Category : Mathematics

 
 

ERA 2010 Mathematics Journal Rankings

ERA 2010 Mathematics Journal Rankings

ERA 2010 Mathematics Journal Rankings 입니다. 어떤 기준인지는 잘 모르겠지만, 저널들마다 A*, A, B, C, NR의 5단계 등급으로 영향력이 나누어져 있습니다. A*급 중에서 제가 친숙한 저널은 Comm. Pure. Appl. Math., Comm. Partial[...]
Introduction to the Free Klein-Gordon Evolution Group

Introduction to the Free Klein-Gordon Evolution Group

In this post, we introduce the factorization technique to analyze nonlinear Klein-Gordon equation proposed by N. Hayashi which is similar to the nonlinear Schrodinger equ[...]
테렌스 타오 (Terence Tao)를 만나다!

테렌스 타오 (Terence Tao)를 만나다!

(예전 네이버 블로그를 정리하면서, 2010년 2월경에 포스팅되어 있던 테렌스 타오 관련 포스트를 이곳에 옮겨 둡니다!)   9살, 호주 수학 경시대회 최연소 우승 11살, 국제 수학 올림피아드 최연소 동메달 12살, 국제 수학 올림피아드 최연소 은메달 13살, 국제 수학 올림피아드 최[...]
근황 : Klein-Gordon Equation 연구 (2013년 4월 13일)

근황 : Klein-Gordon Equation 연구 (2013년 4월 13일)

일단 NLKG(Nonlinear Klein-Gordon Equation)을 연구하기 시작했습니다. 전부 떠벌리기는 어렵고 개략적으로는 아래와 같은 Cauchy Problem을 연구합니다. $$ (square + m) u(t,x) = F (u, partial u), ;;; t geqslant 0, ;[...]
[M2 Seminar I] Week 1~4 : $\sqrt{\log t}$를 달기 위한 여정의 시작

[M2 Seminar I] Week 1~4 : $sqrt{log t}$를 달기 위한 여정의 시작

[latexpage] 블로그 활동이 너무 뜸해서 연구일지나 쓰려고 합니다. 구체적인 연구내용이나, 세미나 발표자료들은 연구가 끝나는 대로 공개하도록 하겠습니다! 2013년 4월 12일 (금) 본격적인 M2 세미나가 시작되었다. 지도교수로부터 1-dimensional Klein-Gordon eq[...]
[Topics in Analysis] Final Report

[Topics in Analysis] Final Report

This report consists of two sections. In the first section, we will prove the global regularity of elliptic partial differential equations. In particular we will extend a[...]
[Differential Equations II] Final Report

[Differential Equations II] Final Report

In this report, we study the global existence of solutions to nonlinear dispersive wave equations begin{eqnarray*}partial_t^2 u + frac{1}{rho^2} |partial_x|^{2rho} u &[...]
교토대학 대학원 수학과 면접후기 [일본 대학원]

교토대학 대학원 수학과 면접후기 [일본 대학원]

제 메일함은 블로그 덕분에 일본대학원 유학에 관한 문의메일로 늘상 붐빕니다만, 이번에 다음달 교토대학 대학원 수학과에 수험한다는 분의 메일에 답장을 하고 있으려니, 작년 이맘 때 교토대학에서의 패배가 다시금 떠오르는군요. 교토대학 수학과 문의메일은 처음이었기에.. 필기시험의 기하문제였나요,[...]
[M1 Seminar II] Week 8 : Lorentz Invariance of the Wave Equations

[M1 Seminar II] Week 8 : Lorentz Invariance of the Wave Equations

Now we will introduce some lemmas for the proof of the global existence theorem for the nonlinear wave equations with quadratic nonlinearities. The exsistence theorem whi[...]
[M1 Seminar II] Week 7 : A Global Existence Theorem for Nonlinear Wave Equations

[M1 Seminar II] Week 7 : A Global Existence Theorem for Nonlinear Wave Equations

We consider the following initial value problem [ y_{tt} - Delta y = f(Dy,nabla Dy);;;;text{with};;;; y(t=0) = y_0,; y_t (t=0) = y_1 tag{P} ] with $f in C^infty ( Bbb R[...]
1옥타브를 왜 하필 12음계로 구분했을까?

1옥타브를 왜 하필 12음계로 구분했을까?

여러분은 음악을 접하면서 왜 하필 한 옥타브(가령 C~C)를 12개의 음(C, C#, D, D#, E, F, F#, G, G#, A, A#, B)으로 나누었는지 궁금해 하셔 본 적이 있나요? 한 옥타브를 5개의 음으로 나누거나, 20개의 음으로 나눌 수도 있을텐데 왜 하필 12음계로 나누었을까요? [...]
편미분 방정식 전공 서적 리뷰 (PDE, Partial Differential Equations Book)

편미분 방정식 전공 서적 리뷰 (PDE, Partial Differential Equations Book)

이 쯤에서 그간 읽거나 접했던 PDE 전공서적을 리뷰 해 볼까 합니다. 아직 책을 평가할 수준이 절대로 안 되기 때문에 나름대로의 감상평에 불과합니다.   1. F. John. Partial Differential Equations. Springer. Courant의 제자 F. [...]
[M1 Seminar II] Week 6 : Weighted a priori Estimates for Small Data

[M1 Seminar II] Week 6 : Weighted a priori Estimates for Small Data

For the proof of the global existence theorem of nonlinear wave equations, this a priori estimates is essential as well as the high energy estimates. Also the high energy[...]
[M1 Seminar II] Week 5 : High Energy Estimates

[M1 Seminar II] Week 5 : High Energy Estimates

We prove the high energy estimates for the nonlinear wave equation on the nonlinearity $alpha =1$. In fact, this problem is a special case of the quasi-linear symmetric h[...]
[M1 Seminar II] Week 4 : Regularities of the Solution and and Improved Existence Theorem

[M1 Seminar II] Week 4 : Regularities of the Solution and and Improved Existence Theorem

We proved the right-continuity of the solution at the initial time in $W^{s,2}$ which implies the continuity of the whole interval. And by the PDE and Sobolev embedding t[...]
[M1 Seminar II] Week 3 : Regularities of the Solution

[M1 Seminar II] Week 3 : Regularities of the Solution

This week we will prove that the solution of the system $u$ is in $C^0([0,T], W^{s,2}) cap C^1([0,T], W^{s-1,2})$ so that $u in C_b^1 ([0,T] times Bbb R^n)$. For this pur[...]
[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)

[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)

This week, we will prove the local existence of quasi-linear symmetric hyperbolic systems by using $u^{k}$ which is iteratively defined by the solution of the linear symm[...]
L^1-L^∞ Decay Estimates for the Odd-Dimensional Wave Equations

L^1-L^∞ Decay Estimates for the Odd-Dimensional Wave Equations

We will show the following estimates. For $n = 1,3,cdots$, let $phi in C_0^{n} (Bbb R^n), ; psi in C_0^{n-1} (Bbb R^n) $. And let $v$ solve begin{equation}partial_t^2[...]
[M1 Seminar II] Week 1 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems

[M1 Seminar II] Week 1 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems

This week, we prove the uniqueness and local existence for quasi-linear symmetric hyperbolic systems using general energy method. M1_Semi2_Week1  [...]
[M1 Seminar] Week 13 : Elementary Inequalities on the Klainerman Vector Field

[M1 Seminar] Week 13 : Elementary Inequalities on the Klainerman Vector Field

We construct the Klainerman vector field with some operators which were defined in the last seminar. And we prove some basic inequalities of these family of operators whi[...]
[M1 Seminar] Week 11,12 : Introduction to the Klainerman Vector Field

[M1 Seminar] Week 11,12 : Introduction to the Klainerman Vector Field

We introduce the Klainerman vector field and prove some commutator relations which will play an important role in the proof of the global existence theorem for the nonlin[...]
Hardy's Inequality

Hardy's Inequality

[latexpage] Theorem 1. (Hardy's Inequality for Sums) Let $p >1$ and $ a_n geqslant 0$ for all $n$. If $ { a_n }_1^{infty} in ell^p $, then $ left { frac{1}{n} sum_[...]
[M1 Seminar] Week 10 : Gagliardo-Nirenberg Inequality

[M1 Seminar] Week 10 : Gagliardo-Nirenberg Inequality

We show the standard Gagliardo-Nirenbergy inequality just by using some elementary inequalities and induction. M1_Semi_Week10  [...]
3 Simple Proofs of the Cauchy-Schwarz Inequality

3 Simple Proofs of the Cauchy-Schwarz Inequality

[latexpage] Theorem. (Cauchy-Schwarz Inequality) Let $X$ be an inner product space with a given inner product $ langle cdot , cdot rangle : X times X to mathbb F ( math[...]
[M1 Seminar] Week 9 : Some Elementary Inequalities

[M1 Seminar] Week 9 : Some Elementary Inequalities

We introduce some elementary inequalities which will be used for proving the Gagliardo-Nirenberg inequality. M1_Semi_Week9  [...]
[M1 Seminar] Week 8 : Some Inequalities with Friedrich Mollifiers

[M1 Seminar] Week 8 : Some Inequalities with Friedrich Mollifiers

We introduce some elementary inequalities with Friedrich mollifiers which will be frequently used later. M1_Semi_Week8  [...]
[M1 Seminar] Week 6,7 : A Global Existence Theorem to the Linear Symmetric Hyperbolic Systems

[M1 Seminar] Week 6,7 : A Global Existence Theorem to the Linear Symmetric Hyperbolic Systems

This week, we prove the global existence theorem to the linear symmetric hyperbolic system by using the standard energy inequality. M1_Semi_Week6-7  [...]
Heptadecagon(17-gon) Construction using GeoGebra 4.

Heptadecagon(17-gon) Construction using GeoGebra 4.

도형을 그려주는 프로그램 지오지브라 4(GeoGebra 4)를 소개합니다. 상당히 좋은 프로그램인 것 같습니다. 매뉴얼 없이도 누구든지 쉽게 사용할 수 있다고 생각합니다.재미삼아 정 17각형을 작도해 보았습니다.갑자기 삼등분가들이 생각나네요...-_-;  [...]
[M1 Seminar] Week 5 : Energy Estimates for the Linear Symmetric Hyperbolic System

[M1 Seminar] Week 5 : Energy Estimates for the Linear Symmetric Hyperbolic System

We derive the corresponding estimates for higher derivatives from the basic energy estimates which was given in the last week. We also prove a simple corollary using this[...]
[M1 Seminar] Week 3,4 : Linear Symmetric Hyperbolic System

[M1 Seminar] Week 3,4 : Linear Symmetric Hyperbolic System

   This week, we derive basic energy estimates for the linear symmetric hyperbolic system. M1_Semi_Week3-4  [...]