Vol 103, No 3-4 (2018)

A decomposable Galton–Watson branching process with two particle types is studied. It is assumed that the particles of the first type produce equal numbers of particles of the first and second types, while the particles of the second type produce only particles of their own type. Under the condition that the total number of particles of the second type is greater than N →∞, a functional limit theorem for the process describing the number of particles of the first type in different generations is proved.

The canonical representation of the Klein group K₄ = ℤ₂⊕ℤ₂ on the space ℂ* = ℂ {0} induces a representation of this group on the ring L = C[z, z⁻¹], z ∈ ℂ*, of Laurent polynomials and, as a consequence, a representation of the group K₄ on the automorphism group of the group G = GL(4,L) by means of the elementwise action. The semidirect product ĜG = GK₄ is considered together with a realization of the group Ĝ as a group of semilinear automorphisms of the free 4-dimensional L-module M⁴. A three-parameter family of representations R of K₄ in the group Ĝ and a three-parameter family of elements X ∈ M⁴ with polynomial coordinates of degrees 2(ℓ − 1), 2ℓ, 2(ℓ − 1), and 2ℓ, where ℓ is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector X is a fixed point of the corresponding representation R. An algorithm for calculating the polynomials that are the components of X was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.

The Zermelo–Fraenkel set theory with the underlying intuitionistic logic (for brevity, we refer to it as the intuitionistic Zermelo–Fraenkel set theory) in a two-sorted language (where the sort 0 is assigned to numbers and the sort 1, to sets) with the collection scheme used as the replacement scheme of axioms (the ZFI2C theory) is considered. Some partial conservativeness properties of the intuitionistic Zermelo–Fraenkel set theory with the principle of double complement of sets (DCS) with respect to a certain class of arithmetic formulas (the class all so-called AEN formulas) are proved. Namely, let T be one of the theories ZFI2C and ZFI2C + DCS. Then (1) the theory T+ECT is conservative over T with respect to the class of AEN formulas; (2) the theory T+ECT+M is conservative over T+M{su−} with respect to the class of AEN formulas. Here ECT stands for the extended Church’s thesis, Mis the strong Markov principle, and M{su−} is the weak Markov principle. The following partial conservativeness properties are also proved: (3) T+ECT+M is conservative over T with respect to the class of negative arithmetic formulas; (4) the classical theory ZF2 is conservative over ZFI2C with respect to the class of negative arithmetic formulas.

It is proved that if X is a normal space which admits a closed fiberwise strongly zero-dimensional continuous map onto a stratifiable space Y in a certain class (an S-space), then IndX = dimX. This equality also holds if Y is a paracompact σ-space and ind Y = 0. It is shown that any closed network of a closed interval or the real line is an S-network. A simple proof of the Kateˇ tov–Morita inequality for paracompact σ-spaces (and, hence, for stratifiable spaces) is given.

In a semi-infinite cylinder, we consider the behavior of generalized solutions of second-order divergence-form elliptic equations satisfying the third boundary condition on the lateral surface of the cylinder.

It is shown by using a model that even a minor change in prices made by the seller is sufficient to coordinate the actions of independent purchasers so that they act as a single economic agent pursuing the aim of maximizing the effective utility function.

We study the second boundary-value problem in a half-strip for a differential equation with Bessel operator and the Riemann–Liouville partial derivative. In the case of a zero initial condition, a representation of the solution is obtained in terms of the Fox H-function. The uniqueness of the solution is proved for the class of functions satisfying an analog of the Tikhonov condition.

This paper introduces the notion of robust hyperbolicity along periodic orbits homoclinically related to p (RNUHP) for conservative diffeomorphisms. It is proved that if f ∈ Diff¹⁺_m (M) is RNUHP, then f is Anosov. It is also shown that f admits a dominated splitting, provided that f is expansive conservative stable.

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ,G,H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ,H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ,G) and study the case of Heisenberg groups.

An asymptotic formula for the conformal module of a doubly connected domain symmetric about the abscissa axis under an unbounded stretching in the direction of this axis is obtained. Thereby, in the case of symmetric domains, a question asked by Vuorinen is answered.

We consider evolution differential equations in sequence spaces and construct a version of the majorant function method to prove existence theorems. We also develop a theory of lower estimates. This theory allows us to study instability and blowups. Applications to stability theory are discussed.

In this note we extend weak-type estimates obtained recently by A. K. Lerner to certain grand square functions by using a simple argument in terms of real variables. In this way, we improve a weak-type L¹-estimate for grand Littlewood–Paley operators due to N. N. Osipov.

A sufficient condition for the hyperbolicity of a group presented in terms of generators and defining relations is considered. The condition is formulated in terms of the negativity of a discrete analog of curvature for the Lyndon–van Kampen diagrams over a presentation of a group and is a generalization of the small cancellation condition.

Nonzero sine series with monotone coefficients tending to zero are considered. It is shown that the measure of the set of those zeros of such a series which belong to [0, π] cannot exceed π/3. Moreover, if this value is attained, then almost all zeros belong to the closed interval [2π/3, π].

Given a continuous function on a closed interval, a sequence of rational interpolation splines is constructed which converges uniformly on this closed interval to the given function for any sequence of grids with step width tending to zero. The derivatives possess this unconditional convergence property as well. Estimates of the rate of convergence are given.

A group is said to be homomorphically stable with respect to another group if the union of the homomorphic images of the first group in the second group is a subgroup of the second group. A group is said to be homomorphically stable if it is homomorphically stable with respect to every group. It is shown that a group is homomorphically stable if it is homomorphically stable with respect to its double direct sum. In particular, given any group, the direct sum and the direct product of infinitely many copies of this group are homomorphically stable; all endocyclic groups are homomorphically stable as well. Necessary and sufficient conditions for the homomorphic stability of a fully transitive torsion-free group are found. It is proved that a group is homomorphically stable if and only if so is its reduced part, and a split group is homomorphically stable if and only if so is its torsion-free part. It is shown that every group is homomorphically stable with respect to every periodic group.