In this paper, we prove a Turán-type inequality for rational functions and thereby extend it to a more general class of rational functions (r(s(z))) of degree mn with prescribed poles, where (s(z)) is a polynomial of degree m. These results not only generalize some Turán-type inequalities for rational functions, but also improve as well as generalize some known polynomial inequalities.
Two problems on the stability of the trivial equilibrium position of floating vessels with cross sections in the form of elliptic and hyperbolic segments are considered. Examples on the stability of floating bodies are reviewed, and the key principles of its investigation by analytical statics methods are outlined. For each of the presented problems, by means of quite serious mathematical constructions, an exact expression for the potential energy is obtained within the accepted configuration, and its quadratic approximation near the equilibrium state under study is calculated. On its basis, the stability conditions of the equilibrium state in terms of three dimensionless parameters are determined and the limiting cases are also analyzed. The intermediate expressions and final results obtained as a result of discussion of each of the problems are compared, and their common and distinctive features are identified. The found solutions are illustrated as families of boundaries of stability regions on the plane of two dimensionless parameters at different values of the third parameter. These results are of fundamental theoretical importance and can prove useful for practical applications.
Analytical representations of the rate of apsidal precession in the planar elliptical restricted three-body problem are considered in the case when the orbit of the disturbing body is external with respect to the orbit of the test particle. The analytical expressions are compared with the numerical data obtained for the apsidal precession rate in the form of a power series with a parameter equal to the ratio of the semi-major axis of the orbit of the test particle to that of the disturbing planet. It is shown that the analytical expressions for the rate of apsidal precession of the particle are reliable only at distances not close to the instability zone near the orbit of the disturbing planet. Near the Wisdom gap, the linear secular theory is no more valid. An empirical correction formula is proposed to calculate the apsidal procession rate at relatively high (however less than 0.5) eccentricities of the particle and disturbing planet. The proposed formulas are applied to describe the precession of orbits in real exoplanetary systems.
The article contains a review of the most important results obtained in the framework of the St. Petersburg State University Research School on Nonlinear PDEs (the O.A. Ladyzhenskaya–N.N. Uraltseva School). The main attention is paid to the works carried out at the university over the past 50 years. The first part of the review concerns the solvability and qualitative properties of solutions to boundary value problems for the second order scalar quasilinear elliptic and parabolic equations, as well as variational problems. The planned second part of the review will include sections on fully nonlinear equations and systems of equations, and on free boundary problems.
Interval (circular arcs) translation mappings, which can be represented as interval exchange transformations with overlap, are studied. It is known that for any mapping of this type there is a Borel probabilistic invariant atomless measure, which is constructed as a weak limit of invariant measures of mappings with periodic parameters. In the latter case, this is simply the normalized Lebesgue measure on some family of subsegments. For such limit measures in the case of shifting arcs of a circle, it is shown that any point of the support of this measure can be made periodic by an arbitrarily small change in the parameters of the system without changing the number of segments. For an arbitrary invariant measure, using the Poincaré recurrence theorem, it is shown that any point can be made periodic with a small change in the parameters of the system, and the number of intervals for mapping increases by no more than two.
The results of research carried out in the 21st century at the Department of Differential Equations of St. Petersburg State University are reviewed. The subject of study is the problem of stability of the zero solution to the second-order equation describing periodic perturbations of the oscillator with a nonlinear restoring force under reversible and conservative perturbations. Such perturbations are classified as transcendental perturbations, for which the solution of the problem of stability requires taking into account all terms in the expansion in a series of the right-hand side of the equation. The problem of stability under transcendental perturbations was formulated in 1893 by A.M. Lyapunov. The results regarding stability presented in this review were obtained using the methods of KAM theory: perturbations of the oscillator with infinitely small and infinitely large oscillation frequencies were considered; conditions for the existence of quasi-periodic solutions in any vicinity of the time axis are determined, from which the stability (not asymptotic) of the zero solution of the perturbed equation follows; and the conditions are found for the stability of the zero solution for a Hamiltonian system with two degrees of freedom, the unperturbed part of which is described by a pair of oscillators (in this case conservative perturbations are considered).
The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of series expansion in special functions or approximate values of the sought original at a number of points. Various handling methods are considered, and their characteristics of accuracy and stability are indicated, which are required when choosing a handling method for solving applied problems. Quadrature inversion formulas adapted for inversion of long-term and slowly occurring processes of linear viscoelasticity were constructed. A method is proposed for deforming the integration contour in the Riemann–Mellin inversion formula, which leads the problem to the calculation of definite integrals and makes it possible to obtain estimates of the error.
A new generalization of the so-called Hegedüs–Szilágyi fixed point theorem is given by introducing a new contractive condition in the framework of complete metric spaces. As an application, we get a new fixed point theorem that generalizes and improves many known results in the literature.
The inverse problem of determining the solution and one-dimensional kernel of the integral term in an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition is considered. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given integro-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral h(t) of the integral term. Using additional information about the solution of the direct problem, we obtain an integral equation of the second kind with respect to the kernel of the integral h(t) of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. In addition the theorem of the conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronwall’s inequality is used.
Damped rotational oscillations of a cylinder equipped with a coaxial disk in the head part and a stabilizer in the tail part are studied. The elongation of the cylinder (the ratio of its length to diameter) is nine. The cylinder is mounted in the test section of the low-velocity wind tunnel with a wire suspension containing steel springs. In the equilibrium position, the cylinder axis is horizontal and parallel to the velocity vector of the incoming flow. A semiconductor strain gauge is attached to one of the suspension springs, which measures the time dependence of spring tension during oscillations. The output voltage of the strain gauge is sent to a PC oscilloscope, the digital signal of which is transmitted to the computer. After calibration of the device, the frequency and amplitude of damped rotational oscillations around the horizontal axis passing through the center of the cylinder and perpendicular to the velocity vector of the incoming flow were determined. The air flow enhances the rate of the damping of rotational oscillations of the cylinder. The air flow effect is described by analogues of rotational derivatives, which, in the case of bluff bodies, depend on the amplitude of the oscillations of the inclination angle of the body and on the amplitude of the angular velocity. A simple model of the effect of a stabilizer on rotational derivatives is proposed.
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