平成 28 年度大学教育再生戦略推進費「大学の世界展開力強化事業」アジア諸国等との大学間交流の枠組み強化

ASEANと日本を繋ぐ

「グローバル・ソフトインフラ基礎人材」育成プログラム

平成 29年度 実施報告サマリー

受入期間 平成２９年５月１５日 — ５月２７日

受入国 ミャンマー

連携大学 ヤンゴン大学

受入学生数 ５名

参加学生数

（派遣先での相手国連携大学の学生、または受入時の本

学の参加学生数）

ティーチング・アシスタントとして２名（大学院博士課程後期学生）

プログラム概要

ヤンゴン大学から５名の学生を受け入れ、基礎的な数学力を充実さ

せるための３名の講師と２名の TA による２週間にわたる集中教育

プログラムを実施した。ヤンゴン大学からは教員も２名が帯同し、

このプログラムについての視察を行った。内容としては３人の教員

が、解析、代数、複素関数論の３分野において、午前中と午後後半

にそれぞれ５コマの講義を行い、午後後半は TA による演習の時間

を設けた。教育効果を測るために中間と期末に試験を行い、学生へ

のフィードバックを行った。週末には名古屋産業博物館、ノリタケ

の森などを見学し、この地域の社会・文化活動にも親しむ機会を設

けた。

スケジュール概要

（事前・事後の教育も含む）

５月１５日 来日、オリエンテーション

５月１６日−２６日 講義と演習

５月２７日 離日

（講義等の委細は別紙に記載）

産学連携：

連携機関、企業、訪問先等

該当なし

成果報告

（学生の成長や相手国との

連携について）

今回のプログラムはミャンマーの中心大学であるヤンゴン大学から

数学の大学院生を日本に招いて集中的な教育を行うという初めての

試みであった。今後の同国との数学教育・研究における確かな基盤

を築くことができたと考える。秋には、今回来日の学生の中から２

名を受け入れて、より長期的な教育・研究指導プログラムを実施す

る。将来的には名古屋大学の学生としての留学も視野に入れる。学

生を軸として研究においても今後の交流を育成していく。

実施部局 多元数理科学研究科

実施責任者 納谷信、大平徹

Program (Tentative) May 15 (Mon) Room: A428 Arrival to Airport 06:55 Arrival to Residence 10:00 Opening & Orientation 15:00 May 16 (Tue) Room: A428 10:00—11:30 Lecture: Okada (1) 13:00—14:30 Lecture: Kanno (1) Complex plane, Power series 14:30—17:00 Exercise: Kadota, Linear Algebra May 17 (Wed) Room: A328 10:00—11:30 Lecture: Okada (2) 13:00—14:30 Lecture: Ito (1) 14:30—17:00 Exercise: Matsuoka, Calculus 17:30—19:00 Reception May 18 (Thu) Room: A317 10:00—11:30 Lecture: Kanno (2) Holomorphic function and Cauchy-Riemann equation 13:00—14:30 Lecture: Ito (2) 14:30—17:00 Exercise: Kadota, Complex Analysis May 19 (Fri) Room: A317 10:30—12:00 G30 Lecture (Room: M109) Small Examination 13:00—14:30 Lecture: Ito (3) 14:30—17:00 Exercise: Matsuoka, Calculus May 20 (Sat) Free May 21 (Sun) Excursion Toyota Museum and Noritake Garden May 22 (Mon) Room: A428 10:00—11:30 Lecture: Okada (3) 13:00—14:30 Lecture: Ito (4) 14:30—17:00 Exercise: Matsuoka, Calculus

May 23 (Tue) Room: A428 7:00—8:30 G30 Lecture (Room: A207) 10:00—11:30 Lecture: Okada (4) 13:00—14:30 Lecture: Kanno (3) Rational function, Exponential and Trigonometric functions 14:30—17:00 Exercise: Kadota, Linear Algebra 17:15— Guidance about G30 Program May 24 (Wed) Room: A328 8:45—9:45 G30 Lecture (Room: M309) 10:00—11:30 Lecture: Okada (5) 13:00—14:30 Lecture: Kanno (4) Line integral and Cauchy’s formula 14:30—17:00 Exercise: Matsuoka, Complex Analysis 18:30—20:00 G30 Lecture (Room: A207) May 25 (Thu) Room: A317 10:00—11:30 Lecture: Kanno (5) Residue calculus 13:00—14:30 Lecture: Ito (5) 14:30—17:00 Exercise: Kadota, Linear Algebra May 26 (Fri) Room: A328 Presentation and/or Examination Closing BBQ party May 27 (Sat) Departure from Airport 10:15

Linear Algebra and Combinatorics

Lecturer : Soichi OKADA

A. Course Description

The goal of this series of lectures is to present a proof of the following Erdos–

Moser conjecture.

Conjecture. Given a finite subset S of positive real numbers and a positive real

number α, let f(S, α) be the number of subsets T of S such that the subset sum∑t∈T t is equal to α.

f(S, α) = #

{T ⊂ S :

∑t∈T

t = α

}.

Then, if #S = 2m+ 1 is odd, then

f(S, α) ≤ f({−m,−m+ 1, · · · ,−1, 0, 1, · · · ,m− 1,m}, 0).

Although the statement is purely combinatorial, the proof requires elementary

linear algebra. In the lectures, we explain how linear algebra can be used effectively

to solve combinatorial problems.

B. Schedule

Lecture 1 (May 16) : Partially ordered sets

Lecture 2 (May 17) : Sperner property

Lecture 3 (May 22) : Boolean lattice

Lecture 4 (May 23) : Weak Erdos–Moser conjecture

Lecture 5 (May 24) : Erdos–Moser conjecture

C. References

[1] R. P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and More,

Undergraduate Texts in Mathematics, Springer, 2013.

[2] P. Erdos, Extremal problems in number theory, in “Theory of Numbers”

(A. L. Whiteman ed.), Amer. Math. Soc. Proc. Symp. Pure. Math. 8

(1965), 181–189.

[3] B. Lindstrom, Conjecture on a theorem similar to Sperner’s, in “Combina-

torical Structures and Their Applications” (R. Guy ed.), 1970, 241.

[4] R. P. Stanley, Weyl group, the hard Lefschetz theorem, and the Sperner

property, SIAM J. Algebraic Discrete Methods 1 (1980), 168–184.

[5] R. A. Proctor, Solution of two combinatorial problems with linear algebra,

Amer. Math. Monthly 89 (1982), 721–734.

1

SMALL EXAMINATION

1. Let V and W be finite dimensional complex vector spaces. Let V ∗ and W ∗

be the dual spaces of V and W respectively. Given a linear map F : V → W , wedefine a map tF : W ∗ → V ∗ by

tF (g) = g ◦ F

for g ∈ W ∗. Show that tF is a linear map.

2. Let V be the vector space consisting of all polynomials of degree at most two:

V = {P (x) = ax2 + bx + c | a, b, c ∈ R}.Let F : V → V be the linear map defined by

F (P (x)) = P ′(x).

Compute the representation matrix A of F with respect to the basis {x2, x, 1}.

3. For a real number a > 0, show that the sequence

cn =(1 + a)n

n2, n = 1, 2, . . .

diverges as n → ∞.

4. Let a > 1.

(1) Show that the sequence

an = n√

a, n = 1, 2, . . .

converges.

(2) Show thatlim

n→∞an = 1.

(Hint: Assume limn→∞ an = 1 + h. Verify h ≥ 0. Show that if h > 0 one getscontradiction.)

Date: May 19, 2017.

1

EXAMINATION

Problem 1.

(1) Let V be the vector space consisting of all polynomials of degree at mosttwo:

V = {P (x) = ax2 + bx + c | a, b, c ∈ R}.Let F : V → V be the linear map defined by

F (P (x)) = P (x + 1).

Find the representation matrix A of F with respect to the basis {x2, x, 1}.

(2) Let V and W be finite dimensional complex vector spaces.(a) Let F : V → W and G : W → V be linear maps. Show that, if the

composition G ◦ F is bijective, then F is injective and G is surjective.

(b) Let H : V → V be a linear map. Show that, if 0 is not an eigenvalueof H, then H is bijective.

(c) Let F : V → W and G : W → V be linear maps. Assume that thereexist a linear map T : V → V with only non-negative real eigenvaluesand a positive real number α such that

G ◦ F = T + αI,

where I denotes the identity operator on V . Show that F is injective.(Hint: Use (a) and (b).)

Problem 2.

(1) Evaluate the following integral:∫∫D

x − 2y

2x − ydx dy D = {(x, y) ∈ R2 : 0 ≤ x − 2y ≤ 1, 1 ≤ 2x − y ≤ 2}

(2) Suppose that f(x, y) on R2 is of class C2 and let (x, y) = (r cos θ, r sin θ)be the polar coordinate expression of (x, y).

(a) Express∂f

∂rand

∂f

∂θusing r, θ,

∂f

∂x,

∂f

∂y.

(b) Express∂2f

∂θ∂rusing r, θ,

∂f

∂x,

∂f

∂y,

∂2f

∂x2,

∂2f

∂x∂y,

∂2f

∂y2.

Date: May 26, 2017.

1

Problem 3.

(1) Let Cr be the circle |z| = r with the counterclockwise orientation. Compute

1

2πi

∮Cr

4z − 7

2z2 − 7z + 3dz

for 0 < r < 1/2, 1/2 < r < 3 and 3 < r.

(2) Let us consider the differential equation

(1 + z)df

dz=

1

2f(z).

Find a power series solution f(z) =∞∑

n=0

cnzn and its radius of convergence.

Problem 4.

(1) Show rigorously that A = {z ∈ C | |z| < r}, where r > 0, is an open subsetof C.

(2) Let B1 and B2 be open subsets of C. Prove that B1 ∩B2 is an open subsetof C.

(3) Let

Cn =

{z ∈ C

∣∣ |z| < 1 +1

n

}, n = 1, 2, . . . .

Show that∞∩

n=1

Cn = {z ∈ C | |z| ≤ 1}.

Is C = ∩∞n=1Cn an open subset of C? (Answer along with the reason.)

2

Dear Kozaki, This is my brief impression of Japan. I do love ''travelling'' where I am interesting to go to new regions for general knowledge. In my firstly foreign trip with education, we went to well-known places of Nagoya in Japan. Among of them, I allowed myself to be persuaded into wondering the tablewares. We can learn in this excursion that Noritake is the world famous tableware brand and its craft center displays through a video presentation on it. The Garden is small and designed as a pleasant place for families. It offers also history, craft tradition, china painting experience, shopping and tasty food in one place. It impresses me that I am always memorable this Garden and our group. By the way, I want to say about Japanese who're kindly, blandly and warmly and they never waste the time. Among of them, our teaching Professors are brilliant and give us many helpful by manpower(include staffs). They are so hospitable. The meeting of this moment I always memorable my all dear. Best wishes, Ei Ei Thaw Dear Kozaki San, I would firstly like to express my gratittude for the amazing hospitality that we were greeted with. My impression of Japan is that it is a beautiful country with very kind and hospitable people. I admire how the university is very spacious and has a peaceful learning environment. In addition, the lectures are very detailed and effective and in my opinion the students are very hardworking and talented. My favorite dish is the different varieties of sushi. Moreover, I think my favorite place would be Asus Kanon and all the shopping places we went to. Thank you. Sincerely, Sa Pai Myint

Certificate of Completion

修　了　証This is to certify that

has participated in, and successfully completedthe 2017 Intensive Education Program

at the Graduate School of Mathematics, Nagoya University, Japanfrom May 15 to May 26, 2017.

上記の者は、2017 年度、名古屋大学大学院多元数理科学研究科　教育プログラム集中セミナー

に参加し、所定のプログラムを修了したことを証する。

Awarded on May 26, 2017平成 29年 5月 26日

NAYATANI Shin, Dean, the Graduate School of Mathematics, Nagoya University

名古屋大学大学院多元数理科学研究科長

Aye Aye Khaing