A commutant property of ternary algebras, defined as linear subspaces of ℬ(ℋ) (bounded operators on some Hubert space ℋ) closed under three-fold multiplication, is demonstrated through the recovery of an operator from its non- orthogonal restrictions.

We denote by F the field R of real numbers, the field C of complex numbers, or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be Hermitian if A = A* and unitary if AA* = In, where In is the n × n identity matrix. An n × n Hermitian matrix A is said to be positive definite (postive semi-definite resp.) if uAu* > 0(uAu* ≥ 0 resp.) for all u (╪ 0) in Fn. Here and in what follows we regard u as a 1 × n matrix and identify a 1 × 1 matrix with its single element. In the following we shall always use A and B to denote two n×n Hermitian matrices with elements in F, and we say that A and B can be diagonalized simultaneously if there exists an n×n non-singular matrix V with elements in F such that VAV* and VBV* are diagonal matrices. We shall use diag {A1, A2} to denote a diagonal block matrix with the square matrices A1 and A2 lying on its diagonal.

1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set

and edge-set E(G(n, k)) to consist of all edges of the form

where i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)

0. Introduction. In (8, 9) certain higher order operations, called K-theory Massey Products, were introduced and developed. These operations were designed to investigate the Kunneth formula spectral sequence in equivariant K-theory, constructed in (4). In that application the important feature of Massey products was that they gave operations on certain Tor-algebras which were well-behaved with respect to the algebraic coboundary, δa.

1. Introduction: In all that follows, by a manifold we shall mean a paracompact, Hausdorff, locally Euclidean topological space. By means of the universal covering construction, the classification problem for connected manifolds splits into two parts, namely (i) classification of simply connected manifolds, (ii) classification of covering actions of groups on simply connected manifolds.

We give a new criterion for an order in a commutative split algebra over the quotient field of a discrete valuation ring R to be self-dual. We derive a result on the ideal classes whose order is an extension by R of a self-dual order. This yields a wide generalization of a theorem of Faddeev(1). We can also answer a question arising from a paper of Fröhlich(2).

Let G be a compact abelian group and let Ĝ be its dual group. We shall denote by M(G) the set of all bounded complex valued Radon measures on G and by M(E) the elements of M(G) with support in a compact subset E of G.

If G is a finite group of order N and ΓN is the Galois group of Q(w) over Q, where w is a primitive Nth root of unity then ΓN acts on the complex representation ring, R(G), of G. The group of co-invariants is denoted by R(G)ΓN = R(G)/W(G).

Let X be a finite CW-complex, and ξ: Eξ → X a complex line-bundle on X. This bundle determines an element c1(ξ) in H2(X:Z) and a class ; let m(ξ), a(ξ) be the orders (possibly infinite) of these elements in their respective groups. How are m(ξ) and a(ξ) related? The answer is contained in the following three propositions.

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the types

and their duals

hold, where pn(x) and qn(x) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e−pxLn(x).

It is proved that every locally convex space is linearly homeomorphic to a subspace of a space which admits no non-trivial continuous linear functionals.

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

In this paper we shall establish some forms of the closed graph theorem for locally convex spaces, using the approach of Pták(17). Our interest is in classifying pairs of locally convex spaces (E, F) which have the property that every closed graph linear mapping T: E → F is continuous; if (E, F) has this property then we shall say that (E, F) is in the class ℒ if is a particular class of locally convex spaces then ℒ() is the class of all E such that (E, F)∈ℒ for all F ∈ .

The objective of this note is to complete in one essential way the study undertaken in (2) of the relation between complex bordism and the connective k-homology theory. Specifically, let us denote by MU*( ) the generalized homology theory associated to the Thorn spectrum MU(6), and by k*( ) the generalized homology theory associated to the connective bu spectrum (2, 4). Recall that

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we mean

and let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfying

For such kernels, the iterated kernels,

are well-defined (1), as are the higher order traces

Carleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinant

the latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

Let G be a locally compact Abelian group.

DEFINITION 1. A compact subset K ⊂ G is called Kroneclcer set if for every continuous function f on K of modulus identically one (|f(x)| = 1, ∀x ∈ K) and for every ε 0 there exists x ∈ Ĝ such that

Throughout this paper a0, a1, a2, l1, l2 denote fixed integers with l1 ≥ 2, l2 ≥ 2. We let l = max(l1, l2) and let P be the set of primes Mordell (4) has shown that for any sufficiently large prime p the congruence

is soluble. Thus there are at most a finite number of such p for which (1·1) is insoluble. If there is at least one prime p ∈ P for which (1·1) is insoluble, we let p0 denote the largest of such p, so that (1·1) is soluble for all p ∈ P with p > p0 but not for p = p0. Otherwise (1·1) is soluble for p ∈ P and we let p0 = 1. From the work of Mordell (4) we have

For p ∈ P with p > p0 (1·1) is thus always soluble and any such solution (x1, x2) can be taken to satisfy

Chalk(2) has posed the problem of estimating a ‘small’ solution of (1·1), at least for p sufficiently large; that is a solution for which p in the inequality (1·3) can be replaced by something less than p. Smith(5) has shown that for p sufficiently large there is always a solution satisfying 1 ≤ xi ≪ p¾log p (i = 1, 2). It is the purpose of this paper to prove the following sharper and more precise result.

Given a polynomial a(λ) with degree n, and polynomials b1(λ), …, bm(λ) of degree not greater than n – 1, then the degree k of the greatest common divisor of the polynomials is equal to the rank defect of the matrix R = [b1(A), b2(A), …, bm(A)], where A is a suitable companion matrix of a(λ). Furthermore, it is shown that if the first k rows of R are expressed as linear combinations of the remaining n – k rows (which are linearly independent) then the greatest common divisor is given by the coefficients of row k + 1 in these expressions. A simple expression is derived for R and a permutation of the columns of this matrix establishes a direct connexion with controllability of a constant linear control system. Finally, when m = 1 a relationship between the corresponding R and Sylvester's matrix is exhibited.

The Liénard–Chipart criterion for determining whether all the zeros of a real nth degree polynomial a(λ) have negative real parts involves calculation of only about half the Hurwitz determinants, the order of the largest being n − 1. It is shown, by using the companion matrix of a polynomial formed from a(λ), that the required determinants are equal to minors of a matrix whose order is ½ n or ½(n−1) according as n is even or odd. In either case this matrix is very easy to obtain A simple application of the result provides a criterion for a(λ) to be stable and aperiodic.

Let {sn} be the sequence of partial sums of the series and let

If , we say that the series is summable (l) to s, and if , the series is said to be summable (l2) to s. It is clear that .