AY 2018 Undergraduate School Course Catalog AY 2018 Undergraduate School Course Catalog AY 2018 Undergraduate School Course Catalog
| 2019/01/30 |
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| 開講学期 ／Semester |
2018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 |
Shigeru Watanabe |
| 担当教員名 ／Instructor |
Takao Maeda, Kazuto Asai, Shigeru Watanabe, Chunhua Su, Cong-Thang Truong |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/05/31 |
|---|---|
| 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 |
2018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 |
Shigeru Watanabe |
| 担当教員名 ／Instructor |
Takao Maeda, Shigeru Watanabe, Yodai Watanabe, Kazuya Matsumoto, Lothar M. Schmitt |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/05/31 |
|---|---|
| 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 |
2018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 |
Hiroshi Kihara |
| 担当教員名 ／Instructor |
Hiroshi Kihara, Yoshiko Ogawa, Nobuyoshi Asai, Peng Li |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/05/31 |
|---|---|
| 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 |
2018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 |
Hiroshi Kihara |
| 担当教員名 ／Instructor |
Hiroshi Kihara, Takeaki Sampe, Takahiro Tsuchiya, Yodai Watanabe, Kenta Ofuji, Xiang Li |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/05/31 |
|---|---|
| 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 |
2018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 |
Takao Maeda |
| 担当教員名 ／Instructor |
Takao Maeda, Takahiro Tsuchiya, Shuxue Ding, Xiang Li |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/05/31 |
|---|---|
| 授業の概要 ／Course outline |
(ICTG class starts in Q3.And Prof. Li, X. is in charge of the class.) 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) |
Handouts by each instructor and the following books: Tsuchiya’s class: Gen-ichiro Sunouchi, Fourier analysis and its applications (SAIENSU-sha) Ding’s class: Handout is a main textbook. As a side reader, Gen-ichiro Sunouchi, Fourier analysis and its applications (SAIENSU-sha) Maeda’s class: Gen-ichiro Sunouchi, Fourier analysis and its applications (SAIENSU-sha) |
| 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: M-3 Calculus I or M-4 Calculus II, M-1 Linear algebra or M-2 linear algebra II Important related courses: M-6 Complex analysis, A-3 Image processing, A-8 Digital signal 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 |
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| 開講学期 ／Semester |
2018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 |
Kazuto Asai |
| 担当教員名 ／Instructor |
Masahide Sugiyama, Kazuto Asai, LI Xiang |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2017/11/10 |
|---|---|
| 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: Sugiyama's class: Nattokusuru Fukusokansu (2000), Kodansha, by Yoshitaka Onodera. Asai's class: Handout is a main textbook. As a side reader, Nattokusuru Fukusokansu (2000), Kodansha, by Yoshitaka Onodera. Kihara's class: Kogakukiso Fukusokansuron (2007), Saiensu-sha, by tetsu yajima, masayuki oikawa. |
| 成績評価の方法・基準 ／Grading method/criteria |
Sugiyama's class: Determine after discussion with students (Last academic year: Mid-term Exam. 30%, Final Exam. 40%, Quiz 10%, Homework 20%) 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). Kihara's class: Each homework 2 points x 13 assignments, Final Exam. 74 points. |
| 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.) |
http://web-int.u-aizu.ac.jp/~sugiyama/Lecture/CA/2018/welcome.html (Home page for Sugiyama's class) ~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 |
2018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 |
Takahiro Tsuchiya |
| 担当教員名 ／Instructor |
Takahiro Tsuchiya, Chunhua Su, Yosuke Kira, Igor Lubashevskiy |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/05/31 |
|---|---|
| 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 |
Test 80 and reports 20 |
| 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 |
2018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 |
Lothar M. Schmitt |
| 担当教員名 ／Instructor |
Lothar M. Schmitt |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
－ |
| 更新日／Last updated on | 2018/02/21 |
|---|---|
| 授業の概要 ／Course outline |
The course has three parts: (1) Review of Boolean Logic and Propositional Logic. ============================================= This serves as a review to bring students with different prerequisites to the same level. Topics discussed include: Mathematical propositions and truth values in {0=false,1=true}; Set theory; Boolean algebras as generalizing principle of the latter two subjects; Stone's Theorem. Some computer applications of the material is discussed as well. (2) Fuzzy Sets and Applications. =========================== Here the structure of (1) is generalized to truth values in the full interval [0,1] in IR. This allows to define fuzzy subsets of a given set, the embedding of regular sets into the set of fuzzy subsets. A special case of fuzzy subsets are fuzzy relations. The main application of this chapter is the rigorous development of the Tamura classification method based upon a given fuzzy relation. An indication how to use this method in image reconstruction is given. See: http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5408605 (3) Model Theory. ================= In mathematics, model theory is the study of (classes of) mathematical structures (e.g., groups, fields, graphs, universes of set theory) using tools from mathematical logic. A theory is built which allows to say in a proper calculus: One can verify a formula or theorem in any set interpretation, if and only if one can find a formal proof of that formula or theorem by means of strict textual manipulations according to a fixed set of proof-rules without referring to special properties of any example. This is known as Goedel's completeness theorem. |
| 授業の目的と到達目標 ／Objectives and attainment goals |
To learn Logic and Applications from three perspectives: classical Boolean Logic, Fuzzy Logic, and Model Theory (logic of proofs, first order logic). |
| 授業スケジュール ／Class schedule |
Boolean Logic I-III. Fuzzy Logic and Relations IV-VII. Model Theory VIII-XIV. |
| 教科書 ／Textbook(s) |
[1] Online Lecture Notes (email L@LMSchmitt.de for the link). [2] Pattern Classification Based on Fuzzy Relations, S. Tamura et al,: IEEE Transactions on Systems, Man and Cybernetics, 1971 [3] Mathematical Logic (Springer Undergraduate Texts in Mathematics) H.-D. Ebbinghaus, J. Flum, W. Thomas |
| 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. |
| 履修上の留意点 ／Note for course registration |
Rather self-contained course. Understanding basic logic and real numbers is welcome. Formal prerequisites: None |
| 参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.) |
Online Lecture Notes (email L@LMSchmitt.de for the link). |
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| 開講学期 ／Semester |
2018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 |
Lothar M. Schmitt |
| 担当教員名 ／Instructor |
Lothar M. Schmitt |
| 推奨トラック ／Recommended track |
－ |
| 履修規程上の先修条件 ／Prerequisites |
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| 更新日／Last updated on | 2018/02/21 |
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| 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 sub-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) |