On the Existence of Linearly Oscillating Galaxies

We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

An Improved Structural Analysis Method for Isolator with Quasi-Zero-Stiffness Characteristic

A typical quasi-zero-stiffness (QZS) vibration isolator consisting of a vertical spring and two oblique springs has been widely researched on its static and dynamic characteristics. A general criterion for determining structural parameters of QZS isolator is to achieve low nondimensional stiffness around the equilibrium position. However, lower nondimensional stiffness of linear isolator means lower isolation frequency, which may be invalid on QZS isolator. Because there is an implicit relationship between geometric parameter and stiffness ratio of QZS isolator, this study presents an improved optimization criterion for determining the optimal structural parameters of the typical QZS isolator. The optimization criterion is that the QZS isolator has the maximum displacement range around the equilibrium position without exceeding given natural frequency, rather than given nondimensional stiffness. The results show that isolator with these optimal parameters can achieve lower stiffness around the equilibrium position and better vibration isolation performance. Furthermore, an extended QZS isolator consisting of vertical spring with fixed stiffness and prestressed oblique springs is discussed to further improve stiffness characteristic. Better stiffness performance can be obtained when the prestressed oblique springs have softening stiffness and the exponent of the nonlinear stiffness is 2. Considering the existence of friction in practical application, the influence of friction on both static and dynamic characteristics is investigated. The analysis reveals that friction has little influence on its stiffness characteristic around the static equilibrium position and friction damping produced by friction affects the response amplitude and resonant frequency in dynamics.

On the possibility of averaging the equations of an electron motion in the intense laser radiation

The problem of averaging of the relativistic motion equations of electron in the intense laser radiation, caused by the decreasing of the rate of wave phase change due to the Dopplers effect, is considered. As a result the phase can go from the fast to slow variables of the motion, so averaging over the phase becomes impossible. An analysis is presented of the conditions which are necessary for averaging of the relativistic equations of motion over the fast phase of the intense laser radiation on the base of the general principles of the averaging method. Laser radiation is considered in the paraxial approximation, where the ratio of the laser beam waist to the Rayleigh length is accepted as a small parameter. It is supposed that the laser pulse duration is of the order if the laser beam waist. In this case first-order corrections to the vectors of the laser pulse field should be taken into account. The general criterion for the possibility of the averaging of the relativistic motion equations of electron in the intense laser radiation is obtained. It is shown that an averaged description of the relativistic motion of an electron is possible in the case of a fairly moderate (relativistic) intensity and relatively wide laser beams. The known in the literature analogical criterion has been obtained earlier on the base of the numerical results.

Physicochemical and Analytical Implications of GATES/GEB Principles

The fundamental property of electrolytic systems involved with linear combination f12 = 2∙f(O) – f(H) of elemental balances: f1 = f(H) for Y1 = H, and f2 = f(O) for Y2 = O, is presented. The dependency/independency of the f12 on Charge Balance (f0 = ChB) and other elemental and/or core balances fk = f(Yk) (k = 3,…,K) is the general criterion distinguishing between non-redox and redox systems. The f12 related to a redox system is the primary form of a Generalized Electron Balance (GEB), formulated for redox systems within the Generalized Approach to Electrolytic System (GATES) as GATES/GEB ⊂ GATES. The set of K balances f0,f12,f3,…,fK is necessary/ sufficient/needed to solve an electrolytic redox system, while the K-1 balances f0,f3,…,fK are the set applied to solve an electrolytic non-redox system. The identity (0 = 0) procedure of checking the linear independency/ dependency property of f12 within the set f0,f12,f3,…,fK (i) provides the criterion distinguishing between the redox and non-redox systems and (ii) specifies Oxidation Numbers (ONs) of elements in particular components of the system, and in the species formed in the system. Some chemical concepts, such as oxidant, reductant, oxidation number, equivalent mass, stoichiometry, perceived as derivative within GATES, are indicated. All the information is gained on the basis of the titration Ce(SO4)2 (C) + H2SO4 (C1) + CO2 (C2) ⇨ FeSO4 (C0) + H2SO4 (C01) + CO2 (C02), simulated with use of the iterative computer program MATLAB.

Generic properties of free boundary problems in plasma physics*

We are concerned with the global bifurcation analysis of positive solutions to free boundary problems arising in plasma physics. We show that in general, in the sense of domain variations, the following alternative holds: either the shape of the branch of solutions resembles the monotone one of the model case of the two-dimensional disk, or it is a continuous simple curve without bifurcation points which ends up at a point where the boundary density vanishes. On the other hand, we deduce a general criterion ensuring the existence of a free boundary in the interior of the domain. Application to a classic nonlinear eigenvalue problem is also discussed.

Using the group analysis transfer operator in the method of similarity for studying the fuel combustion process in engines

In this paper, we study modelling of the process of combustion of small quantities of a pyrotechnic mixture based on potassium perchlorate and aluminium powder in impulse devices (engines). The analysis of combustion processes includes an introduced general criterion (prime criterion) of similarity composed of Wobbe numbers that are determined for the engine and its model in the course of theoretical studies or experiments. Research data show that the transfer operator in the field of group theory is mostly used for describing the process of combustion of metallized pyrotechnic mixtures.

Исследования турбокомпрессоров с общим рабочим колесом для применения в двигателях и энергоустановках

The article deals with the working processes in flowing part of the turbo-compressors with general impeller (TCG) and with two schemes of the flows. The work studies the features of the TCG operation with two schemes of the flows and the gas-dynamic calculation theory development of their flow parts for use in engines and power plants. The tasks to be solved are to investigate two schemes of the flows in the interscapular space of the impeller – direct-flow and counter-flow. If the gas and air flow directions coincide concerning the axis of rotation of the impeller, then the flow pattern in the TCG is direct-flow, with the opposite movement of flows – is counter-flow. The solution to this problem was performed using the developed methods of gas-dynamic calculation of the TCG flow parts. Conclusions. The scientific novelty of the results obtained is as follows: parameters that significantly affect the efficiency of the turbine and compressor parts of the TCG were established and the formula in the form of a general criterion ratio was calculated. The dependence of the correction factor is determined, which considers the effect of the ratio of the impeller width on the average diameter of the working channel of the RK in the compressor section and the turbine section of the TCG. The article contains formulas, with correction factors, for calculating the power factor in the compressor section and the load factor in the turbine section. If the ratio of the grating width to the average diameter of the working channel is overestimated, it is necessary to supplement the formulas with a correction factor that considers the effect of this ratio. Studies have determined that switching from one mode of operation to another – in a certain section of the impeller, accelerates the flow to the required speed. It is especially evident in the compressor working channel of the TCG with a counter-flow pattern. In the turbine section, the gas flow acceleration time does not depend on the flow pattern – the flow is active and accelerates to the required speed in the nozzle apparatus. The given formulas allow calculating the power of the turbine and compressor parts of the impeller to perform an enlarged gas-dynamic calculation of TCG. Studies have determined that TCG can be used in gas turbine engines and the pressurization system of an internal combustion engine.

A New Approach to the Rayleigh–Taylor Instability

In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one—the classical configuration giving rise to the Rayleigh–Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.

MECHANISM OF ROTATION OF GEOSPHERES OF THE PLANET EARTH

In recent decades, a fundamentally new direction of scientific work has appeared in the field of natural sciences, associated with the study of the effect on matter of such physical factors as radiation, electromagnetic radiation, ultrasound, plasma, high pressure, space vacuum, gravity, etc., where the general criterion of extremeness exposure can be the emergence of intermediate highly active states of particles of matter, which ultimately leads to a qualitative change in the micro- and macro characteristics of the processed object, the emergence of new properties. One of the types of complex extreme exposure is the effect of a high-voltage electric discharge, which combines the simultaneous action on a substance of strong mechanical compression, powerful ultrasound, hard X-ray, UV and IR radiation. The electromagnetic fields generated in the course of the discharge also have a strong effect on both the discharge itself and the ionic processes occurring in the surrounding liquid. Under their influence, various physical changes and chemical reactions occur in the processed material.