Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials

We present four types of infinitely many exactly solvable Fokker-Planck equations, which are related to the newly discovered exceptional orthogonal polynomials. They represent the deformed versions of the Rayleigh process and the Jacobi process.Comment: 17 pages, 4 figure

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

Condition for tripartite entanglement

We propose a scheme for classifying the entanglement of a tripartite pure qubit state. This classification scheme consists of an ordered list of seven elements. These elements are the Cayley hyper-determinant, and its six associated $2 \times 2$ subdeterminants. In particular we show that this classification provides a necessary and sufficient condition for separability.Comment: 8 pages, to appear in the Proceedings of "Quantum Theory and Symmetries 7", Prague, Aug 7-13, 201

Multiple Chern-Simons Fields on a Torus

Intertwined multiple Chern-Simons gauge fields induce matrix statistics among particles. We analyse this theory on a torus, focusing on the vacuum structure and the Hilbert space. The theory can be mimicked, although not completely, by an effective theory with one Chern-Simons gauge field. The correspondence between the Wilson line integrals, vacuum degeneracy and wave functions for these two theories are discussed. Further, it is obtained in both of these cases that the two total momenta and Hamiltonian commute only in the physical Hilbert space.Comment: 20 pages, UMN-TH-1128/93, plain Te

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

Reply to Hagen & Sudarshan's Comment

We show that the argument in Phys Rev Lett 70 (1993) 1360 is correct and consistent, and that Hagen & Sudarshan's solution has inconsistency leading to non-vanishing commutators of $[P^1, P^2]$ and $[P^j, H]$ even in physical states. This proves that many of HS's statements in their Comment are based merely on incorrect guess, but not on careful algebra.Comment: one page, UMN-TH-1245/9