All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

We study some properties of Lipschitz exact Lagrangian manifolds isotopic to the zero section. We prove that if such a manifold is invariant under an optical Hamiltonian, then it must be a Lipschitz graph. This extends a recent result of Marie–Claude Arnaud. We also obtain a new geometric description of the Mañé–Mather invariant set.

Let J be a complex Banach space and a complex Jordan algebra equipped with an algebra involution *. Then J is a Jordan C*-algebra if the following conditions are satisfied:

(Ua is defined on page 3).

Several relations between the Cesàro and the Riemann methods of summation are known. For instance, Verblunsky(3) has shown that a series summable (C, k − δ), where k is a positive integer, is also summable (R,k+ 1) and Kuttner (2) has proved that, for k = 1, 2, summability (R, k) implies summability (C, k + δ). In this paper we consider Riesz's typical means and a generalized Riemann summability both of which are intimately connected with almost periodic functions. The result we establish is similar to Verblunsky's, except that we start from a Riesz mean of integral order k*.

The Adams operations {ψk} considered as self-maps of BU have the property that . Id2n on H2n(BU; ℚ). Sullivan in his M.I.T. notes [12] constructed ψk-type self-maps of BU(n) in the case in which k is prime to n!. Later work by Friedlander[5] and Wilkerson[14] gave constructions for all other compact connected Lie groups G, subject to the condition that k be prime to the order of the Weyl group W(G).

We prove a domination theorem for operators on Köthe function spaces by probability measures which includes both the Maurey–Rosenthal domination theorem and the Pisier domination theorem as special cases.

A consideration of Legendre's equation, of order zero, expressed in terms of the angular variable θ, leads heuristically, but methodically, to an infinite integral representation of the delta function in terms of two independent non-periodic solutions Ev−½(θ), E−v−½(θ). This result yields an integral transform and inverse formula, where in the former the kernel is E−v−½(θ) sin θ and the integration over θ runs from − ∞ to ∞. By rigorous contour integral methods, the inverse is verified when the original function is a rectangular pulse, and the standard series expansion in terms of Legendre polynomials is recovered when the original function is even and of period 2π in θ. Analogies with the theory of Fourier transforms are emphasized.

An automorphism σ(G) of a graph G is a permutation of the set of its vertices which preserves adjacency. Under the operation of composition the automorphisms of G form a group Aut(G). The graph G is called asymmetric if Aut(G) is trivial, and symmetric otherwise.

In two recent papers ((l), (2)) I investigated inclusion relations between IF (integral function) methods of summability. In the present paper the investigation is continued with the aid of standard Mellin transform theory and results due to Rogosinski and to Kuttner.

1. Introduction. The subject of this paper is the propagation of small disturbances in a compressible fluid which is also a conductor of electricity, in the presence of a magnetic field. The characteristic feature of the problem is that it combines sound waves and magneto-hydrodynamic waves. The theory is based on linearized equations which generalize the equations of motion of the theory of sound. The disturbances with which it deals might therefore be called magneto-acoustic waves. Energy dissipation by viscosity, heat conduction and Joule heat is neglected, and so is the displacement current. Even so, a hyperbolic system of seven partial differential equations of the first order is obtained, whose solution requires analysis of considerable complexity.

Let β be a braid on n strands, with exponent sum w. Let Δ be the Garside half-twist braid. We prove that the coefficient of vw−n+1 in the Homfly polynomial of the closure of β agrees with (−1)n−1 times the coefficient of vw+n2−1 in the Homfly polynomial of the closure of βΔ2. This coincidence implies that the lower Morton–Franks–Williams estimate for the v–degree of the Homfly polynomial of is sharp if and only if the upper MFW estimate is sharp for the v–degree of the Homfly polynomial of .

A method of deriving the basis functions of the double-valued representations of a point group by the reduction of Kronecker products is described. The method has been used to derive expressions for these bases for cubic point groups for which the results are tabulated.

There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.

The glow discharge between cold aluminium electrodes in air, oxygen, nitrogen and hydrogen has been analyzed by Langmuir's method, for pressures between 0·1 and 0·4 mm. Hg, current densities of from 0·02 to 0·2 mA./sq.cm., and applied potentials between 300 and 700 volts. An annular exploring electrode has been used. It has been found that whilst practically the whole fall of potential is localized across the cathode dark space at the lower pressures, a fall of as much as 40 volts can exist across the remainder of the discharge at the higher pressures. Reversal of the electric field has been found in the negative glow, and in certain cases in the Faraday dark space, when conditions are favourable for passage of an electron current by diffusion against the field. In several instances the negative glow was at a higher potential than the anode. Two groups of electrons occur in the negative glow, together with a single fast group at the anode boundary of the cathode dark space, and a single slow group in the Faraday dark space.

This paper is concerned with the symplectic bordism ring MSp*, whose structure is still largely unknown despite the partial results of Liulevicius(3), Novikov(4) and Stong(lO). It is the first of two elaborating(8), and it relies completely on (7) for notation and prerequisites. In particular we assume the reader to be familiar with (7), section 8, which establishes the relation between the Sp Hattori-Stong conjecture (that KO decides MSp) and the spectral sequence

We are here concerned entirely with computations in this spectral sequence.

In this paper we improve the estimate for the lower bound of the Hausdorff dimension of certain sets of singular n-tuples in ℝn obtained by Baker in [1]. We begin with a brief discussion of the problem. More details can be found in [1]. For ease of comparison with Baker's results we will adopt a similar notation to that of [1].

1. The purpose of this note is to prove a result which includes certain classical theorems generally thought of as being unconnected; in explicit terms, a result about the Fourier series of a periodic Lebesgue-integrable function showing that the series is summable at a point by a Nörlund method (N, pn) defined as usual ((2), p. 64) if pn ↓ 0, Σpn = ∞ and the point is in a certain subset of the Lebesgue set. More precisely, the purpose is to prove Theorem I on the Nörlund summability of Fourier series and to derive from it the well-known Theorems A, B which follow and the recent extension of Theorem A in Theorem A' which appears later and is due to Sahney (8).

This paper is concerned with geometrical properties of null hypersurfaces in Lorentzian manifolds. Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. First, the contravariant metric cannot immediately be defined, so the connection cannot be specified uniquely in the normal way. Secondly, the normal is a null vector lying in the tangent plane, which makes it necessary to look for some other vector to rig the hypersurface, and makes it impossible to normalise the normal in the usual way. These problems are considered in this paper.

This paper is devoted to a preliminary discussion of potential wells in non-linear three-dimensional elasticity. Our interest in the subject arises from the role that potential wells play in the justification of the energy criterion as a sufficient condition for stability of an elastic equilibrium solution. It will be recalled that the energy criterion, which is a simple extension of the original Lagrange-Dirichlet version for finite-dimensional systems, states that an equilibrium solution is stable provided the potential energy achieves its minimum on the solution. No proof of this statement as it applies to three-dimensional elasticity is yet forthcoming, although when the notion of a minimum is replaced by that of a potential well, several authors have proved that the criterion thus modified is sufficient for the Liapunov stability of the equilibrium solution with respect to appropriate measures. Indeed, the proofs are applicable to many other continuum theories, apart from elasticity. (See, for instance, Coleman(5), Gurtin(11) and Koiter(21).) It thus becomes important to determine what constitutive and other conditions, if any, ensure the existence of a potential well. While we present two such conditions, our main purpose is to describe examples in support of the conjecture that non-existence rather than existence of a potential well is likely to be the generic property. In these examples, particular forms of the potential energy are chosen which have positive-definite quadratic part and yet in any W1, P-neighbourhood (1 ≤ p ≤ ∞) of the origin, have a non-positive infimum, thus violating a condition for existence of a potential well in the, Sobolev space W1, P (1 ≤ p ≤ ∞). Related results are also reported by Ball, Knops and Marsden(3) and by Knops(17) for the space W1, ∞. Another conclusion to be drawn from these examples, is that a potential well cannot be ensured by restricting only the quadratic part of the potential energy. Indeed, in our example, we show that the equilibrium (null) solution is unstable in the sense that (non-linear) motions, starting in its neighbourhood, cease to exist after finite time. The same phenomenon has been shown by Knops, Levine and Payne(19) (see also Hills and Knops(14)) to hold for a general class of materials which includes as a special case one possessing the potential energy considered in our example.