なお、https://en.wikipedia.org/wiki/Looman%E2%80%93Menchoff_theorem Looman?Menchoff theorem It is thus a generalization of a theorem by Edouard Goursat, which instead of assuming the continuity of f, assumes its Frechet differentiability when regarded as a function from a subset of R2 to R2.
と書いてあることは、チラ見している ”It is thus a generalization of a theorem by Edouard Goursat ”とあるよｗ 0197１３２人目の素数さん2023/05/18(木) 23:25:41.40ID:kOqY4klp>>195 >昔、Narasimhanのテキストを読んでいて
https://en.wikipedia.org/wiki/Raghavan_Narasimhan Raghavan Narasimhan (August 31, 1937 ? October 3, 2015) was an Indian mathematician at the University of Chicago who worked on real and complex manifolds and who solved the Levi problem for complex manifolds.[1]
https://en.wikipedia.org/wiki/M._S._Narasimhan Mudumbai Seshachalu Narasimhan FRS (7 June 1932 ? 15 May 2021) was an Indian mathematician. His focus areas included number theory, algebraic geometry, representation theory, and partial differential equations. He was a pioneer in the study of moduli spaces of holomorphic vector bundles on projective varieties. His work is considered the foundation for Kobayashi?Hitchin correspondence that links differential geometry and algebraic geometry of vector bundles over complex manifolds. He was also known for his collaboration with mathematician C. S. Seshadri, for their proof of the Narasimhan?Seshadri theorem which proved the necessary conditions for stable vector bundles on a Riemann surface. 0198１３２人目の素数さん2023/05/18(木) 23:31:32.27ID:kOqY4klp>>195 >fとgが正則で|f|^2+|g|^2が定数なら >fとgが定数でなければならないことの証明を読んで
再録>>189より 4）wikipedia Looman-Menchoff theorem >>166 Let Ω be an open set in C and f : Ω → C be a continuous function. Suppose that the partial derivatives ∂f/∂x and ∂f/∂y exist everywhere but a countable set in Ω. Then f is holomorphic if and only if it satisfies the Cauchy-Riemann equation: ∂f/∂z^-＝1/2(∂f/∂x+∂f/∂y)＝0. （注：z^-は、共役複素数） (引用終り)
ここで、証明は二重積分を使う筋>>200とすると ”but a countable set in Ω”としても 測度論からは、二重積分に影響しない そして、仮定側で除外した例外点は 結論側では、”結局holomorphicでした”ってことかな つまり、上記wikipediaの記載も間違いではないのか
さて >>166-167 に戻る > 6.4 等角写像の定義をめぐって >定理 6.12 (メンショフの定理) 領域 D で定義された定数でない連続関数 f(z) が，D に >おいて正則になるための必要十分条件は，D 内の孤立集合を除いて D の各点で f が等角 >写像になることである。
Looman-Menchoff theorem>>189 Let Ω be an open set in C and f : Ω → C be a continuous function. Suppose that the partial derivatives ∂f/∂x and ∂f/∂y exist everywhere but a countable set in Ω. Then f is holomorphic if and only if it satisfies the Cauchy?Riemann equation: ∂f/∂z^-＝1/2(∂f/∂x+∂f/∂y)＝0. （注：z^-は、共役複素数）
手抜きした転記入れます Theorem1(The Looman-Mwenchoff Theorem). Let Ω be an open set in C and let f be a continuous function on Ω. Suppose that ∂f/∂x,∂f/∂y exist at every point of Ω and satisfy ∂f/∂z^- ＝1/2(∂f/∂x+ i∂f/∂y)＝0 on Ω. （注：z^-は、共役複素数） Then f is holomorphic on Ω.
なるほど 再録>>189 4）wikipedia Looman-Menchoff theorem >>166 Let Ω be an open set in C and f : Ω → C be a continuous function. Suppose that the partial derivatives ∂f/∂x and ∂f/∂y exist everywhere but a countable set in Ω. Then f is holomorphic if and only if it satisfies the Cauchy-Riemann equation: ∂f/∂z^-＝1/2(∂f/∂x+∂f/∂y)＝0. （注：z^-は、共役複素数） (引用終り)
対比すると、Narasimhanでは、1/2(∂f/∂x+∂f/∂y)＝0→ f is holomorphic のみか おっと、>>207 では ”if and only if”を、ちょっと誤読しているな。ほんと池沼だった
>>208より ”命題 2.1 z の関数 f(z) が z 平面上の領域 D において正則でつねに f ′(z)≠ 0 なら ば f : z → = f(z) は等角写像である。また, 逆に, f : z → w = f(z) が z 平面上 の領域 D で定義された等角写像ならば D において f(z) は z の正則関数でつねに f ′(z) ≠ 0 である。” ”命題 2.1 は, f(z) が z について正則である場合には Cauchy-Riemann の関係式より f(z) = ux(x, y) + ix(x, y) = y(x, y) ? iuy(x, y) が成り立つので∂(u,v)/∂(x, y) =|ヤコビ行列(原文ご参照)|＝ | f ′(z)|^2 ・・・ (2.2) が言えることによる。”(引用終り) だね
https://en.wikipedia.org/wiki/M-theory One approach to formulating M-theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. 6D (2,0) superconformal field theory ABJM superconformal field theory
https://en.wikipedia.org/wiki/Leech_lattice Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by John Leech (1967). It may also have been discovered (but not published) by Ernst Witt in 1940. The vertex algebra of the two-dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the monstrous moonshine conjectures. 0244あ2023/05/22(月) 20:26:00.15ID:0S5AJj3x ガロアって画像によるけど亀頭みたいな頭してるよね 0245１３２人目の素数さん2023/05/22(月) 23:25:52.25ID:7NpsVkVo>>244 スレ主です ありがとう それは、下記で”弟アルフレッドによるガロアの肖像画”かな？
https://people.math.harvard.edu/~siu/siu_reprints/siu_beijing_paper2005.pdf Science in China Ser. A Mathematics 2005 Vol. 48 Supp. 1?31 Multiplier ideal sheaves in complex and algebraic geometry Yum-Tong Siu Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: siu@math.harvard.edu) Received January 27, 2005
Abstract There are two parts in this article. The first part, which is the main part of the article, discusses the application, by the method of multiplier ideal sheaves, of analysis to complex algebraic geometry. The second part discusses the other direction which is the application of complex algebraic geometry to analysis, mainly to problems of estimates and subellipticity for the ￣∂ operator.
1.2.1 Fujita conjecture.
1.2.2 Effective Matsusaka big theorem. 0266１３２人目の素数さん2023/05/26(金) 12:04:54.19ID:1I7sPBPp 127頁もの 長いので、チラ見もしてないがｗ 貼る
https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/trieste.pdf Multiplier ideal sheaves and analytic methods in algebraic geometry Jean-Pierre Demailly Universit´e de Grenoble I, Institut Fourier Lectures given at the ICTP School held in Trieste, Italy, April 24 ? May 12, 2000 Vanishing theorems and effective results in Algebraic Geometry 0267１３２人目の素数さん2023/05/26(金) 12:26:04.85ID:1I7sPBPp>>266 追加
Introductionだけチラ見したので貼る （文字化けご容赦）
0. Introduction Transcendental methods of algebraic geometry have been extensively studied since a very long time, starting with the work of Abel, Jacobi and Riemann in the nineteenth century. More recently, in the period 1940-1970, the work of Hodge, Hirzebruch, Kodaira, Atiyah revealed still deeper relations between complex analysis, topology, PDE theory and algebraic geometry. In the last ten years, gauge theory has proved to be a very efficient tool for the study of many important questions: moduli spaces, stable sheaves, non abelian Hodge theory, low dimensional topology . . . Our main purpose here is to describe a few analytic tools which are useful to study questions such as linear series and vanishing theorems for algebraic vector bundles. One of the early successes of analytic methods in this context is Kodaira’s use of the Bochner technique in relation with the theory of harmonic forms, during the decade 1950-60. The idea is to represent cohomology classes by harmonic forms and to prove vanishing theorems by means of suitable a priori curvature estimates. The prototype of such results is the Akizuki-Kodaira-Nakano theorem (1954): if X is a nonsingular projective algebraic variety and L is a holomorphic line bundle on X with positive curvature, then Hq (X, ?pX ?L) = 0 for p+q > dim X (throughout the paper we set ? p X = Λ pT ・ X and KX = Λ nT ・ X, n = dim X, viewing these objects either as holomorphic bundles or as locally free OX-modules). It is only much later that an algebraic proof of this result has been proposed by Deligne-Illusie, via characteristic p methods, in 1986.
A refinement of the Bochner technique used by Kodaira led, about ten years later, to fundamental L2 estimates due to H¨ormander [H¨or65], concerning solutions of the Cauchy-Riemann operator. Not only vanishing theorems are proved, but more precise information of a quantitative nature is obtained about solutions of ∂-equations. The best way of expressing these L2 estimates is to use a geometric setting first considered by Andreotti-Vesentini [AV65]. (引用終り) 以上 0269１３２人目の素数さん2023/05/26(金) 18:24:01.56ID:s8fnJ9Ts>>222 >なぜ「共形変換群SO(2,2)は正則関数の等角写像の変換群（無限次元リー群）に拡張される」か 共形変換群SO(2,2)は回転群SO(2)に等しく回転群SO(2)が無限次元リー群だから
