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.

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 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.

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

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.

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.

One of the intriguing complications of constructive analysis is that it does not seem possible to obtain examples of inseparable Banach spaces; more precisely, when one comes to study explicit examples one finds that one cannot assign a constructively denned norm to every vector in the space. One of the objectives of (2) was to make a detailed study of an explicit class of Banach spaces, containing both separable and inseparable examples, in the hope that one might begin better to understand these matters. We chose for this study the class of Orlicz spaces, primarily because the simplest examples (namely the Lebesgue spaces LP) had already been the objective of detailed constructive work in (1).

Recently J. Roe(1) established the striking characterization of the sine and cosine functions given in Theorem 1 (below). His proof was an ingenious application of Fourier transforms of generalized functions.

Let us say that a set points in a Euclidean space has property (2) if no 2 points X, Y ∈ S have X Y > 1. Then an easy observation is the following:

The maximum area (Lebesgue measure) of a plane set S with property(2) is ¼π, attained only when S is (essentially) a disc.

In the projective plane, if H is a harmonic homology (linear transformation with H2 = I), and G a general inversion (quadratic transformation projectively equivalent to an inversion), then under a certain condition there is a pencil of cubics each of which is invariant under G, H separately. These are related to transformations discovered by Mandel, Todd and Lyness. As a near converse, we find that, given a Pascal configuration, there is a quadratic Cremona transformation under which each cubic passing through the vertices of the configuration is invariant. As a by-product, parametric expressions are found for elliptic functions of a fifth of a period.

The rotation number ρ of an orientation preserving homeomorphism of a circle was originally defined by Poincaré It measures the asymptotic rotation of the orbit of each point under iterations of the homeomorphism (see, for example, (2)).

1. Let F be a closed, connected, orientable surface of genus g ≥ 0 smoothly embedded in S4, and let π denote the fundamental group π1(S4 − F). Then H2(π) is a quotient of H2(S4 − F) ≅ H1(F) ≅ Z2g. If F is unknotted, that is, if there is an ambient isotopy taking F to the standardly embedded surface of genus g in S3 ⊂ S4, then π ≅ Z, so H2(π) = 0. More generally, if F is the connected sum of an unknotted surface and some 2-sphere S, then π ≅ π1 (S4 − S), so again H2(π) = 0. The question of whether H2(π) could ever be non-zero was raised in (5), Problem 4.29, and (10), Conjecture 4.13, and answered in (7) and (1). There, surfaces are constructed with H2(π)≅ Z/2, and hence, by forming connected sums, with H2(π) ≅ (Z/2)n for any positive integer n. In fact, (1) produces tori T in S4 with H2(π) ≅ Z/2, and hence surfaces of genus g with H2(π) ≅ (Z/2)g.

A linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

In 1973, V.I.Lomonosov introduced a new technique for finding invariant and hyperinvariant subspaces for certain classes of (continuous, linear) operators on complex Banach spaces. Recall that a closed subspace M of the Banach space X is called hyperinvariant for the operator T if S(M) ⊂ M for every operator S which commutes with T.

We obtain here estimates for the expectation of a restricted quadratic variation of L1 bounded (weak) martingales and positive submartingales, and for related functionals. These inequalities are then used to give a quantitative version of a recent result of D. J. Aldous and D. H. Fremlin concerning the L1 norms of expansions with respect to uniformly integrable L1-normalized martingale difference sequences. Another application relates precisely the degree of norm divergence of an L1-bounded martingale to the degree of non-uniform integrability of that martingale. A partial analogue of this result is proved for certain vector valued martingales with values in L1 which are related to the Radon–Nikodým property.

The existence of solutions is considered for equations of the form

for x ∈ H (a Hilbert space), P a compact linear operator on H; Q(x) a bounded linear operator on H and continuous in x and uniformly bounded; g(x) a continuous uniformly bounded map with range in H. Two situations are considered: Q(x) lies in a weakly compact set of operators for which (a) (I − PQ(x)) is invertible (non-resonance case) or (b) (I − P(Q(x) + λI)) is invertible for 0 < λ ≤ α (resonance case).

Consider an nth-order linear ordinary differential equation

Suppose the αj(x) are holomorphic for x ∈ S, 0 < x0 ≤ |x| < ∞, where S is an open sector of the complex plane with vertex at the origin and positive central angle not exceeding π. Suppose as x → ∞ in each closed subsector of S. The problem of finding a basis for the solutions of (1) can be reduced by a sequence of transformations (see, for example, Gilbert (1)) to the problem of finding a fundamental matrix for a system of the form

where q is a non-negative integer, A(x) is an m by m matrix which is holomorphic for x ∈ S, x0 < ≤ |x| < ∞, and as x → ∞ in each closed subsector of S. If m ≥ 2, A0 is an m by m matrix with elements all zero except for l's on the upper off-diagonal, and the elements of Ar for r ≥ 1 are all zero except possibly in the last row. (There is one such problem (2) for each root, μ of the equation with multiplicity m ≥ 2. Recall that the coefficient αn−r, 0 the first term in the asymptotic expansion of the coefficint αn−r(x) of equation (1).) We denote the elements of the last row of Ar by 1 ≤ k ≤ m. The system (2) has an irregular singular point at infinity.

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.

The Green's functions for semi-infinite and adjacent semi-infinite media are considered in one-speed transport theory. Chandrasekhar's Principles of Invariance are used to formulate integral equations that describe the emergent and interface angular distributions. It is shown that mathematically these are equivalent to subtracting from the transport equation known solutions to related, simpler problems. Duplication of effort is avoided in this way. For adjacent semi-infinite media, a difficulty encountered in the solutions of the integral equations necessitates the application of a Laplace transform for its resolution. Some virtues of using the transform are explored, e.g. in the determination of the internal distribution where it yields concise expressions for the Green's function.