Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

In [8] the notion of a reparameterizing isomorphism of transformation groups, henceforth called an R-isomorphism, is introduced generalizing Ura's type-2 isomorphism (see [13]). We have shown [8] that an R-isomorphism is weaker than a transformation group isomorphism. For example, although R-isomorphisms preserve pointwise almost periodicity and minimality they do not preserve the existence of slices [7] or almost periodicity. This suggests that R-isomorphisms might be a useful classification tool in topological dynamics.

The theory of Plücker coordinates and Grassmann varieties is well-developed and well-known among the algebraic geometers. It gives a one-to-one correspondence between the set of all subspaces of a given dimension in the ambient projective space and the set of points on a certain projective algebraic variety called a Grassmann variety. The unacquainted can find the theory discussed in detail in Hodge-Pedoe [1, Chapters VII and XIV].

Torsion theories have proved a very useful tool in the theory of abelian categories; for example, in one proof of Mitchell's embedding theorem (Bucur and Deleanu [3]) and in ring theory (Lambek [5]). It is the purpose of this paper to initiate an analogous theory for non-abelian categories. Originally we had hoped to prove the non-abelian analogue of Mitchell's theorem this way (Barr, [2, Theorem III (1.3)]), but so far this had not been possible. Nonetheless an interesting variety of examples fit into this theory.

The aim of this paper is to characterize the finite simple group PSp4(3) by the structure of the centralizer of an element of order three contained in the center of its Sylow 3-subgroup. More precisely, we shall prove the following results.

Čebyšev centers have been studied extensively. In this paper an alternate concept of center for infinite bounded point sets is introduced. Some of the results in this paper for this new type of center are similar to previous results for Čebyšev centers.

This paper is concerned with some extensions of the Bures metric d defined on the set of normal states of a W*-algebra [2]. Each subgroup G of the automorphism group of leads naturally to a semi-metric dG. (See Definition 1.1 below.) When G is the identity group dG = d.

It has been proved [3, p. 630] that the martingale convergence theorem obtained by Andersen and Jessen [1, p. 5] follows from the classical theory developed by Doob. By using some results of Yosida and Hewitt [9] on finitely additive set functions, Johansen and Karush [7] proved that the identification of the limit function as a derivative in the approach of Andersen and Jessen can be obtained in the general case. In this paper we sharpen the methods of Andersen and Jessen to obtain a ratio limit theorem for “approximate martingales”.

Let f be a real valued function defined in some neighbourhood of a point x. If there are numbers α1, α2, … αr-1, independent of h such that

then the number αk is called the kth Peano derivative (also called kth de la Vallée Poussin derivative [6]) of f at x and we write αk = fk(x). It is convenient to write α0 = f0(x) = f(x). The definition is such that if the mth Peano derivative exists so does the nth for 0 ≦ n ≦ m.

Let X be a given complex matrix of order n. If f(z) is analytic at the eigenvalues of X, one may define the primary matrix function f(X) with stem function f(z) by using any of several well-known methods: for instance, canonical forms, power series, or interpolating polynomials [9].

R. M. Thrall [10] introduced QF — 1, QF — 2 and QF — 3 rings as generalizations of quasi-Frobenius rings. (For definitions, see section 1. It should be noted that all rings considered are assumed to be left and right artinian.) He proved that QF — 2 rings are QF — 3 and asked whether all QF — 1 rings are QF — 2, or, at least, QF — 3. In [9] we have shown that QF — 1 rings are very similar to QF — 3 rings. On the other hand, K. Morita [6] gave two examples of QF — 1 rings, one of them not QF — 2 and therefore not QF — 3, the other one QF — 3, but not QF — 2.

Let k and A denote commutative rings with identity and assume that A is a k-algebra. A qth order k-derivation δ of A into an A -module V is an element of Homk(A, V) such that for any q + 1 elements a0, … , aq of A, the following identity holds:

Thus, a 1st-order derivation is just an ordinary derivation of A into V.

Let M be an associative *-algebra with complex scalar field. M may be turned into a Lie algebra by defining multiplication by [A, B] = AB - BA. A Lie *-subalgebra L of M is a *-linear subspace of M such that if A, B ∈ L then [A,B] ∈ L. A Lie *-isomorphism ϕ between Lie *-subalgebras L1 and L2 of *-algebras M and N is a one-one, *-linear map of L1 onto L2 such that ϕ[A, B] = [ϕ(A), ϕ(B)] for all A , B ∈ L1.

The determination of the abelian groups in which every neat subgroup is pure is a relatively routine exercise (see [6]). There are numerous problems of this type; for example, the determination of the groups in which every pure subgroup is isotype or the groups in which every subgroup is isotype. These are all special cases of the general problem of determining the abelian groups in which every α-pure subgroup is β-pure for arbitrary ordinal numbers α and β. The solution of this general problem is the object of this paper.

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

In this paper we are chiefly concerned with the chromatic sums we have called l and h, with colour-number 3. In this case h can be interpreted as enumerating the rooted Eulerian triangulations with a given number of faces, and l as enumerating such triangulations with a given number of faces and a given valency for the root-vertex. The series h has been determined already, by summation from the formula enumerating even slicings [3]. However our formula for l does not seem to have been published before, though it could presumably be derived in a similar way.

In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

A singular partial differential equation which occurs frequently in mathematical physics is given by

where is the Laplacian operator on Rn of which the generic point is denoted by x = (x1, … , xn) and s and k are real numbers. The study of solutions of this equation for the case k = 0 was initiated by A. Weinstein [5], who named it ‘Generalized Axially Symmetric Potential Theory'. Numerous references to the literature on this equation can be found in [1; 3; 6]. The analytic theory of equations of the type mentioned above has extensively been treated in [2].

For integral domains of characteristic not 2, we prove here that the symplectic and unitary congruence groups are not isomorphic if the Witt indices are at least 3. This is Theorem 2.1; Theorem 3.3 describes the isomorphisms of unitary congruence groups.