AY 2021 Undergraduate School Course Catalog AY 2021 Undergraduate School Course Catalog AY 2021 Undergraduate School Course Catalog
| 2022/01/30 |
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| 開講学期 ／Semester |
2021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 | 2021/01/20 |
|---|---|
| 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.) 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 |
2021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, 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 | 2021/01/20 |
|---|---|
| 授業の概要 ／Course outline |
(ICTG class starts in Q1.And Prof. Schmitt is in charge of the class.) 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 |
2021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 | 2021/02/03 |
|---|---|
| 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 |
2021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 | 2021/02/03 |
|---|---|
| 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 |
2021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 | 2021/02/04 |
|---|---|
| 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, Kachi class] Gen-ichiro Sunouchi; Fourier analysis and its applications (SAIENSU-sha) and Handouts [Li class] Handouts [Lu class] Handouts |
| 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.) |
[Tsuchiya, Kachi class] 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) [Lu class] Keyzig; Advanced Engineering Mathematics |
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| 開講学期 ／Semester |
2021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 | 2021/01/14 |
|---|---|
| 授業の概要 ／Course outline |
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), Quiz 6%, Final Exam 68%. |
| 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) |
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| 開講学期 ／Semester |
2021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, HASHIMOTO Yasuhiro, ASAI Nobuyoshi |
| 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 | 2021/01/21 |
|---|---|
| 授業の概要 ／Course outline |
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 |
2021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 | 2021/02/02 |
|---|---|
| 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). X. Li class Mainly uses hand out. |
| 成績評価の方法・基準 ／Grading method/criteria |
N. Asai class: Mid-term Exam. 30%, Final Exam. 40%, Quiz 10%, Homework 20% X. Li class: Final Exam 60%, Quiz 15%, Homework 25% |
| 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 is under LMS: https://elms.u-aizu.ac.jp X. Li 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 |
2021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 |
MORI Kazuyoshi |
| 担当教員名 ／Instructor |
MORI Kazuyoshi |
| 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 | 2021/08/27 |
|---|---|
| 授業の概要 ／Course outline |
This course consists of logic, deduction of logic, and formalization of mathematical objects by logic. This course has four parts: (i) Review: We first review basic set theory and its related topics such as binary relations and functions. We then review Boolean logic. They are to bring students with different prerequisites to the same level. (ii) Basic logics: We introduce two different level logics: Propositional Logic and First-Order Logic. Propositional logic is a classical logic with fundamental operations. First-order logic is the fundamental tool to describe many mathematically theoretical materials. (iii) Resolution Principle: Resolution Principle is a deduction method of the first-order logic. We also study Skolem standard forms, Herbrand universe, and Herbrand's theorem. (iv) Logical Formalization of Natural Number Theory: We formalize the natural number theory under Peano axioms by using first-order logic. |
| 授業の目的と到達目標 ／Objectives and attainment goals |
Logic is considered the language of mathematics. This leads that, in computer science and engineering, logic is the foundation and an important tool. In this course, we study mathematical logic and also symbolic logic. The first objective is to understand both propositional logic and first-order logic. The second objective is to understand Resolution Principle of propositional logic and first-order logic. This is mechanical reasoning. The third objective is to formalize a mathematical object by the first-order logic. To do so, we, in this course, employ the Natural number theory with Peano axioms. |
| 授業スケジュール ／Class schedule |
Each class has a lecture and may have exercises and homework. 1-2 Review (Set Theory and its related topics, Boolean Logic) 3-6 Basic Logics (Propositional Logic, First-Order Logic) 7-10 Resolution Principle (Skolem Standard Forms, Herbrand Universe, Herbrand's Theorem, Resolution Principle) 11-14 Logical Formalization of Natural Number Theory under Peano Axioms |
| 教科書 ／Textbook(s) |
Class materials will be distributed. |
| 成績評価の方法・基準 ／Grading method/criteria |
40% Final Examination 30% Midterm Examination 30% Exercises/Homeworks |
| 参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.) |
Japanese References: 福山克 「数理論理学」（培風館） 前原昭二 「記号論理入門」（日本評論社） 山田 俊行「はじめての数理論理学」(森北出版) English References: H. Enderton, H.B. Enderton, "A Mathematical Introduction to Logic (2nd Ed)," Academic Press, 2000. J.R. Shoenfield, "Mathematical Logic," Routledge, 2001. |
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| 開講学期 ／Semester |
2021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 |
Classes are expected to be given in Japanese while will be switched to English if we have ICTG students. |
| 更新日／Last updated on | 2021/08/06 |
|---|---|
| 授業の概要 ／Course outline |
The contents of this course include the fundamental concepts on topology, which serves as the foundation of modern mathematics, and its applications. In topology, also known as "geometry for soft materials," 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 |
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: Development of Closed Surfaces 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 homomorphism 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%) 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." Homework assignments submitted after the deadline will be counted but will not be included in the evaluation. Even if all homework assignments are submitted before the deadline, the credit may not be granted if the term-end exam score is inferior. |
| 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) |