Let $F$ be a perfect field and $M(F)$ the non-associative simple Moufang loop consisting of the units in the (unique) split octonion algebra $O(F)$ modulo the center. Then ${\rm Aut}(M(F))$ is equal to $G_2(F) \rtimes {\rm Aut}(F)$. In particular, every automorphism of $M(F)$ is induced by a semilinear automorphism of $O(F)$. The proof combines results and methods from geometrical loop theory, groups of Lie type and composition algebras; the result being an identification of the automorphism group of a Moufang loop with a subgroup of the automorphism group of the associated group with triality.

An interesting limit formula for the Riemann Zeta function $\zeta (n) (n\in \mathbb{N}\backslash \{1\})$ was contained implicitly in a paper by K. S. Williams [17]. In the case of $\zeta(2n)\ (n\in \mathbb{N})$, we show that Williams' limit formula, and three other analogous limit formulas proven here, involve polynomials of degree $2n$. We also determine these polynomials explicitly and deduce, as an immediate consequence, Euler's celebrated relation between $\zeta(2n)$ and the familiar Bernoulli numbers $B_{2n}$. Each of our closed-form summation formulas, expressing a finite trigonometric sum in terms of higher-order Bernoulli polynomials, is capable of yielding many (new or known) special cases and consequences.

Let $G$ be a group, $g$ an element of $G$ and $n$ a natural number, $n\geqslant 2$. In [8] B.H. Neumann considered the equation $r_0(t)\deq t^ng=1$ and proved its solvablility in an overgroup. A natural generalisation of this equation is the equation $r(t)=1$, where $r(t)$ is defined as follows: let $W\in G\ast\langle t\rangle$ be a cyclically reduced word of length at least 2 which is not a proper power in $G\ast\langle t\rangle$. Define ${(\ast)}\hspace{4.8cm}r(t)\deq r_0(W)=W^ng$. In this paper we show that for $n\geqslant 3$ the equation $r(t)=1$ is solvable in an overgroup with some structural properties, under certain conditions on $g$ and $W$. It is conjectured in [3, p. 131] that this equation is solvable over an arbitrary group, without conditions on $W$ or $g$.

Let $f$ be a real function, continuous over ${\mathbb R}^d$, $d\geq1$, and let $\Gamma(f,D)\,{\hbox{\sun}}\,\{(x,f(x))|x\in D\}$ be its graph over a domain $D\subseteq {\mathbb R}^d$ and $\Gamma{f}$ its graph over ${\mathbb R}^d$. The question naturally arises about which (and how) smoothness properties of $f$ are related to the Hausdorff dimension of $\Gamma(f)$. It is a well-known result that if $f$ is locally $\alpha$-Hölder, which we denote by $f\in\Fspaceloc{C}{\alpha}$, for some $\alpha>0$, then $\mathrm{dim}_{\mathcal{H}}(\Gamma(f))\leq \max(d,d+1-\alpha)$. Of course, in this inequality, one can replace $\alpha$ by the Höolder exponent of $f$, i.e. by $\sup\{s:f\in\Fspaceloc{C}{s}\}$. The question of the sharpness of the obtained bound yields different outputs depending on $f$: one can show that the equality is satisfied over some classes of functions (see [5] and [1, chapter 8]) but it is not difficult to find examples for which the inequality is strict. One should also mention that in the simple case of the well-known Weierstrass function, neither the equality nor a smaller upper bound could be proved (see [11] for related results).

We discuss the exact number of almost periodic solutions of certain ordinary differential equations of the second order. The class of equations under consideration is inspired by a well-known result in the area of elliptic boundary value problems.

It is a well-known result in Diophantine approximation over the reals that the Lebesgue measure of the elements $x \in {\mathbb{R}}$, such that $\Vert qx \Vert < {1}/{q^v}$ for infinitely many $q \in {\mathbb{R}}$, is full for $v \leq 1$ and null otherwise, where $\Vert\cdot\Vert$ denotes the distance to nearest integer. Also, the Hausdorff dimension of the set of these numbers is known to be ${2}({v\,{+}\,1})$ for $v > 1$. Similar results are known for matrices over the reals as well as the $p$-adics. In this paper, we prove the corresponding multi-dimensional result for matrices over the field of Laurent series with coefficients from a given finite field.

In this paper we study the projectivity of Hilbert modules over uniform algebras, first in the category of Banach modules and then in the smaller category of Hilbert modules. We prove that such modules are never projective as Banach modules in the case where the uniform algebras have a metrizable character space without any isolated points. On the other hand, we give an example, whenever the character space of the uniform algebra is metrizable, of a one-dimensional module, which is projective as a Hilbert module.

The variational sequence describes the Helmholtz conditions for local variationality in terms of the Helmholtz map, which is defined on a factor space. We study a tensor modification of this construction and characterize a unique representative, which is called the Helmholtz operator. For the first and the second order cases we prove that, up to a multiplicative constant, the Helmholtz operator is the unique natural operator of the type in question.

We develop some of the basic theory of quaternionic hyperbolic geometry. We give necessary criteria for groups of quaternionic hyperbolic motions to be discrete. We give lower bounds on the volumes of cusped quaternionic hyperbolic manifolds.

By defining new Bryant-type vector fields for foliations on a Riemannian manifold we find necessary and sufficient conditions that a foliation produces $p$-harmonic morphisms. Two applications are given. First, we characterize one-parameter conformal actions using $p$-harmonic morphisms. Then we classify $p$-harmonic morphisms on a constant curvature space with one-dimensional fibres by studying bi-minimal distributions. We also give a description of all conformal foliations which have minimal fibres on a Riemannian manifold in terms of $p$-harmonic morphisms.

A knot in a thickened surface $F^2 \times \mathbb{R}$ is called a global knot if its projection to $F^2$ is transversal to a vector field on this surface, which has at most critical points of index −1. Global knots generalize closed braids. We introduce new knot invariants of finite type which are trivial for all global knots. These invariants are a very effective tool for showing that a given knot in $F^2 \times \mathbb{R}$ is not isotopic to any global knot.

Let $X\,{=}\,\{X(t), \ t \in {\R^N}\}$ be a multiparameter fractional Brownian motion of index $\alpha$ ($0< \alpha < 1$) in $\R^d.$ We prove that if $N < \alpha d\ $, then there exist positive finite constants $K_1$ and $K_2 $ such that with probability 1, $$ K_1 \le \hbox{$\varphi$-$p(X([0,1]^N))$} \,{\le} \hbox{ $\varphi$-$p({\rm Gr}X([0,1]^N))$} \,{\le}\, K_2$$ where $\varphi(s) = s^{N/\alpha}/(\log \log1/s)^{N/(2 \alpha)}$, $\varphi$-$p(E)$ is the $\varphi$-packing measure of $E$, $X([0, 1]^N)$ is the image and ${\rm Gr}X([0, 1]^N) \,{=}\, \{(t, X(t)); \ t \in [0, 1]^N\}$ is the graph of $X$, respectively. We also establish liminf and limsup type laws of the iterated logarithm for the sojourn measure of $X$.

Consider a sphere of radius $\sqrt{n}$ in $n$ dimensions, and consider $\vc{X}$, a random variable uniformly distributed on its surface. Poincaré's Observation states that for large $n$, the distribution of the first $k$ coordinates of $\vc{X}$ is close in total variation distance to the standard normal $N(\vc{0},I_k)$. In this paper we consider a larger family of manifolds, and $\vc{X}$ taking a more general distribution on the surfaces. We establish a bound in the stronger Kullback–Leibler sense of relative entropy, and discuss its sharpness, providing a necessary condition for convergence in this sense. We show how our results imply the equivalence of ensembles for a wider class of test functions than is standard. We also deduce results of de Finetti type, concerning a generalisation of the idea of orthogonal invariance.