Moving Faces to Other Places: Facet~Derangements
By: Gary Gordon and Elizabeth McMahon
Derangements are a popular topic in combinatorics classes. We study a generalization to face derangements of the n-dimensional hypercube. These derangements can be classified as odd or even, depending on whether the underlying isometry is direct or indirect, providing a link to abstract algebra. We emphasize the interplay between the geometry, algebra, and combinatorics of these sequences, with lots of pretty pictures.
On the Number of Regions into Which n Straight Lines Divide the Plane
By: Oleg A. Ivanov
Suppose we have n lines in the plane. In this paper we study the problem of determining which numbers between n + 1 and n(n + 1)/2 + 1 can be realized as the number of regions into which these lines divide the plane.
When Is a Curve an Octahedron?
By: Joel C. Langer and David A. Singer
We consider complex curves of genus 0 and answer the above riddle. Namely, the lemniscate of Bernoulli, which has obvious fourfold symmetry, actually has the octahedral group as its symmetry group, and may in fact be characterized by this symmetry.
On Leonid Gurvits’s Proof for Permanents
By: Monique Laurent and Alexander Schrijver
We give a concise exposition of the elegant proof given recently by Leonid Gurvits for several lower bounds on permanents.
New Wallis- and Catalan-Type Infinite Products for Π , e, and
By: Jonathan Sondow and Huang Yi
We generalize Wallis’s 1655 infinite product for to one for , as well as give new Wallis-type products for , 2, , , and other constants. The proofs use a classical infinite product formula involving the gamma function. We also extend Catalan’s 1873 infinite product of radicals for e to Catalan-type products for e/4, , and . Here the proofs use Stirling’s formula. Finally, we find an analog for of Pippenger’s 1980 product for e/2, and conjecture that they can be generalized to a product for a power of .
A Concise, Elementary Proof of Arzelà’s Bounded Convergence Theorem
By: Nadish de Silva
Arzelà’s bounded convergence theorem (1885) states that if a sequence of Riemann integrable functions on a closed interval is uniformly bounded and has an integrable pointwise limit, then the sequence of their integrals tends to the integral of the limit. It is a trivial consequence of measure theory. However, denying oneself this machinery transforms this intuitive result into a surprisingly difficult problem; indeed, the proofs first offered by Arzelà and Hausdorff were long, difficult, and contained gaps. In addition, the proof is omitted from most introductory analysis texts despite the result's naturality and applicability. Here, we present a novel argument suitable for consumption by freshmen.
Functional Equations and Finite Groups of Substitutions
By: Mihály Bessenyei
Motivated by a method of solving certain functional equations arising in competition problems, we investigate a class of functional equations, giving representations or existence theorems for the solutions. The main tools in the proofs are Cramer’s rule and the inverse function theorem.
On the Regularity of Operators Near a Regular Operator
By: Shibo Liu
Using the Riesz theorem, we give a new proof that the linear operators near a regular operator are regular.
Proof and Other Dilemmas: Mathematics and Philosophy.
Edited by: Bonnie Gold and Roger A. Simons
Reviewed by: Ethan Akin