It was remarked by Liouville in 1844 that there is an obvious limit to the accuracy with which algebraic numbers can be approximated by rational numbers; if α is an algebraic number of degree n (at least 2) then

for all rational numbers h/q, where A is a positive number depending only on α.

In a recent paper [4], I introduced the notion of recursive formal Lie groups (of infinite dimension) over a field of characteristic p > 0, and studied a particular class of such groups, the groups of hyperexponential type; these can be characterized either as being (recursively) isomorphic to a special group of that class, the hyperexponential group, or by simple conditions on their Lie hyperalgebra. An interesting example of a group of that class is the additive Witt group W, whose “infinitesimal” structure can therefore be considered as known, at least “up to an isomorphism”. However, the intrinsic importance of the Witt group (which, as well known, is the “formalization”, so to speak, of the additive group of a p-adic field) leads one to think that it may be worth while to study in greater detail that group itself, instead of being content with the mere existence of an unspecified isomorphism with the hyperexponential group. This is what we intend to do here; it turns out that, although it seems hopeless to write down explicitly the group law of the Witt group, the multiplication table of its hyperalgebra is, on the contrary, as simple as one could hope, and is, in fact, identical to that of the hyperalgebra of the hyperexponential group (although the two groups are distinct). Moreover, this leads to a new and quite unexpected definition of the Witt group, which links it still closer to the hyperexponential group, and provides a well-determined isomorphism between the two groups.

The main object of this paper is to show that an indefinite quadratic form in four or more variables with integral coefficients represents all integers except those excluded by congruence considerations. The subject of the representation of integers by quadratic forms is one on which little is known, and I believe this result to be new, although I can deduce it in an elementary way from the classical theory.

A well-known method of calculating an approximate value of an eigenvalue λ of a self-adjoint operator H employs an approximation, say w, to the corresponding eigenfunction ψ. Since

the exact value of λ is given by

An analogy is drawn between the slow steady rotation of a solid of revolution about its axis in a viscous fluid and the motion, with constant velocity, of the solid along its axis in a perfect fluid. The rotation of a sphere in viscous fluid is then discussed using a method due in this case to Bickley [1] and here further developed.

In this note it is shown that

(i) there is a plane convex set of minimal width λ which does not contain any convex set whose width is constant and greater than λ/(3—31/2),

(ii) every plane convex set of minimal width λ contains a convex set whose width is constant and is equal to λ/(3—31/2).

Certain results ([7], [8], [10], [11]) suggest that there should be some principle of duality in homotopy theory. Among other things one is led to expect that cohomotopy groups will appear as dual to homotopy groups. But the fact that a cohomotopy group πn(X), unlike πn(X), is only defined if dim X ≤ 2n—2 is a serious obstacle to the formulation of such a principle. However, the set of S-maps (i.e.S-homotopy classes [11]) X → Y is a group for every pair of spaces X, Y. Therefore, this difficulty does not appear in S-theory [11].