1. Statement of problem and the method of Barnes and Swinnnerton-Dyer. 1·1. Let ξ1, ξ2, … ξn De n real, homogeneous linear forms in the real variables x1, x2,…, xn and of determinant Δ ǂ 0. Let y1y2, …, yn be any real numbers. We identify the n-tuplet (x1x2, …, xn) with the point P, coordinates (x1, x2, …, xn), of n-dimensional space.We write P≡P0 (mod 1) if x1≡y1, x2≡y2, …, xn≡yn (mod 1), and P0 is the point (y1, …, yn).

Recently Hoyle and Narlikar(2) have put forward a conformal theory of gravitation in which they obtain an explicit solution of the field equations using certain approximations and assumptions of symmetry. To obtain the solution, the metric is assumed to have a special form. In the present paper a more general form of the metric is considered, and it is shown that many of the features of the special solution are shared by the solution in the more general case. In particular, the implications of the special solution as concerns the sign of the gravitational constant remain valid in the more general case under certain assumptions.

Let B ⊆ A be an inclusion of C*-algebras. Then B is said to norm A if, for each X ∈ (A),In this paper we introduce and study the conesThese are shown to coincide with the standard positive cones precisely when B norms A, and we apply this to obtain automatic complete positivity of certain positive maps between C*-algebras.

Hamiltonian circuits with associated word an n-cycle, in Schreier right coset graphs for symmetric groups Sn mod cyclic groups Zn, correspond to change ringing principles on n bells for which the plain course is the extent; that is, neither bobs nor singles are required. This connection is made explicit for the general case, and then specialized to the cases n = 4 (minimus) and n= 5 (doubles). In particular, all 102 no-call doubles principles on three generators are found and catalogued.

The problem of conduction of heat in a solid whose surface is in contact with fluid, which is so well stirred that its temperature is constant throughout its mass, often arises in practice, for example in calorimetry, or in the warming of a closed room. Despite their importance, problems of this type have not been much studied since the presence of the mass of fluid at the surface gives rise to a boundary condition for which the classical methods of solution do not apply without modification; the Laplace transformation method, however, has the advantage that it can be applied in the same way to all cases.

It is shown that a quartic equation may be found which expresses the grid characteristic of a valve accurately over a considerable range, and that for modern dull filament valves the grid current curve is exponential over only a very restricted length.

Expressions are given for calculating rectified current and the amplitude of harmonics produced by a simple grid rectifier when voltages of the following wave-forms are applied: E sin pt, (A + B sin nt), and (A + B cos mt + C cos nt)sin pt. Further, a formula is given from which we may calculate the rectified current produced by a cumulative grid rectifier when a voltage E sin pt is applied.

All these formulae are applicable if the grid characteristic, is parabolic, cubic, quartic or quintic in form.

It is also shown that when an acoustically modulated radio frequency voltage is applied to a cumulative grid rectifier, even if the valve has a parabolic grid characteristic, an infinite number of harmonics of the modulation frequency are produced. These harmonics are due to the. action of the grid leak and condenser. It appears, however, that in practice the amplitude of these harmonics will be very small compared with those produced, primarily, by the curvature of the grid characteristic.

In a previous contribution to these Proceedings (Ramakrishnan(1)) the concept of product density was introduced to describe the statistical distribution of a discrete number of particles in a continuous space E, corresponding to a single point t, where t is the parameter with respect to which the stochastic process evolves. This is extended to densities corresponding to n points on the t axis and correlation problems associated with these density functions are studied with particular reference to electron-photon cascades.

Let ℚp and ℤp respectively denote the p-adic numbers and the p-adic integers for a fixed prime number p. For any subset Y of and n ∈ ℕ we denote by Nn(Y) the cardinality of the image of Y under the natural map . (Thus we have Nn(Y)≤ pnr.)

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove

For r ∈ [0, 1] let μr be the Bernoulli measure on the Cantor set given as the infinite power of the measure on {0, 1} with weights r and 1 − r. For r, s ∈ [0, 1] it is known that the measure μr is continuously reducible to μs (that is, there is a continuous map sending μr to μs) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin:

Is it true that the product measures μrand μsare homeomorphic if and only if each is a continuous image of the other, or, equivalently, each of the numbersrandsis binomially reducible to the other?

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.

Some explicit formulae for the right coaction of A* on H*BO are given. As corollaries, closed formulae for χ(ξk) and other elements in A* are given.

Complementary variational principles are developed for boundary-value problems with field equations φiv = f(φ) in (0, 1) subject to φ(0) = φ(1) = 0, φ′(0) = φ′(1) = 0. The theory is applied in the non-linear case f(φ) = – keφ and accurate variational solutions, which are simple polynomials in x, are obtained.

Lejeune–Jalabert and Reguera computed the geometric Poincaré series $P_{geom}(T)$ for toric surface singularities. They raised the question whether this series equals the arithmetic Poincaré series. We prove this equality for a class of toric varieties including the surfaces, and construct a counterexample in the general case. We also compute the motivic Igusa Poincaré series $Q_{geom}(T)$ for toric surface singularities, using the change of variables formula for motivic integrals, thus answering a second question of Lejeune–Jalabert and Reguera's. The series $Q_{geom}(T)$ contains more information than the geometric series, since it determines the dimension of the tangent space at the singularity. In some sense, this is the only difference between $Q_{geom}(T)$ and $P_{geom}(T)$.

1. Introduction. 1.1 In (5), (6) and (7) I have catalogued the classes of involutions of the orthogonal, symplectic and unitary groups over perfect fields of characteristic 2. So, too, were obtained the involution classes of the well-known relatives of these groups. For each group the relevant classes have simple and explicit descriptions in terms of the geometry of its setting, and an overall pattern can be seen.

We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.

Complex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours with n sides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function . Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.

In continuation of earlier work [7], we study in this paper the structure of cocommutative Hopf algebras of finite representation type. Since every such algebra is the group algebra H([Gscr ]) of a finite algebraic group [Gscr ], this problem has two interrelated aspects. On the one hand, one has to understand the structure of the group scheme [Gscr ], while on the other the Morita equivalence class of H([Gscr ]) has to be determined. Although we shall here be mainly concerned with the latter, structural information on the underlying group schemes enters crucially at several points.

Suppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), where

and we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)

where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.