历史与综述
This paper is a review of the book "Knots" by Alexei Sossinsky. The review includes a short personal history of knot theory at the end of the twentieth century.
Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary. In fact, all we need is the theorem on the dimensions…
We study Smarandache sequences of numbers, and related problems, via a Computer Algebra System. Solutions are discovered, and some conjectures presented.
This expository paper presents the general solution of a quartic equation as a jump off point to introduce Lefschetz fibrations. It should be accessible to a broad audience.
An opiniated essay on what pure mathematics is and why the adjective "pure" in "pure mathematics" is not a good choice.
[Inserted by J. Maurice Rojas] We give a formula for the number of complex roots of a generic system of two polynomial equations in two unknowns. The formula is completely combinatorial, ultimately depending just on the convex hull of the…
The oldest (c. 4000 BC) undeciphered language is the Old European Script known from approximately 940 inscribed objects (82% of inscriptions on pottery) found in excavations in the Vinca-Tordos region Transylvania. Also, it is not known for…
A small and unsystematic selection of my favorite appearances of mathematicians and mathematics in German literature. It includes classic and romantic (Lessing, Goethe, Wezel, F. Schlegel, Kleist, Novalis, Grillparzer, Heine), modern…
We present an elementary self-contained detailed computation of Ramanujan's most famous singular modulus, k210, based on the Kronecker Limit Formula.
The information-theoretic point of view proposed by Leibniz in 1686 and developed by algorithmic information theory (AIT) suggests that mathematics and physics are not that different. This will be a first-person account of some doubts and…
The simultaneous appearance in May 2003 of four books on the Riemann hypothesis (RH) provoked these reflections. We briefly discuss whether the RH should be added as a new axiom, or whether a proof of the RH might involve the notion of…
In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special…
The first approach to the history of mathematics in China led by Li Yan (1892--1963) and Qian Baocong (1892--1974) featured discovering {\it what} mathematics had been done in China's past. From the 1970s on, Wu Wen-tsun and others shifted…
After mathematicians and physicists had learned that the structure of physical space was not necessarily Euclidean, it became conceivable that the global topological structure of space was non-trivial. In the context of the late 19th…
In the 1850s Weierstrass succeeded in solving the Jacobi inversion problem for the hyper-elliptic case, and claimed he was able to solve the general problem. At about the same time Riemann successfully applied the geometric methods that he…
This panel draws on research of the teaching of mathematical proof, conducted in five countries at different levels of schooling. With a shared view of proof as essential to the teaching and learning of mathematics, the authors present…
This article is a part of the report for the research project ``Reform of the Course System and Teaching Content of Higher Mathematics (For Non-Mathematical Specialties)'' in 1995, supported by the National Ministry of Education. There are…
It is rare to succeed in getting mathematics into ordinary conversation without meeting all kinds of reservations. In order to raise public awareness of mathematics effectively, it is necessary to modify such attitudes. In this paper, we…
Linear algebra represents, with calculus, the two main mathematical subjects taught in science universities. However this teaching has always been difficult. In the last two decades, it became an active area for research works in…
Sur l'origine des chiffres arabes A. Boucenna 1 From the pagination of an Algerian Arabic manuscript of the beginning of the 19th century,we rediscover the original shape that the Arabic numerals had before passing in Europe and underwent…