Related papers: PA mapping classes with minimum dilatation and Lan…

In this paper we study the behavior of dilation operators $ D_\lambda \colon f \mapsto f(\lambda\,\cdot) $ with $ \lambda > 1 $ in the context of Triebel-Lizorkin-Morrey spaces $\mathcal{E}^{s}_{u,p,q}(\mathbb{R}^d)$. For that purpose we…

A reduction $\varphi$ of an ordered group $(G,P)$ to another ordered group is an order homomorphism which maps each interval $[1,p]$ bijectively onto $[1, \varphi(p)]$. We show that if $(G,P)$ is weakly quasi-lattice ordered and reduces to…

The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. This result…

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there…

Thomass\'{e} conjectured the following strengthening of the well-known Caccetta-Haggkvist Conjecture: any digraph with minimum out-degree $\delta$ and girth $g$ contains a directed path of length $\delta(g-1)$. Bai and Manoussakis…

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second…

Let $x_{j+1}=\phi(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $\phi$ being a diffeomorphism. Although the state vector $x_{j}$ is often unobservable, the dynamics can be recovered from the delay vector…

A filling curve $\gamma$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(\gamma)$ of $S$ via the process of "point-pushing along $\gamma$." We consider the relationship between the self-intersection number $i(\gamma)$ of…

We consider a class $G(S^n)$ of orientation preserving Morse-Smale diffeomorphisms of the sphere $S^{n}$ of dimension $n>3$ in assumption that invariant manifolds of different saddle periodic points have no intersection. We put in a…

On each nonorientable surface of odd genus $g \geq 5$, we give a mapping class whose dilatation on an invariant subsurface is the golden ratio.

A commuting pair of Hilbert space operators having the closed symmetrized bidisc \[ \Gamma=\{(z_1+z_2, z_1z_2) \in \mathbb C^2 \ : \ |z_1| \leq 1, |z_2| \leq 1\} \] as a spectral set is called a \textit{$\Gamma$-contraction}. A…

A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least…

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper…

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the…

In this paper, we give a uniqueness theorem for the Dirichlet problem of minimal maps into general Riemannian manifolds with non-positive sectional curvature, improving Theorem 5.2 of Lee-Ooi-Tsui's paper published in J. Geom. Anal.. The…

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on $\mathbb{R}l^{1,1}$ describing graphical time-like minimal surfaces embedded in $\mathbb{R}^{1,2}$ enjoy small data global existence for compactly supported initial…

Using a rigidity property of the foliations of $S^2 \times [0, 1]$ that are defined by a non-vanishing closed one-form, we give a rather simple proof of a theorem due J. Cerf, going back to 1968, that the group of direct diffeomorphisms of…

We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and…

Given a point set $P$ in the Euclidean space, a geometric $t$-spanner $G$ is a graph on $P$ such that for every pair of points, the shortest path in $G$ between those points is at most a factor $t$ longer than the Euclidean distance between…

A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) groupoid structure. In this short note we show that when $\pi$ is not…