15,238 research outputs found
New Duality Relations for Classical Ground States
We derive new duality relations that link the energy of configurations
associated with a class of soft pair potentials to the corresponding energy of
the dual (Fourier-transformed) potential. We apply them by showing how
information about the classical ground states of short-ranged potentials can be
used to draw new conclusions about the nature of the ground states of
long-ranged potentials and vice versa. They also lead to bounds on the T=0
system energies in density intervals of phase coexistence, the identification
of a one-dimensional system that exhibits an infinite number of ``phase
transitions," and a conjecture regarding the ground states of purely repulsive
monotonic potentials.Comment: 11 pages, 2 figures. Slightly revised version that corrects typos.
This article will be appearing in Physical Review Letters in a slightly
shortened for
Quaternionic Root Systems and Subgroups of the
Cayley-Dickson doubling procedure is used to construct the root systems of
some celebrated Lie algebras in terms of the integer elements of the division
algebras of real numbers, complex numbers, quaternions and octonions. Starting
with the roots and weights of SU(2) expressed as the real numbers one can
construct the root systems of the Lie algebras of SO(4),SP(2)=
SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the
division algebras. The roots themselves display the group structures besides
the octonionic roots of E_{8} which form a closed octonion algebra. The
automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the
largest crystallographic group in 4-dimensional Euclidean space, is realized as
the direct product of two binary octahedral group of quaternions preserving the
quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such
as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed
as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic
subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192
with different conjugacy classes occur as maximal subgroups in the finite
subgroups of the Lie group of orders 12096 and 1344 and proves to be
useful in their constructions. The triality of SO(8) manifesting itself as the
cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used
to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and
F_{4} respectively
Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals
We have discovered a new family of three-dimensional crystal sphere packings
that are strictly jammed (i.e., mechanically stable) and yet possess an
anomalously low density. This family constitutes an uncountably infinite number
of crystal packings that are subpackings of the densest crystal packings and
are characterized by a high concentration of self-avoiding "tunnels" (chains of
vacancies) that permeate the structures. The fundamental geometric
characteristics of these tunneled crystals command interest in their own right
and are described here in some detail. These include the lattice vectors (that
specify the packing configurations), coordination structure, Voronoi cells, and
density fluctuations. The tunneled crystals are not only candidate structures
for achieving the jamming threshold (lowest-density rigid packing), but may
have substantially broader significance for condensed matter physics and
materials science.Comment: 19 pages, 5 figure
Quintics with Finite Simple Symmetries
We construct all quintic invariants in five variables with simple Non-Abelian
finite symmetry groups. These define Calabi-Yau three-folds which are left
invariant by the action of A_5, A_6 or PSL_2(11).Comment: 18 pages, typos corrected, matches published versio
Estimates of the optimal density and kissing number of sphere packings in high dimensions
The problem of finding the asymptotic behavior of the maximal density of
sphere packings in high Euclidean dimensions is one of the most fascinating and
challenging problems in discrete geometry. One century ago, Minkowski obtained
a rigorous lower bound that is controlled asymptotically by , where
is the Euclidean space dimension. An indication of the difficulty of the
problem can be garnered from the fact that exponential improvement of
Minkowski's bound has proved to be elusive, even though existing upper bounds
suggest that such improvement should be possible. Using a
statistical-mechanical procedure to optimize the density associated with a
"test" pair correlation function and a conjecture concerning the existence of
disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental
Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found
with an asymptotic behavior controlled by . Using the same
methods, we investigate whether this exponential improvement can be further
improved by exploring other test pair correlation functions correponding to
disordered packings. We demonstrate that there are simpler test functions that
lead to the same asymptotic result. More importantly, we show that there is a
wide class of test functions that lead to precisely the same exponential
improvement and therefore the asymptotic form is much
more general than previously surmised.Comment: 23 pages, 4 figures, submitted to Phys. Rev.
Direct observation of irradiation-induced nanocavity shrinkage in Si
Nanocavities in Si substrates, formed by conventional H implantation and thermal annealing, are shown to evolve in size during subsequent Si irradiation. Both ex situ and in situ analytical techniques were used to demonstrate that the mean nanocavity diameter decreases as a function of Si irradiation dose in both the crystalline and amorphous phases. Potential mechanisms for this irradiation-induced nanocavity evolution are discussed. In the crystalline phase, the observed decrease in diameter is attributed to the gettering of interstitials. When the matrix surrounding the cavities is amorphized, cavity shrinkage may be mediated by one of two processes: nanocavities can supply vacancies into the amorphous phase and/or the amorphous phase may flow plastically into the nanocavities. Both processes yield the necessary decrease in density of the amorphous phase relative to crystalline material
Template-based searches for gravitational waves: efficient lattice covering of flat parameter spaces
The construction of optimal template banks for matched-filtering searches is
an example of the sphere covering problem. For parameter spaces with
constant-coefficient metrics a (near-) optimal template bank is achieved by the
A_n* lattice, which is the best lattice-covering in dimensions n <= 5, and is
close to the best covering known for dimensions n <= 16. Generally this
provides a substantially more efficient covering than the simpler hyper-cubic
lattice. We present an algorithm for generating lattice template banks for
constant-coefficient metrics and we illustrate its implementation by generating
A_n* template banks in n=2,3,4 dimensions.Comment: 10 pages, submitted to CQG for proceedings of GWDAW1
Tail resonances of FPU q-breathers and their impact on the pathway to equipartition
Upon initial excitation of a few normal modes the energy distribution among
all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits
exponential localization on large time scales. At the same time resonant
anomalies (peaks) are observed in its weakly excited tail for long times
preceding equipartition. We observe a similar resonant tail structure also for
exact time-periodic Lyapunov orbits, coined q-breathers due to their
exponential localization in modal space. We give a simple explanation for this
structure in terms of superharmonic resonances. The resonance analysis agrees
very well with numerical results and has predictive power. We extend a
previously developed perturbation method, based essentially on a
Poincare-Lindstedt scheme, in order to account for these resonances, and in
order to treat more general model cases, including truncated Toda potentials.
Our results give qualitative and semiquantitative account for the superharmonic
resonances of q-breathers and natural packets
From Operator Algebras to Superconformal Field Theory
We make a review on the recent progress in the operator algebraic approach to
(super)conformal field theory. We discuss representation theory, classification
results, full and boundary conformal field theories, relations to supervertex
operator algebras and Moonshine, connections to subfactor theory and
noncommutative geometry
- …