Omniscopes: Large Area Telescope Arrays with only N log N Computational Cost

We show that the class of antenna layouts for telescope arrays allowing cheap analysis hardware (with correlator cost scaling as N log N rather than N^2 with the number of antennas N) is encouragingly large, including not only previously discussed rectangular grids but also arbitrary hierarchies of such grids, with arbitrary rotations and shears at each level. We show that all correlations for such a 2D array with an n-level hierarchy can be efficiently computed via a Fast Fourier Transform in not 2 but 2n dimensions. This can allow major correlator cost reductions for science applications requiring exquisite sensitivity at widely separated angular scales, for example 21cm tomography (where short baselines are needed to probe the cosmological signal and long baselines are needed for point source removal), helping enable future 21cm experiments with thousands or millions of cheap dipole-like antennas. Such hierarchical grids combine the angular resolution advantage of traditional array layouts with the cost advantage of a rectangular Fast Fourier Transform Telescope. We also describe an algorithm for how a subclass of hierarchical arrays can efficiently use rotation synthesis to produce global sky maps with minimal noise and a well-characterized synthesized beam.Comment: Replaced to match accepted PRD version. 10 pages, 9 fig

Melting and Mixing States of the Earth's Mantle after the Moon-Forming Impact

The Earth's Moon is thought to have formed by an impact between the Earth and an impactor around 4.5 billion years ago. This impact could have been so energetic that it could have mixed and homogenized the Earth's mantle. However, this view appears to be inconsistent with geochemical studies that suggest that the Earth's mantle was not mixed by the impact. Another plausible outcome is that this energetic impact melted the whole mantle, but the extent of mantle melting is not well understood even though it must have had a significant effect on the subsequent evolution of the Earth's interior and atmosphere. To understand the initial state of the Earth's mantle, we perform giant impact simulations using smoothed particle hydrodynamics (SPH) for three different models: (a) standard: a Mars-sized impactor hits the proto-Earth, (b) fast-spinning Earth: a small impactor hits a rapidly rotating proto-Earth, and (c) sub-Earths: two half Earth-sized planets collide. We use two types of equations of state (MgSiO3 liquid and forsterite) to describe the Earth's mantle. We find that the mantle remains unmixed in (a), but it may be mixed in (b) and (c). The extent of mixing is most extensive in (c). Therefore, (a) is most consistent and (c) may be least consistent with the preservation of the mantle heterogeneity, while (b) may fall between. We determine that the Earth's mantle becomes mostly molten by the impact in all of the models. The choice of the equations of state does not affect these outcomes. Additionally, our results indicate that entropy gains of the mantle materials by a giant impact cannot be predicted well by the Rankine-Hugoniot equations. Moreover, we show that the mantle can remain unmixed on a Moon-forming timescale if it does not become mixed by the impact.Comment: Accepted for publication in EPS

Investigation of the Initial State of the Moon-Forming Disk: Bridging SPH Simulations and Hydrostatic Models

According to the standard giant impact hypothesis, the Moon formed from a partially vaporized disk generated by a collision between the proto Earth and a Mars sized impactor. The initial structure of the disk significantly affects the Moon forming process, including the Moons mass, its accretion time scale, and its isotopic similarity to Earth. The dynamics of the impact event determines the initial structure of a nearly hydrostatic Moon forming disk. However, the hydrostatic and hydrodynamic models have been studied separately and their connection has not previously been well quantified. Here, we show the extent to which the properties of the disk can be inferred from Smoothed Particle Hydrodynamic (SPH) simulations. By using entropy, angular momentum and mass distributions of the SPH outputs as approximately conserved quantities, we compute the two dimensional disk structure. We investigate four different models: (a) standard, the canonical giant impact model, (b) fast spinning Earth, a collision between a fast spinning Earth and a small impactor, (c) sub Earths, a collision between two objects with half Earths mass, and (d) intermediate, a collision of two bodies whose mass ratio is 7:3. Our SPH calculations show that the initial disk has approximately uniform entropy. The disks of the fast spinning Earth and sub Earths cases are hotter and more vaporized (80-90% vapor) than the standard case (20%). The intermediate case falls between these values. In the highly vaporized cases, our procedure fails to establish a unique surface density profile of the disk because the disk is unstable according to the Rayleigh criterion. In these cases, we estimate non-unique disk models by conserving global quantities. We also develop a semi analytic model for the thermal structure of the disk, which requires only two inputs: the average entropy and the surface density of the disk.Comment: Accepted for publication in Icaru

Geometric and combinatorial realizations of crystal graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For affine type A, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.Comment: 34 pages, 17 figures; v2: minor typos corrected; v3: corrections to section 8; v4: minor typos correcte

Combinatorial realizations of crystals via torus actions on quiver varieties

Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be realized by elementary combinatorial methods. Here we propose a framework for extracting combinatorial realizations from the geometric picture: We construct certain torus actions on the quiver varieties and use Morse theory to index the irreducible components by connected components of the subvariety of torus fixed points. We then discuss the case of affine sl(n). There the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial models for each highest weight crystal. Applying this construction to the crystal of the fundamental representation recovers a family of combinatorial realizations recently constructed by Fayers. This gives a more conceptual proof of Fayers' result as well as a generalization to higher level. We also discuss a relationship with Nakajima's monomial crystal.Comment: 23 pages, v2: added Section 8 on monomial crystals and some references; v3: many small correction

Are U.S. and Seventh District business cycles alike?

This article explains the recent high levels of residential investment and rates of homeownership.Business cycles

A Theoretical Framework for R-parity Violation

We propose a theoretical framework for R-parity violation. It is realized by a class of Calabi--Yau compactification of Heterotic string theory. Trilinear R-parity violation in superpotential is either absent or negligibly small without an unbroken symmetry, due to a selection rule based on charge counting of a spontaneously broken U(1) symmetry. Although such a selection rule cannot be applied in general to non-renormalizable operators in the low-energy effective superpotential, it is valid for terms trilinear in low-energy degrees of freedom, and hence can be used as a solution to the dimension-4 proton decay problem in the minimal supersymmetric standard model. Bilinear R-parity violation is generated, but there are good reasons why they are small enough to satisfy its upper bounds from neutrino mass and washout of baryon/lepton asymmetry. All R-parity violating dimension-5 operators can be generated. In this theoretical framework, nucleons can decay through squark-exchange diagrams combining dimension-5 and bilinear R-parity violating operators. B-L breaking neutron decay is predicted

Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson Rule

The basic representation of \A is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.Comment: February 1995, 7pages, Using AMS-Te

High-Energy Spin Dynamics in La1.69_{1.69}Sr0.31_{0.31}NiO4_4

We have mapped out the spin dynamics in a stripe-ordered nickelate, La$_{2-x}$Sr$_{x}$NiO$_{4}$ with $x \simeq 0.31$, using inelastic neutron scattering. We observe spin-wave excitations up to 80 meV emerging from the incommensurate magnetic peaks with an almost isotropic spin-velocity: $\hbar c_s\sim 0.32$ eV \AA, very similar to the velocity in the undoped, insulating parent compound, La$_{2}$NiO$_{4}$. We also discuss the similarities and differences of the inferred spin-excitation spectrum with those reported in superconducting high-$T_c$ cuprates.Comment: 4 figure