Optimal Discrete Riesz Energy and Discrepancy

The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the Euclidean distance) governing the point interaction is replaced with the logarithmic potential $\log(1/r)$. In particular, we present a conjecture for the leading term of the asymptotic expansion of the optimal \IL_2-discrepancy with respect to spherical caps on the unit sphere in $\mathbb{R}^{d+1}$ which follows from Stolarsky's invariance principle [Proc. Amer. Math. Soc. 41 (1973)] and the fundamental conjecture for the first two terms of the asymptotic expansion of the optimal Riesz $s$-energy of $N$ points as $N \to \infty$.Comment: 8 page

Discrepancy convergence for the drunkard's walk on the sphere

We analyze the drunkard's walk on the unit sphere with step size theta and show that the walk converges in order constant/sin^2(theta) steps in the discrepancy metric. This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at http://www.math.hmc.edu/~su/papers.htm

Genera Esfera: Interacting with a trackball mapped onto a sphere to explore generative visual worlds

Genera Esfera is an interactive installation that allows the audience to interact and easily become a VJ (visual DJ) in a world of generative visuals. It is an animated and generative graphic environment with a music playlist, a visual spherical world related with and suggested by the music, which reacts and evolves. The installation has been presented at MIRA Live Visual Arts Festival 2015, in Barcelona. Genera Esfera was envisioned, developed and programmed on the basis of two initial ideas: first, to generate our spherical planets we need to work with spherical geometry and program 3D graphics; second, the interaction should be easy to understand, proposing a direct mapping between the visuals and the interface. Our main goal is that participants can focus on exploring the graphic worlds rather than concentrate on understanding the interface. For that purpose we use a trackball to map its position onto sphere rotations. In this paper, we present the interactive installation Genera Esfera, the design guidelines, the mathematics behind the generative visuals and its results.Postprint (published version

Computation of microdosimetric distributions for small sites

Object of this study is the computation of microdosimetric functions for sites which are too small to permit experimental determination of the distributions by Rossi-counters. The calculations are performed on simulated tracks generated by Monte-Carlo techniques. The first part of the article deals with the computational procedure. The second part presents numerical results for protons of energies 0.5, 5, 20 MeV and for site diameters of 5, 10, 100 nm