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We prove a sufficient condition for the existence of explicit first integrals for vector fields which admit an integrating factor. This theorem recovers and extends previous results in the literature on the integrability of vector fields…

It is confirmed in this work that the graph isomorphism can be tested in polynomial time, which resolves a longstanding problem in the theory of computation. The contributions are in three phases as follows. 1. A description graph…

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth…

A sequence of constant mean curvature surfaces $\Sigma_j$ with mean curvature $H_j \to \infty$ in a three-dimensional manifold $M$ condenses to a compact and connected graph $\Gamma$ consisting of a finite union of curves if $\Sigma_j$ is…

In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of $\C^2$. The general problem is also reduced to the study of the combinatorics of…

Consider an analytical function $f:V\subset\mathbb R^2\rightarrow\mathbb R$ having $0$ as its regular value, a switching manifold $\Sigma=f^{-1}(0)$ and a piecewise analytical vector field $X=(X^+,X^-)$, i.e. $X^\pm$ are analytical vector…

In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space $\Gamma$ of finite diameter $\delta$ from Cherlin's catalogue of metrically homogeneous graphs, there exists a finite family $\mathcal…

Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…

In this paper we prove complex bounds, also referred to as a priori bounds, for real analytic (and even C3) interval maps. This means that we associate to such a map a complex box mapping (which provides a kind of Markov structure),…

Ensuring non-interpenetration of matter is a fundamental prerequisite when modeling the deformation response of solid materials. In this contribution, we thoroughly examine how this requirement, equivalent to the injectivity of deformations…

Let $\xi$ be a real analytic vector field with an elementary isolated singularity at $0\in \mathbb{R}^3$ and eigenvalues $\pm bi,c$ with $b,c\in \mathbb{R}$ and $b\neq 0$. We prove that all cycles of $\xi$ in a sufficiently small…

We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus is close to the minimum genus of the primal graph. When the number of vertices is congruent to 5 modulo 12, we further guarantee that…

It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various…

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincar\'e map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits…

We describe a flexible construction that produces triples of finitely generated, residually finite groups $M\hookrightarrow P \hookrightarrow \Gamma$, where the maps induce isomorphisms of profinite completions…

We show that a vector field in $\mathbb{R}^n$ can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve…

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main…

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of…

We present a framework for constructing congruence closure modulo permutation equations, which extends the abstract congruence closure framework for handling permutation function symbols. Our framework also handles certain interpreted…

Let $F$ be a differential field of characteristic zero with algebraically closed constant field $C$. Let $E$ be a Picard--Vessiot closure of $F$, $R \subset E$ its Picard--Vessiot ring and $\Pi$ the differential Galois group of $E$ over…