§ 1. Sir John Herschel gave the condition which must be satisfied in order that a symmetrical optical system, free from spherical aberration for two conjugate axial points, may also be free from spherical aberration for two neighbouring and conjugate points upon the axis of the system; but Herschel's condition applies only to first order aberration, i.e. to aberration depending upon the cube of the inclination of the ray to the axis. Abbe shewed, later, that this condition could be included in a wider result, viz. that the spherical aberration, supposed zero, is stationary for axial variations provided that the incident and emergent rays for two conjugate axial points, associated with modified magnification m, satisfy the relation

where θ and θ′ are their initial and final inclinations to the axis; and by ‘modified’ magnification is meant the ratio of the reduced sizes of the image and object.

It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that these in many cases can be obtained as inverse limits.

We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball Bd and the polydisk ${\mathbb D}$d and of composition operators on the Bergman space on the polydisk.

The objective of this note is to complete in one essential way the study undertaken in (2) of the relation between complex bordism and the connective k-homology theory. Specifically, let us denote by MU*( ) the generalized homology theory associated to the Thorn spectrum MU(6), and by k*( ) the generalized homology theory associated to the connective bu spectrum (2, 4). Recall that

Let K be a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on K exhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

Let {λn} be a sequence of non-negative numbers increasing to infinity. Let be any series. Following the usual terminology, we say that the series is summable (R, λn, k) to s if as u → ∞, where . If, further, u−kA(k)(u) is of bounded variation in (0, ∞), we say that the series is summable s. We say that the series is summable (A, λn) to s if converges for all σ > 0, and tends to s as σ →0+. If A, B are two summability methods, we write A → B (‘A implies B’) if they have the property that any series summable A is necessarily summable B to the same sum.

Domb and Salter (3) have postulated that the Debye characteristic temperature for the rare gas solids should be given by where C involves atomic constants and the atomic mass of the rare gas, r is the mean interatomic distance and Φ(r) is the interatomic potential. This formula provides a good approximation for the heavier rare gas solids but breaks down for the lighter inert elements. On the basis of Born's argument (1) that the dynamic rather than the static equilibrium positions should be considered, Domb (2) suggested that (1) might be replaced by , where the additional term is effectively the zero-point energy contribution to the lattice equilibrium. (R is the gas constant.) From experimental measurements of the Gruneisen parameter γ Domb obtained numerical values which, when substituted in the right-hand side of (2), yielded values of vD that appeared to compare favourably with direct experiments. More recently Hooton (6) considered the problem of lattice dynamics with anharmonic forces (which are the dominant factor in the lighter rare gas solids) and developed a theoretical argument which also led to Domb's equation (2).

If λ > ω then there are 2λ graphs of cardinal λ none of them being a minor of another.

We study the question of when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.

In [8] we studied equivariant bifurcation problems with a symmetry group acting on parameters, from the point of view of singularity theory. We followed the now classical theory originated by Damon [5], using the ideas presented in [5, 13, 14]. We adapted general results about unfoldings, the algebraic characterization of finite determinacy, and the recognition problems, to multiparameter bifurcation problems f(x, λ)=0 with ‘diagonal’ symmetry on both the state variables and on the bifurcation parameters. More precisely, such bifurcation problems satisfy the condition f(γx, γλ)=γf(x, λ) for all γ∈Γ, where Γ is a compact Lie group

In this paper we attack the same problem from a different angle: the path formulation. This idea can be traced back to the first papers of Mather [17] and Martinet [15, 16]. It was used explicitly in Golubitsky and Schaeffer [12] (see also their earlier paper [11]) as a way of relating bifurcation problems in one state variable without symmetry to a miniversal unfolding in the sense of catastrophe theory. At that time the techniques of singularity theory were not powerful enough to handle the full power of the idea efficiently – either in theory or in computational practice. This is why the path formulation was abandoned in favour of contact equivalence with distinguished parameters, as developed in Golubitsky and Schaeffer [12]. Considerable progress has been made since then; for example Montaldi and Mond [19] use the path formulation to apply the idea of [Kscr ]V-equivalence introduced by Damon [6] to equivariant bifurcation theory. Bridges and Furter [3] studied equivariant gradient bifurcation problems using the path formulation, and defined an equivalence relation in the space of paths and their unfoldings that respects contact equivalence of the gradients. Here we describe an algebraic approach to the path formulation that has the advantage of organizing the classification of normal forms. Moreover, it minimizes the calculation involved in obtaining the normal forms (compare with the classical framework in Furter et al. [8]). The geometric approach to the path formulation using [Kscr ]V-equivalence is still open in the context of a symmetry group acting diagonally on parameters.

1. Introduction. In the course of an important memoir on integral functions of finite order†, G. Valiron discusses ‘fonctions orientées’, that is, functions with zeros an such that arg an tends to a limit as n → ∞. He obtains results that include the following two theorems, in which Vρ(x) denotes a function of the form , where α1, …, αν are positive integers, and n(r) has its usual significance in the theory of integral functions:.

If M is any algebraic structure, and R is any Boolean ring, then a structure called the (bounded) Boolean power of M by R, denoted MR, can be defined. This construction, which is also called a bounded Boolean extension, is a sort of generalized direct power, and was introduced by Foster in the 1950's (as a refinement of his previous notion of a Boolean extension). In this paper we shall study isomorphism types and automorphisms of Boolean powers of groups, and obtain information about their characteristic subgroups: we shall be chiefly concerned with Boolean powers of finite groups.

All $\cal {A}$-simple singularities of map-germs from $\bb {R}^n$ to $\bb {R}^p$, where $n\ge p$, of minimal corank (i.e. of corank $n-p+1$) have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the discriminant.

The paper considers the solution of the general variational equation of relativistic dynamics using the methods of Mathisson with a generalization at one stage. The solution gives us two equations describing the translational motion and rotational motion. The rotational equation is also investigated by setting up a variational equation using the conservation of angular momentum, and this is found to be of the same form as that obtained from Mathisson's variational equation.

The Formanek polynomial gives an explicit expression for the calculation of central elements in a prime ring with polynomial identity, in such a form as to provide a set of linear maps of the ring into its centre. This note makes use of properties of these maps with the object of giving a direct proof of the remarkable theorem of M. Artin on the relationship between Azumaya algebras and rings with polynomial identities. We show that such a central map can be chosen to have some of the crucial properties of the reduced trace in matrices, notably that of non-degeneracy, and our proof of the theorem rests on these properties. The method can also be used for the study of prime PI rings in general; this is illustrated by giving new proofs of theorems of Formanek and Cauchon. The changes of proof essentially amount to the elimination of the need for a change of rings to obtain a split extension in simple algebras. The main theorem is the following.

Suppose that f and g are transcendental entire functions of finite order, without wandering domains, and suppose that f∘g=g∘f. Then the Julia sets of f and g coincide.

The purpose of this paper is to establish an embedding of QM (A), the Banach space of all quasi-multipliers on A, in the second conjugate space A** of a Banach algebra A, which extends the well-known embedding of the left multipliers (right multipliers) of A in A**. We also prove that, for a dual B*-algebra, QM(A) is isometrically isomorphic to A**.

Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all (p, v)∈ SM, where p ∈M and v is a unit tangent vector at p, the geodesic through p in the direction of v is dense in M. A theorem of Dani [Dl] says that the set of all (p, v)∈SM for which the corresponding geodesic is bounded (namely those with compact closure in M) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

The problem considered here is to determine the couple required to maintain the steady rotation about its axis of symmetry of an oblate spheroid which is placed in a uniform stream of viscous fluid moving parallel to this axis. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. An expression in terms of spheroidal wave functions is derived for the transverse component of velocity in the fluid, and from this the expression for the couple is deduced. The disk rotating about the normal through its centre is considered as a special case, and some numerical results are given for certain values of a Reynolds number. A comparison is made, and agreement found, with results obtained by Jeffery in a previous paper which treated the case in which the mainstream was at rest.

Let {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such that

as n → ∞, where v0 is the p.d.f. of Vo.