36,065 research outputs found
Brief introduction to tropical geometry
The paper consists of lecture notes for a mini-course given by the authors at
the G\"okova Geometry \& Topology conference in May 2014. We start the
exposition with tropical curves in the plane and their applications to problems
in classical enumerative geometry, and continue with a look at more general
tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise
Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over
It is well known that there is a bijective correspondence between metric
ribbon graphs and compact Riemann surfaces with meromorphic Strebel
differentials. In this article, it is proved that Grothendieck's correspondence
between dessins d'enfants and Belyi morphisms is a special case of this
correspondence. For a metric ribbon graph with edge length 1, an algebraic
curve over and a Strebel differential on it is constructed. It is also
shown that the critical trajectories of the measured foliation that is
determined by the Strebel differential recover the original metric ribbon
graph. Conversely, for every Belyi morphism, a unique Strebel differential is
constructed such that the critical leaves of the measured foliation it
determines form a metric ribbon graph of edge length 1, which coincides with
the corresponding dessin d'enfant.Comment: Higher resolution figures available at
http://math.ucdavis.edu/~mulase
Epitaxial Frustration in Deposited Packings of Rigid Disks and Spheres
We use numerical simulation to investigate and analyze the way that rigid
disks and spheres arrange themselves when compressed next to incommensurate
substrates. For disks, a movable set is pressed into a jammed state against an
ordered fixed line of larger disks, where the diameter ratio of movable to
fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations
is 0.7, where the fixed substrate has the structure of a (001) plane of a
face-centered cubic array. Results obtained for both disks and spheres exhibit
various forms of density-reducing packing frustration next to the
incommensurate substrate, including some cases displaying disorder that extends
far from the substrate. The disk system calculations strongly suggest that the
most efficient (highest density) packings involve configurations that are
periodic in the lateral direction parallel to the substrate, with substantial
geometric disruption only occurring near the substrate. Some evidence has also
emerged suggesting that for the sphere systems a corresponding structure doubly
periodic in the lateral directions would yield the highest packing density;
however all of the sphere simulations completed thus far produced some residual
"bulk" disorder not obviously resulting from substrate mismatch. In view of the
fact that the cases studied here represent only a small subset of all that
eventually deserve attention, we end with discussion of the directions in which
first extensions of the present simulations might profitably be pursued.Comment: 28 pages, 14 figures; typos fixed; a sentence added to 4th paragraph
of sect 5 in responce to a referee's comment
The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve
This article sketches the proofs of two theorems about sphere packings in
Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the
surface area of every bounded Voronoi cell in a packing of balls of radius 1 is
at least that of a regular dodecahedron of inradius 1. The second theorem is L.
Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of
congruent balls such that each ball is touched by twelve others consists of
hexagonal layers. Both proofs are computer assisted. Complete proofs of these
theorems appear in the author's book "Dense Sphere Packings" and a related
preprintComment: The citations and title have been update
A Gaussian Weave for Kinematical Loop Quantum Gravity
Remarkable efforts in the study of the semi-classical regime of kinematical
loop quantum gravity are currently underway. In this note, we construct a
``quasi-coherent'' weave state using Gaussian factors. In a similar fashion to
some other proposals, this state is peaked in both the connection and the spin
network basis. However, the state constructed here has the novel feature that,
in the spin network basis, the main contribution for this state is given by the
fundamental representation, independently of the value of the parameter that
regulates the Gaussian width.Comment: 15 pages, 3 figures, Revtex file. Comments added and references
updated. Final version to appear in IJMP-
Density bounds for outer parallel domains of unit ball packings
We give upper bounds for the density of unit ball packings relative to their
outer parallel domains and discuss their connection to contact numbers. Also,
packings of soft balls are introduced and upper bounds are given for the
fraction of space covered by them.Comment: 22 pages, 1 figur
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