Vol 72, No 3 (2017)

The paper outlines why the spectrum of maximal ideals Spec_ℂA of a countable-dimensional differential ℂ-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ℂ-homomorphism ψ_M: A → ℂ (M ∈ Spec_ℂA) and any a ∈ A the Taylor series \(\widetilde {{\psi _M}}{\left( a \right)^{\underline{\underline {def}} }}\sum\limits_{m = 0}^\infty {\psi M\left( {{a^{\left( m \right)}}} \right)} \frac{{{z^m}}}{{m!}}\) has nonzero radius of convergence depending on the element a ∈ A.

The mean computing time for computation of values of Boolean operators by straight-line programs with a conditional stop and the storage of at most D is studied. An asymptotically exact formula for the mean computation time is obtained for growing number n of variables and for almost all Boolean operators with m components in a wide range of D and m.

A Liouville classification of integrable Hamiltonian systems being geodesic flows on a twodimensional torus of revolution in an invariant potential field is obtained in the case of linear integral. This classification is obtained using the Fomenko–Zieschang invariant (so called marked molecules) of the systems under consideration. All types of bifurcation curves are described. A classification of singularities of the system solutions is also obtained.

A semantics of realizability based on hyperarithmetical predicates of membership is introduced for formulas of the language of set theory. It is proved that the constructive set theory without the extensionality axiom is sound with this semantics.