112 research outputs found
Star Unfolding Convex Polyhedra via Quasigeodesic Loops
We extend the notion of star unfolding to be based on a quasigeodesic loop Q
rather than on a point. This gives a new general method to unfold the surface
of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut
along one shortest path from each vertex of P to Q, and cut all but one segment
of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and
adds references. v3 improves two figures and their captions. New version v4
offers a completely different proof of non-overlap in the quasigeodesic loop
case, and contains several other substantive improvements. This version is 23
pages long, with 15 figure
The connected components of the space of Alexandrov surfaces
Denote by the set of all compact Alexandrov surfaces
with curvature bounded below by without boundary, endowed with the
topology induced by the Gromov-Hausdorff metric. We determine the connected
components of and of its closure
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Closed geodesics in Alexandrov spaces of curvature bounded from above
In this paper, we show a local energy convexity of maps into
spaces. This energy convexity allows us to extend Colding and
Minicozzi's width-sweepout construction to produce closed geodesics in any
closed Alexandrov space of curvature bounded from above, which also provides a
generalized version of the Birkhoff-Lyusternik theorem on the existence of
non-trivial closed geodesics in the Alexandrov setting.Comment: Final version, 22 pages, 2 figures, to appear in the Journal of
Geometric Analysi
Simultaneity as an Invariant Equivalence Relation
This paper deals with the concept of simultaneity in classical and
relativistic physics as construed in terms of group-invariant equivalence
relations. A full examination of Newton, Galilei and Poincar\'e invariant
equivalence relations in is presented, which provides alternative
proofs, additions and occasionally corrections of results in the literature,
including Malament's theorem and some of its variants. It is argued that the
interpretation of simultaneity as an invariant equivalence relation, although
interesting for its own sake, does not cut in the debate concerning the
conventionality of simultaneity in special relativity.Comment: Some corrections, mostly of misprints. Keywords: special relativity,
simultaneity, invariant equivalence relations, Malament's theore
A discrete Laplace-Beltrami operator for simplicial surfaces
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It
depends only on the intrinsic geometry of the surface and its edge weights are
positive. Our Laplace operator is similar to the well known finite-elements
Laplacian (the so called ``cotan formula'') except that it is based on the
intrinsic Delaunay triangulation of the simplicial surface. This leads to new
definitions of discrete harmonic functions, discrete mean curvature, and
discrete minimal surfaces. The definition of the discrete Laplace-Beltrami
operator depends on the existence and uniqueness of Delaunay tessellations in
piecewise flat surfaces. While the existence is known, we prove the uniqueness.
Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides
an optimality criterion for Delaunay triangulations, and this can be used to
prove that the edge flipping algorithm terminates also in the setting of
piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay
triangulations of piecewise flat surfaces revised and expanded. References
added. Some minor changes, typos corrected. v3: fixed inaccuracies in
discussion of flip algorithm, corrected attributions, added references, some
minor revision to improve expositio
Asymptotic behavior of solutions to the -Yamabe equation near isolated singularities
-Yamabe equations are conformally invariant equations generalizing
the classical Yamabe equation. In an earlier work YanYan Li proved that an
admissible solution with an isolated singularity at to the
-Yamabe equation is asymptotically radially symmetric. In this work
we prove that an admissible solution with an isolated singularity at to the -Yamabe equation is asymptotic to a radial
solution to the same equation on . These results
generalize earlier pioneering work in this direction on the classical Yamabe
equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli
et al, we formulate and prove a general asymptotic approximation result for
solutions to certain ODEs which include the case for scalar curvature and
curvature cases. An alternative proof is also provided using
analysis of the linearized operators at the radial solutions, along the lines
of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page
Volumes of polytopes in spaces of constant curvature
We overview the volume calculations for polyhedra in Euclidean, spherical and
hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary
tetrahedron in and . We also present some results, which provide a
solution for Seidel problem on the volume of non-Euclidean tetrahedron.
Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle,
horocycle or one branch of equidistant curve. This is a natural hyperbolic
analog of the cyclic quadrilateral in the Euclidean plane. We find a few
versions of the Brahmagupta formula for the area of such quadrilateral. We also
present a formula for the area of a hyperbolic trapezoid.Comment: 22 pages, 9 figures, 58 reference
Повышение эффективности диагностики и лечения больных с гнойно-воспалительными заболеваниями на основе применения лазерно-флюоресцентной диагностики
In this article the results of using laser-fluorescent diagnostics (LFD) method in patients with purulent diseases are demonstrated. LFD allows making an adequate assessment of the patient's rehabilitation process on the amplitude-spectral objective characteristics; to define dates of convalescence, to reveal complications and to correct an option of drug therapy in time. Such technology permits to prevent complications and reduce the time of treatment for 2-15 days (depends on severity of disease).В статье приведены результаты использования лазерно-флюоресцентной диагностики (ЛФД) в клинике у больных с гнойно-воспалительными заболеваниями. Метод ЛФД основан на объективных амплитудно-спектральных характеристиках, что позволяет проводить адекватную оценку процесса реабилитации больного, определять сроки его выздоровления, выявлять осложнения и своевременно корректировать выбор средств медикаментозной поддержки больных ГВЗ. Применение ЛФД позволяет предотвратить осложнения и сократить сроки лечения больных на 2-15 дней в зависимости от степени тяжести заболевания
Insulator-metal transition in biased finite polyyne systems
A method for the study of the electronic transport in strongly coupled
electron-phonon systems is formalized and applied to a model of polyyne chains
biased through metallic Au leads. We derive a stationary non equilibrium
polaronic theory in the general framework of a variational formulation. The
numerical procedure we propose can be readily applied if the electron-phonon
interaction in the device hamiltonian can be approximated as an effective
single particle electron hamiltonian. Using this approach, we predict that
finite polyyne chains should manifest an insulator-metal transition driven by
the non-equilibrium charging which inhibits the Peierls instability
characterizing the equilibrium state.Comment: to appear at EPJ
- …