Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

Chandrasekhar's integral stability criterion for an equilibrium spherical cloud of a protostar, modified in the framework of non-Gaussian kappa-statistics

Within the framework of the non-extensive statistical mechanics of Kanyadakis, a generalization of the integral stability theorem of Chandrasekhar for the spherically symmetric distribution of matter and black radiation in an exoplanetary cloud in a state of gravitational equilibrium is obtained. For this purpose, the elements of deformed thermodynamics for an ideal gas, deformed canonical Gibbs distribution, as well as the effective gravitational constant, calculated in the formalisms of Kanyadakis and Verlinde, are used. In this, the deformation parameter κ (kappa) measures the so-called degree of nonextensiveness of the cloud system. In addition, the modified thermodynamic properties of blackbody radiation, in particular, the analogue of Stefan's law for radiation energy and generalized expressions for the entropy, heat capacity and radiation pressure, are discussed in the context of κ -statistics. The presented method of combining the indicated anomalous physical processes provides an alternative to the classical procedure of Chandrasekhar's derivation of the well-known integral theorems for gas configurations in gravitational equilibrium, and restores all standard expressions in the limit κ → 0. The results obtained will be able, according to the author, to explain some astrophysical problems of stellar-planetary cosmogony, associated, in particular, with modeling the processes of joint formation and evolution of a protosun and an exoplanetary cloud from a single nebula.

On Asymptotic Standard Normality of the Two Sample Pivot

The asymptotic solution to the problem of comparing the means of two heteroscedastic populations, based on two random samples from the populations, hinges on the pivot underpinning the construction of the confidence interval and the test statistic being asymptotically standard normal, which is known to happen if the two samples are independent and the ratio of the sample sizes converges to a finite positive number. This restriction on the asymptotic behavior of the ratio of the sample sizes carries the risk of rendering the asymptotic justification of the finite sample approximation invalid. It turns out that neither the restriction on the asymptotic behavior of the ratio of the sample sizes nor the assumption of cross sample independence is necessary for the pivotal convergence in question to take place. If the joint distribution of the standardized sample means converges to a spherically symmetric distribution, then that distribution must be bivariate standard normal (which can happen without the assumption of cross sample independence), and the aforesaid pivotal convergence holds. AMS Classification: 62E20, 62G20

Posterior Analysis of State Space Model with Spherical Symmetricity

The present work investigates state space model with nonnormal disturbances when the deviation from normality has been observed only with respect to kurtosis and the distribution of disturbances continues to follow a symmetric family of distributions. Spherically symmetric distribution is used to approximate behavior of symmetric nonnormal disturbances for discrete time series. The conditional posterior densities of the involved parameters are derived, which are further utilized in Gibbs sampler scheme for estimating the marginal posterior densities. The state space model with disturbances following multivariate-tdistribution, which is a particular case of spherically symmetric distribution, is discussed.

Linear Polarization and the Dynamics of Circumstellar Disks of Classical Be Stars

The intrinsic linearly polarized light arising from electron scattering of stellar radiation in a non-spherically symmetric distribution of gas is a characterizing feature of classical Be stars. The distinct polarimetric signature provides a mean for directly probing the physical and geometric properties of the gaseous material enveloping these rapidly-rotating massive stars. Using a Monte Carlo radiative transfer computation and a self-consistent radiative equilibrium solution for the circumstellar gas, we explore the role of this observable signature in investigating the dynamical nature of classical Be star disks. In particular, we focus on the potential for using linearly polarized light to develop diagnostics of mass-loss events and to trace the evolution of the gas in a circumstellar disk. An informed context for interpreting the observed linear polarization signature can play an important role in identifying the physical process(es) which govern the formation and dissipation of the gaseous disks surrounding classical Be stars.

ON A COMMON MISUNDERSTANDING OF THE BIRKHOFF THEOREM AND LIGHT DEFLECTION CALCULATION: GENERALIZED SHAPIRO DELAY AND ITS POSSIBLE LABORATORY TEST

In Newtonian gravity (NG) it is known that the gravitational field anywhere inside a spherically symmetric distribution of mass is determined only by the enclosed mass. This is also widely believed to be true in general relativity (GR), and the Birkhoff theorem is often invoked to support this analogy between NG and GR. Here we show that such an understanding of the Birkhoff theorem is incorrect and leads to erroneous calculations of light deflection and delay time through matter. The correct metric, matching continuously to the location of an external observer, is determined both by the enclosed mass and mass distribution outside. The effect of the outside mass is to make the interior clock run slower, i.e., a slower speed of light for external observer. We also discuss the relations and differences between NG and GR, in light of the results we obtained in this Lettework. Finally we discuss the Generalized Shapiro delay, caused by the outside mass, and its possible laboratory test.