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Vol 211, No 8 (2020)
- Year: 2020
- Articles: 7
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7468
Cocompact lattices in locally pro-$p$-complete rank-2 Kac-Moody groups
Abstract
We initiate an investigation of lattices in a new class of locally compact groups: so-called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order $p$. This statement is still an open question for the Caprace-Remy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume. Bibliography: 22 titles.
Matematicheskii Sbornik. 2020;211(8):3-19
3-19
Waveguide with double threshold resonance at a simple threshold
Abstract
A threshold resonance generated by an almost standing wave occurring at a threshold — a solution of the problem that do not decay at infinity, but rather stabilizes there — brings about various anomalies in the diffraction pattern at near-threshold frequencies. Examples when a simple threshold resonance occurs or does not occur are trivial. For the first time an acoustic waveguide (the Neumann spectral problem for the Laplace operator) of a special shape is constructed in which there is a maximum possible number (namely two) of linearly independent almost standing waves at a threshold (equal to a simple eigenvalue of the model problem on the cross-section of the cylindrical outlets to infinity). Effects in the scattering problem for acoustic waves, which are caused by these standing waves are discussed. Bibliography: 54 titles.
Matematicheskii Sbornik. 2020;211(8):20-67
20-67
Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds
Abstract
We construct a complete topological invariant of foliations of finite type defined by smooth functions on two-dimensional noncompact orientable manifolds. In particular, we describe a complete topological classification of noncompact bifurcations of such foliations. We establish a natural one-to-one correspondence between the set of all such bifurcations and the set of oriented coloured graphs of a special form. As a consequence, we obtain the Liouville and trajectory classifications of Hamiltonian systems of finite type on noncompact two-dimensional manifolds.Bibliography: 25 titles.
Matematicheskii Sbornik. 2020;211(8):68-101
68-101
On $C^m$-reflection of harmonic functions and $C^m$-approximation by harmonic polynomials
Abstract
We obtain several new sharp $C^m$-continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Caratheodory domains in $\mathbb R^N$. These results are based on a new criterion (also obtained in this paper) for $C^m$-continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for $C^m$-approximability of functions by harmonic polynomials on boundaries of simple Caratheodory domains in $\mathbb R^N$. Bibliography: 17 titles.
Matematicheskii Sbornik. 2020;211(8):102-113
102-113
Nondiagonal terms in the second moment of automorphic $L$-functions
Abstract
We investigate the exact formula for the second moment of central $L$-values associated to primitive cusp forms of level one and weight $k\to\infty$, $k\in\mathbb N$, which was proved by Kuznetsov in 1994. Nondiagonal terms in this formula are expressed in terms of several convolution sums of divisor functions. We investigate the main terms of the convolution sums and show that they cancel out.Bibliography: 8 titles.
Matematicheskii Sbornik. 2020;211(8):114-131
114-131
Approximative properties of sets and continuous selections
Abstract
Sets admitting a continuous selection of the operators of best and near-best approximation are studied. Michael's classical continuous selection theorem is extended to the case of a lower semicontinuous metric projection in finite-dimensional spaces (with no a priori convexity conditions on its values). Sufficient conditions on the metric projection implying the solarity of the corresponding set are put forward in finite-dimensional polyhedral spaces. Available results for suns $V$ are employed to establish the existence of continuous selections of the relative (with respect to $V$) Chebyshev near-centre map and of the sets of relative (with respect to $V$) near-Chebyshev points in certain classical spaces. Bibliography: 30 titles.
Matematicheskii Sbornik. 2020;211(8):132-157
132-157
Errata
Matematicheskii Sbornik. 2020;211(8):158-158
158-158