Geometry of the generalized Bloch sphere for qutrits

The geometry of the generalized Bloch sphere $\Omega_3$, the state space of a qutrit, is studied. Closed form expressions for $\Omega_3$, its boundary $\partial \Omega_3$, and the set of extremals $\Omega_3^{\rm ext}$ are obtained by use of an elementary observation. These expressions and analytic methods are used to classify the 28 two-sections and the 56 three-sections of $\Omega_3$ into unitary equivalence classes, completing the works of earlier authors. It is shown, in particular, that there are families of two-sections and of three-sections which are equivalent geometrically but not unitarily, a feature that does not appear to have been appreciated earlier. A family of three-sections of obese-tetrahedral shape whose symmetry corresponds to the 24-element tetrahedral point group $T_d$ is examined in detail. This symmetry is traced to the natural reduction of the adjoint representation of $SU(3)$, the symmetry underlying $\Omega_3$, into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of $T_d$

Lepton flavour violation in The Little Higgs model

Little Higgs models with T-parity have a new source of lepton flavour violation. In this paper we consider the anomalous magnetic moment of the muon \gmtwo and the lepton flavour violating decays \mutoeg and \tautomug in Little Higgs model with T-parity \cite{Goyal:2006vq}. Our results shows that present experimental constraints of \mutoeg is much more useful to constrain the new sources of flavour violation which are present in T-parity models.Comment: LaTeX file with 13 eps figures (included

Mixed Phase in Compact Starts : M-R relations and radial oscillations

It is believed that quark stars or neutron stars with mixed phase in the core have smaller radii compared to ordinary compact stars. With the recent observation of several low radius objects, typically a radius of $<10 Km.$ for star of mass $< 1M_0$ in low mass X-ray binaries (LMXB), it has become very important to understand the nature of these objects. An accurate determination of mass-radius relationship of these objects provide us with a physical laboratory to study the composition of high density matter and the nature of phase transition. We study the effect of quark and nuclear matter mixed phase on mass radius relationship and radial oscillations of neutron stars. We find that the effect of the mixed phase is to decrease the maximum mass of a stable neutron star and to decrease the radial frequencies .Comment: guest contribution at Int. Workshop on Astronomy & Relativistic Astrophysics (IWARA 03)held at Olinda-PE (Brazil) from Oct. 12-17,200

Smoothed Analysis of Tensor Decompositions

Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely to exist because of hardness results. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error -- a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, our main contribution is in analyzing their stability in the framework of smoothed analysis. Our main technical contribution is to show that tensor products of perturbed vectors are linearly independent in a robust sense (i.e. the associated matrix has singular values that are at least an inverse polynomial). This key result paves the way for applying tensor methods to learning problems in the smoothed setting. In particular, we use it to obtain results for learning multi-view models and mixtures of axis-aligned Gaussians where there are many more "components" than dimensions. The assumption here is that the model is not adversarially chosen, formalized by a perturbation of model parameters. We believe this an appealing way to analyze realistic instances of learning problems, since this framework allows us to overcome many of the usual limitations of using tensor methods.Comment: 32 pages (including appendix

Bulk Viscosity of Magnetized Neutron Star Matter

We study the effect of magnetic field on the bulk viscosity of nuclear matter in neutron stars. We employ the framework of relativistic mean field theory to observe the dense nuclear matter in neutron stars. The effects are first studied for the case when the magnetic field does not exceed the critical value to confine the electrons to the lowest Landau levels. We then consider the case of intense magnetic field to evaluate viscosity for the URCA processes and show that the inequality $p_{F}(e)+p_{F}(p)\geq p_{F}(n)$ is no longer required to be satisfied for the URCA processes to proceed.Comment: Latex 2e file with four postscripts figure

Understanding BL Lac objects Structural & kinematic mode changes in the BL Lac object PKS 0735+178

Context. We present evidence that parsec-scale jets in BL Lac objects may be significantly distinct in kinematics from their counterparts in quasars. We argued this previously for the BL lac sources 1803+784 and 0716+714, report here a similar pattern for another well-known BL Lac object, PKS 0735+178, whose nuclear jet is found to exhibit kinematics atypical of quasars. Aims. A detailed study of the jet components' motion reveals that the standard AGN paradigm of apparent superluminal motion does not always describe the kinematics in BL Lac objects. We study 0735+178 here to augment and improve the understanding of the peculiar motions in the jets of BL Lac objects as a class. Methods. We analyzed 15 GHz VLBA (Very Long Baseline Array) observations (2cm/MOJAVE survey) performed at 23 epochs between 1995.27 and 2008.91. Results. We found a drastic structural mode change in the VLBI jet of 0735+178, between 2000.4 and 2001.8 when its twice sharply bent trajectory turned into a linear shape.We further found that this jet had undergone a similar transition sometime between December 1981 and June 1983. A mode change, occurring in the reverse direction (between mid-1992 and mid-1995) has already been reported in the literature. These structural mode changes are found to be reflected in changed kinematical behavior of the nuclear jet, manifested as an apparent superluminal motion and stationarity of the radio knots. In addition, we found the individual mode changes to correlate in time with the maxima in the optical light curve. The last two transitions occurred before a (modest) radio flare. The behavior of this pc-scale jet appears to favor a scenario involving non-ballistic motions of the radio knots, produced by the precession of a continuous jet within the ambient medium.Comment: Accepted for publication in A&A (Abstract reduced for astro-ph