Open Access
Access granted
Subscription Access
Vol 210, No 3 (2019)
- Year: 2019
- Articles: 5
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7451
Banach spaces with shortest network length depending only on pairwise distances between points
Abstract
For a real Banach space realising shortest networks for all finite subsets, we prove that a necessary and sufficient condition for the shortest network length to be expressed as a function only of pairwise distances between its points is that the space is either predual to $L_1$ or a Hilbert space. We give a characterization of spaces predual to $L_1$ and Hilbert spaces in terms of shortest networks.Bibliography: 23 titles.
Matematicheskii Sbornik. 2019;210(3):3-16
3-16
The Fomenko–Zieschang invariants of nonconvex topological billiards
Abstract
Along with a classical planar billiard, one can consider a topological billiard for which the motion takes place on a locally planar surface obtained by an isometric gluing of several planar domains along boundaries that are arcs of confocal quadrics. Here, a point is moving inside every domain along segments of straight lines, passing from one domain into another when it hits the boundary of the gluing. The author has previously obtained the Liouville classification of all such topological billiards obtained by gluings along convex boundaries. In the present paper, we classify all topological integrable billiards obtained by gluing both along convex and along nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics. For all such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules $W^*$) of Liouville equivalence are calculated. Bibliography: 25 titles.
Matematicheskii Sbornik. 2019;210(3):17-74
17-74
Is Zaremba's conjecture true?
Abstract
For finite continued fractions in which all partial quotients lie in the alphabet $\{1,2,3,5\}$, it is shown that the set of denominators not exceeding $N$ has cardinality $\gg N^{0.85}$. A calculation using an analogue of Bourgain-Kontorovich's theorem from 2011 gives $\gg N^{0.80}$. Bibliography: 25 titles.
Matematicheskii Sbornik. 2019;210(3):75-130
75-130
Entropy and renormalized solutions of anisotropic elliptic equations with variable nonlinearity exponents
Abstract
The Dirichlet problem is considered in arbitrary domains for a class of second-order anisotropic elliptic equations with variable nonlinearity exponents and right-hand sides in $L_1$. It is proved that an entropy solution exists in anisotropic Sobolev spaces with variable exponent. It is proved that the entropy solution obtained is a renormalized solution of the problem under consideration. Bibliography: 37 titles.
Matematicheskii Sbornik. 2019;210(3):131-161
131-161
162-188