16,938 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
Quaternions, octonions and Bell-type inequalities
Multipartite Bell-type inequalities are derived for general systems. They
involve up to eight observables with arbitrary spectra on each site. These
inequalities are closely related to the algebras of quaternions and octonions.Comment: 4 pages, no figure
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
Report of the Higgs Working Group of the Tevatron Run 2 SUSY/Higgs Workshop
This report presents the theoretical analysis relevant for Higgs physics at
the upgraded Tevatron collider and documents the Higgs Working Group
simulations to estimate the discovery reach in Run 2 for the Standard Model and
MSSM Higgs bosons. Based on a simple detector simulation, we have determined
the integrated luminosity necessary to discover the SM Higgs in the mass range
100-190 GeV. The first phase of the Run 2 Higgs search, with a total integrated
luminosity of 2 fb-1 per detector, will provide a 95% CL exclusion sensitivity
comparable to that expected at the end of the LEP2 run. With 10 fb-1 per
detector, this exclusion will extend up to Higgs masses of 180 GeV, and a
tantalizing 3 sigma effect will be visible if the Higgs mass lies below 125
GeV. With 25 fb-1 of integrated luminosity per detector, evidence for SM Higgs
production at the 3 sigma level is possible for Higgs masses up to 180 GeV.
However, the discovery reach is much less impressive for achieving a 5 sigma
Higgs boson signal. Even with 30 fb-1 per detector, only Higgs bosons with
masses up to about 130 GeV can be detected with 5 sigma significance. These
results can also be re-interpreted in the MSSM framework and yield the required
luminosities to discover at least one Higgs boson of the MSSM Higgs sector.
With 5-10 fb-1 of data per detector, it will be possible to exclude at 95% CL
nearly the entire MSSM Higgs parameter space, whereas 20-30 fb-1 is required to
obtain a 5 sigma Higgs discovery over a significant portion of the parameter
space. Moreover, in one interesting region of the MSSM parameter space (at
large tan(beta)), the associated production of a Higgs boson and a b b-bar pair
is significantly enhanced and provides potential for discovering a non-SM-like
Higgs boson in Run 2.Comment: 185 pages, 124 figures, 55 table
Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra
The determination of the densest packings of regular tetrahedra (one of the
five Platonic solids) is attracting great attention as evidenced by the rapid
pace at which packing records are being broken and the fascinating packing
structures that have emerged. Here we provide the most general analytical
formulation to date to construct dense periodic packings of tetrahedra with
four particles per fundamental cell. This analysis results in six-parameter
family of dense tetrahedron packings that includes as special cases recently
discovered "dimer" packings of tetrahedra, including the densest known packings
with density . This study strongly suggests that
the latter set of packings are the densest among all packings with a
four-particle basis. Whether they are the densest packings of tetrahedra among
all packings is an open question, but we offer remarks about this issue.
Moreover, we describe a procedure that provides estimates of upper bounds on
the maximal density of tetrahedron packings, which could aid in assessing the
packing efficiency of candidate dense packings.Comment: It contains 25 pages, 5 figures
Adaptation Framework and Strategy Part 3: an adaptation strategy for agriculture in Ningxia, Northwest China
Adaptation Framework and Strategy Part 2: application of the adaptation framework - a case study of Ningxia, Northwest China
Densest local packing diversity. II. Application to three dimensions
The densest local packings of N three-dimensional identical nonoverlapping
spheres within a radius Rmin(N) of a fixed central sphere of the same size are
obtained for selected values of N up to N = 1054. In the predecessor to this
paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305
(2010)], we described our method for finding the putative densest packings of N
spheres in d-dimensional Euclidean space Rd and presented those packings in R2
for values of N up to N = 348. We analyze the properties and characteristics of
the densest local packings in R3 and employ knowledge of the Rmin(N), using
methods applicable in any d, to construct both a realizability condition for
pair correlation functions of sphere packings and an upper bound on the maximal
density of infinite sphere packings. In R3, we find wide variability in the
densest local packings, including a multitude of packing symmetries such as
perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest
local packings of N spheres near a central sphere to minimal-energy
configurations of N+1 points interacting with short-range repulsive and
long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that
they are in general completely different, a result that has possible
implications for nucleation theory. We also compare the densest local packings
to finite subsets of stacking variants of the densest infinite packings in R3
(the Barlow packings) and find that the densest local packings are almost
always most similar, as measured by a similarity metric, to the subsets of
Barlow packings with the smallest number of coordination shells measured about
a single central sphere, e.g., a subset of the FCC Barlow packing. We
additionally observe that the densest local packings are dominated by the
spheres arranged with centers at precisely distance Rmin(N) from the fixed
sphere's center.Comment: 45 pages, 18 figures, 2 table
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