The estimate of the trigonometric sum of the kind
$$
S=\sum_{a<t_s\leq b}e^{2\pi if(t_s)},
$$
where $a\geq 0,a\leq b$ are real numbers, $t_s$ is increasing to infinity of non-negative numbers, $f(t)$ is a smooth real function, is found.

Here also there are proved the analogues of Euler's, Sonin's, Poisson's and van der Corput's formulas for considering sum.

Let be given the sequence of $\Delta$ points
$$
0=t_0<t_1<t_2<\dots<t_s<\dots, \lim\limits_{n\to\infty}t_n=+\infty,
$$
on the positive half-axis of the real line.

For non-negative number $x$ we define the analogue of the integer part $[x]_{\Delta},$ meeting to the sequence $\Delta: [x]_{\Delta}=t_s,$ if $t_s\leq x<t_{s+1}, s\geq 0.$ The fractional part $\{x\}_{\Delta}$ is defined by the equality
$$
\{x\}_{\Delta}=\frac{x-t_s}{t_{s+1}-t_s},
$$
if $t_s\leq x<t_{s+1}, s\geq 0,$ moreover $0\leq\{x\}_{\Delta}<1.$

We define the analogue of the Bernoulli function meeting to the sequence $\Delta: \rho_\Delta(x)=0,5-$ $-\{x\}_\Delta.$

Then is valid the following analogue of the van der Corput's theorem for subdivisions.

{\sl Let $\Delta=\{t_s\}, 0=t_0<t_1<\dots<t_s<\dots, $ be a subdivision of the half-axis $t\geq 0$ of the real line, $\delta_s=t_{s+1}-t_s\geq 1, \delta(a,b)=\max\limits_{a\leq x\leq b}{\rho'_{\Delta}(x)},$ and let be given the sequence $\Delta_0=\{\mu_s\}, \quad \mu_s=0,5(t_s+t_{s+1}), s\geq 0,$ and points $a,b\in\Delta_0,$ let, also, $f'(x)$ be continuous, monotonic sign-constant in the interval $a< x\leq b,$ moreover there exists the constant $\delta$ such that $0<2\delta\delta^{-1}(a,b)<1$ and that for all $x$ from this interval is valid inequality $|f'(x)|\leq\delta.$ Then we have
$$\sum_{a<t_s\leq b}e^{2\pi if(t_s)}=\int\limits_{a}^{b}\rho'_\Delta(x)e^{2\pi if(x)}\,dx+10\theta\frac{\delta}{1-\delta\delta^{-1}(a,b)}, |\theta|\leq 1.
$$

This paper proves that additive categories that occur as hearts of weight structures are
precisely the weakly idempotent complete categories, that is, the categories where all split
monomorphisms give direct sum decompositions. The work also gives several other conditions
equivalent to weak idempotent completeness (some of them are completely new) and discusses
weak idempotent completions of additive categories

The problem studied in the thesis arose from the need to find connections between algebraic
field theory and theory of functions. The Cauchy integral theorem, which is one of the most basic
and classical results of the complex analysis, has a discrete analog in the case of one-dimensional
local fields. The natural question then arises whether it is possible to generalize the same result
to two-dimensional local fields. The present paper contains the definition of Schnirelmann’s
integral and the proof of an analog of Cauchy’s integral theorem for two-dimensional local
fields. As a consequence, links between the Hilbert symbol and Schnirelmann’s integral are
established

We prove that any square matrix over an arbitrary infinite field is a sum of a square-zero
matrix and a diagonalizable matrix. This result somewhat contrasts recent theorem due to
Breaz, published in Linear Algebra & Appl. (2018).

In the work, a mathematical model of digital control of multi-circuit objects is built, taking
into account the real characteristics of a digital controller as an element of a control system.
The problem is formulated that the methods of modeling digital control systems are known and
are widely used in engineering practice, however, in the overwhelming majority, they involve
the formation of models that do not take into account the presence of time intervals between
transactions in a Von Neumann type computer.
To solve the problem, a typical block diagram of complex multi-loop control systems with
digital controllers of the Von Neumann type has been developed, which takes into account the
random nature of the processed data and real time delays between transactions.
It is proposed, taking into account the randomness of the time interval between transactions
and the stochastic nature of switching to conjugate operators, to consider a semi-Markov process
as an adequate model of the algorithm for the functioning of digital control systems.
On the basis of semi-Markov processes, a method is proposed for estimating the parameters
of time intervals between transactions in cyclic control algorithms, which makes it possible to
evaluate the characteristics of the system at the design stage, and therefore is the key to the
rational design of digital control systems for multi-circuit objects with control algorithms of
almost any complexity. An example of mathematical modeling of a two-circuit system with
digital control is presented.

We consider sums of values of the composition of a real periodic arithmetic function and
the number of prime divisors function over integers not exceeding a given bound. The prime
divisors may be counted as with their multiplicity or without it, and we can restrict these
divisors to the additional condition of belonging to some special set. This special set may be,
for example, a sum of several arithmetic progressions with a given difference or imply an analog
of prime number theorem with a power decrement in the remainder term. Moreover, instead of
the number of prime divisors function we can consider an arbitrary real additive function that
equals to one on primes. As an example of the periodic arithmetic function we can consider
the Legendre symbol. In the paper we prove asymptotic formulae for such sums and investigate
their behavior.
The proof uses the decomposition of the periodic arithmetic function into additive characters
of the residue group, so the problem reduces to special trigonometric sums with the number
of prime divisors function in the exponent. In order to establish asymptotic formulae for such
sums we consider the corresponding Dirichlet series, accomplish its analytic continuation and
make use of the Perron formula and complex integration method in specially adapted form.

Let $ g: \mathbb{F}_q[T] \rightarrow \mathbb{R} $ be a multiplicative function which values at the degrees of the irreducible polynomial, depends only on the exponent, such that $g(P^k)=d_k$ polynomial $P$ and for some arbitrary sequence of reals $\{d_k\}_{k=1}^{\infty}$. This paper regards the sum $$ T (N) = \sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}{g (F)}, $$ where $ F $ ranges over polynomials of degree $ N $ with leading coefficient equal to 1 (unitary polynomials). For the sum $ T (N) $, an exact formula is found, and various asymptotics are calculated in cases of \\ $ q \to \infty; \ q \to \infty, \ N \to \infty; \ q ^ N \to \infty $. In particular, the following asymptotic formulas are obtained
$$\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\tau(F^k)=\binom{k+N}{N}q^N+O_{N,k}\left(q^{N-1}\right),\ \ N\ge 1,\ q\to\infty;
$$
$$
\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\dfrac{1}{\tau(F)}=\dfrac{q^N}{4^N}\left(\binom{2N}{N}-\dfrac{2}{3}\binom{2N-4}{N-2}q^{-1}+O\left(\ \dfrac{4^N}{\sqrt{N}}q^{-2}\right)\right),\ N\to\infty,\ q\to\infty;
$$
$$\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\dfrac{1}{\tau(F)}=C_1\cdot\dfrac{\binom{2N}{N}}{4^N}q^N+O\left(\dfrac{q^{N-0.5}}{N^{1.5}}\right),\ \ C_1=\prod_{l=1}^{+\infty}\left(\sqrt{q^{2l}-q^{l}}\ln\dfrac{q^l}{q^l-1}\right)^{\pi_q(l)},\ q^N\to\infty;$$
where $\tau(F)$ is a number of monic divisors of $F$, and $\pi_q(l)$ is a number of monic irreducible polynomials of degree $l$.
The second and third equalities are analogous for polynomials over a finite field of one of Ramanujan's results
$$\sum_{n\leq x}{\dfrac{1}{d(n)}}=\dfrac{x}{\sqrt{\ln x}}\left(a_0+\dfrac{a_1}{\ln{x}}+\ldots+\dfrac{a_N}{(\ln{x})^N}+O_N\left(\dfrac{1}{(\ln{x})^{N+1}}\right)\right),$$
where $d(n)$ is a classical divisor function, and $a_i$ are some constants. In particular,
$$a_0=\dfrac{1}{\sqrt{\pi}}\prod\limits_{p}\ln\dfrac{p}{p-1}\sqrt{p(p-1)}.$$

Corrections to the article “Gorbachev D.V., Martyanov I.A. Interrelation between Nikolskii—
Bernstein constants for trigonometric polynomials and entire functions of exponential
type. Chebyshevskii Sbornik. 2019; 20(3): 143-153. (In Russ.) https://doi.org/10.22405/
2226-8383-2019-20-3-143-153”.

This paper presents an overview of mathematical models that allow us to determine the
effective elastic characteristics of various types of composite materials. The most well-known
models are considered: virial decomposition, self-matching method, correlation approximation,
and singular approximation. Models with a layered structure and matrix systems with a regular
structure are considered.

The paper analyzes the results of laboratory experimental studies of parameters that
characterize the plasticity, strength and wear resistance of P18 tool steel obtained during
material science examinations. New empirical mathematical expert models have been developed
for the dependence of P18 steel ductility and strength on temperature. Empirical mathematical
expert models of the dependence of mechanical properties and wear resistance of P18 steel on
various factors are obtained. The possibility of analytical representation of complex experimental
graphical dependencies of mechanical and tribological properties of P18 steel on various factors
for use in forensic practice is shown.

Today, there is a risk of destruction for a large number of buildings from various emergencies.
Modern regulatory framework for the design and operation of buildings, contains many years of
experience in analyzing the causes of collapse, takes into account a large number of impacts on
structures (dynamic loads, climatic effects, temporary and permanent) during the entire service
life. However, the increasing number of accidents with varying degrees of destruction, both of
individual parts and of the entire structure, suggests that the impact that caused the destruction
was not taken into account in the regulatory documents on the basis of which the object was
designed. Therefore, there is a need for accurate calculation algorithms, modern reliable and
cost-effective methods for the structural strengthening of supporting frames of buildings.
The article considers existing methods for predicting the effects of fracture and solving
problems for determining the stress-strain state based on a specially developed RHT (Riedel-
Hiermaier-Thoma) strength model for high-speed deformation of reinforced concrete under
dynamic loading conditions. A model problem is considered using a variational approach based
on the construction of a functional for calculating the elastic deformation power, taking into
account the power of inertia forces for a spherical explosive charge located directly in front of
the structure. All calculations were performed in the ANSYSLS-DYNA environment, the results
were obtained in the form of graphs of strain rates and stress fields.

In paper the method of spectral elements based on the Legendre polynomial for timeindependent
elastic-plastic plane problems at large strains is proposed. The method of spectral
elements is based on the variational principle (Galerkin’s method). The solution of these
problems has the phenomenon of localization of plastic deformations in narrow areas called
slip-line or shear band. The possibility of using a spectral element for the numerical solution of
these problems with discontinuous solutions is investigated. The yield condition of the material
is the von Mises criterion. The stresses are integrated by the radial return method by backward
implicit Euler scheme. The system of nonlinear algebraic equations is solved by the Newton’s
iterative method. A numerical solution is given of an example of stretching a strip weakened by
cuts with a circular base in a plane stress and plane deformed state. Kinematic fields and limit
load are obtained. Comparisons of numerical results with the analytical solution obtained for
incompressible media constructed by the method of characteristics are presented.

This paper is devoted to the approximation of a quadratic algebraic lattice by an integer

lattice. It calculates the distances between a quadratic algebraic lattice and an integer lattice

when they are given by the numerator and denominator of a suitable fraction to the square root

of the discriminant d — of a square-free natural number.

The results of this work allow us to study questions about the best approximations of

quadratic algebraic lattices by integer lattices.

This paper is devoted to the construction of fast algorithms for calculating the quality

function of rational grids that approximate quadratic algebraic grids in the General case of the

maximum lattice of integer algebraic numbers.

It is shown that the generalized parallelepipedal net approximating the quadratic algebraic

net is parallelepiped.As a consequence, an algorithm for calculating the quality function for

0 (lnN) arithmetic operations is constructed.

The paper continues the author's research on the evaluation of trigonometric sums of an algebraic net with weights with the arbitrary weight function of the $r+1$ order.

For the parameter $\vec{m}$ of the trigonometric sum $S_{M(t),\vec\rho} (\vec m)$, three cases are highlighted.

If $\vec{m}$ belongs to the algebraic lattice $\Lambda (t \cdot T(\vec a))$, then the asymptotic formula is valid
$$
S_{M(t),\vec\rho}(t(m,\ldots, m))=1+O\left(\frac{\ln^{s-1}\det \Lambda(t)} { (\det\Lambda(t))^{r+1}}\right).
$$

If $\vec{m}$ does not belong to the algebraic lattice $\Lambda(t\cdot T(\vec a))$, then two vectors are defined $\vec{n}_\Lambda(\vec{m})=(n_1,\ldots,n_s)$ and $\vec{k}_\Lambda(\vec{m})$ from the conditions $\vec{k}_\Lambda(\vec{m})\in\Lambda$, $\vec{m}=\vec{n}_\Lambda(\vec{M})+\vec{K}_\lambda(\vec{m})$ and the product $q(\vec{n}_\lambda(\vec{m}))=\overline{n_1}\cdot\ldots\cdot\overline{n_s}$ is minimal. Asymptotic estimation is proved
$$
|S_{M(t),\vec\rho}(\vec{m})|\le B_r\left(\frac{1-\delta(\vec{k}_\Lambda(\vec{m}))}{(q(\vec{n}_\Lambda(\vec{m})))^{r+1}}+O\left(\frac{q(\vec{n}_\Lambda(\vec{m}))^{r+1}\ln^{s-1}\det \Lambda(t)}{ (\det\Lambda(t))^{r+1}}\right)\right)

The article considers a variant of the approximation of algebraic lattices by integer ones in
the quadratic case, the set of their local minima is written out explicitly, and it is also shown
that for these integer approximations of algebraic quadratic lattices it is possible to construct
efficient algorithms for calculating the hyperbolic parameter.

For an approximate calculation of a curvilinear integral
$$J(f;\Gamma):=\int\limits_{\Gamma}f(x_1,x_2,\ldots,x_m)dt$$
when the curve $\Gamma$ is given by parametric equations
$$x_{1}=\varphi_{1}(t),
x_{2}=\varphi_{2}(t),\ldots,x_{m}=\varphi_{m}(t), 0\leq t\leq L$$
the quadrature formula is entered into consideration
$$J(f;\Gamma):\approx\sum_{k=1}^{N}p_{k}\, f\Bigl(\varphi_{1}(t_k),\,
\varphi_{2}(t_k), \ldots,\, \varphi_{m}(t_k)\Bigr),$$ where
$P=\left\{p_{k}\right\}_{k=1}^{N}$ and $T:=\left\{t_{k}:0\leq
t_{1}<t_{2}<\cdots<t_{N}\leq L\right\}$-- are arbitrary vector
coefficients and nodes. Let
$H^{\omega_{1},\ldots,\omega_{m}}[0,L]$-- sets of curves $\Gamma$,
whose coordinate functions $\varphi_{i}(t)\in H^{\omega_{i}}[0,L] \
(i=\overline{1,m}),$ where $\omega_{i}(t) \ (i=\overline{1,m})$--
are given moduli of continuity $\mathfrak{M}_{\rho}^{\omega,p}$--
functions class $f(M),$ defined in point $M\in\Gamma,$ such for any
two points
$M^{\prime}=M(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m}^{\prime}),$
$M^{\prime\prime}=M(x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{m}^{\prime\prime})$
belonging to a curve $\Gamma \in
H^{\omega_{1},\ldots,\omega_{m}}[0,L]$ satsify the condition
$$\Bigl|f(M^{\prime})-f(M^{\prime\prime})\Bigr|\le\omega(\rho_{p}(M^{\prime},
M^{\prime\prime})),$$ where $$\rho_{p}(M^{\prime},
M^{\prime\prime})=\left\{\sum_{i=1}^{m}|x^{\prime}_{i}-x_{i}^{\prime\prime}|^{p}\right\}^{1/p},
\ 1\leq p\leq \infty,$$ $\omega(t)$-- given moduls of continuity.
It is proved that among all quadrature formulas of the above from,
the best for a class of functions $\mathfrak{M}_{\rho}^{\omega,p}$
and a class of curves $H^{\omega_{1},\ldots,\omega_{m}}[0,1]$, is
the formula of average rectangles.

The exact error estimate of the best quadrature formula is
calculated for all the functional classes under consideration and
the curves are given a generalization for more general classes of
functions.}

The authors of the article set themselves two main tasks: to characterize the main stages
of life of a professor at Tula State Lev Tolstoy Pedagogical University Vladimir Nikolaevich
Bezverkhnii and to give a brief analysis of his scientific and pedagogical activity, which has a
significant impact on the development of the combinatorial group theory.
The studies of Professor V. N. Bezverkhnii and his students on algorithmic problems in the
theory of groups and semigroups are particularly noted.
V. N. Bezverkhnii, a student of Professor M. D. Grindlinger, conducts the seminar
"Algorithmic Problems of Group and Semigroup Theory"and supervises postgraduate studies
in combinatorial group theory.

Among his students, 8 people defended their Ph.D. theses, and one of them later became
Doctor in Physics and Mathematics.