Existence and physical uniqueness of the restricted evolution problem of the Einstein relativistic mechanics

The existence and physical uniqueness is proved for the restricted evolution problem in general relativity. Precisely, the Einstein equations for the gravitational field are analyzed in harmonic and generic adapted coordinates in order to prove the existence and uniqueness of solution in both systems of coordinates.     

On the restricted homomorphism problem

The restricted homomorphism problem asks: given an input digraph G and a homomorphism g:G→Y, does there exist a homomorphism f:G→H? We prove that if H is hereditarily hard and YH, then is NP-complete. Since non-bipartite graphs are hereditarily hard, this settles in the affirmative a conjecture of Hell and Nešetřil.     

On a restricted cross-intersection problem

Suppose A and B are families of subsets of an n-element set and L is a set of s numbers. We say that the pair (A,B) is L-cross-intersecting if |A∩B|∈L for every A∈A and B∈B. Among such pairs (A,B) we write PL(n) for the maximum possible value of |A||B|. In this paper we find an exact bound for PL(n) when n is sufficiently large, improving earlier work of Sgall. We also determine P{2}(n) and P{1,2}(n) exactly, which respectively confirm special cases of a conjecture of Ahlswede, Cai and Zhang and a conjecture of Sgall.     

On the central configurations of the planar restricted four-body problem

This article is devoted to answering several questions about the central configurations of the planar (3+1)-body problem. Firstly, we study bifurcations of central configurations, proving the uniqueness of convex central configurations up to symmetry. Secondly, we settle the finiteness problem in the case of two nonzero equal masses. Lastly, we provide all the possibilities for the number of symmetrical central configurations, and discuss their bifurcations and spectral stability. Our proofs are based on applications of rational parametrizations and computer algebra.     

The virial theorem in relativistic quantum mechanics

For a class of potentials including the Coulomb potential q = μr^?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem $\text{(u, α ·}\text{p}\text{u) = (u, r(}\text{?q}\text{?r}\text{)u)}$ is shown to hold, u being an eigenfunction of the operator

$\text{Hu = TU : = (α ·}\text{p}\text{+ β + q)u}$

,

$\text{D(H) = {u ¦ u ∈ [H}{}_{\mathrm{loc}}{}^1\text{(}\text{R}\text{+}{}^3\text{)]}{}^4\text{, r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{u, TU ∈ [L}{}^2\text{(R)}{}^3\text{]}{}^4\text{}}$

($\text{R}$₊³ := $\text{R}$?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation.     

On the final evolution of the n-body problem

The Newtonian n-body problem is studied. The main result gives the asymptotic properties of the distances between particles as time (t) approaches infinity. It is shown how particles can separate into subsystems between which the distances are asymptotic to constant multiples of time, and within which the mutual distances are at most of the order $\text{t}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Each subsystem behaves asymptotically like a pure n-body problem in the sense that an energy and an angular momentum relationship are asymptotically satisfied. These techniques allow some three-body results found by Birkhoff and by Sundman to be extended to the n-body problem. Finally, some n-body results are derived for a motion which expands faster than time.     

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT _pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.Partially supported by the grants from TGRC.     

Passing from mechanics of the special theory of relativity to newtonian mechanics, and the relativistic effects

A general method of passing from mechanics of the special theory of relativity to Newtonian mechanics is considered together with various relativistic effects.     

Covariant two-particle world lines in relativistic Hamiltonian mechanics

On the basis of the space-time interpretation of relativistic classical Hamiltonian mechanics for various classes of two-particle models we study the restrictions on the general solution imposed by the condition of invariance of world lines and on the particle interaction for the purpose of guaranteeing that the world lines are timelike, the instantaneous configurations are spacelike, and that the system is predictive.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 100–105.     

A restricted shift completeness problem

We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in $L^2[0,1]$ posed by M. Carlsson and C. Sundberg.     

On quasi Einstein manifolds

In this paper we give some examples of a quasi Einstein manifold (QE)_n. Next we prove the existence of (QE)_n manifolds. Then we study some properties of a quasi Einstein manifold. Finally the hypersurfaces of a Euclidean space have been studied.