基礎數學課程：

Fundamental of Mathematics(Ⅰ)/(Ⅱ)
必修/選修3學分/3學分Freshman

Using a more advanced perspective to organize and integrate elementary mathematics that covers high school math’s basic tools, knowledge, and important concepts in college math. This emphasizes the pattern and connection in elementary and high school math, and consolidates operation and logic in elementary math.

Course(Ⅰ)/(Ⅱ)
必修/必修4學分/4學分Freshman

Calculus is one of the most important fundamental courses in applied mathematics. It is not only used extensively in natural sciences such as physics, chemistry, and biology, but it also has become a very important research tool in economy, finance, and business management in recent years. This curriculum introduces calculus’ fundamental theories and applications. It is a one-year curriculum that has the upper and lower semesters. The upper semester includes single-variable function’s limit, continuum, differentiation, integration, integration techniques and some of their applications. The lower semester covers content on infinite progression, vector calculus, and multi-variable function’s partial and full deviation, _____, and some of their applications.

Linear Algebra(Ⅰ)
必修，3學分，Freshman

The course content is mainly divided into 3 main parts:
（1） “Vector Space”: we will approach first with the definition and introduction of vector space, and then introduce subspace, linearly dependent and linearly independent. Afterwards, we will introduce basis and dimension of vector space.
（2）“Linear Transformation”: first we will introduce the general definition of linear transformation, and then the matrix plotting of linear transformation along with an introduction to matrix calculation. Lastly we will discuss related questions of solution sets in linear homogeneous constant coefficient equations.
（3）“Operation of fundamental matrix and linear system”: this incorporates three fundamental matrixes and their operations, and we will introduce matrix’s rank and its application in solution of linear systems.

Linear Algebra(Ⅱ)
必修，3學分，Freshman

The course content is mainly divided into 3 main parts:
（1） “determinants”: we will start begin with the introduction of second the third order determinants, and then introduce the general equation of n-order determinants before discussing its meaning in geometry.
（2）“Matrix Diagonal”: We will first introduce Eigenvalue and its corresponding Eigenvector in linear transformation. Then we will introduce the general definition and criteria of linear transformation or matrix satisfying matrix diagonal. Lastly, we will introduce the famous Cayley-Hamilton Theorem.
（3）“Inner Product Space”: we will first introduce the general definition of inner product, and then introduce Gram-Schmidt Orthogonalization Process; then we will introduce regular matrix, self-accompanied matrix, and lead to modulized matrix, and orthogonal matrix. Lastly we will introduce the positive and negative matrix.

Introduction to Computer Science(Ⅰ)
必修，3學分，Freshman

Introduces the basic concepts of fundamental computer before the mid-term exam. This includes the fundamental courses such as brief history of the development of computers, basic hardware and introduction to the Internet. The course will also teach the basic operations of Microsoft Office’ Word, Powerpoint, and Excel. After midterm exams, the lessons will be on the basic operation of mathematic calculation software “Matlab,” and this includes introduction to Matlab, 2-D drafting, matrix operation, character/___ process, and basic programming.

Introduction to Computer Science(Ⅱ)
選修，3學分，Freshman

Introduces C programming language and basic information structure. This includes an introduction to basic language of C programming, looping, and index operation, and basic knowledge in programming and practice in basic information structure: they include programming training such as sorting, searching, simple recursion, and duality　

進階數學課程：

Advanced Calculus(Ⅰ)
必修，4學分，Sophomore

Advanced Calculus is the extension of freshman year’s calculus. It mainly discusses the characteristics of space and functions in Rn and metric space. First, we will explore the characteristics of point set topology, define and discuss open set, closed set, connected set and compact set. With a concept on point set topology, we can then begin to define the concepts and basic characteristics of limits, continuity and derivative on metric space.

Advanced Calculus(Ⅱ)
必修，4學分，Sophomore

This extends the curriculum of advanced calculus (I). We will introduce the concept of “bounded variation” and use this concept to define Riemann-Stieltjes calculus which is very useful in probability and statistics. Then, we will introduce “sequence of functions” and the next topic is on discussing its stypticity. Definitions of Multi-variables functions’ differentiations, integrations, inverse function and implicit function are also key topics in advanced calculus (II).

Differential Equations(Ⅰ)
必修 , 3學分 , Sophomore

Many natural phenomenon can be simulated by differential functions. To further explore and predict, we need to be familiar with tools in differential equations. In this course, we will guide you in modeling and learn the techniques in solving basic differential equations. With the help from computer software, we will have a better understanding in the theories and applications.

Differential Equations(Ⅱ)
選修，3學分，Sophomore

This course is focused on solution analysis. It includes the stability in special solutions and equations of linear system, ______, and applications of ________.

Probability and Statistics(Ⅰ)/(Ⅱ)
必修/必修
3學分/3學分
Sophomore

This course wishes mainly to provide students with basic concepts in probability and statistics, but it will emphasize more on probability. Therefore, we wish the students have basic understanding on random variables and common _______ after finishing this course. Thus, this course will start with explaining what random events are, and talks about random variables, and introduce some common ____ and ____ and related probability problems. The later half of the course will talk about some basic statistics concepts that are the fundamentals in the future mathematical statistics.

Numerical Analsis(Ⅰ)/(Ⅱ)
必修/選修
3學分/3學分
Junior
每年開課

1. Course introduction
This course is a introductory course in numerical analysis and ______. The purpose is to:
（1） Introduce all types of ____ techniques.
（2） Explain why these ____ become ______ under certain circumstances.
（3）Training in programming to practice abilities in _______.
（4）To establish a firm foundation in future studies on scientific calculations.

2、 Course content of numerical analysis:
（1）Introductory Concepts and Calculus Review.
（2）A Survey of Simple Methods and Tools , Root Finding.
（3）Interpolation and Approximation.

3、Course content of numerical analysis (II):
（1）Numerical Integration.
（2）Numerical Methods for Ordinary Differential Equations.
（3）Numerical Methods for the Solution of Systems.
（4）Approximate Solution of the Algebraic Eigenvalue Problem.
（5）A Survey of Finite Difference Methods for Partial Differential
Equations.

組合數學(Ⅰ)
必修，3學分，Junior

Combinatorics is one of the fundamental courses in computer science. The course content includes:（1）計數（2）二項式定理及係數（3）鴿籠原理（4）排容定理（5）生成函數（6）遞迴方法（7）特殊數列

組合數學(Ⅱ)
選修，3學分，三下
每年開課

This course extends and connects with combinatorics (I) and becomes a complete combinatorics course. The cour concents includes: （1）基本圖論（2）匹配理論（3）組合設計（4）polya技術原理（5）競賽理論（6）特殊專題

代數學
每二年開課

基礎數論
選修，3學分，Junior

Fundamental mathematical theory is a major part of dispersed mathematics, and is a fundamental course in computer science. It is an important branch of classical mathematics, and this course will introduce the basic concepts of math theories such as ____, ____, _____ , _____ and related applications such as RSA cryptology.

Advanced Confcepts of Mathematics(Ⅰ)
選修，3學分

The purpose of this course to assist the students through means of doing questions and to have a better understanding of subjects such as calculus, advanced calculus, and linear _____. The source of the questions is estimated to cover important textbooks questions and graduate schools exams. The instructors will also conduct the reviews and organizations of important concepts.

專業應用數學課程：

機率論

統計方法

數理統計(Ⅰ)/(Ⅱ)
選修/選修
3學分/3學分
Junior
每年開課

Course introduction:
This course is designed for the 3rd and 4th year students of applied mathematics. The course content covers mainly two parts: _____ theory and statistics _____. ____ is the extension of 2nd year’s probability and statistics courses. And _____ is the main focus of this course. Besides using careful mathematics to prove some statistics theories, this course also focuses on concepts in statistics and explanations of its methods. Through explaining examples and practicing problems, we wish the students to have a good understanding of statistics theories and functions.

數理統計(Ⅰ)Course content of:
（1）Distribution Theory (Review).
（2）Distributions of Functions of Random Variables (Review).
（3）Limiting Distributions : Some modes of convergence , Limiting moment-generating functions , Some theorems on limiting
distributions.
（4）Sampling and Sampling Distributions : Sampling , Sample mean , Sampling from the Normal distributions , Order statistics.
（5）Parametric Point Estimation : Method of finding estimators , Properties of point estimators , Sufficiency.

數理統計(Ⅱ)Course content of :
（1）Parametric Point Estimation : Unbiased estimation , Location or
Scale invariance, Bayes Estimators.
（2）Parametric Interval Estimation : Confidence interval , Sampling
from the Normal distributions , Methods of finding confidence
intervals , Large sample confidence intervals , Baysian interval
estimates.
（3）Test of Hypothesis : Simple and composite hypotheses , Tests of
Hypotheses Sampling from the Normal distributions , Chi-Square
Tests , Test of Hypothesis and confidence intervals , Sequential
tests of hypotheses.
（4）Linear Models or Nonparametric Method (Selected topics if time
permits).

隨機過程

基礎財務數學(Ⅰ)
選修，3學分，四下

This course is mainly using some simple mathematical concepts to define, reason, and explain phenomenon in financial market and model. First, we will introduce the concept of “no arbitrage” and its importance in financial math. By using the premise of “no arbitrage,” we can then introduce financial products’ pricing, investors’ strategies, and risk management. Further, the relationship between utility function and preference order is also a key focus.

資料結構
選修，3學分，Sophomore

This course introduces the basic concepts in C language, and concepts in information structure such as Array, Stack, Queue, Linked Lists, Tree Structure, Sorting, and Searching. The students will also learn and practice in the lectures and practicum sessions. This is a fundamental subjects in the development of information science or information technology.

演算法
選修，3學分，四上

Algorithm and information and information structure are the core of programming. Algorithm itself has become an important branch in computer science. This course will use theories and practicum. Besides introducing and analyzing common algorithm, the students will also learn practical applications through questions.

圖論
選修，3學分，Sophomore

Circular Theory is one of the important fundamental courses in computer science. The course content covers the followings: 1. basic concepts in（1）無向圖及有向圖基本概念（2）樹（3）匹配理論（4）連通性（5）網路及流（6） coloring （7） floor plan, （8）Ramsey theory　

計算工具
選修，3學分，Sophomore

This course is focused on strengthening students’ abilities in using computers to solve problems in applied mathematics. The course will have a brief introduction on the broad perspective of how computers and math should combine, and does a basic introduction on the needed main computational tools. The computer knowledge and programming abilities that the students will need in future jobs will also be strengthened. This is a course that focuses on programming practicum and hand-on learning. Moreover, the course will focus on several topics that are related to applied mathematics and introduce preparatory knowledge, and use the learned computational tools to solve the topic’s small scaled project in order to allow the students to experience the ingeniousness and power of math and computers.

數學規劃(Ⅰ)
選修，3學分，四上
每兩年開課

The course content will be divided into two main parts:
（1）Linear Programming: we will begin with an introduction to simplex method, and then introduce the big-M method and two stage method. Lastly, we will introduce the important ________.
（2） Introduction to typical questions: this mainly introduces modes in linear programming such as _____, _____ and _______

數學規劃(Ⅱ)
選修，3學分，四下

The course content will be divided into 3 main parts:
（1） non-linear programming: we will introduce convex analysis, and then KKT condition. Lastly, we will talk about some issues in geometrical programming.
（2）『動態規劃與整數規劃』：.______: this mainly introduces the problem of the shortest distance, question of _____, and _________ and the problem of resources allocation. Then we will introduce __________ and __________.
（3）『賽局理論』：we will conduct the initial introduction on ____, and then introduce the famous Nash Equilibrium; lastly, we will introduce the famous solutions such as “core” in “n” cooperation games and Shapley Value’s linear programming _____

高效能計算

In this class, we will introduce the high performance computing (HPC) environment to an upper undergraduate in Applied Mathematics including: HPC hardware architectures, software development under Linux environment, code optimization, parallel processing using a Linux based cluster with both MPI and OPENMP directives, parallel algorithms, parallel math libraries, and current developments in both parallel computing libraries or components on constructing a Linux based cluster.

In order to have a much better understanding of the class materials, students are required to hand in four programming exercises ranged from writing up a serial linear algebra program and eventually toward its parallel version. Furthermore, students are required to give 20 minutes in-class presentation based on topics relating to HPC. We hope by both the programming exercises and giving a formal presentation will enhance the students’familiarity on the HPC.

多媒體設計及應用(Ⅰ)
必修，3學分，Sophomore

The main purpose of this course is as follows: understand the concept of application coverage of multimedia; learn to use related multimedia production software; through individual and team effort to develop abilities in planning, execution, cooperation, coordination, finishing, and exhibiting multimedia projects; brief introduction of how math can be applied to multimedia objects; and enhances abilities in appreciating human/cultures. The course content includes brief introduction to multimedia, basic concepts, components, powerpoint skills and practicum, website producing, basic math theories in multimedia, basic principles in designing 2-D media, basic color theory, application in developing multimedia, multimedia case development, managing multimedia cases, and team case development.

高等線性代數
選修，3學分，四上

Course content is divided into two main parts:
（1）. Canonical Form: we mainly extend linear algebra (II)’s second part, _____, and introduce what Jordan Canonical Form and Rational Canonical Form are, and we’ll introduce what minimal polynomial are
（2）We will help the 4th year students by focusing on the questions in graduate school exams – linear algebra, and organize the past exam questions of different schools and discuss and provide simulation exams.

偏微分方程
選修，3學分，四上

This course introduces the basic characteristics and techniques in solving partial differential equations. They include _____, ____, and ____, and we will also discuss their _____ or _____ issues.

實變函數論(Ⅰ)/(Ⅱ)
選修/選修
3學分/3學分
Junior
(I)至少每兩年開課

Course introduction: this courses is mainly on introducing ____ and Lebesgue integration. ______ is not complete, and Lebesgue integration and compensate for the insufficiency in ______. The course content discusses the easier material then the more difficult one, and taking this course will help the students who plan to go into graduate studies to have a good foundation in theories.

Course content of實變函數論(Ⅰ) is as follows:
（1）Set Theory
（2）The Real Number System
（3）The Lebesgue Measure : Outer measure , Measurable sets and Lebesgue measure , Measurable functions , Littlewood’s three
Principles.
（4）The Lebesgue Integral : Lebesgue integral of a bounded function over a set of finite measure , Integral of a nonnegative function , General Lebesgue integral.
（5）Differentation and Integration : Differentiation of monotone functions, Functions of bounded variation , Convex functions.

Course content of 實變函數論(Ⅱ) is as follows:
（1）The Classical Banach Spaces : Lp
spaces , Minkowski and Holder inequalities , Convergence and completeness ,Approximation in Lp, Bounded linear functionals on the Lp spaces.
（2） General Measure and Integration Theory
 Measure and Integration : Measure spaces, Measurable functions , Integration , General convergence theorems , Signed measures , Radon-Nikodym theorem , Lp spaces.
 Measure and Outer measure : Outer measure and measurability, Extension
theorem , Product measure.

複變函數論(Ⅰ)
選修，3學分，三下
至少每兩年開課

This course mainly discusses the characteristics of ____ and its applications. First, we will define analytic function, and then discuss such function’s line integration and its characteristics. The most important part is residue theorem and its estimated applications on general indefinite integral.

泛函分析(導論)
選修，3學分，四下

Teaching objective:
The original meaning of “functionals” is “function of function.” Mathematical modes in applied mathematics can often expressed by functionals. Differential equations, integral equations, and matrix…etc. can all be viewed as functionals. Its application coverage is very broad and includes _________, _______, _____, linear and non-linear programming, and _____...etc. Functionals analysis plays a very important role. In this course, we will discuss the basic concepts and applications of fuctionals analysis in order to enable the students to have the ability in using funcational analysis in getting solutions.

Course content and progress:
（1） Basic concept of metric spces : example of metric spaces, open sets ,
closed sets , concergence and completeness.
（2）Normed spaces and Banach spaces : basic concept, examples ,
compactness , linear operators and linear functionals.
（3）Inner product spaces and Hilbert spaces : orthogonal subspaces ,
operator theory on inner product spaces.
（4）Hahn Banach theorem , open mapping theorem and closed graph
theorem and their applications.
（5）Fixed point theorem and its applications to ODE and approximation
theory.

textbook/reference book
（1）textbook：Erwin Kreyszig : Introductory Functional Analysis with
Applications, John Wiley & Sons.
（2）reference book：A. E. Taylor, Introduction to functional Analysis. John
Wiley & Sons.

分析專題
選修，3學分，Junior

Besides consolidating the content and concepts of advanced calculus, this course will also introduce Vector Calculus (includes ___, ___, Green Theorem, Stoke Theorem, and Gauss Theorem) and Fournier level theory. The more extended and in-depth topics (____ and ___) will be introduced to allow students to prepare for future in-depth research.