AY 2020 Undergraduate School Course Catalog

Mathematics

2021/01/30

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開講学期
／Semester		 2020年度／Academic Year 　1学期・2学期 ／1st & 2nd Quarter
対象学年
／Course for;		 1st year
単位数
／Credits		 2.0
責任者
／Coordinator		 WATANABE Shigeru
担当教員名
／Instructor		 MAEDA Takao, ASAI Kazuto, WATANABE Shigeru, TRUONG Cong Thang, SU Chunhua
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/09/11
授業の概要
／Course outline		 (ICTG class starts in Q3.And Prof. Asai and Prof. Cong-Thang Truong is in charge of the class.)
Asai's class:  Combination of face-to-face classes and remote classes
Truong's class:  Combination of face-to-face classes and remote classes

In the basic course, there are two subjects in mathematics: one is linear algebra
and the other is differential and integral calculus. In this class, the basic part
of linear algebra will be taught. Students are required to study this class with
differential and integral calculus, because these subjects are closely
connected each other.
Linear algebra is a field of mathematics that is based on both addition and scalar
multiple, and vectors in high school mathematics are its basic parts. The main
theme of linear algebra is eigenvalue problem that arose from the theory of
simultaneous linear differential equations, while its historical origin is in
solving simultaneous linear equations. The purpose of this class is to learn
necessary notions and techniques to consider eigenvalue problem.
Exercises will be also given side by side with the lectures. Students are expected
to participate subjectively and positively.
Linear algebra has many applications to computer science and engineering. For example,
it is impossible to understand mechanisms of computer graphics without linear algebra.
Further, some fields of mathematics arose from engineering and physics, and developed
under the influences of them. These are the essential reasons why mathematics is
required to learn.
授業の目的と到達目標
／Objectives and attainment
goals		 The purpose of this class is to learn necessary notions and techniques to consider
eigenvalue problem, and the following contents will be dealt with.
vectors in plane, vectors in space, matrices of (2,2)-type, matrices of (3,3)-type,
general definition of matrix, elementary transformations, simultaneous linear
equations, inverse matrices, determinants, linear spaces, dimension and basis,
subspaces, linear mappings

Attainment targets

Students will be able to understand rank and solve simultaneous linear equations.
Students will be able to understand base and dimension of linear space and deal with
linear subspaces.
授業スケジュール
／Class schedule		 1 Introduction to matrices (1)
2 Introduction to matrices (2)
3 Matrices and linear transformations (1)
4 Matrices and linear transformations (2)
5 Definition of matrix and operations - general theory
6 Square matrices, regular matrices and linear mappings
7 Elementary transformations and rank
8 Simultaneous linear equations
9 Definition of determinant
10 Properties of determinant
11 Expansions of determinant
12 Definition and properties of linear space
13 Basis and dimension
14 Linear subspaces, linear mappings
教科書
／Textbook(s)		 Masahiko Saito Introduction to linear algebra (in Japanese) University of Tokyo Press
Yoshihiro Mizuta Linear algebra (in Japanese) Saiensu-sha
成績評価の方法・基準
／Grading method/criteria		 Asai's class
Final Exam. 100%. (More than 80% of homework assignments should be submitted.)

Watanabe's class
Final 100%

Maeda's class
Quiz 10%, Midterm exam. 30%, Final exam.60%
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Asai's class
Directory for Asai's class: ~k-asai/classes/holm/
Handouts and Exercises for Asai's class:
http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html


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開講学期
／Semester		 2020年度／Academic Year 　3学期・4学期 ／3rd & 4th Quarter
対象学年
／Course for;		 1st year
単位数
／Credits		 2.0
責任者
／Coordinator		 WATANABE Shigeru
担当教員名
／Instructor		 MAEDA Takao, WATANABE Shigeru, WATANABE Yodai, MATSUMOTO Kazuya, SCHMITT Lothar M., HASHIMOTO Yasuhiro
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/08/14
授業の概要
／Course outline		 (ICTG class starts in Q1.And Prof. Schmitt is in charge of the class.)
Watanabe's class  Remote classes
Hashimoto's class  Remote classes
Linear algebra II is a continuation of linear algebra I and deals with eigenvalue problem.
Students are required to study this class with differential and integral
calculus because of the same reason that is described in the syllabus of linear algebra I.
For example, matrices and determinants play important roles in differential and integral
calculus of several variables. And eigenvalue problem gives a strong way to solve recurrence
formulae of sequences. Students must know importance of understanding organic connection
between linear algebra and differential and integral calculus. They will also learn bases of
vector analysis that is necessary to learn electromagnetism. Besides, the fundamental policy
does not change from the case of linear algebra I.
授業の目的と到達目標
／Objectives and attainment
goals		 Eigenvalue problem of matrices
eigenvalues, eigenvectors, diagonalization

Attainment targets

Students will be able to solve eigenvalues and eigenvectors.
Students will be able to deal with diagonalization.
Students will be able to deal with diagonalization of normal matrices by unitary matrices.
授業スケジュール
／Class schedule		 1 Inner products
2 Metric linear spaces
3 Orthogonalization
4 Introduction to eigenvalue problem --- meaning of diagonalization
5 Eigenvalues and eigenvectors (1)
6 Eigenvalues and eigenvectors (2)
7 Diagonalization (1)
8 Diagonalization (2)
9 Diagonalization of normal matrices by unitary matrices (1)
10 Diagonalization of normal matrices by unitary matrices (2)
11 Diagonalization of normal matrices by unitary matrices (3)
12 Diagonalization of real symmetric matrices by orthogonal matrices
13 Quadratic forms
14 Quadratic curves



The order of classes may be changed.
教科書
／Textbook(s)		 Masahiko Saito Introduction to linear algebra (in Japanese) University of Tokyo Press
Yoshihiro Mizuta Linear algebra (in Japanese) Saiensu-sha
成績評価の方法・基準
／Grading method/criteria		 Watanabe's class
Final 100%

Maeda's class
Quiz 10%, Midterm exam. 30%, Final exam.60%


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開講学期
／Semester		 2020年度／Academic Year 　1学期・2学期 ／1st & 2nd Quarter
対象学年
／Course for;		 1st year
単位数
／Credits		 2.0
責任者
／Coordinator		 KIHARA Hiroshi
担当教員名
／Instructor		 KIHARA Hiroshi, OGAWA Yoshiko, LI Peng
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/02/07
授業の概要
／Course outline		 (ICTG class starts in Q4.And Prof. Li, P. is in charge of the class.)
Calculus I is devoted to the former half of (differential and integral) calculus. Calculus and linear algebra are essential in the study of mathematical sciences. Since even the vector calculus is needed in the course of physics, the student should get familiar with calculus and linear algebra and understand the relation between them as soon as possible.

Differential and integral calculus started from an understanding of basic objects such as areas of figures and tangent lines to curves, and was based on the Newton's mechanical investigations. Of course, the basic notions in calculus are constructed using that of a limit.

Calculus I deals with calculus of one variable. The basic calculatioal techniques are reviwed and new notions and results are introduced; the rigorous treatment (epsilon definition) of a limit is also (partly) introduced.

Exercises are also offered.
授業の目的と到達目標
／Objectives and attainment
goals		 Establishing the foundation of the basic calculational techniques studied in high school, we introduce advanced notions and results such as inverse trigonometric functions, expansion of a function, and recurrence relation of integrals.

Calculus I is foundation for an understanding of Calculus II, Probability and Statistics, Fourier analysis, Complex function theory, Physics, and all fields of computer sciences.
授業スケジュール
／Class schedule		 1. Set of real numbers

2. Limit of a sequence

3. Limit of a function and continuous functions

4. Derivative, exponential function, and logarithmic function

5. Trigonometric functions, inverse trigonometric functions, and higher derivatives

6. Euler's formula

7. Derivative, the mean-value theorem, and increase or decrease of a function

8. Taylor's theorem and expansion of a function

9. Indefinite integral and rerecurrence relation

10. Integral of rational functions

11. First and second order linear differential equations

12. Definition and properties of a definite integral

13. Calculation of definite integrals, extention of the definition of a definite integral, and measurement of figures

14. Review of the course
教科書
／Textbook(s)		 Minoru Kurita, Shinkou bisekibungaku, Gakujutsutosho, 1442 yen

Gen Yoneda, Rikoukeinotameno bibunsekibun nyuumon, Science sha, 1890 yen
成績評価の方法・基準
／Grading method/criteria		 Test : Report = 8 : 2
履修上の留意点
／Note for course registration		 None
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Instructed in the lectures


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開講学期
／Semester		 2020年度／Academic Year 　3学期・4学期 ／3rd & 4th Quarter
対象学年
／Course for;		 1st year
単位数
／Credits		 2.0
責任者
／Coordinator		 KIHARA Hiroshi
担当教員名
／Instructor		 KIHARA Hiroshi, SAMPE Takeaki, TSUCHIYA Takahiro, WATANABE Yodai, OFUJI Kenta, LI Xiang
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/02/07
授業の概要
／Course outline		 (ICTG class starts in Q2. And Prof. Li, X. is in charge of the class.)
Calculus II deals with calculus of several variables.

Differential and integral calculus of several variables reduces to calculus of one variable. If you understand it, you can easily master the main part of Calculus II.

See also the syllabuses of Calculus I and Linear algebra II.
授業の目的と到達目標
／Objectives and attainment
goals		 The main objective of this course is to master differential and integral calculus of several variables.

The notion of derivative of a function of one variable is extended to that of partial derivative of a function of several variables. It is applied to solve problems of local minimun/maximum.

The notion of definite integral of a function of one variable is extended to that of multiple integral of a function of several variables. Especially, the technique of transformation of variables, which corresponds to that of substitution integral, is important.

You also study the basics of series. Especially, the notion and results of series of functions are foundation of Fourier analysis and Complex function theory.
授業スケジュール
／Class schedule		 1. Parametrized curves

2. Partial differential coefficients

3. Differentiation of composite functions

4. Total differential and expansion of a function

5. Local minimum/maximum of a function

6. Implicit functions, curves, and surfaces

7. Multiple integrals and their calculations

8. Technique of transformation of variables

9. Areas and volumes

10. Differential 1-forms and integrals

11. Convergence and absolute convergence of a series

12. Sum and product of series and limit of a sequence of functions

13. Power series

14. Review of the course
教科書
／Textbook(s)		 Minoru Kurita, Shinkou bisekibungaku, Gakujutsutosho, 1442 yen

Gen Yoneda, Rikoukeinotameno bibunsekibun nyuumon, Science sha, 1890 yen
成績評価の方法・基準
／Grading method/criteria		 Test : Report = 8 : 2
履修上の留意点
／Note for course registration		 Calculus I, Linear Algebra I
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Instructed in the lectures


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開講学期
／Semester		 2020年度／Academic Year 　1学期 ／First Quarter
対象学年
／Course for;		 2nd year
単位数
／Credits		 2.0
責任者
／Coordinator		 MAEDA Takao
担当教員名
／Instructor		 MAEDA Takao, TSUCHIYA Takahiro, LI Xiang, LYU Guowei
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/03/12
授業の概要
／Course outline		 The origin of Fourier analysis is a study of the heat equation (differential equation) by Fourier in early 19th century.  The basic idea was “express arbitrary functions by using trigonometric functions well!”  This way of thinking led to many useful results in many application fields, but it was uncertain whether the conclusions were mathematically justified, since the concept of convergence had not been discovered.  In the present day, a proper theory has been developed for conventional sciences and technologies.  Fourier analysis is an indispensable basic theory for not only processing differential equations of Fourier’s era, but also modern applications, e.g., signal processing (including for image information and sound information, etc.)  The technique for handling the given function by using trigonometric functions and exponential functions is given though we will not examine of mathematical proofs in this lecture.  Through the answers to the exercises, students will become familiar with calculations and skilled at using basic theorems of Fourier analysis.
授業の目的と到達目標
／Objectives and attainment
goals		 Part1. Fourier series expansion
Students will understand the orthonormal system of the space of the functions defined over finite intervals and Bessel's inequality.    
Students will understand the Fourier series of trigonometric functions and will calculate them for the functions expressed by polynomials, exponential functions or trigonometric functions.

Part2. Properties of Fourier series
Students will understand conditions of convergence of Fourier series, the relationship between a given function and the Fourier series derived from it.  They will understand Parseval’s theorem and apply it. They will also understand Weierstrass' theorem.  

Part3. Fourier integral
Students will understand that the Fourier integral is obtained through a heuristic method that extends finite intervals to infinite ones and calculates Fourier transforms and Fourier integrals (Fourier inverse transforms) of given functions.  Target functions are not only elementary functions described above, but also the functions of the type $e^{-x^2})$.  They will also understand a convolution of the functions, the relationship between the convolutions and the Fourier transforms of the functions and calculate them

Part4. Laplace transform
Students will understand the Laplace transform as a transform based on Fourier transform and apply it to solving a certain kind of ordinary differential equations.  The method by which the solution is expressed by a convolution of the Laplace transform is included.  

Part5. Discrete Fourier transform
Through the heuristic consideration of the stepping functions by applying the theory of Fourier series, students will understand the theory of discrete Fourier transform (DFT) similar to the theory introduced in Part 2.  They will also understand the method of the fast Fourier transform (FFT) as an efficient algorithm of DFT and appreciate its speed.

授業スケジュール
／Class schedule		 1: Part1. Fourier series expansion (Orthogonal system of the function space)
2: Part1. Fourier series expansion (Fourier series of trigonometric functions)
3: Part1. Fourier series expansion (Exercise)
4: Part2. Properties of Fourier series (Convergence condition of Fourier series)
5: Part2. Properties of Fourier series (Parseval's theorem, Weierstrass' theorem)
6: Part2. Properties of Fourier series (Exercise)
7: Part3. Fourier integral (Introduction from Fourier series, Fourier transform)
8: Part3. Fourier integral (Parseval's theorem, convolution)
9: Part3. Fourier integral (Exercise)
10: Part4. Laplace transform (Introduction from Fourier transform)
11: Part4. Laplace transform (Ordinary differential equations of constant coefficients)
12: Part4. Laplace transform (Exercise)
13: Part5. Discrete Fourier transform (Introduction from Fourier series)
14: Part5. Discrete Fourier transform (FFT(Fast Fourier Transform))
教科書
／Textbook(s)		 [Tsuchiya class, Maeda class]
Gen-ichiro Sunouchi; Fourier analysis and its applications (SAIENSU-sha) and Handouts
[Li class]
Handouts
[Lu class]  TBD
成績評価の方法・基準
／Grading method/criteria		 Students will be assessed based on regular quizzes and report with an emphasis on the final examination.
履修上の留意点
／Note for course registration		 Formal prerequisite: MA01 Linear algebra,  MA02 Linear algebra II, MA03 Calculus I, MA04 Calculus II
Important related courses: MA06 Complex analysis, IT03 Image processing, IT08 Signal processing and linear system,  IT09 Sound and audio processing
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Transnational college of Lex (ed.), “Adventure of Fourier” Hippo family club
Ken-ichi Kanaya, “Applied mathematics”, Kyoritsu Syuppan
Nakhle H. Asmar; Partial Differential Equations with Fourier Series and Boundary Value Problems: Third Edition (Dover Books on Mathematics)

[Li class]
Gen-ichiro Sunouchi; Fourier analysis and its applications (SAIENSU-sha)


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開講学期
／Semester		 2020年度／Academic Year 　4学期 ／Fourth Quarter
対象学年
／Course for;		 2nd year
単位数
／Credits		 2.0
責任者
／Coordinator		 ASAI Kazuto
担当教員名
／Instructor		 ASAI Kazuto, LI Xiang, LYU Guowei
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/08/18
授業の概要
／Course outline		 Course implementation methods: K. Asai's class, L. Xiang's class: Combination of face-to-face classes and remote classes, Guo-Wei Lu's class: Remote classes

Although complex functions, in the wider sense, are mappings from complex numbers to themselves, i.e. complex-valued functions of a complex variable, the main objects of Complex Analysis are functions satisfying analyticity. The analyticity is a property of local representability of a function as convergent power series, which is equivalent to the condition that a function is holomorphic (differentiable with respect to a complex variable) in the corresponding domain.

In this course, we introduce complex functions, and learn holomorphy of functions and the Cauchy--Riemann equations. Next we define complex integration along a curve on the complex plane, and learn Cauchy's integral theorem/formula, etc. In virtue of this result, we have the Taylor series expansion and the Laurent series expansion of functions. The former is power series expansion of functions, which is the most fundamental result in Complex Analysis. The latter is applied to the study of singularities and the residue theorem. In addition, we derive many techniques useful for direct applications such as the maximum modulus principle, calculation of solutions to differential equations using the method of power series, and determination of the number of zeros of functions by Rouche's theorem.

When we study Complex Analysis, we are impressed that all of the needed theorems are derived very naturally one after another. Hence it is said that this theory has a beautiful system of mathematics. In particular, the most amazing fact is that every analytic function is completely determined by the behavior in a very small domain. This is similar to the fact that every life can be completely regenerated from a cell.

Analyticity is a property that our familiar many real functions have -- polynomials, rational functions, exponent functions, logarithm functions, trigonometric functions, and all combinations of them have analyticity. Therefore, Complex Analysis is easy to apply to many areas. The knowledge of Complex Analysis is very important for various application areas such as electromagnetism, fluid mechanics, heat transfer, computer system theory, signal processing, etc.
授業の目的と到達目標
／Objectives and attainment
goals		 Objectives: Students understand what is a "holomorphic function", and can apply Cauchy's integral theorem/formula to several problems. They learn analytic functions are expanded by Taylor/Laurent series. Also, they use residue theorem to some integral calculation.

Attainment goals: holomorphic functions, Cauchy--Riemann equations, complex integrals, Cauchy's integral theorem, Cauchy's integral formula, Taylor series, Laurent series, singularities, residue theorem.
授業スケジュール
／Class schedule		 1. complex plane, point at infinity
2. holomorphic functions, Cauchy-Riemann equations
3. harmonic functions
4. exponent functions, trigonometric functions, logarithm functions, roots, complex powers of complex numbers
5. complex integrals
6. Cauchy's integral theorem, integrals of holomorphic functions
7. Cauchy's integral formula, Liouville's theorem, maximum modulus principle
8. complex sequence and series
9. sequence and series of functions, uniform convergence
10. power series and its convergence domain
11. Taylor series expansion
12. Laurent series expansion, zero points, singularities
13. residue theorem
14. application to several (real) definite integrals
(Details depend on each class.)
教科書
／Textbook(s)		 Handouts by each instructor and the following:

Asai's class: Handout is a main textbook. As a side reader, Nattokusuru Fukusokansu, Yoshitaka Onodera, Kodansha, 2000.

Li's class: A first course in Complex Analysis with application, Dennis G. Zill and Patrick D. Shanahan, Jones and Bartlett Publishers, Inc, 2003.
成績評価の方法・基準
／Grading method/criteria		 Asai's class: Final Exam. 100%. (More than 80% of homework assignments should be submitted.) Full score of Final Exam. is approx. 125 points. The raw score p is converted to a scaled score s by the formula: s=80+(p-80)/2 (p>80), s=p (p<80) (in principle).

Li's class: Homework 26% (2 x 13 Assignments, Attendance > 2/3), Final Exam 74%.
履修上の留意点
／Note for course registration		 Preferably prerequisite courses: Differential and Integral Calculus I, Differential and Integral Calculus II.

Other related courses: Fourier analysis, Electromagnetism.
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 ~k-asai/classes/holm/ (Directory for Asai's class)

http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html (Handouts and Exercises for Asai's class)

https://github.com/uoaworks/complex-analysis (Lecture Notes and Assignments for Xiang Li's Class)


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開講学期
／Semester		 2020年度／Academic Year 　1学期 ／First Quarter
対象学年
／Course for;		 2nd year
単位数
／Credits		 2.0
責任者
／Coordinator		 TSUCHIYA Takahiro
担当教員名
／Instructor		 TSUCHIYA Takahiro, SU Chunhua, LUBASHEVSKIY Igor
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/05/19
授業の概要
／Course outline		 (Prof. Lubashevskiy, I. is in charge of ICTG class.)
Probability and Statistics is the most useful area in mathematics. We present an introduction to Probability and Statistics for 2nd year students.
授業の目的と到達目標
／Objectives and attainment
goals		 As a study of random variation and statistical inference, Probability and Statistics are important in computer science and other wide areas of Mathematical sciences. The words of error, mean, variance, correlation, estimation are used most often. But, their definitions are not well known and
badly understood. In this course, we explicitly explain these words and concepts of Probability and Statistics. They are useful knowledge for students. Moreover, statistical analysis is the basis on solve statistic problems in research and business.
授業スケジュール
／Class schedule		 1. basis of statistics
2. 1-dim data
3. 2-dim data
4. probability
5. random variable
6. probability distribution
7. multi-dimensional probability distribution
8. Law of large number
9. sample distribution
10. sample from Gaussian distribution
11. estimation 1(mean)
12. estimation 2(variance)
13. hypothesis test 1 (mean)
14. hypothesis test 2(variance)
教科書
／Textbook(s)		 Tokei kaiseki Nyumon ( Tokyo Univ Press )
成績評価の方法・基準
／Grading method/criteria		 Mini-Tests 60 and reports 40
履修上の留意点
／Note for course registration		 Formal prerequisites:M3 Calculus I or M4 Calculus II
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 An Introductiion to Probability Theory and its Application, Vol 1
Tokeigaku Nyumon (Tokyo Univ Press)


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開講学期
／Semester		 2020年度／Academic Year 　2学期 ／Second Quarter
対象学年
／Course for;		 3rd year
単位数
／Credits		 2.0
責任者
／Coordinator		 ASAI Nobuyoshi
担当教員名
／Instructor		 ASAI Nobuyoshi, LI Xiang
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/01/30
授業の概要
／Course outline		 In this course students study properties of sets which have algebraic operations on two elements. According to the condition of operations, the set is called as group, ring and field. Treating abstract algebraic operations derives general properties for these algebraic objects. As fundamentals in computer science students have already studied ``Algorithms and Data Structures''and ``Discrete Systems''. In these courses abstract algorithms, data structures and discrete systems have been introduced and these abstract thinking can be applied to solve many practical problems. These esprits of algebraic systems are the same, and one of basis of the powerful tools to describe, think and solve many practical problems. Using abstract algebraic discussions, coding theory, public key encryption, random number generation and other interesting applications will be introduced. In this lectures proofs of the theorems will be explained simply and the meaning of the theorems and their applications will be enhanced. Furthermore, abstract thinking and deductive thinking will be taught. In order to make students understand the paper exercises and programming exercises will be given. Also quizzes and homework will be carried out to confirm students understanding.
授業の目的と到達目標
／Objectives and attainment
goals		 Understand algebraic structures and their applications
In this course, we will mainly study the following topics: algebraic operation and structure, semi-group, group, normal subgroup, quotient group, homomorphism theorem, finite group, direct product(direct sum) decomposition, symmetric group, general linear group, ring, matrix ring, ideal and quotient ring, Chinese Remainder theorem, prime and maximal ideal, localization, principal ideal ring, unique factorization ring, Euclidean domain, polynomial ring, field, field extension, algebraic extension, minimal decomposition field, finite field, constructable, M random sequence, coding theory.
授業スケジュール
／Class schedule		 Class schedules and topics might be different by lecturers. The following is a tentative example. For detailed information, please refer to the web page of each class.
01 promenade to algebraic system
02 remainder of integer and polynomial
03 group(1): Lagrange theorem
04 group(2): quotient group and homomorphism theorem
05 group(3): analysis of group structure
06 applications of group
07 Mid-exam
08 ring and field(1): ideal, quotient ring
09 ring and field(2): polynomial ring
10 ring and field(3): reversible
11 application(1): quotient field and operator theory
12 ring and field(4): extension of field
13 application(2): M-sequence random number generation
14 application(3): error correct coding
教科書
／Textbook(s)		 N. Asai class
杉原，今井，工学のための応用代数，共立出版 (1999).

S. Ding class
Mainly uses hand out.
成績評価の方法・基準
／Grading method/criteria		 Mid-term Exam. 30%, Final Exam. 40%, Quiz 10%, Homework 20%
履修上の留意点
／Note for course registration		 Kowledge and skill of the following classes are required:
Linear algebras I, & II, Discrete Systems
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 N.Asai class homepage:
http://hare.u-aizu.ac.jp/AppA/2020

S.DIng class homepage:
Will be announced  in the class


N. Asai's practical work experience:
1997-2000  Researcher, WaveFront Co. Ltd.
2002-2003 Guest researcher, National Institute of Environmental Study
2001-2010 Collaborative Research with Asahi Glass Company

He has practical work experience at Wave Front Co. Ltd for numerical simulations, modelings and high performance computings from 1997 to 2000. After joined to U. of Aizu, he has continued with companies on the mentioned topics.
These experiences of designing models, data structure and algorithms are deeply related with this class topics especially on designing model and algorithms.


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開講学期
／Semester		 2020年度／Academic Year 　3学期 ／Third Quarter
対象学年
／Course for;		 3rd year
単位数
／Credits		 2.0
責任者
／Coordinator		 SCHMITT Lothar M.
担当教員名
／Instructor		 SCHMITT Lothar M.
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/02/19
授業の概要
／Course outline		 We study the ideas of topology from the application perspective of analysis. For this purpose, the definition of the real numbers IR is reviewed and it is shown how the sup-axiom in IR relates to the existence of limits. In the major part of the course, the relationship between
(1) limits in IR^n
(2) the canonical topology in IR^n
(3) continuous functions is discussed.
The relationship between these structures is studied extensively. As an application, we show, e.g., that compactness and associated uniform continuity are the ingredients which make integration work.

The second major part of this course introduces a rigorous treatment of the concept of continuous deformation (homotopy) and the fundamental groups which one can construct using homotopy equivalence classes of continuous functions on a set with a given topology. As an application we show the fundamental theorem of algebra at the end of the course.
授業の目的と到達目標
／Objectives and attainment
goals		 Learn topological concepts in the context of the geometry in finite dimensional vector spaces and the natural (Euclidean) norm including embedded objects such as the torus in IR^3. Relate the concepts of supremum, limit, topology and continuous functions. Show how these concepts and homotopy apply in other mathematical disciplines.
授業スケジュール
／Class schedule		 Review of the Real Numbers IR, the Supremum, and the Vector Space IR^n (I-II).
Limits in IR^n --- Definition, Estimates, Basic Properties and Theorems (III-IV).
Topology (mathematical object) and relation to Limits (V-VI).
Continuous Functions and relation to Limits and Topology (VII-IX).
Applications to concepts in Analysis --- Differentiation and Integration (X).
Homotopy (continuous deformation of functions) and Associated Groups (XI-XIII).
Fundamental Theorem of Algebra and other Applications of Homotopy (XIV).
教科書
／Textbook(s)		 [1] A Geometric Introduction to Topology (Dover Books on Mathematics) C. T. C. Wall
[2] Analysis I (Addison-Wesley) S. Lang
成績評価の方法・基準
／Grading method/criteria		 Required for admission to final exam:
[a] sufficient attendance (2/3), and
[b] participation in online quizzes (2/3, this is for training only).
The final exam determines the grade only.
履修上の留意点
／Note for course registration		 This is a follow-up course on topology and the reader is supposed to be familiar with the introductory course on topology given in UoA.
Formal prerequisites: None
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Lecture Notes can be obtained from L.M.Schmitt on CD. (email: L@LMSchmitt.de)


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開講学期
／Semester		 2020年度／Academic Year 　3学期 ／Third Quarter
対象学年
／Course for;		 3rd year
単位数
／Credits		 2.0
責任者
／Coordinator		 TAKAHASHI Shigeo
担当教員名
／Instructor		 TAKAHASHI Shigeo
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/09/07
授業の概要
／Course outline		 This course will be delivered remotely (online) for countermeasures against COVID-19. Course details will be announced via the mailing list to all registrants prior to the start of the course.

The contents of this course include the fundamental concepts on topology, which serves as the foundation of modern mathematics, and its applications.
In topology, two shapes can be considered as the same if we can transform one to the other smoothly while maintaining the underlying connectivity within the shapes. This idea leads to a new approach to characterizing the global features of shapes and has promoted the development of computer science and engineering so far. Specifically, its contribution ranges from the fundamental representations of graphs and surfaces to the state-of-the-art techniques for extracting features in data analysis. This course will facilitate to study various important concepts for classifying shapes from a topological viewpoint, by employing specific one and two-dimensional shapes as examples.
授業の目的と到達目標
／Objectives and attainment
goals		 The objectives of this course are to study several topological invariants or characteristics that describe the global structures of geometric objects, such as the Euler characteristic and the Betti number, together with the topological classification of closed surfaces using their opened ones along boundaries. This will be followed by understanding the relationship between the geometric objects (e.g., the topological space) and algebraic objects (e.g., the Homology group), which allows us to classify the geometric objects by calculating the associated Homology groups.
授業スケジュール
／Class schedule		 This course will be delivered remotely (online) for countermeasures against COVID-19. Course details will be announced via the mailing list to all registrants prior to the start of the course.

01) Guidance
02) 1-dimensional topology: Seven Bridges of Königsberg and Eulerian paths
03) 1-dimensional topology: Connectivity and Euler-Poincare theorem
04) 1-dimensional topology: Embedding into Euclidean space
05) 2-dimensional topology: Closed surfaces
06) 2-dimensional topology: Surfaces with boundary
07) 2-dimensional topology: Classification of closed surfaces
08) 2-dimensional topology: Connectivity and Euler-Poincare theorem
09) n-dimensional topology: Complexes and polyhedra
10) Homology: Groups and Homomorphisim
11) Homology: Chain complexes
12) Homology: Homology groups
13) Homology: 0-dimensional and 1-dimensional Homology groups
14) Homology: Connectivity and Euler-Poincare theorem
教科書
／Textbook(s)		 瀬山士郎著「トポロジー：柔らかい幾何学」 増補版 日本評論社
"Topology: Yawarakai Kikagaku" by Shiro Seyama
Nippon-Hyouron Publisher
(The text book is written in Japanese)
(ISBN 978-4-535-78405-5)

The course basically follows the contents of the textbook above while you can find almost the same contents in the following book in English:
A Combinatorial Introduction to Topology" by Michael Henle Dover Books on Mathematics
(ISBN 978-0-486-67966-2)
成績評価の方法・基準
／Grading method/criteria		 Term-end exam (50%) + quizzes in class and reports on assignments (homework) (50%)
You are requested to submit your assignments electronically if we have online classes. Detailed instructions on how to submit your work will be provided during the classes.
Assignment submission is due before the next class/session starts.
Those who take the term-end exam are only eligible for credit.
You are requested to attend more than 2/3 of sessions (i.e, 10 out of 14 sessions) and submit all the reports on assignments to take the term-end exam.
Cheating (include proxy submission of quizzes reports) will be strictly penalized.
(Make-up exam will be provided according to the guidelines of absence prepared by the Student Affairs Section.)
履修上の留意点
／Note for course registration		 No formal prerequisites.
It is preferable to have earned credits of "Linear Algebra I" and "Applied Algebra."
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Stephen Barr, "Experiments in Topology," Dover Publications, INC
Michael Henle, "A Combinatorial Introduction to Topology", Dover Books on Mathematics
https://web-int.u-aizu.ac.jp/~shigeo/course/topology/index.html (open to campus only)


Back
開講学期
／Semester		 2020年度／Academic Year 　3学期 ／Third Quarter
対象学年
／Course for;		 3rd year
単位数
／Credits		 2.0
責任者
／Coordinator		 SCHMITT Lothar M.
担当教員名
／Instructor		 SCHMITT Lothar M.
推奨トラック
／Recommended track		 －
履修規程上の先修条件
／Prerequisites		 －
使用言語
／Language		 －

更新日／Last updated on 2020/02/19
授業の概要
／Course outline		 We study the ideas of topology from the application perspective of analysis. For this purpose, the definition of the real numbers IR is reviewed and it is shown how the sup-axiom in IR relates to the existence of limits. In the major part of the course, the relationship between
(1) limits in IR^n
(2) the canonical topology in IR^n
(3) continuous functions is discussed.
The relationship between these structures is studied extensively. As an application, we show, e.g., that compactness and associated uniform continuity are the ingredients which make integration work.

The second major part of this course introduces a rigorous treatment of the concept of continuous deformation (homotopy) and the fundamental groups which one can construct using homotopy equivalence classes of continuous functions on a set with a given topology. As an application we show the fundamental theorem of algebra at the end of the course.
授業の目的と到達目標
／Objectives and attainment
goals		 Learn topological concepts in the context of the geometry in finite dimensional vector spaces and the natural (Euclidean) norm including embedded objects such as the torus in IR^3. Relate the concepts of supremum, limit, topology and continuous functions. Show how these concepts and homotopy apply in other mathematical disciplines.
授業スケジュール
／Class schedule		 Review of the Real Numbers IR, the Supremum, and the Vector Space IR^n (I-II).
Limits in IR^n --- Definition, Estimates, Basic Properties and Theorems (III-IV).
Topology (mathematical object) and relation to Limits (V-VI).
Continuous Functions and relation to Limits and Topology (VII-IX).
Applications to concepts in Analysis --- Differentiation and Integration (X).
Homotopy (continuous deformation of functions) and Associated Groups (XI-XIII).
Fundamental Theorem of Algebra and other Applications of Homotopy (XIV).
教科書
／Textbook(s)		 [1] A Geometric Introduction to Topology (Dover Books on Mathematics) C. T. C. Wall
[2] Analysis I (Addison-Wesley) S. Lang
成績評価の方法・基準
／Grading method/criteria		 Required for admission to final exam:
[a] sufficient attendance (2/3), and
[b] participation in online quizzes (2/3, this is for training only).
The final exam determines the grade only.
履修上の留意点
／Note for course registration		 This is a follow-up course on topology and the reader is supposed to be familiar with the introductory course on topology given in UoA.
Formal prerequisites: None
参考(授業ホームページ、図書など)
／Reference (course
website, literature, etc.)		 Lecture Notes can be obtained from L.M.Schmitt on CD. (email: L@LMSchmitt.de)


Responsibility for the wording of this article lies with Student Affairs Division (Academic Affairs Section).

E-mail Address: sad-aas@u-aizu.ac.jp