Back

開講学期 ／Semester	2014年度／Academic Year 　前期 ／First Semester
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
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	2014/09/27
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 side by side 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 and matrices 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	Objectives and attainment targets 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 Vectors in plane and vectors in space 2 Matrices of (3,3)-type and linear transformations 3 Definition of matrix and operations 4 Square matrices, regular matrices and linear mappings 5 Elementary transformations and rank 6 Simultaneous linear equations (1) 7 Simultaneous linear equations (2) 8 Definition of determinant 9 Properties of determinant 10 Expansions of determinant 11 Definition and properties of linear space 12 Basis and dimension (1) 13 Basis and dimension (2) 14 Linear subspaces 15 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	Final examination (Midterm examination and assignments depend on professors.)
履修上の留意点 ／Note for course registration	Formal prerequisites:None
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	Office hours undecided

Back

開講学期 ／Semester	2014年度／Academic Year 　後期 ／Second Semester
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 , Chikatoshi Honda
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	2014/09/27
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 side by side 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	Objectives and attainment targets 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 15 Exponential mappings 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	Final examination (Midterm examination and assignments depend on professors.)
履修上の留意点 ／Note for course registration	Formal prerequisites:None
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	Office hours undecided

Back

開講学期 ／Semester	2014年度／Academic Year 　前期 ／First Semester
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	Toshiro Watanabe , Hiroshi Kihara , Yoshiko Ogawa , Yodai Watanabe
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	2014/09/27
授業の概要 ／Course outline	本講義では、微積分の前半部分を学習する。もう一方の柱である線型代数と並行して履修することになるが、線型代数の理解なくして微積分の理解はありえない し、その逆もありえないからである。また物理では、早々に微積分が必要となり、さらにはベクトルに対する微積分まで登場する。これらを学習するためにも、 なるべく早い時期に、二つの基礎数学に接し、双方を有機的に結び付けて理解しておくことは重要である。 微積 分は、図形の面積、曲線への接線等の問題を出発点とし、ニュートンによる力学的考察に基づいて一定の基礎が築かれた。極限の概念がその根底にあるのは、い うまでもないことである。その後も、先人達によって整備された微積分は、その扱われ方の違いから、一変数と多変数のふたつにわけられる。本講義では、この うち一変数の微積分を扱う。基本的な計算技術に関しては、高校で学ぶ事柄と大差はなく、扱う関数の範囲を若干ひろげるだけのことである。異なるのは、極限 概念の取り扱いであるが、厳密な議論を行うには無理な点があるので、全面的な導入はしない。受身になりがちな講義と並行して、学生の主体的学習の場として 演習も行う。学生諸君の積極的参加を期待する。
授業の目的と到達目標 ／Objectives and attainment goals	高校で習った微積分に よりしっかりした基礎付けを行いながら、 逆三角関数や関数の展開 積分の漸化式等のより高度な内容も紹介していく。 この教科は このあとで習う微積分II、確率統計学、フーリエ解析、複素関数論だけでなく、物理やコンピュータサイエンスのあらゆる分野に進むための基礎知識となる。
授業スケジュール ／Class schedule	1. 実数の集合 2. 数列の極限 3. 関数の極限と連続関数 4. 導関数および指数関数と対数関数 5. 三角関数と逆三角関数および高次導関数 6. オイラーの公式 7. 微分、平均値の定理および関数の増減 8. テイラーの定理および関数の展開 9. 不定積分と漸化式 10. 有理関数の積分 11. 一階および二階の線型微分方程式 12. 定積分の定義および性質 13. 定積分の計算と定義の拡張、図形の計量
教科書 ／Textbook(s)	栗田稔著「新講 微積分学」学術図書、１、４４２円 米田元 著 「理工系のための微分積分入門」 サイエンス社 １８９０円
成績評価の方法・基準 ／Grading method/criteria	小テスト、期末テスト
履修上の留意点 ／Note for course registration	なし 履修規程上の先修条件：なし
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	授業で指示する

Back

開講学期 ／Semester	2014年度／Academic Year 　後期 ／Second Semester
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	Toshiro Watanabe , Hiroshi Kihara , Yoshiko Ogawa , Takeaki Sampe , Takahiro Tsuchiya
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	2014/09/27
授業の概要 ／Course outline	微積分Iに引続き、多変数関数の微積分を扱う。一変数のときと比べて、趣が異なるけれども、実際の計算は、一変数の場合に帰着してしまう。従って、授業を 大事にして基本さえおさえておけば、何ら恐れることはない。なお、基本的方針は、微積分 I と変わるところはない。また、線型代数 II の項も参照されたい。
授業の目的と到達目標 ／Objectives and attainment goals	多変数の微積分の基礎の修得を目的とする。 １変数関数に対する微分は多変数関数に対する偏微分に拡張されるが その考え方 計算は難しくない。微分を１変数関数の極大極小問題に応用したように偏微分を多変数関数の極大極小問題に応用する。 また １変数関数に対する積分は多変数関数に対する重積分に拡張されるが やはりその考え方 計算は難しくない。特に１変数関数に対する置換積分にあたる変数変換の技法に習熟してほしい。 最後に 級数についても学ぶ。特に関数項級数は フーリエ変換、複素関数論の基礎にもなる。
授業スケジュール ／Class schedule	1. 媒介変数による曲線の表示 2. 偏微分係数 3. 合成関数の微分法 4. 全微分および関数の展開 5. 関数の極大極小 6. 陰関数および曲線と曲面 7. 重積分およびその計算 8. 変数の変換による重積分 9. 面積および体積 10. 一次微分式と積分 11. 無限級数の収束、絶対収束と条件収束 12. 無限級数の和と積および関数列の極限 13. 整級数
教科書 ／Textbook(s)	栗田稔著「新講 微積分学」学術図書、１４４２円 米田元 著 「理工系のための微分積分入門」 サイエンス社 １８９０円
成績評価の方法・基準 ／Grading method/criteria	小テスト、期末テスト
履修上の留意点 ／Note for course registration	線型代数I、微積分I 履修規程上の先修条件：なし
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	授業で指示する

Back

開講学期 ／Semester	2014年度／Academic Year 　前期 ／First Semester
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
推奨トラック ／Recommended track	CF,CM,VD,CN,VH,RC,BM
履修規程上の先修条件 ／Prerequisites	(M1 or M2) & (M3 or M4)

更新日／Last updated on	2014/09/27
授業の概要 ／Course outline	本講義では、一学年に学んだ微分積分の続きであり、解析学の基礎であるフーリエ解析について学習する。フーリエ解析は、フーリエによる熱伝導の研究が出発点であるが、思想的にはギリシャのピタゴラスによる音楽理論にはじまる。音や光などの波動の研究が関係しており、理工学において幅広く利用されている。コンピュータサイエンスにおいてもフーリエ解析（ラプラス変換を含む）は、システム理論、信号処理や通信理論の理解に欠かせない。むしろ、フーリエ解析なくしてこれらを学問的に扱うのは不可能である。フーリエ解析は、信号を多くの正弦波の重ね合わせとしてみる。またフーリエ解析の画像や波形の工学的応用アルゴリズムは、ディジタル信号処理の中で位置づけられる。数学の歴史では、フーリエ解析は、複素関数論とともに、近代解析学が生まれる一つの契機となった。フーリエ変換は、複素関数の演算に利用することも頻繁に行われている。
授業の目的と到達目標 ／Objectives and attainment goals	本授業では、歴史的な発展にはふれないが、できるだけ応用に触れる形で行う。また演習問題を適宜出題するので、それの解答をレポートとして提出してもらう。 学生諸君は、一学年時の線形代数や微分積分を復習しつつ、工学の学習内容と関連させつつ、学習していただきたい。これらを真に学ぶことなくして、コンピュータサイエンスやコンピュータエンジニアリングを深く専攻することは不可能である。本授業は、真剣に大学の学術を学ぼうとする姿勢の学生にとって有意義なものとなるであろう。
授業スケジュール ／Class schedule	0. イントロダクション、 1. 直交関数系、フーリエ級数 2. 任意区間のフーリエ級数、複素フーリエ級数、ベッセルの不等式 3. 演習 4. フーリエ係数、フーリエ級数の収束条件 5. フーリエ級数の収束、パーセバルの等式 6. 演習 7. フーリエ積分、フーリエ変換の性質 、フーリエ積分の収束、 8. 合成積（たたみこみ） 、パーセバルの等式 9. 演習 10. ラプラス変換 11. 定数係数常微分方程式の解法 12. 演習 13. 離散的フーリエ変換 14. 高速フーリエ変換(FFT) 15. 総合演習
教科書 ／Textbook(s)	洲之内源一郎：フーリエ解析とその応用（サイエンスライブラリ、理工系の数学１２）、サイエンス社、及び配布資料
成績評価の方法・基準 ／Grading method/criteria	期末試験を基本とし、これに講義毎の小テスト及び演習毎のレポートを総合的に加味し最終評点をつける。
履修上の留意点 ／Note for course registration	Prerequisites: M1 Linear Algebra I or M2 Linear Algebra II M3 Calculus I or M4 Calculus II Related course: M6 Complex Analysis A8 Digital Signal Processing   A3 Image Processing Formal prerequisites:M1 Linear Algebra I or M2 Linear Algebra II 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.)	「フーリエの冒険」,ヒッポファミリークラブ、3,500円＋税 金谷健一著「これならわかる応用数学教室」,共立出版、2,900円＋税

Back

開講学期 ／Semester	2014年度／Academic Year 　後期 ／Second Semester
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 , Hiroshi Kihara , Kazuto Asai
推奨トラック ／Recommended track	CF,CM,VD,CN,RC
履修規程上の先修条件 ／Prerequisites	M5

更新日／Last updated on	2014/09/27
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	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 targets: 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	(The following is an example, details depend on each class.) 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 15. power series solutions to differential equations
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	Prerequisites: Differential and Integral Calculus I. Other related courses: Differential and Integral Calculus II, Fourier analysis, Electromagnetism.
履修上の留意点 ／Note for course registration	Overall scores of final examination, mini examination and reports. Note: Evaluation method depends on each instructor. Detailed information may be found in his home page. Formal prerequisites:M5 Fourier Analysis
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/2014/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)

Back

開講学期 ／Semester	2014年度／Academic Year 　前期 ／First Semester
対象学年 ／Course for;	2nd year
単位数 ／Credits	2.0
責任者 ／Coordinator	Toshiro Watanabe
担当教員名 ／Instructor	Toshiro Watanabe , Hiroshi Kihara , Takahiro Tsuchiya
推奨トラック ／Recommended track	CF,CM,CN,VH,RC,BM,SE
履修規程上の先修条件 ／Prerequisites	M3 or M4

更新日／Last updated on	2014/09/27
授業の概要 ／Course outline	Statistics is the most useful area in mathematics. However, it is not well known for students. It is important when we learn an elementary course of probability theory and Statistics.
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 understanded. 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 12. estimation 2 13. hypothesis test 1 14. hypothesis test 2 15. exercise
教科書 ／Textbook(s)	Tokei kaiseki Nyumon (Kyoritu Press)
成績評価の方法・基準 ／Grading method/criteria	Test and reports
履修上の留意点 ／Note for course registration	M3 Calculus I or M4 Calculus II 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)

Back

開講学期 ／Semester	2014年度／Academic Year 　後期 ／Second Semester
対象学年 ／Course for;	2nd year
単位数 ／Credits	2.0
責任者 ／Coordinator	Nobuyoshi Asai
担当教員名 ／Instructor	Nobuyoshi Asai , Masahide Sugiyama , Kazuto Asai
推奨トラック ／Recommended track	CF,CM
履修規程上の先修条件 ／Prerequisites	(M1 or M2) & F3

更新日／Last updated on	2014/09/27
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 are 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(2): reversible 11 application(1): quotient field and operator theory 12 ring and field(4): extension of field 13 application(2): constructable in geometry 14 application(3): M-sequence random number generation 15 application(4): error correct coding
教科書 ／Textbook(s)	Textbooks are different by lecturers. - Sugiyama, N.Asai class 杉原，今井，工学のための応用代数，共立出版 (1999). - K.Asai class 平林隆一, 工学基礎 代数系とその応用, 数理工学社 (2006).
成績評価の方法・基準 ／Grading method/criteria	Student evaluation methods are different by lecturers. - Sugiyama, N.Asai class examinations (Mid-term, Final) quizzes/homeworks - K.Asai class examinations (Final) small exmas and reports
履修上の留意点 ／Note for course registration	Prerequisites: Linear algebra I, Discrete Systems Related courses: Information theory Formal prerequisites:M1 Linear Algebra I or M2 Linear Algebra II F3 Discrete Systems
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	Sugiyama class homepage: http://web-int.u-aizu.ac.jp/~sugiyama/Lecture/AA/2013/welcome.html K.Asai class homepage: lecture directory: ~k-asai/classes/aalg handout: http://web-ext.u-aizu.ac.jp/~k-asai/classes/class-texts.html N.Asai class homepage: http://web-int.u-aizu.ac.jp/~nasai/Lecture/AA/2013/welcome.html - D.W.ハーディ，C.L.ウォーカー, 応用代数学入門, ピアソンエデュケーション． - 水野弘文, 情報代数の基礎, 森北出版. - 伊理正夫, 藤重悟, 応用代数, コロナ社. - 小野寛晰, 情報代数, 共立出版. - ヘルマン・ヴァイル, シンメトリー, 紀伊国屋書店. - 一松信, 石取りゲームの数理, 森北出版. - J.ロットマン, ガロア理論, Springer. - 渡辺，草場，代数の世界，朝倉出版. - 細井，情報科学のための代数系入門，産業図書. - 宮崎興二, かたちと空間，朝倉書店. - 内田, 有限体と符号理論, サイエンス社. - 草場, ガロワと方程式, 朝倉書店. - E.アルティン, ガロア理論入門, 東京図書. - ブライアンヘイズ, ベッドルームで群論を, みすず書房. - D.M.デイビス, 美しい数学, 青土社．

Back

開講学期 ／Semester	2014年度／Academic Year 　前期 ／First Semester
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	CF
履修規程上の先修条件 ／Prerequisites	－

更新日／Last updated on	2014/09/27
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-IV. Fuzzy Logic and Relations V-VIII. Model Theory IX-XV.
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 and [b] participation in online quizzes (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).

Back

開講学期 ／Semester	2014年度／Academic Year 　後期 ／Second Semester
対象学年 ／Course for;	2nd year
単位数 ／Credits	2.0
責任者 ／Coordinator	Shuxue Ding
担当教員名 ／Instructor	Shuxue Ding
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	2014/09/27
授業の概要 ／Course outline	The subjects of this course include the fundamental concepts on the topology, which are also the foundation of modern mathematics, and their applications. Some topological invariants or characteristics indexing the global structures of geometric objects, such as the Euler characteristic, the Betti number, etc., are introduced and are further investigated for how to calculate and how to apply. Furthermore, the relationship between the geometric objects, e.g. the topological space, and algebraic objects, such as the Homology group, is introduced. Since this relationship, the Homology group can be used for classifying and investigating the global properties of the geometric object.
授業の目的と到達目標 ／Objectives and attainment goals	Learn about fundamental concepts, basic properties and calculating method about, curve and surface, global structure, Euler characteristic, Betti number, Homology, and other concepts related to topology. Apply them to analysis and classify the global structures of 1-, 2- and 3- dimensional geometric objects.
授業スケジュール ／Class schedule	1) Introduction, 1-dimensional topology 2) 1-dimensional topology: Connectivity and Euler-Poincare theorem (1) 3) 1-dimensional topology: Embed into Euclidean space (1) 4) 2-dimensional topology: Closed surface and polyhedron 5) 2-dimensional topology: Classification of closed surfaces 6) 2-dimensional topology: Connectivity and Euler-Poincare theorem (2) 7) 2-dimensional topology: Embed into Euclidean space (2) 8) Review of the first half and midterm examination 9) Group, Homomorphic, and Isomorphism 10) Part group, Kernel, Image, and Homomorphic theorem 11) Chain group and Chain complex 12) Homology group 13) 0-dimensional Homology group 14) 1-dimensional Homology group 15) Connectivity and Euler-Poincare theorem (3)
教科書 ／Textbook(s)	The text book is written in Japanese: 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
成績評価の方法・基準 ／Grading method/criteria	Attendance and Quizzes: 15% Exercise Reports: 20% Midterm Examination: 30% Final Examination: 35%
履修上の留意点 ／Note for course registration	Linear Algebra I Formal prerequisites:None
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	小宮克弘著 「位相幾何入門」裳華房 川久保勝夫著 「トポロジーの発想」講談社 Stephen Barr 「Experiments in Topology」 Dover Publications, INC

Back

開講学期 ／Semester	2014年度／Academic Year 　前期 ／First Semester
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	2014/09/27
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 IR, sup, IR^n (I-II). Limits (III-IV). Topology and relation to limits (V-VI). Continuous functions and relation to limits and topology (VII-IX). Applications to concepts in Analysis (X). Homotopy and associated groups (XI-XIV). Fundamental Theorem of Algebra and other applications (XV).
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 and [b] participation in online quizzes (this is for training only). The final exam determines the grade.
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)

Back

開講学期 ／Semester	2014年度／Academic Year 　後期 ／Second Semester
対象学年 ／Course for;	3rd year
単位数 ／Credits	2.0
責任者 ／Coordinator	Wenxi Chen
担当教員名 ／Instructor	Wenxi Chen
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	2014/09/27
授業の概要 ／Course outline	地理情報システム（GIS）やコンピュータグラフィックス（CG）、コンピュータ援用設計（CAD）、パターン認識などでは大量の幾何図形データを高速に処理しなければならない。本科目では、コンピュータでこのような幾何学的な問題を取り扱うための効率的なアルゴリズムとデータ構造の設計と解析を勉強する。 本授業は実世界の問題を提起し、計算幾何学と結びつけ、直観的な考え方から出発し、アルゴリズムを導き出し、適切なデータ構造を用いて問題を解く手法を説明する。 プログラミング宿題の他に、プロジェクト研究課題を設け、計算幾何学知識を活かせて、実際の問題解決へのアプローチを身に付けながら計算幾何学の素晴らしさを楽しむ。
授業の目的と到達目標 ／Objectives and attainment goals	計算幾何学の基本となるデータ構造やアルゴリズムを理解し、具体的な問題に対して応用できるようにする。
授業スケジュール ／Class schedule	第一章 基本概念　　主要な幾何学的問題、典型的なデータ構造とアルゴリズムの復習 第二章 線分交差　　直交線分と任意線分の交差検出 第三章 凸包　　　　　定義と計算方法（直接法、包装法、Graham走査法、逐次添加法等） 第四章 ボロノイ図　　定義、性質と構成方法（直接法、逐次添加法、分割統治法、Fortune走査法） 第五章 ドローネ三角形分割　基本概念、関連術語、求め方（正当な三角形分割法、平面双対法、逐次添加法） 第六章 幾何的領域探索　　　直交領域探索の概念、1と2次元領域探索 第七章 多角形の三角形分割　問題の提起、基本概念、平面走査法
教科書 ／Textbook(s)	? Computational Geometry - Algorithms and Applications M. Berg, M. Kreveld, M. Overmars, and O. Schwarzkopf 著 Springer-Verlag 又はその和訳版 ? コンピュータ・ジオメトリ－計算幾何学：アルゴリズムと応用 M. ドバーグ、M. ファン・クリベルド、M. オーバマーズ、O. シュワルツコップ　著 浅野　哲夫訳、近代出版社 又は ? 計算幾何学入門 - 幾何アルゴリズムとその応用 譚学厚、平田富夫 著 森北出版
成績評価の方法・基準 ／Grading method/criteria	? 宿題　　　　　　　　50% プログラミング問題、4回。 プログラムの実装を通じて、計算幾何学のデータ構造とアルゴリズムに関する知識を深める。 又は ? プロジェクト研究　50% １回。 冬休み期間を利用して、実用性のある課題を提出し、色々なリソースから調査活動を展開し、総合的に実際問題解決へのアプローチを勉強する。 及び ? 期末試験　　　　　50% １回。基本概念についての理解程度を検証する
履修上の留意点 ／Note for course registration	プログラミング言語（C又はJAVA） データ構造とアルゴリズムの基礎 履修規程上の先修条件：なし
参考(授業ホームページ、図書など) ／Reference (course website, literature, etc.)	参考ホームページ 授業用 http://i-health.u-aizu.ac.jp/CompuGeo/index.html 和文サイト http://www.ise.chuo-u.ac.jp/ise-labs/imai-lab/ http://www.simplex.t.u-tokyo.ac.jp/kika.html 英文サイト http://compgeom.cs.uiuc.edu/~jeffe/compgeom/compgeom.html http://geometryalgorithms.com/ http://mathworld.wolfram.com/ http://www.diku.dk/hjemmesider/studerende/duff/Fortune/ 参考図書 計算幾何学 浅野哲夫 著 朝倉出版 計算幾何学 今井浩、今井圭子 著 共立出版株式会社 Computational Geometry in C J. O'Rourke 著 Cambridge University Press データ構造（アルゴリズムシリーズ１） 浅野哲夫 著 近代科学社 Algorithm in C, Vol. 2（又は同邦訳版） Robert Sedgewick 著 Addison-Wesley