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This REU in mathematics was hosted by UNICAMP (Universidade de Campinas),
a state University in Sao Paulo. Campinas is located
in a large industrial center in Sao Paulo state and is surrounded by
technological and computational centers.
- Participants:
Lyubov Chumakova, University of Wisconsin
Joel Louwsma , University of Michigan
Sharon Lutz , University of Colorado, Boulder
Alan Tarr, Pomona College
Chris Willmore, Purdue University
Racheal Allen, California State University Northridge
Maria G. Uribe, California State University Northridge
Adilson Eduardo Presoto , Universidade Federal de Sao Carlos(UFSCar)
Anderson Tiago da Silva , Universidade Federal de Vicosa
Anne Caroline Bronzi, Universidade Estadual
de Campinas (UNICAMP)
Jose Regis Azevedo Varao Filho ,
(UNICAMP)
Leonardo Barichello, UNICAMP
Patricia Romano Cirilo, Universidade Federal de Minas
Gerais
Roberta Camelucci Carrocine, (UFSCar)
Welington Vieira Assuncao, Universidade Estadual do Estado de Sao
Paulo, Rio Claro
- Brazilian participants were supported by the University Foundation,
FAEP.
- Organizers: M. Helena Noronha,
California State University Northridge, and Marcelo Firer,
UNICAMP.
- Faculty Advisors: Renato H. L. Pedrosa, Plamen Emilov Koshlukov,
Ana Friedlander (UNICAMP), Yuriko Baldin ,
(UFSCar), and M. Helena Noronha,
California State University Northridge.
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Students and Faculty together
On the first day of the program participants were given the option to select the problem to work
on.
The problems had been presented to them a month before through the REU Website.
On the last day of the program, students gave presentations of
their work. They also wrote papers describing their
results. Click on the title of the papers for a pdf version.
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It is shown that configurations with minimal first eigenvalue for
the 1-periodic composite membrane in \R^2
(the membrane is a strip \Omega=R x [0,1])
are not necessarily the ones invariant under x-translations,
for given basic data, if the period is taken sufficiently large.
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Krakowski and Regev compute the codimension se-
quence of the T-ideal of polynomial identities of the Grassmann
algebra using polynomial relations. We give an elementary proof
of their main result and obtain some corollaries.
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The search for the maximum or minimum of a function is often an essential
problem in both pure and applied mathematics; it can be found anywhere
from finding the best route to the supermarket to finding the best mix of
chemicals for a needed reaction, from maximizing onešs crop yield for the year
to minimizing the stress on an iron beam used to construct a building. The
canonical way of finding the maximum of a real function has been to
find its derivative and then locate its roots, but sometimes that derivative
is not readily available, and sometimes mitigating circumstances can get in
the way of a simple derivative reading as that. This is especially true for more
complex systems which cannot be represented as just an equation from R to
R. We will explore two problems and explore their solutions, investigating
ways to find maxima and minima of complex systems: one maximizing a
complex function along a line, and one minimizing a function from Rn to R
with very large n.
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We show in this paper that if every closed 1-form defined on a domain of
the plane is exact then such a domain is simply connected. We also
show that this result does not hold in dimensions 3 and 4.
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The prospective teachers also worked on research problems and
compared the curriculum of teacher preparation of their
universities. Thier papers as well as the compariosn essay is
below.
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Consider two parallel lines in Euclidean plane
and a transversal intersectin the lines at points A and B. If fix A
and let B go to infinity, the family of rotations given by the
composition of reflections in the transversal and in the line
containing B converges to a translation. We study this problem in the
Hyperbolic plane and show that this result generalizes only when the
two given lines are critically parallel.
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This problem was first explored by Japanese Mathematicians during the
period in which Japan was isolated from the western culture (1603-1867). It is
important to observe that during this era, the Japanese didnt have access to the
progress that was being made in the calculus eld. However, they were still able
to work on geometry problems using their own techniques of Calculus. In this
problem we found that even by using advanced calculus techniques, it was still
difficult to solve by hand. Probably because of this difficulty we never found a
satisfactory answer in the literature.
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