Abstract:
In this paper we prove that any distance-balanced graph $G$ with $\Delta(G)\geq |V(G)|-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $\Delta(T)\geq |V(T)|-3$.

Abstract:
Let $G$ be a graph. A zero-sum flow in $G$ is an assignment of nonzero real number to the edges such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be naturel number. A zero-sum $k$-flow is a flow with value from the set $\{\pm 1, \pm 2,..., \pm(k-1)\}$. It has been conjectured that every $r$-regular graph, $r\geq 3$, admits a zero-sum 5-flow. In this paper we give an affirmative answer to this conjecture, exept for r=5.

Abstract:
Several intersection matrices of $s$-subsets vs. $k$-subsets of a $v$-set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known identities appear as natural consequences of this viewpoint; especially, appearance of the derivative operator $d/dz$ and some related operators reveals some connections between intersection matrices and the "combinatorics of creation-annihilation". As application, the eigenvalues of several intersection matrices including some generalizations of the adjacency matrices of the Johnson scheme are derived; two new bases for the Bose--Mesner algebra of the Johnson scheme are introduced and the associated intersection numbers are obtained as well. Finally, we determine the rank of some intersection matrices.

Abstract:
Many single and three-phase converters are well developed, and covered up in most of electric markets. It is used in many applications in power systems and machine drives. However, an exact definite output signal from the dc side still not recognized. The waveforms of output voltage and current demonstrate an imperfect dc signal and constitute losses, harmonic distortion, low power factor, and observed some ripples. An approximately perfect rectifier bridge is the aim of this research. Perhaps it gives the ability to identify the parameters of the converter to obtain, as much as possible, a perfect dc signal with less ripple, high power factor and high efficiency. Design is implemented by simulation on Power Simulator PSIM, and practically, a series regulator LM723 is applied to provide regulating output voltage. Comparisons of both simulation and hardware results are made to observe differences and similarities.

The paper provides mathematical analysis of sensitivity of different combination rules in the DS/AHP method when an alternative is added to the set of decision alternatives while solving foresight problems. Different cases of rank reversals are defined and two sets of conditions for these cases using the method DS/AHP are considered. Rank reversals are illustrated when the DS/AHP method is used to solve practical problem of critical technologies of energy conservation and power efficiency evaluation in Ukraine. It is shown that the DS/AHP method is not sensitive to exclusion (or addition) of an irrelevant decision alternative from (or to) the set of decision alternatives.

Abstract:
One of the problems in the development of mathematical theory of the genetic code (summary is presented in [1], the detailed—to [2]) is the problem of the calculation of the genetic code. Similar problem in the world is unknown and could be delivered only in the 21st century. One approach to solving this problem is devoted to this work. For the first time a detailed description of the method of calculation of the genetic code was provided, the idea of which was first published earlier [3]), and the choice of one of the most important sets for the calculation was based on an article [4]. Such a set of amino acid corresponds to a complete set of representation of the plurality of overlapping triple gene belonging to the same DNA strand. A separate issue was the initial point, triggering an iterative search process all codes submitted by the initial data. Mathematical analysis has shown that the said set contains some ambiguities, which have been founded because of our proposed compressed representation of the set. As a result, the developed method of calculation was reduced to two main stages of research, where at the first stage only single-valued domains were used in the calculations. The proposed approach made it possible to significantly reduce the amount of computation at each step in this complex discrete structure.

Abstract:
The disclosure of many secrets of the genetic code was facilitated by the fact that it was carried out on the basis of mathematical analysis of experimental data: the diversity of genes, their structures and genetic codes. New properties of the genetic code are presented and its most important integral characteristics are established. Two groups of such characteristics were distinguished. The first group refers to the integral characteristics for the areas of DNA, where genes are broken down in pairs and all 5 cases of overlap, allowed by the structure of DNA, were investigated. The second group of characteristics refers to the most extended areas of DNA in which there is no genetic overlap. The interrelation of the established integral characteristics in these groups is shown. As a result, a number of previously unknown effects were discovered. It was possible to establish two functions in which all the over-understood codons in mitochondrial genetic codes (human and other organizations) participate, as well as a significant difference in the integral characteristics of such codes compared to the standard code. Other properties of the structure of the genetic code following from the obtained results are also established. The obtained results allowed us to set and solve one of the new breakthrough problems—the calculation of the genetic code. The full version of the solution to this problem was published in this journal in August 2017.

Abstract:
This work investigates in-depth the effects of variation of the compositional ratio of the absorber layer in Cu(In,Ga)Se2 (CIGS) thin-film solar cells. Electrical simulations were carried out in order to propose the most suitable gallium double-grading profile for the high efficiency devices. To keep the model as close as possible to the real behavior of the thin film solar cell a trap model was implemented to describe the bulk defects in the absorber layer. The performance of a solar cell with a standard CIGS layer thickness (2 μm) exhibits a strong dependence on the front grading height (decreasing band gap toward the middle of the CIGS layer). An absolute gain in the efficiency (higher than 1%) is observed by a front grading height of 0.22. Moreover, simulation results show that the position of the plateau (the region characterized by the minimum band gap) should be accurately positioned at a compositional ratio of 20% Ga and 80% In, which corresponds to the region where a lower bulk defect density is expected. The developed model demonstrates that the length of the plateau is not playing a relevant role, causing just a slight change in the solar cell performances. Devices with different absorber layer thicknesses were simulated. The highest efficiency is obtained for a CIGS thin film with thicknesses between 0.8 and 1.1 μm.

Abstract:
The results of the long-term investigations in dynamics for study of the conditions of taxonomic groups of microorganisms of irrigated gleyey-yellow soils of the subtropical zone under vegetable crops in rotation with the continuous growing of these cultures have been presented. The results of the researches study demonstrated an important vibration of the microorganisms quantity under growing cultures happens. The most quantity of microorganisms, consuming organic nitrogen is observed in irrigative gleyey-yellow soils of the humid subtropical zone. Intensity of mineralization of organic matters was lower in irrigative gleyey-yellow soils. A quantity of microorganisms was lower, but a coefficient of mineralization was higher under continuous cultures as compared the analogous soils under crop rotation. These data show that a type of the soil influence on the rhizosphere microflora insignificantly, while plant shows a significant effect on its quantity and compositions.

Abstract:
We study the following model: . The aim is to estimate the distribution of X when only ？are observed. In the classical model, the distribution of ？is assumed to be known, and this is often considered as an important drawback of this simple model. Indeed, in most practical applications, the distribution of the errors cannot be perfectly known. In this paper, the author will construct wavelet estimators and analyze their asymptotic mean integrated squared error for additive noise models under certain dependent conditions, the strong mixing case, the β-mixing case and the ρ-mixing case. Under mild conditions on the family of wavelets, the estimator is shown to be -consistent and fast rates of convergence have been established.