On Ryser's maximum term rank formula

A simple proof is given for the maximum term rank of matrices of 0's and 1's with a specified row and column sum vector.     

Some hypergraphs and packing problems associated with matrices of 0's and 1's

We define various classes of hypergraphs associated with m × n matrices of 0's and 1's and determine for which classes the maximum cardinality of a set of pairwise disjoint edges equals the minimum cardinality of a set of nodes that cover all edges independently of the matrix.     

Euler's 1760 paper on divergent series

That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.     

Waring's problem for the ring of polynomials

We investigate the question of which polynomials are not representable as the sum of “few” powers of polynomials. In particular for Waring's problem over the ring of polynomials we show that there exist polynomials which are not the sum of fewer than $\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$nth powers of polynomials.     

On the Dulac's problem for piecewise analytic vector fields

This paper presents a version of Dulac's Problem for piecewise analytic vector fields that states conditions for the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.     

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).     

Ramsey's theorem with sums or unions

Using Hindman's theorem as a strong pigeonhole principle, we prove strengthened versions of Ramsey's theorem and of various generalizations of Ramsey's theorem due to Nash-Williams, Galvin and Prikry, and Silver.     

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.     

Roth's removal rule revisited

The theory of companion matrices is used to give explicit representations for the matrices needed in Roth's removal rule. These are then used to give simple proofs for the cyclic decomposition theorem, as well as for Roth's similarity theorem for matrices over a field.     

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.     

Application of operations research in Israel's water sector

Israel's water sector has moved from a period of development, which ended in the mid-1960's, to an era of scarcity. Over 95% of the natural water potential is already being utilized, and there is severe competition for this scarce resource between economic sectors and geographic regions. Management of development, design and operation of the water systems is therefore an acute problem, with implications ranging from national policy to efficiency in daily operation. Operations research methodologies have been developed and applied quite extensively over the last 15 years in Israel's water sector, dealing with the full range of its problems. The paper is a survey of these applications, aimed at providing a realistic assessment of their value, from which water resources systems analysts in other countries may derive some guidelines for their own work.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.     

Heawood's empire problem

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

Lyapunov's center theorem for resonant equilibrium

Lyapunov's center theorem relative to the existence of families of periodic orbits emanating from an equilibrium is generalized to cases where a resonance occurs between two basic frequencies. Analytical Hamiltonian systems are considered and the theorems depend on the nonannulation of an invariant of the system.The proof is performed in two steps. In a first step the theorems are shown to be valid for some approximation of the Hamiltonian system. These results are described in a previous paper (Henrard, 1970) and are only summarized here. In a second step Poincaré's perturbation theorem is generalized in order to transfer to the original system the conclusions relatives to its approximations.In the conclusion, our results are compared with similar results published recently.     

Zermelo's discovery of the “Russell Paradox”

In his 1908 paper on the Well-Ordering Theorem, Zermelo claimed to have found “Russell's Paradox” independently of Russell. Here we present a short note, written by E. Husserl in 1902, which contains a detailed exposition of Zermelo's original version of the paradox. We add some comments concerning the date of Zermelo's discovery, the circumstances which caused Husserl to write down Zermelo's argument, and the argument itself.     

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.     

Hypergraphs and Whitney's theorem on edge-isomorphisms of graphs

This paper presents a new proof of Whitney's theorem on edge-isomorphisms of graphs and extends Whitney's theorem to hypergraphs. Whitney's theorem asserts that any two edge-isomorphic graphs of order at least 5 have their edge-isomorphism induced by a node-isomorphism isomorphism. Previous results of Gardner and of Berge and Rado are used.     

The Laplace and poisson equations in Schwarzschild's space-time

The method of separation of variables is used to solve the Laplace equation in Schwarzschild's space-time. The solutions are given explicitly in series form and in terms of Legendre functions. Green's function is determined and remarks are made on the solution of Poisson's equation for a point source.     

Carolyn Eisele's place in Peirce studies

Carolyn Eisele's unique, ongoing career as a scholar is sketched, and the importance of her contributions to Peirce Studies and other fields is emphasized. The essay concludes with a series of suggestions about how to interpret Peirce's works based on themes related to the pioneering efforts of Dr. Eisele.