Mathematics Department Research/Writing Projects - Winter 2010
Mathematics Department Research/ Writing Projects—Winter 2010
Below are some ideas for research/writing projects that you can consider as possibilities for extra credit or honors work in mathematics. Check with your instructor.
We continue to publish the best projects in The Right Angle, monthly newsletter of the Mathematics Department. You should feel free to contact the editor, Randy Schwartz, who can offer advice and sources of information such as those mentioned below (office BTC-510, telephone/voicemail 734-462-4400 ext. 5290, e-mail firstname.lastname@example.org).
In writing your paper, you shouldn’t be satisfied with a general description of the topic. You should actually dig into it, explain what is involved, and get into some actual mathematics using examples that help clarify it. Otherwise, what you and your readers learn will be superficial.
Also, remember that writing a research paper, even a brief one, doesn’t mean splicing together phrases and sentences that you find in other people’s books, articles, and websites. Of course, you’ll need to gather information and to list the sources where you found it, but you’ll also need to:
- analyze the information (break the facts and ideas down into their parts in order to see how they relate to one another)
- synthesize the information (put facts and ideas together in a systematic way in order to get a comprehensive understanding of the topic)
- organize and write the paper (figure out the best way, using your own words and understanding, to unfold what you’ve learned so that the readers get a clear understanding, too).
The Writing Fellows website at http://www.schoolcraft.edu/fellows has some useful guidance in how to go about doing this. Also useful is a site created by Gregory McColm (Dept. of Mathematics, University of South Florida), http://shell.cas.usf.edu/~mccolm/pedagogy, especially the page on “Analysis, Synthesis, and Doing Homework”.
MATHEMATICS AND SPORTS
“Mathematics and Sports” is the theme for National Mathematics Awareness Month this April. The wide world of sports offers a cornucopia of instances involving data, strategies, and chance, each of which is perfectly suited to mathematical analysis. Beyond the obvious uses of mathematics for things such as rating players, mathematics is used for everything from scheduling sports tournaments to designing the dimple patterns on golf balls, the composition of racing tires, and the shape of sailboat hulls. In yacht races, every crew keeps an onboard computer with software that advises the best angle to race or tack, the best moment to change sails or direction, etc., based on complex calculations involving wind, water, and the competing boats’ characteristics.
Consider writing a paper that explains one of the following applications:
- Physics and geometry can be used to find the optimal ways to throw or kick a ball, to swing a bat or club, etc.
- Biomechanical models are used to gauge an athlete’s ability to perform a given task, and to predict the ultimate limits in future performances and sports records.
- Mathematics is also used to rank players or teams based on the season’s performance, or to predict their performance in the next season.
- Probability, statistics, and game theory are used to plan the best playing strategy and tactics, to quantify the relative advantages of individual players or teams, to analyze their streaks of success or failure, or to identify long-term trends in the evolution of a sport.
Some of the best resources are as follows.
- The web site http://www.mathaware.org provides lots of interesting articles, videos, and links to other resources.
- Albert, Jim, Teaching Statistics Using Baseball (Washington, DC: Mathematical Association of America, 2003).
- Albert, Jim, and Jay Bennett and James J. Cochran, Anthology of Statistics in Sports (Philadelphia: Society for Industrial and Applied Mathematics, and Alexandria, VA: American Statistical Association, 2005).
- Bradbury, John Charles, and Douglas Drinen, “The Designated Hitter, Moral Hazard, and Hit Batters: New Evidence From Game-Level Data”, Journal of Sports Economics 7:3 (2006), pp. 319-329.
- Freedman, Jonah, “Stats Geek at Bat: Can a ‘Sabermetrics’ Guru Help the Sorry Red Sox?” Popular Science, May 2003, p. 48.
- Pritchard, W. G., “Mathematical Models of Running”, SIAM Review 35:3 (Sep. 1993), pp. 359-379.
- Romer, David, “It’s Fourth Down and What Does the Bellman Equation Say? A Dynamic Programming Analysis of Football Strategy”. National Bureau of Economic Research, Working Paper No. 9024 (June 2002).
- Ross, Kenneth A., A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (New York: Pi Press, 2004).
- Sackrowitz, Harold B., “When to Go for Two”, American Football Coach 6 (March 2000), pp. 46-50.
- Winston, Wayne L., Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football (Princeton: Princeton University Press, 2009).
MATHEMATICS AND THE ENVIRONMENT
Earth Day on April 22 reminds the world of its responsibility to protect the Earth, and to conserve and wisely manage its natural resources. We invite you to investigate one of the following ways in which mathematics is used to study these environmental resources.
Climate change and global warming
How have earth climates changed over geologic time? Are greenhouse gas emissions and other human industrial activities responsible for an overall warming of the earth’s atmosphere in modern times? Answering this controversial question is difficult, in part because climate varies naturally over time, even without human interference, and in part because the underlying mechanisms of climate change are very complex. Mathematical modeling is an important part of answering the challenge of global warming. University of Michigan professors of geological sciences Henry N. Pollack and Chris J. Poulsen are nationally known experts in this field. They and their colleagues have gathered geothermal temperatures at sites around the world and have developed theoretical climate models to reconstruct the history of Earth’s surface temperatures and to help predict future trends.
For information, see:
- Dana Mackenzie, Mathematics of Climate Change: A New Discipline for an Uncertain Century (Berkeley, CA: Mathematical Sciences Research Institute, 2007). This 28-page report is available free at http://www.msri.org/forms/ccbook/ccbookrequest.
- “Climate Change: From Global Models to Local Action” April 2007 symposium: videos can be viewed at http://www.msri.org/specials/climatechange/workshop
- Univ. of Michigan Paleoclimate Research Group publications page, http://www.geo.lsa.umich.edu/paleoclimate/publications.htm
- Henry N. Pollack and Shaopeng Huang, “Underground Temperatures Reveal Changing Climate”, Geotimes 43:8 (August 1998).
- Univ. of Michigan press release, “New Evidence of Global Warming”, San Diego Earth Times, Jan. 1998, available at http://www.sdearthtimes.com/et0198/et0198s2.html.
- Henry N. Pollack and David S. Chapman, “Underground Records of Changing Climate”, Scientific American June 1993, pp. 16-22.
The importance of biodiversity
Why does life in a given area become precarious when the number and variety of species fall below a certain point? Mathematics helps us answer this question. Here are some resources to investigate:
- Lord Robert May, formerly of Oxford University, studies communities of interacting plants and animals, especially their food chains, then applies mathematical concepts to understand how species diversity affects the stability and complexity of these ecosystems. Articles about his work are available at:
- Dr. G. David Tilman at the University of Minnesota uses mathematics extensively in analyzing the results of his field experiments in biodiversity. The New York Times has carried many articles about his work, such as William K. Stevens, “Ecologist Measures Nature’s Mosaic, One Plot at a Time”, New York Times 6 October 1998 (Science section).
Models of environmental pollution
Environmental analysts use a variety of mathematical methods to assess the dangers of, and losses caused by, air, water, and land pollution. These methods include cost-benefit analysis, the environmental Kuznets curve, and quantitative risk assessment (QRA).
- An example of a statistical study of water pollution is that by Song S. Qian and Michael Lavine, “Setting Standards for Water Quality in the Everglades”, Chance 16:3 (Summer 2003), pp. 18-27. Also available at http://www.amstat.org/publications/chance/pdfs/163.lavine.pdf
- The EPA’s use of quantitative risk assessment to gauge the risk of cancer from pollutants emitted in the air by the coke ovens of Ostrava (Czech Republic) is discussed in Jeff Wheelwright, “The Air of Ostrava”, Discover 17:5 (May 1996), pp. 56-67.
- An article that summarizes the use of cost-benefit curves and the environmental Kuznets curve to model the relation between societal wealth and pollution, and the appropriate level of sensitivity to environmental threats, is S. W. Pacala et al., “False Alarm over Environmental False Alarms”, Science 301 (Aug. 29, 2003), pp. 1187-1188.
- A textbook that explains algebraic and computer techniques in investigating the threat of groundwater contamination, air pollution, and hazardous material emergencies is Charles Hadlock, Mathematical Modeling in the Environment (Washington, DC: Mathematical Association of America, 1998).
Sustainable harvests with linear algebra
Models based on matrix algebra are used to predict the growth rate of a population and the long-term trends for its age distribution. This can allow forest and livestock rangers, and the managers of fish, game and wildlife preserves, to design sustainable harvesting policies. A discussion is provided in Howard Anton and Chris Rorres, Elementary Linear Algebra: Applications Version, 9th edition (New York: John Wiley & Sons, Inc. 2005), pp. 648-657, 743-761. [This is the textbook used for Math 230 at Schoolcraft. Similar discussions are provided in other Linear Algebra textbooks]
Balancing predator and prey with differential equations
In 1925-26, the physical chemist Alfred J. Lottka and the population ecologist Vito Volterra independently formulated an important model to describe the population fluctuations between a predator species and a prey species, such as lynxes and hares, or bass and sunfish, or ladybugs and aphids. The Lottka-Volterra model is a system of two nonlinear differential equations, and it predicts that in many cases, the populations of predator and prey will mutually stabilize by cycling between two extremes. See if you can address these questions in your paper:
- What practical problems prompted Lottka and Volterra to formulate their model?
- Explain the assumptions that were made and the variables and equations that were used to define the model.
- Describe the different mathematical solutions of the equations. What ecological behavior is predicted by these solutions?
- See if you can implement this model numerically or graphically with a graphing calculator or a computer.
- What are some of the limitations of this model? How has it been extended to apply to other situations?
Among the many good discussions of this work are:
- John Vandermeer, Elementary Mathematical Ecology (New York: Wiley & Sons, 1981), pp. 158-233.
- C. Henry Edwards, Jr., and David E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, Third edition (Upper Saddle River, NJ: Prentice-Hall, 2004), pp. 393-406. [This is the textbook used for Math 252 at Schoolcraft. Similar discussions are provided in other Differential Equations textbooks]
- Judith R. Goodstein, The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician, 1860-1940 (Providence, RI: American Mathematical Society, 2007)
- E. R. Leigh, “The Ecological Role of Volterra’s Equations”, in Some Mathematical Problems in Biology, Proceedings of the first annual Symposium on Mathematical Biology (Providence, RI: American Mathematical Society, 1968) [a discussion using data on lynxes and hares in Canada from Hudson’s Bay Company, 1847-1903]
- Kenneth Chang, “Mathematics Explains Mysterious Midge Behavior”, New York Times, March 7, 2008
- Anthony R. Ives et al., “High-amplitude Fluctuations and Alternative Dynamical States of Midges in Lake Myvatn”, Nature 452 (March 6, 2008), pp. 84-87
THE 2010 CENSUS
This Spring, the U.S. is launching its 2010 National Census at a cost of about $15 billion— double the cost of the 2000 Census. The census is far more than a simple count of heads. Every 10 years, the Bureau of the Census gathers a wide variety of socioeconomic data bearing on income, occupation and employment status; families, children and marital status; housing and homelessness; race and ethnicity; and more. Such demographic data exert a major influence in shaping governmental policy, in constituting the membership of the U.S. Congress and the Electoral College, and more broadly throughout society.
Sampling and adjustment
The question of whether the Census Bureau should rely on sophisticated sampling techniques, instead of or in addition to the traditional goal of “an actual full count”, has been hotly contested in the mass media, in legislatures and before the U.S. Supreme Court. In the 2000 Census, for the first time, the Census Bureau released two different sets of data: first the raw counts, and then figures adjusted for possible undercounts through the use of statistical sampling. The Obama administration and its Census director, University of Michigan sociology professor Robert M. Groves, plan to discontinue this practice. Write a paper that explores this issue. It should address these questions:
- How are the techniques of sampling and adjustment carried out by the Census Bureau?
- What impact can the adjustment have on the data and conclusions?
- What is meant by “the differential undercount of minorities”? What are its causes and consequences?
- What arguments have been presented by advocates on each side of this debate?
Among the sources available are these:
- An article laying out the key issues is Constance Holden, “America’s Uncounted Millions: The 2010 Census”, Science 324:5930 (22 May 2009), pp. 1008-1009.
- Steven A. Holmes, “Defying Forecasts, Census Response Ends Declining Trend,” New York Times (20 September 2000) summarizes the actual response rates in the 2000 Census.
- Three articles that briefly summarize the mathematics of the census and the statistical issues involved are: Mark Schilling, “Census 2000: Count on Controversy,” Math Horizons (November 1998); Lynne Billard, “Sampling in the Census 2000,” STATS: The Magazine for Students of Statistics 25 (Spring 1999); and the companion article by Barbara Bailar in STATS: The Magazine for Students of Statistics 26 (Fall 1999). The second article is also available at: http://www.amstat.org/publications/stats/billard.pdf.
- The American Statistical Association’s 1997 letter to Congress, which objected to political interference in census methodologies and called for using the most advanced sampling techniques possible, is available at: http://www.amstat.org/outreach/letter-congress.html.
- Martha Riche, former Director of the U.S. Census Bureau, laid out the case for statistical sampling in her article “Should the Census Bureau Use ‘Statistical Sampling’ in Census 2000?,” Insight (August 18, 1997), also available at the Bureau website at http://www.census.gov/dmd/www/insight.html.
- Michael L. Cohen et al., eds., Measuring a Changing Nation: Modern Methods for the 2000 Census (National Academy Press, 1999) is the report of the National Research Council’s Panel on Alternative Census Methodologies, which recommended statistical sampling of census nonrespondents, as well as adjustment for differential undercount. The Executive Summary of the report is available at: http://bob.nap.edu/html/changing_nation.
- The chapter entitled “One, Two, Three, Count It Legally” in Philip J. Davis, The Education of a Mathematician (A. K. Peters, 2000) summarizes the controversy and includes excerpts from testimony given by statisticians on both sides of the issue before Congress.
- Peter Skerry, a Senior Fellow at the Brookings Institution, argues that various political groups have exaggerated the implications of the census undercount, and that there is no scientific consensus in support of statistical adjustment. Skerry summarizes his argument in the article “Sampling Error,” The New Republic (May 31, 1999), p. 18-20, also available at http://www.brook.edu/views/articles/skerry/19990531.htm; in a policy brief entitled “Counting on the Census?” posted on the Brookings website, http://www.brook.edu/comm/PolicyBriefs/pb056/pb56.htm; and in his book Counting on the Census: Race, Group Identity, and the Evasion of Politics (Brookings Institution Press, 2000).
By the U.S. Constitution, the number of representatives in the House varies from state to state, based on the relative populations of each state in the most recent Census. Congressional apportionment is the science of determining the correct number of Representatives for each state. It involves lots of mathematics and also lots of controversy and rival theories. Summarize the following methods of Congressional apportionment—
- the Hamilton-Vinton formula
- the Webster formula
- the Huntington-Hill formula
- the Dean formula
- the Adams formula
- the Jefferson formula.
Among the sources available are these:
- Baker, Peter, “Expand the House?”, New York Times September 17, 2009 (reports on the issues involved in a court challenge to the current apportionment formula)
- Caulfield, Michael J., “Apportioning Representatives in the United States Congress”, Loci: Convergence (an online journal of the Mathematical Association of America), 2008. Available at http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=3163
- Huckabee, David C., The House of Representatives Apportionment Formula (Washington, DC: Library of Congress [Congressional Research Service], August 10, 2001), available at http://www.rules.house.gov/archives/RL31074.pdf
- Wright, Tommy, and George Cobb, “Counting and Apportionment: Foundations of America’s Democracy”, in Roxy Peck et al., eds., Statistics: A Guide to the Unknown, Fourth edition (Belmont, CA: Thomson Brooks/Cole, 2006)
MATHEMATICS AND THE MIDDLE EAST
In conjunction with this year’s Focus Middle East project at Schoolcraft College, we invite you to write about one of the following topics.
Charging interest on most forms of loans is not permitted under shari‘ah, the legal and moral code based on the principles set out in the Qur’an. In recent decades, a new type of “interest-free” loan called shari‘ahcompliant financing has arisen. The largest such Islamic lenders are in the Middle East, but one of the leading ones in North America is University Islamic Financial Corp. in Ann Arbor, a subsidiary of University Bank.
Among other things, this method of financing has made home ownership possible for Muslims all over the world who would not otherwise have become homeowners. The system allows the lender to gain money from the transaction, but the calculation of finance charges differs from the conventional method, which involves a rate of interest. Investigate the mathematics involved in this type of loan.
Middle East demographics
In the Middle East and other Third World regions, there is an acute need to put mathematics and science in the service of tackling problems of poverty and rural and urban development. Statistical techniques and mathematical modeling are used to help answer important questions such as projecting changes in population; the extent and causes of poverty and its geographical distribution; the social status of women and children; and gender inequality. Describe some examples of how mathematics is used in this way.
Representative sources include:
- Gaurav Datt, Dean Jolliffe, and Manohar Sharma. “A Profile of Poverty in Egypt”, African Development Review, 2001. Available at http://users.starpower.net/djolliffe/egypt/adr_v15_i2.pdf.
- Hoda Rashad, Magued Osman, and Farzaneh Roudi-Fahimi, “Marriage in the Arab World”, Washington DC: Population Reference Bureau, 2005. Available at: http://www.iiav.nl/epublications/2005/MarriageInArabWorld.pdf.
The Islamic Calendar
The year 2010 on the Gregorian (Christian) calendar corresponds roughly to the year 1431 on the Islamic calendar. The years given by these two calendars differ from one another not only because their counting began with different events, but also because they are based on the revolution of two different bodies (the Earth and the Moon). Historically, a great deal of mathematics has gone into the design and maintenance of both calendars, especially because they are sued to determine the timing of religious holidays. Today, Muslim holidays and fasting days continue to exert a large impact on commerce in the Middle East and other Muslim regions, an economic phenomenon known as the “Islamic Calendar Effect”. Investigate the historical mathematics of the Islamic calendar, and also how modern researchers use mathematics to analyze the Islamic Calendar Effect.
Some good resources are:
- Berggren, John Lennart, Episodes in the Mathematics of Medieval Islam (New York, NY: Springer-Verlag, 1986).
- Khalid Mustafa, “The Islamic Calendar Effect in Karachi Stock Market”, Proceedings of the Eighth International Business Research Conference, Dubai 2008. Available at: http://www.wbiconpro.com/313A---Mustafa,K.pdf
Mathematicians in the classical Islamic period excelled in the use of geometry to analyze optical phenomena involving reflection and refraction. Geometry was used to study how the path taken by a ray of light is bounced and bent by the surrounding objects. Investigate one or both of the following contributions:
- Ibn al-Haytham and his work on optical phenomena, vision, and the solution of what came to be known as Alhazen’s problem: On the surface of a spherical mirror, to find the point of reflection between two given points, thought of as the eye and the visible object.
- Kamal al-din al-Farisi and his theory of the rainbow, where he analyzed the path of a light ray inside a spherical water droplet in order to predict the angle of rainbows in the sky and the position of colors within them.
Distributing inheritances and alms
One of the major practical applications of arithmetic and algebra in the classical Islamic world was in the distribution of inheritances and alms. Deciding, according to Qur’anic precepts, the correct way to divide a deceased man’s estate among his survivors turns out to often involve some considerable calculations, as did determining one’s share of zakah (alms taxes and tithes) and how to apportion alms among the poor. Write a paper describing these calculations.
Some good resources include:
- Lesser, Lawrence M. “Reunion of Broken Parts: Experiencing Diversity in Algebra.” Mathematics Teacher 93 (January 2000), pp. 62-67.
- Berggren, John Lennart, Episodes in the Mathematics of Medieval Islam (New York, NY: Springer-Verlag, 1986), pp. 63-68.
Mathematics of ancient Egypt
The ancient Egyptians were one of the earliest civilizations to accumulate significant mathematical knowledge. Scribes recorded problems and their solutions on scrolls of papyrus or leather, using hieroglyphic and other scripts. There is evidence that their techniques were applied to land surveying, the design of pyramids and other structures, and for other uses. Write a paper in which you address one of more of the following topics:
- types of evidence available regarding ancient Egyptian mathematics
- Egyptian notation for numerals and counting
- practical arithmetic, including the use of Egyptian or “unit” fractions (fractions of the form 1/n)
- discoveries in plane and solid geometry
- Egyptian applications of mathematics to surveying, building, timekeeping, etc.
A few of the available resources are listed below. The best starting point is the online essay by O’Connor and Robertson, which also includes many references.
- Allen, Don, “Egyptian Mathematics”, Texas A&M Univ., available at http://www.math.tamu.edu/~don.allen/history/egypt/egypt.html
- Eppstein, David, “Ten Algorithms for Egyptian Fractions”, Mathematics in Education and Research 4 (1995), pp. 5-15.
- Eppstein, David, “Egyptian Fractions”, Univ. of California at Irvine, available at http://www.ics.uci.edu/~eppstein/numth/egypt/
- Gillings, Richard J., Mathematics in the Time of the Pharaohs (New York: Dover, 1982; originally published by MIT Press, 1972).
- Gillings, Richard J., “The Volume of a Truncated Pyramid in Ancient Egyptian Papyri”, Mathematics Teacher 57 (1964), pp. 552-555.
- O’Connor, John J., and Edmund F. Robertson, “An Overview of Egyptian Mathematics”, St. Andrews Univ., available at http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_mathematics.html
- Robins, Gay, and Charles Shute, The Rhind Mathematical Papyrus: An Ancient Egyptian Text (London: British Museum Publications, 1987).
Mathematics of ancient Mesopotamia
Over the last few decades, the intensive study of cuneiform tablets and other records has shown that ancient Mesopotamian mathematics was more advanced than previously known. Babylonian mathematicians even learned how to solve the type of problem that we would call a quadratic equation, using a method equivalent to the modern one. Write a paper in which you address one of more of the following topics:
- types of evidence available regarding ancient Mesopotamian mathematics
- the sexagesimal (base 60) number system and its use in timekeeping and astronomy
- methods for calculating square roots
- Babylonian methods for solving linear and quadratic problems
- the use of mathematical tables, including tables of what would later be called Pythagorean triplets
Some resources are as follows:
- Jones, Phillip S., three articles, “Recent Discoveries in Babylonian Mathematics I: Zero, Pi, and Polygons”, “Recent Discoveries in Babylonian Mathematics II: The Earliest Known Problem Text”, and “Recent Discoveries in Babylonian Mathematics III: Trapezoids and Quadratics”. In Frank J. Swetz, ed., From Five Fingers to Infinity: A Journey through the History of Mathematics (Chicago, IL: Open Court Publishing Co., 1994).
- Joseph, George Gheverghese, The Crest of the Peacock: Non-European Roots of Mathematics, New Edition (Princeton, NJ: Princeton University Press, 2000). Especially Chapter 4, “The Beginnings of Written Mathematics: Babylonia”; Chapter 5, “Egyptian and Babylonian Mathematics: An Assessment”; and the final chapter, “Reflections”.
- Katz, Victor J. A History of Mathematics: An Introduction, Third edition. Boston: Addison-Wesley, 2009. Especially Section 1.2, “Mesopotamia”.
- Neugebauer, Otto, “The Exact Sciences in Antiquity”, 2nd edition (New York: Dover, 1969)
- Robson, Eleanor, Mathematics in Ancient Iraq: A Social History (Princeton, NJ: Princeton University Press, 2008).
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