Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix $A$, for any index $k$, one can define an automorphism associated with $A,$ of the field $\mathbf{Q}(u_1, >..., u_n)$ of rational functions of $n$ independent indeterminates $u_1,..., u_n.$ It is an isomorphism between two cluster algebras associated to the matrix $A$ (see section 4 for precise meaning). When $A$ is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.Comment: revised versio

Trions in a periodic potential

The group-theoretical classification of trion states is presented. It is based on considerations of products of irreducible representations of the 2D translation group. For a given BvK period N degeneracy of obtained states is N^2. Trions consist of two identical particles so the symmetrization of states with respect to particles transposition is considered. Completely antisymmetric states can be constructed by introducing antisymmetric spin functions. Two symmetry adapted bases are considered. The third possibility is postponed for the further investigations.Comment: revtex, 5 p., sub. to Physica

On Combining Lensing Shear Information from Multiple Filters

We consider the possible gain in the measurement of lensing shear from imaging data in multiple filters. Galaxy shapes may differ significantly across filters, so that the same galaxy offers multiple samples of the shear. On the other extreme, if galaxy shapes are identical in different filters, one can combine them to improve the signal-to-noise and thus increase the effective number density of faint, high redshift galaxies. We use the GOODS dataset to test these scenarios by calculating the covariance matrix of galaxy ellipticities in four visual filters (B,V,i,z). We find that galaxy shapes are highly correlated, and estimate the gain in galaxy number density by combining their shapes.Comment: 8 pages, no figures, submitted to JCA

The Evolution of Distorted Rotating Black Holes III: Initial Data

In this paper we study a new family of black hole initial data sets corresponding to distorted ``Kerr'' black holes with moderate rotation parameters, and distorted Schwarzschild black holes with even- and odd-parity radiation. These data sets build on the earlier rotating black holes of Bowen and York and the distorted Brill wave plus black hole data sets. We describe the construction of this large family of rotating black holes. We present a systematic study of important properties of these data sets, such as the size and shape of their apparent horizons, and the maximum amount of radiation that can leave the system during evolution. These data sets should be a very useful starting point for studying the evolution of highly dynamical black holes and can easily be extended to 3D.Comment: 16 page

SECOND-ORDER SYSTEMS WITH SINGULAR MASS MATRIX AND AN EXTENSION OF GUYAN REDUCTION

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57864/1/BhatBernsteinGuyanReductionSIMAX.pd

CFH Y402H Confers similar risk of soft Drusen and both forms of advanced AMD

Journal ArticleBACKGROUND: Age-related macular degeneration (AMD) is the most common cause of irreversible visual impairment in the developed world. The two forms of advanced AMD, geographic atrophy and neovascular AMD, represent different pathological processes in the macula that lead to loss of central vision. Soft drusen, characterized by deposits in the macula without visual loss, are considered to be a precursor of advanced AMD. Recently, it has been proposed that a common missense variant, Y402H, in the Complement Factor H (CFH) gene increases the risk for advanced AMD. However, its impact on soft drusen, GA, or neovascular AMD--or the relationship between them--is unclear. METHODS AND FINDINGS: We genotyped 581 Icelandic patients with advanced AMD (278 neovascular AMD, 203 GA, and 100 with mixed neovascular AMD/GA), and 435 with early AMD (of whom 220 had soft drusen). A second cohort of 431 US patients from Utah, 322 with advanced AMD (244 neovascular AMD and 78 GA) and 109 early-AMD cases with soft drusen, were analyzed. We confirmed that the CFH Y402H variant shows significant association to advanced AMD, with odds ratio of 2.39 in Icelandic patients (p = 5.9 x 10(-12)) and odds ratio of 2.14 in US patients from Utah (p = 2.0 x 10(-9)) with advanced AMD. Furthermore, we show that the Y402H variant confers similar risk of soft drusen and both forms of advanced AMD (GA or neovascular AMD). CONCLUSION: Soft drusen occur prior to progression to advanced AMD and represent a histological feature shared by neovascular AMD and GA. Our results suggest that CFH is a major risk factor of soft drusen, and additional genetic factors and/or environmental factors may be required for progression to advanced AMD

FINITE-TIME STABILITY OF CONTINUOUS AUTONOMOUS SYSTEMS

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57815/1/BhatFiniteTimeStabilitySIAM2000.pd

Tight-binding study of structure and vibrations of amorphous silicon

We present a tight-binding calculation that, for the first time, accurately describes the structural, vibrational and elastic properties of amorphous silicon. We compute the interatomic force constants and find an unphysical feature of the Stillinger-Weber empirical potential that correlates with a much noted error in the radial distribution function associated with that potential. We also find that the intrinsic first peak of the radial distribution function is asymmetric, contrary to usual assumptions made in the analysis of diffraction data. We use our results for the normal mode frequencies and polarization vectors to obtain the zero-point broadening effect on the radial distribution function, enabling us to directly compare theory and a high resolution x-ray diffraction experiment