지원사업
학술연구/단체지원/교육 등 연구자 활동을 지속하도록 DBpia가 지원하고 있어요.
커뮤니티
연구자들이 자신의 연구와 전문성을 널리 알리고, 새로운 협력의 기회를 만들 수 있는 네트워킹 공간이에요.
논문 기본 정보
- 자료유형
- 학술저널
- 저자정보
- 저널정보
- 한국산업응용수학회 JOURNAL OF THE KOREAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS Journal of the Korean Society for Industrial and Applied Mathematics Vol.18 No.2
- 발행연도
- 2014.6
- 수록면
- 193 - 207 (15page)
이용수
초록· 키워드
오류제보하기
In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet‘s principle. We emphasize on the intuition in Riemann‘s statement on the principle criticized byWeierstrass‘ requirement of rigor followed by Hilbert‘s restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion.
Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly afterWeierstrass’s famous criticism of Riemann’s use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar´e and Hilbert defended Riemann’s use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.
Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly afterWeierstrass’s famous criticism of Riemann’s use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar´e and Hilbert defended Riemann’s use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.
목차
ABSTRACT
1. INTRODUCTION
2. DIRICHLET’S PRINCIPLE IN THE ABSENCE OF RIGOR
3. SOBOLEV COMPACT EMBEDDINGS AND COMPENSATED COMPACTNESS
4. STONE-WEIERSTRASS, ASCOLI-ARZEL`A AND SHANNON SAMPLING THEOREMS
5. COMPACT OPERATOR AND LAYER POTENTIALS FOR LAPLACE EQUATION
6. Γ-CONVERGENCE AND COMPACTNESS
7. CONCLUSION
REFERENCES
참고문헌 (42)
참고문헌 신청
1. [학술지(정기간행물)] - Alexandrov P / 1923 / Sur les espaces topologiques compacts / Bull. Intern. Acad. Pol. Sci. Ser. A : 5 ~ 8
2. [학술지(정기간행물)] - Arzel`a C / 1895 / Sulle funzioni di linee / Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat 5 (5) : 55 ~ 74
3. [학술지(정기간행물)] - Ascoli G / 1883 / Le curve limiti di una variet`a data di curve / Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat 18 (3) : 521 ~ 586
4. [단행본] - Bolzano B / 1930 / Functionlehre : 1833 ~ 1841
5. [단행본] - Bolzano B / 1842 / Zahlenlehre
2. [학술지(정기간행물)] - Arzel`a C / 1895 / Sulle funzioni di linee / Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat 5 (5) : 55 ~ 74
3. [학술지(정기간행물)] - Ascoli G / 1883 / Le curve limiti di una variet`a data di curve / Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat 18 (3) : 521 ~ 586
4. [단행본] - Bolzano B / 1930 / Functionlehre : 1833 ~ 1841
5. [단행본] - Bolzano B / 1842 / Zahlenlehre
이 논문의 저자 정보
최근 본 자료
댓글(0)
0