Related papers: Flat Norm Decomposition of Integral Currents
We consider the inverse problem of recovering a continuous-domain function from a finite number of noisy linear measurements. The unknown signal is modeled as the sum of a slowly varying trend and a periodic or quasi-periodic seasonal…
We obtain a fine structural result for two-dimensional mod$(q)$ area-minimizing currents of codimension one, close to flat singularities. Precisely, we show that, locally around any such singularity, the current is a…
Let $\mathcal{K}$ be a finite pure simplicial $d$-complex, with oriented facets $\{F_i\}$, which is boundaryless in the sense that $\sum\partial F_i=0$. We call such a $\mathcal{K}$ an \textit{admissible $d$-complex}. Given an admissible…
We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of…
For a function $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$, the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a…
We study the Pareto frontier for two competing norms $\|\cdot\|_X$ and $\|\cdot\|_Y$ on a vector space. For a given vector $c$, the pareto frontier describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition $c=a+b$. The…
Recovering a 3D surface from its surface normal map, a problem known as normal integration, is a key component for photometric shape reconstruction techniques such as shape-from-shading and photometric stereo. The vast majority of existing…
We introduce the Circular Directional Flow Decomposition (CDFD), a new framework for analyzing circularity in weighted directed networks. CDFD separates flow into two components: a circular (divergence-free) component and an acyclic…
In this paper, we investigate in a unified way the structural properties of solutions to inverse problems. These solutions are regularized by the generic class of semi-norms defined as a decomposable norm composed with a linear operator,…
We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm…
Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by…
Normal forms allow the use of a restricted class of coordinate transformations (typically homogeneous polynomials) to put the bifurcations found in nonlinear dynamical systems into a few standard forms. We investigate here the consequences…
The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One…
By studying a nonequilibrium Langevin system, we find that a simple condition determines the decomposition of the coarse-grained force into a dissipative force, an effective driving force and noise. From this condition, we derive a new…
Fluid discontinuities, such as shock fronts and vortex sheets, can reflect waves and become unstable to corrugation. Analytical calculations of these phenomena are tractable in the simplest cases only, while their numerical simulations are…
Flatness of discrete-time systems can be characterized by two simple properties. There exists a map, a submersion, from the flat coordinates and their forward shifts to the state and the input of the discrete-time system, such that the…
The interior regularity of area-minimizing integral currents and semi-calibrated currents has been studied extensively in recent decades, with sharp dimension estimates established on their interior singular sets in any dimension and…
Over the past few years, monocular depth estimation and completion have been paid more and more attention from the computer vision community because of their widespread applications. In this paper, we introduce novel physics…
Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the…
This paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development…