Related papers: Lower-order ODEs to determine new twisting type N …

Based on the CR formalism of algebraically special spacetimes by Hill, Lewandowski and Nurowski, we derive a nonlinear system of two real ODEs, of which the general solution determines a twisting type II (or more special) vacuum spacetime…

$\mathcal{HH}$ spaces of type $[\textrm{N}] \otimes [\textrm{N}]$ with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic…

A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and…

The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan's definition was in terms of differential invariants being first…

A class of metrics solving Einstein's equations with negative cosmological constant and representing rotating, topological black holes is presented. All such solutions are in the Petrov type-$D$ class, and can be obtained from the most…

This paper is the initial part of a comprehensive study of spacetimes that admit the canonical forms of Killing tensor in General Relativity. The general scope of the study is to derive either new exact solutions of Einstein's equations…

We present a large family of twisting and expanding solutions to the Einstein-Maxwell equations of algebraic type D, for which the two double principal null directions (PNDs) of the Weyl tensor are not aligned with the null eigendirections…

This work follows earlier investigations in which the existence of canonical Killing tensor forms and the application of general null tetrad transformations led to a variety of solutions, Petrov types D, III, N, in vacuum with a…

We study a Lorentzian version of the well-known Calder\'{o}n problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In the…

We determine the general form of the solutions of the five-dimensional vacuum Einstein equations with cosmological constant for which (i) the Weyl tensor is everywhere type II or more special in the null alignment classification of Coley et…

For a rotating dust with a 3-dimensional symmetry group all possible metric forms can be classified and, within each class, explicitly written out. This is made possible by the formalism of Pleba\'nski based on the Darboux theorem. In the…

We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity,…

Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a…

We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE…

We present a new class of expanding and twisting solutions to the Einstein-Maxwell equations of algebraic type D, where the null eigendirections of the Faraday tensor are not aligned with PNDs of the Weyl tensor. After deriving this novel…

A class of second-order algorithms is proposed for minimizing smooth nonconvex functions that alternates between regularized Newton and negative curvature steps in an iteration-dependent subspace. In most cases, the Hessian matrix is…

We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional K\"ahler-Einstein equation with symmetry and the one-sided type-D…

We consider a modification of the standard Einstein theory in four dimensions, alternative to R. Jackiw and S.-Y. Pi, Phys. Rev. D 68, 104012 (2003), since it is based on the first-order (Einstein-Cartan) approach to General Relativity,…

Very recently Harada proposed a gravitational theory which is of third order in the derivatives of the metric tensor with the property that any solution of Einstein's field equations (EFEs) possibly with a cosmological constant is…

We briefly discuss explicit compact object solutions in higher-order scalar-tensor theories. We start by so-called stealth solutions, whose metric are General Relativity (GR) solutions, but accompanied by a non-trivial scalar field, in both…