Related papers: Exotic Heat PDE's.II
Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs…
Atmospheric retrievals are now a standard tool to analyze observations of exoplanet atmospheres. This data-driven approach quantitatively compares atmospheric models to observations in order to estimate atmospheric properties and their…
This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We…
We introduce $\Psi \mathrm{ec}$, a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential $r$-form wavelets with flexible directional localization that provide tight…
A Hamiltonian approach to black hole entropy is used to study Riemannian Kerr-AdS solutions in the general, parity-violating Poincar\'e gauge theory. Entropy and the asymptotic charges are entirely determined by the parity-even sector of…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
Direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some…
We prove a version of the Arezzo-Pacard-Singer blow-up theorem in the setting of Poincar\'e type metrics. We apply this to give new examples of extremal Poincar\'e type metrics. A key feature is an additional obstruction which has no…
A particle system with a (2+1)D exotic Newton-Hooke symmetry is constructed by the method of nonlinear realization. It has three essentially different phases depending on the values of the two central charges. The subcritical and…
Spectra of exoplanet atmospheres provide us the opportunity to improve our understanding of these objects just as remote sensing in our own solar system has increased our understanding of the solar system bodies. The challenge is to…
The generalization of Birkhoff's theorem to higher dimensions in Lovelock gravity allows for black hole solutions with horizon geometries of non-constant curvature. We investigate thermodynamic aspects of these `exotic' black hole…
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the…
We study the use of polyhedral discretizations for the solution of heat diffusion and elastodynamic problems in computer graphics. Polyhedral meshes are more natural for certain applications than pure triangular or quadrilateral meshes,…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
In this paper, we discuss the Poincar\'{e} series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincar\'{e} series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type…
In this paper we prove discrete Poincar\'e inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal…
For real $\mathbf{b}$, consider quadratic heat equations like \begin{equation*} \mathbf{w}_t=\mathbf{w}_{\boldsymbol{\xi}\boldsymbol{\xi}} + \mathbf{b}(\boldsymbol{\xi})\,\mathbf{w}^2 \end{equation*} on $\boldsymbol{\xi}\in(0,\pi)$ with…
The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well…
Under Poincar\'e-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional…
This is an elementary introduction to exterior differential systems motivated by two examples: minimal submanifolds and the isometric embedding problem. The two main goals of the lectures are: 1. To explain how to find an appropriate…