In answer to a question posed by J. L Britton in (2), we sketch in this note an effective procedure to decide whether an arbitrary quadratic equation

with rational coefficients, has a solution in integers. A similar procedure will in fact decide whether such an equation over a (suitably specified) algebraic number field k has a solution in any (suitably specified) order in k; but we shall not burden the exposition by giving chapter and verse for this claim.

We begin by describing a concrete example from the class of problems to be considered. A continuous bivariate function f defined on the square |t| ≤ 1, |s| ≤ 1 is to be approximated by a tensor-product form involving univariate functions. For example, the approximation may be prescribed to have the form

in which the Ti are the Tchebycheff polynomials, and the coefficient functions xi(t) and yi(s) are to be chosen to achieve a good or best approximation. Will a best approximation exist? If so, how can it be obtained?

Lagrangian theory (2) and elementary catastrophe theory (5) coincide locally but differ globally. Locally the singularities that occur in the two theories are the same, but globally the manifolds that occur are different. We shall show that Lagrangian manifolds are more general.

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfies

then Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

In 1933(1), Mazur and Orlicz stated that, if a conservative (i.e. convergence preserving) matrix sums a bounded nonconvergent sequence, then it must sum an unbounded sequence. Their proof of this result (2) was one of the early applications of functional analysis to summability theory, and it is based on rather deep topological properties of F K-spaces. In (3) Zeller obtained a proof of this important theorem as a consequence of his study of the summability of slowly oscillating sequences. The purpose of this note is to give a simple direct proof of this theorem using only the well-known Silverman-Töplitz conditions for regularity. In order to reduce the details of the argument, we state and prove the result for regular matrices rather than conservative matrices.

Let U denote the variety of pencils of binary cubics without base point defined over an algebraically closed field k whose characteristic is not equal to 2 or 3. (For full details of the terminology, see § 2.) There is a natural action of SL(2) on U; our main object is to show that U possesses a good quotient for this action in the sense of ((7), Definition 1·5) or ((4), p. 70), and to identify this quotient with the affine line over k. In fact, we prove

Theorem 1. There exists a morphism φ: such that (, φ) is a good quotient of U by SL(2). Moreover, all but one of the fibres of φ are orbits; the exceptional fibre consists of two orbits corresponding to pencils with one or two triple points.

In this note we shall show that the conditions given by G. Torres in (7) do not suffice to characterize the first Alexander polynomial of a link. We shall recall below Torres' results (for the 2-component case) and state the result of J. H. Bailey on which our argument relies, and then prove the following theorem.

Let m be the set of all real sequences x = (xn) with norm . A linear functional L on m is said to be a Banach limit (see Banach(1), p. 32) if it has the following properties:

(i) L(x) ≥ 0, if x ≥ 0 (i.e. xn ≥ 0, for all n ∈ N)

(ii) L{e) = 1, where e = (1, 1, 1,…),

(iii) L(Sx) = L(x), where (Sx)n = xn+1.

We present the following sequence of polynomial identities:

is the Gaussian polynomial denned to be zero for m < 0 or m > N, one for m = 0 or N and

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequality

has a solution in integers x1, x2, …, xr.

Y. Morita proved that, for each prime number p, one can define a p-adic continuous function Γp(x) from p to p, interpolating the sequence

where m runs through the integers m prime to p with 1 ≤ m < n. Our aim is to show how this result is related to Dwork's result on the radius of convergence of

Hardy (3) provides one of the earliest rigorous discussions of the convergence of Riemann multiple integrals of real functions with a general type of algebraic singularity. More recently, and with the advent of Lebesgue integration, various authors such as Garsia(2) have investigated singular integrals involving more general measurable functions. In this paper we shall develop an indirect potential theoretic method to obtain convergence criteria for a class of multiple integral that seems to have been neglected to date.

This note deals with some relations between unmixed local rings and the notion of generalized analytic independence.

In this paper we consider the following problem, which seems to have been brought to light fairly recently by M. Car. Can every sufficiently large integer n be expressed as n = p + ab with p prime and 1 ≤ a, b ≤ n½? Certainly one should expect this to be possible. Taking b = 1, for example, p will be restricted to the range n − n½ ≤ p < n, and this interval is conjectured to contain a prime, for large enough n. Alternatively, providing that n is not a square, we expect n = p + a2 to be solvable for sufficiently large n. However, although the statement that n = p + ab, with a, b ≤ n½, is far weaker than either of the aforementioned conjectures, it is nevertheless rather tricky to show that solutions must in fact exist.

The noncommutative Lp-spaces (1 ≤ p ≤ ∞) of unbounded operators associated with a regular gauge space (a von Neumann algebra equipped with a faithful normal semifinite trace) are studied by many authors ((4), (5) and (7)). It is well-known that the noncommutative Lp-spaces (1 ≤ P < ∞) are Banach spaces and the dual of Lp is Lq (1 ≤ p < ∞, 1/p + 1/q = 1) by means of a Radon-Nikodym theorem.

Wiman, in 1895, found ((5), p. 208) an equation for a 4-nodal plane sextic W that admits a group S of 120 Cremona self-transformations; of these, 24 are projectivities, the other 96 quadratic transformations. S is isomorphic to the symmetric group of degree 5 and Wiman emphasizes that S does permute among themselves 5 pencils (4 pencils of lines and 1 of conics) and 5 nets (4 nets of conics and 1 of lines). But he gives no geometrical properties of W. The omission should be repaired because, as will be explained below, W can be uniquely determined by elementary geometrical conditions. Furthermore: W is only one, though admittedly the most interesting, of a whole pencil P of 4-nodal sextics; every member of P is invariant under S+, the icosahedral subgroup of index 2 in S, while the transformations in the coset S/S+ transpose the members of P in pairs save for two that they leave fixed, W being one of these. When the triangle of reference is the diagonal point triangle of the quadrangle of its nodes the form of W is (7.2) below. Wiman referred his curve to a different triangle.

The main results of this paper show that for any odd primes p and q with q dividing p2 − 1 there exist non-associative Bol loops of order pq. The construction is sufficiently flexible so as to allow the resulting loops to be either Bruck loops or non-Bruck loops. It also permits the determination of isotopy classes among the loops constructed. All Bol loops of order 15 are determined.

Let I be a set and let H(i) (i ∈ I) be non-trivial groups. If J is a subset of I, we denote the free product of the H(j) (j∈J) by H(J). We denote H(I) simply by H. Let R be a cyclically reduced element of Hof length at least two, and let

Let μ: H → G be the natural homomorphism. If J is a subset of I such that R ∉ H(J), we call H(J) a Magnus subgroup, or occasionally the J-Magnus subgroup (of H with respect to R). We will say that the Freiheitssatz holds if μ| M is an injection for each Magnus subgroup M. Magnus (4) showed that the Freiheitssatz holds if the H(i) are free, and this was extended by Pride (5) to the case where the H(i) are locally fully residually free. In this paper we prove Theorem 1. The Freiheitssatz holds if the H(i) are locally residually free.

The aim of this paper is to exhibit a connection between certain types of envelope and the discriminant sets of function singularities. We show how conditions that the envelope has a certain local structure (which arise from the singularity theory) often have pleasant geometric interpretations. Moreover in many cases one can show that these conditions are generically (nearly always) satisfied.

The structure of modules over polycyclic group rings RG is investigated using the idea of a link P ⇝ Q between prime ideals of RG. The representation theory of RG splits into two parts – the part we discuss is determined by the representation theory of certain Noetherian polynomial identity factor rings of subrings of RG.