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American Mathematical Monthly -March 2005

March 2005

Yeuh-Gin Gung and Dr. Charles Y. Hu Award to Gerald L. Alexanderson for Distinguished Service to Mathematics
by Robert E. Megginson


Trisections and Totally Real Origami by Roger C. Alperin Mathematical origami or paper folding methods, their relations to plane geometry constructions, and the corresponding algebraic theory of fields are explored. Special attention is made to the use of trisection and its construction by origami. Applications are made to the classical billiard problem of Alhazan.


Measurable Dynamics of Simple p-Adic Polynomials
by John Bryk and Cesar E. Silva,
Many simple polynomials defined on the p-adic numbers are minimal invertible isometries on balls or spheres of Qp. Minimal invertible isometries defined on compact open subsets of Qp are permutations on balls of Qp, and while they are uniquely ergodic, hence ergodic for Lebesgue measure, they are never totally ergodic. The transformations we consider are translations, multiplications, and monomial mappings. To study their measurable and topological dynamics we start with a review of the p-adic numbers, including their topology and the relevant measures on them, and then define the basic notions from dynamics that we require. We also present a short proof that for invertible isometries on compact open subsets of the p-adics, the properties of minimality, ergodicity, and unique ergodicity are equivalent.


The Ecole Polytechnique, 1794-1850: Differences over Educational Purpose and Teaching Practice
by Ivor Grattan-Guinness
In the first part of this article I review the development of the Ecole Polytechnique in Paris from its founding in 1794 until around 1850. The focus falls upon the organisation and national role of the school, and the place of the mathematical courses. In the second part three manifestations of difference are appraised: the various ways of teaching the calculus, and mechanics; and the balance of civilian, military and educational needs in France.


Quadratic Reciprocity in a Finite Group
by William Duke and Kimberly Hopkins,
In 1872 Zolotarev gave an interpretation of the Legendre symbol that generalizes in a natural way to an arbitrary finite group G. We use this generalization to prove a law of quadratic reciprocity for G that includes the classical law when G is cyclic of prime order. Our proof combines the ideas behind one of the proofs of Gauss of classical quadratic reciprocity with two inventions of Frobenius: the Frobenius automorphism and the character table of G.



A Weighted Erdös-Mordell Inequality for Polygons
by Shay Gueron and Itai Shafrir,

An Inequality for Homogeneous Polynomials on Rn
by Luo Xuebo and Zhu-Jun Zheng

Combinatorial Proofs of Fermat’s, Lucas’s, and Wilson’s Theorems
by Peter G. Anderson, Arthur T. Benjamin, and Jeremy A. Rouse,,

On an Irreducibility Criterion of M. Ram Murty
by Kurt Girstmair

On a "Singular" Integration Technique of Poisson
by Robert J. MacG. Dawson

Problems and Solutions


Linear Algebra, 3rd ed.
by John B. Fraleigh and Raymond A. Beauregard.
Reviewed by Jeffrey L. Stuart

Linear Algebra and its Applications, 3rd ed.
by David C. Lay
Reviewed by Jeffrey L. Stuart

Linear Algebra: A Geometric Approach
by Theodore Shifrin and Malcolm R. Adams
Reviewed by Jeffrey L. Stuart

Introduction to Linear Algebra, 3rd ed.
by Gilbert Strang
Reviewed by Jeffrey L. Stuart