We show that the halting times of infinite time Turing machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of ‘supertasks’ or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Gödel's constructible hierarchy: namely that of Lλ where λ is the supremum of halting times. A number of other open questions are thereby answered.

In 1968 Vizing conjectured that any independent vertex set of an edge-chromatic critical graph G contains at most half of the vertices of G, that is, α(G) [les ] ½[mid ]V(G)[mid ]. Let Δ be the maximum vertex degree in a critical graph. For each Δ, we determine c(Δ) such that α(G) [les ] c(Δ)[mid ]V)[mid ].

If s1, s2, …, st are integers such that n − 1 = s1 + s2 + ··· + st and such that for each i (1 [les ] i [les ] t), 2 [les ] si [les ] n − 1 and sin is even, then Kn can be expressed as the union G1 ∪ G2 ∪ ··· ∪ Gt of t edge-disjoint factors, where for each i, Gi is si-regular and si-connected. Moreover, whenever si = sj, Gi and Gj are isomorphic.

In this paper, we give an elementary proof of a curious identity of elliptic functions. It is very similar to a beautiful proof given by Coates of a different identity. The result was strongly motivated by Wildeshaus' generalisation of Zagier's conjecture.

Keedwell has shown that none of the groups of order less than 5 has a weak uniquely completable set. We prove that a weak uniquely completable set exists in a latin square based on a finite group if and only if the group is of order greater than 5.

We consider the equation

a1x1 + ··· + anxn = 1, (1.1)

where the coefficients a1, …, an are non-zero elements of a field K, to be solved for (x1, …, xn) in a finitely generated multiplicative subgroup G of (K*)n. Evertse [2] and van der Poorten and Schlickewei [6] showed, independently, for the most important case in which K is a number field and G is a group of S-units in K, that (1.1) has only finitely many solutions (x1, …, xn) for which no subsum in (1.1) vanishes. Such solutions are called non-degenerate.

In 1990, Schlickewei [7] was the first to obtain an explicit upper bound for the number of non-degenerate solutions of (1.1) when G is a group of S-units of K. His result went through various successive improvements, and the best result to date is the bound (235n2)n3s by Evertse [3, Theorem 3], where s is the cardinality of S. Very recently, Evertse, Schlickewei and Schmidt [4, 5] obtained a remarkable result. They showed, again for K a number field, that the number of non-degenerate solutions of (1.1) with (x1, …, xn) in a finitely generated subgroup of (K*)n of rank r is at most c(n)r+2, with c(n) = exp((6n)4n). The importance of this result lies with its uniformity with respect to the rank r of the group and its independence of the field K. The proofs of the above-mentioned results are all quite difficult and depend on deep tools from Schmidt's Subspace Theorem and diophantine approximation. In this paper we consider equation (1.1) in the rational function field K = k(t) where k is an algebraically closed field of characteristic 0. Of course, results of this type will follow rather easily from the above-mentioned results by a specialization argument. However, our object is to show that our approach to the function field case is quite elementary and certainly very different, being a simple and direct consequence of the powerful abc-theorem.

Before stating our result, we first define proportional solutions of (1.1). We say that (x1, …, xn) and (x′1, …, x′n) are proportional if xi, /, x′i ∈ k, 1 [les ] i [les ] n. This determines equivalence classes of solutions.

In this note we wish to prove a purely characteristic p > 0 variant of the Kodaira–Akizuki–Nakano vanishing for smooth complete intersections of dimension at least two in projective space. This has some interesting applications; in particular, we show that all Frobenius pull-backs of the tangent bundle of any complete intersection of general type and of dimension at least three in Pn are stable. We also show (see Remark 3.4) that a small modification of the techniques of [5] and a theorem of Mehta and Ramanathan (see [3]) together allow us to extend this stability result to smooth projective hypersurfaces of degree d, where (n + 1)/2 < d < n + 1 (that is, to some Fano hypersurfaces).

It is well known that behaviour of stability under Frobenius pull-backs is a subtle problem of the theory of vector bundles in characteristic p > 0, and hence this result is not without interest. We end with an obvious conjectural form of our variant for a general class of varieties.

We study deformations of zero-dimensional complete intersections in the plane, and prove the following results. (1) Two complex non-singular curves intersecting at r points with multiplicities d1, …, dr can be deformed into curves intersecting (at some points) with multiplicities d′1, …, d′s which are arbitrary prescribed partitions of the numbers d1, …, dr. (2) Two real curves intersecting with multiplicity at most 2 at each of their real common points can be deformed so that all real multiple intersection points split into real simple intersection points.

The aim of this paper is to prove that for certain generalized Weyl algebras A (including the first Weyl algebra A1 over a field of characteristic zero) and for every simple left (right) A-module M, there are infinitely many non-isomorphic simple left (right) A-modules {Ni} such that Ext1A (M, Ni) ≠ 0 (respectively Ext1A(Ni, M) ≠ 0.

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′such that each soluble transitive permutation group of degree n [ges ] 2 can be generated by [c′ n/ √log n] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following.

We show that the Suzuki group Sz(32) is a subgroup of E8(5), and so is its automorphism group. Both are unique up to conjugacy in E8([ ]) for any field [ ] of characteristic 5, and the automorphism group Sz(32)[ratio ]5 is maximal in E8(5).

Answering a question of Gromov [7], we shall present an example of a finitely generated group Γ and two non-principal ultrafilters [Ascr ], [Bscr ] such that the asymptotic cones Con[Ascr ] Γ and Con[Bscr ] Γ are not homeomorphic.

If u is a superharmonic function on ℝ2, then

formula here

for all (x, y) ∈ ℝ2. This follows from the fact that a line segment in ℝ2 is non-thin at each of its constituent points. (See Doob [1, 1.XI] or Helms [7, Chapter 10] for an account of thin sets and the fine topology.) The situation is different in higher dimensions. For example, if u is the Newtonian potential on ℝ3 defined by

formula here

then

formula here

Corollary 2 below will show that, nevertheless, for nearly every vertical line L, the value of a superharmonic function at any point X of L is determined by its lower limit along L at X.

Throughout this paper, we let n [ges ] 3. A typical point of ℝn will be denoted by X or (X′, x), where X′ ∈ ℝn−1 and x ∈ ℝ. Given any function f[ratio ]ℝn → [−∞, +∞] and any point X, we define the vertical cluster set of f at X by

formula here

and the fine cluster set of f at X by

formula here

We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators.

We show that operators on n × n matrices which are representable in the form T(X) = [sum ][lscr ]i=1aiXbi (where ai and bi are n × n matrices) and are k-positive for k = [√[lscr ]] must be completely positive. As a consequence, elementary operators on a C*-algebra with minimal length [lscr ] which are k-positive for k = [√[lscr ]] must be completely positive.

Let X be a separable infinite dimensional Banach space. There exist a closed set A ⊂ X which contains a translate of any compact set in the unit ball of X, and a bi-Lipschitz homeomorphism f of X onto X so that every line in X intersects f(A) in a set of Lebesgue measure zero.