Vol 217, No 5 (2026)

An inclusion generated by a parameter-dependent continuous map and closed convex sets is considered. An implicit function theorem for an inclusion that holds without a priori normality assumptions is established in terms of $\lambda$-truncations. A number of corollaries is deduced, including ones for systems of equations and inequalities with a parameter. The results obtained are compared with the previously known ones.

Toric topology assigns to each simple convex $n$-polytope $P$ with $m$ facets an $n$-dimensional real moment-angle manifold $\mathbb R\mathcal{Z}_P$ with a canonical action of $\mathbb Z_2^m=(\mathbb Z/2\mathbb Z)^m$. We consider (not necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on $\mathbb R\mathcal{Z}_P$. The orbit space $N(P,H)=\mathbb R\mathcal{Z}_P/H$ carries an action of $\mathbb Z_2^m/H$. For general $n$ we introduce the notion of Hamiltonian $\mathcal{C}(n,k)$-subcomplex in the boundary of an $n$-polytope $P$ generalizing the notions of Hamiltonian cycle (for $k=2$), Hamiltonian theta-subgraph (for $k=3$) and Hamiltonian $K_4$-subgraph (for $k=4)$ in the $1$-skeleton of a $3$-polytope. Each $\mathcal{C}(n,k)$-subcomplex $C\subset \partial P$ corresponds to a subgroup {$H_C\subset\mathbb Z_2^m$} such that $N(P,H_C)\simeq S^n$. We prove that in dimensions $n\leqslant 4$ this correspondence is a bijection. Any subgroup $H\subset \mathbb Z_2^m$ defines a complex $\mathcal{C}(P,H)\subset \partial P$. We prove that each Hamiltonian $\mathcal{C}(n,k)$-subcomplex $C\subset \mathcal{C}(P,H)$ inducing $H$ corresponds to a hyperelliptic involution $\tau_C\in\mathbb Z_2^m/H$ on the manifold $N(P,H)$ (that is, an involution with orbit space homeomorphic to $S^n$) and in dimensions $n\leqslant 4$ this correspondence is a bijection. We prove that for the geometries $\mathbb X= \mathbb S^4$, $\mathbb S^3\times\mathbb R$, $\mathbb S^2\times \mathbb S^2$, $\mathbb S^2\times \mathbb R^2$, $\mathbb S^2\times \mathbb L^2$ and $\mathbb L^2\times \mathbb L^2$ there exists a compact right-angled $4$-polytope $P$ with a free action of $H$ such that the geometric manifold $N(P,H)$ has a hyperelliptic involution in $\mathbb Z_2^m/H$, and there are no such polytopes for $\mathbb X=\mathbb R^4$, $\mathbb L^4$, $\mathbb L^3\times \mathbb R$ and $\mathbb L^2\times \mathbb R^2$.

Three characteristic almost Lorentzian geometries in the Grushin plane are viewed as optimal control problems. On the basis of geometric control theory extremal trajectories are studied, the attainability set is calculated, the existence of optimal trajectories is investigated, the longest ‘spheres’ and the distance in the Lorentzian metric are described.