The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ, gΣ) embedded into a complete five dimensional manifold (M5, g) with positive scalar curvature R and nonnegative Ricci curvature. Under a suitable choice, we have $area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ, gΣ) is isometric to a standard sphere ($\mathbb{S}$4, gcan) and in a neighbourhood of Σ, (M5, g) splits as ((-ϵ, ϵ) × $\mathbb{S}$4, dt2 + gcan) and the Riemannian covering of (M5, g) is isometric to $\Bbb{R}$ × $\mathbb{S}$4.

Analytical reflexion coefficients are obtained for horizontally and vertically polarized electromagnetic waves respectively, obliquely incident from below two recently devised models of horizontally stratified isotropic media. It is shown that both models are totally reflecting and the phases of the reflexion coefficients may be expressed in terms of arg Γ(z).

In this paper we obtain various sufficient conditions for the solubility of a finite group. In particular, we show that if G is a finite group and p≥5 is a prime such that all p′-subgroups of G are nilpotent, then G is soluble. We show also that if G is a finite group which has a cyclic Sylow p-subgroup Pand such that for all p′-subgroups H of G, H is nilpotent and H′ is cyclic, then, if p≠3, either P◃G or G has a normal p-complement.

It follows from the Krein-Milman theorem that (c0) is not isomorphic to the dual of a Banach space. Using a technique due to Banach ((4), page 194) we shall extend this result to show that if a subspace of (c0) is isomorphic to the dual of a normed linear space, then it is finite dimensional (Proposition 1). Using this result, we shall show that if E is a normed linear space, the unit ball of which is contained in the closed absolutely convex cover of a weak Cauchy sequence, then Eis finite dimensional (Proposition 2). This result has applications to the Banach-Dieudonné theorem, and to the theory of two-norm spaces.

L'aspect fractal de la suite des diviseurs d'un entier normal peut être révélé de diverses manières, comme l'existence d'une dimension de Hausdorff convenablement définie [5]. Cependant, les critères les plus fins sont liés à la conjecture d'Erdős datant de la fin des années 1930, mentionnée dans [3] et établie dans [6], selon laquelle presque tout entier possède deux diviseurs dont le rapport est dans l'intervalle ]1,2[.

In this paper we shall generalize a theorem of Bogoyavl'enskii (2) showing the equivalence of two apparently different families of integrals of the ‘higher stationary Korteweg–de Vries (KdV)’ equations. We recall that the KdV equations form a hierarchy of evolution equations

for an unknown function u(x, t). The equations can be written in ‘Lax form’ (7)

where L is the Schrödinger operator (δ2/δx2) + u, and P+ runs through all the (ordinary) differential operators such that the commutator on the right of (1·1) has order zero.

The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of , that with respect to the height are “very close” to a given subgroup Γ of finite rank of . Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.

In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and also show that each provides a necessary and sufficient set of conditions for the embeddability of a category in a groupoid. We show that all such conditions, and more besides, may be derived in a uniform way from a particular class of directed graphs which we call quadrangle clubs, and we prove a number of results (extending those of Mal'cev, Lambek, Bush and Krstić) on which families of quadrangle clubs provide sufficient conditions for embeddability.

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$-extension, then it holds for almost all $\mathbb{Z}_{p}$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Let Mn be a smooth closed n-dimensional manifold and let DIFF (Mn) be the group of diffeomorphisms of Mn. Two diffeomorphisms f0, f1 ∈ DIFF(Mn) are said to be concordant (pseudo-isotopic) if there is a diffeomorphism F ∈ DIFF (Mn × I), where I = [0, 1], such that F(x, 0) = f0(x) and F(x, 1) = f1(x) for all x ∈ Mn.

Proposition 2ċ5 of [5] states that a full coaction of a locally compact group on a C*-algebra is nondegenerate if and only if its normalisation is. Unfortunately, the proof there only addresses the forward implication, and we have not been able to find a proof of the opposite implication. This issue is important because the theory of crossed-product duality for coactions requires implicitly that the coactions involved be nondegenerate. Moreover, each type of coaction — full, reduced, normal, maximal, and (most recently) exotic — has its own distinctive properties with respect to duality, making it crucial to be able to convert from one to the other without losing nondegeneracy.

We present infinite analogues of our splinter lemma for constructing nested sets of separations. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.

Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterise those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).

It is well known that there is no static axisymmetric two-body solution of Einstein's gravitational field equations, if it is assumed that the bodies are separated in a certain definite sense. In this paper it is shown, by the construction of a complete physically sensible model, that static two-body solutions do exist for systems in which one body is hollow and contains the other. The stability of the particular system described is briefly discussed.

Selmer(1) conjectured that the Hasse principle holds for all cubic surfaces of the type

that is, such a surface has a rational point whenever it has points defined over every p-adic field Qp; and he proved this assertion in the case that ab = cd.

In a previous paper(1) a number of problems in probability were considered arising out of the finite resolving time of a recording apparatus. It was shown that, although the probability distributions themselves are complicated, their Laplace transforms are relatively simple. Recently Feather(2) has discussed a number of additional problems arising in this connexion, including the number of coincidences, and the number of k-clusters. The method developed in (1) is readily applicable to these problems, and it will be shown how complete probability distributions can be derived. An alternative type of recorder is sometimes considered (3) which remains dead as long as events succeed each other at intervals less than τ. The mathematical problem here is somewhat different, but it can again be effectively dealt with by the use of Laplace transforms.

Series which behave rather like Taylor's series at more than, one point are described and their convergence is discussed briefly.

Let

where z, zs are complex numbers and zs are the points, (the base points)about which a function is to be expanded. They need not be distinct. p{z) is a polynomial of degree n. in z. Assume first that z, zs are real and the zs are distinct. Then polynomials pn-1,t(z) may be chosen so that the series

has contact of order N at the points zs with any specified real function which has N derivatives at these points. This is proved merely by rearranging the simplest polynomial in z which has contact of order N with/(z) at z = zs. The term given by t = 0 in (2) allows the series to have the values f(zs) at z = zs; adding the term for t = 1 does not alter the values of the series at z = zs; it allows the series to have a derivative with values f′(zs) at z = z; similarly adding the term for t = 2 does not alter the value of the series or its first derivative at z = zs, it allows the series to have a second derivative with values f″{zs) at z = zs; and so on. If m of the zs coincide then f(z) has mN + m− 1 derivatives there and contact is of order mN + m− 1. If z, zs are complex the result is seen to be valid subject to the obvious generalization of ‘ contact of the ith order’ to complex variables and functions.