To shed light on the following unsolved problem, several authors have considered related problems. The problem is that of finding commuting approximants to pairs of asymptotically commuting self-adjoint matrices:

Suppose that Hn and Kn are self-adjoint matrices of dimension m(n), with ║Hn║, ║Kn║ ≦ 1, which commute asymptotically in the sense that

Must there then exist commuting self-adjoint matrices H′n and K′n for which

One may alter the conditions imposed on Hn and Kn, for example, by requiring Hn to be normal and Kn to be self-adjoint, and ask whether commuting approximants H′n and K′n can be found satisfying the same conditions. Some of these related problems have been solved. This paper will examine their solutions from a K-theoretic point of view, illustrating the difficulty inherent in modifying them to work for the original problem.

For a given elliptic curve $\mathbf{E}$, we obtain an upper bound on the discrepancy of sets of multiples ${{z}_{s}}G$ where ${{z}_{s}}$ runs through a sequence $Z\,=\,\left( {{z}_{1}},\ldots ,{{z}_{T}} \right)$ such that $k{{z}_{1}},\ldots ,k{{z}_{T}}$ is a permutation of ${{z}_{1}},\ldots ,{{z}_{T}}$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$.

We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.

All finite dimensional indecomposable representations of Ut(Sl (2)) at roots of 1 are determined.

By "positive definite matrices" or, briefly, definite matrices, we mean in this note self-adjoint matrices all the characteristic values of which are positive. Alternatively, they can be defined as matrices, all the hermitian quadratic forms of which are real and positive. It is well known that these two definitions are equivalent and it is the existence of two equivalent definitions which renders the subject particularly interesting. Thus, it follows from the first definition that A-1 is positive definite for positive definite A ; it follows from the second definition that the sum of two positive definite matrices is also positive definite. Similarly, we shall call a matrix positive semidefinite or, briefly, semidefinite if it is self-adjoint and none of its characteristic values are negative, or if all of its hermitian quadratic forms are non-negative. A matrix which is definite is also semidefinite but a semidefinite matrix is, of course, not necessarily definite.

If H is a (complex) Hilbert space and is a collection of (closed linear) subspaces of H it is easily shown that the set of all (bounded linear) operators acting on H which leave every member of invariant is a weakly closed operator algebra containing the identity operator. This algebra is denoted by Alg . In the study of such algebras it may be supposed [4] that is a subspace lattice i.e. that is closed under the formation of arbitrary intersections and arbitrary (closed linear) spans and contains both the zero subspace (0) and H. The class of such algebras is precisely the class of reflexive algebras [3].

We shall assume throughout this paper, unless otherwise specified, that p is a fixed number, 1 < p < ∞.

It is well known that to prove the poin.twise ergodic convergence of a contraction T on an Lp-space it is enough to prove a Dominated Ergodic Estimate (DEE) for T (see e.g. [11]).

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.

Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.

As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

In this paper we are concerned with the fourth order nonlinear differential equation

where the following conditions are always assumed to hold:

(a) r(t) is continuous and positive for t ≠ 0, and

By a projectivityof vector spaces Xand Yover fields F and G is meant an isomorphism Ψ:(X) → (Y) of their lattices of subspaces. A basic theorem of projective geomtry [2, p. 44] asserts that, for spaces of dimension at least 3, any such projectivity is of the form Ψ(M) = SM for a bijection S:X → Y which is semi-linear in the sense that S is an additive mapping for which there exists an isomorphism σ:F→ G such that

S(λx) = σ(λ)Sx for all λ ∈ Fand all x∈ X.

In [12] Mackey obtained a continuous version of this result: for real normed linear spaces Xand Y, the lattices and of closed subspaces are isomorphic if and only if X and Yare isomorphic (i.e., via a bicontinuous linear bijection).

In this paper we shall give an elementary proof of the theorem

(1.1)

where φ(k) denotes Euler's function, and

(1.2)

where p denotes the prime, and and are integers with (,) = 1, positive.

The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups.

For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

It is convenient and not without precedent (see [2], [1], and also [6]) to call a loop which is isomorphic to all of its loop isotopes a G-loop. Since all groups are readily seen to be G-loops, the only interest in such loops, from a loop-theoretic standpoint, resides with those which are not associative. Examples and ad hoc constructions of such loops have appeared sporadically in the literature (see, for instance, [1], [2], [4], [6], [8], [9], and [13]).

Any finite loop of order n < 5 is a group and, hence, must also be a G-loop. R. L. Wilson [11, 12, 13] proved that a finite G-loop of prime order is necessarily a group; he also exhibited for each even integer n > 5 a G-loop of order n which is not associative and then raised questions concerning the existence of finite G-loops which are not groups for every possible composite order n > 5.

Let

denote the Nth Fermat curve. The period lattice of F(N) is contained with finite index in the product of certain lattices Lr,s (see [6]), and to this inclusion of lattices there corresponds an isogeny of the Jacobian of F(N) onto a product of abelian varieties. The purpose of this paper is to determine when two factors in this product are isogenous over C, and whether they are absolutely simple.

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.

Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius

1

of a Hilbert space operator T. Among the many interesting developments, we may mention:

(a) C. Berger's proof of the “power inequality”

2

(b) R. Bouldin's result that

3

for any isometry V commuting with T;

(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;

(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

A space X is in-expandable, where m is an infinite cardinal, if for every locally finite collection {Hα| α ∈ A} of subsets of X with |A| ≦ m(cardinality of A ≦ m) there exists a locally finite collection of open subsets {Gα| α ∈ A} such that Hα ⊆ Gα for every α ∈ A. X is expandable if it is m-expandable for every cardinal m. The notion of expandability is closely related to that of collection wise normality introduced by Bing [1], X is collectionwise normal if for every discrete collection of subsets {Hα|α ∈ A} there is a discrete collection of open subsets {Gα|α ∈ A} such that Hα ⊆ Gα for every α ∈ A. Expandable spaces share many of the properties possessed by collectionwise normal spaces. For example, an expandable developable space is metrizable and an expandable metacompact space is paracompact.

A Riemannian manifold (M, g) is said to be conformally flat if every point has a neighborhood conformai to an open set in Euclidean space. Over the past thirty years, many papers have appeared attacking, with varying degrees of success, the problem of classifying the conformally flat spaces which occur as hypersurfaces in Euclidean space. Most of these start from the following pointwise result of Schouten.

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.

Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.

Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.