A relation between Catalan Numbers and World Series type problems

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Stuck at home because of the cold weather? Warm up your evening with the February edition of the *American Mathematical Monthly*. In our lead article, "A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization," Chi and Lange construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem. Do you like catenaries? In "Two Generalizations of a Property of the Catenary," Coll and Harrison show us how to view them a bit differently. Harold Boas reviews Peter Duren's *Invitation to Classical Analysis* and Peter Lax reviews Victor Moll's *Numbers and Functions: From a Classical Experimental **Mathematician’s Point of View*. Last but not least, don't forget our Problem Section.

Stay tuned for March when we celebrate Pi Day with a special article from *Monthly* Associate Editor Jonathan Borwein and David Bailey.—*Scott Chapman*

Vol. 121, No. 2, pp.95-183.

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

Eric C. Chi and Kenneth Lange

In a recent issue of this *Monthly*, Mordukhovich, Nam, and Salinas pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem, we are given $$k\pm1$$ closed convex sets in $$\mathbb{R}^{d}$$ equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first k sets is minimal. In later work, the authors generalize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the convex case. Here, we revisit the original problem solely from the numerical perspective. By exploiting the majorization-minimization (MM) principle of computational statistics and rudimentary techniques from differential calculus, we are able to construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.095

Vincent Coll and Mike Harrison

A well-known property of the catenary curve is that the ratio of the area under the curve to the arc length of the curve is independent of the interval over which these quantities are concurrently measured. We develop two higher-dimensional generalizations of this invariant ratio, and find that each invariant ratio identifies a class of hypersurfaces connected to classical objects from differential geometry.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.109

Pantelis A. Damianou

Let $$g$$ be a complex simple Lie algebra of rank $$\ell$$, $$h$$ the Coxeter number, $$m_{1},m_{2},\dots,m_{\ell}$$ the exponents of $$\mathfrak{g}$$, and $$C$$ the Cartan matrix. Then $$2^{2\ell}\prod_{i=1}^{\ell}\sin^{2}\frac{m_{i}\pi}{2h}=\det{C}$$

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.120

Ron Evans, Lihua Huang, and Tuan Nguyen

An episode of the television series *Futurama* features a two-body mind-switching machine, which will not work more than once on the same pair of bodies. After the *Futurama* community engages in a mind-switching spree, the question is asked, “Can the switching be undone so as to restore all minds to their original bodies?” Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two “outsiders.” We refine Keeler’s result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences of switches, each sequence effecting a cyclic mind-scrambling permutation in the symmetric group $$S_{n}$$. Finally, we give necessary and sufficient conditions on m and n for the identity permutation to be expressible as a product of $$m$$ distinct transpositions in $$S_{n}$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.136

David Grant

Let $$p$$ be an odd prime and $$\zeta$$ be a primitive $$p$$th-root of unity. For any integer $$a$$ prime to $$p$$, let $$\left(\frac{a}{p}\right)$$ denote the Legendre symbol, which is 1 if $$a$$ is a square mod $$p$$, and is $$-1$$ otherwise. Using Euler’s Criterion that $$a^{(p-1)/2}=\left(\frac{a}{p}\right)\mod p$$, it follows that the Legendre symbol gives a homomorphism from the multiplicative group of nonzero elements $$\mathbb{F}^{*}_{p}$$ of $$\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$$ to $$\{\pm1\}$$. Gauss’s law of quadratic reciprocity states that for any other odd prime $$q$$, $$\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{(p-1)(q-1)/4}$$. A table describing the multitude of proofs of this cherished result over the past two centuries is given in Appendix B of [10], which shows that the starting point of many of the proofs (including one of Gauss’s) is the quadratic Gauss sum, $$g=\sum_{a=1}^{p-1}\left(\frac{a}{p}\right)\zeta^{a}$$, and Gauss’s calculation that $$g^{2}=\left(\frac{-1}{p}\right)p$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.145

Yuly Billig and John D. Dixon

We give a new short proof of a theorem relating solutions of a system of polynomial equations to the eigenvalues of the multiplication operators on the quotient ring, in the case when the quotient ring is finite-dimensional.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.150

Hongwei Lou

By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling’s formula and its refinement.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.154

Richard Ehrenborg

We prove that a Hamiltonian cycle on the faces of an Archimedean solid is *twisting free*, that is, when returning to the first facet of the cycle, it has the same orientation as in the beginning. We also explore a continuous analogue on the unit sphere.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.158

Chua Cheong Siong

Let $$p>3$$ be a prime, and let $$a$$ and $$b$$ be positive integers with $$a\geq b$$. In this article, we give a simple proof of the congruence $$\left(\begin{array}{c}pa\\pb\end{array}\right)\equiv\left(\begin{array}{c}a\\b\end{array}\right)(\mod{p^{3}})$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.162

Peter J. McGrath

This note gives an alternate proof of Clairaut’s theorem—that the partial derivatives of a smooth function commute—using the Stone–Weierstrass theorem.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.165

Hossein Hosseini Giv

The Banach–Alaoglu theorem is an important result in functional analysis whose standard proof relies on Tychonoff’s theorem. In this note, the theorem is proved by assuming the existence of the Stone–Čech compactification for completely regular topological spaces.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.167

Problems 11754-11760

Solutions 11629, 11639, 11640, 11641, 11642, 11647

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.170

*Invitation to Classical Analysis*, by Peter Duren

Reviewed by Harold P. Boas

JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.178

*Numbers and Functions: From a Classical Experimental Point of View*, by Victor H. Moll

Reviewed by Peter Lax

JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.02.183