Let ${\frak M}_{g}$ be the moduli space of smooth, integral curves of genus $g$ over the complex field ${\mathbb C}$ and let $\overline{{\frak M}}_{g}$ be its Mumford–Deligne compactification. If $C$ is a smooth, integral curve of genus $g$ we denote by [$C$] its isomorphism class in ${\frak M}_{g}$.

The Corners theorem states that for any α > 0 there exists an N0 such that for any abelian group G with |G| = N ≥ N0 and any subset A ⊂ G×G with |A| ≥ αN2 we can find a corner in A, i.e. there exist x, y, d ∈ G with d ≠ 0 such that (x,y),(x+d,y),(x,y+d) ∈ A.

Here, we consider a stronger version, in which we try to find many corners of the same size. Given such a group G and subset A, for each d ∈ G we define Sd={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |Sd| is the number of corners of size d. Is it true that, provided N is sufficiently large, there must exist some d ∈G\{0} such that |Sd|>(α3-ϵ)N2?

We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets A with |Sd|>Cα3.13N2 for all d ≠ 0, where C is an absolute constant. We also show that in the special case where $G = {\mathbb{F}}_2^n$, one can always find a d with |Sd|>(α4-ϵ)N2.

We characterise the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

Nuclear C*-algebras having a system of completely positive approximations formed with convex combinations of a uniformly bounded number of order zero summands are shown to be approximately finite dimensional.

The theoretical method for computing the grade of a curve on an algebraic surface is well known. In practice difficulties arise which are not considered in the theory; so that it seems worth while to describe a practical method. This is done in § 1 of this paper. The method is then applied to some examples with the object of discovering whether Noether exceptional curves are necessarily exceptional curves†. In particular, a certain quintic surface with three tacnodes is studied, and our examination leads us to results which differ from those which have been accepted up till now. Another example illustrates the limitations of a practical method for computing grades, because of the possible presence of infinitesimal curves, and leads to the transformation of the quintic surface with two tacnodes into a double plane of order ten of a certain type, which has the singularity known as a (5, 5) point on its branch curve. Light is thrown on the Noether composition of this singularity by the transformation, which also shows the relation between two well-known types of surface for which the Noether relation p(2) = p(1) − 1 does not hold.

Exchange energy between electrons moving in a conductor is assumed to be such that the mean contribution per electron is proportional to the product of the total current within a defined range and the speed of the electron in the direction of the total current.

The ideal resistance of a conductor can be neutralized by such an exchange energy at sufficiently low temperatures.

The surface of the conductor, and internal surfaces of discontinuity if sufficiently marked, are regarded as sites for two-dimensional resonance states in equilibrium with the three-dimensional resonance states in the metal.

The superconducting transition occurs when, at the critical temperature, the neutralization of the ideal resistance of the surface by-passes the residual resistance of the body of the metal.

At the same critical temperature, equilibrium conditions ensure that exactly the right number of electrons enter the surface states to give the whole metal an ideal diamagnetic susceptibility.

These equilibrium conditions are disturbed by a magnetic field, and the transition temperature is shown to depend on the magnetic field in the observed manner.

The intermediate state is explained as one in which the number of electrons in the superconducting surface states is constrained by a non-uniform magnetic field to remain at an intermediate value which is inadequate to produce ideal diamagnetism.

By regarding either the outer surface or internal surfaces of discontinuity as the only regions becoming perfectly conducting, the details concerning restoration of resistance by a current are well explained, as also are the observed size effects in thin films, and the phenomena of ‘non-ideal’ superconductors among the alloys.

Finally, the observed distribution of superconductors in the periodic table receives a satisfactory qualitative explanation.

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the function

in terms of the positions of the singularities of f(z) and g(z).

Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by

Zygmund (1) proved in 1936 that if f(x) ∈Lp(−∞, ∞) (1 ≤ p ≤ 2), the Fourier transform

exists for almost every λ, in the sense that

exists for almost every λ. The purpose of this note is to provide a short proof of this theorem.

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

In this paper we study the cohomology groups Hn(I, I*) and Hn([Uscr ], [Uscr ]*) where [Uscr ] is a Banach algebra with a bounded approximate identity and I is a codimension one closed two-sided ideal of [Uscr ]. This is applied to the case when [Uscr ] is the group algebra L1(G) of a locally compact group G and I={f∈L1(G)[mid ] ∫Gf=0}, the augmentation ideal of G. We show that if G is inner amenable, then I is always weakly amenable, i.e. [Hscr ]1(I, I*)={0}.

Let k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin–Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in (k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) ↦ (xn+1−j,n+1−i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(q) for all prime powers q.

We compute the (uni)versal deformation ring of an odd Galois representation ρ: Gal (M/Q) → Gl2(Fp) with an upper triangular image, where M is the maximal abelian pro-p-extension of F∞ unramified outside a finite set of places S, F∞ being a free pro-p-extension of a subextension F of the field K fixed by Ker ρ. We establish a link between the latter (uni)versal deformation ring and the (uni)versal deformation ring of ρ: Gal (KS/Q) → Gl2(Fp), where KS is the maximal pro-p-extension of K unramified outside S. We then give some examples.

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.

C. Ehresmann (2) has shown that if a leaf L of a smooth foliation has a foliated neighbourhood, then there exists a fibre bundle over L, normal to the leaves, with discrete structure group. Using the concept of a microbundle and the n-isotopy extension theorem, we find a similar result for both PL and TOP categories, and, in addition, show that the structure group can be chosen to be the holonomy group of L. As for applications we show that holonomy characterizes the foliated neighbourhood of a leaf (proved by Haefliger in the differentiable case (3)). In particular, if the holonomy group of a compact leaf L is trivial then the leaf has a trivial foliated neighbourhood, and if it is finite it has a neighbourhood of compact leaves which are covering spaces of L. Another corollary is the known result that a proper submersion with compact fibre is a fibration. Finally we use the fact that the constructed normal microbundle can be chosen to have its fibres contained in the leaves of a transverse foliation to demonstrate isotopy uniqueness of normal microbundles.

It is a classical result of Postnikov [15] that homotopy types X with at most two non-trivial homotopy groups πmX = A and πnX = B, 2 [les ] m < n, are classified by the k-invariant

formula here

Here the cohomology group of the Eilenberg–MacLane space K(A, m) was computed by Eilenberg–MacLane [11] and Cartan [5]. Let p be a prime and let ℤp ⊂ ℚ be the smallest subring of ℚ containing 1/q for all primes q with q ≠ p. We consider finitely generated ℤp-modules A and B and the stable range n < 2m − 1. Hence X is a p-local space with at most two non-trivial homotopy groups in a stable range. Then the homotopy type of X admits a product decomposition

formula here

where all Xi with 1 [les ] i [les ] j are indecomposable and this decomposition is unique up to permutation. We classify in this paper the indecomposable factors in (2) by the following result.

(1) The cataphoresis of ions and colloid particles is discussed in so far as it is affected by the ionic atmosphere.

(2) The ions in the atmosphere which carry a charge opposite in sign to that on the central particle are attracted to the central particle. However, when an electric field is applied to the liquid, they are able to drift away in virtue of their molecular energy, and the migration of the central particle is dependent on this fact. The relaxation force is the resultant of the forces between central particle and ions during this separation.

(3) Such a force draws the ions in the atmosphere after the central particle to some extent. From a consideration of the energy involved in the separation of particle and atmosphere, and of the molecular energies of the ions, we are able to calculate the number of ions which this relaxation force could draw through the liquid as though they were bound to the particle, and hence deduce the magnitude of the force in terms of the friction constant of the ions. The expression is the same as that given by Debye and Hückel.

(4) The cataphoresis equation usually employed for colloid particles takes no account of this relaxation force during migration. The corrected equation is given.

It is a commonplace observation in experiments on cutaneous sensation that a constant stimulus, applied under supposedly constant conditions, evokes a varial response. In this paper it will be shown that the variability is in part a geometri consequence of the fact that the sense organ is not a continuum but an aggregati of discrete, sensitive ‘spots’. The total extent of the ‘spots’ for one modality, acti at a given time, is only a small fraction of the area of the site tested; this is illustrat in all of the classical maps, and is particularly true of the warmth-spots (Rein (8)). I purposes of theoretical discussion, the sensory spots may therefore be treated points that form an array or pattern. This pattern is unlikely to be regular and mt indeed, be highly irregular. No assumptions about its regularity are necessary the present argument.