The theorem which we prove here seems obvious enough when stated, but it appears to have been overlooked by the numerous writers who have discussed the subject, and the proof is less immediate than might be expected.

The Statistical Mechanics which has been developed in accordance with the requirements of the new Quantum Theory is concerned with distribution laws over energy values only—over, that is, the characteristics of Schrödinger's equation. To obtain a space distribution law, even for the Classical limit, some use must be made of the characteristic functions. A formula has been suggested by Fowler, but it has not been shown that this formula gives the Classical law for gases at ordinary temperatures and pressures. In this paper we shall show that this is so, but before doing so we shall sketch the analogous method of obtaining the law, on the Classical theory.

The Schroedinger equation of the helium atom is separable between the radial and angular variables. The separated equations in the radial variables are found by using a new technique of calculation with symmetric traceless tensors. Account is taken of reflexion and permutation characters.

One of the most important techniques in the theory of non-linear differential equations is the direct method of Lyapunov and its extensions. It depends basically on the fact that a function satisfying the inequality

is majorized by the maximal solution of the equation

Using this comparison principle and the concept of Lyapunov's function various properties of solutions of differential equations have been considered (1–11).

1. A Banach space E over the complex field C is said to be a Banach inner-product (BIP) space if there exists a mapping 〈.,.〉 of E × E into C satisfying:

(i) 〈x, x〉 ≥ 0 (x ∈ E) with equality only if x = 0;

(ii) 〈x, y〉 = 〈y, x〉 (x, y ∈ E);

(iii) 〈x + λy, z〉 = 〈x, z〉 + λ〈y, z〉 (x, y, z ∈ E, λ ∈ C);

(iv) 〈x, x〉 ≤ k2 ‖x‖2 (x ∈ E),

where k is a fixed positive number. Thus 〈.,.〉 is an inner product on E, which induces a norm ‖·‖1 by the relation

Suppose that λ > − 1 and that

1. Polarisation measurements are described on the Hg lines λ 4358, 5461, 5770, 5791 emitted from a low-pressure electron-maintained arc to which a longitudinal field of up to 3000 gauss may be applied.

2. Two types of polarisation effect have been found. The first is typified by the line λ 5461. It is a polarisation of the light with the electric vector perpendicular to the lines of force. Its maximum value is 9% but it decreases to zero (a) with decreasing field strength and (b) with decreasing current in the tube. It is suggested that this effect is due to a self-reversal of λ 5461 owing to the metastability of the 2 3p2 state and that the polarisation may be caused through a slight non-uniformity of the magnetic field.

The second type of effect is found for λ5770 and 5791. The polarisation is in a plane parallel to the discharge, and is found in a zero magnetic field. This effect is further discussed elsewhere. The line λ 4358 is unpolarised under all conditions tried.

3. Neither of these effects implies a polarisation of the total light emitted directly from atoms in the Transverse Zeeman Effect when the atoms are excited under isotropic conditions and the experiments give definite evidence against the existence of such a polarisation effect.

This paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form

\begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*}

$\log ((9+\sqrt{105})/2)/\log 16\approx .8166$

In two previous notes I have given the various types of non-singular surfaces with sectional genera 4 and 5. In this paper the methods previously adopted are applied to the non-singular surfaces of sectional genus 6. For purposes of classification these may be divided into the four classes:

(1) Regular, non-rational.

(2) Rational.

(3) Irregular, non-referable.

(4) Referable.

Consider a network linking the points of a rate-1 Poisson point process on the plane. Write Ψave(s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function Ψave(s), and on the analogous “worst-case” function Ψworst(s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent α such that each function has Ψ(s) ≍ (s − 1)−α as s ↓ 1.

Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution L-values L(f,g,s), where g is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's ‘false Tate curve’ congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.

1. Introduction. In a principal ideal domain R, any two-sided ideal is of the form Rc = cR, i.e. it has an invariant element as generator, and the customary development of ideal theory in a principal ideal domain (cf. e.g. (10), ch. III) takes on a more transparent form when expressed in terms of invariant elements. Likewise, the one-sided bounded ideals may be studied in terms of their generators.

The proportion ρk of gaps with length k between square-free numbers is shown to satisfy logρk=−(1+o(1))(6/π2)klogk as k→∞. Such asymptotics are consistent with Erdős's challenge to prove that the gap following the square-free number t is smaller than clogt/log logt, for all t and some constant c satisfying c>π2/12. The results of this paper are achieved by studying the probabilities of large deviations in a certain ‘random sieve’, for which the proportions ρk have representations as probabilities. The asymptotic form of ρk may be obtained in situations of greater generality, when the squared primes are replaced by an arbitrary sequence (sr) of relatively prime integers satisfying [sum ]r1/sr<∞, subject to two further conditions of regularity on this sequence.

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.

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.

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.

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.