ON NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF CARATHÉODORY SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

Given two one-sided Markov chains, the authors illustrate a procedure for ascertaining whether they are essentially the same. Precisely, they show how one can determine whether they are block-isomorphic. An application to hydrology is investigated with an example.     

Spinless Matter in Transposed-Equi-Affine Theory of Gravity

We derive and discuss the equations of motion for spinless matter: relativistic spinless scalar fields, particles and fluids in the model of gravity recently proposed by A. Saa with covariantly constant volume with respect to the transposed connection in Einstein-Cartan spaces. A new interpretation of this theory as a theory with variable Planck constant is suggested. We show that the consistency of the semiclassical limit of the wave equation and classical motion dictates a new definite universal interaction of torsion with massive fields.     

Four-Dimensional Dilatonic Black Holes in Gauss-Bonnet Extended String Gravity

An internal singularity of a string four-dimensional black hole with second order curvature corrections is investigated. A restriction to a minimal size of a neutral black hole is obtained in the frame of the model considered. Vacuum polarization of the surrounding space-time caused by this minimal-size black hole is also discussed.     

Completion of Probabilistic Uniform Limit Spaces

In this article completions of special probabilistic semiuniform convergence spaces are considered. It turns out that every probabilitic Cauchy space under a given t-norm T (triangular norm) has a completion which, in the special case of probabilistic Cauchy spaces with reference to T=min, coincides with the KentRichardson completion for probabilistic Cauchy spaces. Moreover, a completion of probabilistic uniform limit spaces wrt T=min is given which in case of constant probabilistic uniform limit spaces coincides with the Wyler completion.     

Measures of affinity of a sequence for a number

We define strong and weak affinities of a number a for a sequence (x_k ) denoted by L (a,(x_k )) and U (a, (x_k )) respectively. We show U (a,(x_k )) > 0 if and only if the number a is a statistical limit point of the sequence (x_k ). We consider the distribution of sequences with positive weak and strong measures of affinity within the space l ^∞ of bounded sequences. The main result is that the set of bounded sequences with U (a,(x_k )) > 0, that is, the set of sequences with statistical limit points, is a dense subset in l ^∞ of the first category. We also show the set of sequences with positive strong affinities is a nowhere dense subset of l ^∞.     

Modified No-hair Conjecture and the Limiting Process

It is argued that the Schwarzschild black hole solution follows as a unique limit of the Brans-Dicke Class I solutions, provided the correct iterated limit is taken. Such a uniqueness is essential for the validity of a recent version of the no-hair conjecture. A non-trivial modification to this version is proposed in order to exclude Brans-Dicke Class IV solutions which appear to represent scalar hair black holes in general.