We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

In the present paper Bethe's approximation is applied to the statistical treatment of the adsorption of dipoles considering (a) the electrostatic forces alone, and (b) the electrostatic and van der Waals forces jointly. In each case formulae are obtained for the adsorption isotherm and for the variation of the heat of adsorption with the fraction of surface covered. In case (a) the heat curves are compared with those obtained by Wang who used a different approximation to take into account the electrostatic interactions due to particles adsorbed on outer sites. The curves are of the same general shape; sources of differences in them are discussed. In case (b) the results are compared with those obtained by Roberts using a different method. This treatment confirms his result that the electrostatic and van der Waals forces give contributions to the variation of the heat of adsorption which are of opposite sign and almost counterbalance one another, so that the resultant variation in the heat of adsorption is very much less than would be expected from a consideration of forces of one type only. This comparison shows further that the distribution of particles on the surface, which is taken into account in the statistical method used in this paper, has a considerable effect on the variation of the heat of adsorption.

Let M be a compact Riemannian manifold equipped with Laplace–Beltrami operator Δ. We use the Rademacher-Menchoff theorem and the asymptotics of the eigenvalues of Δ to show that if a function belongs to an L2 Sobolev space of positive index then its expansion, in terms of eigenfunctions of Δ, converges almost everywhere on M.

The Dickson algebra Wn+1 of invariants in a polynomial algebra over [ ]2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ]n−2 on a fundamental class in degree 2n – a. This involves a new family of left ideals [Iscr ]a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ]n−2 in the Steenrod algebra.

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.

If A, B are ideals of a commutative ring R, such that B ⊆ A, and, for some positive integer r, Ar = BAr−1 then B is said to be a reduction of A. (This concept was defined and developed by Northcott and Rees in (1).) In this paper, I shall consider commutative Noetherian rings with the property that no non-zero principal ideal is a reduction of an ideal properly containing it.

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the property

a partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

The propagation of electromagnetic waves through isotropic media containing a continuous energy-gain mechanism is considered and the existence of perfect resonance properties is investigated for both homogeneous and inhomogeneous layers.

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the function

in terms of the positions of the singularities of f(z) and g(z).

In the geometry of the cubic curve Γ in space, we have to study the quadrics Q which stand in certain special relations to the curve. We shall find it convenient to refer to these relations as A, B, B′, C, C′; these are defined below.

It is the purpose of this paper to prove a general compactness theorem and deduce from it, as corollaries, the Tychonoff theorem on the product of compact spaces and Ascoli-type theorems giving sufficient conditions for function spaces with the topology of uniform convergence on compact sets to be compact.

The Green's functions for semi-infinite and adjacent semi-infinite media are considered in one-speed transport theory. Chandrasekhar's Principles of Invariance are used to formulate integral equations that describe the emergent and interface angular distributions. It is shown that mathematically these are equivalent to subtracting from the transport equation known solutions to related, simpler problems. Duplication of effort is avoided in this way. For adjacent semi-infinite media, a difficulty encountered in the solutions of the integral equations necessitates the application of a Laplace transform for its resolution. Some virtues of using the transform are explored, e.g. in the determination of the internal distribution where it yields concise expressions for the Green's function.

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and let

be any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):

For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such that

for all integers x1, x2, …, xm not all zero.

The author provides a new and very simple proof of the following theorem of Magnus (2):

Theorem. Let the group G have both a presentation with n + r generators and r relations also a presentation with n generators. Then G is the free group on n generators.

Let ${\frak M}_{g}$ be the moduli space of smooth, integral curves of genus $g$ over the complex field ${\mathbb C}$ and let $\overline{{\frak M}}_{g}$ be its Mumford–Deligne compactification. If $C$ is a smooth, integral curve of genus $g$ we denote by [$C$] its isomorphism class in ${\frak M}_{g}$.

We compute the (uni)versal deformation ring of an odd Galois representation ρ: Gal (M/Q) → Gl2(Fp) with an upper triangular image, where M is the maximal abelian pro-p-extension of F∞ unramified outside a finite set of places S, F∞ being a free pro-p-extension of a subextension F of the field K fixed by Ker ρ. We establish a link between the latter (uni)versal deformation ring and the (uni)versal deformation ring of ρ: Gal (KS/Q) → Gl2(Fp), where KS is the maximal pro-p-extension of K unramified outside S. We then give some examples.

The errors occurring in a large number of Jeffreys's phases for neutral chlorine have been determined. A discussion of the source of these errors is given, from which it appears that errors in the phases of zero order and of order unity are due to a cumulative error over the range of integration. A method of correcting for this error is obtained. This method is of easy practical application, and it reduces the error almost to zero for phases of these orders.

The error occurring in higher order phases is found to be mainly due to errors in the two π/4 terms, which take into account the exponential tails of the waves in the totally reflecting region. For phases of high order these errors in the two π/4 terms cancel each other, and consequently the total error in phases of high order is practically zero.

A correction graph for the zero-order phase is given from which the error in this phase for atoms of atomic number up to 36 can be read off. The graph is such that an extrapolation to higher atomic number would not involve much error.

The solution of problems in two-dimensional potential theory depends, generally speaking, on the discovery of a conformal transformation suitable for the particular region involved, and the degree of success in the applicability of the method seems to be seriously limited by the complexity of the transformation. Only one serious attempt seems to have been made, by W. G. Bickley*, to formulate a general theory coordinating the different types of boundary value problem for general and inclusive classes of curves, but Bickley's analysis is somewhat involved and not very suitable for certain types of problem. In a re-examination of the subject on the basis of certain suggestions made to me by Prof. Livens, I have discovered what appears to be a much simpler and equally general method of formulating directly the solution for a comprehensive class of problem dealing with simple closed cylinders with curved or rectilinear boundaries, and including all those cases for which solutions are known. In its theoretical aspects the method seems to be perfectly general, but to put it into practice it is necessary to determine explicitly the particular form for the conformal transformation of the space outside the cross-section of the cylinder to the space outside a circle with the points at infinity coinciding.

Let A = {Aq}q≥0 be an anti-commutative graded ring with 1. Let X be a space and let G(X; A) be the group of multiplicative units of the ring Πi≥0Hi(X; Ai) of the form l + x1 + x2 +…, where xi∊ Hi (X; Ai), In [6] Segal showed that there is a connective cohomology theory G*(X; A), with G0(X;A) = G(X; A). This was improved by Steiner, who in [7] showed that A→G*(X; A) is a functor from anticommutative graded algebras to cohomology theories.