53,632 research outputs found
Bounding basis reduction properties
The paper describes improved analysis techniques for basis reduction
that allow one to prove strong complexity bounds and reduced basis
guarantees for traditional reduction algorithms and some of their
variants. This is achieved by a careful exploitation of the linear
equations and inequalities relating various bit sizes before and after
one or more reduction steps
Generalized Skein Modules of Surfaces
Frobenius extensions play a central role in the link homology theories based
upon the sl(n) link variants, and each of these Frobenius extensions may be
recast geometrically via a category of marked cobordisms in the manner of
Bar-Natan. Here we explore a large family of such marked cobordism categories
that are relevant to generalized sl(n) link homology theories. We also
investigate the skein modules that result from embedding these marked
cobordisms within 3-manifolds, and arrive at an explicit presentation for
several of these generalized skein modules.Comment: 23 pages, multiple figure
Solution of the Kirchhoff-Plateau problem
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in
which a flexible filament in the form of a closed loop is spanned by a liquid
film, with the filament being modeled as a Kirchhoff rod and the action of the
spanning surface being solely due to surface tension. We establish the
existence of an equilibrium shape that minimizes the total energy of the system
under the physical constraint of non-interpenetration of matter, but allowing
for points on the surface of the bounding loop to come into contact. In our
treatment, the bounding loop retains a finite cross-sectional thickness and a
nonvanishing volume, while the liquid film is represented by a set with finite
two-dimensional Hausdorff measure. Moreover, the region where the liquid film
touches the surface of the bounding loop is not prescribed a priori. Our
mathematical results substantiate the physical relevance of the chosen model.
Indeed, no matter how strong is the competition between surface tension and the
elastic response of the filament, the system is always able to adjust to
achieve a configuration that complies with the physical constraints encountered
in experiments
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