Let a1, a2, … be a sequence of variables. Write

We use the notation Bn = Bn (a1, …, an) to indicate the variables on which Bn depends.

In [8] Partington showed that a Banach space X is uniformly convex if and only if Lp([0, 1], X) has the uniform Kadec–Klee–Huff property with respect to the weak topology (UKKH (weak)), where 1 < p < ∞. In this note we will characterize the Banach spaces X such that HP(D, X) has UKKH (weak), where 1 ≤ p < ∞. Similar results for UKKH (weak*) are also obtained. These results (and proofs) are quite different from Partington's result (and proof).

We investigate the joint distribution of L-functions on the line $ \sigma= {1}/{2} + {1}/{G(T)}$ and $ t \in [ T, 2T]$, where $ \log \log T \leq G(T) \leq { \log T}/{ ( \log \log T)^2 } $. We obtain an upper bound on the discrepancy between the joint distribution of L-functions and that of their random models. As an application we prove an asymptotic expansion of a multi-dimensional version of Selberg’s central limit theorem for L-functions on $ \sigma= 1/2 + 1/{G(T)}$ and $ t \in [ T, 2T]$, where $ ( \log T)^\varepsilon \leq G(T) \leq { \log T}/{ ( \log \log T)^{2+\varepsilon } } $ for $ \varepsilon > 0$.

For a time in the mid-1970s probabilists were tantalized by a seemingly simple problem posed by Araki and Kakutani[3]. An interval is repeatedly divided by points chosen successively at random, the nth point being uniformly distributed over the largest of the n intervals formed by the first n − 1 points. Is this sequence of points asymptotically uniformly distributed?

Let ℝd* = ℝd ∪ {[midast ]} be the one-point compactification of Euclidean space ℝd and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]f(ℝd*) denote the set of all sets C ⊆ ℝd* for which there is a series [sum ]an in ℝd with zero sum such that C is the cluster set in ℝd* of the sequence of partial sums of [sum ]af(n). Every C ∈ [Cscr ]f(ℝd*) is non-empty, connected and closed in ℝd*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝd* belong to [Cscr ]f(ℝd*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]f(ℝd*) which contain [midast ].

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly.

When any metal film is deposited on the surface of a massive copper oxide slab and illuminated with visible light, a current flows in such a direction that the electrons pass from the Cu2O to the metal in greater numbers than from the metal to the Cu2O. This is the reverse direction to the easier flow of electrons under an applied voltage when the contact is allowed to act as a rectifier. This effect is sometimes called the “Vorderwandeffekt” to distinguish it from the similar group of phenomena which occur at the natural interface between massive copper and Cu2O when the surface of the copper has been oxidized in situ (“Hinterwandeffekt”).

The following work arises from the study of a rotating fluid contained in a rectangular vessel whose depth depends on one coordinate only.

Lemma 4·1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4·2, the only result dependent upon it, is true. The following supplies the necessary modification to the proof.

We measure the distance of a general shape from the set of shapes having a certain sequence of their vertices collinear and also describe the behaviour of shapes as they move along a geodesic towards this set. This models problems which arise in computer vision and other applications of shape theory.

Taylor and Fisk have discussed the internal conversion of magnetic multipole radiation by K electrons and have found internal conversion coefficients which are much larger than those for electric multipoles, and which increase with the order of the multipole. The object of the present note is to determine whether this state of affairs holds also for pair production by internal conversion of magnetic multipole radiation.

The two letters now communicated are from G. G. Stokes to W. Thomson, of dates Dec. 12–13, 1848, three years after Faraday's great magneto-optic discovery. They formulated already the permissible types for general equations of propagation, virtually on the basis of the very modern criterion of covariance,—relative to all changes of the spatial frame of reference in the case of active fluids, but having regard to the fixed direction of the extraneous magnetic field in the Faraday case. Their form was elucidated in each case by correlation with a remarkable and significant type of rotational stress in a propagating medium.

Many investigations bave sbown the presence of a “continuous” β-ray spectrum accompanying tbe β-ray lines in tbe naturai spec-trum of Ra B and Ra C. The existence of sucb a spectrum is not easily accounted for if one is to regard the nucleus as existing in sharply quantised states. It has been pointed out that this spectrum may be a spurious effect, arising from scattering from the walls of the apparatus and not from within the atom itself. Therefore it is of importance to obtain accurate information regarding the continuous spectrum and its relation to the line spectrum.

1. I proved in 1908(1) that if A = Σam and B = Σbn are convergent, and

for large m and n, then

is convergent (necessarily to AB); and this theorem has been extended in a number of directions both by other writers and by myself. Thus we may replace (1), when am and bn are real, by

we may use conditions unsymmetrical in am and bn; we may put the same problem for the product of any number of series; and we may consider modes of ‘Dirichlet multiplication’ based on sequences λm and μn, reducing to Cauchy's when λm = m and μn = n. The appropriate references will be found in(2).

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

Let X be a set and let f be a mapping of P(X), the set of all subsets of X, into P(X). Conditions are given which are necessary and sufficient for there to exist a topology on the set X with respect to which, for all S ∈ P(X), the frontier of S is fS.

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).

The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.

I consider functions of numerical magnitudes ast, where s, t may each assume the values 1,2, … n.

Symmetric functions are those which are unaltered by the whole of the substitutions of the n integers, when they are applied simultaneously to both suffixes.