Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of SS-optimization

The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs utilizes random sampling on the uncertainty parameter to substitute the original problem with a representative continuous convex optimization with $N$ convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size $N$ of the sampling relaxation to yield high-likelihood feasible solutions of the chance-constrained problem. They measured the probability of the original constraints to be violated by the random optimal solution from the relaxation of size $N$. This paper has two main contributions. First, we present a generalization of the Calafiore-Campi results to both integer and mixed-integer variables. In fact, we demonstrate that their sampling estimates work naturally for variables restricted to some subset $S$ of $\mathbb R^d$. The key elements are generalizations of Helly's theorem where the convex sets are required to intersect $S \subset \mathbb R^d$. The size of samples in both algorithms will be directly determined by the $S$-Helly numbers. Motivated by the first half of the paper, for any subset $S \subset \mathbb R^d$, we introduce the notion of an $S$-optimization problem, where the variables take on values over $S$. It generalizes continuous, integer, and mixed-integer optimization. We illustrate with examples the expressive power of $S$-optimization to capture sophisticated combinatorial optimization problems with difficult modular constraints. We reinforce the evidence that $S$-optimization is "the right concept" by showing that the well-known randomized sampling algorithm of K. Clarkson for low-dimensional convex optimization problems can be extended to work with variables taking values over $S$.Comment: 16 pages, 0 figures. This paper has been revised and split into two parts. This version is the second part of the original paper. The first part of the original paper is arXiv:1508.02380 (the original article contained 24 pages, 3 figures

Helly numbers of Algebraic Subsets of Rd\mathbb R^d

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was originally the first half of arXiv:1504.00076v

Pattern formation in a predator-prey system characterized by a spatial scale of interaction

We describe pattern formation in ecological systems using a version of the classical Lotka-Volterra model characterized by a spatial scale which controls the predator-prey interaction range. Analytical and simulational results show that patterns can emerge in some regions of the parameters space where the instability is driven by the range of the interaction. The individual-based implementation captures realistic ecological features. In fact, spatial structures emerge in an erratic oscillatory regime which can contemplate predators' extinction.Comment: 5 pages, 4 figure

Magneto--Acoustic Energetics Study of the Seismically Active Flare of 15 February 2011

Multi--wavelength studies of energetic solar flares with seismic emissions have revealed interesting common features between them. We studied the first GOES X--class flare of the 24th solar cycle, as detected by the Solar Dynamics Observatory (SDO). For context, seismic activity from this flare (SOL2011-02-15T01:55-X2.2, in NOAA AR 11158) has been reported in the literature (Kosovichev, 2011; Zharkov et al., 2011). Based on Dopplergram data from the Helioseismic and Magnetic Imager (HMI), we applied standard methods of local helioseismology in order to identify the seismic sources in this event. RHESSI hard X-ray data are used to check the correlation between the location of the seismic sources and the particle precipitation sites in during the flare. Using HMI magnetogram data, the temporal profile of fluctuations in the photospheric line-of-sight magnetic field is used to estimate the magnetic field change in the region where the seismic signal was observed. This leads to an estimate of the work done by the Lorentz-force transient on the photosphere of the source region. In this instance this is found to be a significant fraction of the acoustic energy in the attendant seismic emission, suggesting that Lorentz forces can contribute significantly to the generation of sunquakes. However, there are regions in which the signature of the Lorentz-force is much stronger, but from which no significant acoustic emission emanates.Comment: Submitted to Solar Physic

Plasmon spectroscopy: Theoretical and numerical calculations, and optimization techniques

30 págs.; 22 figs.; 1 tab.; Open Access funded by Creative Commons Atribution Licence 3.0We present an overview of recent advances in plasmonics, mainly concerning theoretical and numerical tools required for the rigorous determination of the spectral properties of complex-shape nanoparticles exhibiting strong localized surface plasmon resonances (LSPRs). Both quasistatic approaches and full electrodynamic methods are described, providing a thorough comparison of their numerical implementations. Special attention is paid to surface integral equation formulations, giving examples of their performance in complicated nanoparticle shapes of interest for their LSPR spectra. In this regard, complex (single) nanoparticle configurations (nanocrosses and nanorods) yield a hierarchy of multiple-order LSPR s with evidence of a rich symmetric or asymmetric (Fano-like) LSPR line shapes. In addition, means to address the design of complex geometries to retrieve LSPR spectra are commented on, with special interest in biologically inspired algorithms. Thewealth of LSPRbased applications are discussed in two choice examples, single-nanoparticle surface-enhanced Raman scattering (SERS) and optical heating, and multifrequency nanoantennas for fluorescence and nonlinear optics.J.A.S.-G. and R.P.-D. acknowledge the Spanish >Ministerio de Economía y Competitividad>, through the Consolider-Ingenio project EMET (CSD2008- 00066) and NANOPLAS+ (FIS2012-31070), for financial support.Peer Reviewe

Seismic Emissions from a Highly Impulsive M6.7 Solar Flare

On 10 March 2001 the active region NOAA 9368 produced an unusually impulsive solar flare in close proximity to the solar limb. This flare has previously been studied in great detail, with observations classifying it as a type 1 white-light flare with a very hard spectrum in hard X-rays. The flare was also associated with a type II radio burst and coronal mass ejection. The flare emission characteristics appeared to closely correspond with previous instances of seismic emission from acoustically active flares. Using standard local helioseismic methods, we identified the seismic signatures produced by the flare that, to date, is the least energetic (in soft X-rays) of the flares known to have generated a detectable acoustic transient. Holographic analysis of the flare shows a compact acoustic source strongly correlated with the impulsive hard X-ray, visible continuum, and radio emission. Time-distance diagrams of the seismic waves emanating from the flare region also show faint signatures, mainly in the eastern sector of the active region. The strong spatial coincidence between the seismic source and the impulsive visible continuum emission reinforces the theory that a substantial component of the seismic emission seen is a result of sudden heating of the low photosphere associated with the observed visible continuum emission. Furthermore, the low-altitude magnetic loop structure inferred from potential--field extrapolations in the flaring region suggests that there is a significant inverse correlation between the seismicity of a flare and the height of the magnetic loops that conduct the particle beams from the corona.Comment: 16 pages, 7 figures, Solar Physics Topical Issue: SOHO 19/GONG 2007 "Seismology of Magnetic Activity", Accepte