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American Mathematical Monthly - June/July 2010


Definitions and Nondefinability in Geometry
By: James T. Smith
Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid's, which handled some basic concepts awkwardly and imprecisely. They would introduce precision required for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By following this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics.


Volterra Functional Differential Equations: Existence, Uniqueness, and Continuation of Solutions
By: Hartmut Logemann and Eugene P. Ryan,
The initial-value problem for a class of Volterra functional differential equations—of sufficient generality to encompass, as special cases, ordinary differential equations, retarded differential equations, integro-differential equations and hysteretic differential equations—is studied. A self-contained and elementary treatment of this over-arching problem is provided, in which a unifying theory of existence, uniqueness and continuation of solutions is developed. As an illustrative example, a controlled differential equation with hysteresis is considered.


Trigonometric Identities à la Hermite
By: Warren P. Johnson
Hermite once observed that a certain product of cotangents can be integrated by breaking it into a sum of cotangents, where the coefficients are themselves products of cotangents. Why should such an identity exist? We give two derivations, one based on the partial fractions expansion of the cotangent. Hermite seems to have used a mixture of the two. We also discuss and extend a second theorem of Hermite, which leads to generalizations of his cotangent identity. The paper veers off into determinants at the end.


Finite Quantum Measure Spacese
By: Stan Gudder
Quantum measure spaces possess a certain "quantum weirdness" and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as "quantum coins" and particle-antiparticle pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don’t need quantum mechanics or measure theory to understand this article.


Lp Norms and the Sinc Function
By: D. Borwein, J. M. Borwein, and I. E. Leonard,,
It’s everywhere! It’s everywhere! Â…

In this note we give elementary proofs of some of the striking asymptotic properties of the p-norm of the ubiquitous sinc function. Based on experimental evidence we conjecture some enticing further properties of the p-norm as a function of p. See, for example,



Indefinite Quadratic Forms and the Invariance of the Interval in Special Relativity
By: John H. Elton
In this note, a simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in special relativity from Einstein's postulate on the universality of the speed of light; students are often rightfully confused by the incomplete or incorrect proofs given in many texts. The result is illuminated and generalized using Hilbert’s Nullstellensatz, allowing one form to be a homogeneous polynomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizability of semi-definite real quadratic forms is given.

Monotone Convergence Theorem for the Riemann Integral
By: Brian S. Thomson
The monotone convergence theorem holds for the Riemann integral, provided (of course) it is assumed that the limit function is Riemann integrable. It might be thought, though, that this would be difficult to prove and inappropriate for an undergraduate course. In fact the identity is elementary: in the Lebesgue theory it is only the integrability of the limit function that is deep. This article shows how to prove the monotone convergence theorem for Riemann integrals using a simple compactness argument (i.e., invoking Cousin’s lemma). This material could reasonably and appropriately be used in classroom presentations where the students are indoctrinated on this antiquated, but still popular, integration theory.

A Proof of a Version of a Theorem of Hartogs
By: Marco Manetti
It is proved that a formal power series in s complex variables is convergent, if it is convergent on each line through the origin.

Discovering and Proving that π Is Irrational
By: Timothy W. Jones
Ivan Niven's proof of the irrationality of π is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of π . This simplifying concept yields a more motivated proof of the irrationality of π and π 2.


Mathematical Modeling: A Case Studies Approach
By: Reinhard Illner, C. Sean Bohum, Samantha McCollum, and Thea van Roode
Reviewed by: Charles R. Hampton