In 1974 the U.S. Coast Guard put into operation its first computerized search and rescue planning system CASP (Computer-Assisted Search Planning) which used a Bayesian approach implemented by a particle filter to produce probability distributions for the location of the search object. These distributions were used for planning search effort. In 2003, the Coast Guard started development of a new decision support system for managing search efforts called Search and Rescue Optimal Planning System (SAROPS).SAROPS has been operational since January, 2007 and is currently the only search planning tool that the Coast Guard uses for maritime searches. SAROPS represents a major advance in search planning technology. This paper reviews the technology behind the tool.

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In the early morning hours of June 1, 2009, during a flight from Rio de Janeiro to Paris, Air France Flight AF 447 disappeared during stormy weather over a remote part of the Atlantic carrying 228 passengers and crew to their deaths. After two years of unsuccessful search, the authors were asked by the French Bureau d'Enqu\^{e}tes et d'Analyses pour la s\'{e}curit\'{e} de l'aviation to develop a probability distribution for the location of the wreckage that accounted for all information about the crash location as well as for previous search efforts. We used a Bayesian procedure developed for search planning to produce the posterior target location distribution. This distribution was used to guide the search in the third year, and the wreckage was found with one week of undersea search. In this paper we discuss why Bayesian analysis is ideally suited to solving this problem, review previous non-Bayesian efforts, and describe the methodology used to produce the posterior probability distribution for the location of the wreck.Comment: Published in at http://dx.doi.org/10.1214/13-STS420 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Let τ ( G ) \tau (G) be the minimum number of complete bipartite subgraphs needed to partition the edges of G G , and let r ( G ) r(G) be the larger of the number of positive and number of negative eigenvalues of G G . It is known that τ ( G ) ⩾ r ( G ) \tau (G) \geqslant r(G) ; graphs with τ ( G ) = r ( G ) \tau (G) = r(G) are called eigensharp. Eigensharp graphs include graphs, trees, cycles C n {C_n} with n = 4 n = 4 or n ≠ 4 k n \ne 4k , prisms C n ◻ K 2 {C_n}\square {K_2} with n ≠ 3 k n \ne 3k , "twisted prisms" (also called "Möbius ladders") M n {M_n} with n = 3 n = 3 or n ≠ 3 k n \ne 3k , and some Cartesian products of cycles. Under some conditions, the weak (Kronecker) product of eigensharp graphs is eigensharp. For example, the class of eigensharp graphs with the same number of positive and negative eigenvalues is closed under weak products. If each graph in a finite weak product is eigensharp, has no zero eigenvalues, and has a decomposition into τ ( G ) \tau (G) stars, then the product is eigensharp. The hypotheses in this last result can be weakened. Finally, not all weak products of eigensharp graphs are eigensharp.

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