It is shown that, under certain conditions, a transformation group is pointwise equicontinuous if and only if it is P-equicontinuous for some replete semigroup P in the phase group.

Fix and elliptic curve y2 = x3 + Ax + B, satisfying A, B ∈ ℤ A ≥ |B| > 0. We prove that the associated quadratic family contains infinitely many elliptic curves with trivial Mordell-Weil group.

Carleson and Hunt proved that the space of functions with almost everywhere convergent Fourier series contains Lp (p > l) as a subspace. We shall give two kinds of subspaces of the spaces of functions with everywhere convergent or uniformly convergent Fourier series.

This paper originated with our interest in the open question “If every pure subgroup of an LCA group G is closed, must G be discrete ?” that was raised by Armacost. The answer was surprisingly easy, but led to some interesting questions. We attempted to characterise those LCA groups that contain a proper pure dense subgroup, and found that every non-discrete torsion-free LCA group contains a proper pure dense subgroup; so does every non-discrete infinite self-dual torsion LCA group. We also give a necessary and sufficient condition for a torsion LCA group to contain a proper pure dense subgroup.

In this paper we study the boundedness of the weighted Bergman projections on the weighted subspaces of Bergman spaces and the Lipschitz spaces on the unit ball and the unit polydisc.

Let f(z) be an n-valued algebroid function of finite lower order μ. In this paper, we give some further results on the deficiencies of f(z). Particularly if 0 ≤ μ ≤ 1/2, the corresponding result is best possible.

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

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.

In this paper, we completely determine the commutativity of two Toeplitz operators on the harmonic Bergman space with integrable quasihomogeneous symbols, one of which is of the form $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}e^{ik\theta }r^{\, {m}}$. As an application, the problem of when their product is again a Toeplitz operator is solved. In particular, Toeplitz operators with bounded symbols on the harmonic Bergman space commute with $T_{e^{ik\theta }r^{\, {m}}}$ only in trivial cases, which appears quite different from results on analytic Bergman space in Čučković and Rao [‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal.154 (1998), 195–214].

We study the behaviour of the transfinite diameter of regular polygons of fixed diameter, as a function of the number of their vertices.

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.

There is a small error in the proof of Lemma 1 [1], namely, that there may not be a subsequence of the sequence {Φn} converging to Φ in the weak*-topology. Thus equation (5) should read: In view of continuity of Pk we have

The same error occurs in the proof of Theorem 1 and Lemma 2 also. Therefore, equation (7) should read

and this leads to lim . Similarly, equation (8) should read

which implies that Ψn (x) → ‖x‖. Then there is no need of taking a subsequence.

All the same the results are not affected by the above error.

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

We prove that for a large class of Segal algebras, the isometric multipliers consist of scalar multiples of translation operators.

Suppose that σ:𝔐→𝔐 is an ultraweakly continuous surjective *-linear mapping and d:𝔐→𝔐 is an ultraweakly continuous *-σ-derivation such that d(I) is a central element of 𝔐. We provide a Kadison–Sakai-type theorem by proving that ℌ can be decomposed into and d can be factored as the form , where δ:𝔐→𝔐 is an inner *-σ𝔎-derivation, Z is a central element, 2τ=2σ𝔏 is a *-homomorphism, and σ𝔎 and σ𝔏 stand for compressions of σ to 𝔎 and 𝔏 , respectively.

In this paper we prove two inequalities relating the warping function to various curvature terms, for warped products isometrically immersed in Riemannian manifolds. This extends work by B. Y. Chen [‘On isometric minimal immersions from warped products into real space forms’, Proc. Edinb. Math. Soc. (2) 45(3) (2002), 579–587 and ‘Warped products in real space forms’, Rocky Mountain J. Math.34(2) (2004), 551–563] for the case of immersions into space forms. Finally, we give an application where the target manifold is the Clifford torus.