与えられた位数を持つ群の個数 シローの定理などの結果から、位数 n の群の構造には n の素因数分解に依存してある制限が加わる。 例えば素数 p , q に対して、 q < p かつ p -1が q で割り切れない場合は、位数 pq の群は必ず巡回群となる。 必要十分条件については巡回数 (群論)（英語版）を参照されたい。
n に平方因子が存在しない場合、位数 n の群はすべて可解である。群の指標の理論（英語版）を用いて証明されたウィリアム・バーンサイド（英語版）の定理によれば、n が2個以下の素因数でのみ割り切れるのであれば、位数 n の群はすべて可解である。 ファイト-トンプソンの定理（英語版）という、証明が長く複雑な定理によると、n が奇数ならば位数 n の群は可解である。
任意の正の整数 n について、位数 n の群のほとんどは可解群である。 特定の位数 n についてこの事実を確認することはそれほど困難なことではない(例えば位数60の群には、同型を除いて非可解なものが1個、可解なものが12個存在する)。 しかし、任意の位数 n についてこの事実を証明するには有限単純群の分類（英語版）を要する。 任意の正の整数 n に対して位数 n の単純群は最大でも2種類しか存在せず、位数 n の同型でない単純群が2種類存在するような正の整数 n は無限に存在する。
>>2 有限単純群の分類（英語版） https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups 抜粋 Gorenstein's program In 1972 Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:
1.Groups of low 2-rank. This was essentially done by Gorenstein and Harada, who classified the groups with sectional 2-rank at most 4. Most of the cases of 2-rank at most 2 had been done by the time Gorenstein announced his program. 2.The semisimplicity of 2-layers. The problem is to prove that the 2-layer of the centralizer of an involution in a simple group is semisimple. 3.Standard form in odd characteristic. If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in "standard form" meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1. 4.Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution the
5 名前：orem. 5.Quasi-standard form 6.Central involutions 7.Classification of alternating groups. 8.Some sporadic groups 9.Thin groups. The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978 10.Groups with a strongly p-embedded subgroup for p odd 11.The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. This was solved by McBride in 1982. 12.Groups of characteristic p type. This is the problem of groups with a strongly p-embedded 2-local subgroup with p odd, which was handled by Aschbacher. 略 []
1 Differential ring 2 Differential field 3 Derivation on a Lie algebra 4 Examples 5 Ring of pseudo-differential operators 6 See also 7 References 8 External links
See also
Differential Galois theory Kahler differential Differentially closed field A D-module is an algebraic structure with several differential operators acting on it. A differential graded algebra is a differential algebra with an additional grading. Arithmetic derivative Differential calculus over commutative algebras Difference algebra Differential algebraic geometry Picard?Vessiot theory
>>13 英語情報ありました https://en.wikipedia.org/wiki/Picard%E2%80%93Vessiot_theory Picard?Vessiot theory In differential algebra, Picard?Vessiot theory is the study of the differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal is to describe when the differential equation can be solved by quadratures in terms of properties of the differential Galois group. The theory was initiated by Emile Picard and Ernest Vessiot from about 1883 to 1904. Kolchin (1973) and van der Put & Singer (2003) give detailed accounts of Picard?Vessiot theory.
Contents 1 History 2 Picard?Vessiot extensions and rings 3 Liouvillian extensions 4 References 5 External links
Abstract We start with the classical definition of Picard-Vessiot extension. We show that the Galois group is an algebraic subgroup of GL(n). Next we introduce the notion of Picard-Vessiot ring and describe the Galois group as spec of a certain subring of a tensor product. We shall also show existence and uniqueness of Picard-Vessiot extensions, using properties of the tensor product. Finally we hint at an extension of the Picard-Vessiot theory by looking at the example of the Weierstras }-function. We use only the most elementary properties of tensor products, spec, etc. We will define these notions and develop what we need. No prior knowledge is assumed.
Magid, Andy R. (1999), "Differential Galois theory" (PDF), Notices of the American Mathematical Society 46 (9): 1041?1049, ISSN 0002-9920, MR 1710665 www.ams.org/notices/199909/fea-magid.pdf
（なおPDFではないが、最新下記） van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 328, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44228-8, MR 1960772
>>28 英語版がかなり章立てが違う。文献が充実している https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%BC%8F 微分方程式 線型微分方程式の研究は歴史が長くヘルマンダー等がそのひとつの頂点であろう[要追加記述]。 それに比して、非線型微分方程式の研究は歴史が浅く比較的簡単な方程式しか解析できていない。 例えばナビエ-ストークス方程式は、流体の支配方程式として重要であるが、その解の存在性は未解決問題でありミレニアム懸賞問題にも選ばれている。 https://en.wikipedia.org/wiki/Differential_equation Differential equation Contents 1 History 2 Example 3 Main topics 3.1 Ordinary differential equations 3.2 Partial differential equations 4 Linear and non-linear 4.1 Examples 5 Existence of solutions 6 Related concepts 7 Connection to difference equations 8 Applications and connections to other areas 8.1 In general 8.2 In physics 8.3 In biology 8.4 In chemistry 8.5 In economics 以下略
関連 https://ja.wikipedia.org/wiki/%E6%93%AC%E5%BE%AE%E5%88%86%E4%BD%9C%E7%94%A8%E7%B4%A0 解析学における擬微分作用素（ぎびぶんさようそ、英: pseudo-differential operator）は、微分作用素の一般化するものである。 1965 年以降、ラース・ヘルマンダー等により急速に研究されて来た。偏微分方程式論の代表的なテーマの一つであるが、マルコフ過程・ディリクレ形式（英語版）・ポテンシャル理論との関わりも深い。 物理学では量子力学や量子統計力学と関係がある。 目次 1 導入 2 定義 3 例 3.1 微分作用素 3.2 熱作用素 3.3 分数的ラプラシアン 3.4 （1?ラプラシアン）の平方根 4 性質 5 擬微分作用素の積分核 6 参考文献 7 関連項目 https://en.wikipedia.org/wiki/Pseudo-differential_operator In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory. Contents 1 History 2 Motivation 2.1 Linear differential operators with constant coefficients 2.2 Representation of solutions to partial differential equations 3 Definition of pseudo-differential operators 4 Properties 5 Kernel of pseudo-differential operator 6 See also 7 Further reading 8 References 9 External links
関連抜粋 https://en.wikipedia.org/wiki/Cotangent_space Cotangent space From Wikipedia, the free encyclopedia
In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors.
Contents 1 Properties 2 Formal definitions 2.1 Definition as linear functionals 2.2 Alternative definition 3 The differential of a function 4 The pullback of a smooth map 5 Exterior powers 6 References
関連抜粋 https://en.wikipedia.org/wiki/Monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'. It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we 'run round' a path encircling a singularity. The failure of monodromy is best measured by defining a monodromy group: a group of transformations acting on the data that encodes what does happen as we 'run round'. Contents 1 Definition 2 Example 3 Differential equations in the complex domain 4 Topological and geometric aspects 4.1 Monodromy groupoid and foliations 5 Definition via Galois theory 6 See also 7 Notes 8 References Definition via Galois theory
Let F(x) denote the field of the rational functions in the variable x over the field F, which is the field of fractions of the polynomial ring F[x]. An element y = f(x) of F(x) determines a finite field extension
45 名前： [F(x) : F(y)].
This extension is generally not Galois but has Galois closure L(f). The associated Galois group of the extension [L(f) : F(y)] is called the monodromy group of f.
In the case of F = C Riemann surface theory enters and allows for the geometric interpretation given above. In the case that the extension [C(x) : C(y)] is already Galois, the associated monodromy group is sometimes called a group of deck transformations.
This has connections with the Galois theory of covering spaces leading to the Riemann existence theorem. []
en.wikipedia.org/wiki/Monodromy Monodromy 抜粋 In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'. It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we 'run round' a path encircling a singularity. The failure of monodromy is best measured by defining a monodromy group: a group of transformations acting on the data that encodes what does happen as we 'run round'.
へへ、英文だと違うね https://en.wikipedia.org/wiki/Hilbert%27s_twenty-first_problem For Riemann?Hilbert factorization problems on the complex plane see Riemann?Hilbert.
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specified singular points and monodromic group.
History
This problem is more commonly called the Riemann?Hilbert problem. There is now a modern (D-module and derived category) version, the 'Riemann?Hilbert correspondence' in all dimensions. The history of proofs involving a single complex variable is complicated. Josip Plemelj published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of G. D. Birkhoff in 1913 also, but the whole area, including work of Ludwig Schlesinger on isomonodromic deformations that would much later be revived in connection with soliton theory, went out of fashion. Plemelj (1964) wrote a monograph summing up his work. A few years later the Soviet mathematician Yuliy S. Il'yashenko and other
49 名前：s started raising doubts about Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.) つづく []
Indeed Andrey A. Bolibrukh (1990) found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind; Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems. (In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists. In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true. Andrey A. Bolibrukh (1992) and independently Vladimir Kostov (1992) showed that for any size, an irreducible monodromy group can be realised by a Fuchsian system. The codimension of the variety of monodromy groups of regular systems of size n with p+1 poles which cannot be realised by Fuchsian systems equals 2(n-1)p (Vladimir Kostov (1992)).) Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on Riemann surfaces. Pierre Deligne proved a precise Riemann?Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means). With work by Helmut Rohrl, the case in one complex dimension was again covered. （引用おわり）
英文 https://en.wikipedia.org/wiki/Sheaf_%28mathematics%29 History The first origins of sheaf theory are hard to pin down ? they may be co-extensive with the idea of analytic continuation[clarification needed]. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.
1936 Eduard ?ech introduces the nerve construction, for associating a simplicial complex to an open covering. 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochains. 1943 Norman Steenrod publishes on homology with local coefficients. 1945 Jean Leray publishes work carried out as a prisoner of war, motivated by proving fixed point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences. 1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with Andre Weil (see De Rham-Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the later carapaces). 1948 The Cartan seminar writes up sheaf theory for the first time. 1950 The "second edition" sheaf theory from the Cartan seminar: the sheaf space (espace etale) definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables. つづく
1951 The Cartan seminar proves the Theorems A and B based on Oka's work. 1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre, as is Serre duality. 1954 Serre's paper Faisceaux algebriques coherents (published in 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Hirzebruch, who writes a major 1956 book on topological methods. 1955 Alexander Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors. 1956 Oscar Zariski's report Algebraic sheaf theory 1957 Grothendieck's Tohoku paper rewrites homological algebra; he proves Grothendieck duality (i.e., Serre duality for possibly singular algebraic varieties). 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them, local cohomology, derived categories (with Verdier), and Grothendieck topologies. There emerges also his influential schematic idea of 'six operations' in homological algebra. 1958 Godement's book on sheaf theory is published. At around this time Mikio Sato proposes his hyperfunctions, which will turn out to have sheaf-theoretic nature.
At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology. It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke?Joyal semantics, but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as Leib
>>30 マスロフ https://en.wikipedia.org/wiki/Victor_Pavlovich_Maslov Victor Pavlovich Maslov From Wikipedia, the free encyclopedia This name uses Eastern Slavic naming customs; the patronymic is Pavlovich and the family name is Maslov.
Viktor Pavlovich Maslov (Russian: Виктор Павлович Маслов; born 15 June 1930, Moscow) is a Russian physicist and mathematician. He is member of the Russian Academy of Sciences. He obtained his doctorate in physico-mathematical sciences in 1967. His main fields of interest are quantum theory, idempotent analysis, non-commutative analysis, superfluidity, superconductivity, and phase transitions. He is editor-in-chief of Mathematical Notes and Russian Journal of Mathematical Physics.
The Maslov index is named after him. Selected books Karasev, M. V.; Maslov, V. P.: Nonlinear Poisson brackets. Geometry and quantization. Translated from the Russian by A. Sossinsky [A. B. Sosinski?] and M. Shishkova. Translations of Mathematical Monographs, 119. American Mathematical Society, Providence, RI, 1993. Kolokoltsov, Vassili N.; Maslov, Victor P.: Idempotent analysis and its applications. Translation of Idempotent analysis and its application in optimal control (Russian), "Nauka" Moscow, 1994. Translated by V. E. Nazaikinskii. With an appendix by Pierre Del Moral. Mathematics and its Applications, 401. Kluwer Academic Publishers Group, Dordrecht, 1997. Maslov, V. P.; Fedoriuk, M. V.: Semiclassical approximation in quantum mechanics. Translated from the Russian by J. Niederle and J. Tolar. Mathematical Physics and Applied Mathematics, 7. Contemporary Mathematics, 5. D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. This book was cited over 700 times at Google Scholar in 2011.
Maslov, V. P. Operational methods. Translated from the Russian by V. Golo, N. Kulman and G. Voropaeva. Mir Publishers, Moscow, 1976.
Maslov index https://en.wikipedia.org/wiki/Lagrangian_Grassmannian#Maslov_index 抜粋 In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V.
Maslov index A path of symplectomorphisms of a symplectic vector space may be assigned a Maslov index, named after V. P. Maslov; it will be an integer if the path is a loop, and a half-integer in general.
If this path arises from trivializing the symplectic vector bundle over a periodic orbit of a Hamiltonian vector field on a symplectic manifold or the Reeb vector field on a contact manifold, it is known as the Conley-Zehnder index. It computes the spectral flow of the Cauchy-Riemann-type operators that arise in Floer homology[citation needed].
It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization and in symplectic geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.
In the mathematical discipline of set theory, forcing is a technique discovered by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo?Fraenkel set theory. Forcing was considerably reworked and simplified in the following years, and has since served as a powerful technique both in set theory and in areas of mathematical logic such as recursion theory.
Descriptive set theory uses the notion of forcing from both recursion theory and set theory. Forcing has also been used in model theory but it is common in model theory to define genericity directly without mention of forcing.
Contents
1 Intuitions 2 Forcing posets 2.1 P-names 2.2 Interpretation 2.3 Example 3 Countable transitive models and generic filters 4 Forcing 5 Consistency 6 Cohen forcing 7 The countable chain condition 8 Easton forcing 9 Random reals 10 Boolean-valued models 11 Meta-mathematical explanation 12 Logical explanation 13 See also 14 References 15 External links
97 名前：だね では、君にはこの言葉を贈ろう http://proverbes.kitakama-france.com/index.php?%E8%AB%BAIAK フランス語のことわざI-1 Au royaume des aveugles, les borgnes sont rois. 【逐語訳】「めくらの国ではめっかちが王様だ」
