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1996
美国大学生数模竞赛题
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)?
1997
年美国大学生数模竞赛题
Problem A: The Velociraptor Problem
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 or larger packs. Unfortunately, there is no way to
observe its hunting behavior in the wild as can be done with modern
mammalian predators. A group of paleontologists has approached your 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 that Velociraptor prey on Thescelosaurus neglectus, a herbivorous
biped approximately the same size as the Velociraptor. A biomechanical
analysis of a fossilized thescelosaurus indicates that if could run at a
speed of about 50km.hr. for long periods of time.
Part1
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 thecelosaurus can always detect the velociraptor when
in 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 and limitations given in Part 1.
Problem B: Mix Well For Fruitful Discussions
Small group meetings for the discussion of important issues, particularly
long-rang 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 issues before
meeting as a whole. These smaller groups still run 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-horse 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
hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each
morning session will consist of six discussion group with each discussion
group led by one of the corporation's six senior officers. None of these of
officers are board members. Thus each senior officer will lead three
different discussion groups. The sessions will consist of only four
discussion groups.
The president of the corporation wants a list of board-member assignments
to discussion group for each of seven sessions. The assignments should
achieve as much of a mix of 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 precious 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 user to make assignments for future
meetings involving different levels of participation for each type of
attendee.
Problem A:
Introduction:
Industrial and medical diagnostic machines known as Magnetic Resonance
Imagers (MRI) scan a three-dimensional object such as a brain, and deliver their results
in the
form of a three-dimensional array of pixels. Each pixel consists of one
number indicating a color or a shade of gray that encodes a measure of water concentration
in a
small region of the scanned object at the location of the pixel. For
instance, 0 can picture high water concentration in black (ventricles, blood vessels), 128
can picture a
medium water concentration in gray (brain nuclei and gray matter), and 255
can picture a low water density in white (lipid-rich white matter consisting of myelinated
axons). Such MRI scanners also include facilities to picture on a screen
any horizontal or vertical slice through the three-dimensional array (slices are parallel
to any of the
three Cartesian coordinate axes). Algorithms for picturing slices through
oblique planes, however, are proprietary. Current algorithms are limited in terms of the
angles and
parameter options available; are implemented only on heavily used
dedicated workstations; lack input capabilities for marking points in the picture before
slicing; and tend
to blur and "feather out" sharp boundaries between the original
pixels.
A more faithful, flexible algorithm implemented on a personal computer
would be useful (1) for planning minimally invasive treatments, (2) for calibrating the
MRI
machines, (3) for investigating structures oriented obliquely in space,
such as postmortem tissue sections in animal research, (4) for enabling cross-sections at
any angle
through a brain atlas consisting of black-and-white line drawings. To
design such an algorithm, one can access the values and locations of the pixels, but not
the initial data
gathered by the scanner.
Problem:
Design and test an algorithm that produces sections of three-dimensional
arrays by planes in any orientation in space, preserving the original gray-scale values as
closely as possible.
Data Sets:
The typical data set consists of a three-dimensional array A of numbers
A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)_{ijk}
. Typically,
A(i,j,k) can range from 0 through 255. In most applications; the data set
is quite large. Teams should design data sets to test and demonstrate their algorithms.
The data sets
should reflect conditions likely to be of diagnostic interest. Teams
should also characterize data sets that limit the effectiveness of their algorithms.
Summary:
The algorithm must produce a picture of the slice of the
three-dimensional array by a plane in space. The plane can have any orientation and any
location in space. (The
plane can miss some or all data points.) The result of the algorithm
should be a model of the density of the scanned object over the selected plane.
Problem B:
Background:
Some college administrators are concerned about the grading at A Better
Class (ABC) college. On average, the faculty at ABC have been giving out high grades (the
average grade now given out is an A-), and it is impossible to distinguish
between the good and mediocre students. The terms of a very generous scholarship only
allow
the top 10% of the students to be funded, so a class ranking is required.
The dean had the thought of comparing each student to the other
students in each class, and using this information to build up a ranking. For example, if
a student obtains
an A in a class in which all students obtain an A, then this student is
only "average" in this class. On the other hand, if a student obtains the only A
in a class, then that
student is clearly "above average". Combining information from
several classes might allow students to be placed in deciles (top 10%, next 10%, etc.)
across the college.
Problem:
Assuming that the grades given out are (A+, A, A-, B+, ... ) can the
dean's idea be made to work? Assuming that the grades given out are only (A, B, C, ... )
can the dean's
idea be made to work? Can any other schemes produce a desired ranking? A
concern is that the grade in a single class could change many student's deciles. Is this
possible?
Data Sets:
Teams should design data sets to test and demonstrate their algorithms.
Teams should characterize data sets that limit the effectiveness of their algorithms.