RELATIVIZING CHAITIN'S HALTING PROBABILITY

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

On Some Extremal Problems on Linearly Invariant Classes

In present paper the definition of linearly invariant class of analytical in the right half-plane is given and some extremal problems on introduced class are solved. For solving we use method based on variational formulas with specially introduced omega-operator, defined on these classes. In case when domain is unit disk similar linearly invariant classes were considered by Ch. Pommerenke, V. Starkov, E.G. Kiriyatzkii.