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The AMATYC Review
A refereed publication of the American Mathematical Association
of Two-Year Colleges
Editor: Barbara
S. Rives, Lamar State College
Production Manager: John
C. Peterson
Abstracts
Spring 2004 issue, Vol. 25, No.2
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Sky-High
i’s
Andrew J. Berry
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Andrew
J. Berry received his BS and MS degrees in mathematics at
the University of Illinois at Urbana-Champaign, and his PhD at New
York University. He is Associate Professor of mathematics at LaGuardia
Community College, City University of New York.
E-mail: ajberry@nyc.rr.com |
When
discussing the topic of elementary complex variables, students are
often mystified by the fact that ii is real. After seeing a proof
of this statement, a standard question is “well, what about
iii or iiii etc., are these real or complex?”
In this paper the meaning of the infinite power-tower iii... is considered
both from the “bottom-up” and the “top-down.”
Some surprising graphical representations are revealed and issues
of convergence and evaluation are discussed.
Finally, we introduce the elementary properties of the Lambert’s
W-function and it’s relationship to the function f(x) = xxxx...
which Euler explored in 1783. (Back to top) |
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Poiseuille’s
Law - Showing that p Is Inversely
Proportional to R4 Using the Shell Method
John P. Drost, and Rachel A. Georges
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John
P. Drost earned his PhD in Education Administration with
emphasis in Mathematics from the University of Utah in 1975. He
is currently a Professor of Mathematics at the University of Wisconsin-Eau
Claire. His interests are in teaching undergraduate mathematics,
M.C. Escher, symmetric groups, and planar and vortex tessellations.
In recent years, he has been integrating technology, graphing calculators
and Maple into his teaching.
E-mail: drostjp@uwec.edu
Rachel
A. Georges is a junior at the University of Wisconsin-Eau
Claire, pursuing a mathematics major in actuarial science. She is
interested in applied mathematics and found Poiseuille’s Law
of Resistance to be intriguing. This article is a result of her
research into why p is inversely proportional to fourth
power of the resistance, R4. Her hobbies include horseback riding,
spending time with friends and family, and active sports.
E-mail: georgera@uwec.edu
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Jean
Poiseuille, a physician, developed a mercury filled U-tube to measure
blood pressure in 1828. He discovered that pressure in veins is significantly
lower than pressure in arteries. As a result, he studied liquid flow
in small tubes. A few years later he established Poiseuille’s
Law, which states the resistance, p, of the flow of blood
as
p = K(L/R4) where L and R are the length and radius
of the vessel respectively. K is a positive constant determined by
the viscosity of the blood.
It is curious that the resistance is inversely proportional to the
fourth power of the radius R. At first glance, most individuals may
assume the resistance should be inversely proportional to the second
power of the radius since the area of a circle is R2. In this article,
mathematics, including finding volume using cylindrical shells, is
developed to establish that the resistance is indeed inversely proportional
to the fourth power of the radius. |
Problems
on Divisibility of Binomial Coefficients
Thomas J. Osler, and James Smoak
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Tom
Osler, professor of mathematics at Rowan University, is
the author of 58 mathematical papers. In addition to teaching university
mathematics for 43 years, Tom has been competing in long distance
running races for the past 50 consecutive years. He is the author
of two books on running.
E-mail: Osler@rowan.edu
Jim Smoak is a retired mathematician with an insatiable
interest in number patterns. Jim served as a ballistic meteorologist
in Viet Nam from 1968–70, receiving a bronze star for his
efforts. He maintains an active correspondence with some of America’s
leading mathematicians, including George Andrews, the world famous
number theorist from Penn State University.
E-mail:jsmoak@worldnet.att.net
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Twelve unusual
problems involving divisibility of the binomial coefficients are
represented in this article. The problems are listed in “The
Problems” section. All twelve problems have short solutions
which are listed in “The Solutions” section. These problems
could be assigned to students in any course in which the binomial
theorem and Pascal’s triangle are presented. This includes
courses in precalculus mathematics, real analysis, and number theory.
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