Hilbert’s Nullstellensatz proved by him in 1890 is one of the basic results in modern algebraic geometry. We give various statements and proofs of this theorem all of which are used in algebraic geometry. All notions and facts outside the basic algebra course are explained in the paper.

We refined the axiomatics of asymmetric logics. For logics X(km, k) of family subsets of the km-element set X, which cardinal numbers are multiples of k we completely described the cases in which X(km, k) a) is symmetric or b) is asymmetric. For an infinite set Ω and a natural number n ≥ 2 we constructed the concrete logics EΩn and completely described the cases in which these logics are asymmetric. For asymmetric logics E we determine when both the set A ∈ E and its complement Ac are atoms of the logic E. Let a symmetric logic E of a finite set Ω be not a Boolean algebra, and let A be an algebra of subsets from Ω, and assume that E ⊂ A. Then there exists a measure on E, that does not admit an extension to a measure on A.

We study the phenomenon of phase trajectories of a Hamiltonian system going to infinity in a finite time, the phase space of which is a separable Hilbert space. It is shown that if the Hamiltonian is a densely defined quadratic form on the phase space, which is not majorized either from below or from above by the quadratic form of the Hilbert norm, then the phase trajectories allow going to infinity in a finite time. To describe the phase flow of such Hamiltonian systems, an extended phase space is introduced, which is a locally convex space to which the Hamiltonian function, trajectories of the Hamiltonian system, and the symplectic form defined on the original Hilbert space can be extended. Flowinvariant measures on extended space are also studied. The properties of the Koopman unitary representation of the extended phase flow in the Hilbert space of functions that are quadratically integrable with respect to an invariant measure are investigated.

We prove that every coinfinite set is a characteristic transversal of a suitably computably separable equivalence relation, over which only locally finite, locally finite separable and finitely approximable unary algebras are represented. Similar properties for uniformly computable separable equivalences are considered.

We prove the existence of 2ω pairwise non-Σ-embeddable into each other (and henceforth non-Σ-isomorphic) Σ-presentations of the additive group of the real numbers in the hereditarily finite superstructure over the ordered field of the real numbers.

We prove that for each u ⩾ 2 the class of all single-valued Σ0 ucomputable numberings of any infinite family of total functions is effectively infinite and the class of all its Σ0 u-1-computable numberings is generated by the downward closure with respect to the reducibility of the set of all infinite direct sums of uniformly Σ0 u-1-computable sequences of its single-valued numberings. It is established that if u > 2, then the class of all Σ0 u-computable numberings of any infinite family is generated by infinite direct sums of uniformly Σ0 u-computable and uniformly Σ0 u-minimal sequences of its numberings.