A case of mathematical eponymy: the Vandermonde determinant

Abstract

We study the historical process that led to the worldwide adoption, throughout mathematical research papers and textbooks, of the denomination "Vandermonde determinant". The mathematical object can be related to two passages in Vandermonde's writings, of which one inspired Cauchy's definition of determinants. Influential citations of Cauchy and Jacobi may have initiated the naming process. It started during the second half of the 19 th century as a teaching practice in France. The spread in textbooks and research journals began during the first half of 20 th century, and only reached full acceptance after the 1960's. The naming process is still ongoing, in the sense that the volume of publications using the denomination grows significantly faster than the overall volume of the field. RésuméNous étudions le processus historique qui a conduit à l'adoption dans le monde entier de la dénomination « déterminant de Vandermonde ». L'objet mathématique peut être relié à deux passages dans les écrits de Vandermonde, dont l'un a inspiré Cauchy pour sa définition des déterminants. Les citations de Cauchy et Jacobi ont pu déclencher le processus de dénomination. Celui-ci a démarré au cours de la seconde moitié du xix e siècle comme une pratique pédagogique. Cette pratique a précédé, plutôt que suivi, la diffusion dans les livres et les articles de recherche, qui a commencé pendant la première moitié du xx e siècle, et n'a atteint un réel consensus qu'après les années 1960. Le processus de dénomination est encore en cours, au sens où l'usage du nom croît significativement plus vite que le volume global de publications du domaine.

Taylor Polynomials in High Arithmetic Precision as Universal Approximators

“…For example if we apply the previous for data of Newtons second law [30] of motion s(t) = t 2 , with s indicating space and t time, we may calculate vectors aand solve Equation ( 4) for b, with c = 1, and p = 1000bits precision, we derive that b = {0, 0, 1/2} + O(10 −270 ), and hence 1 2 s = 1 → s = 2, which is equivalent with s = a, where a = F m = 2, which represents the external source which produces s(t) = t 2 .…”

“…n [22,16,28]. The square Vandermonde matrix for distinct x i is invertible, with det(V) = 1≤i<j≤n (x j − x i ) [30], and inverse matrix V −1 =U −1 L −1 , where the elements l ij of the L −1 , and u ij of U −1 , are given by [24]. Hence we have closed-form formulas for the matrix V −1 , and for det(V), which will be later used for the comparison among the various digits utilized in the calculations.…”