純粋・応用数学・数学隣接分野（含むガロア理論）12

[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）12 (1002ﾚｽ)
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545: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/01/08(日) 17:04:02.18 ID:9zXu/9tz

つづき

Applications
The Vandermonde determinant is used in the representation theory of the symmetric group.[8]

When the values αk belong to a finite field, then the Vandermonde determinant is also called a Moore determinant and has specific properties that are used, for example, in the theory of BCH code and Reed?Solomon error correction codes.

The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix, where the numbers αi are chosen to be roots of unity. Using the Fast Fourier Transform it is possible to compute the product of a Vandermonde matrix with a vector in O(n(log n)^2) time.[9]

https://en.wikipedia.org/wiki/Alexandre-Th%C3%A9ophile_Vandermonde
Alexandre-Theophile Vandermonde (28 February 1735 ? 1 January 1796)
Biography
Vandermonde was a violinist, and became engaged with mathematics only around 1770. In Memoire sur la resolution des equations (1771) he reported on symmetric functions and solution of cyclotomic polynomials; this paper anticipated later Galois theory (see also abstract algebra for the role of Vandermonde in the genesis of group theory).
The same year he was elected to the French Academy of Sciences. Memoire sur des irrationnelles de differents ordres avec une application au cercle (1772) was on combinatorics, and Memoire sur l'elimination (1772) on the foundations of determinant theory.
The Vandermonde determinant does not make an explicit appearance.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1671460269/545

792: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/01/15(日) 20:01:20.18 ID:fdSQKtbP

>>773
>Π_{k=1}^{5}(x-1/cos(2kπ/11)).

p=11ね
下記のGaloisは、Chevalierへの手紙で
楕円曲線の等分問題で、p = 11の解法を取り上げている
英文によるfulltextを探すと、下記がヒットしたので貼る

彼は、20歳で亡くなったという
存命ならば、ここらは論文として出版されたろうに
なお、GaloisのChevalierへの手紙については
下記高木先生の近世数学史談でも、これは取り上げられている

https://www.ias.ac.in/describe/article/reso/004/10/0093-0100
The Last Mathematical Testament of Galois Indian Academy of Sciences
Classics Volume 4 Issue 10 October 1999 pp 93-100

https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100
The Last Mathematical Testament of Galois
Evariste Galois's last mathematical testament in the form ofa letter to his friend Auguste Chevallier is
reproduced here in English translation I.

P3
The last application of the theory of equations is related to the modular equation of elliptic functions.

P5
For p = 7 we find a group of (p + 1) (p - 1) /2 permutations, where
∞ 1 2 4
are respectively related to
0 3 6 5.
This group has its substitutions of the form
略
b being the letter corresponding to c, and a a letter which is a residue or non-residue
according as c.

For p = 11, the same substitutions take place with the same notations,
∞ 1 3 4 5 9
are respectively related to
o 2 6 8 10 7.
Thus, for the case of p = 5,7,11, the modular equation is reduced to degree p.
In all rigor, this reduction is not possible in the higher cases.

The third paper concerns the integrals.
We know that a. sum of terms of the same elliptic function is always reduced to a
single term plus algebraic or logarithmic quantities.

https://www.アマゾン
近世数学史談 (岩波文庫) Paperback Bunko ? August 18, 1995
by 高木 貞治

http://rio2016.5ch.net/test/read.cgi/math/1671460269/792

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