February 2013 Contents
Our feature articles for February include a discussion of how Dedekind’s Cut Theorem is equivalent to the four “cornerstone theorems” of single-variable Real Analysis, an explanation of how Lehmer obtained π as a limit of a sequence of interesting series, a proof of a theorem from which well-known sequence theorems of Moessner, Passache and Long can be had as corollaries, an argument that each Heronian tetrahedron can be positioned with integer coordinates, and an analysis using Graph Theory of the popular game Rubik’s Slide. Our Notes feature an elementary proof of Hilbert’s Inequality, an alternate proof of a classical result of Cauchy concerning zeros of polynomials, a novel way to evaluate , and a proof of Fermat’s Little Theorem using Dynamical Systems. After our world famous Problem Section, we close with a review by Michael Nathanson of Elements of Information Theory by Thomas M. Cover.
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Toward a More Complete List of Completeness AxiomsHolger Teismann
We first discuss the Cut Axiom, due to Dedekind, which is one of the many equivalent formulations of the completeness of the real numbers. We point out that the Cut Axiom is equivalent to four “cornerstone theorems” of single-variable Real Analysis, namely, the Intermediate, Extreme, and Mean Value Theorems, as well as Darboux’s Theorem.
We then describe some general properties of ordered fields, in particular the Archimedean Property and its consequences, and provide a list of statements that are equivalent to completeness and may thus serve as alternate completeness axioms.
Minimum Degree, Leaf Number, and Hamiltonicity
Let G be a finite connected graph with minimum degree . The leaf number L(G) of G is defined as the maximum number of leaf vertices contained in a spanning tree of G. We show that if , then G is Hamiltonian. This confirms, and improves, a conjecture of the computer program Graffiti.pc.
Lehmer’s Interesting Series
Freeman J. Dyson, Norman E. Frankel, and M. Lawrence Glasser
The series is evaluated in a nonrecursive and closed process. It can be analytically continued beyond its domain of convergence when k = 0, 1, 2, …. From this we provide a firm basis for Lehmer’s observation that π emerges from the limiting behavior of as k 1.
Volumes of Generalized BallsFuchang Gao
A concise proof is presented for the known formula (see , , ) of the volume of
The derivation assumes a familiarity with the properties of both the gamma function and the exponential distribution.
On Moessner’s TheoremDexter Kozen and Alexandra Silva
Moessner’s theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1n , 2n , 3n , . . . . Paasche’s theorem is a generalization of Moessner’s; by varying the parameters of the procedure, we can obtain the sequence of factorials 1!, 2!, 3!, . . . or the sequence of superfactorials 1!, 2! 1!, 3! 2! 1!, . . . . Long’s theorem generalizes Moessner’s in another direction, providing a procedure to generate the sequence Proofs of these results in the literature are typically based on combinatorics of binomial coefficients or calculational scans. In this note, we give a short and revealing algebraic proof of a general theorem that contains Moessner’s, Paasche’s, and Long’s as special cases. We also prove a generalization that gives new Moessner-type theorems.
Heronian Tetrahedra Are Lattice TetrahedraSusan H. Marshall and Alexander R. Perlis
Extending a similar result about triangles, we show that each Heronian tetrahedron may be positioned with integer coordinates. More generally, we show the following: if an integral distance set in R3 can be positioned with rational coordinates, then it can in fact be positioned with integer coordinates. The proof, which uses the arithmetic of quaternions, is tantamount to an algorithm.
Rubik’s on the Torus
Jeremy Alm, Michael Gramelspacher, and Theodore Rice
Rubik’s Slide is an electronic puzzle in the same style as other puzzles in the Rubik’s family. It will be of interest to puzzle enthusiasts as well as to teachers and students of mathematics. This paper analyzes the puzzle using tools from group theory and graph theory. We discuss solution heuristics and model game-play using group theory. We briefly give results concerning solutions requiring a minimal number of moves.
A Simple Elementary Proof of Hilbert’s InequalityDavid C. Ullrich
We give a very simple proof of Hilbert’s inequality.
The Twin of a Theorem by CauchyAaron Melman
A classical result by Cauchy determines a disk containing all the zeros of a given polynomial. We derive this result using only linear algebra techniques, and, in the process, discover a twin disk that also contains all the zeros of the polynomial.
How Would Riemann Evaluate ?Marco Dalai
Driven by an inspiring comment by Prof. H. M. Edwards, we present a method of evaluation of , apparently unnoticed before, that follows easily from Riemann’s own representation of the zeta function.
Polynomial Dynamics and a Proof of the Fermat Little TheoremVladimir Dragovi?
We present a dynamical proof of the Fermat little theorem, based on the Chebyshev polynomials and their extraordinary composition property.