Vol. 52 No. 1 (245) (2018)

The following paper introduces a computationally effective way of finding and utilizing maximum entropy problem boundaries for up to three dimensional cases. The application of the results is concentrated on financial options pricing and reverse distribution calculation. Based on market information in form of current option prices a distribution of future states is constructed. Using the suggested approach it will be possible to identify cases, where no solution to the maximum entropy problem exists, and parameters, for which a feasible solution can be reached.

It is determined minimum number of cosets over linear subspaces in $ \mathbb{F}_q $ necessary to cover following two sets of $ A (n \mathclose{\times} n) $ matrices. For one of the set of matrices $ \det{A} = 0 $ and for the other set $ \det{A} \neq 0 $. It is proved that for singular matrices this number is equal to $ 1 \mathclose{+} q \mathclose{+} q^2 \mathclose{+} \ldots \mathclose{+} q^{n-1} $ and for the nonsingular matrices it is equal to $ (q^n \mathclose{-} 1)(q^n \mathclose{-} q)(q^n \mathclose{-} q^2) \cdots (q^n \mathclose{-} q^{n-1}) / q^{\large{\binom{n}{2}}} $.

Let $ \{ W_k (x) \} _{k = 0}^{\infty} $ be either unbounded or bounded Vilenkin system. Then, for each $ 0 < \varepsilon < 1 $, there exist a measurable set $ E \subset [0,1)^2 $ of measure $ |E| > 1 \mathclose{-} \varepsilon $, and a subset of natural numbers $ \Gamma $ of density 1 such that for any function $ f(x,y) \in L^1 (E) $ there exists a function $ g(x,y) \in L^1 [0,1)^2 $, satisfying the following conditions: $ g(x, y) = f(x,y) $ on $ E \ ; $ the nonzero members of the sequence $ \{ |c_{k, s}(g)| \} $ are monotonically decreasing in all rays, where $ c_{k, s} (g) = \int\limits_{0}^{1} \int\limits_{0}^{1} g(x, y) \overline{W_k}(x) \overline{W_s}(y) dx dy \ ; $ $ \lim\limits_{R \in \Gamma,\ R \to \infty} S_R((x,y),g) = g(x,y) $ almost everywhere on $ [0,1)^2 $, where $ S_R((x,y),g) = \sum\limits_{k^2+s^2 \leq R^2} c_{k, s}(g) W_k(x) W_s(y) $.

In the article the questions of solvability of boundary value problem for a homogeneous pseudoparabolic-pseudohyperbolic type integro-differential equation with degenerate kernels are considered. The Fourier method based on separation of variables is used. A criterion for the one-valued solvability of the considering problem is found. Under this criterion the one-valued solvability of the problem is proved.

Problem of the piezoceramic plate polarized along the normal of the middle plane of the plate is solved, based on the assumptions of the hypothesis of Kirchhoff, taking into account the components characterizing the electric field. The equations of planar and bending vibrations are obtained. Localized bending vibrations are considered, and the effect of the electric field on the frequency of localized vibrations is investigated.

In the paper the result of a research done by using the automated system xRandNet is presented, which is designed and implemented for generating and analyzing the main topological properties of some hierarchical models of random networks. The research is related to the connected component distribution of random block-hierarchical networks, which are quite new objects in the random network theory.

This study aims to provide arguments for an increase in the stability of bilayer lipid membranes (BLM) in an electrical field in presence of $ \mathrm{CuO} $ nanoparticles. Moreover, the sustainability of BLM becomes more stable with increasing of nanoparticles concentration. It is shown that an increase in the stability of BLM in an electrical field is mainly due to an increase in the coefficient of linear tension of the edge of a pore, which is forming in BLM. It is shown as well, that the presence of nanoparticles of $ \mathrm{CuO} $ in the solution surrounding BLM leads to a decrease in the conductivity of the BLM.

The main goal of this paper is the investigation of helix–coil transition in heteropolymers by means of straight calculation of partition function, free energy, helicity degree and other characteristics of transition on the bases of generalized model of polypeptide chain. Product of random matrices was applied to solve the problem for the comparison with constrained annealing method in the future.

In the work the influence of the laser beam on the optical properties of highly absorbing dye-doped cholesteric liquid crystalline (CLC) structure is investigated. By means of polarizing microscope studies step-like variation of the CLC helical pitch and formation of the disclination defect lines (oily streaks) is observed.

We consider the automorphisms description question for the semigroups End($G$) of a group $G$ having only cyclic centralizers (CC) of nontrivial elements. In particular, we prove that each member of the automorphism group Aut($G$) of a group $G$ from this class is uniquely determined by its action on the elements from the subgroup of inner automorphisms Inn($G$). Note that, typical examples of CC groups are absolutely free groups, free periodic groups of large enough odd periods, $n$-periodic and free products of CC groups.

Southwest Power Pool (SPP) is a regional power transmission organization in the United States. Virtual transactions on the SPP markets are sold at auction. There are Real-Time (RT) and Day-Ahead (DA) markets, where you can buy electricity by winning an auction. Virtual transactions are a valuable component of the bilateral market. All market participants have DA and RT series of prices in hourly basis. In this article, we take a large amount of data of DA and RT prices and try to see which distribution function have the returns of RT prices. Then we make a regression between DA and RT prices, to see how they are related to each other.