Space-like and time-like invariant space-time intervals are used to analyse measurements of spatial and temporal distances. The former are found to be Lorentz invariant –there is no 'relativistic length contraction', whereas the latter exhibit the experimentally confirmed 'time dilatation' effect, without, however, 'relativity of simultaneity'. It is shown how the spurious predictions of length...

Let Σ be a closed, orientable surface of genus g. It is known that the SU(2, 1) representation variety of π1(Σ) has 2g− 3 components of (real) dimension 16g− 16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety th...

The Klein-Gordon equation in the Rindler space-time is studied carefully. It is shown that the stable properties depend on using what time coordinate to define the initial time. If we use the Rindler time, the scalar field is stable. Alternatively, if we use the Minkowski time, the scalar field may be regarded unstable to some extent. Furthermore, the complete extension of the Rindler space tim...

We consider nonnegative solutions of the initial-boundary value problems in cone domains for the reaction-diffusion systems with inhomogeneous terms dependent on space coordinates and times. In our previous paper the conditions for the nonexistence of global solutions in time were shown. In this paper we show the condition of existence of global solutions in time.

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the latent space is discovered using Diffusion Maps (a learning technique) on time series data. A second round of those coordinates allows approximation models. This enables mapping back full ambient (what called lifting); it also state functions interest in terms coordinates. In our work, we dev...

The coupling of the metric to an incoherent dust introduces into spacetime a privileged dynamical reference frame and time foliation. The comoving coordinates of the dust particles and the proper time along the dust worldlines become canonical coordinates in the phase space of the system. The Hamiltonian constraint can be resolved with respect to the momentum that is canonically conjugate to th...

This article defines a new way to perform intuitive and geometrically faithful regressions on histogram-valued data. It leverages the theory of optimal transport, and in particular the definition of Wasserstein barycenters, to introduce for the first time the notion of barycentric coordinates for histograms. These coordinates take into account the underlying geometry of the ground space on whic...

Parametric cubic spline curves are commonly used to model the geometry of road surfaces in real-time driving simulators. Roads are represented by space curves that define a curvilinear frame of reference in which three-dimensional points are expressed in coordinates of distance along the curve, offset from the central axis, and loft from the road surface. Simulators must map from global Cartesi...

4 emitters broadcasting an increasing electromagnetic signal generate a system of relativistic coordinates for the space-time, called emission coordinates. Their physical realization requires an apparatus similar to the one of the Global Navigation Satellite Systems (GNSS). Several relativistic corrections are utilized for the current precisions, but the GNSS are conceived as classical (Newtoni...

We pursue the problem of step size determination for tessellating arbitrary degree polynomial and rational Bézier patches, under highly varying modeling and viewing transformations, to within post-viewing size and/or deviation thresholds specified in display coordinates. The technique involves the computation of derivative bounds of surfaces in modeling coordinates, and the mapping of these bou...