In this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

Let n≥5. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in (n+2)-space. There are two inequivalent embeddings of an n-manifold in (n+2)-space such that the singular point set of the projection of each embedding consists of double points and that the number of components of the singular point set of each embedding is one.

We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g ≥ 4.

Some applications of Fourier series in the analytic theory of numbers.

Page 589, equation (3–10), after “k>0” insert “and 0<R(s)< 1,” and for “2nπi/k” read “2nπix/k.”

Page 589, equation (3·11), for

Add also ‘The evaluation of the integrals given in (3·11) is obvious when 0 < R (s) < 1, and then holds also for 0 < R (s) < 2 by the theory of analytic continuation.”

Let [Mfr ]g be the moduli space of smooth curves C of genus g[ges ]4 over the complex field [Copf ] and, as in [4], denote by [Mfr ](n)g⊂[Mfr ]g the locus of points representing curves having n vanishing theta-null, i.e. n points of order two in Sing (Θ).

I. V. Dolgachev proved the rationality of [Mfr ](1)4 (see [4, p. 13]) and of [Mfr ](2)5 (see [4, p. 11]). The aim of this paper is to prove the

Theorem. [Mfr ](1)tis rational.

The Tamm-Dancoff theory of meson-nucleon scattering (pseudoscalar coupling) leads to a series of singular integral equations. The asymptotic behaviour of the solutions of these equations is obtained here for all scattering states. For attractive states there is a critical coupling constant beyond which no normalizable solutions exist for these equations; for repulsive states another critical coupling constant appears, beyond which the solutions oscillate infinitely often but are still normalizable. It is concluded that the renormalization procedures proposed previously (3) are consistent and successfully define finite vertex-functions, and that the equations obtained for these vertex-functions have satisfactory properties for numerical work.

In this paper, the general theory of a Cosserat surface given by Green, Naghdi and Wainwright(1), has been applied to the problem of a thermo-elastic Cosserat plate containing a circular hole of radius a. We assume that the major surfaces of the plate and the boundary of the hole are thermally insulated and that a uniform temperature gradient τ exists at infinity. In the limiting case, when h/a → 0, where h is the thickness of the plate, the thermal stresses at the circular hole reduce to those obtained by Florence and Goodier (4), by means of the classical plate theory. Results for the other limiting case h/a → ∞ are also given.

A monothetic semigroup is a topological semigroup with jointly continuous multiplication which contains a dense cyclic subsemigroup. These semi-groups arise in a natural way in the study of semi-algebras. In (4) we showed that a compact monothetic semigroup in a Banach algebra can be characterized in terms of the spectral properties of a generating element. In this paper these spectral theorems are linked with the well-known structure theory of compact semigroups.

1. The reflexion of a simple harmonic train of surface waves by a rigid plane barrier inclined at an angle α to the undisturbed free surface of the water is considered in the particular case in which 2sα = π, where s is an integer. The solution if s = 2 has been given by Gourret(1), while the solution if s = 1 is well known; the solution for any integral value of s is given here.

In 1953, Calabi proved a rigidity theorem for Kählerian submanifolds in complex space forms [3]. The Calabi rigidity theorem, since then, has been successfully applied to various areas in geometry. Among them is the study of minimal surfaces in real space forms; see [4] for example.

Isomorphism classes of dihedral Artin representations of ℚ can be counted asymptotically using Siegel's asymptotic averages of class numbers of binary quadratic forms. Here we consider the analogous problem for quaternionic representations. While an asymptotic formula is out of our reach in this case, we show that the asymptotic behaviour in the two cases is quite different.

Internal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

In (7), we investigated some elements in the Adams spectral sequence for the stable homotopy groups of spheres, and proved that they were never boundaries, for any differential. This paper extends and generalizes these results: we consider more elements than in (7), and also prove that many of them do not survive to the E∞ term, so that they fail to be cycles for some dr, and this differential is therefore non-zero.

Recently Ragab ((3), p. 538) evaluated an integral involving the product of an associated Legendre function and an E-function. In this note we derive an integral which is more general than the one given by Ragab.

After some manipulation from the general formula, Racah (1,2) obtained a closed formula for his V coefficient in the special case of all three mj- components zero. He also showed that the integral of the product of three Legendre polynomials followed at once from these special V coefficients. In this case the general formula ((2), eq. 10–14) becomes

where

and a, b, c are all non-negative integers. By changing the parameter of summation to (a+ b−c−p) it follows that is zero unless 2g = a+b+c is even (2).

The flow pattern past a thin wedge-like profile set at a small angle of attack in a gas flowing with high subsonic or sonic velocity is discussed within the order of the transonic approximation. In the model considered the flow has a stagnation point at the nose of the wedge and breaks away, with velocity equal to that of sound, from the shoulders. The velocity is subsonic throughout the whole field of flow. The solution of the boundary-value problem for the wedge in a channel is formulated as a pair of dual integral equations. The complete solution is given for the wedge in a free stream and the dimensions of the profile, together with the lift coefficient, are computed as functions of the transonic similarity parameter.

The following well-known theorem has been proved by Iwasawa(1) and can also be deduced without much difficulty from earlier work of Levi(2), Magnus and Witt.

The number of Seifert circuits in a diagram of a link is well known 9 to be an upper bound for the braid index of the link. The -breadth of the so-called P-polynomial 3 of the link is known 5, 2 to give a lower bound. In this paper we consider a large class of links diagrams, including all diagrams where the interior of every Seifert circuit is empty. We show that either these bounds coincide, or else the upper bound is not sharp, and we obtain a very simple criterion for distinguishing these cases.

The usual proof of the adiabatic invariance of the quantum integrals of a multiply periodic dynamical system rests on the assumption that the Hamiltonian equations remain valid for the adiabatically changing system, with the same Hamiltonian function H, or at any rate with a new Hamiltonian function differing from the old one only by terms that vanish with ȧ, (a) being the varying parameter.