Selection Lemmas for various geometric objects

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set $P$. This question has been widely explored for simplices in $\mathbb{R}^d$, with tight bounds in $\mathbb{R}^2$. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from $P$. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in $\mathbb{R}^d$. In the second selection lemma, we consider an arbitrary $m$ sized subset of the set of all objects induced by $P$. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in $\frac{m^3}{24n^4}$ rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when $m$ is almost quadratic

Periodic-Orbit Theory of Universality in Quantum Chaos

We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K(\tau)$ as power series in the time $\tau$. Each term $\tau^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.Comment: 31 pages, 17 figure

3D simulations of the early stages of AGN jets: geometry, thermodynamics and backflow

We investigate the interplay between jets from Active Galactic Nuclei (AGNs) and the surrounding InterStellar Medium (ISM) through full 3D, high resolution, Adaptive Mesh Refinement simulations performed with the FLASH code. We follow the jet- ISM system for several Myr in its transition from an early, compact source to an extended one including a large cocoon. During the jet evolution, we identify three major evolutionary stages and we find that, contrary to the prediction of popular theoretical models, none of the simulations shows a self-similar behavior. We also follow the evolution of the energy budget, and find that the fraction of input power deposited into the ISM (the AGN coupling constant) is of order of a few percent during the first few Myr. This is in broad agreement with galaxy formation models employing AGN feedback. However, we find that in these early stages, this energy is deposited only in a small fraction (< 1%) of the total ISM volume. Finally we demonstrate the relevance of backflows arising within the extended cocoon generated by a relativistic AGN jet within the ISM of its host galaxy, previously proposed as a mechanism for self-regulating the gas accretion onto the central object. These backflows tend later to be destabilized by the 3D dynamics, rather than by hydrodynamic (Kelvin- Helmholtz) instabilities. Yet, in the first few hundred thousand years, backflows may create a central accretion region of significant extent, and convey there as much as a few millions of solar masses.Comment: Accepted in MNRAS - 16 pages, 12 figures - Multimedia available on the author's webpage: http://www.mpia.de/~ciel

From a (p,2)-Theorem to a Tight (p,q)-Theorem

A family F of sets is said to satisfy the (p,q)-property if among any p sets of F some q have a non-empty intersection. The celebrated (p,q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in R^d that satisfies the (p,q)-property for some q >= d+1, can be pierced by a fixed number (independent on the size of the family) f_d(p,q) of points. The minimum such piercing number is denoted by {HD}_d(p,q). Already in 1957, Hadwiger and Debrunner showed that whenever q > (d-1)/d p+1 the piercing number is {HD}_d(p,q)=p-q+1; no exact values of {HD}_d(p,q) were found ever since. While for an arbitrary family of compact convex sets in R^d, d >= 2, a (p,2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner (1965) and (independently) Dol\u27nikov (1972) used a (p,2)-theorem for axis-parallel rectangles to show that {HD}_{rect}(p,q)=p-q+1 holds for all q>sqrt{2p}. These are the only values of q for which {HD}_{rect}(p,q) is known exactly. In this paper we present a general method which allows using a (p,2)-theorem as a bootstrapping to obtain a tight (p,q)-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we show that {HD}_{d-box}(p,q)=p-q+1 holds for all q > c\u27 log^{d-1} p, and in particular, {HD}_{rect}(p,q)=p-q+1 holds for all q >= 7 log_2 p (compared to q >= sqrt{2p}, obtained by Wegner and Dol\u27nikov more than 40 years ago). In addition, for several classes of families, we present improved (p,2)-theorems, some of which can be used as a bootstrapping to obtain tight (p,q)-theorems. In particular, we show that any family F of compact convex sets in R^d with Helly number 2 admits a (p,2)-theorem with piercing number O(p^{2d-1}), and thus, satisfies {HD}_{F}(p,q)=p-q+1 for all q>cp^{1-1/(2d-1)}, for a universal constant c

The Ocean Observatories Initiative

Author Posting. © The Oceanography Society, 2018. This article is posted here by permission of The Oceanography Society for personal use, not for redistribution. The definitive version was published in Oceanography 31, no. 1 (2018): 16–35, doi:10.5670/oceanog.2018.105.The Ocean Observatories Initiative (OOI) is an integrated suite of instrumented platforms and discrete instruments that measure physical, chemical, geological, and biological properties from the seafloor to the sea surface. The OOI provides data to address large-scale scientific challenges such as coastal ocean dynamics, climate and ecosystem health, the global carbon cycle, and linkages among seafloor volcanism and life. The OOI Cyberinfrastructure currently serves over 250 terabytes of data from the arrays. These data are freely available to users worldwide, changing the way scientists and the broader community interact with the ocean, and permitting ocean research and inquiry at scales of centimeters to kilometers and seconds to decades.Funding for the OOI is provided by the National Science Foundation through a Cooperative Support Agreement with the Consortium for Ocean Leadership (OCE-1026342)