A space is said to be subparacompact if every open covering of it has a σ-discrete closed refinement. Subparacompactness is equivalent to Fσ -screenability of McAuley and also to σ-paracompactness of Arhangel'skiĭ. Some properties of these spaces have been obtained in this note. Countably subparacompact spaces, which can be defined in an analogous manner, have also been studied.

We study weak convergence of tensor products of vector measures with values in nuclear spaces, such as the space of all rapidly decreasing, infinitely differentiable functions, the space of all test functions, and the strong duals of those spaces. It is shown that the weak convergence of a net of tensor products of vector measures follows from that of the corresponding net of real product measures.

Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$

$t_{j}$

${\it\varepsilon}$

In this paper we show that the method of calculating the Gaussian period polynomial which originated with Gauss can be replaced by a more general method based on formulas for Lagrange resolvants. The period polynomial of cyclic sextic fields of arbitrary conductor is determined by way of example.

Characterizations are obtained of rings R such that the only torsion classes (respectively, hereditary torsion classes) of left unital R-modules are {0} and the class of all modules.

We construct a counterexample to a theorem of Jon Wolfson concerning the existence of globally smooth solutions of the second boundary value problem for Monge-Ampère equations in two dimensions, or equivalently, on the existence of minimal Lagrangian diffeomorphisms between simply connected domains in R2.

In the usual Krull-Schmidt-Azumaya theorem in abelian categories it is essential that each of the direct summands has a local ring of endomorphisms. A partial answer is given here to the case where this last condition is not satisfied by the indecomposable direct summands. It is found that those summands with local endomorphism rings are determined up to isomorphism and cardinality of occurrence.

In this paper we obtain criteria of stability for ηEinstein k-contact manifolds, for Sasakian manifolds of constant ϕ-sectional curvature and for 3-dimensional Sasakian manifolds. Moreover, we show that a stable compact Einstein contact metric manifold M is Sasakian if and only if the Reeb vector field ξ minimises the energy functional. In particular, the Reeb vector field of a Sasakian manifold M of constant ϕ-holomorphic sectional curvature +1 minimises the energy functional if and only if M is not simply connected.

We prove the following theorem: Let R be a ring, l a positive integer, and n a non-negative integer. If for each x, y ∈ R, either xy = yx or xy = xn f(y)x1 for some f(X) ∈ X2Z[X], then R is commutative.

We characterise restricted weak upper semicontinuity of the subdifferential of convex functions in terms of the Fenchel biconjugate mapping.

We consider 3-dimensional conformally flat hypersurfaces of E4 with constant Gauss-Kronecker curvature. We prove that those with three different principal curvatures must necessarily have zero Gauss-Kronecker curvature.

In 1996, Cromwell and Nutt conjectured that α(L) − c (L) = 2 for a link L if and only if L is alternating. In this paper we calculate that α(L) = c (L) for some non-alternating pretzel links L, define a new invariant ρ(L) of adequate links L and show that for each non-negative integer n, there is a prime adequate knot K such that α(K) − c (K) = −2n. We conjecture that α(L) − c (L) = 2ρ(L) for any adequate link L.

The proof of Lemma 2 in [1] invoked elementary analytic number-theory. I have just realized that there is a proof which is entirely elementary. It is doubtless “well-known” (in the usual technical sense that it appears somewhere in the literature) and it is certainly “well-knowable” in Conway's terminology. However, as it renders the entire argument of my paper elementary, I give it here.

Let M be a compact complex manifold and ∇ an arbitrary complex (not necessarily Riemannian) connection. In this paper we study the relation between the geometry of (M, ∇) and the topology of M, that is, we are interested in the following problem: To what extent does the topology of M determine the relations between the group of holomorphically projective transformations, the group of projective transformations and the group of affine transformations on M? Under assumptions on the Ricci-type tensors of ∇ and Chern numbers of M we show that a holomorphically projective transformation and a projective transformation are in fact affine transformations on M. A family of interesting examples of connections of this kind are constructed. Also, the case when M is a Kähler manifold is studied.

For a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.

Let S be a topological semigroup with separately continuous multiplication and H a uniformly closed invariant subspace of LUC(S) (the space of left uniformly continuous bounded functions on S ) that contains the constants. It is shown that if H is left introverted and H admits a tight two-sided invariant mean m, then for each h ∈ H, m(h) is the unique constant function in the norm closed convex hull of the left orbit of h; consequently, H has a unique left invariant mean. (In fact, it is enough for H to admit a tight right invariant mean and a left invariant mean. ) For certain S, a similar result is obtained when H is a left compact-open introverted subspace of LCC(S) (the space of left compact-open continuous functions on S ).

Let K be a closed convex set in the plane containing no nonzero point of the integral lattice. We show that if the area A(K) of K is equally distributed amongst the four principal quadrants of the plane, then A(K) < 4.

We show that if X is a separable Banach space such that X* fails the weak* convex point-of-continuity property (C*PCP), then there is a subspace Y of X such that both Y* and (X/Y)* fail C*PCP and both Y and X/Y have finite dimensional Schauder decompositions.

A ring R is said to be an AE-ring if every endomorphism of its additive group R+ is a ring endomorphism. Clearly, the zero ring on any abelian group is an AE-ring. In a recent article, Birkenmeier and Heatherly characterised the so-called standard AE-lings, that is, the non-trivial AE-rings whose maximal 2-subgroup is a direct summand. The present article demonstrates the existence of non-standard AE-rings. Four questions posed by Birkenmeier and Heatherly are answered in the negative.