Narrow Results

We are concerned with the equivalence problem of third-order ordinary differential equations (ODEs) under bundle diffeomorphisms of three-dimensional real space. For those ODEs, we define canonical structure equations of a bundle on the second jet space, and introduce four torsion parts. We characterize linear third-order ODEs and projective linear third-order ODEs defined on tangent scrolls of space curves by vanishing of some torsion parts.

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves "almost" like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of "semi-flat" manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection.

In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

The teleparallel equivalent of general relativity (TEGR) is an alternative formulation of Einstein's equations in the framework of Riemann-Cartan spacetimes. The gravitational field can be described either by the curvature of the torsion-free connection of general relativity (GR) or by the torsion of the curvature-free connection of the TEGR. Both in GR and TEGR the freedom in the choice of coordinates gives rise to the equivalence problem of deciding whether two solutions of the field equations are the same. This problem is solved by means of a invariant description of the gravitational field. We investigate whether the equivalence between GR and TEGR also holds at the level of these invariant descriptions. We show that the GR description assures equivalence in TEGR only in very special situations. These results are illustrated on teleparallel spacetimes with torsion and Gödel metric.

We investigate the computational complexity of deciding whether or not a given polynomial, presented as the sum of monomials, is identically 0 over a ring. It is proved that if the factor by the Jacobson-radical is not commutative, then the problem is coNP-complete.

General relativistic spacetimes in which at every point $p$ in some neighbourhood $W$ the tangent space $T_p$ is invariant under the same continuous subgroup $g$ of the Lorentz group are considered. Some previously open problems for such spacetimes are discussed and solved, and a comprehensive survey of such spacetimes is provided. For most cases, invariance of the curvature and its first covariant derivative under $g$ imply the existence of an isometry group in $W$ containing $g$ as an isotropy group. In the remaining cases, invariance of second and third derivatives may be required for this implication, and in particular, the necessity of invariance of the third derivative is proved for the locally rotationally symmetric spacetimes discussed by Ellis and by Stewart and Ellis.

We study knowledge bases as a certain algebraic and model-theoretical structure. Algebraic model of a knowledge base allows to distinguish some algebraic and logical invariants of knowledge bases. In particular, our interest is to find out how these invariants are related to the informational equivalence of knowledge bases.

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the examples the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.