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Vol 216, No 5 (2025)
- Year: 2025
- Articles: 9
- URL: https://journal-vniispk.ru/0368-8666/issue/view/20348
Tribute to Anatoly Timofeevich Fomenko
3-4
Geodesically equivalent metrics, Nijenhuis operators, geodesic flows, symmetries, conservation laws
Abstract
We show how concepts, methods and results from Nijenhuis geometry can be used to study geodesically equivalent metrics. We propose a new method of the presentation and proof of many facts in the classical theory of geodesically equivalent metrics and develop methods for the further development of this theory.
5-32
Polynomials of complete spatial graphs and Jones polynomial of the related links
Abstract
A spatial $K_n$-graph is an embedding of a complete graph $K_n$ with $n$ vertices in a $3$-sphere $S^3$. Knots in a spatial $K_n$-graph corresponding to cycles of $K_n$ are called constituent knots. We consider the case $n=4$. The boundary of the orientable band surface constructed from a spatial $K_4$-graph and having the zero Seifert form is a $4$-component link, which is referred to as the associated link. We obtain formulae relating the normalized Yamada and Jaeger polynomials of spatial $K_4$-graphs, their $\theta$-subgraphs and cyclic subgraphs with the Jones polynomials of constituent knots and related links.
33-63
If a Minkowski billiard is projective, then it is the standard billiard
Abstract
In the recent paper [5] the first-named author proved that if a billiard in a convex domain in $\mathbb R^n$ is simultaneously projective and Minkowski, then it is the standard Euclidean billiard in an appropriate Euclidean structure. The proof was quite complicated and required high smoothness. Here we present a direct simple proof of this result which works in $C^1$-smoothness. In addition, we prove the semi-local and local versions of this result.
64-82
Magic billiards: the case of elliptic boundaries
Abstract
We introduce a novel concept of magic billiards, which can be viewed as an umbrella unifying several well-known generalisations of mathematical billiards. We analyse the properties of magic billiards in the case of elliptic boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic and polynomial forms. A topological description of these billiards is given using Fomenko graphs.
83-105
Normalization of rationally integrable systems
Abstract
It is well known that any analytic vector field near a singular point admits a normalization à la Poincare-Birkhoff, but this normalization is only formal in general, and the problem of analytic (convergent) normalization is a difficult one. In [26] and [27] we proposed a new approach to the normalization of vector fields, via their intrinsic associated torus actions: an analytic vector field is analytically normalizable near a singular point if and only if its associated torus action is analytic (and not just formal). We then showed that if a vector field is analytically integrable, then its associated torus action is analytic, and therefore the vector field is analytically normalizable [26], [27]. In this paper we extend this analytic normalization result to the case of rationally integrable systems, where the first integrals and commuting vector fields are not required to be analytic, but just rational (that is, quotients of analytic functions or vector fields by analytic functions). For example, any vector field of the type $X = f Y$, where Y is an analytically diagonalizable vector field and f is an analytic function such that $Y (f ) = 0$, is rationally integrable but not necessarily analytically integrable.
106-122
Contact line bundles, foliations and integrability
Abstract
We formulate the definition of the noncommutative integrability of contact systems on a contact manifold $(M,\mathcal H)$ using the Jacobi structure on the space of sections $\Gamma(L)$ of a contact line bundle $L$. In the cooriented case, if the line bundle is trivial and $\mathcal H$ is the kernel of a globally defined contact form $\alpha$, the Jacobi structure on the space of sections reduces to the standard Jacobi structure on $(M, \alpha)$. We therefore treat contact systems on cooriented and non-cooriented contact manifolds simultaneously. In particular, this allows us to work with dissipative Hamiltonian systems, where the Hamiltonian does not have to be preserved by the Reeb vector field.
123-150
Maupertuis's principle for systems with Lagrangians linear in velocities
Abstract
Maupertuis's variational principle is discussed for Lagrangian systems with Lagrangian which is linear in generalized velocities. In particular, this includes Hamiltonian systems in the Poincare-Helmholtz representation. Our approach allows us to indicate a new variational principle for Lagrange's systems whose Lagrangians are degenerate in velocities. In its derivation we use Dirac's generalized Hamiltonian formalism.
151-160
Probabilistic morphisms and Bayesian supervised learning
Abstract
We develop the category theory of Markov kernels to the study of categorical aspects of Bayesian inversions. As a result, we present a unified model for Bayesian supervised learning, including Bayesian density estimation. We illustrate this model with Gaussian process regressions.
161-180