In a previous paper we discussed the duals of generic hypersurfaces: both smooth hypersurfaces in ℝn and algebraic hypersurfaces in real or complex projective space ℙn. In this note we show how to extend the methods of [1] to cover the case of complete intersections in ℙn and preface this with a brief discussion on the contact of space curves in ℝn with planes. We shall use the notation of [1].

A necessary and sufficient condition for confluence of two poles of a class of meromorphic solutions of the KdV equation is introduced and proved.

In this paper we generalize the folding process initiated by Stallings for graphs to a class of generalized covering spaces. These spaces are called pinched coverings or pinched cores, depending on the particular situation. We then apply our generalized folding process to manipulate these spaces into actual coverings. By using elementary homotopy arguments, we can calculate the fundamental groups of these spaces. As a corollary to our main result we obtain a generalization of a result due to Gersten concerning monomorphisms between free products of groups.

The complex Stiefel manifold Wn,k, where n≦k≦1, is a space whose points are k-frames in Cn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

The rational treatment of Geometry has this important disadvantage, that for want of suitable demonstrations it seems impossible to preserve the natural grouping of the facts developed. The study of Rational Geometry, in fact, should always be supplemented by a systematic attempt to array the facts demonstrated according to their subject-matter; for it will hardly be denied that a direct and systematic knowledge of the Properties of Geometrical Figures has an intrinsic value apart from the knowledge of their demonstrations. In pursuing such a retrospective scheme as this in connection with the Second Book of Euclid, I have found that a very comprehensive view of the subject-matter is obtained by adding a Third Mode of Section of a straight line to the two which are already recognised. This third mode of section, for which I have not been able to find a more suitable name than “Circuitous Section,” along with the other two known as Internal and External Section respectively, exhausts the possible modes of section of a line—for three-dimensional space at any rate. From this point of view, the elementary treatment of the subject may be arranged as follows. It will be observed that several important properties of triangles and polygons acquire a new-interpretation as cases of circuitous section.

Let {Ai; i ∈ Ω} be a family of C*-algebras acting on a Hubert space H and let A be the C*-algebra that they generate. We shall assume throughout that C*-algebras always contain the identity operator. Let M(A) denote the space of characters, that is, multiplicative linear functionals, acting on A, with the weak star topology. We obtain here a natural characterisation of M(A) as a subset of the product space determined by {M(Ai); i ∈ Ω}. In the case of singly generated C*-algebras this characterisation is related to the joint normal spectrum (5) of a family of operators.

In this article some properties on subseries convergent sequences in locally convex topological vector spaces are studied and some open questions of (2) are answered.

In (1), § 6.2, a multiplying factor method has been used to solve certain dual integral equations. The results are then used to solve a single integral equation of the Wiener-Hopf type. In this note we indicate how a related technique can be used to solve Wiener-Hopf integral equations directly. Consider

where

Define

where α = σ+iτ, and F+(α) is regular for τ>q; K(α) is regular and non-zero in −p < τ < p. For simplicity we restrict ourselves to the case where

The purpose of this note is to extend the results of Reilly and Scheiblich (6) (see also Scheiblich (7) and Hall (2)) on the θ-class decomposition of the congruence lattice of a regular semigroup and, at the same time, to provide an alternative proof of these results.