This leads to the echelon factorization A = CaB, in which B contains the nonzero rows of rref(A), C contains the nonzero columns of (rref(A'))', of A. This factorization tells everything about the echelon bases for the fundamental subspaces. It was pointed out to us by Hans Schneider, and we learned that Albert Tucker and R. H. Bruck knew it too (but did not publish it).
Constructing Kaleidoscopic Tiling Polygons in the Hyperbolic Plane
by S. Allen Broughton
allen.broughton@rose-hulman.edu
Tiling the hyperbolic plane by iterated reflection in the sides of a
kaleidoscopic polygon can be employed to make many beautiful and artistic
patterns in the plane. These constructions may be greatly assisted by
computer methods, provided we can construct the sides and vertices of these
polygons, and hence the reflections in the sides of these polygons. We show
how simple analytic geometry may be used to construct kaleidoscopic
triangles and quadrilaterals. This work was motivated by work with
undergraduates Dawn Haney, Lori McKeough, and Brandi Smith at the
Rose-Hulman NSF-REU Tilings project site
(http://www.tilings.org/index.html).
Indeed, the methods of the paper have been used for visualization in a
classification project of all divisible tilings of the hyperbolic plane. A
description of the project is available at the same site. More importantly,
figures depicting the divisible tilings in the classification?using the
methods of this paper?are given in the file
http://www.tilings.org/images/divquad/table_all.pdf.
A Generalized Approach to the Fundamental Group
by Daniel K. Biss
daniel@math.mit.edu
We begin with a description of a basic phenomenon of algebraic topology:
the correspondence between subgroups of the fundamental group of a space X
and connected covers of X. The essential theorems in this subject are given
historical and intuitive motivation, but proofs are mostly omitted. In the
second half of the paper, we summarize recent work that uncovers a new piece
of structure in the fundamental group, and exploit that structure to
demonstrate that the conditions under which the correspondence holds are
actually far weaker than was once believed.
The Sixtieth William Lowell Putnam Mathematical Competition
by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson
Putnam Trivia for the 90s
by Joseph A. Gallian
jgallian@d.umn.edu
The annual Putnam competition has a long and glorious history of identifying
extraordinary mathematical talent. Indeed, three Putnam Fellows (top 5
finishers) have won the Fields Medal and two have won a Nobel prize in
Physics. In fact, the 1954 Harvard Putnam team included a future Fields
Medalist and a future Nobel Laureate! And of course many Putnam Fellows
have had distinguished careers starting with the very first winner of the
Putnam Fellowship to Harvard?Irving Kaplansky. In the early days (1930s)
only a few hundred students competed in the competition whereas by the 1990s
over 2000 per year took part. In a 1989 issue of the MONTHLY I gave a list
of trivia questions based on the first fifty years of the competition. In
this article I offer trivia questions based on the competitions of the
1990s.
NOTES
On Envy-Free Cake Division
by Oleg Pikhurko
o.pikhurko@dpmms.cam.ac.uk
Calculating Higher Derivatives of Inverses
by Tom Apostol
apostol@caltech.edu
The Modulus of Polynomials with Zeros at the Roots of Unity
by Holly Carley and Xin Li
xli@math.ucf.edu, hkc7t@virginia.edu
On the Norm of Idempotent Operators in a Hilbert Space
by Vladimir Rakocevic
vrakoc@bankerinter.net
A Simple Proof of a Theorem of Block and Hart
by Guang Yuan Zhang
gyzhang@mail.tsinghua.edu.cn
PROBLEMS AND SOLUTIONS
REVIEWS
The Interactive Geometry Software Cinderella
By J. Richter-Gebert and U.
H. Kortenkamp
Reviewed by H. Burgiel
burgiel@math.uic.edu
The Nature of Mathematical Modeling
By Neil Gershenfeld
Reviewed by Shirley B. Pomeranz
pomeranz@euler.mcs.utulsa.edu