Many investigations bave sbown the presence of a “continuous” β-ray spectrum accompanying tbe β-ray lines in tbe naturai spec-trum of Ra B and Ra C. The existence of sucb a spectrum is not easily accounted for if one is to regard the nucleus as existing in sharply quantised states. It has been pointed out that this spectrum may be a spurious effect, arising from scattering from the walls of the apparatus and not from within the atom itself. Therefore it is of importance to obtain accurate information regarding the continuous spectrum and its relation to the line spectrum.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings–Jaco theorem which established a similar result for the Poincaré Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

We fill a gap in the construction of the extended Johnson homomorphism, stated in Section 2 of the paper The abelianization of the congruence IA-automorphism group of a free group.

We study the Hausdorff dimensions of certain sets of non-normal numbers defined in terms of the exact rate of convergence of digits in their N-adic expansions. As an application of our results we analyse the rate of convergence of local dimensions of multinomial measures.

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).

The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.

We prove that if G is a centre-by-metabelian pro-p group of type FPm over , so every homomorphic image of G is of type FPm over .

We study the space of slice torus invariants. In particular we characterise the set of values that slice torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local Thom conjecture implies the existence of slice torus invariants without having to appeal to any explicit construction from a knot homology theory.

The LMO invariant is a very strong invariant such that it is expected to classify integral homology 3-spheres. In this paper we identify the set of the degree ≤ 6 parts of the logarithm of the LMO invariant for integral homology 3-spheres. As an application, we obtain a complete set of relations which characterize the set of Ohtsuki's invariants {λi(M)} for i ≤ 6. For any simple Lie algebra , we also obtain a complete set of relations which characterize the set of perturbative P invariants {(M)} for i ≤ 3.

Cosmic censorship hypothesis [1] is a major unsolved problem in classical general relativity. According to this hypothesis singularities occurring in generic space-times should not be naked. This means that there should not be singularities to the future of a regular initial surface that are visible to observers at infinity A mathematically precise statement of the hypothesis is that a space-time should be future asymptotically predictable from a partial Cauchy surface ([2] p. 310).

Using class field theory we prove an explicit result of André–Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the special points of $\mathbb{G}_m(\mathbb{C})$ are defined to be the roots of unity.

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power q. For fixed d, we restrict to moduli q so that there is a unique subgroup of invertible classes modulo q of order d. We study distribution properties of these families of sums as q grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of such regions.

An asymptotic estimate is obtained for the error in approximation of functions by partial sums of spherical harmonic expansions. The precise constant in the main term is found and the order of growth of the remainder term is given.

An identity for a rational function is used, in conjunction with generating functions for a certain class of polynomials, to derive infinite sets of identities for these polynomials. Identities are given for the polynomials of Legendre, Gegenbauer, Hermite, Tchebycheff and L. Carlitz. A novel type of classification for these polynomials is indicated. Next, infinite sequences of identities for various Bessel functions and for power functions are derived. New identities for trigonometric functions are also given, as well as brief miscellanea on: functions resembling Dirac's delta function, integral transforms and a functional equation.

It is shown that, in a simply connected four dimensional Riemannian space, an arbitrary divergence-free vector generates a one-parameter group of point transformations which leaves Maxwell's equations unchanged. This result is used to show that, if the metric tensor of a simply connected vacuum Einstein–Maxwell space-time admits a group of motions which is also an invariance group of the electromagnetic field tensor, then there exists a one-parameter family of metric tensors all of which satisfy the Einstein–Maxwell equations with the invariant electromagnetic field as source.

Singer [7] proved that in a Banach space all basic sequences are shrinking if and only if all of them are boundedly complete. Afterwards; Zippin [2] proved a similar result for Schauder bases, and it was extended [1] to Schauder decompositions.

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.

In this paper, we construct the Ext-quivers of the principal blocks of F[Sfr ]11 and F[Sfr ]12, where F is an algebraically closed field of characteristic 3. We also obtain the Loewy structures of three principal indecomposable modules of the principal block of F[Sfr ]11.

Assume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.

A convolution formula is established for Bell polynomials. This is expressed in seven equivalent ways and used to derive further properties of these polynomials. The application of these results to some twenty-seven special polynomial sets is shown and illustrated in the case of binomial, Hermite, Gegenbauer and generalized Bernoulli sets.

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).