AY 2023 Undergraduate School Course Catalog AY 2023 Undergraduate School Course Catalog AY 2023 Undergraduate School Course Catalog
| 2024/01/30 |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
ASAI Kazuto, WATANABE Shigeru, SU Chunhua, KACHI Yasuyuki, VIGLIETTA Giovanni |
| 推奨トラック ／Recommended track |
－ |
| 先修科目 ／Essential courses |
－ |
| 更新日／Last updated on | 2023/09/25 |
|---|---|
| 授業の概要 ／Course outline |
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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency Codes] C-MS-001, C-MS-002 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 Viglietta (ICTG) class Lipschutz-Lipson, Schaum's Outlines Linear Algebra (sixth edition), McGraw-Hill Education |
| 成績評価の方法・基準 ／Grading method/criteria |
Asai class Final exam. 100% Watanabe class Final 100% Kachi class Final exam. 100% Viglietta (ICTG) class Final exam. 100% |
| 参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.) |
Course website for Asai's class: http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
WATANABE Shigeru, WATANABE Yodai, MATSUMOTO Kazuya, HASHIMOTO Yasuhiro, KACHI Yasuyuki, SU Chunhua |
| 推奨トラック ／Recommended track |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses) MA01 Linear algebra I |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 授業の概要 ／Course outline |
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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency Codes] C-MS-003 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 class Final 100% Kachi class Final exam. 100% Hashimoto class Final exam. 100% |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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, FUJIMOTO Yusuke |
| 推奨トラック ／Recommended track |
－ |
| 先修科目 ／Essential courses |
－ |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency Codes] C-MS-007, C-MS-008 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 |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses) MA03 Calculus I |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency Codes] C-MS-009, C-MS-010 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. Spatial curves and polar equations 3. Partial differential coefficients 4. Differentiation of composite functions 5. Total differential and expansion of a function 6. Local minimum/maximum of a function 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 |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
KACHI Yasuyuki |
| 担当教員名 ／Instructor |
TSUCHIYA Takahiro, LI Xiang, LYU Guowei, KACHI Yasuyuki |
| 推奨トラック ／Recommended track |
－ |
| 先修科目 ／Essential courses |
Prerequisite: MA01 Linear algebra I, MA02 Linear algebra II, MA03 Calculus I, MA04 Calculus II |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 授業の概要 ／Course outline |
The origin of Fourier analysis goes back to the study of heat transmission (mathematically, "the heat equation") by Joseph Fourier in the early 19th century. Fourier's idea was to “express an arbitrary function using trigonometric functions”. This very school of thought has led to a cornucopia of useful results in applied science, with one major caveat: Whether one can justify the conclusions as Fourier's method dictates has long remained murky, as it hinged on the notion of convergence (limits), which mathematicians in Fourier's time fell short of understanding in rigorous terms (if judged retrospectively by today's standards). Today, mathematically sound mathematical theories are available that underpin the rigor of those applied science fields in which Fourier's theory plays a part. Fourier analysis has earned an enduring status as one of the most important (as in indispensable) and basic mathematical theories. Fourier analysis is a sine qua non not only in studying differential equations, but also in modern applications, e.g., signal processing (including, but not limited to, image information and sound information). Emphasis will be on practical skills. We may not necessarily dwell too much on mathematical proofs. We thoroughly teach you how to convert a given function into a sum of trigonometric/exponential functions - as a starter. Through solving exercise problems, students will become conversant with calculations and basic theorems of Fourier analysis. |
| 授業の目的と到達目標 ／Objectives and attainment goals |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency code] C-MS-011 Part1. Fourier series expansion Students will understand what an orthonormal system of the space of functions over finite intervals are. Bessel's inequality. Students will understand the notion of Fourier series. Students will learn how to calculate Fourier series for polynomials, exponential/trigonometric functions, etc. Part2. Properties of Fourier series Students will learn sufficient conditions for a Fourier series to converge, as well as the relationship between a given function and the Fourier series derived from it. Parseval’s theorem and applications. Weierstrass' theorem. Part3. Fourier integral Students will understand that the notion of Fourier integrals naturally comes to the fore through a heuristic observation, i.e., by way of shifting gears from functions over a finite interval to functions over (-∞, +∞). Students will learn how to calculate Fourier transforms and Fourier inverse transforms of a given function. Some of the more archetypal examples involve an antiderivative of $e^{-x^2})$ and the like. Students will also understand the notion of convolutions (of the functions), as well as the relationship between the convolutions and the Fourier transforms of the functions. Students will learn how to calculate them. Part4. Laplace transform Students will understand that Laplace transforms arise as a special case of Fourier transforms. Students will learn how to apply the notion of Laplace transforms to solve a certain type of ordinary differential equations. The case the solution is expressed by a convolution of Laplace transforms will be addressed. Part5. Discrete Fourier transform Students will understand that, through some heuristic observation on step functions, paired with a scrupulous use of the theory of Fourier series, one naturally arrives at the theory of discrete Fourier transforms (DFT). Students will understand the similarity between DFT and the theory introduced in Part 2. Students will also understand that fast Fourier transforms (FFT) are a souped-up version of DFT. Students will appreciate how fast FFTs are. |
| 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, Fourier transform) 8: Part3. Fourier integral (Parseval's theorem, convolution) 9: Part3. Fourier integral (Exercise) 10: Part4. Laplace transform (Introduction) 11: Part4. Laplace transform (Ordinary differential equations with constant coefficients) 12: Part4. Laplace transform (Exercise) 13: Part5. Discrete Fourier transform (Introduction) 14: Part5. Discrete Fourier transform (FFT(Fast Fourier Transform)) |
| 教科書 ／Textbook(s) |
[Tsuchiya's section, Kachi's section] Gen-ichiro Sunouchi; Fourier analysis and its applications (Science-sha) and Handouts [Li's section] Handouts [Lu's section] Handouts |
| 成績評価の方法・基準 ／Grading method/criteria |
Regular quizzes,* homework ** and the final exam. * ** Whether these are given is up to the instructor. |
| 履修上の留意点 ／Note for course registration |
Prerequisite: MA01 Linear algebra I, 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's section, Kachi's section] 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's section] Gen-ichiro Sunouchi; Fourier analysis and its applications (Science-sha) [Lu's section] Kreyszig; Advanced Engineering Mathematics |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
| 推奨トラック ／Recommended track |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses) MA03 Calculus I |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. 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. polynomials, rational functions, 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: A textbook is given as a handout. All necessary materials can be downloaded. Although not necessary for the class, famous side readers are as follows: Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis), Elias M. Stein, Rami Shakarchi, Princeton University Press, 2011. Sinban Hukusokaiseki (kisosuugaku 8), Reiji Takahashi, University of Tokyo Press, 1990. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, Lars Ahlfors, AMS Chelsea Publishing, 385, 2021. 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 completed.) 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 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.) |
Home page of Asai's class: http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses) MA03 Calculus I MA04 Calculus II MA01 Linear algebra I MA02 Linear algebra II |
| 更新日／Last updated on | 2023/12/19 |
|---|---|
| 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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. 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. C-DS-001-1,C-DS-002-1, C-DS-007, C-MS-013, C-MS-014 |
| 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 |
| 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) |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses) MA01 Linear algebras I FU03 Discrete Systems |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. 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 |
Related course: FU02 Information Theory |
| 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. |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023年度／Academic Year 1学期 ／First 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 |
－ |
| 先修科目 ／Essential courses |
－ |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 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 as follows: (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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency Codes] C-AL-004-4. C-AL-005-5, C-CN-001-2, C-DS-001-3, C-DS-002-2, C-MS-005 Logic is considered the language of mathematics. This leads that, in computer science and engineering, the 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 the 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/Others |
| 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) 鈴木 登志雄 「ろんりの相談室 」(日本評論社) 鈴木 登志雄 「例題で学ぶ集合と論理」(森北出版) 小島 寛之 「証明と論理に強くなる」(日本評論社) English References: H. Enderton, H.B. Enderton, "A Mathematical Introduction to Logic (2nd Ed)," Academic Press, 2000. J.R. Shoenfield, "Mathematical Logic," Routledge, 2001. |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023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 |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses) MA01 Linear Algebra I |
| 更新日／Last updated on | 2023/03/20 |
|---|---|
| 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 |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. [Competency Codes] C-DS-003, C-DS-006-1, C-MS-002, C-MS-004, C-MS-006 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) |
"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 |
Final exam (50%) + quizzes in class (50%) Those who take the final exam are only eligible for credit. Cheating (including proxy attendance and proxy submission of quizzes reports) will be strictly penalized. (We will provide a make-up exam according to the guidelines of absence prepared by the Student Affairs Section.) |
| 履修上の留意点 ／Note for course registration |
Note: It is preferable to have acquired credits for "M08 Applied Algebra" also. Even if answers to all quizzes are submitted during the classes, the credit may not be granted if the final exam score is inferior. This class is basically given in Japanese, but we will consider conducting the class in English if ICTG students plan to attend all class sessions. Please be sure to consult with the instructor before registering for the course if you wish to participate in the class in English. |
| 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 Please refer to the course page maintained on the Learning Management System (LMS). |
| Open Competency Codes Table Back |
| 開講学期 ／Semester |
2023年度／Academic Year 4学期 ／Fourth Quarter |
|---|---|
| 対象学年 ／Course for; |
3rd year |
| 単位数 ／Credits |
2.0 |
| 責任者 ／Coordinator |
VIGLIETTA Giovanni |
| 担当教員名 ／Instructor |
VIGLIETTA Giovanni |
| 推奨トラック ／Recommended track |
－ |
| 先修科目 ／Essential courses |
Courses preferred to be learned prior to this course (This course assumes understanding of entire or partial content of the following courses): PL01 Intro. Programming, FU03 Discrete Systems, MA01 Linear Algebra I, MA10 Introduction to Topology |
| 更新日／Last updated on | 2023/08/15 |
|---|---|
| 授業の概要 ／Course outline |
This course covers a selection of modern applications of geometry and topology to computer science, with particular emphasis on computer graphics and data analysis. In the first part of the course, we explore some fundamental techniques revolving around the concepts of "Ray Marching" and "Signed-Distance Field", which are standard in both real-time and pre-processed 3D rendering. As an application of these techniques, we learn how to "paint with math" and create high-quality graphics with purely mathematical formulas, using shadertoy.com as our main platform. A sophisticated example is found here: https://www.shadertoy.com/view/ld3Gz2 In the second part of the course, we cover an emerging technique in topological data analysis called "Persistent Homology". This technique extracts information from a data set by viewing it as an approximation of a topological space and studying its topological properties that remain persistent across several levels of approximation. We begin by reviewing homology's basic definitions and properties, then delve into algorithms for calculating homology groups of simplicial complexes. As a practical application, we construct a simple software tool to perform Persistent Homology on reasonably sized datasets. |
| 授業の目的と到達目標 ／Objectives and attainment goals |
[Corresponding Learning Outcomes] (C)Graduates are able to apply their professional knowledge of mathematics, natural science, and information technology, as well as the scientific thinking skills such as logical thinking and objective judgment developed through the acquisition of said knowledge, towards problem solving. Objectives: Part 1. Students will learn applications of linear algebra and analytic geometry to real-time and pre-processed 3D rendering. Students will understand the concepts of "Ray Marching" and "Signed-Distance Field" and apply them to create original demos on shadertoy.com. Part 2. Students will learn Persistent Homology as an application of topology to data analysis. Students will understand how to represent a simplicial complex as a data structure and how to compute its homology groups. Students will also be able to implement a rudimentary software tool to perform Persistent Homology on data sets. |
| 授業スケジュール ／Class schedule |
Part 1: Lesson 1) Introduction to GLSL and shadertoy.com Lesson 2) Basic 2D shaders and fractals Lesson 3) Ray Marching and Signed-Distance Fields Lessons 4-7) Sculpting and painting 3D scenes with a variety of mathematical techniques Part 2: Lesson 8) Review of group theory Lesson 9) Review of topology and homology Lesson 10) Persistent Homology algorithms Lessons 11-13) Implementing Persistent Homology algorithms Lesson 14) Experimenting with Persistent Homology |
| 教科書 ／Textbook(s) |
Lecture notes and other study materials will be provided by the teacher. |
| 成績評価の方法・基準 ／Grading method/criteria |
Each student can choose between two options: Option 1: Programming project. The student will implement an original shader on shadertoy.com by utilizing the theory learned in the course and combining it with features found in other shaders. The student should be able to explain the purpose of each line of code in his/her project. Each project is individual; it can be started at any time during the course and has to be completed by the end of the term. Option 2: Written exam. In a classroom setting, the student will answer written comprehension questions about the theories discussed in the course. Additionally, there will be exercises to test the student's practical understanding of the concepts learned. |
| 履修上の留意点 ／Note for course registration |
Students are expected to have a working knowledge of at least one programming language, simple data structures, discrete mathematics, and basic linear algebra. Some familiarity with computer graphics, elementary group theory, basic topology and homology is preferred but not required. |
| 参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.) |
Most of Part 1 is based on articles and tutorials by Inigo Quilez, a software engineer and 3D artist who worked for Pixar, Oculus and Adobe: https://iquilezles.org/articles/ Some study materials will also be taken directly from https://www.shadertoy.com For Part 2, a good reference are the two articles https://www.math.uri.edu/~thoma/comp_top__2018/stag2016.pdf https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf A good introduction to homology theory is Hatcher's book: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html |