On the evolution and gravitational collapse of a toroidal vortex

The evolution and collapse of a gaseous toroidal vortex under the action of self-gravitation are considered using the Hamiltonian mechanics approach. It is shown that evolution occurs in three main stages separated by characteristic time scales. First, a compression along the small radius to a quasi-equilibrium state takes place, followed by a slower compression along the large radius to a more stable compact vortex object. In the latter stage, the possibility of effective scattering and ejection of particles along the vortex axis (jet formation) is detected.As a result, mass, energy, and momentum losses take place, and the vortex collapses.     

An exact solution of the kompaneets equations for a strong point explosion in a medium with quadratic density decrease

In the Kompaneets approximation, we study the propagation of a shock front from the explosion point in an inhomogeneous medium with a power-law variation of the density characterized by the exponent n=2, which is important and corresponds to the external parts of the solar and stellar coronas. An unexpectedly simple exact solution, which allows us to clarify the structure of the general solution under an arbitrary monotonous law of the density variation, was obtained. The results for the plane case are compared with Korikanskii’s results for a noncentral point explosion in a radially stratified medium. A correspondence between the exact solutions is established in these cases, and a new solution for the noncentral explosion is obtained for the case of a density with a singularity at a finite radius, which decreases quadratically at infinity. Radioastronimical Institute of the National Academy of Sciences, Khar’kov, Ukraine; Research Institute of Physics, State University, Rostov-on-Don, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 6, pp. 683–698, June, 1998.     

A note on sensitivity of semigroup actions

It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey (Am. Math. Mon. 99, 332–334, 1992), Silverman (Rocky Mt. J. Math. 22, 353–375, 1992) and Glasner and Weiss (Nonlinearity 6, 1067–1075, 1993). In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner (Ergodic Theory via Joinings, 2003). Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.     

Why do the microstructures of the main pulse and the interpulse of the pulsar in the Crab nebula differ so dramatically?

Theoretical and Mathematical Physics - Pulsars are magnetized neutron stars. They are not resolved by modern radio telescopes and are studied only by radiation coming from the regions of the...     

On the connection between the gamma and radio spectra of pulsars

We suggest a model of pulsar radio emission in which a coherent radio emission is excited in a vacuum gap above the polar cap of a neutron star. The pulsar X-ray and gamma radiation is considered as the result of inverse Compton scattering of radio emission by ultrarelativistic electrons accelerated in the gap. The influence of the pulsar magnetic field on the inverse Compton scattering is taken into account. The relationship between the radio and gamma spectra has been found in this model.     

Response to Mahmood and Mahmood (2015)

In the article published in the International Journal of Mathematics Education in Science and Technology in 2015, Mahmood and Mahmood suggested an explanation for defining 0! as 1. In this response, I argue that their reasoning is flawed.     

Theory of weak turbulence of waves on a fluid surface

The nonlinear interaction of waves in a fluid of finite depth is discussed. Forbidden decay processes in the gravitational portion of the spectrum are eliminated from the Hamiltonian by means of a canonical transformation. This provides an opportunity to obtain a kinetic equation which takes into account scattering of capillary waves by gravitational waves, in addition to decays in the subsystem of gravitational waves. The distribution N_k P^1/2h^1/4k^–4 is obtained for capillary waves in shallow water with constant flow of energy P with respect to the spectrum in the space of the wave numbers k. The interaction of the gravitational and capillary turbulence spectra is discussed. An induced distribution of gravitational waves is found which results from their interaction with capillary waves. It is an increasing function of the wave numbers q in the region bounded by the capillary constant k_o, N_q q^9/4 (q < k_o). The coupling of spectra in the gravitational and capillary regions and the conversion from slightly turbulent distributions to universal distributions are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 97–106, November–December, 1974.     

Why do students not check their solutions to mathematical problems? A field-based hypothesis on epistemological status

ABSTRACT

This study embarks on the question in the title with the construct of epistemological status, which pertains to the solver’s satisfaction with the way and the degree to which their solution had fulfilled their intellectual and psychological needs in a particular problem situation. The construct is used to hypothesize that a solver’s decision to check the solution and the act of checking itself may be shaped by the epistemological status of a developed solution. Driven by the abduction methodology, this hypothesis is supported by two empirical illustrations. The first one comes from the final exam in a large first-year course for non-mathematics majors, where many students accompanied their solutions of a low epistemological status by written checks of final answers as a way to elevate it. No such checks were found in solutions with high epistemological status. The second illustration revisits some previously reported findings to propose that an application of conventional problem-solving methods may not be sufficient for students to endow their solutions with high epistemological status.