On particle dynamics near the singularity inside the Schwarzschild black hole and T-spheres

The problem of the speed of the objects inside the Schwarzschild black hole is considered. The general result is that the value of the relative speed of the objects following their non-zero angular momentum trajectories, both of geodesic and non-geodesic character, when approaching the ultimate singularity, tends to the value of speed of light. There is only one exception when both objects move in the same plane and have parallel angular momenta. This outcome appears to have a deeper sense: it reflects the anisotropic character of the dynamics of interior of this particular black hole. The result in question means that near the singularity, collisions of two particles lead to an indefinitely large energy in the center of mass frame. Aforementioned properties have their counterpart in the phenomenon of an indefinitely large blueshift near the singularity. Thus the angular momentum of a particle turns out to be an important feature that affects the final behavior of particle near the singularity. Motivated by this fact, we generalize the Lema\^{\i}tre frame under the horizon in such a way that reference particles themselves have nonzero angular momentum. Our results apply not only to the Schwarzschild singularity but also to other space-like ones for which the scale factor $g\rightarrow \infty$. We also analyze another type of singulairites for which the circumference radius vanishes but $g$ remains finite.Comment: Matches published versio

Vegetation diversity on coal mine spoil heaps-how important is the texture of the soil substrate?

© 2019, The Author(s). The relationship between the size of the particle fractions of the soil substrate and the diversity of the spontaneously developing vegetation was investigated on coal mine spoil heaps in Upper Silesia (Southern Poland). The analyses were based on 2567 research plots of developed spontaneous vegetation and their associated soil substrate samples collected from 112 coal mine spoil heaps. For each research plot the prevailing particle size fraction was determined (stones, gravel, sand, silt), the species composition and abundance was recorded and the species richness (S), Shannon-Wiener diversity index (H′), Simpson (C) and Evenness (E) indices were used to determine species diversity. From a total of 119 research plots (in all particle size fraction categories), the values of 15 physicochemical properties (pH, electrical conductivity, water holding capacity, moisture, carbon content, total N, available P, Mg and exchange cations Ca, Mg, K, Na, fine particles (%), gravel (%), stone (%)) were obtained to asses their impact on the floristic composition of vegetation patches using Canonical Correspondence Analysis (CCA). Additionally, functional traits of the dominant species of each vegetation patch (life forms, life strategies and socio-ecological groups), were selected to analyse their relation to substrate texture. It was shown that the highest species richness and the highest values for Shannon-Wiener diversity index, as well as Simpson and Evenness indices, were obtained in plots formed on stones. Moreover, the greatest variation in the participation of species representing different habitats, life forms, and life strategies was found on gravelly substrates. Contrary to our expectations, the vegetation diversity (in terms of both species and their functional traits) was not highest in habitats with a high composition of fine size particles.Published versio

Absence of bound states for waveguides in 2D periodic structures

We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting models a so-called "soft-wall" waveguide problem. We show that there are no bound states, i.e., the spectrum of the operator under study contains no point spectrum.Comment: This is an updated version of our paper (in slightly different form in Journal of Mathematical Physics). An anonymous reviewer kindly made us aware that ref. 10 is not applicable in our situation. An application of the theorem in ref. 10 would have proved the absence of singular continuous spectrum also. Our result on the absence of point spectrum is not affected by thi

Thermodynamics of entropy-driven phase transformations

Thermodynamic properties of one-dimensional lattice models exhibiting entropy-driven phase transformations are discussed in quantum and classical regimes. Motivated by the multistability of compounds exhibiting photoinduced phase transitions, we consider systems with asymmetric, double, and triple well on-site potential. One finds that among a variety of regimes, quantum versus classical, discrete versus continuum, a key feature is asymmetry distinguished as a "shift" type and "shape" type in limiting cases. The behavior of the specific heat indicates one phase transformation in a "shift" type and a sequence of two phase transformations in "shape"-type systems. Future analysis in higher dimensions should allow us to identify which of these entropy-driven phase transformations would evolve into phase transitions of the first order

A new diversity index

We introduce here a new index of diversity based on consideration of reasonable propositions that such an index should have in order to represent diversity. The behaviour of the index is compared with that of the Gini-Simpson diversity index, and is found to predict more realistic values of diversity for small communities, in particular when each species is equally represented and for small communities. The index correctly provides a measure of true diversity that is equal to the species richness across all values of species and organism numbers when all species are equally represented, as well as Hill's more stringent 'doubling' criterion when they are not. In addition, a new graphical interpretation is introduced that permits a straightforward visual comparison of pairs of indices across a wide range within a parameter space based on species and organism numbers