Subspace procrustes analysis

Postprint (author's final draft

Semiclassical Mechanics of the Wigner 6j-Symbol

The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano-Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.Comment: 64 pages, 50 figure

Semiclassical analysis of Wigner 3j3j-symbol

We analyze the asymptotics of the Wigner $3j$-symbol as a matrix element connecting eigenfunctions of a pair of integrable systems, obtained by lifting the problem of the addition of angular momenta into the space of Schwinger's oscillators. A novel element is the appearance of compact Lagrangian manifolds that are not tori, due to the fact that the observables defining the quantum states are noncommuting. These manifolds can be quantized by generalized Bohr-Sommerfeld rules and yield all the correct quantum numbers. The geometry of the classical angular momentum vectors emerges in a clear manner. Efficient methods for computing amplitude determinants in terms of Poisson brackets are developed and illustrated.Comment: 7 figure file

Chaotic motion and spiral structure in self-consistent models of rotating galaxies

Dissipationless N-body models of rotating galaxies, iso-energetic to a non-rotating model, are examined as regards the mass in regular and in chaotic motion. The values of their spin parameters $\lambda$ are near the value $\lambda=0.22$ of our Galaxy. We obtain the distinction between the sets of particles moving in regular and in chaotic orbits and we show that the spatial distribution of these two sets of particles is much different. The rotating models are characterized by larger fractions of mass in chaotic motion ($\thickapprox 65$) compared with the fraction of mass in chaotic motion in the non-rotating iso-energetic model ($\thickapprox 32$). Furthermore, the Lyapunov numbers of the chaotic orbits in the rotating models become by about one order of magnitude larger than in the non-rotating model. Chaotic orbits are concentrated preferably in values of the Jacobi integral around the value of the effective potential at the corotation radius. We find that density waves form a central rotating bar embedded in a thin and a thick disc with exponential surface density profile. A surprising new result is that long living spiral arms are exited on the disc, composed almost completely by chaotic orbits. The bar excites an $m=2$ mode of spiral waves on the surface density of the disc, emanating from the corotation radius. These spiral waves are deformed, fade, or disappear temporarily, but they grow again re-forming a well developed spiral pattern. Spiral arms are discernible up to 20 or 30 rotations of the bar (lasting for about a Hubble time).Comment: 30 pages, 17 figures (low resolution). Revised version. Accepted for publication in MNRAS. For high resolution figures please send email to [email protected]

Three-dimensional Simulations of Disk Accretion to an Inclined Dipole: I. Magnetospheric Flow at Different Theta

We present results of fully three-dimensional MHD simulations of disk accretion to a rotating magnetized star with its dipole moment inclined at an angle Theta to the rotation axis of the disk. We observed that matter accretes from the disk to a star in two or several streams depending on Theta. Streams may precess around the star at small Theta. The inner regions of the disk are warped. The warping is due to the tendency of matter to co-rotate with inclined magnetosphere. The accreting matter brings positive angular momentum to the (slowly rotating) star tending to spin it up. The corresponding torque N_z depends only weakly on Theta. The angular momentum flux to the star is transported predominantly by the magnetic field; the matter component contributes < 1 % of the total flux. Results of simulations are important for understanding the nature of classical T Tauri stars, cataclysmic variables, and X-ray pulsars.Comment: 26 pages, 22 figures, LaTeX, macros: emulapj.sty, avi simulations are available at http://www.astro.cornell.edu/us-rus/inclined.ht

Deformation compatibility in a single crystalline Ni superalloy

Deformation in materials is often complex and requires rigorous understanding to predict engineering component lifetime. Experimental understanding of deformation requires utilization of advanced characterization techniques, such as high spatial resolution digital image correlation (HR-DIC) and high angular resolution electron backscatter diffraction (HR-EBSD), combined with clear interpretation of their results to understand how a material has deformed. In this study, we use HR-DIC and HR-EBSD to explore the mechanical behaviour of a single-crystal nickel alloy and to highlight opportunities to understand the complete deformations state in materials. Coupling of HR-DIC and HR-EBSD enables us to precisely focus on the extent which we can access the deformation gradient, F, in its entirety and uncouple contributions from elastic deformation gradients, slip and rigid body rotations. Our results show a clear demonstration of the capabilities of these techniques, found within our experimental toolbox, to underpin fundamental mechanistic studies of deformation in polycrystalline materials and the role of microstructure

The Logic behind Feynman's Paths

The classical notions of continuity and mechanical causality are left in order to refor- mulate the Quantum Theory starting from two principles: I) the intrinsic randomness of quantum process at microphysical level, II) the projective representations of sym- metries of the system. The second principle determines the geometry and then a new logic for describing the history of events (Feynman's paths) that modifies the rules of classical probabilistic calculus. The notion of classical trajectory is replaced by a history of spontaneous, random an discontinuous events. So the theory is reduced to determin- ing the probability distribution for such histories according with the symmetries of the system. The representation of the logic in terms of amplitudes leads to Feynman rules and, alternatively, its representation in terms of projectors results in the Schwinger trace formula.Comment: 15 pages, contribution to Mario Castagnino Festschrif

A Self-Consistent Dynamical Model for the COBE Detected Galactic Bar

A 3D steady state stellar dynamical model for the Galactic bar is constructed with 485 orbit building blocks using an extension of Schwarzschild technique. The weights of the orbits are assigned using non-negative least square method. The model fits the density profile of the COBE light distribution, the observed solid body stellar rotation curve, the fall-off of minor axis velocity dispersion and the velocity ellipsoid at Baade's window. We show that the model is stable. Maps and tables of observable velocity moments are made for easy comparisons with observation. The model can also be used to set up equilibrium initial conditions for N-body simulations to study stability. The technique used here can be applied to interpret high quality velocity data of external bulges/bars and galactic nuclei.Comment: submitted to MNRAS; 37 page AAS latex file with 2 tables and no figures; complete uuencoded compressed PS file with 9 figs is available at ftp://ibm-1.mpa-garching.mpg.de/pub/hsz/cobe_bar_dynamics.uu Hardcopies are available by reques