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MMATH
( RESEARCH ) 





The items here are things I have had a great
curiosity about but did not find in the literature.
They are here so I can point them out to or request
feedback from interested
colleagues. Because many manuscripts are
symbolically complex and I do not have math typing
software attached to this website, I may just scan
the pages and thumbnail them here for all to enjoy
and maybe manipulate for printing in readable
format. Go ahead and print what you want  no
copyright restrictions on these pages!
(Latest update: December
26, 2008.) 





e
contains a paper, "Transcendental Meditations",
which treats the existence of the classical limit
defining e and the more modern "area" definition.
It makes the connection in the simplest way I know
of, one which my calculus students can grasp
visually and intuitively. 





Integrals of Products are Products of Integrals
deals with the
classical student error of integrating products by
multiplying antiderivatives of its factors (a source
of anguish to any calculus instructor)! It
turns out there is a curious existence theorem that
can be made to sound like this, and in fact there
are constructive methods that produce numerous
examples whose verification is an interesting and
challenging excursion through the "legitimate"
methods of the subject. This talk has been
presented at regional meetings of the MAA and
TexMATYC. 





Pythagoreana is a collection of results
related to Pythagorean triplets, including tilings
of rectangles and even a square with pairwise
nonsimilar Pythagorean triangles. A difficult
related problem is generation of triples of
Pythagorean tangents where the sum of two equals the
third. Families of solutions are discussed,
but because they lie on a
surface that is not (Mordell theoretically) simple, it is unlikely any
exhaustive parametrization of solutions will be
found. 





Bertrand
refers to a proof of "Bertrand's Theorem", a
brilliant result of celestial mechanics.
Simply put, the premise is that periodicity of
bounded orbits for the two body problem implies Newton's
inversesquare gravitation law. The Nineteenth
Century proof offered by Bertrand lacked rigor; and
indeed the statement had to be tightened up a
little; but his methods seem to be repairable, and
repairable using
only mathematics known at the time the result was
first proposed (one point has been shown using Lebesgue theory, and that came later;
but I think even that can be conformed).
(Final manuscript to appear
when done, presumably in 2009. 





Sarkovskii
refers to a result in chaos theory, regarding what
periods periodic points of endomorphisms of the real
line or an interval can have. It was
astonishing that the implication is not chaotic at
all  in fact, it linearly orders the positive
integers and partitions the set of endomorphisms
corresponding to "cuts" in the ordering. The
papers here will show how the problem of computing
periods is related to lengths of cycles in directed
graphs and, once this is understood, the
constructions make the reason for the "Sarkovskii
ordering" clear. A secondary paper
examines a particular "chaotic" quadratic polynomial
whose periodic points can be explicitly computed. 





Minimal Polynomials for Trig Functions
is now published in primitive
form. Click
on
Minpolys to access it. It
includes a familiar derivation of the minimal
polynomials for cosines of angles rationally
commensurate with π.
From these, we have derived minimal polynomials for
all the other basic functions, but the manuscript is
not yet complete. Check this page occasionally
for developments. 






