20,354 research outputs found
Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical
code) is called rigid or jammed if it is isolated within the space of packings.
In other words, aside from applying a global isometry, the packing cannot be
deformed. In this paper, we systematically study the rigidity of spherical
codes, particularly kissing configurations. One surprise is that the kissing
configuration of the Coxeter-Todd lattice is not jammed, despite being locally
jammed (each individual cap is held in place if its neighbors are fixed); in
this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic
lattice in three dimensions. By contrast, we find that many other packings have
jammed kissing configurations, including the Barnes-Wall lattice and all of the
best kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations
known in 25 through 31 dimensions. Motivated by this phenomenon, we find new
kissing configurations in these dimensions, which improve on the records set in
1982 by the laminated lattices.Comment: 39 pages, 8 figure
An Efficient Linear Programming Algorithm to Generate the Densest Lattice Sphere Packings
Finding the densest sphere packing in -dimensional Euclidean space
is an outstanding fundamental problem with relevance in many
fields, including the ground states of molecular systems, colloidal crystal
structures, coding theory, discrete geometry, number theory, and biological
systems. Numerically generating the densest sphere packings becomes very
challenging in high dimensions due to an exponentially increasing number of
possible sphere contacts and sphere configurations, even for the restricted
problem of finding the densest lattice sphere packings. In this paper, we apply
the Torquato-Jiao packing algorithm, which is a method based on solving a
sequence of linear programs, to robustly reproduce the densest known lattice
sphere packings for dimensions 2 through 19. We show that the TJ algorithm is
appreciably more efficient at solving these problems than previously published
methods. Indeed, in some dimensions, the former procedure can be as much as
three orders of magnitude faster at finding the optimal solutions than earlier
ones. We also study the suboptimal local density-maxima solutions (inherent
structures or "extreme" lattices) to gain insight about the nature of the
topography of the "density" landscape.Comment: 23 pages, 3 figure
Random Sequential Addition of Hard Spheres in High Euclidean Dimensions
Employing numerical and theoretical methods, we investigate the structural
characteristics of random sequential addition (RSA) of congruent spheres in
-dimensional Euclidean space in the infinite-time or
saturation limit for the first six space dimensions ().
Specifically, we determine the saturation density, pair correlation function,
cumulative coordination number and the structure factor in each =of these
dimensions. We find that for , the saturation density
scales with dimension as , where and
. We also show analytically that the same density scaling
persists in the high-dimensional limit, albeit with different coefficients. A
byproduct of this high-dimensional analysis is a relatively sharp lower bound
on the saturation density for any given by , where is the structure factor at
(i.e., infinite-wavelength number variance) in the high-dimensional limit.
Consistent with the recent "decorrelation principle," we find that pair
correlations markedly diminish as the space dimension increases up to six. Our
work has implications for the possible existence of disordered classical ground
states for some continuous potentials in sufficiently high dimensions.Comment: 38 pages, 9 figures, 4 table
A Novel Symmetric Four Dimensional Polytope Found Using Optimization Strategies Inspired by Thomson's Problem of Charges on a Sphere
Inspired by, and using methods of optimization derived from classical three
dimensional electrostatics, we note a novel beautiful symmetric four
dimensional polytope we have found with 80 vertices. We also describe how the
method used to find this symmetric polytope, and related methods can
potentially be used to find good examples for the kissing and packing problems
in D dimensions
On kissing numbers and spherical codes in high dimensions
We prove a lower bound of on the
kissing number in dimension . This improves the classical lower bound of
Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a
similar linear factor improvement to the best known lower bound on the maximal
size of a spherical code of acute angle in high dimensions
Seven Staggering Sequences
When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison
gave it an enthusiastic review in the Scientific American and Martin Gardner
was kind enough to say in his Mathematical Games column that "every
recreational mathematician should buy a copy forthwith." That book contained
2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains
117000 sequences. This paper will describe seven that I find especially
interesting. These are the EKG sequence, Gijswijt's sequence, a numerical
analog of Aronson's sequence, approximate squaring, the integrality of n-th
roots of generating functions, dissections, and the kissing number problem.
(Paper for conference in honor of Martin Gardner's 91st birthday.)Comment: 12 pages. A somewhat different version appeared in "Homage to a Pied
Puzzler", E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters,
Wellesley, MA, 2009, pp. 93-11
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