Vol 83, No 1 (2019)

A hyperbolic lattice is said to be $(1{,}{\kern 1pt}2)$-reflectiveif its automorphism group is generated by $1$- and $2$-reflections up to finite index.We prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmeticcocompact reflection group in three-dimensional Lobachevsky space contains an edge with sufficiently small distance between its framing faces.Using this fact, we obtain a classification of $(1{,}{\kern 1pt}2)$-reflective anisotropic hyperbolic lattices of rank $4$.

In this paper, we study the existence of positive radialsolutions for a class of quasilinear systems of the form$$\begin{cases}\Delta_{\phi_1}u=a_1(|x|)f_1(v),\Delta_{\phi_2}v=a_2(|x|)f_2(u),\end{cases}\quad x\in \mathbb{R}^N, \quad N\ge 3,$$where $\Delta_{\phi}w:=\operatorname{div}(\phi(|\nabla w|)\nabla w)$,under appropriate conditions on the functions $\phi_1$, $\phi_2$,the weights $a_1$, $a_2$ and the non-linearities $f_1$, $f_2$.The conditions imposed for the existence of such solutions are different from those in previous results.

This is the final paper in a series devoted to generic singularitiesof geodesic flows for two-dimensional pseudo-Riemannian metrics of changingsignature and metrics induced from the Euclidean metric of the ambient spaceon surfaces with a cuspidal edge. We study the local phase portraitsand the properties of geodesics at degenerate points of a certain type.This completes the list of singularities in codimensions $1$ and $2$.

We present theorems on common fixed points and coincidences for familiesof multi-valued mappings of ordered sets. This generalizes some recent resultsof the author and Podoprikhin as well as the classical theorems of Knaster–Tarski, Smithson and Zermelo. We also considerconnections with the Caristi fixed-point theorem and other metricalresults.

Deficiency graphs arise in the problem of decomposing a tropical vectorinto a sum of points of a given tropical variety.We give an application of this concept to the theory of extendedformulations of convex polytopes, and we show that the chromatic numberof the deficiency graph of a special tropical matrix is a lower boundfor the extension complexity of the corresponding convex polytope.We compare our new lower bound for extended formulations with existingestimates and make several conjectures on the relations betweendeficiency graphs, extended formulations, and rank functions of tropicalmatrices.