Vol 88, No 2 (2024)

The main purpose of this paper is to study an abstract system which consists of a non-linear differential inclusion with $C_0$-semigroups and history-dependent operators combined with an evolutionary non-linear inclusion involvingpseudomonotone operators, which contains several interesting problems as special cases. We first introduce a hybrid iterative system by using the Rothe method, pseudomonotone operators theory,and a feedback iterative technique. Then, the existence and a priori estimates for solutions to a series of approximating discrete problems are established. Furthermore, through a limiting procedure for solutions of the hybrid iterative system, we show that the existence of solutions to the original problem.

Kleene algebras are structures with addition, multiplication and constants $0$and $1$, which form an idempotent semiring, and the Kleene iterationoperation. In the particular case of $*$-continuous Kleene algebras,Kleene iteration is defined, in an infinitary way, as the supremum of powersof an element. We obtain results on algorithmic complexityfor Horn theories (semantic entailment from finite sets of hypotheses)of commutative $*$-continuous Kleene algebras. Namely,$\Pi_1^1$-completeness for the Horn theory and $\Pi^0_2$-completenessfor its fragment, where iteration cannot be used in hypotheses, is proved.These results are commutative counterparts of the corresponding theoremsof D. Kozen (2002) for the general (non-commutative) case.Several accompanying results are also obtained.

In the proposed paper, the process of propagation of shock waves intwo-dimensional media without its own pressure drop is studied. Themodel of such media is a system of equations of gas dynamics, whereformally the pressure is assumed to be zero. From the point of viewof the theory of systems of conservation laws, the system ofequations under consideration is in some sense degenerate, and,consequently, the corresponding generalized solutions have strongsingularities (evolving shock waves with density in the form ofdelta functions on manifolds of different dimensions). We willdenote this property as the evolution of the hierarchy of strongsingularities or the evolution of the hierarchy of shock waves. Inthe paper, in the two-dimensional case, the existence of such an interaction of strong singularities with density delta functionalong curves in the space $\mathbb{R}^2$ is proved, at which a density concentration occurs at a point, that is, a hierarchy ofshock waves arises. The properties of such dynamics of strongsingularities are described. The results obtained provide a startingpoint for moving on to a much more interesting multidimensionalcase in the future.

It is proved that the Grothendieck standard conjecture $B(X)$of Lefschetz type holds for a smooth complex projective4-dimensional variety $X$ provided that there exists a morphismof $X$ onto a smooth projective curve whose generic scheme fibreis an Abelian variety with bad semi-stable reductionat some place of the curve.

The problem of finding the first integrals of the Newton equations inthe $n$-dimensional Euclidean space is reduced to that of findingtwo integrals of motion on the Lie algebra $\mathrm{so}(4)$which are invariant under $m\geqslant n-2$ rotation symmetry fields.As an example, we obtainseveral families of integrable and superintegrable systems with first,second, and fourth-degree integrals of motion in the momenta.The corresponding Hamilton–Jacobi equationdoes not admit separation variables in any of the known curvilinear orthogonal coordinate systemsin the Euclidean space.