Evolution of the equations of dynamics of the Universe: From Friedmann to the present day

Celebrating the centenary of general relativity theory, we must recall that Friedmann’s discovery of the equations of evolution of the Universe became the strongest prediction of this theory. These equations currently remain the foundation of modern cosmology. Nevertheless, data from new observations stimulate a search for modified theories of gravitation. We discuss cosmological aspects of theories with two dynamical metrics and theories of massive gravity, one of which was developed by Logunov and his coworkers.     

Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.     

Generalized Kondo lattice model and its spin-polaron realization by the projection method for cuprates

The spin–fermion model, which is an effective low-energy realization of the three-band Emery model after passing to the Wannier representation for the p_x and p_y orbitals of the subsystem of oxygen ions, reduces to the generalized Kondo lattice model. A specific feature of this model is the existence of spin-correlated hoppings of the current carriers between distant cells. Numerical calculations of the spectrum of spin-electron excitations highlight the important role of the long-range spin-correlated hoppings.     

Families of exact solutions for linear and nonlinear wave equations with a variable speed of sound and their use in solving initial boundary value problems

We propose a procedure for multiplying solutions of linear and nonlinear one-dimensional wave equations, where the speed of sound can be an arbitrary function of one variable. We obtain exact solutions. We show that the functional series comprising these solutions can be used to solve initial boundary value problems. For this, we introduce a special scalar product.     

Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal <Emphasis Type="Italic">B</Emphasis>-spline

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ _m as the weight functions on [0, m] (m ∈ ?) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.     

Stokes constants of an oblate ellipsoid of revolution with equidensites homothetic to its surface

The theory of equilibrium figures was actively developed in the 19th century, when it was found that the observed massive celestial bodies (the Sun, planets, and satellites) had an almost ellipsoidal form. The existence of exactly ellipsoidal figures was also established. The gravitational potential of these figures is represented by a Laplace series with its coefficients (Stokes’ constants I _n ) determined by some integral operator. The general term of the series was found for a homogeneous ellipsoid of revolution and the first terms of the series were found for some other mass distributions. Here, we have obtained the general term of the series for an arbitrary mass distribution given that the equidensites (surfaces of equal density) are homothetic to the outer surface of the ellipsoid of revolution. Simple estimates and an asymptotics of I _n have also been obtained.     

A Geometric Characterization of a Class of Poisson Type Distributions

Tempered distributions on \(\mathbb {R}^{n}\) which have Dirac comb structure are characterized in terms of sparseness properties of its support and spectrum.