A Liouville classification of a natural Hamiltonian system on the projective plane with a rotation metric and a linear integral is obtained. All Fomenko–Zieschang invariants (i. e., labeled molecules) of the system are calculated.

The notion „intermediate bordism group“, which was introduced by P. J. Eccles to investigate filtrations in stable homotopy group of spheres, is considered. А new notion „cobordism groups of stable-framed immersions“ is introduced. The classifying space for the new groups is constructed, ranks of the groups are calculated. Hopf invariants and the Kahn-Priddy homomorphism are generalized for cobordism groups of stable-framed immersions.

We study how quantum field theory models are modified under the reparametrizations
of the space-time coordinates and some simultaneous transformations of the field function.
The transformations that turn the action of the massive field in the Minkowski space-time into
the action of the massless field in some curved space are presented.

On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint
representations, and the Pontryagin maximum principle for the time-optimal problem are given
an independent foundation for methods of geodesic vector field to search for normal geodesics
of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally
optimal controls in (sub-)Riemannian case, as well as some their applications.

In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky
space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the
number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces
are computed. It is shown that the minimum volumes are realized on antiprisms and twisted
antiprisms. The first 248 values of volumes of ideal right-angled polyhedra are presented.
Moreover, the class of polyhedra with isolated triangles is introduces and there are obtained
combinatorial bounds on their existence as well as minimal examples of such polyhedra are
given.

On flat n-dimensional torus we study stochastic differential inclusions with mean derivatives,
for which the right-hand sides have, generally speaking, not convex (aspherical) values.
A subclass of such inclusions is distinguished for which there exists a sequence of $\varepsilon$-approximations,
converging point-wise to a Borel measurable selector. On this base a solution existence
theorem is obtained.

Let $\mathcal G$ be the family of periodic groups of period either $2$ or $4$, and $\Bar \Sigma^m$ be a homotopy $m$-space form where $\pi_1(\Bar \Sigma^m)\in \mathcal G$.
For $m=3$, we study the set $D(\Bar \Sigma_1^m, \Bar \Sigma_2^m)$ of degrees of the maps from $\Bar \Sigma_1^m$ to $\Bar \Sigma_2^m$.

In the last hundred years many significant gravitational phenomena have been predicted and
discovered by General Relativity (GR), which is still the best theory of gravity. Nevertheless,
due to the great observational discoveries of 20th century some (quantum) theoretical and
(astrophysical and cosmological) phenomenological difficulties of modern gravity have been
motivation to search more general theory of gravity than GR. As a result, many modifications
of GR have been considered. One of promising recent investigations is Nonlocal Modified
Gravity. In this article we present a review of some nonlocal gravity models with their
exact cosmological solutions, in which nonlocality is expressed by an analytic function of the
d’Alembert–Beltrami operator. Some of obtained solutions contain effects which are usually
assigned to the dark matter and dark energy.

Coercive continuous injective mappings acting from one linear finite-dimensional space to another are considered. It is proved that the images of these mappings are retracts of linear spaces.

An optimal control problem with a feedback is considered for an initial boundary problem
describing a motion of non-linearly viscous liquid. An existence of an optimal solution
minimising a given quality functional is proved. A topological approximation approach to study
of mathematical problems of hydrodynamics is used in the proof of existence of an optimal
solution.

The investigation of arithmetic properties of the values of the generalized hypergeometric
functions is often carried out by means of known in the theory of transcendental numbers Siegel’s
method. The most general results in this field have been obtained precisely by this method.
But the possibilities of Siegel’s method in case of hypergeometric functions with irrational
parameters are restricted. This is connected with the fact that such hypergeometric functions are
not E-functions and for that reason one is unable to construct linear approximating form with
large order of zero by means of pigeonhole method. To consider problems connected with the
investigation of arithmetic properties of the values of hypergeometric functions with irrational
parameters it is possible in some cases to use the method based on the effective construction
of linear approximating form but the possibilities of this method are also limited because of
the absence of too general effective constructions. There are some difficulties also in the cases
when such constructions are available. The peculiarities of these constructions often hinder the
realization of arithmetic part of the method.
For that reason of some interest are situations when one is able to realize the required
investigation by means of specific properties of concrete functions. Sometimes it is possible to
choose the parameters of the functions under consideration in such a way that one receives
the possibility to overcome the difficulties of the general case. In this paper we consider
hypergeometric function of a special kind and its derivatives. By means of effective construction
it is possible not only to prove linear independence of the values of this function and its
derivatives over some imaginary quadratic field but also to obtain corresponding quantitative
result in the form of the estimation the modulus of the linear form in the aforesaid values.

Relations between Gromov–Hausdorff distance and Discrete Optimisation problems are
discussed. We use the Gromov–Hausdorff distances to single-distance metric space for solving
the following problems: calculation of lengths of minimum spanning tree edges of a finite metric
space; generalised Borsuk problem; chromatic number and clique cover number of a simple
graph calculation problems.

In 2018, at the World Economic Forum in Davos it was presented a new countries’ economic
performance metric named the Inclusive Development Index (IDI) composed of 12 indicators.
The new metric implies that countries might need to realize structural reforms for improving
both economic expansion and social inclusion performance. That is why, it is vital for the
IDI calculation method to have strong statistical and mathematical basis, so that results are
accurate and transparent for public purposes.

In the current work, we propose a novel approach for the IDI estimation — the Ranking
Relative Principal Component Attributes Network Model (REL-PCANet). The model is based
on RELARM and RankNet principles and combines elements of PCA, techniques applied
in image recognition and learning to rank mechanisms. Also, we define a new approach for
estimation of target probabilities matrix TRnet to reflect dynamic changes in countries’ inclusive
development. Empirical study proved that REL-PCANet ensures reliable and robust scores and
rankings, thus is recommended for practical implementation.

Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a
tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and
combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted
as a result about a critical point of a discrete Morse function on the Bier sphere Bier(K) of an
associated simplicial complex K. We illustrate the use of “standard discrete Morse functions”
on generalized chessboard complexes by proving a connectivity result for chessboard complexes
with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for j-wise disjoint partitions of a simplex.

The paper presents an algorithm for topological classification of nondegenerate saddle-focus
singularities of integrable Hamiltonian systems with three degrees of freedom up to semilocal
equivalence. In particular, we prove that any singularity of saddle-focus type can be represented
as an almost direct product in which the acting group is cyclic. Based on constructed algorithm,
a complete list of singularities of saddle-focus type of complexity 1, 2, and 3, i. e., singularities
whose leaf contains one, two, or three singular points of rank 0, is obtained. Earlier, both
singularities of saddle-focus type of complexity 1 were also described by L. M. Lerman.

On a surface homeomorphic to 2-sphere, we study a natural mechanical system with a
magnetic field that is invariant under the $S^1$-action. For singular points of rank 0 of the
momentum mapping, a criterion for non-degeneracy is obtained, the type of non-degenerate
singular points (center-center and focus-focus) is determined, bifurcations of typical degenerate
singular points are described (integrable Hamiltonian Hopf bifurcation of two types). For families
of singular circles of rank 1 of the momentum mapping (consisting of relative equilibriums of the
system) their parametric representation is obtained, nondegeneracy criterion is proved, the type
of nondegenerate (elliptic and hyperbolic) and typical degenerate (parabolic) singular circles
is determined. The parametric representation of the bifurcation diagram of the momentum
mapping is obtained. Geometric properties of the bifurcation diagram and the bifurcation
complex are described in the case when the functions defining the system are in general position.
The topology of nonsingular isoenergy 3-dimensional manifolds is determined, the topology of
the Liouville foliation on them is described up to the rough Liouville equivalence (in terms of
Fomenko’s atoms and molecules). The “splitting” hyperbolic singularities of rank 1 are described,
which are topologically unstable bifurcations of the Liouville foliation.

Abel's theorem claims that the Lemniscate can be divided into n equal arcs by ruler and compass iff $n=2^kp_1\ldots p_m$, where $p_j$ are pairwise distinct Fermat primes. The proof is based on the fact that the lemniscate can be parametrised by rational functions and the arc length is a first type elliptic integral of the parameter. Joseph Alfred Serret proposed a method to describe all such curves in [1]. In papers [1, 2, 3] he found series of such curves and described its important properties. Since then no new examples of curves with rational parametrisation such that arc length is a first type elliptic integral of the parameter are known. In this note we describe new example of such a curve.

We describe in breaf the complete table of closed irreducible orientable 3-manifolds of
complexity <=13, and method of its creation and verification. Also we formulate a conjectures
concerning the growth of the number of some kinds of manifolds. The appendix contains a short
explanation of used terminology.

In the paper the topological characteristics of multivalued mappings that can be represented
as a finite composition of mappings with aspherical values are considered. For such random
mappings, condensing with respect to some abstract measure of noncompactness, a random
index of fixed points is introduced, its properties are described and applications to fixed-point
theorems are given. The topological coincidence degree is defined for a condensing pair consisting
of a linear Fredholm operator of zero index and a multivalued mapping of the above class. In
the last section possibilities of extending this theory to random condensing pairs are shown.

The main purpose of this paper is to investigate commuting flows and integrable systems
on the configuration spaces of planar linkages. Our study leads to the definition of a natural
volume form on each configuration space of planar linkages, the notion of cross products of
integrable systems, and also the notion of multi-Nambu integrable systems. The first integrals
of our systems are functions of Bott-Morse type, which may be used to study the topology of
configuration spaces.

We consider a system of an infinite number of absolutely elastic particles on a straight
line, the masses and initial distances between which are periodically repeated. We study the
conditions under which solutions such as traveling waves can exist in such systems.

In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on $\R^n$),
understood as description of the orbit space of the standard $\GL(n, \R)$-action on $\Lambda^k \R^{n*}$
(resp. on $\Lambda ^k \R^n$). We discuss the existence of related geometry defined
by differential forms on smooth manifolds. This paper also contains an Appendix by Mikhail Borovoi on
Galois cohomology methods for finding real forms of complex orbits.

Deformations of the Kepler problem and the harmonic oscillator are considered for which
additional integrals of motion are the coordinates of the reduced divisor, according to the
Riemann–Roch theorem. For this family of non-commutative integrable systems the validity of
the Mishchenko–Fomenko hypothesis about the existence of integrals of motion from a single
functional class, in this case polynomial integrals of motion, is discussed.

The mean-value theorem for trigonometric sums on the sequence of binomial type polynomials was proved.
As known, the classical I. M. Vinogradov mean-value theorem belong to the sequence of polynomials of the form $\{x^n, n\geq 0\}.$ Estimates of sums of the kind
$$
\sum_{m\leq P}e^{2\pi if(m)}, f(m)=\sum_{k=0}^n\alpha_kp_k(m),
$$
are the important application of the finding mean-value theorem.
Here $p_k(x)$ is the sequence integer-valued polynomials of the binomial type, but a set of numbers $(\alpha_1\alpha_1,\dots,\alpha_n)$ represents a point of the $n$-fold unit cube $\Omega: 0\leq \alpha_1,\dots,\alpha_n<1.$