历史与综述

In Interwar France, Henri Villat became the true leader of theoretical researches on fluid mechanics. Most of his original work was done before the First World War; it was highly theoretical and its applicability was questioned. After…

This paper shows that the main features of Turing's thesis derived from those of the expression system (Turing machines) it inaugures, and in particular its conformity. The notions of inaugural statements and texts are defined and…

This is a missing chapter from Hans Magnus Enzensberger's mathematical adventure The Number Devil (Henry Holt and Company, New York, 1997). In the book, a math-hating boy named Robert is visited in his dreams by the clever Number Devil, who…

This paper has been withdrawn by the author

The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\"ottingen in 1854 entitled "\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie…

We investigate the reviews of a Comptes rendus note by Andr\'e Weil in 1940 in the three journals Jahrbuch \"uber die Fortschritte der Mathematik, Zentralblatt f\"ur Mathematik und ihre Grenzgebiete and Mathematical Reviews, together with…

This paper appeals to the figure of \'Evariste Galois for investigating the gates between mathematics and their "publics." The figure of Galois draws some lines of/within mathematics for/from the outside of mathematics and these lines in…

During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled vigorously over the organisation of the theory of bilinear forms. That theory promised a "general" and "homogeneous" treatment of numerous questions arising in…

We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…

Using an identity arising in the known Banach probability problem on matchboxes, we prove an unexpected congruence for odd prime $p:$ for $1\leq k\leq \frac{p-1}{2},\enskip \sum_{i=1}^{p-2k-1}2^{i-1}\binom{k-1+i}{k}\equiv 0\pmod p.$

We throw a brief glance at Galois' life, on the occasion of his 200th anniversary (written in German).

This is the material for two lectures given at Ecole Polytechnique in May 2011 for the math teachers of "classes pr\'eparatoires"(parallel to the undergraduate classes in universities). The introduction is a personal overview on Fourier…

The article reviews some of the (fairly scattered) information available in the mathematical literature on the subject of angles in complex vector spaces. The following angles and their relations are considered: Euclidean, complex, and…

An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…

Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence,…

The Jordan measure, the Jordan curve theorem, as well as the other generic references to Camille Jordan's (1838-1922) achievements highlight that the latter can hardly be reduced to the "great algebraist" whose masterpiece, the Trait\'e des…

This paper reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory (ergodic theory). In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter…

We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a…

We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of ten equivalent statements} borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss…