Vol 60, No 2 (2019)

We give an example of a simplicial complex whose corresponding moment-angle complex is homotopy equivalent to a wedge of spheres, but there is a sphere that cannot be realized by any linear combination of iterated higher Whitehead products. Using two explicitly defined operations on simplicial complexes, we prove that there exists a simplicial complex that realizes any given iterated higher Whitehead product. Also we describe the smallest simplicial complex that realizes an iterated product with only two pairs of nested brackets.

We examine a boundary value problem for a mixed type equation of the second order in a cylindrical domain. Under certain conditions, the existence of regular solutions in a suitable weighted Sobolev space is proven by regularization. The uniqueness of solutions is also established under the same conditions. A few estimates of the second derivatives of a generalized solution are obtained in the elliptic-parabolic domain.

We study the Lorentzian manifolds M₁, M₂, M₃, and M₄ obtained by small changes of the standard Euclidean metric on ℝ⁴ with the punctured origin O. The spaces M₁ and M₄ are closed isotropic space-time models. The manifolds M₃ and M₄ (respectively, M₁ and M₂) are geodesically (non)complete; M₁ are M₄ are globally hyperbolic, while M₂ and M₃ are not chronological. We found the Lie algebras of isometry and homothety groups for all manifolds; the curvature, Ricci, Einstein, Weyl, and energy-momentum tensors. It is proved that M₁ and M₄ are conformally flat, while M₂ and M₃ are not conformally flat and their Weyl tensor has the first Petrov type.

Attempting to solve the Four Color Problem in 1940, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P₅ of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Not many precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P₅. Given a 3-polytope P, by w(P) denote the minimum of the maximum degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol’ and Madaras showed that if a polytope P in P₅ is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5, ∞)-star), then w(P) can be arbitrarily large. For each P* in P₅ with neither vertices of degree 6 and 7 nor minor (5, 5, 5, 5, ∞)-star, it follows from Lebesgue’s Theorem that w(P*) ≤ 51. We prove that every such polytope P* satisfies w(P*) ≤ 42, which bound is sharp. This result is also best possible in the sense that if 6-vertices are allowed but 7-vertices forbidden, or vice versa; then the weight of all minor 5-stars in P₅ under the absence of minor (5, 5, 5, 5, ∞)-stars can reach 43 or 44, respectively.

We establish the criteria of validity of some integral weighted inequalities for a class of quasilinear integral operators.

We introduce and study the class of holographic models which can be defined by copying of some of its finite parts by means of automorphisms. We prove this class to differ from the class of countably categorical models. Characterizations of the classes of holographic Boolean algebras, abelian groups, linear orderings, fields, and equivalences are given.

We consider the problem with free (unknown) boundary for the one-dimensional diffusion–convection equation. The unknown boundary is found from the additional condition on the free boundary. A dilation of the variables reduces the problem to an initial-boundary value problem for a strictly parabolic equation with unknown coefficients in the known domain. These coefficients are found from an additional boundary condition, which makes it possible to construct a nonlinear operator whose fixed points define the solution to the initial problem.

In this article we study the Gieseker–Maruyama moduli spaces ℬ(e, n) of stable rank 2 algebraic vector bundles with Chern classes c₁ = e ∈ {−1, 0} and c₂ = n ≥ 1 on the projective space ℙ³. We construct the two new infinite series Σ₀ and Σ₁ of irreducible components of the spaces ℬ(e, n) for e = 0 and e = −1, respectively. General bundles of these components are obtained as cohomology sheaves of monads whose middle term is a rank 4 symplectic instanton bundle in case e = 0, respectively, twisted symplectic bundle in case e = −1. We show that the series Σ₀ contains components for all big enough values of n (more precisely, at least for n ≥ 146). Σ₀ yields the next example, after the series of instanton components, of an infinite series of components of ℬ(0, n) satisfying this property.