The so-called AF + BΦ theorem of Max Noether refers to a whole collection of results. In most of these, the object is to determine conditions under which a form H(x, y, z) will belong to the ideal generated by two given forms F(x, y, z) and Φ(x, y, z). This problem is of obvious importance in the theory of plane curves, and therefore it is customary to state the conditions in geometric language. To give one example, a particularly important form of the theorem may be stated as follows.*

Second-order perturbation theory, in conjunction with the usual treatment of electron-phonon interaction, allows us to calculate the correction to the first-order result for the scattering of phonons by electrons. It is shown that the second-order term is much smaller, and therefore negligible. This justifies the use of the first-order theory in the treatment of the interaction in metals and semi-conductors.

In this paper we apply pointwise subsequence ergodic theory to show that certain sequences of integers constructed randomly are multiply intersective. Analogous results for sequences constructed from Hardy fields are also indicated.

Criteria for singularity and absolute continuity analogous to Zygmund's for Riesz products, are demonstrated for those generalized products introduced by Modica and Mortola.

Let unity and the irrational components ri of a d-dimensional vector r be linearly independent, and consider the integers n for which

where {x} denotes the fractional part of x. It is a trivial restriction to assume that ri and φi lie between 0 and 1. This note is concerned with the question: what are the gaps between the successive n?

1. The object of this note is to deduce some self-reciprocal functions and kernels for Hankel transforms. We denote by Rμ the class of functions which are self-reciprocal for Hankel transforms of order μ; that is, those that satisfy

where Jμ(x) is a Bessel function of order μ. In particular we write R8 for R½ and Rc for R−½.

In [R2] we showed how elementary considerations involving geometry on ruled surfaces may be used to obtain recursive enumerative formulae for rational plane curves. Here we show how similar considerations may be used to obtain further enumerative formulae, as follow. First some notation. As usual we denote by Ngd the number of irreducible plane curves of degree d and genus g through 3d+g−1 general points. Also, we denote by Ngd→ (resp. Ngd×) the number of such curves passing through general points A1, …, A3d+g−2 and having a given tangent direction (resp. a node) at A1. As is well known and easy to see, we have

formula here

For any d, g, these numbers are computed in [R1] as part of a more general recursive procedure. For N0d, N1d, relatively simple recursions have been given by Kontsevich–Manin (see [FP]) and Eguchi–Hori–Xiong–Getzler (see [P1]), respectively.

A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic K-theory of such manifolds:

Conjecture. Let Γ be the fundamental group of a closed aspherical manifold. Then Whi(Γ) = 0 for i ≥ 0 where Whi(Γ) is the i-th higher Whitehead group of Γ.

Let G be a nilpotent, connected and simply connected real Lie group and let Γ be a lattice in G. A necessary and sufficient condition is given for a Lie subgroup of G to act ergodically on Γ[setmn ]G. As an application, we characterize the irreducible unitary representations of G for which the restriction to Γ remains irreducible.

The forced torsional motion of a semi-infinite isotropic right-circular elastic cylinder is studied in this paper, the vibrations being excited by a harmonically oscillating rigid disc, securely attached to the plane face of the cylinder. Two distinctsituations are considered, namely, when the curved surface of the cylinder is clamped and when it is stress free. In both cases the problem is reduced to the solution of a Fredholm integral equation of the second kind which can be solved iteratively when the frequency of oscillation is small and the cylinder radius large. Using the iterative solutions, approximate expressions are deduced for the distribution of shear stress under the disc and the driving couple required. In an appendix a discussion is given of a mathematically similar problem in the theory of the swirling flow of a perfect fluid.

In the last two decades a family of closely related functions have been introduced for the solution of quantum-mechanical problems. In dealing with atomic and molecular systems Pauling (12) introduced in 1928 linear combinations of the eigenfunctions of a single atom, which were called hybrids. These have specific symmetry properties which were utilized by Kimball (6) to construct a group-theoretical method for their determination. More recently Lennard-Jones (8) defined linear combinations of molecular eigenfunctions which he called equivalent orbitals. These are formally similar to the hybrids and Lennard-Jones took over the Kimball method for their determination.

We show that if μ is a measure on the LCA group G whose Gelfand transform vanishes off the set Σ of symmetric maximal ideals, then μ µ M0(G), that is, then the Fourier-Steiltjes transform of μ vanishes at infinity. This result is then used to show μ µ L1(G)½.

As is known, on a locally compact group G, the mere assumption of pointwise convergence of a sequence (n) of continuous positive definite functions implies uniform convergence of (n) to on compact subsets of G. This result was first proved in 1947 by Raikov8 (and independently by Yoshizawa9). An interesting discussion of the relationship between such theorems and various Cramr-Lvy theorems of the 1920s and 1930s, concerning the Central Limit Problem of probability, is given by McKennon(7, p. 62).

We have called a linear partial differential operator P almost hypoelliptic in some of the variables if the distribution u is strongly regular in those variables in every open subset where this is true of Pu. In an earlier paper (l) we have characterised almost hypoelliptic operators with constant coefficients. Following techniques of Hörmander (2) and Yoshikawa(5) we obtain in this paper some a priori inequalities for almost hypoelliptic operators and derive results about the solvability of the adjoint operator P*. Our notation for various standard distribution spaces and multi-indices follows Hörmander (2).

Volume 98 (1985), 403–412

‘On the mobility of bodies in ℝn’

The proof of the following theorem which I gave in the above paper is incorrect. A correct proof is given here.

Theorem 2·1. Let ℬ be an intersection regular, finite collection of bodies, and ℱ a manoeuvre on ℬ. ℱ is strongly separating if and only if every component of S(ℬ, ℱ) is positive; and, if any S(i, j, p) (ℬ, ℱ) is negative, ℱ is colliding.

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

Our previous discussion ((4), (5)) is here extended to an ellipsoid of arbitrary conductivity. Again, the flow is ultimately cylindrical and the magnetic field undisturbed. An ellipsoidal insulator allows fluid to pass through its circumscribing cylinder, but also forces some to go around. An ellipsoidal conductor behaves like a very tall insulator, irrespective of its own height or conductivity: all the fluid goes round. If it is very wide an unexpectedly large amount of fluid is carried with it.

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and let

The purpose of this note is to prove that the Jacobian of these systems is given by the equivalence

where Kh is Eger's operator, defined as follows:

A condition of frequency domain type on the vector polynomial G(z) is obtained which is both necessary and sufficient for the Hurwitzian set of G(z) to include the special Hurwitzian set , where E(z) = col (1, z, …, zn). This result is extended to Hurwitzian sets arising from scalar delay-differential equations. Analogous conditions are also given for each of the inclusions where denotes the Schur set of G(z).

An action of a group, G, on a surface, F, consists of a homomorphism

ø: G → Homeo (F).

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.