[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)

純粋・応用数学・数学隣接分野（含むガロア理論）12 http://rio2016.5ch.net/test/read.cgi/math/1671460269/

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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

546: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2023/01/08(日) 17:04:29.39 ID:9zXu/9tz >>545 つづき （ついでに） https://en.wikipedia.org/wiki/Vandermonde_polynomial Vandermonde polynomial It is also called the Vandermonde determinant, as it is the determinant of the Vandermonde matrix. The value depends on the order of the terms: it is an alternating polynomial, not a symmetric polynomial. https://en.wikipedia.org/wiki/Lagrange_polynomial Lagrange polynomial https://ja.wikipedia.org/wiki/%E3%83%A9%E3%82%B0%E3%83%A9%E3%83%B3%E3%82%B8%E3%83%A5%E8%A3%9C%E9%96%93 ラグランジュ補間 名称はジョゼフ＝ルイ・ラグランジュに因んだものだが、ラグランジュの発表する1795年よりも以前に、この方法を初めて発見したのは1779年のエドワード・ワーリングである。ラグランジュの結果はレオンハルト・オイラーが1783年に発表したより複雑な形の公式の簡単な帰結となるものであった[1] (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1671460269/546

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