Quantum and Braided Linear Algebra

Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail, distinguishing between two forms of this algebra, $V(R)$ (vectors) and $V^*(R)$ (covectors). A(R)\to V(R_{21})\tens V^*(R) is an algebra homomorphism (i.e. quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. We show that if $V(R)$ and $V^*(R)$ are endowed with the necessary braid statistics $\Psi$ then their braided tensor-product V(R)\und\tens V^*(R) is a realization of the braided matrices $B(R)$ introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from $B(R)$ act on themselves by conjugation in a way impossible for the quantum groups obtained from $A(R)$.Comment: 27 page

The Determination Of Reddening From Intrinsic VR Colors Of RR Lyrae Stars

New R-band observations of 21 local field RR Lyrae variable stars are used to explore the reliability of minimum light (V-R) colors as a tool for measuring interstellar reddening. For each star, R-band intensity mean magnitudes and light amplitudes are presented. Corresponding V-band light curves from the literature are supplemented with the new photometry, and (V-R) colors at minimum light are determined for a subset of these stars as well as for other stars in the literature. Two different definitions of minimum light color are examined, one which uses a Fourier decomposition to the V and R light curves to find (V-R) at minimum V-band light, (V-R)_{min}^F, and the other which uses the average color between the phase interval 0.5-0.8, (V-R)_{min}^{\phi(0.5-0.8)}. From 31 stars with a wide range of metallicities and pulsation periods, the mean dereddened RR Lyrae color at minimum light is (V-R)_{min,0}^F = 0.28 pm 0.02 mag and (V-R)_{min,0}^{\phi(0.5-0.8)} = 0.27 pm 0.02 mag. As was found by Guldenschuh et al. (2005) using (V-I) colors, any dependence of the star's minimum light color on metallicity or pulsation amplitude is too weak to be formally detected. We find that the intrinsic (V-R) of Galactic bulge RR Lyrae stars are similar to those found by their local counterparts and hence that Bulge RR0 Lyrae stars do not have anomalous colors as compared to the local RR Lyrae stars.Comment: accepted by A

Lower limit in semiclassical form for the number of bound states in a central potential

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}{\pi}\int_0^\infty dr\sqrt{-V(r)\theta[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge {{N^{\text{(semi)}}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\ell$-wave potential $V_\ell^{\text{(eff)}}(r)=V(r)+\frac{\ell(\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.Comment: 9 page

Energy Extraction and Particle Acceleration Around Rotating Black Hole in Horava-Lifshitz Gravity

Penrose process on rotational energy extraction of the black hole (BH) in the original non-projectable Ho\v{r}ava-Lifshitz gravity is studied. The strong dependence of the extracted energy from the special range of parameters of the Ho\v{r}ava-Lifshitz gravity, such as parameter $\Lambda_W$ and specific angular momentum $a$ has been found. Particle acceleration near the rotating BH in Ho\v{r}ava-Lifshitz gravity has been studied. It is shown that the fundamental parameter of the Ho\v{r}ava-Lifshitz gravity can impose limitation on the the energy of the accelerating particles preventing them from the infinite value.Comment: 6 pages, 3 figures, accepted for publication in Physical Review

Klein-Gordon lower bound to the semirelativistic ground-state energy

For the class of attractive potentials V(r) <= 0 which vanish at infinity, we prove that the ground-state energy E of the semirelativistic Hamiltonian H = \sqrt{m^2 + p^2} + V(r) is bounded below by the ground-state energy e of the corresponding Klein--Gordon problem (p^2 + m^2)\phi = (V(r) -e)^2\phi. Detailed results are presented for the exponential and Woods--Saxon potentials.Comment: 7 pages, 4 figure

Embedding variables in finite dimensional models

Global problems associated with the transformation from the Arnowitt, Deser and Misner (ADM) to the Kucha\v{r} variables are studied. Two models are considered: The Friedmann cosmology with scalar matter and the torus sector of the 2+1 gravity. For the Friedmann model, the transformations to the Kucha\v{r} description corresponding to three different popular time coordinates are shown to exist on the whole ADM phase space, which becomes a proper subset of the Kucha\v{r} phase spaces. The 2+1 gravity model is shown to admit a description by embedding variables everywhere, even at the points with additional symmetry. The transformation from the Kucha\v{r} to the ADM description is, however, many-to-one there, and so the two descriptions are inequivalent for this model, too. The most interesting result is that the new constraint surface is free from the conical singularity and the new dynamical equations are linearization stable. However, some residual pathology persists in the Kucha\v{r} description.Comment: Latex 2e, 29 pages, no figure

Quasinormal modes of a black hole in the deformed Ho\v{r}ava-Lifshitz gravity

We study the quasinormal modes of the massless scalar perturbation in the background of a deformed black hole in the Ho\v{r}ava-Lifshitz gravity with coupling constant $\lambda=1$. Our results show that the quasinormal frequencies depend on the parameter in the Ho\v{r}ava-Lifshitz gravity and the behavior of the quasinormal modes is different from those in the Reissner-Norstr\"{om} and Einstein-Born-Infeld black hole spacetimes. The absolute value of imaginary parts is smaller and the scalar perturbations decay more slowly in the deformed Ho\v{r}ava-Lifshitz black hole spacetime. This information can help us understand more about the Ho\v{r}ava-Lifshitz gravity.Comment: 9 pages, 3 figures and 4 table