abstract

We discuss a Brane World scenario where we live on a 3-brane with massive gravity in the infinite-volume bulk. The bulk graviton can be much heavier than the inverse Hubble size, as heavy as the bulk Planck scale, whose lower bound is roughly the inverse of 0.1 mm. The 4D Einstein–Hilbert term on the brane shields the brane matter from both strong bulk gravity and large bulk graviton mass. Gravity on the brane does not become higher-dimensional at large distances. Instead, at distance scales above the bulk Planck length, gravity on the brane behaves as 4D gravity with small graviton mass roughly of the order of or below the inverse Hubble size. Unlike the massless case, with massive gravity in the bulk one can have: (i) 4D tensor structure on a codimension-1 brane; and (ii) no infrared tachyon for smoothed-out higher codimension branes. The effects of the brane dynamics on the bulk are exponentially suppressed away from the brane. One consequence is that there are no “self-accelerated” solutions. In codimension-2 cases, there exist nonsingular solutions with a flat 3-brane for a continuous positive range of the brane tension. In higher codimension cases, as in the massless case, higher curvature terms are required to obtain such solutions.

abstract

Exactly solvable potentials of the position-dependent mass Schrödinger equation are generated by taking ‘Hulthén plus hyperbolic cotangent potential’ as a parent system. We apply a simple transformation method that includes a co-ordinate transformation followed by a functional transformation of wave function, and also a set of plausible ansatze. The mass function of the parent system gets transformed to a new mass function of the generated system. The generated potentials are mostly Sturmian, which are energy dependent for non-power law potentials. Some of the generated Sturmian potentials can be converted into normal potentials by regrouping various potential parameters. The wave functions of generated systems are normalizable in most cases.

abstract

This is a consecutive paper on the timelike geodesic structure of static spherically symmetric spacetimes. First, we show that for a stable circular orbit (if it exists) in any of these spacetimes all the infinitesimally close to it timelike geodesics constructed with the aid of the general geodesic deviation vector have the same length between a pair of conjugate points. In Reissner–Nordström black hole metric, we explicitly find the Jacobi fields on the radial geodesics and show that they are locally (and globally) maximal curves between any pair of their points outside the outer horizon. If a radial and circular geodesics in R–N metric have common endpoints, the radial one is longer. If a static spherically symmetric spacetime is ultrastatic, its gravitational field exerts no force on a free particle which may stay at rest; the free particle in motion has a constant velocity (in this sense the motion is uniform) and its total energy always exceeds the rest energy, i.e.  it has no gravitational energy. Previously, the absence of the gravitational force has been known only for the global Barriola–Vilenkin monopole. In the spacetime of the monopole, we explicitly find all timelike geodesics, the Jacobi fields on them and the condition under which a generic geodesic may have conjugate points.

abstract

Recently proposed by Korsch and Glück [Eur. J. Phys. 23, 413 (2002)] an extremely simple method for computation of eigenvalues via direct representation of position and momentum operators in matrix form is successfully applied to the calculation of energies of the ground and excited states of the \(x^{2}y^{2}\) Hamiltonian and its supersymmetric quantum matrix extensions.

abstract

The one-dimensional attractive Hubbard model (\(U\ll 0\)) is discussed, assuming periodic boundary conditions and the half-filling case. The considered chains have \(N\) nodes, the same number of electrons, where \(N-1\) of them have the same spin projection. The exact diagonalization was performed for any number \(N\) of atoms. The eigenvectors and eigenvalues in some cases are constructed based on the Golden Number.

abstract

The transition network for \(RR\)-increments is a directed and weighted graph, where the vertices represent \(RR\)-increments and the edges correspond to subsequent increments. We show that based on the transition matrix of this network, the entropy of heart rhythm can be calculated. We compare the entropy of the distributions of eigenvalues of the transition matrix for heart transplant patients and for healthy young subjects. We show the regulatory effect of the autonomic nervous system on the entropy values and evaluate the effects of the progression of graft reinnervation on the entropy values.