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In everyday mathematics, we expect that a matrix with a two-sided inverse will be a square matrix. However, when the entries of the matrix lie in an arbitrary ring, the situation may be very different. In this note, we exhibit some rings over which there are genuinely rectangular invertible matrices. We also show that, for a given ring R, the permitted sizes of the invertible matrices over R are determined by a congruence relation on the natural numbers, and we give an elementary description of such congruences and their connection with the more familiar congruences on the integers. This description of congruences on the natural numbers leads in turn to a classification of rings according to the sizes of their invertible matrices.
Equal Pay for all Prisoners
Maarten C. Boerlijst, Martin A. Nowak and Karl Sigmund
There exist simple strategies for the iterated Prisoner's Dilemma game that yield the same payoff to every opponent, no matter which strategy she is using.
The Long and the Short on Counting Sequences
Jim Sauerberg and Linghsueh Shu
To form a counting sequence one forever repeats the process of forming a new integer sequence that describes the previous integer sequence. This description can be made in many different ways. For instance, Conway's well-known counting sequences are built using "local" descriptions. In this paper we study the periodic behavior of several types of "globally" constructed counting sequences.
Daniel J. Velleman
It is well known that a function from the reals to the reals is continuous in the epsilon-delta sense if and only if the inverse image under the function of every open set is open. Recently, when I was teaching a topology class, a student asked if continuity could be characterized using images instead of inverse images. More precisely, is there a family of sets of reals such that a function from the reals to the real is continuous if and only if the image under the function of every set in the family is also in the family? The answer is "almost, but not quite," and the proof involves some surprising uses of the Cantor set and the Cantor-Lebesgue function.
Probabilistic Pursuits on the Grid
A.M. Bruckstein, C.L. Mallows, and I.A. Wagner
The paths of a sequence of a(ge)nts engaged in a chain of continuous pursuits converge to the straight line between the origin and destination. We consider a discrete setting where the a(ge)nts are only allowed to visit grid points and chase each other according to a probabilistic rule of motion, and prove a similar result: the average paths of ants in a chain of probabilistic pursuit converge rapidly to a straight line. This discrete model of pursuit leads to interesting results also in the context of linear and cyclic pursuits.
Hipparchus, Plutarch, Schrder, and Hough
Richard P. Stanley
In Plutarch's "Table-Talk", written in the first century A.D., the following statement appears: "Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.)" For centuries no sense had been made of this statement. We report on how David Hough, a graduate student at George Washington University, discovered in 1994 the combinatorial significance of the number 103,049. The number 310,952 remains an enigma.