7,134 research outputs found
Inflations of ideal triangulations
Starting with an ideal triangulation of the interior of a compact 3-manifold
M with boundary, no component of which is a 2-sphere, we provide a
construction, called an inflation of the ideal triangulation, to obtain a
strongly related triangulations of M itself. Besides a step-by-step algorithm
for such a construction, we provide examples of an inflation of the
two-tetrahedra ideal triangulation of the complement of the figure-eight knot
in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the
figure-eight knot exterior. As another example, we provide an inflation of the
one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven
tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary.
Several applications of inflations are discussed.Comment: 48 pages, 45 figure
Euler characteristic and quadrilaterals of normal surfaces
Let be a compact 3-manifold with a triangulation . We give an
inequality relating the Euler characteristic of a surface normally embedded
in with the number of normal quadrilaterals in . This gives a relation
between a topological invariant of the surface and a quantity derived from its
combinatorial description. Secondly, we obtain an inequality relating the
number of normal triangles and normal quadrilaterals of , that depends on
the maximum number of tetrahedrons that share a vertex in .Comment: 7 pages, 1 figur
Quantization in geometric pluripotential theory
The space of K\"ahler metrics can, on the one hand, be approximated by
subspaces of algebraic metrics, while, on the other hand, can be enlarged to
finite-energy spaces arising in pluripotential theory. The latter spaces are
realized as metric completions of Finsler structures on the space of K\"ahler
metrics. The former spaces are the finite-dimensional spaces of Fubini--Study
metrics of K\"ahler quantization. The goal of this article is to draw a
connection between the two. We show that the Finsler structures on the space of
K\"ahler potentials can be quantized. More precisely, given a K\"ahler manifold
polarized by an ample line bundle we endow the space of Hermitian metrics on
powers of that line bundle with Finsler structures and show that the resulting
path length metric spaces recover the corresponding metric completions of the
Finsler structures on the space of K\"ahler potentials. This has a number of
applications, among them a new approach to the rooftop envelopes and
Pythagorean formulas of K\"ahler geometry, a new Lidskii type inequality on the
space of K\"ahler metrics, and approximation of finite energy potentials, as
well as geodesic segments by the corresponding smooth algebraic objects
Hungry in the “Land of Pleasant Living”: Combating the Effects of Baltimore’s Food Deserts on Childhood Education Through Eminent Domain
Love and Suffering: Adolescent Socialization and Suicide in Micronesia
Youth suicide has reached epidemic proportion in Micronesia over the past two
decades. Suicides display remarkable cultural patterning in the typical actors,
methods, motivational themes, and precipitating social scenarios. The focus of
contemporary high rates is among young men aged fifteen to twenty-four, who
hang themselves following incidents of conflict with parents. Predominant
themes invoked in adolescent suicide accounts involve anger and suffering at the
hands of their parents, and feelings of familial rejection juxtaposed with reaffirmations
of filial love. Less frequent are themes involving personal shame over
violations of fundamental social rules. In situations of both "anger" and
"shame" suicides, the primary locus of conflict is within close family relations.
The suicides appear as an extreme form of an accustomed pattern of resolving
conflict with senior family members by withdrawing from the scene.
In this article I employ one paradigmatic case history to provide a description
of the cultural construction and social dynamics of contemporary adolescent suicide
in Micronesia. The suicide phenomenon is situated within recent changes in
the stage of adolescent male socialization in Micronesian societies. For adolescent
males of earlier generations, social involvement at the level of lineage and
clan activities provided important support. The recent rapid shift from subsistence
exchange to cash economy has severely attenuated lineage and clan structures
and, by undermining the process of adolescent socialization, has set the
stage for high rates of suicide among young men. Finally, I explore the potential
for suicide modeling and contagion among Micronesian youth
A state variable for crumpled thin sheets
Despite the apparent ease with which a sheet of paper is crumpled and tossed
away, crumpling dynamics are often considered a paradigm of complexity. This
complexity arises from the infinite number of configurations a disordered
crumpled sheet can take. Here we experimentally show that key aspects of
crumpling have a very simple description; the evolution of the damage in
crumpling dynamics can largely be described by a single global quantity, the
total length of all creases. We follow the evolution of the damage network in
repetitively crumpled elastoplastic sheets, and show that the dynamics of this
quantity are deterministic, and depend only on the instantaneous state of the
crease network and not at all on the crumpling history. We also show that this
global quantity captures the crumpling dynamics of a sheet crumpled for the
first time. This leads to a remarkable reduction in complexity, allowing a
description of a highly disordered system by a single state parameter. Similar
strategies may also be useful in analyzing other systems that evolve under
geometric and mechanical constraints, from faulting of tectonic plates to the
evolution of proteins
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