Saaty's priority theory and the nomination of a senior professor in operations Research

This paper briefly sketches Saaty's priority theory which has been developed to assign weights (priorities) to a multiple of decision criteria via pairwise comparison of the criteria. Next, there is an extensive treatment of an attempt to apply the ideas in the nomination procedure for the chair of a senior professor in operations research. The reasons why the operations-research specialists in the nomination committee ignored Saaty's priority theory (although they faced a crucial, long-term decision) is analyzed in depth, in order to clarify the experiences of operations-research specialists in industry. They frequently find that operations research is dismissed as inadequate for long-term industrial decisions, on the ground of similar reasons. The paper has therefore been written as a basis for discussion rather than to illustrate effective or ineffective handling of a nomination procedure.     

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).     

Application of operations research in agriculture decision making

A survey is given of applications of operations research in the area of agriculture in China, which includes farming, forestry, stock-raising, fishery, etc.     

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.     

Peirce's place in mathematics

This article compares treatments of the infinite, of continuity and definitions of real numbers produced by the German mathematician Georg Cantor and Richard Dedekind in the late 19th century with similar interests developed at virtually the same time by the American mathematician/philosopher C. S. Peirce. Peirce was led, not by the internal concerns of mathematics which had motivated Cantor and Dedekind, but by research he undertook in logic, to investigate orders of infinite sets (multitudes, in his terminology), and to introduce the related concept of infinitesimals. His arguments in support of the mathematical and logical validity of infinitesimals (which were rejected by such eminent mathematicians as Cantor, Peano, and Russell at the turn of the century) are considered. Attention is also given to the connections between Peirce's mathematics, his philosophy, and especially his interest in continuity as it was related to his Pragmatism.     

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.     

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.     

On Dumont's polynomials

We derive two generating functions and an explicit formula for the polynomials {H_n(x)} studied by Dumont.     

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.