Let A = {a1, a2, …} be an increasing sequence of positive integers whose upper density, θ, is less than 1, Then the counting function of the subsequence does not exceed .

An empire map is a map where the set of countries is divided into disjoint subsets which we call empires. If one empire consists of a single country then the remaining map is called an empire island map, the single country being referred to as the ocean. If every empire consists of at most M countries then it is called an M-pire map.

It is proved that an M-pire map is colorable by 6M – 2 colors. Further there exists such a map which has 6M – 2 mutually adjacent empires.

If an M-pire island map has every country adjacent to the ocean, then it is colorable in 4M colors. Such a map exists with 4M mutually adjacent empires where every country is adjacent to the ocean.

If A is a Dedekind domain and f generates a prime ideal of A[X] which is not maximal, then the domain A[X]/(f) is Dedekind if and only if f is not contained in the square of any maximal ideal of A[X]. This criterion is used find the ring of integers of a cyclotomic field, and to determine when a plane curve is normal.

An investigation is made of the stability of block LU-decomposition of matrices A arising from boundary value problems of differential equations, in particular of ordinary differential equations with separated boundary conditions. It is shown that for such matrices the pivotal growth can be bounded by constants of the order of ‖A‖ and, if solution space is dichotomic, often by constants of order one. Furthermore a method to estimate the growth of the pivotal blocks is given. A number of examples support the analysis.

We give a definition of any discrete dynamical system in compact ultrametric space we study the behaviour of its trajectories.

The classic HNN-embedding theorem for groups does not transfer to associative rings or algebras. In its first part this paper presents constructions which provide such a theorem if an additional condition is put on the isomorphic subalgebras or if one restricts to algebras over fields and drops the associativity. The main part of the paper deals with applications of these results. For example, it is known that every existentially closed group is ω-homogeneous. It is shown that the corresponding is false for existentially closed associative Δ-algebras but true for existentially universal nonassociative K-algebras. Further-more, orthogonal sequences of idempotents in existentially closed associative Δ-algebras over a regular ring Δ are investigated. It is shown that the conjugacy class of such a sequence depends only on a corresponding order sequence. In particular, in every existentially closed K-algebra all idempotents different from 0 and 1 are conjugated.

A system of parabolic equations is considered: Lui = uit − uix = fi (x, t, u, uix) on Q, Biui(j, t) = ωij(t), t ∈ (−∞, ∞), j = 0, 1, i = 1, 2, …, n, where Bi is one of the boundary operators Biui = ui or Biui = ∂ui/∂v + βi(x, t)ui, x = 0, 1, Ω = (0, 1), Q = Ω × R, u(=(u1, …, un)): Q → Rn, v(x) is the outward normal to the boundary ∂Ω, f, u, ω0, ω1 are n-valued functions and f, ω0, ω1 are periodic in t with period T and Bi is a positive function.

The paper is classified into two parts. The first part deals with the existence and uniqueness of periodic solutions of the above system of parabolic equations. The second part deals with a monotone iterative method which develops a monotone iterative scheme for the solution of the above system of equations. In this paper we establish the existence of coupled quasi-solutions of the above equation. Also we give a monotone iterative scheme for the construction of such a solution.

We develop some comparison theorems that connect the solutions of a differential system and its perturbation in a manner useful in the practical stability.

It is shown that if a McLain group is perfect, then it is super-perfect. The proof involves demonstrating that any dense linearly ordered set has the apparently stronger property of being superdense.

A norm |·| of a Banach space x is called locally uniformly rotund if lim|xn−x| = 0 whenever xn, x ∈ X, and . It is shown that such an equivalent norm exists on every Banach space x which possesses a projectional resolution {pα} of the identify operator, for which all (pα+1−pα)X admit such norms. This applies, for example, for the dual space of a space with Fréchet differentiable norm.

In this lecture we describe, discuss and compare the two classes of methods most commonly used for the numerical solution of boundary value problems for partial differential equations, namely, the finite difference method and the finite element method. For both of these methods an extensive development of mathematical error analysis has taken place but individual numerical analysts often express strong prejudices in favor of one of them. Our purpose is to try to convey our conviction that this attitude is both historically unjustified and inhibiting, and that familiarity with both methods provides a wider range of techniques for constructing and analyzing discretization schemes.