Related papers: Optimal Synthesis in a Radially Symmetric Grushin …
We consider higher-dimensional generalizations of the $\alpha$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using…
A solution to the left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is obtained. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first…
Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove…
This paper explicitly constructs the complete set of optimal sub-Riemannian geodesics starting from a point for certain Carnot groups of step two. These are groups of dimension 2n+1 equipped with a left-invariant distribution of dimension…
The properties of some locally rotationally symmetric (LRS) perfect fluid space-times are examined in order to demonstrate the usage of the description of geometries in terms of the Riemann tensor and a finite number of its covariant…
This work deals with intersection points: conjugate points and cut points, of timelike geodesics emanating from a common initial point in special spacetimes. The paper contains three results. First, it is shown that radial timelike…
In this paper we study the geometry of $\varphi$-static perfect fluid space-times ($\varphi$-SPFST, for short). In the context of Einstein's General Relativity, they arise from a space-time whose matter content is described by a perfect…
We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph $G$. The sampling algorithm assumes strong spatial mixing together with subexponential growth…
In this paper, we study Riemannian zeroth-order optimization in settings where the underlying Riemannian metric $g$ is geodesically incomplete, and the goal is to approximate stationary points with respect to this incomplete metric. To…
We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Y. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we…
The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted…
This paper contains locally rotationally symmetric kinematic self-similar perfect fluid and dust solutions. We consider three families of metrics which admit kinematic self-similar vectors of the first, second, zeroth and infinite kinds,…
We propose a criterion for finding the minimum distance at which an interior solution of Einstein's equations can be matched with an exterior asymptotically flat solution. It is based upon the analysis of the eigenvalues of the Riemann…
We generalize a Bernstein-type result due to Albujer and Al\'ias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $\Sigma_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete…
We study isometric $G$-spaces and the question of when their maximal equivariant compactification is the Gromov compactification (meaning that it coincides with the compactification generated by the distance functions to points). Answering…
The scope of this survey is to give a self-contained introduction to synthetic timelike Ricci curvature bounds for (possibly non-smooth) Lorentzian spaces via optimal transport $\&$ entropy tools, including a synthetic version of Hawking's…
The Riemann, Ricci and Einstein tensors for N-dimensional spherically symmetric spacetimes in various systems of coordinates are studied, and the general metric for conformally flat spacetimes is given. As an application, all the…
We propose and analyse a numerical method for time-harmonic acoustic scattering in $\mathbb{R}^n$, $n=2,3$, by a class of inhomogeneities (penetrable scatterers) with fractal boundary. Our method is based on a Galerkin discretisation of the…
We exhibit a simple and explicit formula for the metric of an arbitrary static spherically symmetric perfect fluid spacetime. This class of metrics depends on one freely specifiable monotone non-increasing generating function. We also…
We analyze the indicial roots of the self-dual deformation complex on a cylinder $(\mathbb{R} \times Y^3, dt^2 + g_Y)$, where $Y^3$ is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual…