Massless spin 3/2 field, spherical solutions, eliminating of the gauge degrees of freedom

Relativistic system for a vector-bispinior describing a massless spin 3/2 field is studied in the spherical coordinates of Minkowski space. Presentation of the equation with the use of the covariant Levi-Civita tensor exhibits existence of the gauge solutions in the form of the covariant 4-gradient of an arbitrary bispinor. Substitution for 16-component field function is based on the use of Wigner functions, it assumes diagonalization of the operators of energy, square and third projection of the total angular momentum, and space reflection. We derive radial system for eight independent functions. General structure of the spherical gauge solutions is specified, and it is demonstrated that the gauge radial functions satisfy the derived system. It is proved that the general system reduces to two couples of independent 2-nd order and nonhomogeneous differential equations, their particular solutions may be found with the use of the gauge solutions. The corresponding homogeneous equations have one the same form, they have three regular singularities and one irregular of the rank 2. Frobenius types solutions for this equation have been constructed, and the structure of the involved power series with 4-term recurrent relations sre studied. Six remaining radial functions may be straightforwardly found by means of the simple algebraic relations. Thus, we have constructed two types of solutions with opposite parities which do not contain gauge constituents.

Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation

In this paper, we study the sections of the generating series for solutions to a linear multidimensional difference equation with constant coefficients and find recurrent relations for these sections. As a consequence, a multidimensional analogue of Moivre's theorem on the rationality of sections of the generating series depending on the form of the initial data of the Cauchy problem for a multidimensional difference equation is proved. For problems on the number of paths on an integer lattice, it is shown that the sections of their generating series represent the well-known sequences of polynomials (Fibonacci, Pell, etc.) with a suitable choice of steps.

About the calculation of hypergeometric function F4 (1,2;2,2; z1, z2 ) by the branched continued fraction of a special kind

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2.

Study of Synergistic Effects in Complex Stochastic Systems

In this paper, a method for detecting synergistic effects of the interaction of elements in multi-element stochastic systems of separate redundancy, multi-server queuing, and statistical estimates of nonlinear recurrent relations parameters has been developed. The detected effects are quite strong and manifest themselves even with rough estimates. This allows studying them with mathematical methods of relatively low complexity and thereby expand the set of possible applications. These methods are based on special techniques of the structural analysis of multi-element stochastic models in combination with majorant asymptotic estimates of their performance indicators. They allow moving to more accurate and rather economical numerical calculations, as they indicate in which direction it is most convenient to perform these calculations.

Controlling the correctness of retention parameters variations in reversed phase HPLC using recurrent relations

Recurrent approximation of analytes’ net retention times (tR) in reversed phase high performance liquid chromatography (RP HPLC) at different contents of organic constitu­ent in the eluent (C) is recommended as a method of revealing the reversible hydrate for­ma­tion. The criterion of that are the deviations of the dependences tR(С + DС) = atR(С) + b (*) from linearity, where DС is the constant increment of concentration variations; in our case DС = 5%. However, such deviations are rather small and, hence their measuring requires high robustness of the equipment involved. Besides hydrate formation, there are additional reasons for deviations, namely discrepancies between the real and the selected flows of the eluent. Compa­ring tR values obtained for the same analytes using the same chromatogra­phic column at the same conditions, but with different HPLC instruments using the systems methanol – water as the eluent confirms that tR values of one data set are equal only to approx. 76-98% tR values of another data set. Therefore, the eluent flow in the second case exceeds that in the first regime at the same pro­por­tion. The simple method for revealing such flow deviations is proposed. It is based on the recurrent approximation of tR = f(C) data sets for any compounds forming no hydrates in RP HPLC conditions (chlorobenzene was selected). The absence of the influ­en­ce of any distorted factors is confirmed with values of correlation coefficients for re­cur­rent depen­dencies (*) exceeding 0.999.

Spinless Particle with Darwin–Cox Structure in an External Coulomb Field

Generalized Klein–Fock–Gordon equation for a spinless particle with the Darwin–Cox structure, which takes into account distribution of the electric charge of a particle inside a finite spherical region is studied in presence of an external Coulomb field. There have been constructed exact Frobenius type solutions of the derived equations, convergence of the relevant power series with 8-term recurrent relations has been studied. As an analytical quantization rule is taken the so-called transcendency conditions. It provides us with a 4-th order algebraic equation with respect to energy values, which has four sets of roots. One set of roots, 0 < En;k < 1, depending on the angular momentum n = 0; 1; 2; : : : and the main quantum number n = 0; 1; 2; : : : may be interpreted as corresponding to some bound states of the particle in a Coulomb field. In the same manner, a generalized nonrelativistic Schr¨odinger equation for such a particle is studied, the final results are similar.

ROUND SLABS ON A RESILIENT BASE WITH VARIABLE BEDDING RATIO

The paper considers the application of the analytical method - the method of direct integration - to the calculation of building structures in the form of round and annular plates and slabs lying on a continuous variable elastic basis. The application of the proposed approach allowed to obtain solutions of a wide class of problems, the mathematical summary of which are differential equations with variable coefficients or systems of such equations, and at the same time to evaluate the possibilities and accuracy of calculation of finite elements. The base reaction is described by the Winkler model with a variable bed ratio. With respect to the bending of round and annular plates, formulas for the function of deflections and its derivatives, transverse force and bending moments are obtained. The method is applicable under any given boundary conditions on the contours. The calculation is reduced to determining from the given boundary conditions of the unknown constants of integration and numerical realization of the obtained solutions. The found formulas of the general form are transformed for practically important case when the factor of a bed and loading have the form of polynomials. It is shown that in this case dimensionless fundamental functions are represented by static series. To calculate the coefficients of static series, the corresponding recurrent relations are derived. The calculations show that the discrepancy in the results of the calculation of the deflections of the FEA and the author’s method (AM) is insignificant (not more than 1%), and the results of the calculation of radial and circumferential moments differ significantly, and this difference sometimes reaches 12-14%. However, when the grid is condensed in the circumferential direction, the picture changes, there is a convergence of the results obtained by two methods. This indicates the inaccuracy of the finite element analysis performed on the basis of automatic partitioning of the finite element grid. And this, in turn, leads to "blind" reinforcement of reinforced concrete slabs, in which it is possible, both re-reinforcement of the structure and its insufficient reinforcement.