Let $k\ge 3$ be an integer. We study the possible existence of finite sets of positive integers such that the product of any two of them increased by 1 is a $k$th power.

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.

A sufficient condition for the equisingularity of inverse images of plane curve singularities, as well as a formula for their Milnor numbers, are given.

Filtrations of modules over wreath products of algebras are studied and corresponding multiplicity formulas are given in terms of Littlewood–Richardson coefficients. An example relevant to Jantzen filtrations in Schur algebras is presented.

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.

On page 209 of his lost notebook, Ramanujan records an unusual product formula, reminiscent of a product of theta functions. The formula involves hypergeometric functions and has a connection with elliptic functions. In this paper, we prove the formula, offer some generalizations, and indicate some further connections with Ramanujan's work.

The aim of this paper is to answer a question of Coates and Greenberg: let $F$ be a commutative $m$-dimensional formal group over the ring of integers of a local field $k$, and let $K$ be an algebraic extension of $k$ with infinite ramification index. Denote by ${\cal M}_{\Mbar}$ the maximal ideal in the ring of integers of the separable closure of $K$. Suppose that the height of $F$ is greater than $m$. Does $H^1 (K, F ({\cal M}{}^m_{\Mbar}))=0$ imply that $K$ is deeply ramified? The answer is positive.

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 $g$.

We identify all non-elementary Kleinian groups which can be generated by two elliptic elements whose commutator is also elliptic. Among those, we identify the finite sets which have finite covolume.

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).

For any $x \in (0,1]$, let the series $ \sum_{n=1}^{\infty}1/d_n(x)$ be the Sylvester expansion of $x$. In this paper, we consider the Hausdorff dimension of the set $$B(\alpha, \beta)= \bigg\{x \in (0,1]: \lim\limits _{n \to \infty} \frac{d_{n+1}(x)}{d^{\beta}_n(x)}=\alpha\bigg\}$$ for any $\alpha \geq 0$ and $\beta \geq 2$. As a corollary, we answer the question posed by Goldie and Smith in [6].

In this paper we study Iwasawa modules arising from Galois cohomology over general $p$-adic Lie extensions both in the local and global case. In particular we calculate their $\Lambda$-ranks. We then apply the results to abelian varieties.

Let ${\bb D}_p$ be a central simple ${\bb Q}_p$-division algebra of index 2, with maximal ${\bb Z}_p$-order $\Delta_p$. We give an explicit formula for the number of subalgebras of any given finite index in the ${\bb Z}_p$-Lie algebra $\mathcal{L}\colone \spl_1(\Delta_p)$. From this we obtain a closed formula for the zeta function $\zeta_\mathcal{L}(s) \colone \sum_{M \leq \mathcal{L}} |\mathcal{L}:M|^{-s}$. The results are extended to the $p$-power congruence subalgebras of $\mathcal{L}$, and as an application we obtain the zeta functions of the corresponding congruence subgroups of the uniform pro-$p$ group $\SL_1^2(\Delta_p)$.

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.

We prove that a point $x$ in R is out of the support of a pseudomeasure $T$ on R if and only if the Fourier integral of $T$ at $x$ is summable with Cesàro means of order 1 to zero.

We study the pointwise behavior of Birkhoff sums $S_n\phi(x)$ on subshifts of finite type for Hölder continuous functions $\phi$. In particular, we show that for a given equilibrium state $\mu$ associated to a Hölder continuous potential, there are points $x$ such that $S_n\phi(x) - n \mathbb{E}_\mu \phi \sim a n^\beta$ for any $a>0$ and $0< \beta <1$. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of $\mu$ and it is attained by some maximizing equilibrium state $\nu$ such that $\mathbb{E}_\nu \phi = \mathbb{E}_\mu \phi$. On such points the ergodic average $n^{-1}S_n\phi(x)$ converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different choices $(a, \beta)$, are distinct but have the same dimension. This reveals a rich multifractal structure of the symbolic dynamics. As a consequence, we prove that the set of uniform recurrent points, which are close to periodic points, has full dimension. Applications are also given to the study of syndetic numbers, Hardy–Weierstraß functions and lacunary Taylor series.

Let $G$ be a universal Chevalley group defined over an algebraically closed field $F$ of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of $G$ of the irreducible $FG$-modules corresponding to minuscule highest weights via a combinatorial scheme of “chamber systems” indexed over certain cosets in the Weyl group.

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.

For the dual object of any compact, non-abelian group, the theory of $I_0$ sets (sets of interpolation by Fourier transforms of discrete measures) is quite different from that for abelian compact groups. In general, infinite $I_0$ sets need not exist. However, when the dual object contains an infinite (local) Sidon set, we prove that this set itself has an infinite $I_0$ subset.

The proof is constructive and includes some key examples of $I_0$ sets: certain sets of representations of bounded degree and, for products of simple Lie groups, the set of self-representations of the factor groups and their conjugates (the FTR sets).

The concept of ‘varieties of Banach algebras’ was introduced by the author (Quart. J. Math. Oxford (2), 27 (1976), 481–487). Here we develop the theory, producing analogues of the ‘relatively free’ objects in varieties of universal algebras. We take as a test question the problem of describing all varieties which are closed under bicontinuous isomorphisms (i.e. varieties which are also semivarieties), and show that this may be answered easily with the aid of these new concepts. (The answer is, as expected, just those varieties which are ‘algebraically defined’.)