This paper derives some sufficient conditions for the global validity of projective relativity theory. The basic object of study is the infinitesimal bundle, which is a generalization of a principal circle bundle. A method ofconstructing infinitesimal bundles is developed. The main result is that ifspacetime is non-compact and simply connected, and has a finite homology basis, then projective relativity is globally valid.

We give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$ . We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$ . For $(1/2,1]$ , we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$ , where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$ .

Let a1, a2, … be a sequence of variables. Write

We use the notation Bn = Bn (a1, …, an) to indicate the variables on which Bn depends.

In this note we first show, using results of McDuff, that for a Cr-manifold S, diffeomorphic to the interior of a compact manifold with boundary, the class of all Cr-diffeomorphisms lying in a one parameter group of S generates the connected component of 1S in Diffr (S, S). Then we use this result to obtain two extension theorems for stratified maps defined on some strata of a stratified space [Xscr ]. Our extension theorems hold for Mather's abstract stratified sets, for Whitney (b)-regular, Bekka (c)-regular, Verdier (w)-regular and Lipschitz-regular stratified spaces.

The theory of large elastic deformation of reinforced elastic materials is applied to discuss the rotation of a circular tube of incompressible elastic material, reinforced with inextensible cords lying along helical paths.

We study the variation of the φ-Selmer groups of the elliptic curves y2 = x3 − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integer D is constrained to lie in a family for which the relative Tamagawa number of the isogeny φ is fixed.

In an article on the philosophy of chance, Poincaré[7] showed that the distribution of the stopping position of a needle pivoted about its centre tends to the uniform distribution on the circle as the distribution of the initial push becomes spread out along the real line. This result was formalized by Feller [2] and strengthened by Mardia [6] as follows.

It is trivial that a group all of whose elements except the identity have order two is Abelian; and F. Levi and B. L. van der Waerden(1) have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R. Baer(3) has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.

In this paper, we use the Shintani decomposition, known to number theorists, to describe an effective method of finding a resolution of the cusps of a Hilbert modular variety, in any dimension.

The incitement, or provocation, to assemble the following pages was the recognition in a recent paper (5) of a geometrical property characterizing those pairs, in a pencil of quadric surfaces, such that the quadrics of a pair are both outpolar and inpolar to each other: symmetrically apolar let it be said. That there are three such pairs was known (8), but their identification by means of the principal chords of their common curve had apparently not been remarked upon. The question naturally arose as to the number and identification of such pairs of quadrics of a pencil in n-dimensional projective space [n]; it is easy to show that the number is ½n(n–1), but geometrical identification seems quite another matter and no success whatever has yet been achieved even when n is 4.

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown that

can be expressed in the form

where Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In addition

where is the partition of n conjugate to (λ). In fact, Littlewood (5) showed that

where the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and

.

In the preceding paper, Blackett and Williams described an automatic curve follower for use with the differential analyser. The device would only operate if the x-coordinate was either continuously increasing or continuously decreasing; it could not handle input functions where the x value pulsated, as may often happen when the x-coordinate is not the independent variable.

In [1], J. Bruna studied the Beurling classes EM(I) of infinitely differentiable functions f defined on an interval I such that for every positive ε, there exists a constant Cε > 0 with the property that

where M = {Mn} is a given sequence of positive numbers. With the hypothesis that the class EM(ℝ) is differentiable, he proved that it is inverse-closed (in the sense that if fεEM(ℝ) and if f is bounded away from zero on ℝ, then its inverse f-1 lies in EM(ℝ)) if and only if the associated sequence A = {An = (Mn/n!)1/n} is almost increasing (i.e. Am ≤ KAn for all m ≤ n, where K > 0 is a constant independent of m and n).

We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and between wheeled properads and modular operads.

A systematic analysis is made of the equation ∇. [k(r) ∇ψ] = 0 for useful profiles k(r) in spherically stratified media which enable the radial variation of the potential function ψ to be expressed exactly in terms of hypergeometric functions or confluent hypergeometric functions. Previously known profiles are shown to emerge as special cases of the present analysis.

It is well known that the trace X([0, ∞) of the 3-dimensional Brownian motion X has positive capacity and, therefore, the harmonic measure is well defined on this set. It is shown that a.s., this harmonic measure is singular with respect to the occupation measure Lebesgue ο X−1.

We fill a gap in the construction of the extended Johnson homomorphism, stated in Section 2 of the paper The abelianization of the congruence IA-automorphism group of a free group.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings–Jaco theorem which established a similar result for the Poincaré Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

We prove that for any non-planar graph H, we can choose a two-colouring G of H such that G is intrinsically chiral, and if H is 3-connected and is not K3,3 or K5, then G is intrinsically asymmetric. No such asymmetric two-colouring is possible for K3,3 or K5.

Let m ≥ 0 and p, p0, …, pm be integers such that p, pj are relatively prime and 1 ≤ Pj < p for 0 ≤ j ≤ m. In the (2m + 2)-dimensional Euclidean space R2m+2 with points (x0, …, x2m+1), the (2m + l)-sphere S2m+1