Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong set-theoretical concepts.

The preceding paper ‘Strong statements of analysis’ by A. R. D. Mathias defends a so-called full-blooded set theory without full detail [3]. He again objects to a weak set theory which he calls ‘Mac’, in which the usual Zermelo–Fraenkel separation scheme is required only for formulas with suitably ‘restricted’ quantifiers. I had proposed that such separation is adequate for all standard uses of set theory in mathematics. But Mathias has not produced any counter examples of actual mathematics which requires the use of a stronger separation.

For a transcendental meromorphic function f, various properties of the set

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were obtained in [8] and [9]. Here we establish analogous properties for the smaller sets

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introduced in [5], and

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We deduce a symmetry result for Julia sets J(f), and also indicate some techniques for showing that certain invariant curves lie in I′(f), Z(f) and J(f).

Let X be a smooth complex projective curve of genus g [ges ] 1. If g [ges ] 2, then assume further that X is either bielliptic or with general moduli. Fix integers r, s, a, b with r > 1, s > 1 and as [les ] br. Here we prove the existence of an exact sequence

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of semistable vector bundles on X with rk(H) = r, rk(Q) = s, deg(H) = a and deg(Q) = b.

Two new families of commutative semifields of order q4 with middle nucleus GF(q2), where q is an odd prime power, are defined. One family occurs when q ≡ 3 (mod 4) and the other when q ≡ 1 (mod 4).

Let v be a henselian valuation of a field K. In this paper it is proved that any finite extension (K′, v′) of (K, v) is tame if and only if there exists α ≠ 0 in K′ such that v′(α) = v(TrK′/K(α)) using elementary results of valuation theory. A special case of this result, when the characteristic of the residue field of v is p > 0 and (K′, v′)/(K, v) is an extension of degree p, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math. 404 (1990) 1–38).

It is shown that a rational function of degree [ges ] 2 admits an invariant line field with respect to some measure μ, which is an equilibrium state of a Hölder continuous potential whose topological pressure is greater than its supremum, only in very special cases when the Julia set is either a geometric circle or an interval, or totally disconnected and contained in a real-analytic curve.

Let Tt be the semigroup of linear operators generated by a Schrödinger operator − A = Δ − V, where V is a non-negative polynomial, and let ∫∞0λdEA(λ) be the spectral resolution of A. We say that f is an element of HpA if the maximal function [Mscr ]f(x) = supt>0[mid ]Ttf(x)[mid ] belongs to Lp. We prove a criterion of Mihlin type on functions F which implies boundedness of the operators F(A) = ∫∞0F(λ)dEA(λ) on HpA, 0 < p [les ] 1.

In this paper, we define a Hardy–Littlewood maximal function operator M associated with the Jacobi transform, and prove that M is of weak (1, 1) and strong (p, p) (p > 1) type.

Let Ω⊆ℝd be an open set of measure 1. An open set D⊆ℝd is called a ‘tight orthogonal packing region’ for Ω if D−D does not intersect the zeros of the Fourier transform of the indicator function of Ω, and D has measure 1. Suppose that Λ is a discrete subset of ℝd. The main contribution of this paper is a new way of proving the following result: D tiles ℝd when translated at the locations Λ if and only if the set of exponentials EΛ = {exp 2πi〈λ, x〉 [ratio ] λ ∈ Λ} is an orthonormal basis for L2(Ω). (This result has been proved by different methods by Lagarias, Reeds and Wang [9] and, in the case of Ω being the cube, by Iosevich and Pedersen [3]. When Ω is the unit cube in ℝd, it is a tight orthogonal packing region of itself.) In our approach, orthogonality of EΛ is viewed as a statement about ‘packing’ ℝd with translates of a certain non-negative function and, additionally, we have completeness of EΛ in L2(Ω) if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier analytic language, and use this to prove our result.

Given a principal ideal domain R of characteristic zero, containing 1/2, and a two-cone X of appropriate connectedness and dimension, we present sufficient algebraic conditions for the Hopf algebra FH(ΩX; R) to be isomorphic with the universal enveloping algebra of an R-free graded Lie algebra; here, F stands for free part (that is, quotient by the R-torsion), H for homology, and Ω for the Moore loop space functor.

We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re-derive Milnor's result that the mod 2 Steenrod algebra is copolynomial by means of a more general result that is also applicable to a number of other related Hopf algebras.

Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f [ratio ] mod Λ → mod Γ between the categories of finitely generated modules induces a bijection M [map ] Mf between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of Mf is bounded by the endolength of M up to a scalar which depends only on f. Using Crawley-Boevey's characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved.