The finiteness of K-rational torsion points of a Drinfeld module of rank 2 over a locally compact complete field K with a discrete valuation is proved.

In an earlier paper we provided a counterexample to an old conjecture of Hopf. In this note we show that the "strong sweeping out property" obtains for the Hopf operators (Tt) both when t —> +∞ and when t —> 0+, that is a.e. convergence fails in the worst possible way.

Let M2n+1 be a compact contact manifold and 𝓐 the set of associated metrics. Using the scalar curvature R and the *-scalar curvature R*, in [5] we defined the "total scalar curvature", by and showed that the critical points of I(g) on 𝓐 are the K-contact metrics, i.e. metrics for which the characteristic vector field is Killing. In this paper we compute the second variation of I(g) and prove that the index of I(g) and of —I(g) are both positive at each critical point. As an application we show that the classical total scalar curvature A(g) = ∫M R dVg restricted to 𝓐 cannot have a local minimum at any Sasakian metric.

Let F be a relatively closed subset of a domain G in the complex plane. Let f be a function that is the limit, in the Lip α norm on F, of functions which are holomorphic or meromorphic on G (0 < α < 1). We prove that, under the same conditions that give Lip α-approximation (0 < α < 1 ) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that f — m can be extended to a function belonging to lip

In this note we extend a theorem of Kwong and Zettl concerning the inequality

The Kwong-Zettl result holds for 1 ≤ p < ∞ and real numbers α, β, γ such that the conditions (i) β = (α + γ)/2, (ii) β > - 1 , and (iii) γ - 1 - p hold. Here the inequality is proved with β satisfying (i) for all α, γ except p — 1,-1 — p. In this case the inequality is false; however u is shown to satisfy the inequality

In this paper, several characterizations are given of the group of eigenvalues of a rank one transformation. One of these is intimately related to the corresponding expression for the maximal spectral type of a rank one transformation given in an earlier paper.

We prove that a free centred extension R[E] is a semisimple ring if R is a semisimple ring and C[E] is semisimple for every field C which is the extended centroid of a primitive factor of R.

Let S = k[[x,y]] be a formal power series ring in two variables x, y over a field k and I an (x, y)-primary ideal of S. We show that S/I is selfinjective if Hi(S/I, S/I ⊗k S/I) = 0 for i = 1 and 2.

Cet article de survol est le résumé de la conférence Coxeter-James de l'auteur, prononcée à la réunion d'hiver 1993 de la Société Mathématique du Canada.

La théorie de Morse décrit les liens entre la topologie d'une variété et la topologie des points critiques d'une fonction sur cette variété. La fonctionnelle d'énergie pour les applications d'une surface dans une variété, dont les points critiques seront des applications harmoniques et parfois holomorphes, et la fonctionnelle de Yang-Mills pour des connections sur une variété de dimension quatre sont deux cas en dimension infinie pour lesquels la théorie de Morse ne tient pas. Néanmois, dans les deux cas, on peut récupérer une quantité étonnante d'information, pourvu qu'on stabilise par rapport à un degré ou une charge qui sont des données du problème. Les preuves recyclent des résultats de la théorie de l'homotopie des années '70, et les combinent à des idées de géométrie complexe pour donner de jolis modèles des espaces en cause en termes de "particules". Nous espérons donner un survol général et accessible des idées utilisées.

A description is given of the unit group for the two groups G = D12 and G = D8 × C2. In particular, it is shown that in both cases the bicyclic units generate a torsion-free normal complement. It follows that the Bass-cyclic units together with the bicyclic units generate a subgroup of finite index in for all n ≥ 3.

Let ϕ0 be an Anzai transformation on the 2-torus T2 defined by ϕ0(x,y) = (e2πiθx,xy) and ϕy a Furstenberg transformation on T2 defined by ϕf(x,y) = (e2πiθx,e2πif(x)xy) where θ is an irrational number and f is a real valued continuous function on the 1-torus T. In the present note we will show that ϕf has topologically quasi-discrete spectrum if and only if ϕf is topologically conjugate to ϕ0. Furthermore we will show that for any irrational number θ there is a real valued continuous function f on T such that ϕf does not have topologically quasi-discrete spectrum but is uniquely ergodic.

Let E be one of the spaces C(K) and L1, F be an arbitrary Banach space, p > 1, and (X, σ) be a space with a finite measure. We prove that E is isometric to a subspace of the Lebesgue-Bochner space LP(X; F) only if E is isometric to a subspace of F. Moreover, every isometry T from E into Lp(X; F) has the form Te(x) = h(x)U(x)e, e ∊ E, where h: X —> R is a measurable function and, for every x ∊ X, U(x) is an isometry from E to F

Our aim is to give an arithmetical expression of the class number formula of real quadratic fields. Starting from the classical Dirichlet class number formula, our proof goes along arithmetical lines not depending on any analytical method such as an estimate for

For the trace map on an irreducible semigroup of n × n matrices over a field, I. N. Herstein presented a theorem in [3] which enables us to limit the nature of matrix groups of a certain kind. However, this is incorrect in general. For the theorem, we shall present a counter example, a revision, and some generalizations to non-irreducible semigroups.

The unit group of the modular group algebra of a finite p-group in characteristic p is nilpotent. The p-groups for which it is of nilpotency class 3 were determined in work of Coleman, Passman, Shalev and Mann when p ≥ 3. We resolve the p = 2 case here which completes the classification.

Let Mn be an n-dimensional compact oriented connected Riemannean manifold. It is proved that either of the following conditions is sufficient to insure that the flow defined by a generalized gradient vector field in Mn has either a stationary point or a periodic orbit:

• a)Mn is the product of a circle with an (n — 1 ) dimensional manifold of non-zero Euler characteristic.

• b)The (n — 1) dimensional Stiefel-Whitney class of Mn is different from zero and in addition Mn possesses no one-dimensional 2-torsion.

We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.