The article develops the theory of integral transforms in order to obtain operational calculus for the study of transient events. An analogue of the Laplace transform is introduced, which can
be applied to expressions with a piecewise constant factor before the differentiation operator.
Concepts such as original function, the Laplace transform, convolution are defined. Theorems on differentiation of the original, on differentiation of the Laplace transform and others have been proved. A generalized convolution definition is given and a formula for calculation such convolution is proved. Based on the concept of convolution, a fractional integral is defined.
The transmutation operators method is the main tool in the theory of generalized operational calculus. The generalized Laplace integral transforms introduced in the article and the classical Laplace integral transforms are connected with its help. The solution to the heat problem with piecewise constant coefficients for the semi-infinite rod is found.

The paper studies the question of representing numbers as the sum of two primes from an arithmetic progression, that is, the binary Goldbach problem, when primes are taken from an
arithmetic progression. New estimates are proved for the number of even natural numbers that are (possibly) not representable as a sum of two primes from an arithmetic progression and for a number representing a given natural number, as a sum of two primes from an arithmetic progression.

The paper shows that a linear manifold of matrices of the form: Q=Q0+Σ︀a𝑖P𝑖, can consist of projectors only. It turns out that for this it is necessary and sufficient that P𝑖 =Q𝑖-Q0 and all the matrices Q𝑖 be projectors, moreover: (Q𝑖-Q𝑗)2=0 for any pair i and j. It is established that all projectors that make up this linear manifold have one rank and any pair A, B of these projectors satisfies (A-B)2=0.
Several conditions were found equivalent to the fact that two projectors A,B satisfy (AB) 2=0, one of them in terms of the subspaces defining these projectors.
Let n be the order of the projectors Q𝑖, r be their rank, then it is shown that the maximum number of linearly independent matrices P𝑖=Q𝑖-Q0 such that the conditions (Q𝑖-Q𝑗)2=0 are
satisfied is r(n-r). Therefore, any projector of rank r can be represented as the sum of an orthoprojector Q0 and a linear combination of at most r(n-r) projectors Q𝑖 so that (Q𝑖-Q𝑗)2=0,
i,j=0,1,..,r(n-r).
The paper calculates the minimum distance between two projectors of ranks k and l - |𝑘 − 𝑙|1/2. The maximum distance between two orthoprojectors of the same rank k is (2𝑘)1/2.
It is established that the polynomial h(p,q)=(p-q)2 plays a special role for the algebra 𝒜(𝑝, 𝑞) generated by the projectors p,q,I. The polynomial h generates the center of this algebra — the set of elements commuting with all elements of 𝒜(𝑝, 𝑞).

The paper considers four new concepts: a modified basic measure of the quality of a set of coefficients, absolutely optimal coefficients of the index 𝑠, the mathematical expectation of
the local deviation of the parallelepipedal grid and the variance of the local deviation of the parallelepipedal grid.
It is shown that at least ((𝑝−1)^𝑠)/2 of different sets (𝑎1, . . . , 𝑎𝑠) integers mutually prime with the module 𝑝 will be absolutely optimal sets of the index 𝑠 with the constant 𝐵 = 2𝑠.
It is established that any absolutely optimal set of optimal coefficients of the 𝑠 index is an optimal set of optimal coefficients of the 𝑠 index, while any subset of its 𝑠1 coefficients is an
optimal set of optimal coefficients of the 𝑠1 index.
For the finite deviation introduced by N. M. Korobov in 1967, new formulas and estimates are obtained for parallelepipedal grids.
In this paper, for the first time, the concept of the mathematical expectation of a local deviation is considered and a convenient formula for its calculation is found.
The concept of local deviation variance is also considered for the first time.
The paper outlines the directions of further research on this topic.

Linkages can be represented as devices consisting of solid bodies, for example, rods, some pairs of which are connected to each other by hinges, in other words they have a common point
around which they can freely rotate. Linkages became widespread along with the development of instrumentation. One of the important first problems was to design a mechanism in which one of the hinges would move along a straight line segment. This issue has received several solutions, some of which were proposed by Peaucellier, Lipkin, Watt, Garth. After it became
clear how to draw a segment, the next big problem was to describe all possible curves that could be the trajectories of one of the hinges of a linkage. The solution to this problem was King’s theorem, which says that a set can be drawn if and only if it is either an ambient space or a semi-algebraic compact [16], [17].
The issues investigated by the author of this paper continue the exploration of previous tasks related to linkages, since they consider the possibilities of solving optimization problems
using linkages, for example, finding the shortest network connecting a set of points in Euclidean space. The main result of this work describes the construction of a mechanism that builds a
minimal parametric network in a Euclidean space of dimension 𝑑 > 2. In the author’s previous work, a proof of the existence of a linkages that builds a minimal Steiner network is given, and
a variant of constructing such a mechanism is also proposed. Since the main task was to prove the existence of such a mechanism, without minimizing it. The described assembly method can obviously be optimized and the results obtained in this work allows us to do that.

Gelfond proved the uniformity of distribution of the sums of binary digits expansions of natural numbers in arithmetic progressions. Later, this result was generalized to many other
numeration systems, including Fibonacci numeration system.
Eminyan find an asymptotic formula for the number of natural 𝑛, not exceeding a given one, such that 𝑛 and 𝑛 + 1 have a given parity of the sum of digits of their binary expansions.
Recently, this result was generalized by Shutov to the case of Fibonacci numeration system.
In the paper we consider quite more general problem about the number of natural 𝑛, not exceeding 𝑋, such that 𝑛 and 𝑛 + 𝑙 have a given parity of the sum of digits of their representations in Fibonacci numeration system. A method is presented that allows to obtain asymptotic formula for a given quantity for all 𝑙. It is based on the study of some special sums associated with the problems and recurrence relations for these sums. It is shown that for any 𝑙 and all variants of parity the leading term of the asymptotic is different from the expected value 𝑋/4 . Als it is proved that the remainder has the order 𝑂(log𝑋). For 𝑙 ≤ 10 constants in the leading term of asymptotic formulas are found explicitly.
In the conclusion of the work, some open problems for further research are formulated.

The paper provides a definition of the hinge mechanism, taking into account its kinematic nature. This definition differs significantly from that adopted by a number of mathematicians
in recent works. If we use the definition accepted today, which does not take into account the kinematic background, then the classical result of A. B. Kempe [1] about the possibility of
drawing by parts of an arbitrary plane algebraic curve with hinges of suitably chosen plane hinge mechanisms cannot be considered sufficiently substantiated by Kempe himself. This has
been noted in the modern literature [6], and even led to accusations of Kempe in error. The development and modern substantiation of Kempe’s result proposed in the works [6, 7] is, in essence, a modification of Kempe’s method for constructing the required mechanism from brick mechanisms performing algebraic actions. However, it is based on the use of a complex language of modern algebraic geometry, which leads to the replacement of Kemp’s short and transparent reasoning by an order of magnitude longer and difficult to understand texts. In our definition of the hinge mechanism, we can give a rigorous formulation of Kempe’s theorem, for the proof of which Kempe’s arguments with minimal refinements are sufficient. This updated proof is provided in the paper. The paper discusses the modern development of Kemp´e’s result, and the claims against Kemp´e’s reasoning. It also gives general ideas about mathematics that the
author has in connection with the Kemp´e theorem and its modern development.

The paper is an attempt both to give an overview of the results of OM Kasim-Zade, the largest specialist in discrete mathematics and mathematical cybernetics, and to understand his
scientific legacy in fields such as research measures the circuit complexity of Boolean functions related to the operation of the circuits, the problems of implicit and parametric expressibility
in finite-valued logics, the questions of the depth and the complexity of Boolean functions and functions of multivalued logics in infinite bases.

The article provides a statistical analysis of the properties of lexical and n-gram models of the Russian language based on the news text corpus. A specialized corpus of political news articles of recent years has been created, reflecting a narrow area of language use. The token and n-gram dictionaries are compiled, the coverage values are found, as well as the values of
entropy. Lemmatization of the original text corpus and extrapolation of the dictionary volumes are performed.

The Chebyshev inquality is one of important inequalities in mathematics. It’s a necessary tool in probability theory. The item of Chebyshev’s inequality may also refer to Markov’s
inequality in the context of analysis.
In[6, 7], using the usual Riemann–Liouville fractional integral operator 𝐼𝛼, were established and proved some new integral inequalities for the Chebyshev fonctional

$$𝑇(𝑓, 𝑔) :=1/(𝑏 − 𝑎)∫︁𝑎𝑏 𝑓(𝑥)𝑔(𝑥)𝑑𝑥 −1/(𝑏 − 𝑎)∫︁𝑎𝑏 𝑓(𝑥)𝑑𝑥 1/(𝑏 − 𝑎)∫︁ 𝑎𝑏 𝑔(𝑥)𝑑𝑥.$$

In this work, we give some generalizations of Chebyshev-type integral inequalities by using Riemann—Liouville fractional integrals of function with respect to another function.

Given a list 𝐿(𝑣) for each vertex 𝑣, we say that the graph 𝐺 is 𝐿-colorable if there is a proper vertex coloring of G where each vertex 𝑣 takes its color from 𝐿(𝑣). The graph is uniquely 𝑘-list
colorable if there is a list assignment 𝐿 such that |𝐿(𝑣)| = 𝑘 for every vertex 𝑣 and the graph has exactly one 𝐿-coloring with these lists. If a graph 𝐺 is not uniquely 𝑘-list colorable, we also
say that 𝐺 has property 𝑀(𝑘). The least integer 𝑘 such that 𝐺 has the property 𝑀(𝑘) is called the 𝑚-number of 𝐺, denoted by 𝑚(𝐺). In this paper, first we characterize about the property of
the complete tripartite graphs when it is uniquely 𝑘-list colorable graphs, finally we shall prove that 𝑚(𝐾2,2,𝑚) = 𝑚(𝐾2,3,𝑛) = 𝑚(𝐾2,4,𝑝) = 𝑚(𝐾3,3,3) = 4 for every 𝑚 > 9, 𝑛 > 5, 𝑝 > 4.

In the article a generalized empirical mathematical model of the dynamics of changes in the friction force at rest and the beginning of sliding is presented. Using the example of the friction of a ball made of ShKh15 steel over 𝑆𝑖𝑂2 coatings deposited on flat surfaces made of polycarbonate and polyethylene terephthalate, it is shown that there are deviations from the stationary value of
the friction force when sliding over short distances. The developed mathematical model describes the frictional interaction both at a stationary value of the friction force and at deviations from it.

The paper presents the results of a study of the influence of biological lubricants on the tribological properties of the friction pair "steel - titanium alloy". Tribological tests were carried out with indentation (scratching) on a friction path of 2 mm with an increase in load from 0.030 to 10 N on a Micro Indentation Tester CSM. The studies were carried out under various conditions: dry rubbing, with hyaluronic acid and with biological oil. It was found that lubricating media of biological origin create boundary lubricating layers on friction surfaces and are able to reduce wear due to microcutting. The development and experimental verification of a mathematical model expressing the dependence of the penetration depth on the friction path and other parameters has been implemented.

In this paper the analog of A.G.Postnikov formula for a primitive Dirichlet’s character on modulo equals a prime-power of number two is found. The deduction is based on the detail consideration the algebraic structure of a reducing of a residues system modulo of a primepower of the number two.

In the paper an interconnection of ortodox laws belief and the notion of infinity in
mathematical analysis is described. Essentially use the statement of evangelists and apostles. Hence the comtability of mathematics and christian teology is established.

This work is dedicated to the ninetieth anniversary of Doctor of Physical and Mathematical Sciences, Professor Martin Davidovich Grindlinger, the founder of the algebraic school and his role in the revival of the number-theoretic school in the city of Tula. Biographical data, a brief overview of his scientific, pedagogical, organizational and publishing activities are given.
Particular attention is paid to the role of M. D. Grindlinger in the revival of the scientific school of number theory at the TSPU L. N. Tolstoy.

This article is devoted to the life and scientific activity of the famous historian of mathematics Margarita Babkenovna Nalbandyan (1931-2004). Whose 90th anniversary is celebrated on September 3. The key points of her biography are considered against the background of the history of the development of the Rostov mathematical school, which began to form in 1915 after the Warsaw University moved to Rostov-on-Don. The Faculty of Physics and Mathematics
(phizmat), which survived the revolution and the civil war, numerous reorganizations and various reforms, was replenished with its own talented graduates in the 20s-30s, and after returning from evacuation in 1944, it successfully began to restore status of one of the leading soviet mathematical centers. 50s of the XX century, which coincided with the student youth of
M. B. Nalbandyan, are considered one of the best in the history of phizmat (mehmat).
The article analyzes the main works on the history of the development of the theory of elliptic functions in Russia (they were published in the late 60s-early 70s) and materials related to the biography of the famous mathematician, the founder of the Rostov mathematical school, D.D. Mordukhai-Boltovsky, which M. B. Nalbandyan collected and published for many years.
Special attention is paid to the cooperation of Margarita Babkenovna with leading Soviet historians of mathematics, among whom we can mention G.P.Matviyevskaya and E.P. Ozhigova.
In conclusion the fate of the scientific heritage and the home archive is considered.