The velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs of larger packs. Unfortunately, there is no way to oversee its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached our team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.

The average adult velociraptor was 3 meters long with a hip height of 0.5 meters and an approximate mass of 45 kg. It is estimated that the animal could run extremely fast, at speeds of 60 km/hr., for about 15 seconds. After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles.

Suppose the velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr. for long periods of time.

Part 1. Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15 meters, but may detect the predator at even greater ranges (up to 50 meters) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 meters.

Part 2. Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions given in Part 1.

Small group meetings for the discussion of important issues, particularly long- range planning, are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings the board will meet in small groups to discuss issue before meeting as a whole. These smaller groups still run the risk of control by a dominant personality. In an attempt to reduce this danger it is common to schedule several sessions with a different mix of people in each group.

A meeting of An Tostal Corporation will be attended by 29 Board Members of which nine are in-house members (i.e., corporate employees). The meeting is to be an all-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hours from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each morning session will consist of six discussion groups with each discussion group led by one of the corporation's six senior officers. None of these officers are board members. Thus each senior officer will lead three different discussion groups. The senior officers will not be involved in the afternoon sessions and each of these sessions will consist of only four different discussion groups.

The president of the corporation wants a list of board-member assignments to discussion groups for each of the seven sessions. The assignments should achieve as much of a mix of the members as much as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions. The assignments should also satisfy the following criteria:

1. For the morning sessions, no board member should be in the same senior officer's discussion group twice.
2. No discussion group should contain a disproportionate number of in-house members.

Give a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the previous paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an hour's notice would be appreciated. It would be ideal if the algorithm could also be used to make assignments for future meetings involving different levels of participation for each type of attendee.