with the study of mathematical methods used in biomathematics. This course includes basic theory and methods of biomathematics, and topics on using methods in combinatorial mathematics and graph theory

Optional courses

Theory and method of variable structure control

The concepts and methods of the variable structure control (VSC); VSC of the linear systems; VSC of the nonlinear linear systems; some specialized problems of the VSC (following system, adaptive model following system, uncertainly systems, model arrived system, large-scale system, nonlinear large-scale systems, delay systems); the applications of the VSC (saturation control method, robot dynamics model, VSC method, state observer, fault-tolerance VSC, VSC of the discrete time systems ).

Linear system theory

Linear system theory: state space description, motion analysis, controllability and observability, stability analysis, and approaches of time-domain synthesis

Discrete control systems

The basic concepts of the discrete systems, the basic properties of the discrete control systems, the feed-back control of the linear systems, optimal control method (variational method and maximum volume principle, dynamic programming), linear quadratic form Gauss problems.

Graph Theory

The main task of Graph Theory is to discuss the structures and properties of systems with 2-elements relations by making use of deep and elegant techniques. This course presents basic theory and main technique, together with a wide variety of applications both to other branches of mathematics and to practical problems. The topics included are basic definitions, trees, Euler tour and Hamilton cycles, matching, independent sets and cliques, coloring, planar graphs and so on. In addition, the application part includes connection problem, assignment problem, Chinese postman problem and ranking in a tournament

Generating functionnlogy

Generating functions are extensively used in discrete mathematics. They are a bridge between discrete mathematics, on the one hand, and continuous analysis on the other. It is possible to study them as tools for solving discrete problems. As such there is much that is powerful and magical in the way generating functions give unified methods for handling such problems. It is concerned with the academic study of Generating function, and some applications to discrete mathematics.

Complex Network Theory

This course is mainly in the study of complex network theory in general, including the topology character and model of network, such as communicative behavior in complex network, cascading failure, search algorithm and community structure, as well as synchronism and control of complex network.

Theory of computational complexity

This course contains basic definitions concerning NP-completeness and NP-hardness, theoretical results with proof methodology, some general approaches for the design of approximation algorithms for NP-hard problems, and some non-approximability results.

Approximation algorithms

Approximation algorithms have developed in response to the impossibility of solving a great variety of important optimization problems. Too frequently, when attempting to get a solution for a problem, one is confronted with the fact that the problem is NP-hard, which means that there unlikely exists an efficient algorithm for the problem unless P=NP. Charting the landscape of approximability of NP-hard problems, via polynomial time algorithms, therefore becomes a compelling subject of scientific inquiry in computer science and mathematics. This course covers the theory of approximation algorithms as it stands today. It includes combinatorial algorithms, linear programming and semi-definite programmingbased algorithms for a number of important classical problems as well as very recent ones.

Matrix analysis

Matrix analysis: linear space and linear transformation, inner space, standard form of matrix and some decompositions, matrix functions and their applications, eigenvalue estimate and general converse matrix.

Principle of Automatic Control

General concept of automatic control;mathematics model of control system; the time-domain analysis method, root locus method, the method of frequency domain and calibration methods of the linear system; analysis and correction of the linear discrete system; analysis of the nonlinear control systems; analysis and synthesis of the state space for the linear system.