Related papers: Helmholtz-Hodge Decomposition on Graphs

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field…

A string graph is the intersection graph of curves in the plane. We prove that there exists an absolute constant $c>0$ such that if $G$ is a string graph on $n$ vertices, then $G$ contains either a clique or an independent set of size at…

In this work, we prove the vanishing of the two cohomological group of the higher local field. This generalize the well-known propriety of finite field and one dimensional local field. We apply this result to study the arithmetic of curve…

Drawing on work of Berndtsson and of Lempert and Sz\H{o}ke, we define a kind of complex analytic structure for families of (possibly finite-dimensional) Hilbert spaces that might not fit together to form a holomorphic vector bundle but…

We consider periodic Schr\"{o}dinger operators on the hexagonal lattice with self-adjoint vertex conditions that allow discontinuity and concentrated mass at the vertices. This model generalizes the periodic Schr\"{o}dinger operator on the…

Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an {\em atom} if it has no clique separator. A {\em hole} is a chordless cycle with at least five vertices, and…

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves…

When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant…

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node…

For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the…

Cores are, besides connectivity components, one among few concepts that provides us with efficient decompositions of large graphs and networks. In the paper a generalization of the notion of core of a graph based on vertex property function…

We study dismantlability in graphs. In order to compare this notion to similar operations in posets (partially ordered sets) or in simplicial complexes, we prove that a graph G dismants on a subgraph H if and only if H is a strong…

We obtain global analytic hypoellipticity for a class of differential operators that can be expressed as a zero-order perturbation of a sum of squares of vector fields with real-analytic coefficients on compact Lie groups. The key…

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A…

We propose a number of variational regularisation methods for the estimation and decomposition of motion fields on the $2$-sphere. While motion estimation is based on the optical flow equation, the presented decomposition models are…

We prove that the equational theory of the positive calculus of relations with transitive closure (PCoR*) is EXPSPACE-complete. Here, PCoR* terms consist of the following standard operators on binary relations: identity, empty,…

Given a generically smooth stable curve over a discrete valuation ring such that its special fibre is irreducible with one double point, we construct a moduli stack over that descrete valuation ring which is a model for the moduli stack of…

The pedal of a curve in the Euclidean plane is a classical subject which has a singular point at the inflection point of the original curve or the pedal point. The primitive of a curve is a curve given by the inverse construction for making…

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in $\mathbb{R}^3$. This is a purely geometric construction in which all of the properties of the entire knotted…

If $H$ is (or is isomorphic to) a subgraph of $G$, $H$ is said to {\it divide} $G$ if there is an edge-decomposition of $G$ by copies of $E(H)$, the edge set of $H$. A more restrictive version of this is when there is a subgroup ${\cal H}$…