Related papers: The Bubble Transform and the de Rham Complex
In this thesis we examine a set of foundational questions concerning closed forms in superspace. By reformulating a number of definitions through the use of a new ring of (anti-)commuting variables and the concept of an exact Bianchi form,…
Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these…
Breakup of drop/bubble can be viewed as a result of fundamental force balance when the disruptive force is greater than the restorative force. A disruptive force acting on the drop/bubble tries to deform it, whereas a restorative force…
The Funk-Radon transform assigns to a function defined on the unit sphere its integrals along all great circles of the sphere. In this paper, we consider a frame decomposition of the Funk-Radon transform, which is a flexible alternative to…
The aim of this article is to construct a specific Poisson transform mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. The transforms we construct have values…
In this work we use the deformation procedure and explore the route to obtain distinct field theory models that present similar stability potentials. Starting from systems that interact polynomially or hyperbolically, we use a deformation…
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously…
Given a minimum measurable length underlying spacetime, the latter may be effectively regarded as discrete, at scales of order the Planck length. A systematic discretization of continuum physics may be effected most efficiently through the…
The deformation of finite-sized bubbles in intense turbulence exhibits complex geometries beyond simple spheroids as the bubbles exchange energy with the surrounding eddies across a wide range of scales. This study investigates deformation…
A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous…
Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable…
This is the first in a series of papers where we study the dynamics of a bubble wall beyond usual approximations, such as the assumptions of spherical bubbles and infinitely thin walls. In this paper, we consider a vacuum phase transition.…
This paper is the second in a series of works dedicated to studying non-linear partial differential equations via derived geometric methods. We study a natural derived enhancement of the de Rham complex of a non-linear PDE via…
We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields,…
A bicomplex is a simple mathematical structure, in particular associated with completely integrable models. The conditions defining a bicomplex are a special form of a parameter-dependent zero curvature condition. We generalize the concept…
This paper introduces a notion of differential forms on closed, potentially fractal, subsets of Euclidean space by defining pointwise cotangent spaces using the restriction of $C^1$ functions to this set. Aspects of cohomology are…
This paper considers the cohomology and bounded interpolation of nonstandard finite element complexes, e.g. Stokes, Hessian, Elasticity, divdiv. Compared to the standard finite element exterior calculus, the main challenge is the existence…
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…
The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth order…