A method of growing single-crystal resistance specimens free, to a large extent, from strains and impurities is described. The magneto-resistance effect in cadmium single crystals has been studied in some detail at the temperature of liquid nitrogen, using sufficiently high magnetic fields to observe the linear effect found by Kapitza.

Although it proved impossible, in general, to determine the orientation of individual crystals, the experiments suggest that the “critical field” of the linear effect is dependent on the orientation of the crystal with respect to the current and magnetic field, and not on the perfection of the crystal lattice.

It has long been realized that a Stieltjes integral can be defined and developed on the same lines as a Lebesgue integral, using in place of the measure of a set E the variation of an increasing function ø (x) upon E; but only the more obvious—and less useful—properties of such an integral seem to have been stated. The original integral defined by Stieltjes possesses the remarkable Integration by Parts property that if either of exists, so does the other, and their sum is equal to

In this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.

As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

We prove that a point $x$ in R is out of the support of a pseudomeasure $T$ on R if and only if the Fourier integral of $T$ at $x$ is summable with Cesàro means of order 1 to zero.

Let L = K1 ∪ K2 be a 2-component link in the 3-sphere such that K1 is a trivial knot. In this paper, we introduce a new bridge index, denoted by bK1 = 1([L]), for L. Roughly speaking, bK1 = 1([L]) is the minimum of the bridge numbers of the links ambient isotopic to L under the constraint that all of the bridge numbers of the components corresponding to K1 are 1. We provide a lower bound estimate of bK1 = 1([L]) in the case when L is a non-split satellite link. By using this result, we show that for each integer n(≥ 2), there exists a link Ln = K1n ∪ K2n with K1n a trivial knot such that bK1n = 1([Ln])−b([Ln]) = n−1, where b([Ln]) is the bridge index of Ln.

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) by

and, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

Let K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

The object of this paper is to give a method of finding the units of a set of algebraical numbers that shall be convenient practically. Little attention is given to theoretical considerations. The case of numbers composed with the cube root of an integer is treated in § 2, and a few theoretical remarks are added in § 3. In § 4 we show how to test whether the unit found is a fundamental one or not. In § 5 the method is extended to other cases.

A random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

Summary. The main result of the present paper is contained in the following: Theorem ℜ α ≠ −1, −2, −3,…

where2F1[.,.;.;.] and 1f1[.;.;.] (l).

Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which preserves the symmetric monoidal structure. Since I is not closed, this makes it difficult to see how S might be closed.

The concept of arcbody was introduced in [4] as a generalization of prime tangles used by Lickorish[3] and Bleiler[l]. The pair (A, l), where A is a compact 3-manifold with boundary ∂A, and l is a 1-submanifold properly embedded in A, is an arcbody if:

(i) the inclusion ∂ A – l ↪ A – l is monic, i.e. A – l has incompressible boundary;

(ii) no component of (A, l) is homeomorphic to (D2, 0) × I or (D2, 0) × S1; and

(iii) every sphere of ∂A intersects l.

We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses, defined below) for the automorphism group of V. We use Brauer characters and computer calculations. Our algorithm for finding permutation submodules of modules for p-groups may be of independent interest.

We present an example of an entire function with a Baker domain such that the complement of the set of singular values of the function contains annuli of arbitrarily large width. This is the first known example of a function with this property.

In analogy to the classical isomorphism between and , we show that a large class of moderate linear mappings acting between the space of Colombeau rapidly decreasing generalized functions and the space of temperate ones admits generalized integral representations, with kernels belonging to . Furthermore, this result contains the classical one.

Every real number α admits many rational approximations in the sense that for every natural number q,

for some integer p. Perhaps surprisingly, numbers α which have k distinct rational approximations with given denominators q1,…,qk,

for a fixed ε > 0, are quite sparse; their measure in U = [0,1) is at most C(ε)/k.

The Hopf-James invariants are a series of homomorphisms

(see (3) or (13)). I. M. James (7) and H. Toda (13), have shown effectively that in the case n even and r a prime p the restrictions to the p-primary components may be embedded in exact sequences

In this note we show that there are similar exact sequences if n is even and r = pq or if n is odd and r = 2pq, and we establish connexions in certain cases between the homomorphisms Hmn and the composite homomorphisms HmHn. We are able to compute the action of the invariants on the groups of the r-stem for r ≤ 9 with the partial exception of π6(S2) and π7(S2). The exact sequences yield useful criteria for determining the reduced product filtrations (see (9)) of elements.

Suppose that X is any vector space on which it is possible to recognize a class of sets of such a nature that it is natural to call them ‘bounded’. (Precise conditions for such a class of sets are given in § 2.) Let L be any vector space of linear functionals on X which map each ‘bounded’ set into a bounded set. We say that a filter is boundedly-weakly convergent if it is convergent in the weak topology of the linear system [X, L] and contains a ‘bounded’ set. If M is a vector space of boundedly-weakly continuous linear functionals on X which includes L, we say that a subset S of M is limited if 〈 , f〉 converges uniformly for f ε S whenever is a boundedly-weakly convergent filter.