A modification of the alternating field method of measuring ionic mobility in a gas gives an experimental curve showing upper and lower limits to k. From this a distribution curve is derived, which has a calculable resolving power.

The mobility of negative ions in dry air shows a continuous distribution between the limits 2·15 and 1·45, with a peak value about 1·8.

At low pressures the current is resolved into ions and free electrons. From the relative numbers reaching the electrometer it is found that the electron makes an average number 9·4. 104 collisions before capture, independent of field strength and pressure, and therefore independent of the electron speed over a range W = 2×105 to W = 7×105.

We present the first anisotropic cosmological solutions of d = 11 supergravity. The solutions given are of Bianchi types III, V and VIh coupled with the seven-dimensional torus. The cosmological implications are briefly discussed.

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.

The Harada-Norton group is one of the twenty-six sporadic simple groups. It has order 273, 030, 912, 000, 000 = 214.36.56.7.11.19. In this paper our main objective is:

Theorem 1. The Harada-Norton group acts as a group of linear automorphisms of a 133-dimensional commutative, non-associative algebra defined over F5.

Given disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

One of the most celebrated theorems of mathematical analysis is the intermediate value theorem of Bolzano which, in a simple case, states that a real-valued continuous map f of a closed interval [a, b], such that f(a) and f(b) have different signs, has a zero in (a, b). Recently, Shih in [5] observed that without loss of generality we may suppose that a 7 < 0 < b and f(a) < 0 < f(b) and, consequently, the condition f(a).f(b) < 0 becomes x.f(x) > 0 for x∈∂Ω where ∂Ω denotes the boundary of the interval Ω = (a, b); then the conclusion is that f has at least one zero in ω. It is a remarkable fact that Shih extends this form of Boizano's theorem to analytic maps in ℂ [5] and, subsequently, in ℂn [6]. He proved that if Ω is a bounded domain in ℂn containing the origin, is continuous in and analytic in Ω and Re for z∈∂Ω, then f has exactly one zero in Ω. In this paper we extend Shih's result to Banach spaces.

The nonsingular model of Kummer's surface, or its birational equivalent the Weddle surface, is an octavic surface F in [5], projective space of dimension 5 (11), ((10), p. 53), ((1), pp. 218, 219). F is the base variety of a net N of quadrics with a common self-polar simplex S, and has on it 32 lines. These form a single orbit under the group G, of order 32, consisting of the harmonic homologies fixing S.

1. The following results are special cases of theorems of Irwin and Zeeman, respectively (see (4), (9), (10)).

(a) Let V be a (2n − m + 1)−connected piecewise-linear m−manifold (bounded or not), where m − n ≥ 3. Then any element of πn(V) can be represented by a piecewise-linear imbedding of Sn in V.

(b) Let M be a (2n − m + l)-connected closed piecewise-linear (m − 1)-manifold, where m − n ≥ 3. Then two piecewise linear imbeddings of Sn−1 in V are isotopic if and only if they are homotopic.

For any locally compact topological group G let M(G) denote the topological semigroup of all probability (Borel) measures on G, furnished with the weak topology and with convolution as the multiplication. A Gauss semigroup on G is a homomorphism t→ μt of the strictly positive reals (under addition) into M(G) such that

(i) no μt is a point mesaure,

(ii) for each neighbourhood V of 1 in G we have

We identify Morita cohomology, which is a categorification of the cohmology of a topological space X, with the category of homotopy locally constant sheaves of perfect complexes on X.

The problem of wave generation by a line source of sinusoidally varying strength situated in water above a beach of arbitrary angle α(0 < α ≤ π) is solved by the use of a Laplace-integral representation of the solution. It is shown that a solution can be constructed which is regular at the shoreline and gives an outgoing wave-train at infinity.

The two-sided Laplace transform is used to obtain a solution of a certain integral equation. The equation is that associated with the transport equation for isotropic scattering, when there is a point source of radiation giving a concentrated ray pencil, in a spherically symmetric homogeneous medium extending to infinity. The solution is then used to find the asymptotic form of the intensity in the forward and backward directions.

Let k ∈ R, k ≥ 3. Consider the curve of class C3 in R3 defined by

Let σk(t) = t(k−3)/6dt be the affine arclength measure on γk. Define the convolution operator T (= Tk) by

For V a complex algebraic threefold with k( V) = 0 and having at least one minimal model (for definition, see below), we show that (except possibly in one rather special case) the components of an effective divisor mKv are birationally ruled surfaces.

The purpose of this paper is to determine the optimal order of decrease of the L2 norm for the following degenerate and singular oscillatory integral operators defined by

formula here

for f∈C∞0(ℝ), where ψ(x, y) is a smooth compactly supported function on ℝ2, S(x, y) is a homogeneous polynomial of degree n

formula here

and K(x, y) is a distribution kernel which satisfies the following hypotheses:

K is a C2 function away from the diagonal and the estimates

formula here

hold for 0<μ<1 and i=1, or 2.

This paper basically uses the methods of [6], which are very general and can be widely used. The main idea of these methods is that the sizes of operator norms after the decomposition are estimated in different ways and the decomposition is then summed back by balancing the estimates of two types. The first type estimate takes into account the support sizes of the integrand after the decomposition. The second type estimate is delicate, depending on how to keep track of the distance to the singular varieties of phase function. In our case, we have to include the singular variety arising from distribution kernel as well. The analytic tool of the second type estimate is the operator version of the Van der Corput lemma [7]. We notice that the crux of matters is to locate the singular varieties of phase function and kernel. For higher dimensional situations, the problem is very difficult. One expects that the powerful theory of algebraic geometry may play a significant role. Here we would like to remark that Mather's results [2] may be useful although the geometrical shapes of singular varieties are basically undecidable.

By an iteration of a regular summation matrix B = (bmn) with a regular matrix A = (amn), or B.A, we shall mean the summation method defined by the matrix B operating on the sequence {tn} of A transforms of a sequence {sn}. Thus the sequence {τn} of B. A is given by

This paper is a continuation of (8), and is concerned with geometrical properties of special null hypersurfaces. In particular, on a one-parameter family of null hypersurfaces in four-dimensional Lorentzian manifold V4, we consider the relation between their normal and the Debever vectors, especially repeated ones. Throughout this paper, the same notations as those in (8) are used.

Let CPn denote the complex projective space of complex dimension n. If k < n, CPk−1 is embedded as the (2k –1)-skeleton of CPn and we denote by , the stunted projective space, the quotient of CPn. by CPk−1.

Statement of results. Let A and B be two matrix summability methods, and S a given set of sequences. We shall say that B is S-stronger than A if B sums every sequence which belongs to S and is summed by A. If each of A, B is S-stronger than the other, A and B will be called S-equivalent. If B is S-stronger than A, but A is not S-stronger than B, we say that B is strictly S-stronger than A.

1. degenerate iterative processes. A well-known class of iterative processes is specified by the equation