Finding Good Bets in the Lottery, and Why You Shouldn't Take Them
By: Aaron Abrams and Skip Garibaldi
Everyone knows that buying lottery tickets is a bad investment. But do you know why? You will most likely lose your dollar, but there is a small chance that you will win big. In some lottery drawings, your expected rate of return is spectacularly high. Do you know how to identify a drawing that has a positive expected rate of return? When you find one, how do you decide if it is a good investment? In a sense, such a drawing is a caricature of a typical investment, for which both the risks and the rewards are more modest. In general, how should an investor compare such alternatives? And can the lottery ever be a good investment?
Voting in Agreeable Societies
By: Deborah E. Berg, Serguei Norine, Francis Edward Su, Robin Thomas, and Paul Wollan
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When is agreement possible among a group of voters? Given preferences on a political spectrum, when can we guarantee that some fraction (say, a majority) of the population will agree on some candidate? By "agree," we mean in the sense of approval voting, in which voters declare which candidates they find acceptable. This article grew out of the observation that Helly's theorem, a classical result in convex geometry, has an interesting voting interpretation. We develop a generalization that produces conditions under which, in a society of voters, a majority or a specified plurality exists.
Agreement in Circular Societies
By: Christopher S. Hardin
Approval voting gives rise to numerous combinatorial questions. For instance, consider an election in which the candidates are arranged along a linear political spectrum, with each voter approving of some interval of candidates. What conditions might guarantee that there is a candidate with the approval of at least half the voters? Recent work by Berg et al. in this issue of the MONTHLY shows that if, out of every 3 voters, one can find 2 who approve of a common candidate, then there must be a single candidate with the approval of at least 1/2 the voters; more generally, if out of every m voters there are k who approve of a common candidate, then there is a candidate with the approval of at least (k-1)/(m-1) of the voters.
What happens under a different political spectrum? We consider the case with candidates arranged circularly and each voter approving of an arc of candidates. We show that a similar result holds (though the proof is quite different), but with a strict lower bound of (k-1)/m.
Revisiting the Pascal Matrix
By: Barry Lewis
Pascal’s triangle still has many surprises in store and this article explores some that relate to its role as a lower triangular (infinite) matrix. The key tool is not matrix algebra but the generating functions of combinatorics. These lead to fascinating results about matrix functions whose arguments involve Pascal’s triangle. Along the way, generalised telescoping sums emerge and a number of well-known sequences are encountered Â– Bernoulli, Bell, and one first known to Jacob Bernoulli.
Newton's Identities and the Laplace Transform
By: Mircea I. Cîrnu
The well-known Newton identities express the sums of powers of the roots of a polynomial in terms of its coefficients. In this note a new proof of these identities, based on the Laplace transform, is given. For this, an explicit formula for the solution of the initial value problem for homogeneous linear differential equations with constant coefficients is obtained by the Laplace transform method.
Farey Sums and Dedekind Sums
By: Kurt Girstmair
If we arrange all reduced fractions a/b with 0\le a\le b, 1\le b\le n, in ascending order, we obtain the Farey sequence of order n. Among the many remarkable properties of this sequence we find the following: the sum of the quotients b/b'of the denominators of any two consecutive fractions a/b equals 3N/2-2, where Nis the length of the whole sequence. This fact can be proved in an elementary way, but it gives rise to a number of interesting observations about Dedekind sums, which, in turn, play a role in quite different fields of mathematics (such as statistical distributions and topology).
Euler's Constant and Averages of Fractional Parts
By: Friedrich Pillichshammer
In 1898, de la Vallée Poussin showed that if one divides an integer m by all integers less than it, then the average of the fractional parts of these fractions tends to 1-\γ when m approaches infinity. Here γ is Euler's constant. In this note we present an easy proof of this fact and we show a new formula of the same kind, but where the divisors are only allowed to be a fixed power of integers. We conclude with an open question.
A Simple Proof that Γ(1) = -γ
By: Richard Bagby
The gamma function and Euler’s constant are both introduced in undergraduate calculus classes, but the relationship between them given in the title is generally left to the study of analytic functions in the complex plane. Here we use calculus methods to derive it, requiring only the basic definition of Euler’s constant and the integral formulas for the gamma function and its derivative. We do so by making use of a simple observation about weighted averages with respect to different weights.
The Shape of Content: Creative Writing in Mathematics and Science
Edited by: Chandler Davis, Marjorie Wikler Senechal, and Jan Zwicky
Reviewed by: Amir Alexander