Vol. 52 No. 2 (246) (2018)

In the present paper the connection between a class of systems of differential equations and integral operators with semi-separable kernel is established. Using a matrix of the system the inverse to a given integral operator is constructed. Moreover by putting some additional conditions on the kernel of integral operator and by the help of inverse of integral operator a fundamental matrix of the system is constructed.

In the present short note we prove a version of the classical fixed-point theorem, as well as present its application.

In the paper an answer to a problem posed by A.I. Sozutov in the Kourovka Notebook is given. The solution is based on some modification of the method that was proposed for constructing a non-abelian analogue of the additive group of rational numbers, i.e. a group whose center is an infinite cyclic group and any two non-trivial subgroups of which have a non-trivial intersection.

In this paper we prove that there exist a nontrivial Franklin series and a sequence $ M_n $ such that the partial sums $ S_{M_n} (x) $ of that series converge to 0 almost everywhere and $ \lambda \cdot \text{mes} \{ x : \sup\limits_{n}{\left| S_{M_n} (x) \right|} > \lambda \} \to 0 $ as $ \lambda \to +\infty $. This shows that the boundedness assumption of the ratio $ \dfrac{ M_{n+1}}{M_n} $, used for the proofs of uniqueness theorems in earlier papers, can not be omitted.

On the basis of the refined theory of orthotropic plates of variable thickness, the equations of the beam bending problem are obtained with the simultaneous action of compressive forces and transverse load. It is accepted that the edges of the beam have an elastically clamped support and the reduction of the compressive force by the support due to friction is taking into account. Passing to dimensionless quantities, a certain problem is solved. The stability of a beam is discussed. Based on the results obtained, conclusions are drawn.

In the paper the interpretation algorithms of typed functional programs are considered. The interpretation algorithm is based on substitution, $ \beta $-reduction and canonical $ \delta $-reduction. It is shown that seven known interpretation algorithms (FS (of full substitution), PES (of parallel external substitution), LES (of left external substitution), PIS (of parallel inner substitution), LIS (of left inner substitution), ACT (active algorithm), PAS (passive algorithm)) are pairwise incomparable with respect to undefined value ($ \perp $-incomparable).

In this paper the interpretation algorithms for typed and untyped functional programs are considered. Typed functional programs use variables of any order and constants of order $ \leq 1 $, where constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The basic semantics of the typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. The interpretation algorithms of typed functional programs are based on substitutions, $ \beta $-reduction and canonical $ \delta $-reduction. The basic semantics of the untyped functional program is the untyped $ \lambda $-term, which is defined by means of the fixed point combinator. The interpretation algorithms of untyped functional programs are based on substitutions and $ \beta $-reduction. Interpretation algorithms are examined for completeness and comparability. It is investigated how the “behavior” of the interpretation algorithm changes after translation of typed functional program into untyped functional program.

The results of temperature measurements in the Argon discharge in plasma with acoustic perturbation are presented. Possibility of, as decreasing, so increasing of discharge temperature in several times is shown.

This work presents development of a method for multi-criteria decision making under uncertainty conditions based on single-attribute value functions and probabilistic distributions. The values of different criteria are modeled using normal distributions, i.e. the value of the $ i $-th criteria of the $ j $-th option is given by $ x^j_i \sim \mathcal{N} ({\mu}^j_i, {\sigma}^j_i) $ distribution. The method is evaluated and the results are analyzed on a simple example.

Glucose concentration in the aqueous solution was measured using a microwave metamaterial sensor based on the modified Hilbert curve of the first fractal order. In order to increase the system sensitivity at fixed meta-element sizes, the design of the sensor was optimized by using finite element method. It was observed that $ S $ parameters of the metamaterial microstrip line sensor at resonant frequencies strongly depend on the glucose concentration. Consequently, by measuring the microwave responses for the different concentrations one can detect the glucose concentration in the solution. Such structure can be useful in the biological and medical applications.