The equations satisfied by perturbations of solutions of the Euler-Lagrange equations corresponding to a Lagrangian density L are derived from the first and second variations of L and these new Lagrangian densities are used to construct Noether and Euler identities. It is shown how these identities can be used in general relativity to construct conserved quantities and local uniqueness theorems.

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

In the expression of Taylor's theorem

there are three well-known forms of the remainder R(n, a, b). The first is the Schlömilch form

It is shown that the analogue of the Helson–Reiter theorem for ideals in group algebras of locally compact abelian groups holds for a class of groups including all divisible nilpotent discrete groups.

In this paper we shall consider differential operators that admit a periodic structure (cf. [1, 2])

(formula here),

where

(formula here),

are Hölder continuous for some α > 0 and periodic, so that, say, without loss of generality, we have

(formula here).

We shall assume that Q is elliptic, i.e. that the matrix A = (aij) satisfies

(formula here),

for some

(formula here).

The simplest definition of transversality in the PL or Top categories is the purely local one: Manifolds M and N are transverse in Q if, for each point x of their intersection, there is a (closed) neighbourhood U of x in Q and a (PL) homeomorphism

for suitable p, q, and r

1. Let P = 0, Q = 0, U = 0, V = 0 represent four independent quadric surfaces, and consider the equation

The wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.

The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.

1. In a former paper we solved the following problem (Problem I). Given k + 1 real numbers co, …, ck, to find necessary and sufficient conditions that there shall exist a function f(x) in (0, 1) which satisfies the conditions

It is a classical and well-established result that a unirational algebraic variety of dimension ≪ 2 is necessarily rational. It is also generally agreed, though not so well established, that this result is no longer true for varieties of dimension three. In this connexion, the critical example has been the Enriques threefold E, asserted by Roth (1) to be unirational but not rational. The purpose of this note is to point out a fallacy in the proof of the non-rationality of E and, incidentally, to resolve a difficulty raised by Serre (2) in which E would appear to feature as a counter-example to a well-established general theorem. As far as the author is aware, there are no other examples of nonrational unirational threefolds, so that the question of their existence is still open.

It is known, (see [7], theorem 4 or [8], corollary II, 10.1), that a positive generated, ordered Banach space X is order-isomorphic to l1 iff X has a Schauder basis which generates its positive cone X+ and X+ has a bounded base.

Let A be a strictly increasing sequence of integers exceeding 1 and let

denote its set of multiples. We say that A is a Behrend sequence if M(A) has asymptotic density 1. The theory of sets of multiples was first developed in the late thirties, under the influence of Erdős, Besicovitch, and others. An account of the classical notions is presented in the book of Halberstam and Roth (1966), and recent progress in the area may be found in Hall and Tenenbaum [7], Erdős and Tenenbaum [4], Ruzsa and Tenenbaum [9]. As underlined by Erdős in [3], one of the central problems in the field is that of finding general criteria to decide whether a given sequence A is Behrend.

The motion of a pendulum on a periodically accelerated support has been studied by means of an electronic computing machine. The results have been applied to the problem of calculating the second-order correction to measurements of gravity made at sea with the Vening Meinesz three-pendulum apparatus. It is found that the correction for a single horizontal acceleration under typical conditions is likely to be more than twice that found analytically by Browne. His correction for vertical acceleration is valid for a wide range of frequency of the disturbing motion.

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.

Every nonsingular projective real algebraic curve C has a unique, up to isomorphism over ℝ, nonsingular projective complexification V. If V\C is disconnected, then C is said to be dividing (we identify C with the set of real points of V). Classical results supply numerous examples of dividing real curves. This class of curves was first studied by Felix Klein [11]. A characterization of dividing real curves is given in [6].

In higher dimensions the situation is more complicated. First of all, every nonsingular projective real algebraic variety X of dimension d always has several nonisomorphic nonsingular projective complexifications, provided that d[ges ]2. Furthermore, if d[ges ]2 and W is a nonsingular projective complexification of X, then W\X is connected (we identify X with the set of real points of W). What does it then mean for X to be dividing? An answer can be given in terms of homology theory. Let K be a principal ideal domain. Assume that X, regarded as a topological manifold, is orientable over K. We say that X is dividing over K if there exists a fundamental homology class of X over K whose image in Hd(W, K) under the homomorphism induced by the inclusion map X[rarrhk ]W is zero. In the present paper we prove that this definition does not depend on the choice of W, and give a characterization of real varieties dividing over ℚ. For more information on real varieties dividing over ℤ/2 the reader may consult [18] (please note that terminology in [18] is different than here). We also discuss the relationship between real varieties dividing over ℚ (or ℤ) and dividing over ℤ/2. It follows from well-known facts that a real curve is dividing over K if and only if it is dividing.

It is well known that an n-dimensional crystallographic group can be reconstructed from its point group, the integral representation of the point group which arises from its action on the translation lattice, and the 2-cocycle which glues the point group to the lattice ([2]). In practice, this constitutes a complicated list of invariants. When confronted with the classification of objects possessing a rich structure, the algebraic geometer first attempts to find more coarse birational invariants. We begin such a programme for torsion-free crystallographic groups. More precisely, if Γ is a torsion-free crystallographic group and k is a field then the group algebra k[Γ] is a non-commutative domain (see [6], chapter 13). It can be localized at its centre to yield a division algebra k(Γ) which is a crossed product; the Galois group is the point group and it acts on the rational function field generated by k and the lattice (regarded multiplicatively), which is a maximal subfield ([3]). What are thecommon invariants of Γ1 and Γ2 when k(Γ1) and k(Γ2) are isomorphic k-algebras?

The work of Heitler and London has shown that the first order perturbation energies of two atoms taken as a function of the separation between the nuclei gives a fairly good description of the behaviour of the atoms with regard to possible molecular formation. For example, the dissociation energy of the hydrogen molecule calculated in this way is of the same order of magnitude as the accepted observed value, being in fact about 25% too small. The theory has been generalised by Heitler, and also by London and others, to the case where each atom contains many electrons, but detailed calculations do not appear to have been made for systems more complex than that of two hydrogen atoms.

In spite of a large amount of general theory, it remains a surprising fact that the number of known irregular surfaces in space of three dimensions is quite small. Moreover, although the evaluation of the arithmetical genus pα is comparatively direct, it is often a matter of considerable difficulty to calculate the geometrical genus pg. This paper deals with a number of surfaces for which the equations of the canonical system can be written down and their number counted. An extension to primals in higher space is also suggested for the simplest case.