There are 432 strongly squarefree symmetric bilinear forms of signature (2, 1) defined over $\mathbb{Z}[\sqrt{2}]$ whose integral isometry groups are generated up to finite index by finitely many reflections. We adapted Allcock's method (based on Nikulin's) of analysis for the 2-dimensional Weyl chamber to the real quadratic setting, and used it to produce a finite list of quadratic forms which contains all of the ones of interest to us as a sub-list. The standard method for determining whether a hyperbolic reflection group is generated up to finite index by reflections is an algorithm of Vinberg. However, for a large number of our quadratic forms the computation time required by Vinberg's algorithm was too long. We invented some alternatives, which we present here.

Introduction and summary: In the distribution of the sum of n random variables

for which we suppose , the tail probability will be called an extreme tail probability if the normal approximation

is poor, when is asymptotically normal and n is large. This paper concerns the specialization of (1.1) in which

with a random permutation of nλ ones and n − nλ zeros, that is, in which T is the total for a simple random sample from the population with sampling fraction λ. In this case, we will always write T as Tλ. For brevity, we may write {c1,…, cn} for .

Suppose that K = {K0, K1} is a partition of some finite combinatorial power [S]r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]S consisting of those functions ƒ whose supports Sƒ = {ƒ > 0} are 0-homogeneous i.e., [Sƒ]r ⊆ K0. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K1 must be very small. Hence K0 is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace Xm(K) of X(K) consisting of those ƒ for which Sƒ are maximal 0-homogeneous subsets of S. Unfortunately, while Xm(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ1 are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)+ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ0. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)+ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.

Based on the characterization of periodic eigenvalues using rotation numbers, we analyse the second and the third periodic eigenvalues of one-dimensional Schrödinger operators with certain step potentials. This gives counter-examples to the Alikakos–Fusco conjecture on the second periodic eigenvalues. Using this simple model, we can also construct infinitely many resonance pockets, which are much like calabashes emanating from a cane, of one-parameter Hill's equations.

It is well known that the planes which meet four given lines in a space of four dimensions meet a fifth line, determined by the first four. Also the trisecant planes of a rational quartic curve in [4] which meet a line meet another rational quartic curve. These two theorems are of the same type and may conveniently be called “fifth incidence theorems”.

The success of Stoner's extensions of the theory of collective electron ferromagnetism(1) opens up new possibilities of considerable importance. The background of the theory can be found in standard works on the theory of metals(2) and statistical mechanics(3) and will not be given here.

Some time ago, Dr I. B. S. Passi pointed out to me the connexion between the structure of the terms of the α-central series of a wreath product and that of polynomial groups which he had defined and studied. In discussing this problem with him, we looked closely at a paper (1) which I had written on nilpotent wreath products. In the process of this, I found a mistake in Theorem 3·7, the main result of (1). The results in (2) depend onthis theorem. Fortunately, it is possible to obtain a correct version of Theorem 3·7 and to modify the proof in (2) which depends on this theorem in such a way that all the results of (2) remain valid.

Let X be a Banach space. In this paper we show that, assuming that X does not contain a copy of [lscr ]∞, then ba (Σ,X) contains a complemented copy of c0 whenever X contains a copy of c0. Conversely, if ba (Σ,X) contains a complemented copy of c0, then X contains a copy of c0.

Some orthogonal sets of Bessel functions of real order v are identified using the equation Δ2u = utt of the vibrating plate. Our main concern is with the L2 completeness of such sets, and we prove that the well known ‘clamped edge’ type is complete for v > -1, thus completing a result of E. Dahlberg. We also study a very closely related set and show that it needs an extra (non-Bessel) function for completeness.

Our method for proving the completeness is based on one given by H. Hochstadt in connection with Dini functions. We have found it necessary to reorganize Hoch-stadt's method and correct some errors contained in it.

Certain isolated values of v require special attention and we treat these by subjecting the Dalzell completeness criterion to a continuity argument.

We show that the non-existence of elements in the p-stem πSp of Hopf invariant one implies that: there exists no smooth map f[ratio ]M→N with only fold singularities when M is a closed n-dimensional manifold with odd Euler characteristic and N is an almost parallelizable p-dimensional manifold (n[ges ]p), provided that p≠1, 3, 7. In fact, the result itself is originally due to Kikuchi and Saeki [25, 34]. Our proof clarifies the relationship between the two problems and gives a new insight to the problem of the global singularity theory. Furthermore we generalize the above result to maps with only Morin singularities of types Ak with k[les ]3 when p≠1, 2, 3, 4, 7, 8.

For a nonsingular projective threefold of general type X over the field of complex numbers, we show that the fourth pluricanonical map ϕ4 is not birational onto its image if and only if X is birationally fibred by (1,2)-surfaces, provided that vol(X) ≥ 303. We also have similar characterization of birationality of ϕ3.

A non-singular solution of the bosonic SU(1, 1)/U(1) no-scale supergravity field equations is constructed for an anisotropic cosmological model which is of Bianchi type VIIh.

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* if

for each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

1. The problem which I propose to solve is that of finding the number of quartic curves of intersection of two quadrics which pass through p points and have q lines as chords, where p + q = 8. There are ∞16 elliptic quartics in space; to contain a line as a chord is two conditions, and to pass through a point is two conditions, so we should expect a finite number of solutions. Throughout this paper I shall refer to an elliptic quartic curve in space of three dimensions simply as a “quartic.” I shall denote the number of solutions for a particular value of p by np.

The general arrangement of the apparatus is shown in Fig. 1. A vertical steel rod AA of circular cross-section is fixed to a heavy cast-iron base Z shown diagrammatically in the figure. The rod AA is bent by a load applied to the end of the horizontal bar BC carried by the upper end of A A. The point of application of the load is defined by a transverse groove cut in the bar. The rod AA passes through the two metal plates DD, FF, which are soldered to the rod, the faces of these plates being horizontal.

The theorem that, if four arbitrary lines be taken in a plane, the four circles about the triangles formed by threes of these lines, meet in a point, can be generalised to space of any even number of dimensions, as was recognised by Mr J. H. Grace in 1897. In the plane case the centres of the four circles lie on another circle passing through the point of concurrence of these four; it has been sought to prove that the centres of the n + 2 spheres, similarly arising in space of an even number, n, of dimensions, also lie on a sphere†.

The modified Thomas-Fermi method reported in (2) has been applied numerically to calculate the distribution of valence electrons in CC14. Slater wave functions have been used to describe the Cl inner-shell electrons, whilst a description of the C ls electrons based on the Thomas-Fermi model for atoms has been adopted. The electron distribution thus found is compared with the two approximations discussed in part I of this series (3). The first of these, in which valence and inner-shell electrons are treated in the same way, is shown to give a distribution of valence electrons in good agreement with the present results. The second approximation, that of effective nuclear charges, is less satisfactory and could only lead to a distribution of valence electrons in reasonable agreement with the present work if the effective nuclear charges were considerably reduced. This seems to throw some light on the results of Bowers for the force constants of tetrachlorides and tetrabromides.

The overall distribution of electrons obtained here seems to be the inevitable result of assuming the valence electrons to move in any plausible central field. It is, of course, by no means certain that such an approximation is near to the truth. In fact, the results of (3) and the present paper leave little doubt that the electron cloud obtained by such an approximate method will be too diffuse. This seems at least partly responsible for our finding that rather more than 20 of the 28 Cl valence electrons lie outside the sphere drawn through the Cl nuclei.

It was in a paper bearing this title that Cayley(1) first considered the problem of representing a curve in projective space of three dimensions by means of the complex of lines which meet the curve. He took the conic given by the equations

and found that the line

with dual Grassmann coordinates (…,pij,…), where

intersects the conic if, and only if,

where F(u0, u1) is homogeneous and of degree 2 in both sets of indeterminates u0 and u1 and G(…,pij,…) is a form of degree 2 in the pij. Both F(u0, u1) and G(…,pij,…) are easily determined in this case.

Some data obtained at Leiden on relaxation effects in a diluted potassium chrome alum are discussed in the light of a paper by Temperley (10), and excellent agreement is found with the conclusions of that paper. The primary effect of diluting the salt is to reduce the magnetic interaction, and the observed effects are a big increase in the relaxation time in zero field, and a decrease of the relaxation time with increasing field instead of an increase. It is predicted that a decrease should set in with an ordinary salt at liquid helium temperatures at fields somewhat higher than have so far been used. Van Vleck's criticisms of Temperley's paper are discussed, and it is concluded that they are not valid in their present form.

The aim of this paper is to make some remarks about the theory of parallel immersions in ℝn. We show that the existence of parallels to a curve in ℝn depends on the values of the integrals of the n − 1 curvature functions along the curve. We then give examples of immersions of S1 in ℝ3 and of immersions of codimension 2 with local parallel rank zero, thereby answering certain questions in [1].