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 $t \mapsto e^{-\alpha t^2}$ with $0 < \alpha < 4\pi/e$, we show that no point configuration in $\mathbf{R}^n$ of density $\rho$ can have energy less than $(\rho+o(1))(\pi/\alpha)^{n/2}$ as $n \to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho (\pi/\alpha)^{n/2}$ 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 $\rho (\pi/\alpha)^{n/2}$ is no longer asymptotically sharp when $\alpha > \pi e$. As a consequence of our results, we obtain bounds in $\mathbf{R}^n$ for the minimal energy under inverse power laws $t \mapsto 1/t^{n+s}$ with $s>0$, and these bounds are sharp to within a constant factor as $n \to \infty$ with $s$ 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

Number of right ideals and a qq-analogue of indecomposable permutations

We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field $\mathbb F\_q$ is equal to $(q-1)^{n+1} q^{\frac{(n+1)(n-2)}{2}}\sum\_\theta q^{inv(\theta)}$, where the sum is over all indecomposable permutations in $S\_{n+1}$ and where $inv(\theta)$stands for the number of inversions of $\theta$.Comment: submitte

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