*This article is published in the April/May 2012 issue of *MAA FOCUS.

Mathematical Reviews and its online version MathSciNet provide timely reviews and abstracts of articles and books that contribute to research in the mathematical sciences. Published monthly in print and continuously updated online, they are produced by the Mathematical Reviews Division of the American Mathematical Society.

MR2732245 (2011k:05005) Conger, Mark A.(1-MI); Howald, Jason(1-SUNYPD)
A better way to deal the cards. (English summary) Amer. Math. Monthly 117 (2010), no. 8, 686–700. 05A05
Most models of the randomization of a deck of cards by shuffling assume that the cards in the deck are distinct, and measure the deviation from the uniform distribution on all orderings of the permutation. For example, the estimates from the standard model of riffle shuffling [D. Bayer and P. W. Diaconis, Ann. Appl. Probab. 2 (1992), no. 2, 294−313; MR1161056 (93d:60014)] use these assumptions. The authors consider decks with repeated cards, or decks in which the cards are dealt into hands as in most card games. They compute asymptotic formulas for both cases. In particular, the amount of shuffling needed for randomization is strongly dependent on how the cards are dealt; cutting a poker deck properly improves the randomization significantly, dealing bridge cards cyclically improves the randomization by a factor of 13 over cutting the deck into four piles (asymptotically; Monte Carlo simulations show that the asymptotic is close after five riffles), and dealing 1-2-3-4-4-3-2-1 improves the randomization by a further factor of 13 (asymptotically).
Reviewed by David J. Grabiner |

The first issue of Mathematical Reviews appeared in 1940, published jointly with the MAA (though MAA later withdrew from the partnership). MathSciNet (

www.ams.org/mathscinet) went online in 1996. MathSciNet includes the latest reviews (nearly 75,000 added in 2011) and also serves as a compendium of all reviews. It provides tools to search the database, track citations, create bibliographic references, and find collaboration distances between authors.

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In recent years, Mathematical Reviews has been paying closer attention to MAA journals and is now routinely reviewing articles in

*The American Mathematical Monthly*,

*Mathematics Magazine*, and

*The College Mathematics Journal* (see sidebar).

### The Reviewing Process

Reviewing for Mathematical Reviews offers faculty members a way to serve the mathematical community and, at the same time, enhance their research knowledge.

All content sent out for review has been published, which means that reviewing is quite different from refereeing. While a referee’s task is to determine whether an article is correct, interesting, and suitable to a journal, a reviewer’s goal is to summarize mathematical content in an effort to help other researchers determine if they should spend time reading the item. The most helpful reviewers also place the mathematics into context and provide links to other reviews in the MathSciNet database.

The styles of reviews, as well as length, have evolved, so that different fields of mathematics have their own styles. Typically, a review is less than a page long.

### Advantages of Reviewing

Reviewers specify the number and subject areas of the items they wish to review. Most reviewers agree to have up to three to six articles to review at one time. However, beginning reviewers often decide to receive only one or two items at a time. Reviewers designate Mathematics Subject Classification (MSC 2010) numbers and provide a short written description of the types of items that they wish to review (e.g., graphs and their spectra).

In this way, reviewers can use MathSciNet to stay up to date on research in areas that they specify. They can also target areas about which they would like to learn more. It is not uncommon for reviewers to write follow-up papers after completing a review.

Reviewing can also have a positive impact on teaching. For example, after reviewing a paper on the relationship between an enumeration of the positive rational numbers and optimal play in the combinatorial game Euclid, I introduced this to high school students in the 2009 Michigan Math and Science Scholars summer program.

*Adapted from an article by Michael A. Jones that appeared in the April 2010 Michigan Section−MAA Newsletter (pdf).*

*Michael A. Jones has been an associate editor at Mathematical Reviews since 2008, after 14 years as a faculty member at various institutions. He is a member of the editorial board of the College Mathematics Journal.*
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