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

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

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

Home equity withdrawal in retirement

The authors study empirically and theoretically the patterns of home equity withdrawal among retirees, using a model in which retirees are able to own or rent a home, save, and borrow against home equity, in the face of idiosyncratic risks concerning mortality, health, medical expenditures, and household size and observed house price changes. The estimated model is found to successfully replicate the patterns of homeownership and the saving/borrowing decisions of retirees. They use the estimated model for several counterfactual experiments. There are three main findings. First, the model predicts that a house price boom suppresses homeownership and increases borrowing, while a decline in house prices has the opposite effect. Second, the costs of home equity borrowing restrict the borrowing of retirees, and thus a reduction of such costs (e.g., lower costs of reverse mortgage loans) might significantly raise home equity borrowing. Third, there are two implications for the retirement saving puzzle. Although the cost of borrowing against equity in the house affects the borrowing of retirees, it does not affect total asset holding, implying that equity borrowing costs do not seem to offer a quantitatively significant contribution to resolving the retirement saving puzzle. On the other hand, the magnitude of the retirement saving puzzle might be exaggerated, because a sizable part of "retirement saving" is due to house price appreciation.Home equity loans ; Retirement

The Refined Topological Vertex

We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the self-dual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a two-parameter (equivariant) instanton expansion of gauge theories which reproduce the results of Nekrasov. The refined vertex is also expected to be related to Khovanov knot invariants.Comment: 70 Pages, 23 Figure