Familiarity with the category of simplicial sets and the Milnor realization functor, as in e.g. (5), is assumed. We say f: |Y| → |Z| is realized if f = |g| for some g.

We are concerned in this work with the following question:

Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal number

We observe two easy results.

1. Suppose that f is a homeomorphism of the Euclidean plane E2 onto itself. The set M ⊂ E2 is said to be invariant if f(M) = M and minimal if it is non-void, closed, invariant and irreducible with respect to these properties. In general, invariant and minimal sets in E2 can have a finite or infinite number of components.

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.

J. R. Paktington. ‘On nearly uniformly convex Banach spaces.’

Theorem 1 was printed incorrectly. It should read

If X is (UKK) then so is lp(x) (1 ≤ p < ∞).

A similar replacement of L by l should also be made on p. 127, 1. 4* and p. 128, 1. 13, while p. 128, ll. 11*–10* should read

Also the l2 direct sum of lp, p = 2, 3, … is not even isomorphic to any (UKK) space but is (BS).

The spontaneous disappearance of an electron and proton in the form of radiation has often been suggested as a possible source of energy to maintain the emission of radiation from hot stars. It can be calculated that the energy of a single quantum formed in this way should be about 1000 million electron-volts. In order to account for the absorption of the ultra-penetrating radiation observed by Regener in deep lakes, and interpreted on the basis of the Klein-Nishina formula, a quantum of still higher energy is required. Sir James Jeans has recently calculated that the transformation of a helium atom into a single quantum would provide a radiation of about the required penetrating power.

Let [Mfr ]g be the moduli space of smooth curves of genus g [ges ] 4 over the complex field [Copf ] and let [Tfr ]g ⊆ [Mfr ]g be the trigonal locus, i.e. the set of points [C] ∈ [Mfr ]g representing trigonal curves C of genus g [ges ] 4. Recall that each such curve C carries exactly one g13 (respectively at most two) if g [ges ] 5 (respectively g = 4). Let |D| be a g13 on C and suppose that it has a total ramification point at P (t.r. for short), i.e. that there is on C a point P such that 3P ∈ |D|. Such a P is a Weierstrass point whose first non-gap is three. In the present paper we study some sub-loci of [Tfr ]g related to curves possessing such points.

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb{P}$r of dimension n and degree d ⩽ n(r − n) + 2 is Calabi–Yau, and give an example that shows this bound is also sharp.

This investigation is to discuss some of the conditions which influence the absorption spectra of various “saturated” and “unsaturated” organic compounds. A condensed cadmium spark was used as the source of radiant energy; alcoholic solutions of the substances were employed, except for stearic acid, which was dissolved in ether. The vapours were examined in a quartz tube 200 mm. long, placed in a bath which could be heated to various temperatures.

The problem of the transmission of power over a transmission line from an alternating current generator at one end to a synchronous motor at the other is easily solved when the line is uniform or made up of a small number of uniform line segments. The object of the present note is to apply the theory of integral equations to the general case of the non-uniform line. The solution obtained for the current at any point on the line in the form of a rational function of the frequency of the two alternators is of advantage when resonance effects are being considered; the latter effects are hard to trace in the case of the uniform line by the usual methods. It seems probable that numerical methods and computing devices can be developed for the solution of the integral equations given. The effect of armature inductance on the power limit of the system may then be more easily studied than at present.

(Volume 125 (1999), 463–509)

Case (x) in Theorem 6 is missing. The correct statement of Theorem 6 in the paper should read as follows.

THEOREM 6. Let x: Mn → Hn+1 (−1) ⊂ [ ]n+21 (n > 2) be an isometric immersion of a conformally flat n-manifold. Then it satisfies equality (1·2) if and only if one of the following ten cases occurs.

Since the publication of a previous paper on chemical constants, some further experimental data upon the dissociation of chlorine by Wohl have appeared, from which a new and rather more satisfactory value of the chemical constant of diatomic chlorine can be calculated. Wohl concludes that Q0, the heat of dissociation at absolute zero, is – 57,000 calories. This is in good agreement with Henglein's value – 54,000 but differs from that of Trautz and Stackel, namely – 71,000, which is the value adopted in the previous paper. Q0 is necessarily an adjustable constant, and since Wold's value gives results which are a good deal more concordant with each other and with the theory, we now take Q0 = − 57,000. Wohl also uses hv0/k = 902 instead of 1093, but as the difference made by this change is very small compared with other disagreements, we retain the value 1093. The following table gives the new values of Γ (Cl2), calculated in the same way as before. The initials at the heads of the columns refer to the results of Henglein, Trautz and Wohl respectively.

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.

The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]:

Let G be a compact group, and suppose that

\[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\]

$G^{k+1}$

We give a positive answer for $k = 2$ .

We prove that the set of all endpoints of the Julia set of $f(z)=\exp\!(z)-1$ which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space $\mathfrak E$ . As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.

A procedure is described to determine phase velocities of a circular helix. Computations are made for three helix configurations using the method of internal constraints. The results so obtained are compared with the results obtained by using the elementary theory proposed by J. H. Michell, and the extended elementary theory proposed by A. B. Basset.

We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.

We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.

1. The perturbations of a focal elliptic orbit are as easily found by Quaternions as the orbit itself.

Let L be a finite-dimensional Lie algebra and U(L) its universal envelope. It is known that U(L) is a Noetherian domain (see (5), theorem v. 3·4) and therefore U(L) has a field of fractions. (Throughout the paper we use the term ‘field’ in the sense of skew field.) We prove in this article the following theorem.