Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'_α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

In this paper we obtain a classical solution of the one-dimensional wave equation with conditions on the characteristics for different areas this problem is considered in. The analytical solution is constructed by the method of characteristics. In addition, the uniqueness of the obtained solution is proved. The necessity and sufficiency of the matching conditions for given functions of the problem are proved. When these conditions are satisfied and the given functions are smooth enough, the classical solution of the considered problem exists.

Herein, the approximation properties of the Abel – Poisson means of rational conjugate Fourier series on the system of the Chebyshev–Markov algebraic fractions are studied, and the approximations of conjugate functions with density | x |^s , s ∈(1, 2), on the segment [–1,1] by this method are investigated. In the introduction, the results related to the study of the polynomial and rational approximations of conjugate functions are presented. The conjugate Fourier series on one system of the Chebyshev – Markov algebraic fractions is constructed. In the main part of the article, the integral representation of the approximations of conjugate functions on the segment [–1,1] by the method under study is established, the asymptotically exact upper bounds of deviations of conjugate Abel – Poisson means on classes of conjugate functions when the function satisfies the Lipschitz condition on the segment [–1,1] are found, and the approximations of the conjugate Abel – Poisson means of conjugate functions with density | x |^s , s ∈(1, 2), on the segment [–1,1] are studied. Estimates of the approximations are obtained, and the asymptotic expression of the majorant of the approximations in the final part is found. The optimal value of the parameter at which the greatest rate of decreasing the majorant is provided is found. As a consequence of the obtained results, the problem of approximating the conjugate function with density | x |^s , s ∈(1, 2), by the Abel – Poisson means of conjugate polynomial series on the system of Chebyshev polynomials of the first kind is studied in detail. Estimates of the approximations are established, as well as the asymptotic expression of the majorants of the approximations. This work is of both theoretical and applied nature. It can be used when reading special courses at mathematical faculties and for solving specific problems of computational mathematics.

In this article the relationship between the conditions of p-differentiability, p-holomorphycity, and the existence of the derivative of a function of a p-complex variable is considered. The general form of a p-holomorphic function is found. The sufficient conditions for p-analyticity and local invertibility are obtained. The open mapping theorem and the principle of maximum of the norm for a p-holomorphic function and the uniqueness theorem are proved.

The classical results by J. Fell, J. Tomiyama, M. Takesaki describe n-homogeneous С*-algebras as algebras of all continuous sections for an appropriate algebraic bundle. By using this realization, several authors described the set of n-homogeneous С*-algebras with different spaces of primitive ideals. In 1974 F. Krauss and T. Lawson described the set of all n-homogeneous С*-algebras whose space Prim of primitive ideals is homeomorphic to the sphere S². Suppose the space PrimA of primitive ideals is homeomorphic to the sphere S3 for some n-homogeneous С*-algebra A. In this case, these authors showed that the algebra A is isomorphic to the algebra C(S³,C^n×n). If n ≥ 2 then there are countably many pairwise non-isomorphic n-homogeneous С*-algebras A such that PrimA ≅ S 4. Further, let n ≥ 3. There is only one n-homogeneous С*-algebra A such that PrimA ≅ S 5. There are two non-isomorphic 2-homogeneous С*-algebras A and B with space PrimA ≅ S 5. On the other hand, algebraic bundles over the torus T 2 are described by a residue class [p] in Z/nZ = π₁(PU_n). Two such bundles with classes [p_i] produce isomorphic С*-algebras if and only if [p₁] = ±[p₂]. An algebraic bundle over the torus T 3 is determined by three residue classes in Z/nZ. Anatolii Antonevich and Nahum Krupnik introduced some structures on the set of algebraic bundles over the sphere S2. Algebraic bundles over the compact connected two-dimensional oriented manifolds were considered by the author. In this case, the set of non-equivalent algebraic bundles over such space is like the set of algebraic bundles over the torus T2. Further advances could be in describing the set of algebraic bundles over the 3-dimensional manifolds.

Herein, some classes of linear two-dimensional difference equations of Volterra type are considered. Representations of solutions using analogs of the resolvent and the Riemann matrix are obtained.

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.

Inorganic scintillation glasses form a domain of rapidly evolving detector materials used to measure various types of ionizing radiation. The most widespread are lithium-silicate glasses enriched with the ⁶Li isotope, which are used to register thermal neutrons. At the same time, due to the specificity of the energy dependence of the neutron cross-section of light nuclei, such materials are of little use for the evaluation of epithermal and more highly energetic neutrons. The use of rare earth elements in the composition of glasses makes it possible to increase the sensitivity to neutrons. In the BaO–Gd₂O₃–SiO₂ system, doped with Ce ions, a scintillation glass with a yield of at least 2500 photons / MeV was created for the first time, which permits to create inexpensive detector elements of a significant volume for registering neutrons. It has been shown that a detector based on BaO–Gd₂O₃–SiO₂ glass has satisfactory properties when detecting neutrons in a wide spectrum of their energies.

Herein, using the femtosecond absorption spectroscopy method, the dynamics of the nonstationary induced absorption spectra of diflavonoid 3,7-dihydroxy-2,8-di(4-methoxyphenyl)-4H, 6H-pyrano[3,2-g]chromene-4,6-dione (DFV) in solvents of different polarities is studied. It is found that the rapid transformation of the transient absorption spectra of DFV in time is due to the processes of intramolecular protons transfer in excited singlet states. For a nonpolar solvent, two protons are transferred in two stages. Initially, during the sub-picosecond times, a form with a single transferred proton is formed from the Frank-Condon state. From this transition state, in a time range of about 9 ps, the second proton is transferred and the two proton transfer tautomer with a high quantum yield of fluorescence ~0.66 is formed, which has the gain band in the transient absorption spectra. For the polar solvent dimethylformamide only the short-lived form with a single proton transferred is formed also during the subpicosecond times practically the same ones as for the nonpolar solution and has a lifetime of about 20 ps. The polarity of the medium, which affects the formation of a set of the “closed” and “open” forms of DFV in the ground state, differing in relative positions in the space of hydroxyl and carbonyl groups, largely determines the mechanism of the intramolecular proton transfer process in the DFV molecule, which consists in the sequential transfer of two protons in a non-polar solvent to form a fluorescent long-lived tautomer and the transfer of one proton in polar solvents to form a short-lived non-fluorescent form.

Herein, the temperature dependences of the static current gain (β) of bipolar n-p-n-transistors, formed by similar process flows (series A and B), in the temperature range 20–125 °С was investigated. The content of uncontrolled technological impurities in the A series devices was below the detection limit by the TXRF method (for Fe < 4.0 · 10⁹ at/cm²). In series B devices, the entire surface of the wafers was covered with a layer of Fe with an average concentration of 3.4 ∙ 10¹¹ at/cm²; Cl, K, Ca, Ti, Cr, Cu, Zn spots were also observed. It was found that in B series devices at an average collector current level (1.0 ∙ 10^–6 < I_c <1.0 ∙ 10^–3 A) the static current gain was greater than the corresponding value in A series devices. This was due to the higher efficiency of the emitter due to the high concentration of the main dopant. This circumstance also determined a stronger temperature dependence of β in series B devices due to a significant contribution to its value from the temperature change in the silicon band gap. At I_c < 1.0 ∙ 10^–6 A β for B series devices became significantly less than the corresponding values for A series devices and practically ceases to depend on temperature. In series B devices, the recombination-generation current prevailed over the useful diffusion current of minority charge carriers in the base due to the presence of a high concentration of uncontrolled technological impurities. For A series devices at I_c < 10^–6 A, the temperature dependence of β practically did not differ from the analogous dependence for the average injection level.

Herein, multidirectional quasiperiodic air flows in an exhaust shaft above a four-order horizontal bundle consisting of bimetallic finned tubes used to remove heat in heat exchangers are considered. Modeling of the air movement is carried out on the basis of equations for thermogravitational convection in the Boussinesq approximation. It takes into account the viscosity of the air and the dependence of the air density on the temperature. An interpretation of quasiperiodic airstreams is proposed on the basis of Rayleigh – Bénard convection, as a result of which regular structures, called Rayleigh – Bénard cells, are formed in a liquid or gas. Rayleigh – Bénard cells are an analytical solution to the problem of the stability of hydrodynamics flows in the linear approximation. The appearance of two-dimensional (convective rolls) and threedimensional (rectangular cells) is possible. To estimate the number of emerging structures, the critical Rayleigh numbers were calculated, which characterizes the transition from an unstable mode of the convective fluid flow to a stable mode. For two experiments, the experimental Rayleigh numbers are compared with their critical values. The differences between the experimental conditions and the ideal boundary conditions used in the calculations and the partial destruction of quasiperiodic structures as a result of this are also discussed.