AUGUST/SEPTEMBER 1998

Two Functions Whose Powers Make Fractals
by Marc Frantz
mfrantz@math.iupui.edu
A common counterexample in real analysis is the function f defined on (0,1) by letting f(x)=0 if x is irrational, and f(x)=1/n if x=m/n with m and n relatively prime. Variations of this function can be constructed by raising 1/n to a power p>1, or by replacing the rationals with the dyadic rationals (fractions with denominators that are powers of 2). In each case the graph looks like a fractal , and indeed the graphs are easily generated by simple iterated function systems like those commonly used in fractal geometry. For sufficiently large values of p, the nondifferentiability sets of these functions have nonintegral Hausdorff dimensions, and this can be proved by exploiting interesting connections with rational approximation theory. By thinking of the functions as step-width functions for Devil's staircases, connections can also be made with important dynamical systems.

Pinhole Cameras, Perspective, and Projective Geometry
by M. H. Eggar
m.eggar@maths.ed.ac.uk
In art, perspective is often used to convey depth cues and enhance the reality of a picture. Suppose collinear points in three dimensions are always represented as collinear points in the picture. When is such a picture obtained by the projection from an artist's eye onto a canvas? When it is, how can one reconstruct from the picture the positions of the artist's eye and the canvas? This is the same problem as locating from a small part of a photo taken by a pinhole camera the position from which the photo was taken. The connection between these questions and theorems of (projective) geometry, such as Desargues' theorem and the invariance of cross-ratio theorem, is made precise.

Functions Redefined
by Kosta Dosen
This article is about definitions of the general notions of function, onto function, and one-one function that exhibit clearly the regularities and symmetries of these notions. These definitions match a Galois connection.

Unique Developments in Non-Integer Bases
by Vilmos Komornik and Paola Loreti
komornik@math.u-strasbg.fr, loreti@vaxiac.iac.rm.cnr.it
According to a surprising theorem of Erd&oumls, Horv&aacuteth and Joó—, there exist numbers 1 <q <2 for which there is only one sequence , of zeroes and ones such that We determine the smallest number having this property: it turns out that Whether q is irrational or not, remains an open question.

Mathematics on a Distant Planet
by R. W. Hamming
The paper examines the question of how much of standard mathematics is arbitrary and how much is fixed.

NOTES
A Theorem of Burnside on Matrix Rings
by T. Y. Lam
lam@math.berkeley.edu

A Proof of the Change of Variable Formula for d-Dimensional Integrals
by Peter Dierolf and Volker Schmidt
dierolf@uni-trier.de, schmidt@math39.uni-trier.de

On the Diophantine Equation
by Hong Bing Yu
yuhb@math.ustc.edu.cn

Wolstenhome Revisited
by Ira M. Gessel
gessel@math.brandeis.edu

THE EVOLUTION OF...
Nonstandard Analyses and the History of Classical Analysis
by F. A. Medvedev

PROBLEMS AND SOLUTIONS
REVIEWS
Introduction to Calculus and Classical Analysis.
By Omar Hijab.

Introduction to Mathematical Structures and Proofs.
By Larry J. Gerstein.
Reviewed by Steven G. Krantz
sk@math.wustl.edu

Mathematical Reflections; In a Room with Many Mirrors.
By Peter Hilton, Derek Holton, and Jean Pedersen.
Reviewed by Joby Milo Anthony
janthony@pegasus.cc.ucf.edu

Interactive Differential Equations.
By Beverly West, Steven Strogatz, Jean Marie McDill, and John Cantwell.
Reviewed by Robert L. Devaney
bob@bu.edu

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