The increase in the angle of reflexion from a powdered crystal varies from (1 − n) tan θ to (1 − n) (tan θ + cot θ) as its absorption increases. In either case the true lattice spacing of a cubic crystal is obtained by increasing the extrapolated lattice spacing by a fraction 1 − n of itself. The divergence of the reflected beam by refraction is small in comparison with other causes of line width.

The wave function φ(x, y) satisfies the equation

in the infinite region

and satisfies the boundary condition Kφ + φy = 0 on the two line segments (y = 0, −∞ < x < −R) and (y = 0, R < x < ∞). We are not concerned with the properties of φ(x, y) inside the circle r = R. It is shown that outside the circle r = R the wave function φ(x, y) can be written as the sum of a wave source, a wave dipole, regular waves and wave-free potentials. (These basic functions are defined in section 2.) The proof makes use of analytic continuation into the image region (r > R, y ≤ 0) cut along the negative y-axis.

Two methods are given for this step. The first method uses Schwarz's symmetry principle to define the auxiliary wave function Φ = Kφ + φy throughout r > R, and φ can then be found by solving differential equations in the y or x direction. The second method uses Green's theorem. Applications in the theory of water waves are suggested.

The interaction between gas atoms and solid surfaces has been studied experimentally for many years. The first detailed study of the nature of this interaction was perhaps due to Knudsen, who in his classical researches introduced the idea of the accommodation coefficients for energy or momentum exchanges between gas atoms and a solid surface, when the gas atoms and the solid are at different temperatures or possess different mass motions. Knudsen and other investigators have given numerical values for these accommodation coefficients for various gases and solid surfaces, which seem to indicate that the accommodation coefficients are never small and are often of the order unity. This means that the gas molecules before reflection accommodate themselves almost completely to equilibrium with the temperature or motion of the wall. Before the recent work of Roberts referred to below, which has inspired this paper, no special precautions had been taken in the preparation of the wall surface, and as we now know the walls used by all previous investigators must have been completely covered with at least a mono-molecular film of gas. Thus the old observations of the accommodation coefficient do not determine it under precise conditions, and find in fact a value many times larger than that found by Roberts for the energy exchanges between the gas and a clean tungsten surface. For helium and metal with a dirty surface the value of the accommodation coefficient is some 6 times as large as the true value for the clean surface. The older values of the coefficients were so large that there were apparently grave difficulties in the way of any simple theory, but this is so no longer.

Let Xnn ≥ 1, be i.i.d.r.v.'s and Sn = X1+…+Xn. Let be Sn minus the r terms of largest absoluete value. Maller proved that if coverages in distribution to N(0, 1) then so does (Sn/bn)−an, assuming that Xn have a continuous symmetric distribution. We show that his resul;t is true without these extra assumptions. Some related results are also given.

Consideration of the direction of the normal at any point on a continuous sheet of a surface yields a sufficient condition for the existence of parabolic points on that sheet. This condition has been used to derive some simple inequalities between elastic constants, whose fulfilment determines the existence of parabolic points upon the inverse surfaces of media of orthorhombic, tetragonal, cubic or hexagonal symmetry; in virtue of the polar reciprocal relation between inverse and wave surfaces, the existence of cusp points on the latter is thereby simultaneously established. It is also pointed out that conditions for external conical refraction may prevail in hexagonal media.

One of the most widely studied concepts in graph theory is that of the colouring of graphs. Much of the research in this area has dealt with the chromatic number χ(G) of a graph G, defined as the minimum number of colours which can be assigned to the points of G so that adjacent points are coloured differently. One can obtain a natural generalization of this concept as follows: we define the n-chromatic number χn (G) to be the smallest number of colours which one can associate with the points of G so that not all points on any path of length n are coloured the same. The 1-chromatic number is then simply the chromatic number.

When impact occurs between clean metal surfaces, plastic flow of the metal usually takes place at the points of real contact, so that the pressures developed are as high as the dynamic yield pressure of the metal concerned. Early experiment show that if the surfaces are covered with a thin film of a highly viscous liquid, the pressures developed and transmitted through the liquid film may be sufficiently great to produce plastic deformation of the metal, even though no metallic contact occurs through the film (1). The existence of these high pressures in the liquid layer means that extremely high rates of flow and shear may be developed in the liquid film, and that the energy dissipated in overcoming viscous flow may lead to an appreciable temperature rise in the liquid. Even in much gentler impacts, where plastic deformation of the metal surfaces does not occur, very high pressures, rates of flow and shear, etc., may be developed in the liquid film. These effects are of great interest in any study of collisions through liquids; they are of particular significance in the study of the mechanism of detonation of liquid explosives by impact.

This paper was initiated by a question of Eisenbud who asked whether the entries of the matrices in a minimal free resolution of a monomial ideal (which, after a suitable choice of bases, are monomials) divide the least common multiple of the generators of the ideal. We will see that this is indeed the case, and prove it by lifting the multigraded resolution of an ideal, or more generally of a multigraded module, keeping track of how the shifts ‘deform’' in such a lifting; see Theorem 2·1 and Corollary 2·2.

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection

is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

Throughout this paper we shall suppose that denotes a set of elements x in which a Lebesgue measure is defined and that itself is measurable and has finite measure. A (1, 1) transformation T of into itself is called an equimeasure transformation if the transform T E of any measurable subset E of is measurable and has measure equal to that of E. Then, if f(x) is integrable in , it is plain that f(Tx) is also integrable and that

We prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

We denote by tm(n) the number of partitions of the positive integer n into non-decreasing parts which are positive or zero powers of a fixed integer m > 1 and we call tm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula for tm(n), the first term of which is

Mahler's result was later improved by de Bruijn (2).

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way that

where the integral is considered in Dinculeanu's sense.

Asymptotic expansions of the Barnes double zeta-function

formula here

and the double gamma-function Γ2(α, (1, w)), with respect to the parameter w, are proved. An application to Hecke L-functions of real quadratic fields is also discussed.

For random walks on finite graphs, we record some equalities, inequalities and limit theorems (as the size of graph tends to infinity) which hold for vertex-transitive graphs but not for general regular graphs. The main result is a sharp condition for asymptotic exponentiality of the hitting time to a single vertex. Another result is a lower bound for the coefficient of variation of hitting times. Proofs exploit the complete monotonicity properties of the associated continuous-time walk.

Experiments are described for measuring the lateral deviation of wireless waves after reflection from the E and F regions of the ionosphere. It was found that the greatest lateral deviation observed, 20° or more, was that due to the e region, and the least, about 0·5°, was due to the normal E region in the case of a distant transmitter.

The time variation of amplitude of a reflected wave was found to be consistent with a random scattering at the ionosphere.

In the theoretical discussion it is shown that changing horizontal irregularities, ion clouds, are a very important cause of fading. Values are calculated for the average fading periods which would result from the horizontal winds in the neighbourhood of the E region known to exist from other evidence. These calculated periods agree with the observed and it is inferred that horizontal winds are a very important cause of fading.

For functions F satisfying a certain submultiplicativity condition and polynomials Q1, . . ., Qk in [X], Nair and Tenenbaum obtained an upper bound on the short sum with an implicit dependency on the discriminant of Q1 . . . Qk. We obtain a similar upper bound uniform in the discriminant.

The object of this paper is to obtain some properties of non-degenerate quadric primals in the projective geometry in [s] over a Galois field of order n. It will be shown that, if s is even, there is only one type of quadric, but that, if s is odd, there are two types of quadric. The number of points on a quadric of each type, and the number of quadrics of each type, will be found. Finally, a possible application to statistics will be indicated.

A modification of the MO method is described which leads to the same polynomial secular equation as does the LCAO method, but via a simpler secular determinant. The basic idea is to divide the whole molecule into parts, calculate the MO's for these by the usual method, and then use linear combinations of these MO's as trial eigen-functions in a variational treatment of the whole molecule.

Equations are developed for a number of cases, and it is shown that the computations of energy levels and MO's from them is often much simpler than from the corresponding LCAO equations.

Let L be an alternating link and be its reduced (or proper) alternating diagram. Let w() denote the writhe of [3], i.e. the number of positive crossings minus the number of negative crossings. Let VL(t) be the Jones polynomial of L [2]. Let dmaxVL(t) and dminVL(t) denote the maximal and minimal degrees of VL(t), respectively. Furthermore, let σ(L) be the signature of L [5].