Boundary value problem for the four-dimensional Gellerstedt equation

In this work, the solvability of the problem with Neumann and Dirichlet boundary conditions for the Gellerstedt equation in four variables is investigated. The energy integral method is used to prove the uniqueness of the solution to the problem. In addition to it, formulas for differentiation, autotransformation, and decomposition of hypergeometric functions are applied. The solution is obtained explicitly and is expressed by Lauricella’s hypergeometric function.

Self-rotation of emitting galaxies without dark matter

Temporal derivatives of the attracting mass in Newton’s law of distant interactions can balance the centripetal and centrifugal accelerations for the rotating periphery of a spiral galaxy. Thermal losses of the mass-energy integral inside the circle of rotation are the cause of the mega-vortex organization of the emitting galaxy. To reject dark matter in cosmic distributions, a conceptual modification of the Euler/Navier–Stokes hydrodynamics is required using adaptive tensor responses with metric waves but not gravimagnetic corrections from General Relativity.

Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated.

Is Jacobi theorem valid in the singly averaged restricted circular Three-Body-Problem?

C. Jacobi found that in the General N-Body-Problem (including N = 3) for the Lagrangian stability of any solution necessary is the negativity of the total energy of the system. For the restricted three-body-problem, this statement is trivial, since a zero-mass body introduces zero contribution to the energy of the system. If we consider only the equations describing the movement of the zero mass point, then the energy integral disappears. However, if we average the equations over the longitudes of the main bodies, the energy integral appears again. Is the Jacobi theorem valid in this case? It turned out not. For arbutrary large values of total energy, there exist bounded periodic orbits. At the same time the negative energy is sufficient for the boundedness of an orbit in the configuration space.

Investigation of the motion of a satellite, according to general theory of relativity

In this paper we investigate the energy integral which is obtained in the problem of the motion of a material point in a central field within the frame of the general theory of relativity. Applied is the method of the small parameter in a combination with the balance method. Derived is a compact formula, describing the trajectory of the motion. This formula gives a correct quantitative description of the basic relativistic effects. We prove the shortening of the major axis of the orbit in comparison with the case where we do not take into account relativistic effects. This result can be useful for analysing the structure of planet systems around massive stars.