Vol 74, No 4 (2019)

Systems of Boolean functions inducing algebras of Bernoulli distributions whose universal set has a single limit point are considered. A criterion for an algebra generated by a given distributions set to have a single limit point is proved.

About four centuries ago, considering flat sections of cone x² + y² = z² (along the axis of revolution on the plane Oxy), Robert Hooke wrote one fundamental differential equation \((x,y,z)^{\prime\prime} = - {{4{\pi ^2}k} \over {{{(\sqrt {{x^2} + {y^2} + {z^2}})}^3}}}\; \cdot \;(x,y,z)\), which thereafter set the foundation of the law of universal gravitation and explanation of movement of charged particle in the classical stationary Coulomb field. In this paper, differential-algebraic models arising as the result of replacement of a cone by an arbitrary quadric surface F(x, y, z) = 0 with respect to (as we call it) the Kepler parametrization of quadratic curves {F(x, y, α · x + β · y + δ)=0 | α, β, δ ∈ K}, K = ℝ, ℂ, are proposed and studied.

The following theorem is proved: the union of countably many closed U_r-sets (r > 2) for the Vilenkin-Dzhafarli system is again a U_r-set for this system.

Let L be an extension of the language of arithmetic, F be a class of number-theoretical functions. A notion of the V-realizability for L-formulas is defined in such a way that indexes of functions in V are used for interpreting the implication and the universal quantifier. It is proved that the semantics for L based on the V-realizability coincides with the classic semantics if and only if V contains all L-definable functions.

It is proved that the set of closed classes containing some minimal classes in the partially ordered set \({\cal L}_2^3\) of closed classes in the three-valued logic that can be homomorphically mapped onto the two-valued logic has the cardinality of continuum.