Discipline code: 0701

Ⅰ. ADMISSION
The department of Mathematics offers master degree in Mathematics, and aims to develop the student's innovative ability on carrying out original research projects. Students who want to pursue their master degrees in the above mentioned majors of our faculty should:
--- A bachelor degree in public administration or in the relevant fields/majors, such as Economics, Management Science etc., from an accredited university with a transcript at least 3.0 of the GPA.
--- Two recommendation references to support your application from persons who know you well in your professional career.
--- English capability to meet the minimum requirements for a degree study at JU, including the skills in reading, oral communication, as well as writing, is also required.
--- Meet other university’s admission requirements. (Surf on http://oec.ujs.edu.cn/pub/eng/Admission/ChoosingAProgram/DegreeProgram/).

Ⅱ. GOAL
The graduates have a solid and broad mathematical basis, mainly engaged in scientific research and applied mathematics, teaching mathematics and computer to solve practical problems in the engineering, economic, financial and other departments with work.

Ⅲ. OBJECTIVES
In order to achieve the goal of this major, following objectives are to be accomplished by the time the candidate completes the 3 years courses.
A. With correct outlook on life and values, good moral character, strict style of study, strong sense of enterprise and pioneering spirit.
B. With firm grasp of the basis theory of Mathematics, professional knowledge and practical skill, professional development and frontiers; being competent of reading and writing scientific papers; with the ability to independently carry out scientific research work, being competent in the specialized field of teaching, research, and mathematics-related enterprise survey; with a better understand of the theories, techniques, and methods in the field of Applied Mathematics, with new insights in scientific or specialized mathematical technique.
C. With physical and mental health.

Ⅳ. RESEARCH FIELDS
The Master Degree Program in Mathematics focuses on:
● Nonlinear Differential Equations
● Limit Theorem in Probability, Information Theories
● Complex System Modeling, Analysis and Controls
● Theory and Applications of Dynamical System

Ⅴ. DEGREE REQUIREMENTS
The graduate students for Mathematics major are required to accomplish at least 20 course credits which should be earned from the courses you will take, and the credits for degree courses should be more than 14, The completion of 20 credits-courses is usually within one year, while the additional 1.5 years is used to complete the dissertation research and oral examination in thesis defense.

Ⅵ.CURRICULUM

Ⅶ. COURSE CONTENT
Fundamentals for Real and Functional Analysis
Goal
This course introduces the analysis fundamentals which required from modern mathematics. The main contents are the theory of functional analysis and harmonic analysis.

Essential Objectives
All masters from mathematics can grasp the general measure theory, integral theory, L^p space theory, distribution theory and so on.

Content Coverage
　　Chapter I Integral theory
　　　　1．Sets algebra and s algebra
　　　　2．Summable function and measure
　　　　3．The properties of measure
　　　　4．Borel measure
　　　　5．Distribution function
　　　　6．Integration
　　Chapter II L¹ Space
　　　　1. The measure properties of L¹
　　　　2. Lebesgue differential theorem
　　Chapter III L^p Space
　　　　1. The basic theory of L^p space
　　　　2. The dual space of L^p
　　　　3. Basic inequalities
　　　　4. Distribution function and weak L^p space
　　　　5. Interpolation formula in L^p space
　　Chapter IV Sobolev space
　　　　1. Regularity
　　　　2. Distribution
　　　　3. Integer order Sobolev space
　　　　4. Sobolev inequality and imbedding theorem

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Fundamentals of Algebraic
Goal
Abstract algebra is a foundation course for many research fields of mathematics. The main objective of this course is to get the basis knowledge of abstract algebras, including groups, rings, fields and modules.

Content Coverage

　　Chapter I Groups and Subgroups
　　　　1．Introduction and Examples
　　　　2．Binary operations
　　　　3．Isomorphic Binary Structures
　　　　4．Groups
　　　　5．Subgroups
　　　　6．Cyclic groups
　　　　7．Generating sets and diagraphs (*)

　　Chapter II Permutations, Cosets and Direct Products
　　　　8．Groups of permutations
　　　　9．Orbits, cycles and alternating groups
　　　　10．Cosets and the Theorem of Lagrange
　　　　11．Direct products and finitely generated abelian groups
　　　　12．Plane isometries (*)

　　Chapter III Homomorphisms and factor groups
　　　　13．Homomorphism
　　　　14．Factor groups
　　　　15．Factor-group computations and simple groups
　　　　16．Group action on a set (*)
　　　　17．Applications of G-sets to counting (*)

　　Chapter IV Rings and Fields
　　　　18．Rings and fields
　　　　19．Integral domains
　　　　20．Fermat’s and Euler’s Theorems
　　　　21．The fields of quotients of an integral domain
　　　　22．Rings of polynomials
　　　　23．Factorization of polynomials over a field (*)
　　　　24．Noncommutative examples (*)
　　　　25．Ordered rings and fields (*)

　　Chapter V Ideals and Factor Rings
　　　　26．Homomorphisms and factor rings
　　　　27．Prime and maximal ideals
　　　　28．Gröbner basis for ideals (*)

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Functional Analysis
Goal
Make students with a new understanding of axiomatic method in modern mathematics, the ties and the difference between special and general as well as specific and abstract.

Essential Objectives
　　▪ Equip students with the basic theories and ideas about spaces and operators to improve students' ability to abstract thinking and logical reasoning skills.

Content Coverage
　　Chapter 1 Abstract Integration
　　　　1．Set notations and terminology
　　　　2．The concept of measurability
　　　　3．Simple functions
　　　　4．Elementary properties of measures
　　　　5．Integration of functions
　　　　6．The role played by sets of measure zero
　　Chapter 2 Abstract Integration
　　　　1．Set notations and terminology
　　　　2．The concept of measurability
　　　　3．Simple functions
　　　　4．Elementary properties of measures
　　　　5．Integration of functions
　　　　6．The role played by sets of measure zero
　　Chapter 3 Abstract Integration
　　　　1．Set notations and terminology
　　　　2．The concept of measurability
　　　　3．Simple functions
　　Chapter 4 Elementary Hilbert Space Theory
　　　　1．Inner products and linear functionals
　　　　2．Orthonormal sets
　　　　3．Trigonometric series
　　Chapter 5 Examples of Banach Space Techniques
　　　　1．Banach space
　　　　2．Baire’s theorem
　　　　3．Fourier series of continuous functions

Evaluation
_ Assignments: Homework and class presentation
_Final exam

The Basis of Dynamical System
Goal
　　Aim to equip students with the basic concepts and ideas about Dynamical System, and to improve students' ability to apply Dynamics Theories and method to analysis continuous or discontinuous model.

Content Coverage
　　Chapter 1 Scalar Autononous Equations and Continuous Dynamical System
　　　　1. Existence and Uniqueness
　　　　2. Geometry of Flows
　　　　3. Stability of Equilibrium
　　Chapter 2 Elementary Bifurcations
　　　　1．Bifurcation and Bifurcation Diagram
　　　　2．The Implicit Function Theorem and Local Perturbations Near Equilibrium
　　Chapter 3 Scalar Maps and Discontinuous Dynamical System
　　　　1．Euler’s Algorithm and Maps
　　　　2．Geometry of Scalar Maps and The Logistic Map
　　　　3. Equivalence of Flows and Bifurcations of Scalar Maps
　　Chapter 4 Planar Autonomous Systems
　　　　1．First Integrals and Conservative System
　　　　2．Examples of Elementary Bifurcations
　　　　3. Qualitative Equivalence in Linear System
　　　　4. Linear System with 1-periodicity Coefficients
　　Chapter 5 Stability for Equilibrium of Planar Autonomous Systems
　　　　1．Linearization Method to Test Stability of Nonlinear Equilibrium
　　　　2．Lyapunov Method to Test Stability of Nonlinear Equilibrium
　　　　3. Hopf bifurcation

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Foundations of Modern Probability

Goal
　　Initially grasp the basic concepts of measure theory-based analysis of probability theory, the basic theory and methods, establish the foundation for modern stochastic analysis.

Content Coverage
　　Chapter 1 Measure Theory
　　　　1．Set classes and σ domain
　　　　2．Probability measure and its distribution function
　　Chapter 2 Random variables，Expectations，Independence
　　　　1．General theorems
　　　　2．Some properties of mathematical expectation
　　　　3．Independence
　　Chapter 3 The concept of convergence
　　　　1．Any mode of convergence
　　　　2．Almost surely convergence，Borel-cantelli Lemma
　　　　3．Light convergence
　　　　4．Uniform integrability, Moment convergence
　　Chapter 4 Independent random variables sequences
　　　　1．Independence and 0-1 law
　　　　2．Independent Progression
　　　　3．Law of Large Numbers
　　　　4．Optional Times and Wald formula
　　Chapter 5 Conditional expectation and martingale
　　　　1．Generalized measure
　　　　2．Conditional expectation
　　　　3．The definition of Martingale and basic inequality
　　　　4．Martingale convergence theorem and applications

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Elements of Measure Theory

Goal
Through this course, students should understand the basic concepts of σ -algebra, measure, measurable functions etc.; be familiar with the convergence of several different ways , such as: almost everywhere convergence, uniform convergence, convergence in measure; grasp the definition and basic points nature,the application of Radon-Nikodym theorem, construction product space; measure theory and the typical method of product measure. Finally, master the basic theory and method of Measure

Theory and Probability Theory to provide a basis for further scientific research.

Content Coverage
　　Chapter 1 Measurable space and measurable mapping
　　　　1 Set and its operations, the collection of set
　　　　2 Generation. ofσ-algebra
　　　　3.Measurable mapping and measurable functions, application of measurable functions
　　Chapter 2 Measurable space
　　　　1. Definition and properties of measure, outer measure
　　　　2. Expansion of the measure, Completion of a measure space
　　　　3 Convergence of measurable function
　　Chapter 3 Integration
　　　　1. The definition and property of integration
　　　　2. Space L_p(X,F,m)
　　　　3. Integration of probability space
　　Chapter 4 Signed measure
　　　　1. Symbol measure
　　　　2.Hahn and Jordan decomposition
　　　　3. Radon-Nikodym Theorem
　　　　4. Lebesgue decomposition
　　　　5. Conditional expectation and conditional probability
　　Chapter 5 Product space
　　　　1. A finite-dimensional product space
　　　　2. Lebesgue-Stieltjes measure
　　　　3. Probability measure of countable product space
　　　　4. Probability measure of infinite dimensional product space
　　Chapter 6 Independent random variable sequence
　　　　1. Zero one law and three series theorem
　　　　2. Strong law of large numbers, the characteristic function
　　　　3. Weak law of large numbers, central limit theorem

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Theory of Matrices

Goal
　▪ Make students to master eigenvalues and eigenvectors of matrices and explain their applications; geometrical concepts related to orthogonality and least squares solutions and perform calculations related to orthogonality; spectral properties of the following classes of matrices: Hermitian,unitary, normal, and positive definite; algorithms that are important in matrix computations including:QR-factorization, Schur's triangularization, Gram-Schmidt methods; the Jordan canonical forms of matrices under similarity and perform computations associated with Jordan form.

Content Coverage
　　Chapter 1 Eigenvalues, Eigenvectors and Similarity
　　　　1．Eigenvalues and eigenvectors of matrices
　　　　2．Similarity
　　　　3. Diagonalization
　　Chapter 2 Unitary Equivalenc and Normal Matrices
　　　　1．Unitary similarity and Unitary equivalence
　　　　2． Schur's triangularization theorem
　　　　3. Normal matrices
　　　　4. QR-factorization
　　Chapter 3 Jordan canonical form and Triangular Factorizations
　　　　1．Jordan canonical form
　　　　2．Minimal polynomial and the companion matrx
　　　　3. Triangular factorizations
　　Chapter 4 Hermitian matrices and Symmetric matrices
　　　　1．Hermitian matrices
　　　　2．Symmetric matrices
　　　　3. Spectral properties

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Mathematical Modeling

Goal
The aim of the course is to familiarize students with the principles of mathematical modeling. Students will acquire the review of applying mathematical modeling methods in biology.

Content Coverage
　　Chapter 1 Introduction to mathematical modeling　　　　（credits）
　　　　1．Disciplines and their historical development
　　　　2．Principles of modeling
　　Chapter 2 Mathematical Methods　　　　（credits）
　　　　1．Modeling using graphs
　　　　2．Modeling using differential equations
　　　　3. Modeling using Statistics
　　Chapter 3 Models and their applications in biology.
　　　　1．Population models
　　　　2．Epidemic models

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Numerical methods for differential equations

Goal

　▪ By learning courses, the students are able to master certain numerical methods, including the difference and finite element methods. The course could lay the foundation for the study of scientific problems.

Content Coverage
　　Chapter I Numerical solution for ordinary differetial equations
　　　　1．Introduction
　　　　2．Euler’s method
　　　　3. Higher-order Taylor methods
　　　　4. Runge-Kutta methods
　　Chapter II Finite difference methods
　　　　1．Shooting method
　　　　2．Linear boundary value problem
　　　　3. Finite difference methods for linear problems
　　　　4. Finite difference methods for Elliptic equations
　　Chapter III Finite element methods
　　　　1．Variational form of boundary value problems
　　　　2．Ritz method
　　　　3. Galerkin method

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Stochastic Processes

Goal

Stochastic process is an important branch of probability theory, the study which dependented on the nature and regularity of changes in the parameters of a family of random variables is an important science and engineering graduate students in basic courses.

Essential Objectives
(1) Enable students to master the basic concepts of stochastic processes, the basic theory and methods;
(2) Have the ability to use the knowledge to analyze and solve practical problems in random process ;
(3) Make good preparations of the probability theory and stochastic processes studied in the future.

Content Coverage
　　Chapter 1 Basic concept
　　　　1.Moments and common inequality and convergence concepts
　　　　2.Integrability consistent and mean square convergence; random vectors, random sequences　and random function
　　Chapter 2 Conditional probability and conditional expectation
　　　　1.Definition and properties of conditional expectation
　　　　2.Independence
　　　　3.Regular conditional probability
　　Chapter 3 Some basic concepts of random function
　　　　1.The general properties of the random function
　　　　2. Properties of separability, separable functions
　　　　3. Continuity
　　　　4. Stopping time
　　Chapter 4 Independent increments process
　　　　1. Partial sum of independent random variables; series of independent random variables
　　　　2. Sample Properties independent increments process
　　　　3 Random measure generated by probability continuous independent incremental process (separable)
　　　　4. Decomposition of independent increment process, the nature of the sample and its independent increments characteristic function
　　Chapter 5 Martingale
　　　　1. Definitions of martingale and martingale inequality
　　　　2. Convergence problems of martingale
　　　　3 .Decomposition on the super-martingale
　　　　4.Continuous parameter martingale
　　　　5.Decomposition on Doob-Meyer martingale, square integrable martingale
　　Chapter 6 Brown motion and stochastic differential equations
　　　　1. Definitions and sample properties
　　　　2. Asymptotic properties of the sample
　　　　3.Strong Markov property of Brown motion and its application
　　　　4 Local property of Brown motion
　　　　5.Stochastic differential equation

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Stochastic Differential Equations

Goal

Through the study of the course, students master the basic theory of stochastic differential equations, the principle and basic methods; have some simple knowledge of the practical application, prepare for the future research in the aspect of theory of probability and stochastic differential equations.

Content Coverage
　　Chapter 1 Introduction
　　Chapter 2 Some Mathematical Preliminaries
　　　　1．Probability Spaces, Random Variables and Stochastic processes
　　　　2．An Important Example: Brownian Motion
　　Chapter 3 Ito Integrals
　　　　1．Construction of the Ito Integral
　　　　2．Some properties of the Ito Integral
　　　　3. Extensions of the Ito Integral
　　Chapter 4 The Ito Formula and the Martingale Representation Theorem
　　　　1．The 1-dimensional Ito formula
　　　　2．The Multi-dimensional Ito formula
　　　　3. The Martingale Representation Theorem
　　Chapter 5 Stochastic Differential Equations
　　　　1．Examples and Some Solution Methods
　　　　2．An Existence and Uniqueness Result
　　　　3. Weak and Strong Solutions

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Economic Mathematical Methods and Models

Goal

The main feature of this course is to introduce a variety of major mathematical methods commonly used in micro-economic and macro-economic theory in the modern mathematical view of the topology, manifolds and some others, combined with mathematical models. The course also puts an emphasis on the optimization of the model, discusses various nonlinear dynamic optimization method and its application. This course is to enable students to understand the content of modern mathematical, based on the knowledge of calculus to get a knowledge of concepts commonly used in micro-economic analysis, such as convex set, upper semi-continuity, lower semi-continuity, concave function, and know the basics of nonlinear dynamic optimization .

Content Coverage
　　Chapter 1 Static Models and Comparative Statics
　　　　1. Linear Models
　　　　2. Comparative Statics and the Implicit-Function Theorem
　　　　3. Existence of Equilibrium
　　　　4. Problems
　　Chapter 2 Convex Sets and Concave Function
　　　　1. Convex Sets and Separation Theorem in Rⁿ
　　　　2. Concave Function
　　　　3. Quasiconcave Functions
　　Chapter 3 Static Optimization
　　　　1. Nonlinear Programming
　　　　2. Comparative Statics and Value Functions
　　　　3. Problems and Applications
　　Chapter 4 Some Applications to Microeconomics
　　　　1. Consumer Preferences and Utility
　　　　2. Consumer Theory
　　　　3. Walrasian General Equilibrium in a Pure Exchange Economy
　　　　4. Games in Normal Form and Nash Equilibrium
　　Chapter 5 Dynamical Systems Ⅰ: Basic Concepts and Scalar Systems
　　　　1. Difference and Differential Equations:Basic Concepts
　　　　2. Autonomous Systems
　　　　3. Autonomous Differential Equations
　　　　4. Autonomous Difference Equations
　　　　5. Solution of Nonautomation Linear Equations
　　　　6. Solution of Continuous-Time Systems
　　Chapter 6 Dynamical Systems Ⅱ: Higher Dimensions
　　　　1. Some General Results on Linear Systems
　　　　2. Solution of Linear Systems with Constant Coefficients
　　　　3. Autonomous Nonlinear Systems
　　　　4. Problems
　　Chapter 7 Dynamical Systems Ⅲ: Some Applications
　　　　1. A Dynamic IS-LM Model
　　　　2. An Introduction to Perfect-Foresight Models
　　　　3. Neoclassical Growth Models
　　　　4. Some Useful Techniques

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Mathematical Modeling

Goal

The aim of the course is to familiarize students with the principles of mathematical modeling. Students will acquire the review of applying mathematical modeling methods in biology.

Content Coverage
　　Chapter 1 Introduction to mathematical modeling　　　　（credits）
　　　　1．Disciplines and their historical development
　　　　2．Principles of modeling
　　Chapter 2 Mathematical Methods　　　　（credits）
　　　　1．Modeling using graphs
　　　　2．Modeling using differential equations
　　　　3. Modeling using Statistics
　　Chapter 3 Models and their applications in biology.
　　　　1．Population models
　　　　2．Epidemic models

Evaluation
_ Assignments: Homework and class presentation
_Final exam

Ⅷ. PRACTICE PROCESS AND REQIREMENTS
Graduate students are required to participate in the professional teaching practice, including the guidance of undergraduate experiment, marking the lab reports and homework to complete at least 60-70 hours of work. The practice process will be reviewed by the teachers in charge of teaching. The graduate students are also required to attend related seminars and experts forum, and if possible, to participate in social investigation and research or technology promotion. See also <Training Program of Jiangsu University for Graduate Students (general)>, <Jiangsu University graduate training process Introduction>

Ⅸ ADVISORY COMIMITEE AND SUPERVISORY
The master postgraduate students should be directed by a qualified major supervisor (Ph.D.graduate faculty), as well as a committee panel containing several qualified co-advisors/committee members. The Advisory Committee initially consists of at least 4 members of the Graduate Faculty, including the Major Advisor, who acts as the chair. At least 2 members must be from the Faculty of Science, with at least 1 of these being a full member of the Graduate Faculty. The remaining 2 members may be from Faculty of Science or another college, with 1 being a full member of the Graduate Faculty. If the student declares a minor, 1 committee member must be from the minor department. The committee should be established by the end of the second semester of the student’s graduate career.

Ⅹ. OTHER ISSUES AND REQUIREMENTS
Graduate students are required to initiate their thesis study projects prior to the end of the second semester. The medium-term examination for thesis study is scheduled in the fourth semester. Other following important schedules relevant to your graduate study could be found from the Overseas Education College (OEC) at Jiangsu University. In general, a Master’s student is required to have a research proposal and a plan of study accepted by his or her Graduate Advisory Committee by the end of the second semester of study. A list of completed courses and those proposed to meet school requirements should also be prepared. A meeting of the Advisory Committee should be convened by the student to discuss his/her proposal and course work.
Every graduate student is required to publish at least one research paper in a relevant INTERNATIONAL JOURNAL prior to being eligible to apply a dissertation defense. The thesis must demonstrate a mastery of research techniques, ability to perform original and independent research, and skill in formulating conclusions that enlarge upon or modify accepted ideas.
The above achievements are required to be with the first unit of Jiangsu University.

Ⅺ. FINANCIAL ASSISTANCE
Applicants from a foreign country can apply for a variety of Chinese government scholarship that may fully or partially support your degree study at JU. For further information regarding these scholarships provided by Chinese government, you can surf on the website of Overseas Education College (OEC), JU, at http://oec.ujs.edu.cn/pub/eng/Scholarship/GS/. In addition to apply these funding supports, you can also apply a financial assistance provided by your supervisor during your degree study, which depends on your performance in academic research.