Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).     

Hypergraphes de chaines d'aretes d'un arbre

We introduce the ‘edges-paths hypergraph of a tree’ and study relations of this notion with graphic geometries, chordable graphs. As particular case, we give a simple characterization of intervals hypergraphs.     

On Schmerl's effective version of Brooks' theorem

A new proof is given of Schmerl's recent result that a highly recursive graph G with χ(G) ≤ k according to Brooks' theorem, has a recursive k-colouring.     

D'Alembert's letter to Euler of 3 march 1766

Just before Leonhard Euler resigned his post at the Berlin Academy to return to St. Petersburg, where he spent the rest of his life, he received a letter from d'Alembert urging him to remain in Berlin. This letter is here published for the first time with commentary on the historical context.     

The Ladies' Diary or Woman's Almanack, 1704–1841

The existence of the Ladies' Diary or the Woman's Almanack, an 18th century English magazine devoted largely to problems and puzzles in mathematics, indicates that stereotypes about the inability of women to understand and enjoy mathematics were less strongly believed in the 18th century than they are today. The beginning of the Ladies' Diary coincides with the popularization of mathematics and the growth of mathematical literacy. However, as mathematical literacy spread in response to developing technology's requirements for more mathematically sophisticated workers, women, not part of this need, were left behind. This effect is reflected in the decline in the number of women contributors over the life of the publication.     

Sur les extensions lineaires d'une famille particuliere d'ordres partiels

Corners are defined as ideals of an ordered integer half-dihedron; the paper develops a method of enumeration of the linear extensions of a given corner by means of an alternating sum of products of trinomials. The main result substantially generalizes previously known results and is by itself the starting point of generalizations to some further ordered sets.     

Mersenne's solution for Galileo's problem of the rotating earth

More than twenty years before Huygens and Newton developed formulas for centrifugal acceleration, Mersenne contrived a statisfactory solution for Galileo's problem of the extrusion of bodies from the earth as a result of its daily rotation. Mersenne was able to overcome an error in Galileo's approach without the use either of an explicit notion of infinitesimals or of any clear concept of force. His solution depends on comparing the lengths of two lines, a technique that several historians have claimed to be inadequate for this problem.     

On Axelsson's perturbations

A nice perturbation technique was introduced by Axelsson and further developed by Gustafsson to prove that factorization iterative methods are able, under appropriate conditions, to reach a convergence rate larger by an order of magnitude than that of classical schemes. Gustafsson observed however that the perturbations introduced to prove this result seemed actually unnecessary to reach it in practice. In the present work, on the basis of eigenvalue bounds recently obtained by the author, we offer an alternative approach which brings a partial confirmation of Gustafsson's conjecture.     

Heawood's empire problem

Heawood's map color theorem is generalized to maps of empires on surfaces.     

On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

Matrices of 0's and 1's with total support

We determine when there exists a matrix of 0's and 1's with total support (a pattern of a doubly stochastic matrix) having a specified row sum vector and column sum vector.