In this paper we will be concerned with studying operators T: C(K, X) → Y defined on Banach spaces of continuous functions. We will be particularly interested in studying the classes of strictly singular and strictly cosingular operators. In the process, we obtain answers to certain questions recently raised by Bombal and Porras in [5]. Specifically, we study Banach space X and Y for which an operator T: C(K, X) → Y with representing measure m is strictly singular (strictly cosingular) whenever m is strongly bounded and m(A) is strictly singular (strictly cosingular) for each Borel subset A of K. Along the way we establish several results dealing with non-compact operators on continuous function spaces, and we consolidate numerous results concerning extension theorems for operators defined on these same spaces. Also, we join Saab and Saab [25] in demonstrating that if l1 does not embed in X* then the adjoint T* of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [2].

Let (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

The free field equations for particles with spin are invariant under a group SL(2, c) whose transformations correspond to changes of representation of the twocomponent spinor algebra. The generalization of the equations which extends this invariance to a guage invariance in the Yang–Mills sense necessitates the introduction of auxiliary fields (which are also necessary to maintain Lorentz covariance). These fields can be interpreted as the potentials of a spin-2 field, just as the auxiliary fields for the charge gauge group are the potentials of a spin-l field (electromagnetism); this spin-2 field is then self-interacting. The Bargmann–Wigner formulation of the linear spin-2 field, when modified by the proposed self-interaction, yields a non-linear theory of a spin-2 field which is shown to be identical with Einstein's gravitational theory. With this interpretation the auxiliary fields take on an extra role of Yang–Mills field for the general coordinate transformation group – that is, they are the components of the affine connexion.

The following result has at least the appeal of intuitive plausibility. Let U and V be subvarieties of an algebraic variety X; let x ∈ X be an isolated point of the intersection of U and V, and let I(X, U. V, x) denote the intersection multiplicity (in some sense to be made precise) of U and V at x. Now let Tx be the tangent space to X at x, within which lie the tangent cones Cu, Cv to U and V at x. Then

Let us denote the group of based homotopy classes of seif-homotopy equivalences of a space X by E(X). We consider E0(X), the subgroup of E(X) consisting of elements which induce the identity map on homology. Dror and Zabrodsky have shown that E0(X) and the subgroup E#(X) consisting of elements inducing the identity on homotopy are both nilpotent groups for finite-dimensional nilpotent spaces, or finite-dimensional spaces respectively ([4], theorem D, theorem A). The theory of localization for nilpotent groups has been developed by several authors (see [8]). The aim of this paper is to prove the following theorem. The corresponding result for E#(X) is obtained in [9], theorem 0·1.

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [4] for basic definitions and elementary results.

In a recent paper, G. N. Lewis and D. F. Smith have discussed the problem of the velocity of chemical reactions, especially that of the only gas-reaction, the decomposition of N2O5, which had then been proved with sufficient accuracy to be kinetic unimolecular, i.e. unimolecular in the purely empiric sense of the word.

Valiron[3] showed that if f(z) is an entire function for which

where A(r) = log(sup{|f(z)|:|z| = r}), then there is a function x(r) = rρ(r) satisfying both

Although examples have been known since 1969 [Si], which reveal a quite similar failure of the differentiable and piecewise-linear s-cobordism theorems, these did not immediately suggest any topological failure.

The main result of the present paper is the construction of a very oscillatory inner function F in the unit disc. The existence of such an F leads to a certain extension of Fatou's classical theorem concerning the boundary behaviour of H∞-functions.

We prove an equivariant version of Galin's theorem on versal deformations of infinitesimally symplectic matrices. Matrix families of codimension zero and one are classified, and the results are used to study the movement of eigenvalues in one parameter families.

Generalities. Suppose that

is real and L2(0, ∞), that

is its Mellin transform, and that

Then K(1)(x) generates a “Watson transformation”, with a “Fourier theorem”

valid for every F(x) of L2(0, ∞).

We show that for every $n\,{\geq}\,2$ there exists closed hyperbolic n-manifolds for which the full group of orientation preserving isometries is trivial.

Improved sufficient conditions are developed for the functional completeness of a set F of functions with variables and values ranging over N = {1, 2, …, n}, where n ≥ 3. In particular, F is complete if it generates a triply transitive group of permutations of N and contains either (i) only a single function, or (ii) at least one function satisfying the Slupecki conditions, the latter apart from certain exceptional cases. These are shown by a detailed investigation to occur only when n = 3 or when n is a power of 2 and all functions of F are linear in each variable, relative to some representation of N as a vector space over Z2.

In (3) W. Gaschütz introduced the concept of a formation of finite soluble groups, generalizing the results of R. W. Carter on the existence and conjugacy of nilpotent self-normalizing subgroups in finite soluble groups (2). Carter's results have already been extended to two classes of infinite groups in (9), (10), and the object of the present work is to extend the results of Gaschütz to a class of infinite groups, thereby generalizing (9).

A new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Let L be a Lie ring and denote the product of x and y in L by [x, y]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, yεL there is an integer k = k(x, y)such that

If k(x, y) can be taken equal to a fixed integer n for all x, y ε L then L is said to satisfy the n-th Engel condition.

A number of methods are at present available for the numerical solution of linear ordinary differential equations over a restricted range of the independent variable. Of these perhaps the most commonly used are methods based on finite differences which have as their aim the construction of a table of values of the dependent variable at stated intervals of the independent variable. The appropriate method to be used depends upon the boundary conditions. Thus for second-order equations, for example, if boundary conditions occur at each end of the range of integration the relaxation method of Southwell (6) is suitable, whereas if this is not the case resort may be made to step-by-step methods or the recently proposed method of Allen and Severn(1). This method, applicable to equations of any order, brings the problem within the scope of normal relaxation methods at the expense of an increase in the order. In all such methods the accuracy attainable for a given (small) amount of labour is a worth-while consideration, and methods have been suggested by Fox (3) and Fox and Goodwin (4) for improving this by the use of comparatively large intervals of the independent variable.

In the course of some calculations on the lifetimes of excited states of mercury and potassium(9) and photo-ionization of excited potassium, wave functions were derived for various excited states of these atoms. Tables of these wave functions are presented in this paper.