This paper contains some basic relations between Ganea strong category and Lusternik Schnirelmann category in proper homotopy theory. We focus our interest on the case of category 2 in order to show that ℚn is the unique open n-manifold with proper Lusternik-Schnirelmann category 2 (n ≠ 3).

1. If θ be the eccentric angle of a point P on an ellipse whose semi-axes are a and b and if x, y be the rectangular co-ordinates of a point Q, then PQ2 = (x − acosθ)2 + (y − bsinθ)2. When a = b and x2 + y2 = f2, PQ2 = f2 2afcosθ + a2, if the line from which θ is measured passes through Q. The integral

is a well-known one. I propose in the present paper to consider the more general form which the integral takes when P lies on an ellipse.

This paper deals with Schauder decompositions of Banach spaces X2π of 2π-periodic functions by projection operators Pk onto the subspaces Vk, k = 0,1,…, which form a multiresolution of X2π,. The results unify the study of wavelet decompositions by orthogonal projections in the Hilbert space on one hand and by interpolatory projections in the Banach space C2π on the other. The approach, using “orthogonal splines”, is constructive and leads to the construction of a Schauder decomposition of X2π and a biorthogonal system for X2π, and its dual X2π. Decomposition and reconstruction algorithms are derived from the construction.

It is known that an induced matrix of an induced matrix is expressible as the direct sum of invariant matrices, or more generally that an invariant matrix of an invariant matrix can be expressed as a direct sum of invariant matrices. The spurs of the irreducible invariant matrices of a given matrix A = [ast], are the S-iunctions of the latent roots of A.

In der Theorie der Invarianten binärer Formen gilt der Satz, dass die projektiven Komitanten eines vollen Systems durch sukzessives Ueberschieben berechnet werden können. Hierauf beruht der Originalbeweis für die Endlichkeit, den. P. Gordan 1868 zuerst gegeben hat. Konstruktiv wird dieser Beweis aber erst dann, wenn man von vorneherein eine obere Grenze für den Grad der Komitanten eines vollen Systems in den Koeffizienten der Grundformen angeben kann. Also eine natürliche Zahl N, die die Sicherheit gibt, dass alle Komitanten eines vollen Systems gefunden sind, wenn man die Ueberschiebungen soweit berechnet hat, dass ihr Grad N nicht übersteigt.

Throughout this paper any near-ring N will be left distributive, zero symmetric and have an identity. Furthermore, all N-groups will be unitary. All groups considered will be written additively. This does not imply commutativity.

L. J. Mordell has recently considered (1) the squared modulus of a complex difference-product, namely

under the conditions

and also under the quite different condition

He proves that under (2) the maximum of δ is nn, and is attained when and only when the zr are vertices of a regular n-gon on the unit circle.

We consider the Dirichlet problem

where g is continuous and h ∈ L2(0, π).

This note is an exposition of the simple and elegant approach to Grothendieck's inequality given in [2] and lsqb;4], with one further simplification. The process of factorizing through L2 ([2], p. 21) introduces a factor of into the final constant. We show that this step can be avoided.

A group G is called a Bn-group, if all its subnormal subgroups are of defect n. If all infinite subnormal subgroups of G have defect n, we say that G is an IBn group. DeGiovanni and Franciosi [2] have considered soluble IBn groups and characterized them.

We show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).