7,872 research outputs found
The Gaussian core model in high dimensions
We prove lower bounds for energy in the Gaussian core model, in which point
particles interact via a Gaussian potential. Under the potential function with , we show that no point
configuration in of density can have energy less than
as with and
fixed. This lower bound asymptotically matches the upper bound of obtained as the expectation in the Siegel mean value
theorem, and it is attained by random lattices. The proof is based on the
linear programming bound, and it uses an interpolation construction analogous
to those used for the Beurling-Selberg extremal problem in analytic number
theory. In the other direction, we prove that the upper bound of is no longer asymptotically sharp when . As
a consequence of our results, we obtain bounds in for the
minimal energy under inverse power laws with , and
these bounds are sharp to within a constant factor as with
fixed.Comment: 30 pages, 1 figur
Optimality and uniqueness of the (4,10,1/6) spherical code
Linear programming bounds provide an elegant method to prove optimality and
uniqueness of an (n,N,t) spherical code. However, this method does not apply to
the parameters (4,10,1/6). We use semidefinite programming bounds instead to
show that the Petersen code, which consists of the midpoints of the edges of
the regular simplex in dimension 4, is the unique (4,10,1/6) spherical code.Comment: 12 pages, (v2) several small changes and corrections suggested by
referees, accepted in Journal of Combinatorial Theory, Series
Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets
H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which
is conjectured to be a universally optimal code. We show that this scheme has a
generalization in terms of Kerdock codes, as well as in terms of maximal real
mutually unbiased bases. These schemes also related to extremal line-sets in
Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam
constructed two infinite series of formally dual 3-class association schemes.
We explain this formal duality by constructing two dual abelian schemes related
to quaternary linear Kerdock and Preparata codes.Comment: 16 page
Characterizing simulated galaxy stellar mass histories
Cosmological galaxy formation simulations can now produce rich and diverse ensembles of galaxy histories. These simulated galaxy histories, taken all together, provide an answer to the question ‘How do galaxies form?’ for the models used to construct them. We characterize such galaxy history ensembles both to understand their properties and to identify points of comparison for histories within a given galaxy formation model or between different galaxy formation models and simulations. We focus primarily on stellar mass histories of galaxies with the same final stellar mass, for six final stellar mass values and for three different simulated galaxy formation models (a semi-analytic model built upon the dark matter Millennium simulation and two models from the hydrodynamical OverWhelmingly Large Simulations project). Using principal component analysis (PCA) to classify scatter around the average stellar mass history, we find that one fluctuation dominates for all sets of histories we consider, although its shape and contribution can vary between samples. We correlate the PCA characterization with several z = 0 galaxy properties (to connect with survey observables) and also compare it to some other galaxy history properties. We then explore separating galaxy stellar mass histories into classes, using the largest PCA contribution, k-means clustering, and simple Gaussian mixture models. For three component models, these different methods often gave similar results. These history classification methods provide a succinct and often quick way to characterize changes in the full ensemble of histories of a simulated population as physical assumptions are varied, to compare histories of different simulated populations to each other, and to assess the relation of simulated histories to fixed time observations
A palindromization map for the free group
We define a self-map Pal: F_2 --> F_2 of the free group on two generators a,
b, using automorphisms of F_2 that form a group isomorphic to the braid group
B_3. The map Pal restricts to de Luca's right iterated palindromic closure on
the submonoid generated by a, b, and is continuous for the profinite topology
on F_2. The values of Pal are palindromes and coincide with the elements g of
F_2 such that abg is conjugate to bag.Comment: 14 pages. Introduction expanded. Two references added. Sections 3 and
6 of Version 1 have been restructured, now forming Sections 3-
A computational model of human-robot spatial interactions based on a qualitative trajectory calculus
In this paper we propose a probabilistic sequential model of Human-Robot Spatial Interaction (HRSI) using a well-established Qualitative Trajectory Calculus (QTC) to encode HRSI between a human and a mobile robot in a meaningful, tractable, and systematic manner. Our key contribution is to utilise QTC as a state descriptor and model HRSI as a probabilistic sequence of such states. Apart from the sole direction of movements of human and robot modelled by QTC, attributes of HRSI like proxemics and velocity profiles play vital roles for the modelling and generation of HRSI behaviour. In this paper, we particularly present how the concept of proxemics can be embedded in QTC to facilitate richer models. To facilitate reasoning on HRSI with qualitative representations, we show how we can combine the representational power of QTC with the concept of proxemics in a concise framework, enriching our probabilistic representation by implicitly modelling distances. We show the appropriateness of our sequential model of QTC by encoding different HRSI behaviours observed in two spatial interaction experiments. We classify these encounters, creating a comparative measurement, showing the representational capabilities of the model
Evolution of Multi-mass Globular Clusters in Galactic Tidal Field with the Effects of Velocity Anisotropy
We study the evolution of globular clusters with mass spectra under the
influence of the steady Galactic tidal field, including the effects of velocity
anisotropy. Similar to single-mass models, velocity anisotropy develops as the
cluster evolves, but the degree of anisotropy is much smaller than isolated
clusters. Except for very early epochs of the cluster evolution, nearly all
mass components become tangentially anisotropic at the outer parts. We have
compared our results with multi-mass, King-Michie models. The isotropic King
model better fits to the Fokker-Planck results because of tangential
anisotropy. However, it is almost impossible to fit the computed density
profiles to the multi-mass King models for all mass components. Thus if one
attempts to derive global mass function based on the observed mass function in
limited radial range using multi-mass King models, one may get somewhat erratic
results, especially for low mass stars. We have examined how the mass function
changes in time. Specifically, we find that the power-law index of the mass
function decreases monotonically with the total mass of the cluster. This
appears to be consistent with the behaviour of the observed slopes of mass
functions for a limited number of clusters, although it is premature to compare
quantitatively because there are other mechanisms in contributing the
evaporation of stars from the clusters. The projected velocity profiles for
anisotropic models with the apocenter criterion for evaporation show
significant flattening toward the tidal radius compared to isotropic model or
anisotropic model with the energy criterion. Such a behaviour of velocity
profile appears to be consistent with the observed profiles of collapsed
cluster M15.Comment: 13 pages including 18 figures in mn styl
Cataclysmic Variables and a New Class of Faint UV Stars in the Globular Cluster NGC 6397
We present evidence that the globular cluster NGC 6397 contains two distinct
classes of centrally-concentrated UV-bright stars. Color-magnitude diagrams
constructed from U, B, V, and I data obtained with the HST/WFPC2 reveal seven
UV-bright stars fainter than the main-sequence turnoff, three of which had
previously been identified as cataclysmic variables (CVs). Lightcurves of these
stars show the characteristic ``flicker'' of CVs, as well as longer-term
variability. A fourth star is identified as a CV candidate on the basis of its
variability and UV excess. Three additional UV-bright stars show no photometric
variability and have broad-band colors characteristic of B stars. These
non-flickering UV stars are too faint to be extended horizontal branch stars.
We suggest that they could be low-mass helium white dwarfs, formed when the
evolution of a red giant is interrupted, due either to Roche-lobe overflow onto
a binary companion, or to envelope ejection following a common-envelope phase
in a tidal-capture binary. Alternatively, they could be very-low-mass
core-He-burning stars. Both the CVs and the new class of faint UV stars are
strongly concentrated toward the cluster center, to the extent that mass
segregation from 2-body relaxation alone may be unable to explain their
distribution.Comment: 11 pages plus 3 eps figures; LaTeX using aaspp4.sty; to appear in The
Astrophysical Journal Letter
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