Two constructions of cohomology classes for congruence subgroups of unit groups of quadratic forms over totally real number fields are given and shown to coincide. One is geometric, using cycles, and the other is analytic, using the oscillator (Weil) representation. Considerable background material on this representation is given.

The purpose of this paper is to generalize the Lindelöf-Čirka theorem.

Residual canonical correlations are defined and are derived in terms of canonical correlations. Some measures of residual association are also defined, in terms of the residual canonical correlations and some possible applications are suggested.

Let C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.

B. Grünbaum, J. N. Lillington and lately R. J. Gardner, S. Kwapien and D. P. Laurie have considered inequalities defined by three concurrent straight lines in the interior of a planar compact convex set. In this note we prove two elegant conjectures by R. J. Gardner, S. Kwapien and D. P. Laurie.

Let R be a prime ring with σ ∊ Aut (R). We determine the extended centroid of the skew polynomial ring R[x, σ] when (i) 〈σ〉 is X-outer of finite order, (ii) 〈σ〉 is X-outer and infinite, (iii) σm is X-inner and no smaller power of σ fixes the extended centroid of R.

Let (𝓜,g) be a Riemannian manifold and let 1, 2, 3 be mutually orthogonal distributions on 𝓜 of dimensions p1, p2,P3 such that p1 + p2 + p3 = dim 𝓜. We assume that , and 1, ⊕ 2 are integrable and that all the geodesies of 𝓜 with initial tangent vector in 2 remain tangent to 2. Then, we prove that Pontk(2, ⊕ 3) = 0 for k > p2 + 2p3, where Pontk(2, ⊕ 3) is the subspace of the Pontrjagin algebra of 2 ⊕ 3 generated by forms of degree k.

The object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

In this paper we study when an integral domain is a finite intersection of PID or factorial overrings. We show that any Krull domain is the intersection of a PID and a field. We give several sufficient conditions for a Krull domain to be an intersection of two PID or factorial overrings.

It is proved that an integral domain R is a Bézout domain if (and only if) R is integrally closed and I J = {ij|i ∊ I, j ∊ J} for all ideals I and J of R; that is, if (and only if) R is an integrally closed condensed domain. The article then introduces a weakening of the "condensed" concept which, in the context of the k + M construction, is equivalent to a certain field-theoretic condition. Finally, the field extensions satisfying this condition are classified.

A space S has property P-1 if S is nonempty. For n > — 1, S has property Pn if it is locally connected, has property Pn-1 and if whenever it is written as a union, S = A ∪ B where each of A and B is closed and has property Pn-1, then A ∩ B also has property Pn-1. The purpose of this paper is to establish that for locally compact spaces, each of the properties Pn is both cyclicly extensible and reducible.

In this article we give a sufficient condition for quasisimilar analytic Toeplitz operators to be unitarily equivalent. We also use a result of Deddens and Wong to give a sufficient condition for an operator intertwining two analytic Toeplitz operators to intertwine their inner parts too. Analytic Toeplitz operators with univalent symbols satisfying a suitable normalization that are quasisimilar are shown to have equal symbols.

The Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.

Let R be a periodic ring, N the set of nilpotents, and D the set of right zero divisors of R. Suppose that (i) N is commutative, and (ii) every x in R can be uniquely written in the form x = e + a, where e2 = e and a ∊ N. Then N is an ideal in R and R/N is a Boolean ring. If (i) is satisfied but (ii) is now assumed to hold merely for those elements x ∊ D, and if 1 ∊ R, then N is still an ideal in R and R/N is a subdirect sum of fields. It is further shown that if (i) is satisfied but (ii) is replaced by: "every right zero divisor is either nilpotent or idempotent," and if 1 ∊ R, then N is still an ideal in R and R/N is either a Boolean ring or a field.

In this paper we show that the homotopy-theoretic fibre of the double suspension map E2:S2n-1 → Ω2S2n+1 is an H-space.