Introduction and Notation. In this paper all the scalars are real and all matrices are, if not stated to be otherwise, p-rowed square matrices. The diagonal and superdiagonal elements of a symmetric matrix, and the superdiagonal elements of a skew-symmetric matrix, will be called the distinct elements of the respective matrices. Σ will denote both the set of all symmetric matrices and the ½p(p + 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. K will denote both the set of all skew-symmetric matrices and the ½p(p – 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. Any sub-set of Σ(K) will mean both the sub-set of symmetric (skew-symmetric) matrices and the set of points of Σ(K). Any point function defined in Σ(K) will be written as a function of a symmetric (skew-symmetric) matrix. Dα will denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

In two-dimensional discontinuous fluid motion one point of considerable importance has not hitherto been given sufficient attention. I raise it formally in a paper to be published soon (Fluid Motion past Circular Barriers, Scripta Universitatis atque Bibliothecae Heirosolymitanarum, 1923, Vol. I., XI., 1–14) in the following manner. Given the form of the barrier by means of, say, the radius of curvature in terms of the angle of contingence, how does the solution take into account the angular extent of the barrier? Clearly barriers which are defined by the same curve, but differ in the extent of curve used, must necessarily give rise to different solutions. Further, there must be a limiting extent of barrier, so that if it extends beyond this limit the part of the barrier in excess must lie in the “dead” fluid.

In this paper we show that away from umbilic points certain measures of the local reflectional symmetry of a surface in Euclidean 3-space are detected by the extrema of the sectional curvatures along lines of curvature. There are two types of reflectional symmetry, with one detected by the contact between the surface and spheres, and in this case the result is due to Porteous and is 20 years old. We show that an analogous result remains true for the second type of symmetry.

If an oriented manifold M immerses in codimension k, then the normal bundle has dimension k such that its Euler class χ є Hk(M; Z) and χ2 є H2k(M; Z). (Cf. (3)).

If M is the complex Grassmann manifold G2(Cn) of 2-planes in Cn (n = 4, 5,…, 15, 17), then dim M = 4n – 8 ≡ d and we shall show that although M immerses in R2d–1 by classical results (3), M does not immerse in Rd+d/2.

The same result was obtained for n = 4 and 5 by Connell (2) and for n = 6 and 7 by the author (6). The nonimmersion results of this paper are new for n = 8, 9, …, 15, 17 and they are an improvement over the result for the general G2(Cn) obtained in (5). In this paper, we use generators of the cohomology ring of G2(Cn) different from those used in (2) and (6) and this simplifies the calculations considerably.

With any twisted curve of order six is associated a system of planes, usually finite in number, which touch the curve at three distinct points. The curve with its system of tritangent planes possesses properties which recall the properties of a plane quartic curve and its system of bitangent lines; and this is specially true of the sextic which is the intersection of a cubic and a quadric surface. But whereas the properties of the plane curve were discovered by geometrical methods, such methods have only recently been applied with success to the space-curve; the earliest properties were obtained by Clebsch from his Theory of Abelian Functions. In the absence of any one place to which reference can conveniently be made, an account of these properties in their geometrical aspect will be useful.

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

In this note, we derive a necessary and sufficient condition for a flat map of (commutative) rings to be a flat epimorphism. Flat epimorphisms φ:A → B(i.e.φ is an epimorphism in the category of rings, and the ring B is flat as an A-module) have been studied by several authors in different forms. Flat epimorphisms generalize many of the results that hold for localizations with respect to a multiplicatively closed set (see, for example [6]).In a geometric formulation, D. Lazard [3, Chapitre IV, Proposition 2.5] has shown that isomorphism classes of flat epimorphisms from a ring A are in 1-1 correspondence with those subsets of Spec A such that the sheaf structure induced from the canonical sheaf structure of Spec A yields an affine scheme. N. Popescu and T. Spircu [4, Théorème 2.7] have given a characterization for a ring homomorphism to be a flat epimorphism, but our characterization, under the assumption of flatness is easier to apply. For corollaries, we can obtain known results due to D. Lazard, T. Akiba, and M. F. Jones, and generalize a geometric theorem of D. Ferrand.

Last session (1910–11) I contributed two papers investigating the Focal properties of Circular Cubics and Bi-Circular Quartics by the methods of pure geometry. I find on further investigation that not only can the properties of the individual Focal Circles be directly found by the method there given, but the mutual properties of the Focal Circles can be obtained by an extension of the same method. Several geometrical results, which as far as I know are new, are obtained incidentally in the discussion.

Let be a set of disjointly supported, positive functions in the Banach envelope of weak L1. We prove that each fi can be written as ei + gi where ei and gi, are disjointly supported and satisfy these additional properties: the ei's are isometrically the basis in the envelope norm; the envelope norm of a linear combination of the gi's is equal to the envelope norm of the corresponding combination of the fi's.

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),

defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

In studying the cohomology of the symmetric groups and its applications in topology one is led to certain questions concerning the representation rings of special subgroups of . In this note we calculate the Chern classes of the regular representation of (Z/p)n where p is a fixed odd prime in terms of certain modular invariants first described by L. E. Dickson in 1911. In a later paper [9] we apply these results to study the odd primary torsion in the PL cobordism ring. Some indications of this application are given in Sections 10–12 where we apply the result above to obtain information about the cohomology of . After circulation of this note in preprint form we learned that H. Mui [10], has also proved Theorem 6.2.