Multiple sampling and interpolation in the classical Fock space

We study multiple sampling, interpolation and uniqueness for the classical Fock space in the case of unbounded mul-tiplicities

Discovery of a Proto-cluster Associated with a Ly-α\alpha Blob Pair at z=2.3

Bright Ly-$\alpha$ blobs (LABs) --- extended nebulae with sizes of $\sim$100kpc and Ly-$\alpha$ luminosities of $\sim$10$^{44}$erg s$^{-1}$ --- often reside in overdensities of compact Ly-$\alpha$ emitters (LAEs) that may be galaxy protoclusters. The number density, variance, and internal kinematics of LABs suggest that they themselves trace group-like halos. Here we test this hierarchical picture, presenting deep, wide-field Ly-$\alpha$ narrowband imaging of a 1$^\circ$ $\times$ 0.5$^\circ$ region around a LAB pair at $z$ = 2.3 discovered previously by a blind survey. We find 183 Ly-$\alpha$ emitters, including the original LAB pair and three new LABs with Ly-$\alpha$ luminosities of (0.9--1.3)$\times$10$^{43}$erg s$^{-1}$ and isophotal areas of 16--24 arcsec$^2$. Using the LAEs as tracers and a new kernel density estimation method, we discover a large-scale overdensity (Bo{\"o}tes J1430+3522) with a surface density contrast of $\delta_{\Sigma}$ = 2.7, a volume density contrast of $\delta$ $\sim$ 10.4, and a projected diameter of $\approx$ 20 comoving Mpc. Comparing with cosmological simulations, we conclude that this LAE overdensity will evolve into a present-day Coma-like cluster with $\log{(M/M_\odot)}$ $\sim$ $15.1\pm0.2$. In this and three other wide-field LAE surveys re-analyzed here, the extents and peak amplitudes of the largest LAE overdensities are similar, not increasing with survey size, implying that they were indeed the largest structures then and do evolve into rich clusters today. Intriguingly, LABs favor the outskirts of the densest LAE concentrations, i.e., intermediate LAE overdensities of $\delta_\Sigma = 1 - 2$. We speculate that these LABs mark infalling proto-groups being accreted by the more massive protocluster

The Hirsch conjecture holds for normal flag complexes

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus the dimension, as in the Hirsch conjecture. This proves the Hirsch conjecture for all flag polytopes, and more generally, for all (connected) flag homology manifolds.Comment: 9 pages, 1 figure; to appear in Mathematics of Operations Researc

The art of finding the optimal scattering center(s)

The efficient use of a multipole expansion of the far field for rapid numerical modeling and optimization of the optical response from ordered and disordered arrays of various structural elements is complicated by the ambiguity in choosing the ultimate expansion centers for individual scatterers. Since the multipolar decomposition depends on the position of the expansion center, the sets of multipoles are not unique. They may require constrained optimization to get the compact and most efficient spatial spectrum for each scatterer. We address this problem by finding {\em the optimal scattering centers} for which the spatial multipolar spectra become unique. We separately derive these optimal positions for the electric and magnetic parts by minimizing the norm of the poloidal electric and magnetic quadrupoles. Employing the long-wave approximation (LWA) ansatz, we verify the approach with the theoretical discrete models and realistic scatterers. We show that the optimal electric and magnetic scattering centers, in all cases, are not co-local with the centers of mass. The optimal multipoles, including the toroidal terms, are calculated for several structurally distinct scattering cases, and their utility for low-cost numerical schemes, including the generalized T-matrix approach, is discussed. Expansion of the work beyond the LWA is possible, with a promise for faster and universal numerical schemes