It is well known that a solution of the wave equation

for which u = ∂u/∂t = 0 initially outside a surface S0, vanishes at time t in the exterior of a surface St parallel to, and at normal distance ct from S0, so that the wave fronts of disturbances represented by the solutions of the wave equation obey the laws of geometrical optics. Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary value conditions of the ‘Cauchy’ type. But the wave fronts of solutions of problems in which some of the boundary conditions are of the type representing reflexion do not seem to have been treated, and in particular the case of diffraction, when there is a ‘shadow’, does not seem to have been considered at all.

We determine the automorphism group for a large class of affine quadrics over a field, viewed as affine algebraic varieties. The proof uses a fundamental theorem of Karpenko's in the theory of quadratic forms [13], along with some useful arguments of birational geometry. In particular, we find that the automorphism group of the n-sphere {x02+···+xn2=1} over the real numbers is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. This result is reminiscent of Wood's theorem that when n is a power of 2, every real polynomial mapping from the n-sphere to a lower-dimensional sphere is constant [22].

In (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Möbius transformations.

First-order calculations are given concerning the possibility of designing a β-particle spectrograph in which the correlation between the photoelectrons of the stronger “natural β-ray lines” and the disintegration particles of the “continuous spectrum” might be investigated. It has been shown that, with a coincidence counting circuit of resolving time 10−6 sec., and with a source of suitable strength, this investigation should be possible in the case of any natural β-ray line the absolute intensity of which is sensibly greater than 0.01. A successful investigation of this kind would provide important data for the analysis of a complex continuous spectrum into the “partial spectra” corresponding to different modes of transformation (and might furnish direct information regarding the internal conversion coefficients appropriate to the stronger γ-radiations involved).

The famous Haken–Kneser–Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3-manifolds. The analogous statement for 3-orbifolds has been part of the folklore for several years, and it was commonly believed that slight variations on the argument used for manifolds would be sufficient to establish it. We demonstrate in this paper that this is not the case, proving that the apparently natural notion of “essential” system of spherical 2-orbifolds is not adequate in this context. We also show that the statement itself of the theorem must be given in a substantially different way. We then prove the theorem in full detail, using a certain notion of “efficient splitting system.”

We study different localization techniques both for classical and motivic spectra. In particular, we show that the motivic model structures of Morel and Voevodsky are cellular in the sense of Hirschhorn and thus may be further localized. As a possible application, we discuss a motivic version of the chromatic filtration and the chromatic spectral sequence.

1. Introduction. The wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p (Baumslag (1)). When A is an Abelian p-group of exponent pk, and B is the direct product of cyclic groups of orders pβ1, …, pβn and β1 ≥ β2 ≥ …, ≥ βn, then Liebeck has shown that the nilpotency class c of A wr B is

The conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.

The velocity distribution in the wake behind a body of revolution calculated by Miss Swain on the momentum transfer theory with l constant over a cross-section of the wake is not in accordance with experiment.

On the generalized vorticity transfer theory the equation for the first approximation to the velocity defect u far downstream is

There are two methods of simplifying this equation on the analogy of the successful simplification of the corresponding equation for the two-dimensional case. One method gives the same equation as if the turbulence were symmetrical; the other gives the equation of the modified vorticity transfer theory.

For the former there is no real solution if l is constant over a cross-section; for a real solution l must be infinite on the axis of symmetry. The equation was integrated with l ∝ r−1 over a cross-section; this variation of l was suggested by putting the change in the area enclosed by a circular vortex-line proportional to the area of the section of the wake. The agreement with experiment is fairly good over the middle portion of the wake, but not near the edge, where the theoretical value of ∂u/∂r is not zero. If l2 ∝ rp, ∂u/∂r vanishes at the edge of the wake only if p = − 3. The solution is carried through for this case; it shows only fair agreement with experiment over the middle of the wake, and none at the edge.

The calculation on the modified vorticity transfer theory is carried out with l constant over a section; in this case there is no definite edge to the wake, nor does u fall off sufficiently rapidly to make the momentum integral for the drag converge. (u ∝ r−2 for large r.) Even over the middle of the wake there is a discrepancy with experimental results, rather like that shown by the momentum transfer theory.

The temperature distribution behind a heated body was calculated according to each of the foregoing theories.

The temperature distribution was also calculated from the experimental velocity distribution, without any assumption being made concerning the kinematic coefficient of turbulent diffusion, which was eliminated between the equations for the velocity and temperature. The vorticity transfer theory with symmetrical turbulence gives an impossible result. The modified vorticity transfer theory gives θ/θmax < u/umax when the experimental velocity distribution is used.

1. An expression in finite terms is found in §§2,3 for the small displacement of a thin elastic plate due to the application of a transverse force at an arbitrary point; the plate is infinite and is bounded internally by an ellipse along which it is clamped. The limiting case in which the ellipse degenerates to a straight line is considered in §4, and the other limiting case of a circular boundary in §5.

In a series of papers [13, 14, 15], Hilton introduced the terminology relative group to denote a group epimorphism ∈:G↠Q, and relative space to denote a map ƒ:E→B between connected spaces inducing an epimorphism of fundamental groups. He pointed out the desirability of relativizing the theory of P-localization of nilpotent groups and spaces developed in [17], and carried out the algebraic part of this project in [14, 16]. The homotopy-theoretic part was settled by Llerena in [18, 19].

We introduce non-homogeneous equilibrium states which include the classical equilibrium states for subshifts of finite type, Riesz products and G-measures. We prove a rather precise estimate for ∥Pnf∥∞ where Pn are the averaging operators. Applications are given to estimate ∥Lngf∥∞ (Lg being a transfer operator on a subshift of finite type) and then to prove a central limit theorem for f(Tnx) (T being the shift), to study the almost everywhere convergence of some lacunary series with respect the equilibrium state and then to compute the Hausdorff dimension of the equilibrium state etc.

If R is a Banach algebra and ø ∈ R′, the dual space, then we may define a bounded linear map by

We shall show that for suitable p the requirement that each be p-absolutely summing constrains R to be an operator algebra, or even, in certain cases, a uniform algebra. In this way we are able to give generalizations of results of Varopoulos (12) and Kaijser (4).

The calculation of the strengths of forbidden lines in the 1s22s22p2 configuration is extended to include the effect of interaction with the 1s22p4 configuration. For this case it is shown that if observed energy levels are used instead of theoretical energies in line-strength formulae which neglect configuration interaction the major part of the effect of configuration interaction is nevertheless taken into account. The results are illustrated by detailed computations on the Ca xv spectrum. The 1s22s22p4 configuration is also discussed.

Small variations in the elastic parameters of a solid medium give rise to scattering of elastic waves and hence to a reduction in their amplitudes. The attenuation coefficients associated with this process have been calculated for surface waves.

Expressions have been derived for waves with wavelengths long compared with the mean wavelength of the variation of the elastic parameters in space. The attenuation coefficients were found to be the same as if the attenuating mechanism were a particular type of visco-elasticity.

The problem with which this paper is concerned is that of packing non-overlapping plane figures into a given plane figure. It is of a different type from the usual packing problem in which equal figures are used in the whole plane, and the aim is to calculate the limit of the ratio of covered area to the total area inside a large circle, as the radius of the circle tends to infinity. A closest packing in this problem is an arrangement of the figures in such a way that this limit attains its largest value.

We consider the scattering of time-harmonic electromagnetic plane waves by a piecewise homogeneous obstacle. In the interior of the obstacle there exists a core which may be a dielectric, a perfect conductor or an imperfect conductor. First, we obtain integral representations for the scattered fields incorporating all the boundary and transmission conditions. These representations are then used to derive the corresponding electric far field patterns. We prove reciprocity and scattering theorems and study the spectrum of the electric far field operator for these problems. Finally, we prove the optical theorem, that is a connection of the electric far field pattern to the scattering cross-section.

In recent years new materials of industrial importance have become known whose rheological properties cannot be adequately characterized by the classical Newtonian working model, many of these materials exhibiting both viscous and elastic properties. The large-scale flow behaviour of some of these new materials can be strikingly different from that of purely viscous liquids. For example, when sheared between coaxial cylinders, some liquids are observed to ‘climb up’ the inner cylinder, i.e. the free surface rises spectacularly near the inner boundary. This phenomenon has been observed by many authors (see, for example (1–4)), and is now widely known as the Weissenberg effect. Again, Merrington (5) observed that a stream of rubber solution ‘swells’ radially immediately on emerging from a capillary tube. In order to attempt to explain behaviour such as this, we must examine the consequences of a working model which is more complex than that for Newtonian liquids.

1. Introduction. The object of this paper is to obtain some generating functions for Appell functions with the help of fractional derivatives. The results are general in character and include as particular cases some of the results given earlier by Mexiner (3) and the author (4, 5, 6).