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Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method…
We review an algorithm developed for parameter estimation within the Continuous Data Assimilation (CDA) approach. We present an alternative derivation for the algorithm presented in a paper by Carlson, Hudson, and Larios (CHL, 2021). This…
The C-metric solution of conformal gravity with a conformally coupled scalar field is presented. The solution belongs to the class of Petrov type D spacetimes and is conformal to the standard AdS C-metric appeared in vacuum Einstein…
We introduce a fully-corrective generalized conditional gradient method for convex minimization problems involving total variation regularization on multidimensional domains. It relies on alternatively updating an active set of subsets of…
A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the…
The Hamiltonian approach to the General Relativity is formulated as a joint nonlinear realization of conformal and affine symmetries by means of the Dirac scalar dilaton and the Maurer-Cartan forms. The dominance of the Casimir vacuum…
We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a conformal deformation of the metric. We derive some existence results using a variational approach, either by…
Conformal electrodynamics is a particularly interesting example of power Maxwell non-linear electrodynamics, designed to possess conformal symmetry in all dimensions. In this paper, we propose a regularized version of Conformal…
The requirements of conformal invariance for two and three point functions for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented…
This paper is devoted to a detailed convergence analysis of the method of codifferential descent (MCD) developed by professor V.F. Demyanov for solving a large class of nonsmooth nonconvex optimization problems. We propose a generalization…
A lack of regularity in the solution of the porous medium equation poses a serious challenge in its theoretical and numerical studies. A common strategy in theoretical studies is to utilize the pressure formulation of the equation where a…
We use covariant and first-order formalism techniques to study the properties of general relativistic cosmology in three dimensions. The covariant approach provides an irreducible decomposition of the relativistic equations, which allows…
We show the traditional rocket problem, where the ejecta velocity is assumed constant, can be reduced to an integral quadrature of which the completely non-relativistic equation of Tsiolkovsky, as well as the fully relativistic equation…
This article develops mathematical formalisms and provides numerical methods for studying the evolution of measures in nonsmooth dynamical systems using the continuity equation. The nonsmooth dynamical system is described by an evolution…
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and…
This study presents a fractional-order continuum mechanics approach that allows combining selected characteristics of nonlocal elasticity, typical of classical integral and gradient formulations, under a single frame-invariant framework.…
We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data…
Low density plasmas in curved spacetimes, such as those found in accretion flows around black holes, are challenging to model from first principles, owing to the large scale separation between the characteristic scales of the microscopic…
The ADM formalism together with a constant mean curvature (CMC) temporal gauge is used to derive the monotonic decay of a weak Lyapunov function of the Einstein dynamical equations in an expanding universe with a positive cosmological…
We consider the Klein-Gordon and sine-Gordon type equations with a point-like potential, which describes the wave phenomenon in disordered media with a defect. The singular potential term yields a critical phenomenon--that is, the solution…