


Mathematics from Ancient to Modern Times
Org: Richard O'Lander and Ronald Sklar (St. John's) [PDF]
 ROBERT BARONE, Buffalo State College, 1300 Elmwood Avenue, Buffalo,
NY 14222
History of Floquet's Characteristic Equation
[PDF] 
A second order ordinary differential equation with real periodic
coefficients mathematically describes synthetic or naturally occurring
systems that involve periodic structure. The solution to this type of
equation was fully investigated by G. Floquet in 1883. Floquet's
characteristic equation is an eigenvalue equation that arises in the
determination of a solution to the differential equation. Floquet's
theorem states that when the roots of the characteristic equation are
distinct, the solutions satisfying the differential equation are
linearly independent. Floquet's characteristic equation has found
wide application in stability theory and wave propagation in periodic
structure spanning many fields of science and engineering. We trace
the history of Floquet's characteristic equation beginning in the late
nineteenth century through the twentieth century and finally into the
new millennium.
 ROBERT DAWSON, Saint Mary's University, Halifax, Nova Scotia
"The Hardest Set of Arithmetical Problems in Existence"
[PDF] 
Workman's Tutorial Arithmetic was a high school textbook written
around 1900 in Britain that contained what the author referred to as
"... I believe, the hardest set of arithmetical problems in
existence." This talk will examine some aspects of this textbook,
with special consideration of some of the infamous problems, and of
what constituted "arithmetic" at that time.
 JAMES GUYKER, Buffalo State College, 1300 Elmwood Avenue, Buffalo, New
York 14222, USA
Vector Spaces of Magic Squares
[PDF] 
An interesting example to use to illustrate the notions of linear
algebra is the vector space M(n) of real magic squares of order n.
In 1910 E. Bergholt obtained a formula for generating magic squares of
order four from eight arbitrary real numbers, thereby extending the
solution to the ancient problem of constructing one with sixteen
consecutive integers. J. Chernick generalized this formula in 1938 in
describing a basis for the space M(n) for any n. We show that the
orthogonal projection of any n×n matrix onto M(n) has a simple,
closed form and its complementary projection may be explicitly
described in terms of the projections onto the spaces of affine,
coaffine, and doubly affine matrices.
 SANFORD SEGAL, University of Rochester
Helmut Hasse
[PDF] 
After World War I, Helmut Hasse was shunned by many North American
mathematicians as a Nazi collaborator. Hasse did support Hitler, and
was praised by Nazi functionaries, but the full story of his
involvement, which will be sketched, is much more complicated.
 GEORGE STYAN, McGill University, Montreal
Issai Schur (18751941) and the Early Development of the
Schur Complement: Photographs, Documents and Biographical
Remarks
[PDF] 
We comment on the development of the Schur complement from 1812
through 1968 when it was so named and given a notation by Emilie
Virginia Haynsworth (19161985). The adjectival noun "Schur" was
chosen because of the Hilfssatz in the paper by Issai Schur
(18751941) published in 1917 in the Journal für die reine und
angewandte Mathematik (vol. 147, pp. 205232), in which the
Schur determinant formula was introduced.
Following some biographical remarks about Issai Schur, we present the
Banachiewicz inversion formula for the inverse of a nonsingular
partitioned block matrix which was introduced in 1937 by the
astronomer Tadeusz Banachiewicz (18821954). We note, however, that
closely related results were obtained earlier in 1923 by the geodesist
Hans Boltz (18831947) and in 1933 by Ralf Lohan (b. 1902).
We illustrate our findings with several photographs and copies of
original documents. [Joint research with Simo Puntanen (University of
Tampere) supplementing Chapter 0 in The Schur Complement and Its
Applications, Series: Numerical Methods and Algorithms 4 (Fuzhen
Zhang, ed.), Springer 2005; see also Issai Schur (18751941) and
the Early Development of the Schur Complement, with Applications to
Statistics and Probability by Simo Puntanen & George P. H. Styan,
Report A 346, Dept. of Mathematics, Statistics & Philosophy,
University of Tampere, July 2004.]
 ROBERT THOMAS, University of Manitoba, Winnipeg, MB R3T 2N2
Euclid's NonEuclidean Geometry
[PDF] 
One sometimes hears it said that Euclid's Elements is a
compendium of the mathematical or geometrical results that were known
at the time of writing. While Euclid's Conics ought to be a
counterexample, it is not extant and so is useless. Another of
Euclid's books, his astronomy book the Phaenomena, is not
itself a counterexample either because it contains no significant
geometrical results. But it makes definite presuppositions of
geometrical theorems well outside the material of the Elements.
The paper illustrates some of these.

