The local formula of representation of a solution for a functional differential equation with the mixed initial condition considering perturbations of delays containing in the phase coordinates and in controls

For the perturbed controlled nonlinear delay functional differential equation with the mixed initial condition a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of main interval. In the formula the effects of perturbations of the delay parameters containing in the phase coordinates and controls, the initial vector, the initial and control functions are detected.

Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail

The closed forms of the non-resonant thermonuclear function in the Maxwell–Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their future work in this area.

On the behaviour of analytic representation of the generalized Pascal snail

We find an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail. Special cases as the Pascal snail, Both leminiscate, conchoid of the Sluze and a disc are included. The behaviour of functions related to generalized Pascal snail is studied.

Analytic representation of all planar two-loop five-point Master Integrals with one off-shell leg

We present analytic expressions in terms of polylogarithmic functions for all three families of planar two-loop five-point Master Integrals with one off-shell leg. The calculation is based on the Simplified Differential Equations approach. The results are relevant to the study of many 2 → 3 scattering processes of interest at the LHC, especially for the leading-color W + 2 jets production.

Generalizations of Certain Representations of Real Numbers

In the present paper, real number representations that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are proven. The theorem on the representation of real numbers from a certain interval is formulated.One of the peculiarities of the research presented in this paper, is introducing numeral systems with mixed bases (i.e., with bases containing positive and negative numbers). In 2016, an idea of a corresponding analytic representation of numbers was presented in [14, Serbenyuk, S.: On some generalizations of real numbers representations, arXiv:1602.07929v1]. These investigations were presented in [15, Serbenyuk, S.: Generalizations of certain representations of real numbers, arXiv:1801.10540] in January 2018.Also, an idea of such investigations was presented by the author of this paper at the conference in 2015 (see [9, Serbenyuk, S.: Quasi-nega-\tilde QQ-representation as a generalization of a representation of real numbers by certain sign-variable series, https://www.researchgate.net/publication/303255656]).

Analytic Representation of the Optimal Flow for Gravity Irrigation

The aim of this study is the deduction of an analytic representation of the optimal irrigation flow depending on the border length, hydrodynamic properties, and soil moisture constants, with high values of the coefficient of uniformity. In order not to be limited to the simplified models, the linear relationship of the numerical simulation with the hydrodynamic model, formed by the coupled equations of Barré de Saint-Venant and Richards, was established. Sample records for 10 soil types of contrasting texture were used and were applied to three water depths. On the other hand, the analytical representation of the linear relationship using the Parlange theory of infiltration proposed for integrating the differential equation of one-dimensional vertical infiltration was established. The obtained formula for calculating the optimal unitary discharge is a function of the border strip length, the net depth, the characteristic infiltration parameters (capillary forces, sorptivity, and gravitational forces), the saturated hydraulic conductivity, and a shape parameter of the hydrodynamic characteristics. The good accordance between the numerical and analytical results allows us to recommend the formula for the design of gravity irrigation.

Analytic Representation of Relation between Solar Altitude Angle and Local Time for Calculating Daylight Irradiance and Illuminance of the Earth Surface

The analytic expression which sets relation between the solar altitude angle and local time at a random point of the Earth surface on a random day of a year is obtained. The obtained expression and derived relations allow one to conduct calculations of daylight irradiance and illuminance of the Earth surface in analytic form.