Weak Field Expansion of Gravity: Graphs, Matrices and Topology

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor (\pl)^d h_\ab, which generally appears in the weak field expansion around the flat space: g_\mn=\del_\mn+h_\mn. Making use of this representation, we explain 1) Generating function of graphs (Feynman diagram approach), 2) Adjacency matrix (Matrix approach), 3) Graphical classification in terms of "topology indices" (Topology approach), 4) The Young tableau (Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of \pl\pl h-, {and} (\pl\pl h)^2- invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).Comment: 41pages, 26 figures, Latex, epsf.st

Semi-Invariant Terms for Gauged Non-Linear Sigma-Models

We determine all the terms that are gauge-invariant up to a total spacetime derivative ("semi-invariant terms") for gauged non-linear sigma models. Assuming that the isotropy subgroup $H$ of the gauge group is compact or semi-simple, we show that (non-trivial) such terms exist only in odd dimensions and are equivalent to the familiar Chern-Simons terms for the subgroup $H$. Various applications are mentioned, including one to the gauging of the Wess-Zumino-Witten terms in even spacetime dimensions. Our approach is based on the analysis of the descent equation associated with semi-invariant terms.Comment: section 6 expande

Cosmokinetics: A joint analysis of Standard Candles, Rulers and Cosmic Clocks

We study the accelerated expansion of the universe by using the kinematic approach. In this context, we parameterize the deceleration parameter, q(z), in a model independent way. Assuming three simple parameterizations we reconstruct q(z). We do the joint analysis with combination of latest cosmological data consisting of standard candles (Supernovae Union2 sample), standard ruler (CMB/BAO), cosmic clocks (age of passively evolving galaxies) and Hubble (H(z)) data. Our results support the accelerated expansion of the universe.Comment: PDFLatex, 15 pages, 12 pdf figures, revised version to appear in JCA

Exact asymptotic behavior of magnetic stripe domain arrays

The classical problem of magnetic stripe domain behavior in films and plates with uniaxial magnetic anisotropy is treated. Exact analytical results are derived for the stripe domain widths as function of applied perpendicular field, $H$, in the regime where the domain period becomes large. The stripe period diverges as $(H_c-H)^{-1/2}$, where $H_c$ is the critical (infinite period) field, an exact result confirming a previous conjecture. The magnetization approaches saturation as $(H_c-H)^{1/2}$, a behavior which compares excellently with experimental data obtained for a $4 \mu$m thick ferrite garnet film. The exact analytical solution provides a new basis for precise characterization of uniaxial magnetic films and plates, illustrated by a simple way to measure the domain wall energy. The mathematical approach is applicable for similar analysis of a wide class of systems with competing interactions where a stripe domain phase is formed.Comment: 4 pages, 4 figure

Gain-scheduled H∞ control via parameter-dependent Lyapunov functions

Synthesising a gain-scheduled output feedback H∞ controller via parameter-dependent Lyapunov functions for linear parameter-varying (LPV) plant models involves solving an infinite number of linear matrix inequalities (LMIs). In practice, for affine LPV models, a finite number of LMIs can be achieved using convexifying techniques. This paper proposes an alternative approach to achieve a finite number of LMIs. By simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality can be formed. Hence, the LMIs need only to be evaluated at all vertices of such a symmetric matrix polytope. In addition, a construction technique of the intermediate controller variables is also proposed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameters. Computational results on a numerical example using the approach were compared with those from a multi-convexity approach in order to demonstrate the impacts of the approach on parameter-dependent Lyapunov-based stability and performance analysis. Furthermore, numerical simulation results show the effectiveness of these proposed techniques