Mesh ratios for best-packing and limits of minimal energy configurations

For N-point best-packing configurations ωN on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ρN,A), which is the quotient of the covering radius of ωN relative to A and the minimum pairwise distance between points in ωN . For best-packing configurations ωN that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ωN,A)≤1 and this bound can be attained even for the sphere. In the particular case when N=5 on S1 with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid ω∗5, that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, γ(ω∗5,S2)=1

Fractal Functions and Wavelet Expansions Based on Several Scaling Functions

AbstractWe present a method for constructing translation and dilation invariant functions spaces using fractal functions defined by a certain class of iterated function systems. These spaces generalize the C0 function spaces constructed in [D. Hardin, B. Kessler, and P. R. Massopust, J. Approx. Theory71 (1992), 104-120] including, for instance, arbitrarily smooth function spaces. These new function spaces are generated by several scaling functions and their integer-translates. We give necessary and sufficient conditions for these function spaces to form a multiresolution analysis of L2R

Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently

Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently

Quasi-Monte Carlo rules for numerical integration over the unit sphere S2\mathbb{S}^2

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$

Point sets on the sphere S2\mathbb{S}^2 with small spherical cap discrepancy

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given

Scoping prediction of re-radiated ground-borne noise and vibration near high speed rail lines with variable soils

This paper outlines a vibration prediction tool, ScopeRail, capable of predicting in-door noise and vibration, within structures in close proximity to high speed railway lines. The tool is designed to rapidly predict vibration levels over large track distances, while using historical soil information to increase accuracy. Model results are compared to an alternative, commonly used, scoping model and it is found that ScopeRail offers higher accuracy predictions. This increased accuracy can potentially reduce the cost of vibration environmental impact assessments for new high speed rail lines. To develop the tool, a three-dimensional finite element model is first outlined capable of simulating vibration generation and propagation from high speed rail lines. A vast array of model permutations are computed to assess the effect of each input parameter on absolute ground vibration levels. These relations are analysed using a machine learning approach, resulting in a model that can instantly predict ground vibration levels in the presence of different train speeds and soil profiles. Then a collection of empirical factors are coupled with the model to allow for the prediction of structural vibration and in-door noise in buildings located near high speed lines. Additional factors are also used to enable the prediction of vibrations in the presence of abatement measures (e.g. ballast mats and floating slab tracks) and additional excitation mechanisms (e.g. wheelflats and switches/crossings)

Field testing and analysis of high speed rail vibrations

This paper outlines an experimental analysis of ground-borne vibration levels generated by high speed rail lines on various earthwork profiles (at-grade, embankment, cutting and overpass). It also serves to provide access to a dataset of experimental measurements, freely available for download by other researchers working in the area of railway vibration (e.g. for further investigation and/or the validation of vibration prediction models). First, the work outlines experimental investigations undertaken on the Belgian high speed rail network to investigate the vibration propagation characteristics of three different embankment conditions. The sites consist of a 5.5 m high embankment, an at-grade section and a 7.2 m deep cutting. The soil material properties of each site are determined using a ‘Multichannel Analysis of Surface Waves’ technique and verified using refraction analysis. It is shown that all sites have relatively similar material properties thus enabling a generalised comparison. Vibration levels are measured in three directions, up to 100 m from the track due to three different train types (Eurostar, TGV and Thalys) and then analysed statistically. It is found that contrary to commonly accepted theory, vertical vibrations are not always the most dominant, and that horizontal vibrations should also be considered, particularly at larger offsets. It is also found that the embankment earthworks profile produced the lowest vibration levels and the cutting produced the highest. Furthermore, a low (positive) correlation between train speed and vibration levels was found. A selection of the results can be downloaded from www.davidpconnolly.com

MIinimal N-point diameters and f-best-packing constants in Rd[superindex]

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