Volume 105, Nº 3-4 (2019)

The objective of this paper is to trace the increase in the complexity of the description of classes of analytic complexity (introduced by the author in previous works) under the passage from the class Cl₁ to the class Cl₂. To this end, two subclasses, Cl₁⁺ and Cl₁⁺⁺, of Cl₂ that are not contained in Cl₁ are described from the point of view of the complexity of the differential equations determining these subclasses. It turns out that Cl₁⁺ has fairly simple defining relations, namely, two differential polynomials of differential order 5 and algebraic degree 6 (Theorem 1), while a criterion for a function to belong to Cl₁⁺⁺ obtained in the paper consists of one relation of order 6 and five relations of order 7, which have degree 435 (Theorem 2). The “complexity drop” phenomenon is discussed; in particular, those functions in the class Cl₁⁺ which are contained in Cl₁ are explicitly described (Theorem 3).

The present paper deals with estimates of the chromatic number of a space with forbidden one-color triangles. New lower bounds for the quantity under study, which are sharper than all bounds obtained so far, are presented.

Let K ⊂ ℂ be a polynomially convex compact set, f be a function analytic in a domain \(\overline{\mathbb{C}} \backslash K\) with Taylor expansion \(f(z) = \sum\nolimits_{k = 0}^\infty {{a_k}/{z^{k + 1}}} \) at ∞, and \({H_i}(f): = {\rm{det}}({a_{k + l}})_{k,l = 0}^i\) be the related Hankel determinants. The classical Polya theorem [11] says that \(\mathop {{\rm{lim\; sup}}}\limits_{i \to \infty } \;{\rm{|}}{H_i}(f){{\rm{|}}^{1/{i^2}}} \le d(K),\) where d(K) is the transfinite diameter of K. The main result of this paper is a multivariate analog of the Polya inequality for a weighted Hankel-type determinant constructed from the Taylor series of a function analytic on a ℂ-convex (= strictly linearly convex) domain in ℂⁿ.

Let G be a finite group, and let A and B be, respectively, an Abelian and a nilpotent subgroup in G. In the present paper, we complete the proof of the theorem claiming that there is an element g of G such that the intersection of A with the subgroup conjugate to B by g is contained in the Fitting subgroup of G.

The notion of the quality of approximate solutions of ill-posed extremum problems is introduced and a posteriori estimates of quality are studied for various solution methods. Several examples of quality functionals which can be used to solve practical extremum problems are given. The new notions of the optimal, optimal-in-order, and extra-optimal qualities of a method for solving extremum problems are defined. A theory of stable methods for solving extremum problems (regularizing algorithms) of optimal-in-order and extra-optimal quality is developed; in particular, this theory studies the consistency property of a quality estimator. Examples of regularizing algorithms of extra-optimal quality for solving extremum problems are given.

We consider the following elliptic problem of exponential-type growth posed in an open bounded domain with smooth boundary B₁ (0) ⊂ ℝ⁴: \((P_\lambda)\begin{cases}\Delta^{2}u = \lambda(u^{-\delta}+h(u)e^{u^{\alpha}}),\;\;u>0\;in\;B_{1}(0),\\\;\;\;\;\;u=\Delta{u}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\partial{B}_{1}(0).\end{cases}\) Here Δ²(.):= −Δ(−Δ)(.) denotes the biharmonic operator, 1 < α < 2, 0 < δ < 1, λ > 0, and h(t) is assumed to be a smooth “perturbation” of \({e^{{t^\alpha }}}\) as t→∞ (see (H1)–(H4) below). We employ variational methods in order to show the existence of at least two distinct (positive) solutions to the problem (P_λ) in \({H^2} \cap H_0^1({B_1}(0))\).

Let \(\mathbb{F}_{q}\) be a finite field, let \(\mathbb{X}\) be a subset of the projective space ℙ^s−1 over \(\mathbb{F}_{q}\) parametrized by rational functions, and let I(\((\mathbb{X})\)) be the vanishing ideal of \(\mathbb{X}\). The main result of this paper is a formula for I(\((\mathbb{X})\)) that will allow us to compute (i) the algebraic invariants of I(\((\mathbb{X})\)) and (ii) the basic parameters of the corresponding Reed–Muller-type code.

An upper bound for the multiplicative energy of the spectrum of an arbitrary subset of \(\mathbb{F}_{p}\) is obtained. Apparently, at present, this is the best bound.

Let L be a bounded 2 × 2 block operator matrix whose main-diagonal entries are self-adjoint operators. It is assumed that the spectrum of one of these entries is absolutely continuous, being presented by a single finite band, and the spectrum of the other main-diagonal entry is entirely contained in this band. We establish conditions under which the operator matrix L admits a complex deformation and, simultaneously, the operator Riccati equations associated with the deformed L possess bounded solutions. The same conditions also ensure a Markus–Matsaev-type factorization of one of the initial Schur complements analytically continued onto the unphysical sheet(s) of the complex plane of the spectral parameter. We prove that the operator roots of this Schur complement are explicitly expressed through the respective solutions to the deformed Riccati equations.

We prove that a periodic group is locally finite, given that each of its finite subgroups lies in a subgroup isomorphic to a finite simple group G₂ of Lietypeovera field of odd characteristic.

The behavior of the modulus of the curvature tensor and of the holomorphic sectional curvature on Ricci-flat Kähler manifolds is investigated.

Properties of functions from the Sobolev orthogonal system \(\mathfrak{W}_{r}\) generated by the Walsh system are studied. In particular, recurrence relations for functions from \(\mathfrak{W}_{1}\) are obtained. The uniform convergence of Fourier series in the system \(\mathfrak{W}_{r}\) to functions f from the S obolev spaces \(W_{{L^p}}^r\), p ≥ 1, r = 1, 2,… is proved.

The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

We obtain new estimates for the number of edges in induced subgraphs of some distance graph.

The closure of the set of smooth compactly supported functions in a weighted Hölder space on ℝⁿ, n ≥ 1, with a weight controlling the behavior at the point at infinity is described. As an application, a solvability criterion for operator equations generated by de Rham differentials both in these spaces and on the closure of the set of smooth compactly supported functions in them for n ≥ 2 is obtained.