Artin-Schelter regular algebras of dimension five with two generators

We study and classify Artin-Schelter regular algebras of dimension five with two generators under an additional $\mathbb Z^2$-grading by Hilbert driven Gr\"{o}bner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen-Macaulay. One of the results provides an answer to Fl{\o}ystad-Vatne's question in the context of $\mathbb Z^2$-grading. Our results also achieve a connection between Lyndon words and Artin-Schelter regular algebras.Comment: 32 page

Regularity criterion and classification for algebras of Jordan type

We show that Artin-Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}^r$-graded algebra. We prove that there is exactly one class of four-dimensional Artin-Schelter regular algebras with two generators of degree one in the Jordan case. This class is strongly noetherian, Auslander regular, and Cohen-Macaulay. Their automorphisms and point modules are described.Comment: 24 page

The structure of connected (graded) Hopf algebras

In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic $0$ by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions. The approach to the structure theorem is constructive, based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a surprising consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic $0$ are all iterated Hopf Ore extensions of the base field. In addition, some keystone facts of connected Hopf algebras over a field of characteristic $0$ are observed as corollaries of the structure theorem, without the assumptions of having finite Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base field is algebraically closed.Comment: 25 pages, add more details on some proof

Nakayama automorphisms of twisted tensor products

In this paper, we study homological properties of twisted tensor products of connected graded algebras. We focus on the Ext-algebras of twisted tensor products with a certain form of twisting maps firstly. We show those Ext-algebras are also twisted tensor products, and depict the twisting maps for such Ext-algebras in-depth. With those preparations, we describe Nakayama automorphisms of twisted tensor products of noetherian Artin-Schelter regular algebras

The Black Hole Mass and Magnetic Field Correlation in Active Galactic Nuclei

The observed optical luminosity in 5100 angstorm and black hole mass correlation is used to probe the magnetic field of black holes harbored in active galactic nuclei(AGNs). The model is based on the assumption that the disk is heated by energy injection due to the magnetic coupling(MC) process and the gravitational dissipation due to accretion. The MC process can transfer energy and angular momentum from a rotating Kerr black hole to its surrounding disk. The relation of optical luminosity in 5100 angstorm and black hole mass as functions of the spin and magnetic field of the black hole is modelled. The model predicts that optical luminosity in 5100 angstorm emitted from the disk is sensitive to the strength of the poloidal component of the magnetic field on the BH horizon. Based on the observations of optical luminosity in 5100 angstorm for 143 AGN sources, we obtain the correlation between mass and magnetic field of black hole. And we compared out result with the approximate result between mass and magnetic field of black hole derived from the condition in the standard accretion disc theory.Comment: 7 pages, 4 figures, accepted by Chinese Journal of Astronomy and Astrophysics, 2004 Supplement

Behavior of the Auslander condition with respect to regradings

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen-Macaulay property from the same perspective of that for the Auslander condtion. A key step of our approach is to establish homological relations between a graded ring $R$, its quotient ring modulo the ideal $\hbar R$ and its localization ring with respect to the Ore set $\{\, \hbar^i\, \}_{i\geq0}$, where $\hbar$ is a homogeneous regular normal non-invertible element of $R$.Comment: 23 page

Quantum resonances and decay of a chaotic fractal repeller observed using microwaves

The quantum resonances of classically chaotic n-disk geometries were studied experimentally utilizing thin 2-D microwave geometries. The experiments yield the frequencies and widths of low-lying resonances, which are compared with semiclassical calculations. The longtime or small energy behavior of the wave-vector auto-correlation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. The intermediate energy behavior shows non-universal oscillations determined by periodic orbits.Comment: 5 pages, 3 eps figs include

Skew Calabi-Yau property of normal extensions

We prove that the skew Calabi-Yau property is preserved under normal extension for locally finite positively graded algebras. We also obtain a homological identity which describes the relationship between the Nakayama automorphisms of skew Calabi-Yau locally finite positively graded algebras and their normal extensions. As a preliminary, we show that the Nakayama automorphisms of skew Calabi-Yau algebras always send a regular normal element to a multiple of itself by a unit.Comment: 14 page

Homological Integral of Hopf Algebras

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to study noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.Comment: 30 page

Radial Velocity Studies of Close Binary Stars.V

Radial-velocity measurements and sine-curve fits to the orbital velocity variations are presented for the fifth set of ten close binary systems: V376 And, EL Aqr, EF Boo, DN Cam, FN Cam, V776 Cas, SX Crv, V351 Peg, EQ Tau, KZ Vir. All systems are double-lined spectroscopic contact binaries (KZ Vir may be a low inclination, close, non-contact binary), with seven (all except EL Aqr, SX Crv and EQ Tau) being the recent photometric discoveries of the Hipparcos satellite project. The most interesting object is SX Crv, a contact system with an unprecedently low mass ratio, q=0.066+/-0.003, whose existence challenges the current theory of tidal stability of contact systems. Several of the studied systems are prime candidates for combined light and radial-velocity synthesis solutions.Comment: aastex5.0, 3 figures in PS; submitted to Astron.