We survey the recent results on algorithmic learning for families of countable algebraic structures. Within this framework, at each step a learner obtains a finite amount of data about a given countable structure S (which is supposed to be learned), and then the learner outputs a conjecture describing the isomorphism type of S. If the sequence of conjectures converges to the correct answer, then the learning procedure is successful. The paper discusses the results connecting learnability with syntactic properties of structures S. We also give some results on the new approach to learnability which uses equivalence relations on the Cantor space.

For topological T0-spaces X and Y, we prove that the space Y is H-sober if and only if the function space CT (X, Y) endowed with a certain topology T is H-sober.

The paper contains a series of examples of admissible sets A in which the family of all A-c.e. sets has neither negative, nor positive computable A-numbeings.

We observe the unitary equivalence up to reordering on the set of tight frames in a finite-dimensional space. We show that Parseval frames in Rn in general position are unitarily equivalent up to permutation if and only if certain permutational unitary invariants are equal. This is proved by giving an algorithm to recover a Parseval frame up to permutational unitary equivalence from a some subset of these invariants.

The main result is proved by using the action of the symmetric group on the space of symmetric matrices. More precisely, we show algebraically independent generators of the field of invariants for this action.

The Beurling – Malliavin Theorem on the multiplier and its various versions give several variants of conditions for the function f on the real axis R, under which this function can be multiplied by an entire bounded on R function h of arbitrarily small exponential type > 0 so that the product of fh is bounded on R. We consider a new version for the function f = exp(u − M), where u and M are a pair of subharmonic functions of finite type with finite logarithmic integrals over R.

We study a special splitting of 2-c.e. Turing degrees. The special splitting means a splitting which has a c.e. splitting part. We investigate an existence of a special splitting, including a special splitting with upper cone avoidance. We prove that there exists a 2-c.e. degree without special splitting and that there is a 2-c.e. degree which has no special splitting avoiding upper cone of some incomputable c.e. degree.