This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\mathbf{u}};\lambda)}$,
defined by a family of nonsingular algebraic curves of genus~$2$, where ${\mathbf{u}} = (u_1,u_3)$ and
$\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})$.
It is an analogue of the Weierstrass sigma function $\sigma({{u}};g_2,g_3)$ of a family of elliptic curves. Logarithmic derivatives
of order 2 and higher of the function ${\sigma({\mathbf{u}};\lambda)}$ generate fields of hyperelliptic functions of ${\mathbf{u}} = (u_1,u_3)$
on the Jacobians of curves with a fixed parameter vector $\lambda$.
We consider three Hurwitz series $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) =
\sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$
and $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$.
The survey is devoted to the number-theoretic properties of the functions $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ and $\mu_k(u_3;\lambda)$.
It includes the latest results, which proofs use the fundamental fact that the function ${\sigma ({\mathbf{u}};\lambda)}$ is determined
by the system of four heat equations in a nonholonomic frame of six-dimensional space.

The study of algebraic group actions on varieties and coordinate algebras is a major area
of research in algebraic geometry and ring theory. The subject has its connections with the
theory of polynomial mappings, tame and wild automorphisms, the Jacobian conjecture of
O.-H. Keller, infinite-dimensional varieties according to Shafarevich, the cancellation problem
(together with various cancellation-type problems), the theory of locally nilpotent derivations,
among other topics. One of the central problems in the theory of algebraic group actions has
been the linearization problem, formulated and studied in the work of T. Kambayashi and
P. Russell, which states that any algebraic torus action on an affine space is always linear with
respect to some coordinate system. The linearization conjecture was inspired by the classical
and well known result of A. Bia lynicki–Birula, which states that every effective regular torus
action of maximal dimension on the affine space over algebaically closed field is linearizable.
Although the linearization conjecture has turned out negative in its full generality, according
to, among other results, the positive-characteristic counterexamples of T. Asanuma, the
Bia lynicki–Birula has remained an important milestone of the theory thanks to its connection to
the theory of polynomial automorphisms. Recent progress in the latter area has stimulated the
search for various noncommutative analogues of the Bia lynicki–Birula theorem. In this paper, we
give the proof of the linearization theorem for effective maximal torus actions by automorphisms
of the free associative algebra, which is the free analogue of the Bia lynicki–Birula theorem.

In the paper representation varieties of two classes of finitely generated groups are investigated.
The first class consists of groups with the presentation
\begin{gather*}
G = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g\mid\\
a_1^{m_1}=\ldots=a_s^{m_s}= x_1^2\ldots x_g^2 W(a_1,\ldots,a_s,b_1,\ldots,b_k)=1\rangle,
\end{gather*}
where $g\ge 3$, $m_i\ge 2$ for $i=1,\ldots,s$ and $W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ is an element in normal form in the free product of cyclic groups
$$
H=\langle a_1\mid a_1^{m_1}\rangle\ast\ldots\ast\langle a_s\mid a_s^{m_s}\rangle\ast\langle b_1\rangle\ast\ldots\ast
\langle b_k\rangle.
$$

The second class consists of groups with the presentation
$$
G(p,q) = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g,t\mid a_1^{m_1}=\ldots=a_s^{m_s}=1,\ tU^pt^{-1}=U^q \rangle,
$$
where $p$ and $q$ are integer numbers such that $p>|q|\geq1$, $(p,q)=1$, $m_i\ge 2$ for $i=1,\ldots,s$, \linebreak $g\ge3$,
$U=x_1^2\ldots x_g^2W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ and $W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ is an above defined element.

Irreducible components of representation varieties $R_n(G)$ and $R_n(G(p,q))$ are found, their dimensions are calculated and it is proved, that every irreducible component is a rational variety.

The article deals with finite cyclic semirings with a semilattice addition which are defined as finite cyclic multiplicative monoids~$\langle S, \cdot \rangle$ with an operation of addition~$(+)$ such that the algebraic structure~$\langle S, + \rangle$ is an upper semilattice and laws of distributivity of multiplication over addition are satisfied.

The structure of finite cyclic semirings with a semilattice additive operation defined by a two-generated semiring of nonnegative integers is described.

The result of the work is a theorem about a structure of cyclic semirings with the semilattice additive operation defined by a two-generated ideal of non-negative numbers.
This fact, in particular, allows to calculate the number of cyclic semirings corresponding to each two-generated ideal of non-negative integers.

The method of ideals of a semiring of nonnegative integers is used in the article.
Some properties of ideals of semirings of nonnegative integers determining the structure of finite cyclic semirings are obtained.

This work complements the research of E.\,M. Vechtomova and I.\,V. Orlova where the structure of finite cyclic semirings with~idempotent noncommutative addition is described in terms of cyclic semifields and~finite cyclic semirings with semilattice addition.

The paper provides a survey of the modern state of the theory of fast multiplication of
numbers and polynomials. We consider the development of multiplication methods from the
initial block algorithms of 1960s due to Karatsuba and Toom to the methods of 1970s based
on the Discrete Fourier transform (DFT) and further to the novel methods invented in 2007–
2009. Modern multiplication methods combine exploiting of special algebraic structures, the use
of approximate computations, special forms of Fourier transforms, namely, multidimensional
DFT, additive analogue of DFT. These and other concepts essential for the fast multiplication
algorithms are thoroughly discussed in the present survey. In particular, we provide an
introduction to the theory of DFT with derivation of facts necessary for the exposition.
The final part of the survey contains a brief discussion of results on parallel multiplication
algorithms, accurate complexity bounds of the basic methods, online multiplication algorithms,
multiplicative complexity of the multiplication of polynomials over finite fields. We mention
computational models where multiplication has either linear, or quadratic complexity.

The examples of algebraic identities between solution matrices of generalized
hypergeometric equations are found in paper. These identities generate
all the algebraic identities between components of solutions of
hypergeometric equations in some cases.

Generalized hypergeometric functions (see [1-5]) are defined as
$$
{}_l\varphi_{q}(z)={}_l\varphi_{q}(\vec \nu;\vec\lambda;z)=
{}_{l+1}F_{q}\left(\left.{1,\nu_1,\dots,\nu_l\atop\lambda_1,\dots,\lambda_q}\right|z\right)=
\sum_{n=0}^\infty \frac{(\nu_1)_n\dots (\nu_l)_n}{(\lambda_1)_n
\dots(\lambda_{q})_n} z^n,
$$
where $0\leqslant l\leqslant q$, $(\nu)_0=1$, $(\nu)_n=\nu(\nu+1)\!\dots\!(\nu+n-1)$,
$\vec\nu=(\nu_1,\dots,\nu_l)\in {\mathbb C}^l$, $\vec \lambda\in
({\mathbb C}\setminus{\mathbb Z^-})^q$.

The function ${}_l\varphi_{q}(\vec \nu;\vec\lambda;z)$ satisfies the
(generalized) hypergeometric differen\-tial equation
$$
{L}(\vec \nu;\vec\lambda;z)\;y =(\lambda_1-1)\dots(\lambda_q-1),
$$
where
$$
{L}(\vec \nu;\vec\lambda;z)
\equiv \prod_{j=1}^q(\delta+\lambda_j-1)-
z\prod_{k=1}^l(\delta+\nu_k), \quad \delta=z\frac{d}{dz}.
$$

The Siegel-Shidlovskii method (see [4], [5]) is one of the main
methods in the theory of transcendental numbers. It permits
to establish the transcendency
and the algebraic independence of the values of entire functions
of some class, which contains the functions
${}_l\varphi_{q}(\alpha z^{q-l})$, provided that these
functions are algebraically independent over ${\mathbb C}(z)$.

F. Beukers, W.D. Brownawell and G. Heckman introduced in paper [6]
notions of cogredience and contragredience of differential equations,
which are important for determination of algebraic dependence and
independence of functions (these notions appeared firstly in paper
[7] of E. Kolchin really).

This work contains detailed proof and further development of
results connected with cogredience and contragredience, that have been
published in notes [8], [9]. Some results in
[6] have been revised particularly.

In this paper we consider some questions of the theory and practice of mean first passage time quasi-metric, a generalized metric structure closely related to ergodic homogeneous Markov chains.

The introduction considers the history of the problem and provides an overview of the main ideas and results presented in the article.

The first section gives the basic concepts of the theory of Markov chains.
In fact, a Markov chain is a mathematical model of some random process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.

The second section collects the basic definitions needed to consider the role of graph models
in the presentation and study of ergodic homogeneous Markov chains.
The Markov chain can be depicted as an oriented weighted graph of transitions whose vertices
correspond to the states of the chain and the arcs correspond to the transitions between them.
The process will be ergodic if this weighted oriented graph is weakly connected, and the largest common divisor of the lengths of all its cycles is equal to 1.
On the other hand, any connected graph can be used as a basis for building a model of the simplest Markov chain: if a vertex $i$ has
degree $k$, all incident edges are converted into arcs with the weights $\frac{1}{k}$.

In the third section the definition of the mean first passage time for an ergodic homogeneous Markov chain is given. There are several ways to build the corresponding matrix. The algorithm of finding the mean first passage time is analyzed in detail by using converging trees of the oriented graph, related to the transition matrix of the ergodic homogeneous Markov chain. Related recurrent procedure is described.

In the fourth section, a mean first passage time is analyzed as the quasi-metric $m$
of mean first passage time on the vertices $V =\{1, 2,..., n\}$ of the oriented graph corresponding to the transition matrix of a given ergodic homogeneous Markov chain: $m(i, j)$
is the expected number of steps (arcs) for random wandering on the oriented graph $\Gamma$, starting at $i$, to reach $j$ for the first time.
In particular, the quasi-metric of mean first passage time for the simple random walking on a connected unweighted graph $G$, in which there is an equal probability of moving from any vertex to any adjacent vertex,
is a  weighted quasi-metric, i.e., there exists a weight function $w: V\rightarrow\mathbb {R}_ {\ge 0}$,
%$w=(w_1, w_2,\ldots, w_n)$,
such that
$
m(i,j) + w_i=
m(j,i)+ w_j
$
for all $i, j\in V$.

We consider also some connections of the mean first passage time quasi-metric to other metric structures on graphs (in particular to $\alpha $-metric forest and its relatives), which are less studied, but not less interesting.

Finally, the fifth section deals with examples of the construction of mean first passage time quasi-metrics. In addition to illustrating the ''graphical`` procedure of building the matrix $M$, recurrent research algorithms are presented and the resulting generalized metric structures are analyzed.

In this paper, for an arbitrary monoid ${M(PE)}$ with an exponential sequence of primes $PE$ of type $q$, the inverse problem is solved, that is, finding the asymptotic for the distribution function of elements of the monoid ${M(PE)}$, based on the asymptotic distribution of primes of the sequence of primes $PE$ of type $q$.

To solve this problem, we introduce the concept of an arbitrary exponential sequence of natural numbers of the type $q$ and consider the monoid generated by this sequence. Using two homomorphisms of such monoids, the density distribution problem is reduced to the additive Ingham problem.

It is shown that the concept of power density does not work for this class of monoids. A new concept of $C$ logarithmic $\theta$-power density is introduced.

It is shown that any monoid ${M(PE)}$ for an arbitrary exponential sequence of primes $PE$ of type $q$ has $C$ logarithmic $\theta$-power density with $C=\pi\sqrt{\frac{2}{3\ln q}}$ and $\theta=\frac{1}{2}$.

In paper for any algebraically closed group $G$, as well as for the class of the algebraically closed groups, we prove algorithmic undecidability of the positive $\forall^2 \exists^{24}$-theory and $\forall^3 \exists^{2}$-theory. For an arbitrary $g\in G$, we also prove the decidability of the equation of the type
$$
w(x_1, \ldots , x_n) = g,
$$
where $w(x_1, \ldots , x_n)$ is a non-empty irreducible word in the unknowns $x_1,\ldots x_n\in G$.

Generalized hypergeometric function is defined as a sum of the power series whose coefficients
are the products of the values of some fractional rational function. Taken with a minus sign
roots of a numerator and denominator of this rational function are called parameters of the
corresponding hypergeometric function. For the investigation of the arithmetic nature of the
values of hypergeometric functions and their derivatives (including derivatives with respect to
parameter) one often makes use of Siegel’s method. The corresponding reasoning begins as
a rule by the construction of the functional linear approximating form. If parameters of the
hypergeometric function are rational one is able to use pigeonhole principle for the construction
of this form. In addition the construction is feasible not only for the hypergeometric functions
themselves but also for the products of their powers. By this is explained the generality of
results obtained by such method. But if there are irrational numbers among the parameters
the application of a pigeonhole method is impossible and for carrying out the corresponding
investigation it is necessary to employ some additional considerations.
One of the methods of surmounting the difficulty connected with the irrationality of
some parameters of a hypergeometric function consists in the application of the effective
construction of the linear approximating form from which the reasoning begins. Primarily
effective constructions of such approximations appeared for the functions of a special kind
(the numerator of the rational function by means of which the coefficients of hypergeometric
functions are defined was to be equal to unity). The investigation of the properties of these
approximations revealed the fact that they can be useful in case of rational parameters as well
for the quantitative results obtained by effective methods turned out to be more precise than
their analogs obtained by Siegel’s method. Subsequently the methods of effective construction
of linear approximating forms were generalized in diverse directions.
In this paper we propose a new effective construction of approximating form in case when
for the hypergeometric functions derivatives with respect to parameter are also considered. This
construction is made use of for the sharpening of the lower estimates of the linear independence
measure of the values of corresponding functions.

Rational numbers are uniformly distributed, even though distances between rational neighbors in a Farey sequence can be quite different. This property doesn't hold for algebraic numbers.
In 2013 D. Koleda [6, 7] found the distribution function for real algebraic numbers of an arbitrary degree under their natural ordering.

It can be proved that the quantity of real algebraic numbers $ \alpha $ of degree $n$ and height $H( \alpha ) \le Q$ asymptotically equals $c_{1}(n)Q^{n+1}$. Recently it was proved that there exist intervals of length $Q^{- \gamma }, \gamma >1$, free of algebraic numbers $ \alpha , H( \alpha ) \le Q$, however for $0 \le \gamma <1$ there exist at least $c_{2}(n)Q^{n+1- \gamma }$ algebraic numbers in such intervals.

In this paper we show that special intervals of length $Q^{-\gamma }$ may contain algebraic numbers even for large values of $ \gamma $, however their quantity doesn't exceed $c_{3}Q^{n+1-\gamma }$. An earlier result by A. Gusakova \cite{Gus15} was proved only for the case $\gamma = \frac{3}{2}$.

In the paper, we continue to study the distribution of inverse residues to given modulus. Earlier, the author obtained a series of non-trivial estimates for incomplete Kloosterman sums over prime numbers with an arbitrary modulus $q$. One of the applications of such estimates are some assertions concerning the distribution of inverse residues $\overline{p}$ to prime numbers lying in a "short" segment: $p\overline{p}\equiv 1\pmod{q}$, $1<p\leqslant N$, $N\leqslant q^{1-\delta}$, $\delta>0$, and, more general, concerning the distribution of the quantities $g(p) = a\overline{p}+bp$ with respect to modulus $q$, where $a,b$ are some integers, $(ab,q)=1$.

Another application is connected with the problem of the representation of a given residue $m\pmod{q}$ by the sum $g(p_{1})+\ldots+g(p_{k})$ for fixed $a,b$ and $k\geqslant 3$, in primes $1<p_{1},\ldots,p_{k}\leqslant N$. For the number of such representations, the author have found the formula, where the behavior of the expected main term is controlled by some analogous of the "singular series" that appears in classical circle method, that is, by some function $\kappa$ depending on $q$ and the tuple $k,a,b,m$. For fixed $k,a,b,m$, this function is multiplicative with respect to $q$. In the case when $q$ is not divisible by 2 or 3, this function is strictly positive, and therefore the formula for the number of the representations becomes asymptotic.

In this paper, we study the behavior of $\kappa$ for $q = 3^{n}$. It appears that, for any $n\geqslant 1$, $k\geqslant 3$, there exist the "exceptional" triples $a,b,m$ such that $\kappa = 0$.
The main purpose is to describe all such triples and to obtain the lower estimate for $\kappa$ for all non-exceptional triples.

We study the Nikolskii constant (or the Jackson-Nikolskii constant) for
complex trigonometric polynomials in the space $L_{\alpha}^{p}(\mathbb{T})$ for
$p\ge 1$ with the periodic Gegenbauer weight $|\!\sin x|^{2\alpha+1}$:
$$
\mathcal{C}_{p,\alpha}(n)=\sup_{T\in \mathcal{T}_{n}\setminus \{0\}}
\frac{\|T\|_{\infty}}{\|T\|_{p}},
$$
where $\|{\,\cdot\,}\|_{p}=\|{\,\cdot\,}\|_{L_{\alpha}^{p}(\mathbb{T})}$.
D. Jackson (1933) proved that $\mathcal{C}_{p,-1/2}(n)\le c_{p}n^{1/p}$ for all
$n\ge 1$. The problem of finding $\mathcal{C}_{p,-1/2}(n)$ has a
long history. However, sharp constants are known only for $p=2$. For $p=1$, the
problem has interesting applications, e.g., in number theory. We note the
results of Ja. L. Geronimus, L. V. Taikov, D. V. Gorbachev, I. E. Simonov,
P. Yu. Glazyrina. For $p>0$, we note the results of I. I. Ibragimov, V. I. Ivanov,
E. Levin, D. S. Lubinsky, M. I. Ganzburg, S. Yu. Tikhonov, in the weight case -
V. V. Arestov, A. G. Babenko, M. V. Deikalova, A. Horvath.

It is proved that the supremum here is achieved on a real even trigonometric
polynomial with a maximum modulus at zero. As a result, a connection is
established with the Nikolskii algebraic constant with weight
$(1-x^{2})^{\alpha}$, investigated by V. V. Arestov and M. V. Deikalova (2015).
The proof follows their method and is based on the positive generalized
translation operator in the space $L^{p}_{\alpha}(\mathbb{T})$ with the periodic
Gegenbauer weight. This operator was constructed and studied by D. V. Chertova
(2009). As an application, we propose an approach to computing
$\mathcal{C}_{p,\alpha}(n)$ based on the Arestov-Deikalova duality relations.

The Hankel matrices considered in the article arose at one reformulation of the Riemann
hypothesis proposed earlier by the author.
Computer calculations showed that in the case of the Riemann zeta function the eigenvalues
and the eigenvectors of such matrices have an interesting structure.
The article studies a model situation when instead of the zeta function function one takes
a function having a single zero. For this case we indicate the first terms of the asymptotic
expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding
eigenvectors.

A study of the periodicity problem of functional continued fractions of elements of elliptic and hyperelliptic fields
was begun about 200 years ago in the classical papers of N.~Abel and P.~L.~Chebyshev.
In 2014 V.~P.~Platonov proposed a general conceptual method based on the deep connection between three classical problems:
the problem of the existence and construction of fundamental $S$-units in hyperelliptic fields,
the torsion problem in Jacobians of hyperelliptic curves,
and the periodicity problem of continued fractions of elements of hyperelliptic fields.
In 2015-2019, in the papers of V.~P.~Platonov et al. was made great progress in studying the problem of periodicity of elements in hyperelliptic fields,
especially in the effective classification of such periodic elements.
In the papers of V.~P.~Platonov et al, all elliptic fields $\mathbb{Q}(x)(\sqrt{f})$ were found
such that $\sqrt{f}$ decomposes into a periodic continued fraction in $\mathbb{Q}((x))$,
and also futher progress was obtained in generalizing the indicated result, as to other fields of constants,
and to hyperelliptic curves of genus $2$ and higher.
In this article, we provide a complete proof of the result announced by us in 2019
about the finiteness of the number of elliptic fields $k(x)(\sqrt{f})$ over an arbitrary number field $k$
with periodic decomposition of $\sqrt{f}$,
for which the corresponding elliptic curve contains a $k$-point of even order not exceeding $18$ or
a $k$-point of odd order not exceeding $11$.
For an arbitrary field $k$ being quadratic extension of $\mathbb{Q}$ all such elliptic fields are found,
and for the field $k = \mathbb{Q}$ we obtained new proof about of the finiteness of the number of periodic $\sqrt{f}$, not using the parameterization of elliptic curves and points of finite order on them.

The article gives proof of the completeness of the list of one class of convex symmetric polyhedra in three-dimensional Euclidean spac. This class belongs to the class of so-called RR-polyhedra. The RR-polyhedra are characterized by the following symmetry conditions: each polyhedron of the class RR has symmetric rhombic vertices and there are faces that do not belong to any star of these vertices; and each face that does not belong to the star of the rhombic vertex is regular. The vertex rhombicity here means that the vertex star is composed of n equal, equally spaced rhombuses. The symmetry of the vertex means that the rotation axis of the order n of its star passes through it. Previously, the author found all polyhedra with rhombic or deltoid vertices and locally symmetric faces. Moreover, locally symmetric faces do not belong to any of the rhombic or deltoid stars. The class of RR-polyhedrons is obtained from the previously considered replacement of the local symmetry condition of the non-rombic faces by the condition of their regularity.

Thus, the considered class RR is connected with the well-known result of D. Johnson and V. Zalgaller on the enumeration of all convex polyhedra with the condition of the regularity of the faces. But as shown in this article, RR-polyhedra cannot simply be obtained from the class of regular-faced, but require a special method. The proof of completeness of the class of RR-polyhedra with two isolated symmetric rhombic vertices V, W is given in this article. Wherein, the rhombuses converge at the vertices of V, W not necessarily at its acute angles, and V, W are not necessarily separated by only one belt of regular faces.

The article considers a boundary value problem for a system of linear second-order ordinary
differential equations constructed for determining the displacement field in a continuously
inhomogeneous elastic coating of the plate when a plane sound wave passes through.
It is believed that a homogeneous isotropic elastic plate with inhomogeneous in thickness
elastic coating borders on ideal liquids.
An approximate analytical solution of the boundary value problem by the power series
method is obtained . The boundary-value problem is reduced to problems with initial conditions.
The solution of the boundary value problem is presented in the form of a linear combination
of fundamental decisions. Found analytical solution boundary value problem is valid for a wide
class of heterogeneity laws of the coating material.
The numerical calculations of the dependences of the components of the displacement vector
at the boundaries of the coating from the angle of incidence of the plane wave are presented.

One of the pressing contemporary problems of algebra and number theory is the problem
of the existence and searching for fundamental S-units in hyperelliptic fields. The problem of
the existence and searching of S-units in hyperelliptic fields is equivalent the solvability of the
norm equation - the functional Pell equation - with some additional conditions on the form
of this equation and its solution. There is a deep connection between points of finite order in
Jacobian variety (Jacobian) of hyperelliptic curve and nontrivial S-units of hyperelliptic field.
This connection formed the basis of the algebraic approach proposed by V. P. Platonov to the
well-known fundamental problem of boundedness of torsion in Jacobian varieties of hyperelliptic
curves. For elliptic curves over a field of rational numbers, the torsion problem was solved by
Mazur in the 1970s. For curves of genus 2 and higher over the field of rational numbers, the
torsion problem turned out to be much more complicated, and it is far from its complete solution.
The main results obtained in this direction include to the description of torsion subgroups of
Jacobian varieties of specific hyperelliptic curves, and also to the description of some families
of hyperelliptic curves of the genus f >= 2.

In this article, we have found a new method for studying solvability. functional norm
equations giving a full description hyperelliptic curves over the field of rational numbers, whose
Jacobian varieties possess torsion points of given orders. Our method is based on an analytical
study of representatives finite order divisors in a divisor class group of degree zero and their
Mumford representations. As an illustration of the operation of our method in this article,
we directly found all parametric families of hyperelliptic curves of genus two over the field of
rational numbers, whose Jacobian varieties have rational torsion points of orders not exceeding
five. Moreover, our method allows us to determine which parametric family found this curve
belongs, whose Jacobian has a torsion point of order not exceeding five.

A mean-value theorem for multiple trigonometric generalizing from the G. I. Arkhipov’s
theorem [12, 13] was proved. The first theorem of the similar type lies in the core of the
I. M. Vinogradov’s method [2]. In the paper the version of theorem with “similar” lengths of
changing intervals of variables. Estimates of zeta-sums of the form

$$
\sum_{n\leq P}n^{it}.
$$
are the interesting application of the I. M. Vinogradov's method. The similar application of the mean-value theorem proving by us serve the estimate of sums of the form
$$
\sum_{n\leq P_1}\dots\sum_{n\leq P_r}(n_1\dots n_r+k)^{it}, \sum_{n\leq P}\tau_s(n)(n+k)^{it}, \sum_{p\leq P}(p+k)^{it}.
$$

For the first time  in the article [1] was established non-trivial lower bounds on the size of the set of products of rational numbers, the numerators and denominators of which are limited to a certain quantity $Q$. Roughly speaking, it was shown that the size of the product deviates from the maximum by no less than $$\exp \Bigl\{(9 + o(1)) \frac{\log Q}{\sqrt{\log{\log Q}}}\Bigl\}$$ times. In the article [7], the index of $ \log{\log Q} $ was improved from $ 1/2 $ to $ 1 $, and the proof of the main result on the set of fractions was fundamentally different. This proof, its argument was based on the search for a special large subset of the original set of rational numbers, the set of numerators and denominators of which were pairwise mutually prime numbers. The main tool was the consideration of random subsets. A lower estimate was obtained for the mathematical expectation of the size of this random subset. There, it was possible to obtain an upper bound for the multiplicative energy of the considered set. The lower bound for the number of products and the upper bound for the multiplicative energy of the set are close to optimal results. In this article, we propose the following scheme. In general, we follow the scheme of the proof of the article [1], while modifying some steps and introducing some additional optimizations, we also improve the index from $1/2$ to $1-\varepsilon$ for an arbitrary positive $\varepsilon>0$.

In the paper, we consider $N_d(x)=N(x;\alpha,\beta;d,a)$, $x\in\mathbb{N}$, which is the number of values of Beatty sequence $[\alpha n+\beta]$, $1\leqslant n\leqslant x$, for $\alpha>1$ irrational and with bounded partial quotients, $\beta\in[0;\alpha)$, in an arithmetic progression $(a+kd)$\textup, $k\in\mathbb{N}$. We prove the asymptotic formula $N_d(x) = \frac{x}{d} + O(d\ln^3 x)$ as $x\to\infty,$
where the implied constant is absolute. For growing difference~$d$ the~result is non-trivial provided $d\ll \sqrt{x}\ln^{-3/2-\varepsilon}x$, $\varepsilon>0$.

In this paper, we construct Hensel–Shafarevich generating set in Honda formal modules over a higher dimensional field. Later, that should allow us to compute Hilbert symbol in this case.

In paper we discuss the solution of mean value general form problem in case of all variables
symmetry absence. In 1930 A. N. Kolmogorov proved the formula for general form of mean
value. He formulated four axioms: continuity and monotony on each variable, symmetry on
each variable, mean value of equal variables is equal to these variables, any substitution of any
group of variables with their mean value does not change the mean value. In Kolmogorov’s
theorem all arguments are equitable, this means that the mean value is symmetric on each
variable. V. N. Chubarikov set the task of generalization to this result in case of all variables
symmetry absence. We divide all the variables on groups and the mean value is a symmetric
function for variables in each group separately. For example, if we have only one group the mean
value will be Kolmogorov’s mean value, so we have a generalization of Kolmogorov’s theorem.
In paper we show the general form of mean value in our case and we note the connection with
uniform distribution modulo 1.

In this paper we give an explicit construction of morsifications with the smallest topologically
possible number of real critical points for functions of two variables with smooth level-set
branches, as well as for semiquasihomogenous functions of two real variables.

January 11, 2020 marks the centenary of the birth of Vassiliy Ilyich Nechaev, a well-known
Soviet mathematician, Doctor of Physical and Mathematical Sciences, Professor, Head of the
Department of Number Theory of Moscow Pedagogical State University (MSPU).

Vassiliy Ilyich Nechaev has made a significant contribution to Development of Russian
scientific school on number theory. The research of Professor V. I. Nechaev and his disciples in
the field of analytical number theory and its applications is very considerable.

Moreover, the scientific and educational activities of Professor V.I. Nechaev have been very
influential in the development of methods of teaching mathematics at colleges for teacher
education. V. I. Nechaev was a leading figure at the scientific school and the organizer of
the analytic number theory research seminar at the Moscow State Pedagogical University.
Among his many students received their PhD and doctoral degrees. Vassiliy Ilyich Nechaev
has published a large number of scientific and methodical works, was a translator of several
fundamental scientific monographs. He was for many years a member of the editorial board of
the journal ”Mathematical Notes“ and served as a member of the program committees of many
international conferences on algebra and number theory.