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Vol 83, No 1 (2019)
- Year: 2019
- Articles: 9
- URL: https://bakhtiniada.ru/1607-0046/issue/view/7536
Articles
Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4
Abstract
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 $4$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):3-24
3-24
Dual and almost-dual homogeneous spaces
Abstract
We study homogeneous spaces $G/H$ such that the transitive action of theLie group $G$ on $G/H$ preserves the structure of a dual or almost-dualmanifold. We consider general homogeneous spaces of this kind as wellas compact or lower-dimensional ones.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):25-58
25-58
Existence theorems for a class of systems involving two quasilinear operators
Abstract
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.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):59-74
59-74
Real Kummer surfaces
Abstract
This paper is devoted to the deformation classification of arbitraryreal Kummer surfaces and real Kummer surfaces of the Jacobians of realcurves of genus 2 as well as the rigid isotopy classification of real Kummerquartics.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):75-118
75-118
Completion of the classification of generic singularities of geodesicflows in two classes of metrics
Abstract
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 $2$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):119-139
119-139
The main classes of invariant Banach limits
Abstract
We study functional characteristics and extreme points of the set of Banachlimits on the space of bounded sequences. We also consider Banach limitswith additional invariance properties with respect to the dilationoperator and the Cesàro operator.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):140-167
140-167
Fixed points and coincidences of families of mappings betweenordered sets and some metrical consequences
Abstract
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.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):168-191
168-191
On integers whose number of prime divisors belongs to a given residue class
Abstract
We consider positive integers whose number of prime divisors is congruent to $l$modulo $k$. In this case, the calculation of prime divisors can be made either withor without taking into account the multiplicity, and the divisors themselves can besubjected to the additional requirement of belonging to some special set. We showthat for $k\geqslant3$, the distribution pattern of these numbers, in dependenceon the value of $l$, differs fundamentally from that in the case $k=2$, which wasstudied earlier.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):192-202
192-202
Tropical lower bound for extended formulations. II. Deficiency graphs of matrices
Abstract
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.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(1):203-216
203-216