[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）19 (1002ﾚｽ)

純粋・応用数学・数学隣接分野（含むガロア理論）19 http://rio2016.5ch.net/test/read.cgi/math/1725190538/

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4: １３２人目の素数さん [] 2024/09/01(日) 20:38:48.95 ID:Dvgug1+6 つづき https://ja.wikipedia.org/wiki/%E8%B6%85%E5%BC%A6%E7%90%86%E8%AB%96 超弦理論 基本的な説明 超弦理論には5つのバリエーションがあり、それぞれタイプI、IIA、IIB、ヘテロSO(32)、ヘテロE8×E8と呼ばれる。この5つの超弦理論はいずれも理論の整合性のために10次元時空を必要とする。 https://en.wikipedia.org/wiki/Leech_lattice Leech lattice Applications 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. (引用終り) つづく http://rio2016.5ch.net/test/read.cgi/math/1725190538/4

739: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/04/04(金) 10:26:20.63 ID:nFnX0O4C >>734 補足 >4 Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402. これの海賊版PDFが見つかった P62 Theorem 1.3.12. Let A, B ∈ Mn be diagonalizable. Then A and B commute if and only if they are simultaneously diagonalizable. Proof. Assume that A and B commute, perform a similarity transformation on both A and B that diagonalizes A (but not necessarily B) and groups together any repeated eigenvalues of A. Ifμ1,...,μd are the distinct eigenvalues of A and n1,...,nd are their respective multiplicities, then we may assume that 略す P63 Observation 1.3.18. Suppose that n ≥ 2. A given A ∈ Mn is similar to a block triangular matrix of the form (1.3.17) if and only if some nontrivial subspace of Cn is A-invariant. Moreover, if W ⊆ Cn isanonzero A-invariantsubspace,thensomevector in W is an eigenvector of A. A given family F ⊆ Mn is reducible if and only if there is some k ∈{2,...,n −1} and a nonsingular S ∈ Mn such that S−1AS has the form (1.3.17) for every A ∈ F. The following lemma is at the heart of many subsequent results. Lemma1.3.19. Let F ⊂ Mn beacommutingfamily. Then some nonzero vector in Cn is an eigenvector of every A ∈ F. Proof. 略す P64 Lemma 1.3.19 concerns commuting families of arbitrary nonzero cardinality. Our next result shows that Theorem 1.3.12 can be extended to arbitrary commuting families of diagonalizable matrices. Definition 1.3.20. A family F ⊂ Mn is said to be simultaneously diagonalizable if there is a single nonsingular S ∈ Mn such that S−1AS is diagonal for every A ∈ F. Theorem 1.3.21. Let F ⊂ Mn be a family of diagonalizable matrices. Then F is a commuting family if and only if it is a simultaneously diagonalizable family. Moreover, for any given A0 ∈ F and for any given ordering λ1,...,λn of the eigenvalues of A0, there is a nonsingular S ∈ Mn such that S−1A0S = diag(λ1,...,λn) and S−1BS is diagonal for every B ∈ F. つづく http://rio2016.5ch.net/test/read.cgi/math/1725190538/739

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