1. Let X be a normed linear space of periodic functions (of period 2π), which includes the class of trigonometric polynomials. We restrict ourselves throughout to functions f ∈ X. Let D) = (dn, k) be the matrix of a regular sequence-to-sequence transformation. (We could also consider sequence-to-function transformations, in which case n is replaced by a continuous variable.) We suppose that, for all n,

Let {Ln(x)} be the D transform of the Fourier series of f(x). If f(x) is a constant, it follows from (1) that, for all n, Ln(x) = f(x). However, it is often found that, roughly speaking, except in this trivial case, Ln(x) cannot tend to f(x) (in the topology given by the norm) with more than a certain degree of rapidity. This leads to the concept of ‘saturation’, which was first introduced by Favard (1) and Zamanski (8) and which has since been investigated by many authors (see (2), (3), (4), (5), (6), (7)). This is defined as follows.

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.

This note is devoted to the construction of a 〈Q, EΦ〉-closed Fischer class which is not a formation. It provides a negative answer to the Questions 3–7 raised by Cossey in (1). All groups considered are finite and soluble.

On a smooth Banach manifold $M$, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle $T^{2}M$ of $M$. This is a vector bundle in the presence of a linear connection $\nabla$ on $M$ and the corresponding local structure is heavily dependent on the choice of $\nabla$. In this paper we study the extent of this dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on $T^{2}M$ remains invariant under conjugate connections with respect to diffeomorphisms of $M$.

Let X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x ∈ X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Ex → r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

Let { = V × ℝl : V ∈ G(n−l,m−l)} be the family of m-dimensional subspaces of ℝn containing {0} × ℝl, and let : ℝn → be the orthogonal projection onto . We prove that the mapping V ↦ Dim (B) is almost surely constant for any analytic set B ⊂ ℝn, where Dim denotes either Hausdorff or packing dimension.

The present paper is a companion to one which appeared in these Proceedings some months ago. Since that time the characteristics of the Hyper-Octohedral group of order 2n.n! have been calculated by Dr Alfred Young, who has kindly allowed me to make use of his results.

The following result is due to J. H. Grace (see [4], p. 356, also see [9], §3).

Theorem A. if p is a polynomial such that p(–1) = p(1) then the derivative p' has a zero in each of the half-planes {Rez ≤0} and {Rez ≥0}.

1. In this paper I consider various properties of the area of a surface including the relationship between Hausdorff measure and the Lebesgue area. The results here obtained will enable me, in a subsequent paper, to investigate the tangential properties of parametric surfaces. For simplicity of exposition I confine my attention to surfaces homoeomorphic to a disk, but the general case follows easily from cyclic element theory.

In many cases of practical interest there is some kind of symmetry associated with an ordinary second-order differential equation of the Schrödinger type. This symmetry is reflected in certain relations between different F-matrices or different elements of the same F-matrix of the phase-integral method. These symmetry relations imply in turn further relations between the Stokes constants, in addition to those which exist for a general cluster of transition points, and thereby reduce the number of independent Stokes constants. The present paper deals with two important kinds of symmetry relations.

The main aim of the present note is to evaluate certain definite integrals involving Meijer's G-function. The results proved are of general character and include as particular cases some of the results given earlier by MacRobert ((3), p. 81). For the definitions and properties of the G-functions see (1), §§5·3 and 5·3·1.

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

(1) In this paper we are concerned with normalized integer-valued length functions on a group G, that is, mappings l:G→ℤ satisfying three axioms:

(A1′)l(1) = 0

(A2) l(x) = l(x−1)for all x∈G

(A4) d(x, y) > d(x, z) implies that d(x, z) = d(y, z) for all x, y and z in G, where d(x, y) = ½(l(x) + l(y)−l(xy−1)).

These axioms were first considered by Lyndon(6), where their significance is discussed. Lyndon's axiom A1, which we shall not use, stated that l(x) = 0 if and only if x = 1. His axiom A 3 was that d(x, y) ≥ 0 for all x and y in G, but it was noted in (2) that this follows from A1′, A2 and A4. In particular, taking x = y, we find that l(x) ≥ 0 for all x in G. The other major axioms used in (6) were:

(A0) l(x2) > l(x), provided that x ≠ 1, and (A5) d(x, y) + d(xl, y1) > l(x) = l(y) implies that x = y.

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

Kirby and Siebenmann (in (7)) proved the annulus conjecture in dimensions five and above, by proving the stable homeomorphism conjecture in those dimensions, that is, by proving that every orientation-preserving homeomorphism of Rn is stable, if n ≥ 5. Although this result apparently lessens the importance of stable homeomorphisms, the concept of stability should still be of use in settling the fate of the annulus conjecture in dimension 4, or for uncovering a simple (i.e. non-surgical) proof in other dimensions.

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

Some time ago Besicovitch(l) gave an example of a linear Gδ-set E of Besicovitch dimension 1, with the property that E – A also has dimension 1, for each Fσ.-set A contained in E. This shows that an Fσ-set contained in E cannot be a very close approximation to E, but it leaves plenty of scope for Fσ-sets contained in E to be reasonably good approximations to E.

In this paper we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measures of Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalised moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a “linear response formula” at almost every parameter of the perturbation.

The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.

In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.

Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.

Let τ be the formally self-adjoint differential operator denned by

where the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.