http://www.numdam.org/articles/10.24033/bsmf.1409/ Idéaux et modules de fonctions analytiques de variables complexes Cartan, Henri Bulletin de la Société Mathématique de France, Volume 78 (1950), pp. 29-64. http://www.numdam.org/item/10.24033/bsmf.1409.pdf0119132人目の素数さん2023/12/25(月) 19:10:59.03ID:ND/nzYaN きみがため 春の野にいでて若菜摘む わが衣手に雪は降りつつ 0120132人目の素数さん2023/12/25(月) 22:43:05.36ID:7HvkKwKX 歌会始は、日本の宮中で行われる歌会の一つで、毎年1月15日に天皇陛下が主宰されます。歌会始の入選作品は、昭和22年から令和5年までのお題をPDF形式で閲覧できます1. ただし、入選作品の内容については、著作権の関係上、公開されていません。ご了承ください。 0121132人目の素数さん2023/12/26(火) 09:48:50.85ID:S5czeSxx これやこの 行くも帰るも分かれては 知るも知らぬも逢坂の関
https://en.wikipedia.org/wiki/Nakano_vanishing_theorem Nakano vanishing theorem References Original publications ・Akizuki, Yasuo; Nakano, Shigeo (1954). "Note on Kodaira-Spencer's proof of Lefschetz theorems". Proceedings of the Japan Academy. 30 (4): 266–272. doi:10.3792/pja/1195526105. ISSN 0021-4280. ・Nakano, Shigeo (1973). "Vanishing theorems for weakly 1-complete manifolds". Number theory, algebraic geometry and commutative algebra — in honor of Yasuo Akizuki. Kinokuniya. pp. 169–179. ・Nakano, Shigeo (1974). "Vanishing Theorems for Weakly 1-Complete Manifolds II". Publications of the Research Institute for Mathematical Sciences. 10 (1): 101–110. doi:10.2977/prims/1195192175. 0312132人目の素数さん2024/01/10(水) 22:45:39.22ID:9Ar19oBn>>311 Nadel vanishing theorem Demailly vanishing theorem 0313132人目の素数さん2024/01/10(水) 23:21:45.63ID:9Ar19oBn Shokurov nonvanishing 0314132人目の素数さん2024/01/11(木) 05:57:45.18ID:b6kSf205>>311 コピペじゃない何か書けるまで、ROMでお願いします 0315132人目の素数さん2024/01/11(木) 07:14:25.94ID:GWyUET7U Girbau vanishing 0316132人目の素数さん2024/01/11(木) 09:01:20.30ID:GWyUET7U Kobayashi-Ochiai vanishing 0317132人目の素数さん2024/01/11(木) 10:18:50.05ID:91XlFR8b>>312 ふむ
>Nadel vanishing theorem
https://www.math.kyoto-u.ac.jp/~fujino/kollar-nadel.pdf Southeast Asian Bulletin of Mathematics (2018) 42: 643–646 Koll´ar–Nadel Type Vanishing Theorem∗ Osamu Fujino Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka,
>Demailly vanishing theorem
https://arxiv.org/abs/alg-geom/9410022 L^2$ vanishing theorems for positive line bundles and ... arXiv JP Demailly 著 · 1994 · 被引用数: 203 — L^2 vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic
https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/vanishing.pdf vanishing theorems for tensor powers of an ample vector bundle Institut Fourier JP DEMAILLY 著 · 被引用数: 45 — Demailly. — Vanishing theorems for tensor powers of a positive vector bundle,. Proceedings of the Conference Geometry and Analysis on Manifolds held in ... 18 ページ
https://link.springer.com/chapter/10.1007/BFb0094302 L2 vanishing theorems for positive line bundles and ... Springer JP Demailly 著 · 1996 · 被引用数: 203 — Demailly, JP. (1996). L2 vanishing theorems for positive line bundles and adjunction theory. In: Catanese, F., Ciliberto, C. (eds) ... 0318132人目の素数さん2024/01/11(木) 10:25:49.58ID:91XlFR8b>>315 >girbau vanishing theorem
中身を見てないが、メモ貼りますね
おお K Takegoshi 著 · 1981がヒット https://www.jstage.jst.go.jp/article/kyotoms1969/17/2/17_2_723/_pdf A Vanishing Theorem for on Weakly 1 -Complete Manifolds J-Stage K Takegoshi 著 · 1981 · 被引用数: 8 — Girbau's work [4], O. Abdelkader [1] proved the following. Theorem 1. Let X be a weakly \-complete Kahler manifold and let B be a semi-positive
2023か、新しい文献を見ておくことは大事だね https://academic.oup.com/imrn/article-abstract/2023/16/13501/6650269 Vanishing Theorems for Sheaves of Logarithmic Differential ... Oxford Academic C Huang 著 · 2023 · 被引用数: 2 — ... theorems, including Norimatsu's vanishing theorem, Girbau's vanishing theorem, Le Potier's vanishing theorem, and a version of the Kawamata–
これは、ご当人のJ Girbau 氏 https://link.springer.com/article/10.1007/BF02761365 Vanishing cohomology theorems and stability of complex ... Springer J Girbau 著 · 1981 · 被引用数: 1 — Girbau,Sur le théorème de stabilité de feuilletages de Hamilton, Epstein et Rosenberg, C. R. Acad. Sci. Paris291 (1980), A-41-44. J. Girbau and M. 0319132人目の素数さん2024/01/11(木) 11:08:33.82ID:91XlFR8b>>316 >Kobayashi-Ochiai vanishing theorem
中身を見てないが、メモ貼りますね
おお S Nakano 著 · 1974 "Kobayashi, S. and Ochiai, T" Kobayashi, S 小林 昭七 Ochiai, T 落合卓四郎 かな (”Kobayashi-Ochiai vanishing theorem”にヒットしているか不明ですが)
https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0819-14.pdf Iitaka's conjecture based on Severi's theorem. ness if $X$ RIMS, Kyoto University K MAEHARA 著 · 1993 — Socond, Kobayashi-Ochiai ([KO])proved finiteness of the set of the generically ... Iitaka's conjecture based on Severi's theorem. Is the set fnite2. Thanks to ...
https://www.mathsoc.jp/assets/pdf/publications/pubmsj/Vol15.pdf DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR ... 日本数学会 2011/03/04 — In retrospect, we need mostly vanishing theorems for holomorphic sections for the purpose of this book, but I decided to include cohomology ... 289 ページ 0320132人目の素数さん2024/01/11(木) 12:17:28.01ID:OmSwUFPK Malgrange vanishing 0321132人目の素数さん2024/01/11(木) 13:39:04.54ID:91XlFR8b>>320 >Malgrange vanishing
Malgrange (6 July 1928 – 5 January 2024) ”Malgrange died on 5 January 2024, at the age of 95.[2]” 知らなかったな。”His advisor was Laurent Schwartz”か。そうでしたね
”Malgrange vanishing”は、中身見てないが貼ります
(参考) https://en.wikipedia.org/wiki/Bernard_Malgrange Bernard Malgrange (6 July 1928 – 5 January 2024) was a French mathematician who worked on differential equations and singularity theory. He proved the Ehrenpreis–Malgrange theorem and the Malgrange preparation theorem, essential for the classification theorem of the elementary catastrophes of René Thom. He received his Ph.D. from Université Henri Poincaré (Nancy 1) in 1955. His advisor was Laurent Schwartz. He was elected to the Académie des sciences in 1988. In 2012 he gave the Łojasiewicz Lecture (on "Differential algebraic groups") at the Jagiellonian University in Kraków.[1] Malgrange died on 5 January 2024, at the age of 95.[2]
https://www-fourier.ujf-grenoble.fr/sites/default/files/ref_404.pdf the malgrange vanishing theorem with support conditions Institut Fourier THE MALGRANGE VANISHING. THEOREM WITH SUPPORT CONDITIONS. C. Laurent-Thibebaut and J. Leiterer. 0 . Introduction. Let X be a complex manifold of dimension n ...
https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/25395/1/1367-16.pdf Vanishing Theorems in Hyperasymptotic Kyoto University Research Information Repository PDF H Majima 著 · 2004 — Malgrange proved also the vanishing theorem of commutative case in asymptoticanalysis and, Malgrange and Deligne showed that it was usefull to study the ... 0322132人目の素数さん2024/01/11(木) 18:12:15.21ID:b6kSf205>>318 >中身を見てないが、メモ貼りますね >>319 >中身を見てないが、メモ貼りますね >>321 >中身見てないが貼ります
2021/03/29 I present the Akizuki-Nakano formula for the Laplacian of a Hermitian line bundle. Then I discuss cases for the positivity of the right hand side. As an application I prove the Nakano vanishing theorem for positive line bundles. Reference: Demailly agbook sections VI.5, VII.1-3. 0325132人目の素数さん2024/01/13(土) 09:19:42.66ID:d5SAamBZ>>324 追加
下記は、偏微分方程式の基本解とか書いてあった記憶あり (参考) https://en.wikipedia.org/wiki/Malgrange%E2%80%93Ehrenpreis_theorem Malgrange–Ehrenpreis theorem In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956).
This means that the differential equation 略 where P is a polynomial in several variables and δ is the Dirac delta function, has a distributional solution u.
Proofs The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn–Banach theorem. Since then several constructive proofs have been found. There is a very short proof using the Fourier transform and the Bernstein–Sato polynomial, as follows. By taking Fourier transforms the Malgrange–Ehrenpreis theorem is equivalent to the fact that every non-zero polynomial P has a distributional inverse. 0326132人目の素数さん2024/01/13(土) 09:51:25.29ID:d5SAamBZ Bernstein–Sato polynomialか
(参考) https://en.wikipedia.org/wiki/Bernstein%E2%80%93Sato_polynomial Bernstein–Sato polynomial In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Joseph Bernstein (1971) and Mikio Sato and Takuro Shintani (1972, 1974), Sato (1990). It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory, and quantum field theory.
Severino Coutinho (1995) gives an elementary introduction, while Armand Borel (1987) and Masaki Kashiwara (2003) give more advanced accounts.
Definition and properties Definition and properties If f(x) is a polynomial in several variables, then there is a non-zero polynomial b(s) and a differential operator P(s) with polynomial coefficients such that 略 The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials b(s). Its existence can be shown using the notion of holonomic D-modules.
Kashiwara (1976) proved that all roots of the Bernstein–Sato polynomial are negative rational numbers.
Nero Budur, Mircea Mustață, and Morihiko Saito (2006) generalized the Bernstein–Sato polynomial to arbitrary varieties.
Note, that the Bernstein–Sato polynomial can be computed algorithmically. However, such computations are hard in general. There are implementations of related algorithms in computer algebra systems RISA/Asir, Macaulay2, and SINGULAR.
Applications ・The Malgrange–Ehrenpreis theorem states that every differential operator with constant coefficients has a Green's function. By taking Fourier transforms this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above. ・The Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in quantum field theory Fyodor Tkachov (1997).
Local cohomology, Grothendieck local residues and algorithms, Tajima,Shinichi Workshop on Rresidues, dynamics and hyperfunctions, Jul 24, 2017 Invited
An algorithm for computing Grothendieck local residues I -- shape basis case -- Tajima, Shinichi, Ohara, K Applications of Computer Algebra 2017, Jul 16, 2017
https://en.wikipedia.org/wiki/Local_cohomology Local cohomology In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005). Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function 1/x, for example, 0335132人目の素数さん2024/01/14(日) 09:56:01.54ID:nCpmxPMj 多変数留数の計算とか言う前に 一変数留数の計算について何か書けるまで ROMでお願いします 0336132人目の素数さん2024/01/14(日) 12:43:24.75ID:JdlgwxhN 解析が専門とは思えない不明教授
128 名前:132人目の素数さん[] 投稿日:2024/01/14(日) 20:54:51.00 ID:qnrEEgUG [2/2] 例えばトネリ関数がどうしても必要な理論は? 0344132人目の素数さん2024/01/14(日) 21:59:39.50ID:qnrEEgUG>>343 例えばトネリ関数の概念がどうしても必要な理論は? 0345132人目の素数さん2024/01/14(日) 22:03:04.89ID:qnrEEgUG 台がコンパクトなC∞級関数のなす L2ノルムに関するプレヒルベルト空間の 0346132人目の素数さん2024/01/14(日) 22:04:40.41ID:qnrEEgUG 完備化があれば ルベーグ可測な二乗可積分関数のなす ヒルベルト空間は必要ない。 0347132人目の素数さん2024/01/15(月) 02:53:03.40ID:I3zcDmzY コーシー列の極限が今まで考えてた意味で積分できなかったり 関数にすらならなくても構わないわけですか 0348132人目の素数さん2024/01/15(月) 06:04:36.11ID:JEVrqZGt 目的次第。 トネリ関数がないと 多様体論が停滞するわけではないだろう。 0349132人目の素数さん2024/01/15(月) 06:47:56.66ID:JEVrqZGt いたるところ微分不可能な連続関数を 極限として構成するために 関数空間が必要なわけではない。 0350132人目の素数さん2024/01/15(月) 06:52:10.38ID:w4XPejWv お前が書き込んだのは偏微分方程式のスレだが 0351132人目の素数さん2024/01/15(月) 06:59:38.63ID:JEVrqZGt 滑らかな解の存在を示すために必要なのは ソボレフ空間などであって 個々の関数ではない 0352132人目の素数さん2024/01/15(月) 08:34:09.50ID:I3zcDmzYhttps://en.wikipedia.org/wiki/Meyers–;Serrin_theorem Hだけで構わないらしい 0353132人目の素数さん2024/01/15(月) 09:09:55.61ID:JEVrqZGt Originally there were two spaces: W^{{k,p}}(\Omega ) defined as the set of all functions which have weak derivatives of order up to k all of which are in L^{p} and H^{k,p}(\Omega ) defined as the closure of the smooth functions with respect to the corresponding Sobolev norm (obtained by summing over the L^{p} norms of the functions and all derivatives). The theorem establishes the equivalence W^{k,p}(\Omega )=H^{k,p}(\Omega ) of both definitions. It is quite surprising that, in contradistinction to many other density theorems, this result does not require any smoothness of the domain Ω\Omega . According to the standard reference on Sobolev spaces by Adams and Fournier (p 60): "This result, published in 1964 by Meyers and Serrin ended much confusion about the relationship of these spaces that existed in the literature before that time. It is surprising that this elementary result remained undiscovered for so long." 0354132人目の素数さん2024/01/15(月) 10:02:12.19ID:YRXYpIL/ 完備化に入ってる関数の境界挙動は非自明 それをうまく使って面白いことを言う人もいる 大抵の人には意外性など無意味 0355132人目の素数さん2024/01/15(月) 10:51:20.50ID://W0c+B+>>333 ありがとうございます。 検索: 13:30〜14:30 特別講演 田島慎一 (新潟大⋆) 特異点の複素解析, 代数解析とアルゴリズム の結果で見繕い
(参考) これちょっと面白い https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1927-08.pdf 数理解析研究所講究録 第 1927 巻 2014 年 66-76 ニュートン非退化孤立特異点と局所コホモロジー類 田島慎一 筑波大学数理物質系数学域 梅田陽子 理科大理工学部数学 1 Introduction X を {C}^{n} の原点 O の開近傍,f を X 上定義された正則関数とする.f が定める複素解析的超曲面 S= {x in X|f(x)=0 } は,原点を孤立特異点として持つとする.幕級数環 O_{X,O} における f のヤコビイデアル J_{f} やその剰余 {O}_{X,O}/J_{f} には,超曲面 S の特異点に関する様々な情報が含まれており,特異点の複素解析的 諸性質を考える際に最も基本的な対象であると言える.さて,寡級数環 O_{X,O} の局所凸位相ベクトル空間と しての双対ベクトル空間は,原点に台を持つ局所コホモロジー群として実現できることが知られている.こ の双対性に注目すると,「局所コホモロジーを積極的に使うことで,ヤコビイデアル J_{J} を具体的に扱い、特 異点の複素解析的な諸性質を調べる」 という発想は自然である.剰余 {O}_{X,O}/J_{f} の双対ベクトル空間となる 有限次元ベクトル空間を,局所コホモロジー類のなす集合 (以下,H_{J_{f}} で表す) として求めるアルゴリズム を構成した.このベクトル空間 H_{J_{f}} を用いると,Grothendieck local duality により,瓢級数環におけるヤコ ビイデアルみに対するイデアルメンバーシップが容易に判定可能となる.また,Tjurina 数の計算,射数的 ベクトル場の構造の決定や具体的な構成等への応用がある.