350 research outputs found
When is .999... less than 1?
We examine alternative interpretations of the symbol described as nought,
point, nine recurring. Is "an infinite number of 9s" merely a figure of speech?
How are such alternative interpretations related to infinite cardinalities? How
are they expressed in Lightstone's "semicolon" notation? Is it possible to
choose a canonical alternative interpretation? Should unital evaluation of the
symbol .999 . . . be inculcated in a pre-limit teaching environment? The
problem of the unital evaluation is hereby examined from the pre-R, pre-lim
viewpoint of the student.Comment: 28 page
Bolza quaternion order and asymptotics of systoles along congruence subgroups
We give a detailed description of the arithmetic Fuchsian group of the Bolza
surface and the associated quaternion order. This description enables us to
show that the corresponding principal congruence covers satisfy the bound
sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the
Bolza group as a congruence subgroup, and calculate out a few examples of
"Bolza twins" (using magma). Like the Hurwitz triplets, these correspond to the
factoring of certain rational primes in the ring of integers of the invariant
trace field of the surface. We exploit random sampling combined with the
Reidemeister-Schreier algorithm as implemented in magma to generate these
surfaces.Comment: 35 pages, to appear in Experimental Mathematic
Relative systoles of relative-essential 2-complexes
We prove a systolic inequality for the phi-relative 1-systole of a
phi-essential 2-complex, where phi is a homomorphism from the fundamental group
of the complex, to a finitely presented group G. Indeed we show that
universally for any phi-essential Riemannian 2-complex, and any G, the area of
X is bounded below by 1/8 of sys(X,phi)^2. Combining our results with a method
of Larry Guth, we obtain new quantitative results for certain 3-manifolds: in
particular for Sigma the Poincare homology sphere, we have sys(Sigma)^3 < 24
vol(Sigma).Comment: 20 pages, to appear in Algebraic and Geometric Topolog
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