# American Mathematical Monthly -August - September 2000

## August - September 2000

Defendens Imperium Romanum: A Classical Problem in Military Strategy
by Charles S. ReVelle and Kenneth E. Rosing
[email protected], [email protected]
In the Fourth Century of the Common Era, the forces of the Roman Empire had been reduced to just 25 legions, an apparently insufficient number to secure the far-flung provinces that nourished the empire. The Emperor Constantine (Constantine the Great, 274-337) re-deployed the regular troops of the empire in groups of six legions, each of these groups constituting a field army. Thus, four field armies were to be deployed amongst the eight provinces of the empire.

The problem that Constantine was trying to solve is how to station the field armies so that, to the greatest extent possible, the empire's provinces were completely "defended", given special rules on the proximity of a field army that define "defended". Two fundamental mathematical questions are "What is the fewest number of field armies needed to defend all of the provinces?" and "If the number available is less than this minimum, how should the limited resources be deployed to defend the largest number of provinces?"

Constantine's problem, extended to any network and any number of field armies, is, in reality, a discrete location problem. As such it can be represented in an optimization framework-in this case as two zero-one mathematical programming problems. Although such problems are usually NP-complete, certain programming formulations allow researchers to solve medium-size representations using a combination of linear programming and branch and bound. Such optimization formulations are offered here and applied to Constantine's problem not only to provide a solution but also to develop all of the equal alternative deployments.

The Books Studied by Ramanujan in India
by Bruce C. Berndt and Robert A. Rankin
[email protected], [email protected]
Since Ramanujan had access to so few books in India before coming to Cambridge, it is natural to ask, "From which texts did he learn mathematics while in India?" The authors provide evidence that Ramanujan knew certain books, and that he likely knew a few others. Short biographies of the authors and descriptions of the books are also given.

Pinching, Trimmin, Truncating, and Averaging of Matrices
by Rajendra Bhatia
[email protected]
Each n-by-n matrix A represents a linear operator on the n-dimensional Euclidean space. We use the term "the norm of A" to mean the operator (bound) norm of this linear operator. This is equal to the largest singular value of A, and is called the spectral norm in the numerical analysis literature.

When some of the entries of A are replaced by numbers that are smaller in magnitude, its norm does not necessarily become smaller. One of the interesting phenomena, well-known to operator theorists, is that there exists an n-by-n matrix A the norm of whose upper triangular part is as large as log n times the norm of A.

The goal of this article is to present interesting theorems that give bounds for the norms of some matrices obtained from A by replacing some of its entries by zeros.

One theme that runs through the article is that such matrices can be realized as averages of unitary conjugates of A. The second theme is that these averages involve trigonometric polynomials and Fourier integrals, and results on Fourier series can be used to obtain good bounds for norms of matrices.

When Is a Family of Sets a Family of Bounded Sets?
by Gerald Beer
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Let X be a metrizable space and let a be a family of subsets of X. We give necessary and sufficient conditions for the existence of an unbounded metric d compatible with the topology such that each member of the family a is d-bounded.

Statistics and Mathematics: Tension and Cooperation
by David S. Moore and George W. Cobb
[email protected], [email protected]
The gradual distancing of statistics from mathematics, due mainly to the fact that statistics has changed much more rapidly than mathematics, carries risks for both disciplines. Statistics risks dissipating back into the many fields from which it coalesced or being swallowed by broader information technology. Mathematics risks following academic philosophy into irrelevant profundity. In fact, statistics has cultural strengths that might greatly assist mathematics, while mathematics has organizational strengths that can provide shelter for academic statistics. Moreover, mathematics and statistics are natural intellectual allies in the debate over the relative roles of thought and automation. Increased cooperation between professional societies could benefit both fields. We offer some specific suggestions for such cooperation.

Notes

Nilpotent Numbers
by Jonathan Pakianathan and Krishnan Shankar
[email protected], [email protected]

Triangular Triples from Ceilings to Floors
by Tom Jenkyns and Eric Muller
[email protected], [email protected]

Normal Numbers Without Measure Theory
by Rodney Nillsen
[email protected]

Evolution of. . . The Problem of Squarable Lunes
M.M. Postnikov

Problems and Solutions

Reviews

Life by the Numbers
By Keith Devlin

Reviewed by Colm Mulcahy
[email protected]

Small Worlds: The Dynamics of Networks between Order and Randomness
By Duncan J. Watts

Reviewed by Jerrold W. Grossman
[email protected]

Telegraphic Reviews

Lester R. Ford Awards for1999