4-Holes in point sets

We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft

Compatible 4-Holes in Point Sets

Counting interior-disjoint empty convex polygons in a point set is a typical Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let $P$ be a set of $n$ points in the plane and in general position. A subset $Q$ of $P$, with four points, is called a $4$-hole in $P$ if $Q$ is in convex position and its convex hull does not contain any point of $P$ in its interior. Two 4-holes in $P$ are compatible if their interiors are disjoint. We show that $P$ contains at least $\lfloor 5n/11\rfloor {-} 1$ pairwise compatible 4-holes. This improves the lower bound of $2\lfloor(n-2)/5\rfloor$ which is implied by a result of Sakai and Urrutia (2007).Comment: 17 page

4-Holes in Point Sets

Abstract We consider a variant of a question of Erdős on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be nonconvex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes

Evolution of Neel order and localized spin moment in the doped two-dimensional Hubbard model

We investigate effects of doped holes' hopping on Neel order in the two-dimensional Hubbard model. Semiclassical staggered moments are computed by solving saddle point equations derived from a path-integral formalism. Effects of quantum fluctuations are taken into account by the Schwinger boson mean field theory. We argue that hopping of doped holes is ineffective in suppressing Neel order compared to rapid supprestion of Neel order in high-temperature superconductors. After destruction of Neel order, the quantum disordered phase sets in. Taking the strong coupling limit in the quantum disordered phase leads to a model of spinless fermions and bosons but no gauge field interaction.Comment: 6 pages, 4 figure

Moduli Stabilization, Large-Volume dS Minimum Without anti-D3-Branes, (Non-)Supersymmetric Black Hole Attractors and Two-Parameter Swiss Cheese Calabi-Yau's

We consider issues of moduli stabilization and "area codes" for type II flux compactifications, and the "Inverse Problem" and "Fake Superpotentials" for extremal (non)supersymmetric black holes in type II compactifications on (orientifold of) a compact two-parameter Calabi-Yau expressed as a degree-18 hypersurface in WCP^4[1,1,1,6,9] which has multiple singular loci in its moduli space. We argue the existence of extended "area codes" [1] wherein for the same set of large NS-NS and RR fluxes, one can stabilize all the complex structure moduli and the axion-dilaton modulus (to different sets of values) for points in the moduli space away as well as near the different singular conifold loci leading to the existence of domain walls. Using techniques of [3] we explicitly show that given a set of moduli and choice of a gauge(the superpotential) corresponding to an extremal black hole, one can actually work out the corresponding charges (of the extremal black hole) - the so-called "inverse problem". We also show the existence of "fake superpotentials" [4] corresponding to non-BPS extremal black-hole solutions corresponding to the aforementioned Calabi-Yau three-fold. By including non-perturbative alpha' and instanton corrections in the Kaehler potential and superpotential [2], we show the possibility of getting a large-volume non-supersymmetric (A)dS minimum - a dS minimum without the addition of anti-D3 branes a la KKLT. The chosen Calabi-Yau has been of relevance also from the point of other studies of stabilization of the Kaehler moduli via nonperturbative instanton contributions [5] and the possibility of getting non-supersymmetric AdS vacua (and their subsequent dS-uplifts) using (alpha')^3 corrections to the Kaehler potential [6,7,8].Comment: 1+32 pages, LaTeX; published (in NPB) version - title changed and some cosmetic changes made on the reviewer's suggestion

SDSS J092712.64+294344.0: recoiling black hole or merging galaxies?

We report long-slit spectroscopic observations of SDSS J092712+294344 carried-out at the recently commissioned 2m telescope in IUCAA Girawali Observatory, India. This AGN-like source is known to feature three sets of emission lines at zem = 0.6972, 0.7020 and 0.7128. Different scenarios such as a recoiling black hole after asymmetric emission of gravitational waves, binary black holes and possible merging systems are proposed for this object. We test these scenarios by comparing our spectra with that from the Sloan Digital Sky Survey (SDSS), obtained 4 years prior to our observations. Comparing the redshifts of [OIII]4960,5008 we put a 3 sigma limit on the relative acceleration to be less than 32 km s^-1 yr^-1 between different emitting regions. Using the 2D spectra obtained at different position angles we show that the [OIII]5008 line from the zem = 0.7128 component is extended beyond the spectral point spread function. We infer the linear extent of this line emitting region is ~8 kpc. We also find a tentative evidence for an offset between the centroid of the [OIII]5008 line at zem = 0.7128 and the QSO trace when the slit is aligned at a position angle of 299 degrees. This corresponds to the zem = 0.7128 system being at an impact parameter of ~1 kpc with respect to the zem = 0.6972 in the north west direction. Based on our observations we conclude that the binary black hole model is most unlikely. The spatial extent and the sizes are consistent with both black hole recoil and merging scenarios.Comment: accepted for publication in MNRAS Letter

4D Localization in Randall-Sundrum 2 Supergravity and in Vasiliev Theories

We discuss the problem of localization of 4D massless states in Randall-Sundrum 2 (one-brane) models. A Randall-Sundrum 2 construction starting from N=8 gauged supergravity in 5D Anti de Sitter space gives rise to an N=4 supergravity-matter system. We explicitly show that only the modes of the N=4 graviton supermultiplet localize on the 4D brane, streamlining and generalizing previous works. We also point out that while charged 1/4 BPS black holes do exist in the 4D theory, they are always produced in sets of total charge zero. This zero-charge configuration uplifts to a 5D metric without naked singularities, thus avoiding the curvature singularity of the 5D uplift of an isolated charged BPS black hole. Finally, we resolve a puzzle with localization of massless high spin fields on a (putative) Randall-Sundrum 2 construction based on Vasiliev's high spin theories. We show that while high spin fields do localize, the gauge symmetry that ensures decoupling of their unphysical polarizations is anomalous. This implies that the high spin fields must acquire a mass.Comment: 15 pages, one reference added. To appear in Physics Letters