2,550 research outputs found
Geodesic Universal Molecules
The first phase of TreeMaker, a well-known method for origami design, decomposes a planar polygon (the “paper”) into regions. If some region is not convex, TreeMaker indicates it with an error message and stops. Otherwise, a second phases is invoked which computes a crease pattern called a “universal molecule”. In this paper we introduce and study geodesic universal molecules, which also work with non-convex polygons and thus extend the applicability of the TreeMaker method. We characterize the family of disk-like surfaces, crease patterns and folded states produced by our generalized algorithm. They include non-convex polygons drawn on the surface of an intrinsically flat piecewise-linear surface which have self-overlap when laid open flat, as well as surfaces with negative curvature at a boundary vertex
Black holes as generalised Toda molecules
In this note we compare the geodesic formalism for spherically symmetric
black hole solutions with the black hole effective potential approach. The
geodesic formalism is beneficial for symmetric supergravity theories since the
symmetries of the larger target space leads to a complete set of commuting
constants of motion that establish the integrability of the geodesic equations
of motion, as shown in arXiv:1007.3209. We point out that the integrability
lifts straightforwardly to the integrability of the equations of motion with a
black hole potential. This construction turns out to be a generalisation of the
connection between Toda molecule equations and geodesic motion on symmetric
spaces known in the mathematics literature. We describe in some detail how this
generalisation of the Toda molecule equations arises.Comment: 19 pages, references adde
Crystalline Order on a Sphere and the Generalized Thomson Problem
We attack generalized Thomson problems with a continuum formalism which
exploits a universal long range interaction between defects depending on the
Young modulus of the underlying lattice. Our predictions for the ground state
energy agree with simulations of long range power law interactions of the form
1/r^{gamma} (0 < gamma < 2) to four significant digits. The regime of grain
boundaries is studied in the context of tilted crystalline order and the
generality of our approach is illustrated with new results for square tilings
on the sphere.Comment: 4 pages, 5 eps figures Fig. 2 revised, improved Fig. 3, reference
typo fixe
The influence of the cosmological expansion on local systems
Following renewed interest, the problem of whether the cosmological expansion
affects the dynamics of local systems is reconsidered. The cosmological
correction to the equations of motion in the locally inertial Fermi normal
frame (the relevant frame for astronomical observations) is computed. The
evolution equations for the cosmological perturbation of the two--body problem
are solved in this frame. The effect on the orbit is insignificant as are the
effects on the galactic and galactic--cluster scales.Comment: To appear in the Astrophysical Journal, Late
From Unruh temperature to generalized Bousso bound
In a classical spacetime satisfying Einstein's equation and the null
convergence condition, the same quantum mechanical effects that cause black
holes to have a temperature are found to imply, if joined to the macroscopic
nature of entropy, the covariant entropy bound in its generalized form. This is
obtained from thermodynamics, as applied across the local Rindler causal
horizon through every point p of the null hypersurfaces L the covariant entropy
bound refers to, in the direction of the null geodesics generating L.Comment: 5 pages. v2: some changes to clarify the path to the obtained
results; two (final) paragraphs, the acknowledgments and a reference adde
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