Positive-definite means that gp(v, v) > 0 for every v 6= 0, v ∈ TpM . Smoothly depending on p means that for every pair Xp,Yp of C ∞-smooth vector fields on M the expression gp(Xp, Yp) defines a C ∞-smooth function of p ∈M . Alternatively, consider a coordinate neighbourhood on M containing p and let x, i = 1, . . . , dimM be the local coordinates. Then any two tangent vectors u, v ∈ TpM may be...

We re-express the Kerr metric in standard Bondi-Saches’ coordinate near null infinity I+. Using the uniqueness result of characteristic initial value problem, we prove the Kerr metric is the only asymptotic flat, stationary, axial symmetric, Type-D solution of vacuum Einstein equation. The Taylor series of Kerr space-time is expressed in terms of B-S coordinates and the N-P constants have been ...

In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane H C . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a su...

We formulate a multi-scale perturbation technique to asymptotically solve weakly nonlinear hyperbolic equations. The method is based on a set of scaled characteristic coordinates. We show that this technique leads to a simplified system of ordinary differential equations describing the weakly nonlinear interaction between left and right running waves. Using this method, a uniformly valid first ...

Abstract. Let F be a field of characteristic not 2 and assume all algebras are over F. We establish several conjugacy theorems for the special linear Lie algebra sl2 over an F -algebra which is a UFD. We find the structure of the full automorphism group as well as conditions for when two such algebras are isomorphic. We also study the structure of their derivation algebras. Often we work with m...

Nasal cavity is an important component of respiration system for the various physiological functions (Elad et al., 2008; Lee, 2010). Due to its complicated geometry, flow patterns are difficult to visualize inside 3D spaces. By adopting the ISOMAP algorithm (Tenenbaum et al., 2000) implemented in Matlab, the surface of a nasal cavity, together with characteristic lines serving as visual markers...

We have studied binary two-dimensional (2D) mixtures of superparamagnetic colloidal particles interacting through magnetic dipole moments, which were induced by an external magnetic field B. By tuning B the effective system temperature could be widely adjusted. Time-dependent particle coordinates measured by video-microscopy provide radial pair-distribution functions, mean-square displacements ...

In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane H C . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a su...

For FPGA based coprocessors for elliptic curve cryptography, a significant performance gain can be achieved when hybrid coordinates are used to represent points on the elliptic curve. We provide a new area/performance tradeoff analysis of different hybrid representations over fields of characteristic two. Moreover, we present a new generic cryptoprocessor architecture that can be adapted to var...

Quasi-spherical light cones are lightlike hypersurfaces of the Kerr geometry that are asymptotic to Minkowski light cones at infinity. We develop the equations of these surfaces and examine their properties. In particular, we show that they are free of caustics for all positive values of the Kerr radial coordinate r. Useful applications include the propagation of high-frequency waves, the defin...