The purpose of this paper is to investigate the locally nilpotent radical of a Lie algebra L over a field of characteristic zero, its behaviour under derivations of L, and its behaviour with regard to finite-dimensional nilpotent subinvariant and ascendant subalgebras of L.

An explicit formula is derived for exp (iβJz) as a finite sum of irreducible tensor components. With this formula, a technique is developed to obtain the matrix elements of exp (iβJy).

Let Q (x1, …, xn) be a real indefinite quadratic form in n-variables x1,…, xn with signature (r, s),r + s = n and determinant D ≠ 0. Then it is known (see Blaney (2)) that there exists constant Cr, s depending only on r, s such that given any real numbers c1, …,cn we can find integers x1, …, xn satisfying

A well-known theorem of Minkowski on the product of two linear forms states that if

are two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that

Let Ωn be the set of all n-square doubly stochastic matrices, i.e. matrices with non-negative entries all of whose row sums and column sums are 1. If A = (aij) is an n-square matrix let per (A) denote the permanent of A, that is.

Suppose that E is a compact subset of a topological linear space ℒ. Then the convex kernel K, of E, is such that a point k belongs to K if every point of E can be seen, via E, from k. Valentine (l) has asked for conditions on E which ensure that the convex kernel K, of E, consists of exactly one point, and in this note we give such a condition. If A, B, C are three subsets of E, we use (A, B, C) to denote the set of those points of E, which can be seen, via E, from a triad of points a, b, c, with a ∈ A, b ∈ B, c ∈ C. We shall say that E has the property if, whenever A is a line segment and B, C are points of E which are not collinear with any point of A, the set (A, B, C) has linear dimension of at most one, and degenerates to a single point whenever A is a point.

It is well known that if M is a vector bundle over a space Y, and r: X → Y is a map, then there is an ‘induced’ vector bundle r*M over X. It is also known that a vector bundle L over X determines a ‘direct image’ vector bundle r* L over Y provided that r is a finite covering map ((1), § 1).

We extend the notion of topological degree to a suitable class of continuous mappings of a locally convex Hausdorff topological vector space into itself which are not completely continuous. This extension and the resulting existence theorems for non-linear equations are further extended to product spaces. Our extension is developed with application to quantum field theory in mind; this application is made elsewhere.

The nth symmetric product of a topological space, X, is defined to be the quotient of the Cartesian product Xn by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S1, and to show that they are manifolds with boundary.

If L is a subcomplex of a simplicial complex K, we say that L is obtained from K by an elementary simplical collapse if K − L consists of a simplex σ of some dimension d together with one ‘free’ face of σ, i.e. a face τ of dimension d − 1 which is a face of no other simplex of K except σ. Such a collapse is said to take place through σ from τ. If L can be obtained from K by a finite sequence of elementary simplicial collapses we say that K simplicially collapses (s-collapses) onto L, denoted by K ↘ L

If K is regarded as being embedded in some Euclidean space we shall for convenience of notation fail to distinguish between K and its underlying polyhedron.

The purpose of this paper is to study the mapping properties of certain operators which depend on a variable parameter. We prove a general theorem which is applied to some special cases. Among results obtained are new inequalities involving the Poisson operator and its conjugate.

Mr Denis Lichtman has kindly pointed out to us that an added hypothesis is needed in Theorem (4·1). The revised theorem follows.

Let Q1,…, Q5 be five fixed points (no four coplanar) of the real projective space S3: let s be a variable quadric surface through these points. The set of all such quadrics can be represented by the points of a real S4, in which there is a quartic primal that represents cones. The geometry of this threefold is well known in the complex case, but has hardly been considered at all in the real case: and one object of the present paper is to describe the real threefold and determine its homology groups.

A composite number n is called a pseudoprime if n|2n− 2.

Theorem 1. If a and b are natural numbers such that (a, b) = 1, then there exist infinitely many pseudoprimes of the form ax + b, where x is a natural number.

The proof of this theorem is given by the author in (5). This proof is based on the following two lemmas.

Ingham (3) discusses the following summation method:

A series ∑an will be said to be summable to s if

where, as usual, [x] indicates the greatest integer ≤ x. (An equivalent method was introduced somewhat earlier by Wintner (8), but the notation (I) for the above method and the attachment to Ingham's name seem to have become usual following [(1), Appendix IV].) The method (I) is intimately connected with the prime number theorem and the fact that the Riemann zeta-function ζ(s) has no zeros on the line σ = 1. Ingham proved, among other results, that (I) is not comparable with convergence but, nevertheless, for every δ > 0, (I) ⇒ (C, δ) and for every δ, 0 < δ < 1, (C, −δ) ⇒ (I), where the (C, k) are Cesàro means of order k.

The series is said to be summable (L) to sif

where sn = a0 + a1 + … + an, and it is said to be summable (l) to s if

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us set

The sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformation

The series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, if

and is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).

By introducing complex variable techniques in the theory of shallow shells, the problem is reduced to the solution of one fourth-order differential equation, and the displacement components and stress resultants and couples are expressed in terms of four complex potentials. Two of these complex potentials are those of thin plate theory, thus admitting of deducing the solution of a shallow shell problem from the limiting case of the circular flat plate. These results are applied to a case of non-rotational symmetry, a special case of which has practical significance in the design of tanks and boilers.

1. Making use of the familiar abbreviation

let us adopt a contracted notation for the generalized hypergeometric function AFB[x] and write

where (a) denotes the sequence of parameters

In this paper we obtain generating functions of Jacobi polynomials. The results obtained are of general character and include as particular cases of the results given earlier by Bailey ((1), p. 102, Ex. 19) and Brafman (2).