历史与综述

We review the extraordinary fertility and proliferation in mathematics and physics of the concept of a surface with constant and negative Gaussian curvature. In his outstanding 1868 paper Beltrami discussed how non-Euclidean geometry is…

We describe the links between group theory and psychology, in particular through the works of Piaget. We show that groups appear universally in his description of children's intelligence, and that the notion of groupoid, which was little…

In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of…

In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form $\frac{3n^2 \pm n}{2}$. He recalls that the infinite product $(1-x)(1-x^2)(1-x^3)...$ expands into an infinite series with exponents the…

There are compelling historical and mathematical reasons why we ended up, among others in Physics, with using the scalars given by the real or the complex numbers. Recently, however, infinitely many easy to construct and use other algebras…

The Law of Quadratic Reciprocity was conjectured by Euler and Legendre who both found an incomplete proof. Gauss called this law "Theorema Fundamentale", and he was the first who gave a complete proof, he also highlighted the equivalence of…

We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…

Stephen Toulmin once observed that `it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate'. Might the application of Toulmin's layout of arguments to mathematics remedy this oversight?…

This essay, originally published in the Sept 1990 Notices of the AMS, discusses problems of our mathematical education system that often stem from widespread misconceptions by well-meaning people of the process of learning mathematics. The…

This is an English translation from the Latin original of Leonhard Euler's ``Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur''. In this paper, Euler…

We write down the functional equation of the zeta function of a global field. This equation is implicit in Weil's ``Basic Number Theory''.

We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.

This seems to be the first English translation of this paper from the French original, ``Sur les rentes viageres''. In the paper, Euler gives a general formula for calculating the price of a life annuity that yields a certain amount per…

We survay some nice result concerning the irrationals with a metric space point of view.Here is ofcourse nothing new may be or an expert in this field.

This article is based on the author's inaugural lecture at the University of Cologne on 24 January 2003.

This book is a textbook for the basic course of differential geometry. It is recommended as an introductory material for this subject.

This is a translation into English from the original Latin of Leonhard Euler's Exercitatio analytica, Nova Acta Academiae Scientarum Imperialis Petropolitinae 8 (1794), 69-72; E664 in the Enestrom index. In it Euler uses the infinite…

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871-1956).

This is an essay on the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of…

By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for…