A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.

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.

In the light of the hypothesis that the helium atom can exist in a hydrogen-like form in which one electron is relatively far removed from the nucleus with respect to the other, a series of experiments have been carried out with a view to determine whether or not helium could be obtained in the form of a compound similar to certain known hydrides. The mass of evidence points to a positive result. A gaseous compound was shown to form when excited helium was passed over a strong source of the active deposit of radium. Compound formation was proved by counting scintillations caused by the transportation of the radioactive material from the discharge tube into a bulb containing a zinc sulphide screen. That the effect observed in the bulb was due to the presence of a gaseous radioactive helide and not to radium emanation or solid particles in suspension was definitely proved; that it was due to hydrogen impurities in the helium was shown to be very unlikely. The compound in hydrogen was more efficiently condensed by a carbon dioxide-ether cooling mixture than that in helium. The catalytic effects of tap grease and mercury vapour on the reaction were noted.

In conclusion the writer wishes to thank Professor Sir Ernest Rutherford, O.M., P.R.S., for his interest in the work; Dr J. Chadwick, F.R.S., for advice and helpful suggestions; Mr J. A. Ratcliffe, for the wireless set used in producing the electrodeless discharge; Mr Crowe, for the preparation of the radioactive sources; and Mr Brown, for help in counting scintillations.

1. Introduction. In (2) we associated with a commutative ring R with identity the lattice ℒ(R) of rings S which satisfy (i) R ⊆ S ⊆ T, where T is the total quotient ring of R, and (ii) dS ⊆ R for some d ∈ R which is not a zero-divisor. The rings of ℒ(R) can be constructed by blowing up ideals of the set of ideals A of R which satisfy (i) 0: A = 0, and (ii) there exist integers m1n ≥ 1 and α ∈ An such that αAm = Am+n (see Propositions 2·3 and 2·5 of (2)).

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.

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. We also give a general formula for the Poincaré series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.

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.

We study the weak star accumulation points of sequences of probability measures of the form [XKn(x) ρn(x)dσ(x)]/ [∫Kn ρn(t)dσ(t)], where ρ(x)>0 is continuous on Rκ, σ denotes Lebesgue measure in Rκ and the sequence of compact sets Kn⊂Rκ converges in the sense of Hausdorff towards a compact set K. The motivation of our study was given by a result relating interval averages with the winding number. Similar probability measures are considered in partial differential equation problems and we extend our study to this case.

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

Let X be a (based) space of the homotopy type of a CW-complex. Let H(X) denote the classical (ungraded) cohomology ring Πi≥0Hi (X;Z/2). In [1] Atiyah and Hirzebruch described the group of natural ring automorphisms of H(X) (‘cohomology automorphisms’) with group operation given by composition. They showed that is isomorphic to the group of formal power series of the form with group operation given by ‘substitution’ of one power series into another. In particular the most famous ‘cohomology automorphism’, the total Steenrod Square, corresponds to x + x2.

An investigation by G. L. Clark is found to provide exceptionally instructive illustrations of the calculation of the variously defined ‘densities’ and ‘masses’ occurring in general relativity theory and of their physical significance. In particular, the effects of motion and stress upon gravitational mass are elucidated. An explicit example shows how kinetic energy is included in the gravitational mass.

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 this paper we prove an existence result for local and global isometric immersions of semi-Riemannian surfaces into the three dimensional Heisenberg group endowed with a homogeneous left–invariant Lorentzian metric. As a corollary, we prove a rigidity result for such immersions.

Some theorems in the operational calculus, taking definite integration as the fundamental operator, are proved for discrete systems. It is suggested that it is physically more satisfactory to regard the solution for a continuous system as the limit of the solutions for a set of discrete systems, rather than as the solution of a partial differential equation. The necessary and sufficient condition for the validity in this sense of the usual operational method for continuous systems appears to be that the discrete systems considered shall not tend to infinite instability; that is, that all poles of the Bromwich integrand shall always have real parts less than some fixed positive quantity. The correction for finiteness of the number of degrees of freedom is examined for a case analogous to heat conduction and found to be unimportant.

Let P(α, β)n (x) be the Jacobi polynomials which are defined by

formula here

or given as hypergeometric series by

formula here

They satisfy the orthogonality relations

formula here

The object of this paper is to draw attention to a series of five types of rational quartic primal in [4]. Two of these are already known. The greater part of this paper deals with one of the other three types of primal and with a new symmetrical quarto-quartic Cremona transformation of [4] determined by a homaloidal system of primals of this type passing through a certain surface of order nine. It is hoped in a subsequent paper to continue the investigation into the remaining two types of primal.

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.

The aim of this paper is to answer a question of Coates and Greenberg: let $F$ be a commutative $m$-dimensional formal group over the ring of integers of a local field $k$, and let $K$ be an algebraic extension of $k$ with infinite ramification index. Denote by ${\cal M}_{\Mbar}$ the maximal ideal in the ring of integers of the separable closure of $K$. Suppose that the height of $F$ is greater than $m$. Does $H^1 (K, F ({\cal M}{}^m_{\Mbar}))=0$ imply that $K$ is deeply ramified? The answer is positive.

Based on recent work by Futer, Kalfagianni and Purcell, we prove that the volume of sufficiently complicated positive braid links is proportional to the signature defect Δσ = 2g−σ.