Problem A

The world's oceans contain an ambient noise field. Seismic disturbances,
surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, developa method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed freqency and amplitude.

Problem B

When determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest constains both the number of judges that can be obtained and amount of time that they can judge. For eample if P=100, then J=8 is typical.

Ideally, each judge would read paper and rank-order them, but there are too
many papers for this. Instead, there will be a number of screening rounds in which each judge will read some number of papers and give them scores. Then some selection scheme is used to reduce the number of papers under
consideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from 1 to 100),
then all papers below some cut-off level could be rejected.

The new pool of papers is then passed back to the judges, and the process is repeated. A concern is then the total number of papers that judge reads must be substantially less than P. The process is stopped when there are only W papers left. There are the winners. Typically for P=100, W=3.

Your task is to determine a selection scheme, using a combination of
rank-ordering, numerical scoring, and other methods, by which the final W
papers will include only papers from among the "best" 2W papers. (By "best",we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers. Among all such methods, the one that required each judge to read the least number of papers is desired.

Note the possibility of systematic bias in a numerical scoring scheme. For
example, for a specific collection of papers, one judge could average 70
points, while another could average 80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)?