H1. Equip ℝ3 with the group law

(1.1)

where (z, t) stands for (x, y, t). This is a nilpotent Lie group, usually referred to as the first Heisenberg group, H1. In general Hk denotes ℝ2k+1 equipped with a similar group law, namely

In this note I investigate smoothness and regularity of the ring the ring of integers. These results do not seem to be well known, especially those dealing with regularity. At any rate, several of my colleagues knew that was not smooth, but thought that it might be regular. I was surprised by the number of possibilities that can occur, as well as by the fact that A can be regular but not smooth. As expected the prime 2 plays a special role. Smoothness depends on a, b, c mod 2, and regularity depends on a, b, c mod 4.

The classical theorem that a complete separable metric space is the image under a one-to-one continuous function of a closed subset of the irrational numbers has been extended in two directions, the first leading to various characterizations in descriptive set theory of Borel and analytic sets or generalizations of them as continuous images of certain subsets of the irrationals, or generalizations of them (see, e.g. [3] and references cited there; [4]; [6]). The second direction originates in the observation that a closed subset of the irrationals is a complete 0-dimensional metric space (under a suitable metric), and leads to the general question asked by Alexandroff [1], "Which spaces can be represented as images of 'nice' (e.g. metric, 0-dimensional) spaces under 'nice' [e.g. one-to-one, open, closed, perfect] continuous mappings?" (See, e.g. [7], [9] and the survey articles [1], [2] and [11].)

It is well known that complex analytic functions and harmonic functions of two real variables are closely related, so that from methods and results in complex function theory one can easily obtain theorems on those harmonic functions. This is the prototype of a relation between complex analysis and a partial differential equation (Laplace's equation in two variables). In the case of more general linear partial differential equations, one can establish similar relations.

The existence of a first countable Lindelöf extension L of an uncountable discrete space D for which L-D is countable is considered. Assuming CH, such extensions exist; however it is also consistent that no such spaces exist, as follows from MA + ¬ CH

Characterizations of the upper semi-continuity of the subdifferential mapping of a continuous convex function are given.

A function M is defined which maps the plane onto a square region in such a way that the planar graphs In, exp, X, —X, 1/X, and all compositions formed from them are transformed into straight lines. One can then solve for their intersections. It also provides a natural definition for the repeated composition of In with itself t times, where t can be a non-integer.

This note considers the question of the connectedness of the set of limit points of the A-transforms of a sequence, where A is a conservative Hausdorff, quasi-Hausdorff or Meyer-König- Ramanujan type of matrix. New proofs of some known results, as well as some new results are obtained.

It is shown that if V=C(X) or V = L1(μ) then the subspace of compact (resp. weakly compact) operators from V into itself is not an M-ideal in the space of bounded operators. This is the contrary to what happens when V= Co(ℕ) or lp(ℕ). The main result is proved via the best approximation properties of M-ideals and some results concerning norm one projections in C(X) and L1(μ) are deduced from this fact.

Sufficient conditions for a sequence of numbers to be the degree sequence of a graph are derived from the Erdos-Gallai theorem on degree sequences of graphs.

Let , where ai, s, and x are real, and II denotes the product over cyclic rearrangements of the subscripts. We show that, in five special cases, Dn, s(x)Dn, s(y) is greater than a fixed multiple of Dn, s(x + y).

La portée et les buts de cet article sont signalés au début des différents paragraphes et dans les remarques qui suivent le théorème 3, le lemme 5 et le théorème 21. Limitons-nous à mentionner ici que le sujet traité est à la fois en relation avec la théorie des systèmes d'inéquations linéaires (théorie des polyèdres coniques), avec le concept de point Λ-extrême, avec les diverses notions d'efficacité par rapport à un système d'objectifs linéaires et avec la programmation à objectifs linéaires multiples.

Let R be a ring with identity. The study of rings in which every left (right) ideal is quasi-injective was begun by Jain, Mohamed, and Singh (3). They called these rings left (right) q-rings. A number of structure theorems have been proved for q-rings. See, for example, (1), (2), and (5). A ring with the dual property (rings in which every homomorphic image of R as a left (right) R-module is quasi-projective) is called left (right) q*. These rings were first studied by Koehler (4), where some results connecting q* -rings with q-rings were obtained.

Let Z be the additive group of integer numbers with discrete topology, the space of complex-valued integrable functions on Z with respect to normalized Haar measure, the space of bounded functions on Z.

Nous nous emploierons dans cet article à démontrer la convergence presque partout pour les moyennes de projections orthogonales croissantes sur L2, qui sont ou bien u.a.c. ou bien positives, ce qui nous donnera une double généralisation du théorème classique de Fejér-Lebesgue pour les séries de Fourier. Enfin nous terminerons en énonçant un résultat général qui englobe tous ceux connus jusqu'à maintenant avec la condition u.a.c. (cf. R. Duncan, Can. Math. Bull. 20 (1977) 277-284).

It is shown that if 1<n ≡ 0 or 1 (mod 2 m), then the edges of Kn may be partitioned into isomorphic copies of a graph D3(m) and also of a graph D4(m), graphs consisting respectively of a triangle with an attached path of m - 3 edges or a quadrilateral with an attached path of m - 4 edges. If m is a power of 2 then the above condition is shown to be necessary and sufficient for the existence of such a partition.

In a paper by Wilansky and the writer [4] there were five questions left open, four of which have been answered by Beekman and the writer, [1], [3]. We shall consider the fifth one, namely, “If , must for every matrix D with cD = cA?” Here A is a conservative summability matrix with column limits .

It is proved that if u is an element of a faithful algebra over a commutative ring R, then u satisfies a polynomial over R which has unit content if and only if the extension R ⊂ R[u] has the imcomparability property. Applications include new proofs of results of Gilmer-Hoffmann and Papick, as well as a characterization of the P-extensions introduced by Gilmer and Hoffmann.

Let (n|k) denote the number of k-choices 1≤x1<x2<…<xk≤n satisfying xi-xi-1≥2, i = 2,…, k, n + x1-xk≥2; let (m, n | k) = Σi+j=k (m | i)(n | j). Several elementary proofs of the new identity (m, n|k) = (m + n | k) if 0≤k<m≤n. and

if 0≤m≤n, m≤k, are given. Generalizations and applications are considered.

In this paper the notion of inverse cluster set, which was recently introduced and studied for functions by T. R. Hamlett and P. E. Long (Proc. Amer. Math. Soc, 53 (1975), 470-476), is extended to and investigated for multifunctions. We generalize the notion of inverse cluster set, extend to multifunctions and generalize some known results for inverse cluster sets of functions and offer some new results. In the latter sections, compactness generalizations are characterized in terms of inverse cluster sets and some results on connected and conectivity functions are extended to multifunctions.