If (a1, b1, c1), (a2, b2, c2), (a3, b3, c3) be three sets of numbers subject to the six relations such as and with a positive determinant, then the four tetrahedra, M, M1, M2, M3, whose angular points have coordinates given by the rows of the four matrices

are such that every two are mutually inscribed. The sixteen points so arising are the nodes of a Kummer surface having for one form of its equation

1. In this paper we shall use the methods of (6) to give a short proof of Wunderlich's result (7, 8) on the distribution of twin ‘primes’ for the Hawkins random sieve. We shall also obtain a sharp estimate on the gaps between successive ‘primes’.

For every $N > 0$ there exists a group of deficiency less than ${-}N$ that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold of the $n$-sphere, $n > 4$, if and only if it is the fundamental group of a homology $n$-sphere. There exist fundamental groups of homology $n$-spheres, $n > 4$, that cannot arise as the fundamental group of the complement of a contractible submanifold of the 4-sphere.

An existence theorem is proved for Robin's integral equation for the density of electric charge on a closed surface, under the assumptions that the surface is convex, smooth and twice continuously differentiable. The technique is essentially Neumann's method of the arithmetic mean, used by Robin himself to show that the solution, assumed to exist, can be successively approximated by a sequence. In order to facilitate the main argument of the proof, it is assumed initially that the Gaussian curvature is everywhere positive, but this restriction is subsequently removed.

Let ω be the space of all (complex) sequences. If E, F are subspaces of ω and if A is any (infinite) normal matrix, we set

and

If A is the matrix of a sequence–sequence summability transform, Ā and  shall denote the series–sequence and series–series forms of the transform, respectively. The multiplier spaces M(c(Ā), c(Ā)), M(lp(Â), l1(Â)) and M(lp(Â), c(Ā)) are characterized (1 ≤ p < ∞). Partial results are given for the spaces M(c(Ā), lp(Â)) and M(lp(Â), lp(Â)).

For any $x \in (0,1]$, let the series $ \sum_{n=1}^{\infty}1/d_n(x)$ be the Sylvester expansion of $x$. In this paper, we consider the Hausdorff dimension of the set $$B(\alpha, \beta)= \bigg\{x \in (0,1]: \lim\limits _{n \to \infty} \frac{d_{n+1}(x)}{d^{\beta}_n(x)}=\alpha\bigg\}$$ for any $\alpha \geq 0$ and $\beta \geq 2$. As a corollary, we answer the question posed by Goldie and Smith in [6].

The calculation of the interaction between a point multipole moment and an external charge distribution is considered in elliptic coordinates. It is complicated by the singularity at the point moment. The results obtained are useful in calculating hyperfine interactions in diatomic molecules where the elliptic coordinates are the most natural system of coordinates. The results lead to an extension of Heine's expansion of (λ + μ)−1.

A differential invariant of a G-structure is a function which depends on the r-jet of the G-structure and such that it is invariant under the natural action of the pseudogroup of diffeomorphisms of the base manifold. The importance of these objects is clear, since they seem to be the natural obstructions for the equivalence of G-structures. Hopefully, if all the differential invariants coincide over two r–jets of G-structure then they are equivalent under the action of the pseudogroup. If all the differential invariants coincide for every r it is hoped that the G-structures are formally equivalent, and so equivalent in the analytic case. This is the equivalence problem of E. Cartan. In this paper we deal with the problem of finding differential invariants on the bundles of ℝ*-structures, following the program pointed out in [3]. There are several reasons that justify the study of this type of G-structures. The first one is that it is a non-complicated example that helps to understand the G-structures with the property for the group G of having a vanishing first prolongation (i.e. of type 1). The simplicity comes from the fact that the algebraic invariants of ℝ* are very simple. The differential geometry of this type of structure, however, has much in common with general G-structures of type 1. Also, ℝ*-structures are objects of geometrical interest. They can be interpreted as ‘projective parallelisms’ of the base manifold and they can also be interpreted as a generalization of Blaschke's notion of a web.

The groups of rotations that transform the regular polygons and polyhedra into themselves have. been studied for many years. Lately, increasing interest has been shown in the “extended” groups, which include reflections (and other congruent transformations of negative determinant). Todd has proved that every such group can be defined abstractly in the form

This group is denoted by [k1, k2, …, kn−1], and is the complete (extended) group of symmetries of either of the reciprocal n.-dimensional polytopes {k1, k2,…, kn−1}, {kn−1, kn−2,…, k1}. There is a sense in which these statements hold for arbitrarily large values of the k's. But here we are concerned only with the cases where the groups and the polytopes are finite. The finite groups are

[k] is simply isomorphic with the dihedral group of order 2k (e.g. [2], the Vierergruppe). [3, 3,…, 3] with n − 1 threes, or briefly [3n−1], is simply isomorphic with the symmetric group of order (n + 1)!.

We shall first give some definitions concerning parametric surfaces. Denote by H a closed circle (disk) and by M a variable point on it. Let P = Ф(M) be a continuous function on H whose value P is a point in three-dimensional space. The symbols Ф(E), Ф−1(P), where E is a set of points on H and P a point in the three-dimensional space, will have their usual meaning. Ф−1(P) is a closed set. Any saturated continuum in Ф−1(P) or any point of Ф−1(P) that does not belong to such continua is called a Ф-element of H. Thus to any continuous function Ф(M) corresponds a representation of H in the form of the sum σQ of Ф-elements. The set of the pairs (P, Q), where Q runs through all Ф-elements of H and, for any Q, P = Ф(Q), is called a parametric surface, and any pair (P, Q) is called a point of the parametric surface. We shall often speak of a point Ф(M) of the parametric surface, by which we shall mean either the point (P, Q), where P = Ф(M) and Q is the Ф-element containing M, or the point P = Ф(M) of the three-dimensional space. The exact meaning will always be clear from the context. If there are exactly k points of the parametric surface whose first member is P0 we say that P0 is a point of multiplicity k. If k = 1, P0 is a simple point.

An innovation by D. R. Heath-Brown on square-full numbers in short intervals is applied to establish an asymptotic formula for the number of cube-full numbers in the interval

In this paper we have evaluated an integral involving Fox's H-function and Jacobi polynomial and employed its particular cases to establish two expansion formulae for the H-function involving Jacobi polynomials.

In this paper, all numbers are real and all radicals are positive.

Let f(x, y) = ax2 + bxy + cy2 be an indefinite quadratic form, and let d = Δ2 = b2 − 4ac, where Δ > 0. A well-known theorem of Minkowski states that, if (x0, y0) is any pair of numbers, then there exists a pair (x, y), x ≡ x0 (mod 1), y ≡ y0 (mod 1), say (x, y) ≡ (x0, y0) (mod 1), such that

for k ≥ ¼, and Davenport has shown how this result may be improved if we know a value assumed by f(x, y) for coprime integral values, (x, y) = (m, n) ≠ (0, 0). In this paper, we discuss the more general inequality

where R and S are constants, and use the method developed in the first paper of this series to obtain a sharper and more general theorem than Davenport's. We give an application to the theory of real Euclidean quadratic fields and to a problem in Diophantine approximation discussed by Khintchine.

We prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter ≤5. For 7–systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7–systolic groups over finite subgroups are also 7–systolic.

Experiments are described in which observations were made of the motion of electrically charged cloud particles past a sphere. The cloud particles were moving vertically up in an air stream, and there was a vertical electric field. This gave conditions similar to those surrounding a falling rain drop in a thundercloud, and the observations are in accordance with the theory proposed by Wilson to account for the mechanism of thunderclouds.

The object of this paper is to make certain calculations which will be required in the estimation of the quantum defect. Hartree has defined certain solutions Gi(σ, ζ), Hi(σ, ζ) of the wave equation for an electron in a Coulomb field, viz.

Introduction. Dans cet article, K désigne un corps commutatif quelconque. Toutes les algèbres de Lie considérées sont des algèbres de Lie de dimension finie sur K. Si g est une algèbre de Lie, on désigne par (D0g, D1g, D2g, …) la suite des algèbres dérivées de g (D0g = g; Dig = [Di−1g, Di−1g]). Si h est un idéal de g, et si x ↦ g, on désigne par adh, x l'application y → [x, y] de h dans h.

In [10] Loeb and Osswald develop a Daniell–Stone integration theory in topological vector lattices. As an application they obtain an extension of the Bochner integral for functions with values in a topological vector lattice. The Daniell–Stone approach requires an order structure for definition of a null function. Here we build on the ideas in [10], but abandon the lattice structure. The internal representatives of the extended integrable functions in [10] turn out to be S-integrable liftings. This is our starting point: we define the class of extended integrable functions as the class of functions that have S-integrable liftings; a definition that is independent of any order structure. As a consequence, several of the proofs in the construction of the Banach space of extended integrable functions are almost identical to the analogous proofs in [10] and we omit them. After defining the integral we show that each extended integrable function induces a countably additive vector measure of bounded variation. This permits us to use the tools of the theory of vector measures. If the nonstandard hull of the Banach space has the Radon–Nikodym property, then the vector measure induced by an extended integrable function has a Bochner integrable Radon–Nikodym derivative; this yields a characterization of the Banach space of extended integrable functions. On the other hand for some spaces we can construct internal functions and standardize them to get very ‘spiky’ extended integrable functions whose induced measure is not differentiable on any set of positive measure. With this method we show that if a Banach space is not superreflexive, then it does not have the super-Radon–Nikodym property.

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the fixed point groups of metabelian endomorphisms. In particular, we show that the rank of the fixed point group of an IA-endomorphism of the free metabelian group of rank n[ges ]2 can be either equal to 0, 1, or greater than (n−1) (in particular, it can be infinite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.