The problem of finding an axiomatic characterisation of dimension was first tackled by Menger, who gave a set of five independent axioms characterising the dimension (in the sense of dim, ind, or Ind since they are all equal on separable metric spaces) of subsets of the plane [7, p. 156]. The question of whether Menger's axioms characterise the dimension of more general spaces is still unsettled. Recently, Nishiura “11” obtained a set of seven independent axioms characterising the dimension of separable metric spaces. By modifying one of Nishiura's axioms, Aarts [1] then obtained an axiomatic characterisation of the strong inductive dimension (Ind) of metric spaces. Also, Ščepin [12] and Lokucievskiĭ [9] have obtained different axioms for dim on the class of compact metric and compact spaces, respectively. We present here four sets of independent axioms that characterise the dimension function u-dim, which is defined on the class of all uniform spaces.

§0. Einführung. Durch die bekannten Croftonschen Integrale können bekanntlich die Minkowskischen Quermaβintegrale konvexer Körper dargestellt werden. In der vorliegenden Note betrachten wir gewisse Erweiterungen dieser klassischen integralgeometrischen Formeln, durch die allgemeinere invariante Eikörperfunktionale gegeben sind. Es handelt sich hierbei um kinematische Integrale mit beweglichen unterdimensionalen Teilräumen, wobei passend gewahlte Funktionen ihrer Abstände vom Eikörper eingehen.

Here we give necessary and sufficient conditions foi a prime ι to divide the class number of the Galois closure of a pure field of degree ι over the rationals. The work extends that of Honda in [4] and that of the first author in [8].

Let K be a field and * an involution of K. Let V and V′ be vector spaces over K of dimensions greater than or equal to 3, and let P(V) and P(F′) denote the projective spaces associated with V and V′ respectively. The fundamental theorem of projective geometry states that a bijection σ between P(V) and P(V′), which preserves the collinearity of points in P(V) and P(V′), is induced by a semi-linear bijection ø between V and V′ with respect to an automorphism α of K. We now consider the following additional structure on V and V′. Let f and f′ be hermitian forms with respect to * on V and V′ respectively; they define twisted polarities in P(V) and P(V′). We prove the following theorem. If there are no self-polar points in P(V) and P(V′) and σ preserves the twisted polarities, then ø is “almost” an isomorphism of the hermitian forms in the sense that

for some non-zero A in K such that A = A*. We give examples of forms f and f′ satisfying the above condition and we illustrate our theorem.

The 2-adic density of a quadratic form is a factor in the expression for the weight of a genus (of positive forms) and so has been investigated by a number of writers. As pointed out by Pall [1], the most complete investigation, that by Minkowski, omitted many details and errors resulted. Unfortunately, Pall's paper also contains errors: the last — in his formula (23) should be +, and 8, 4 in the second and third lines after (47) should be 3, 2. In the fourfold distinction of cases (with nine sub-cases) at the end, the possibility ei+l – ei = 1, øi and øi+1 both pp., seems to have been overlooked.

In the above paper [1] there is an error in the statements of Proposition 1 and Theorem 1 which we shall now correct. The principal results, Theorems 2 and 3, are unaffected.

We give sufficient conditions for the spectra and essential spectra of certain classes of operators to be contained in or coincide with an interval of the form [μ, ∞).

Let M be a finite extension field of the rational numbers Q, and let CM denote the ideal class group of M. Let l be a rational prime, and let AM denote the l-class group of M (i.e., the Sylow l-subgroup of CM). If G is any finite abelian l-group, we define

where ℤl is the ring of l-adic integers and Fl is the finite field of l elements. One of the classical results of algebraic number theory is the specification of rank AM when M is a quadratic extension of Q and l = 2. This result was obtained by means of Gauss's theory of genera. A generalization of this result can be found in [1], where A. Fröhlich has obtained upper and lower bounds for rank AM when M is a cyclic extension of Q of degree l. His methods also show how to compute rank AM exactly when l = 3. In [4], G. Gras has described a procedure for analyzing the l-class groups of relatively cyclic extensions of degree l. However when l > 3, the computations can be very difficult.