Spinor Bose Condensates in Optical Traps

In an optical trap, the ground state of spin-1 Bosons such as $^{23}$Na, $^{39}$K, and $^{87}$Rb can be either a ferromagnetic or a "polar" state, depending on the scattering lengths in different angular momentum channel. The collective modes of these states have very different spin character and spatial distributions. While ordinary vortices are stable in the polar state, only those with unit circulation are stable in the ferromagnetic state. The ferromagnetic state also has coreless (or Skyrmion) vortices like those of superfluid $^{3}$He-A. Current estimates of scattering lengths suggest that the ground states of $^{23}$Na and $^{87}$Rb condensate are a polar state and a ferromagnetic state respectively.Comment: 11 pages, no figures. email : [email protected]

Hierarchical interpolative factorization for elliptic operators: differential equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF-DE is based on the multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB codes are freely available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math. arXiv admin note: substantial text overlap with arXiv:1307.266

Hierarchical interpolative factorization for elliptic operators: integral equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF-IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher-dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF-IE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat

Boson Mott insulators at finite temperatures

We discuss the finite temperature properties of ultracold bosons in optical lattices in the presence of an additional, smoothly varying potential, as in current experiments. Three regimes emerge in the phase diagram: a low-temperature Mott regime similar to the zero-temperature quantum phase, an intermediate regime where MI features persist, but where superfluidity is absent, and a thermal regime where features of the Mott insulator state have disappeared. We obtain the thermodynamic functions of the Mott phase in the latter cases. The results are used to estimate the temperatures achieved by adiabatic loading in current experiments. We point out the crucial role of the trapping potential in determining the final temperature, and suggest a scheme for further cooling by adiabatic decompression

A fast semi-direct least squares algorithm for hierarchically block separable matrices

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an $M \times N$ HBS matrix with $M \geq N$ having bounded off-diagonal block rank, the algorithm has optimal $\mathcal{O} (M + N)$ complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes $\mathcal{O} (M + N)$ in 1D, $\mathcal{O} (M + N^{3/2})$ in 2D, and $\mathcal{O} (M + N^{2})$ in 3D. We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App

Two-dimensional gases of generalized statistics in a uniform magnetic field

We study the low temperature properties of two-dimensional ideal gases of generalized statistics in a uniform magnetic field. The generalized statistics considered here are the parafermion statistics and the exclusion statistics. Similarity in the behaviours of the parafermion gas of finite order $p$ and the gas with exclusion coefficient $g=1/p$ at very low temperatures is noted. These two systems become exactly equivalent at $T=0$. Qumtum Hall effect with these particles as charge carriers is briefly discussed.Comment: Latex file, 14 pages, 5 figures available on reques

Stellar Velocity Dispersion and Black Hole Mass in the Blazar Markarian 501

The recently discovered correlation between black hole mass and stellar velocity dispersion provides a new method to determine the masses of black holes in active galaxies. We have obtained optical spectra of Markarian 501, a nearby gamma-ray blazar with emission extending to TeV energies. The stellar velocity dispersion of the host galaxy, measured from the calcium triplet lines in a 2"x3.7" aperture, is 372 +/- 18 km/s. If Mrk 501 follows the M-sigma correlation defined for local galaxies, then its central black hole has a mass of (0.9-3.4)x10^9 solar masses. This is significantly larger than some previous estimates for the central mass in Mrk 501 that have been based on models for its nonthermal emission. The host galaxy luminosity implies a black hole of 6x10^8 solar masses, but this is not in severe conflict with the mass derived from the M-sigma relation because the M_BH-L_bulge correlation has a large intrinsic scatter. Using the emission-line luminosity to estimate the bolometric luminosity of the central engine, we find that Mrk 501 radiates at an extremely sub-Eddington level of L/L_Edd ~ 10^-4. Further applications of the M-sigma relation to radio-loud active galactic nuclei may be useful for interpreting unified models and understanding the relationship between radio galaxies and BL Lac objects.Comment: To appear in ApJ Letters. 5 pages, 2 figure