(p,q)-Rogers-Szego polynomial and the (p,q)-oscillator

A (p,q)-analogue of the classical Rogers-Szego polynomial is defined by replacing the q-binomial coefficient in it by the (p,q)-binomial coefficient. Exactly like the Rogers-Szego polynomial is associated with the q-oscillator algebra it is found that the (p,q)-Rogers-Szego polynomial is associated with the (p,q)-oscillator algebra.Comment: 13 pages; Dedicated to the memory of Prof. Alladi Ramakrishnan; To appear in: "The Legacy of Alladi Ramakrishnan in the Mathematical Sciences'' (K. Alladi, J. Klauder, C. R. Rao, Editors) Springer, 201

A Diagonal Representation of Quantum Density Matrix Using q-Boson Oscillator Coherent States

A q-analogue of Sudarshan's diagonal representation of the Quantum Mechanical density matrix is obtained using q-boson coherent states. Earlier result of Mehta and Sudarshan on the self reproducing property of rho(z',z) is also generalized and a self-consistent self-reproducing kernel {K-tilde}(z',z) is constructed.Comment: 9 Pages, Latex versio

On the q-analogues of the Zassenhaus formula for dientangling exponential operators

Katriel, Rasetti and Solomon introduced a $q$-analogue of the Zassenhaus formula written as $e_q^{(A+B)}$ $=$ $e_q^Ae_q^Be_q^{c_2}e_q^{c_3}e_q^{c_4}e_q^{c_5}...$, where $A$ and $B$ are two generally noncommuting operators and $e_q^z$ is the Jackson $q$-exponential, and derived the expressions for $c_2$, $c_3$ and $c_4$. It is shown that one can also write $e_q^{(A+B)}$ $=$ e_q^Ae_q^Be_{q^2}^{\C_2}e_{q^3}^{\C_3}e_{q^4}^{\C_4}e_{q^5}^{\C_5}.... Explicit expressions for \C_2, \C_3 and \C_4 are given.Comment: 12 Pages. New references have been added. Title and Abstract have been modified in view of an earlier work of Katriel, Rasetti and Solomon on a different form of the q-Zassenhaus formula. The text is modified only slightly since the result of the paper is unchange

A statistical formulation of the estimation and the combined identification and control problems. Worst case initial condition Fifth quarterly report

Statistical formulation of estimation and combined identification and control problems of nonlinear dynamical system

Light beams with general direction and polarization: global description and geometric phase

We construct the manifold describing the family of plane monochromatic light waves with all directions, polarizations, phases and intensities. A smooth description of polarization, valid over the entire sphere S^2 of directions, is given through the construction of an orthogonal basis pair of complex polarization vectors for each direction; any light beam is then uniquely and smoothly specified by giving its direction and two complex amplitudes. This implies that the space of all light beams is the six dimensional manifold S^2 X C^2, the Cartesian product of a sphere and a two dimensional complex vector space. A Hopf map (i.e mapping the two complex amplitudes to the Stokes parameters) then leads to the four dimensional manifold S^2 X S^2 which describes beams with all directions and polarization states. This product of two spheres can be viewed as an ordered pair of two points on a single sphere, in contrast to earlier work in which the same system was represented using Majorana's mapping of the states of a spin one quantum system to an unordered pair of points on a sphere. This is a different manifold, CP^2, two dimensional complex projective space, which does not faithfully represent the full space of all directions and polarizations. Following the now-standard framework, we exhibit the fibre bundle whose total space is the set of all light beams of non-zero intensity, and base space S^2 X S^2. We give the U(1) connection which determines the geometric phase as the line integral of a one-form along a closed curve in the total space. Bases are classified as globally smooth, global but singular, and local, with the last type of basis being defined only when the curve traversed by the system is given. Existing as well as new formulae for the geometric phase are presented in this overall framework.Comment: 19 pages; submitted to Journal of Physics

Stellar Dynamics around a Massive Black Hole II: Resonant Relaxation

We present a first-principles theory of Resonant Relaxation (RR) of a low mass stellar system orbiting a more massive black hole (MBH). We first extend the kinetic theory of Gilbert (1968) to include the Keplerian field of a black hole of mass $M_\bullet$. Specializing to a Keplerian stellar system of mass $M \ll M_\bullet$, we use the orbit-averaging method of Sridhar & Touma (2015; Paper I) to derive a kinetic equation for RR. This describes the collisional evolution of a system of $N \gg 1$ Gaussian Rings in a reduced 5-dim space, under the combined actions of self-gravity, 1 PN and 1.5 PN relativistic effects of the MBH and an arbitrary external potential. In general geometries RR is driven by both apsidal and nodal resonances, so the distinction between scalar-RR and vector-RR disappears. The system passes through a sequence of quasi-steady secular collisionless equilibria, driven by irreversible 2-Ring correlations that accrue through gravitational interactions, both direct and collective. This correlation function is related to a `wake function', which is the linear response of the system to the perturbation of a chosen Ring. The wake function is easier to appreciate, and satisfies a simpler equation, than the correlation function. We discuss general implications for the interplay of secular dynamics and non-equilibrium statistical mechanics in the evolution of Keplerian stellar systems toward secular thermodynamic equilibria, and set the stage for applications to the RR of axisymmetric discs in Paper III.Comment: Accepted for publication in MNRAS (final version

Migration into a Companion's Trap: Disruption of Multiplanet Systems in Binaries

Most exoplanetary systems in binary stars are of S--type, and consist of one or more planets orbiting a primary star with a wide binary stellar companion. Gravitational forcing of a single planet by a sufficiently inclined binary orbit can induce large amplitude oscillations of the planet's eccentricity and inclination through the Kozai-Lidov (KL) instability. KL cycling was invoked to explain: the large eccentricities of planetary orbits; the family of close--in hot Jupiters; and the retrograde planetary orbits in eccentric binary systems. However, several kinds of perturbations can quench the KL instability, by inducing fast periapse precessions which stabilize circular orbits of all inclinations: these could be a Jupiter--mass planet, a massive remnant disc or general relativistic precession. Indeed, mutual gravitational perturbations in multiplanet S--type systems can be strong enough to lend a certain dynamical rigidity to their orbital planes. Here we present a new and faster process that is driven by this very agent inhibiting KL cycling. Planetary perturbations enable secular oscillations of planetary eccentricities and inclinations, also called Laplace--Lagrange (LL) eigenmodes. Interactions with a remnant disc of planetesimals can make planets migrate, causing a drift of LL mode periods which can bring one or more LL modes into resonance with binary orbital motion. The results can be dramatic, ranging from excitation of large eccentricities and mutual inclinations to total disruption. Not requiring special physical or initial conditions, binary resonant driving is generic and could have profoundly altered the architecture of many S--type multiplanet systems. It can also weaken the multiplanet occurrence rate in wide binaries, and affect planet formation in close binaries.Comment: The published version of the paper in compliance with Nature's embargo policy is available at http://nature.com/articles/doi:10.1038/nature1487

Stellar Dynamics around a Massive Black Hole III: Resonant Relaxation of Axisymmetric Discs

We study the Resonant Relaxation (RR) of an axisymmetric low mass (or Keplerian) stellar disc orbiting a more massive black hole (MBH). Our recent work on the general kinetic theory of RR is simplified in the standard manner by ignoring the effects of `gravitational polarization', and applied to a zero-thickness, flat, axisymmetric disc. The wake of a stellar orbit is expressed in terms of the angular momenta exchanged with other orbits, and used to derive a kinetic equation for RR under the combined actions of self-gravity, 1 PN and 1.5 PN relativistic effects of the MBH and an arbitrary external axisymmetric potential. This is a Fokker-Planck equation for the stellar distribution function (DF), wherein the diffusion coefficients are given self-consistently in terms of contributions from apsidal resonances between pairs of stellar orbits. The physical kinetics is studied for the two main cases of interest. (1) `Lossless' discs in which the MBH is not a sink of stars, and disc mass, angular momentum and energy are conserved: we prove that general H-functions can increase or decrease during RR, but the Boltzmann entropy is (essentially) unique in being a non-decreasing function of time. Therefore secular thermal equilibria are maximum entropy states, with DFs of the Boltzmann form; the two-Ring correlation function at equilibrium is computed. (2) Discs that lose stars to the MBH through an `empty loss-cone': we derive expressions for the MBH feeding rates of mass, angular momentum and energy in terms of the diffusive flux at the loss-cone boundary.Comment: Submitted to MNRAS; 28 preprint pages, 3 figure

Stellar Dynamics around a Massive Black Hole I: Secular Collisionless Theory

We present a theory in three parts, of the secular dynamics of a (Keplerian) stellar system of mass $M$ orbiting a black hole of mass $M_\bullet \gg M$. Here we describe the collisionless dynamics; Papers II and III are on the (collisional) theory of Resonant Relaxation. The mass ratio, $\varepsilon = M/M_\bullet \ll 1$, is a natural small parameter implying a separation of time scales between the short Kepler orbital periods and the longer orbital precessional periods. The collisionless Boltzmann equation (CBE) for the stellar distribution function (DF) is averaged over the fast Kepler orbital phase using the method of multiple scales. The orbit-averaged system is described by a secular DF, $F$, in a reduced phase space. $F$ obeys a secular CBE that includes stellar self-gravity, general relativistic corrections up to 1.5 post-Newtonian order, and external sources varying over secular times. Secular dynamics, even with general time dependence, conserves the semi-major axis of every star. This additional integral of motion promotes extra regularity of the stellar orbits, and enables the construction of equilibria, $F_0$, through a secular Jeans theorem. A linearized secular CBE determines the response and stability of $F_0$. Spherical, non-rotating equilibria may support long-lived, warp-like distortions. We also prove that an axisymmetric, zero-thickness, flat disc is secularly stable to all in-plane perturbations, when its DF, $F_0$, is a monotonic function of the angular momentum at fixed energy.Comment: Accepted for publication in MNRAS (final version

E-Speed Governors For Public Transport Vehicles

An accident is unexpected, unusual, unintended and identifiable external event which occurs at any place and at any time. The major concern faced by the government and traffic officials is over speeding at limited speed zones like hospitals, schools or residential places leading to causalities and more deaths on the roads. Hence the speed of the vehicles is to be regulated and confined to the limits as prescribed by the traffic regulations. In this paper we propose a solution in the form of providing E-speed governor fitted with a wireless communication system consisting of a Rx which receives the information regarding the speed regulation for their zones. The TX will be made highly intelligent and decide when receiver should be made active to regulate the speed and unwarranted honking from the vehicles which can be deactivated in the silent zones.Comment: IEEE Publication format, International Journal of Computer Science and Information Security, IJCSIS, Vol. 8 No. 1, April 2010, USA. ISSN 1947 5500, http://sites.google.com/site/ijcsis