We show that any hyperbolic flow (X, π) on a metric space X is topologically stable by showing that it is expansive and has the chain-tracing property.

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

We show that an essentially amenable Banach algebra need not have an approximate identity. This answers a question posed by Ghahramani and Loy [‘Generalized notions of amenability’, J. Funct. Anal.  208 (2004), 229–260]. Essentially Connes-amenable dual Banach algebras are introduced and studied.

In a paper dealing with trans-sonic jet flows Frankl (Bull. Acad. Sci. URSS Sér. Math. [Izv. Akad. Nauk SSSE] 9 (1945), 121–143) considered the following problem (T) by applying the condition

where k = k(y) is a monotone increasing function with a continuous second derivative, k(0) = 0, F(0) > 0, k′(y) ≠ 0 for y < 0. Consider an equation of the form

which is elliptic for y > 0, hyperbolic for y < 0, and parabolic for y = 0. Consider equation (2) in a bounded simply connected region D ⊂ R2 which is bounded by the following three curves: a piecewise smooth curve Γ0 lying in the half-plane y > 0 which intersects the line y = 0 at the points A(0, 0) and B(l, 0); for y < 0 by a smooth curve Γ2 through B which meets the characteristic of (2) issuing from A(0, 0) at the point P; and the curve Γ1 which consists of the portion PA of the characteristic through A. The problem (T) (or problem of Tricomi-Frankl) consists of finding a solution u = u(x, y) ∈ C2(D) assuming prescribed values on Γ0 ∪ Γ2. In the present paper we generalize Frankl's uniqueness theorem; our uniqueness theorem includes cases where F(y) may be negative.

The Cauchy representation is not valid for every Schwartz distribution f ∈ (D(R))′ because (t – x)−l ∉ D(R). Since for all z ∈ {z: Im z ≠ 0} the kernel (t – z)−l belongs to (R) (1 > p > ∞), the Cauchy representation of the distributions in ((R))′ seems possible.

In this paper, we prove this fact. It is also proved that every probability density defines a generalised function on the space (R)(1 < p < ∞), of test functions. Applications of these results in probability theory are discussed.

In 1969, Ligh proved that distributively generated (d-g) Boolean near-rings are rings, and hinted that some of the more complicated polynomial identities implying commutativity in rings may turn d-g near-rings into rings. In the present paper we investigate the following conditions: (1) xy = (xy)n(x, y); (2) xy = (yz)n (xy); (3) xy = ym (x, y)xn (x, y); (4) xy = xy n(x, y)x; (5) xy = xn(x, y)ym (x, y); finally prove that under appropriate additional hypotheses a d-g near-ring must be a commutative ring. Indeed the theorem proved here is a wide generalisation of many recently established results.

We find topological conditions on a space X under which the left (right, or two-sided) uniformity of the free topological group F(X) induces the universal uniformity or the product uniformity ux × ux on the square of X. Special attention is given to kω metrisable spaces. The main technical tool in the paper is an extension of certain continuous pseudometrics from X2 to F(X) considered by the author in the previous volume of this journal.

We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and $\Delta _c^n f$ omit zero in the whole complex plane, then either $f(z)=\exp (h_1(z)+C_1 z)$, where $h_1$ is an entire function of period c and $\exp (C_1 c)\neq 1$, or $f(z)=\exp (h_2(z)+C_2 z)$, where $h_2$ is an entire function of period $2c$ and $C_2$ satisfies

$$ \begin{align*} \bigg(\frac{1+\exp(C_2c)}{1-\exp(C_2 c)}\bigg)^{2n}=1. \end{align*} $$

Inserting new knots into B-spline curves is a well-known technique in CAGD to gain extra flexibility for design purposes. However, from a user's point of view, the insertion of knots is somewhat unsatisfactory since the newly generated control points sometimes show up in unexpected locations. The aim of this note is to show that these problems can be circumvented by inserting the control vertices directly, thus also providing a more natural user interface.

An associative ring R is called a BT-ring if R is strongly right bounded, but not right duo, and not strongly left bounded. We show that the order of the smallest BT-rings (without unity) is 16, while we prove earlier that the order of the smallest unitary BT-rings is 32.

We study in this paper the relations existing between Joly's rectangular constant (µ) and the degree of asymmetry of Birkhoff-James's orthogonality relation (β). New bounds on the variation of µ in terms of β and estimation of the values taken by β in the case of uniformly convex Banach spaces are given.

In this note, the Morse-Smale characteristic of a differentiable manifold is defined and certain of its properties are studied.

By defining and applying the restricted topology, we have investigated certain connections between the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum (see Mouton and Harte [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745]). We now solve a remaining problem regarding the restricted connected hull.

We prove metric regularity results for both single-valued maps and set-valued maps defined between metric spaces.

Completely semi-stable trees are characterised by the complete absence of three types of subtrees.