Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in {\mathbb{C}^{n},n\geq 2}, is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.

Moving and reactive boundary conditions in moving-mesh hydrodynamics

ABSTRACT We outline the methodology of implementing moving boundary conditions into the moving-mesh code manga. The motion of our boundaries is reactive to hydrodynamic and gravitational forces. We discuss the hydrodynamics of a moving boundary as well as the modifications to our hydrodynamic and gravity solvers. Appropriate initial conditions to accurately produce a boundary of arbitrary shape are also discussed. Our code is applied to several test cases, including a Sod shock tube, a Sedov–Taylor blast wave, and a supersonic wind on a sphere. We show the convergence of conserved quantities in our simulations. We demonstrate the use of moving boundaries in astrophysical settings by simulating a common envelope phase in a binary system, in which the companion object is modelled by a spherical boundary. We conclude that our methodology is suitable to simulate astrophysical systems using moving and reactive boundary conditions.

Синхронизация в турбулентном сферическом течении Куэтта под действием неравномерного вращения

Turbulent flows of viscous incompressible fluid in rotating spherical layer in the presence of synchronization are under consideration. Numerical results are presented. Synchronization of turbulent flow is due to the action of periodical modulation of the angular velocity of inner spherical boundary. The angular velocity of outer spherical boundary is constant. Obtained results were compared with experimental data. The interval of modulation amplitudes was determined where synchronization is followed by intermittency “chaos – chaos”.

Casimir Energies for Isorefractive or Diaphanous Balls

It is familiar that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with $\varepsilon\mu=1$, a so-called isorefractive or diaphanous ball. Here we re-examine that example, and attempt to extend it to an electromagnetic $\delta$-function sphere, where the electric and magnetic couplings are equal and opposite. Unfortunately, although the energy expression is superficially ultraviolet finite, additional divergences appear that render it difficult to extract a meaningful result in general, but some limited results are presented.