We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X\,\backslash \,\left\{ 0 \right\}$ . More generally, if $X$ is an infinite dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$ , and $X/F$ is infinite-dimensional, then $X$ and $X\backslash F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$ -torus on certain Banach spaces.

In this paper we extend a well-known theorem of M. Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

If an operator $T$ satisfies a modular inequality on a rearrangement invariant space ${{L}^{\rho}}\left( \Omega ,\mu\right)$ , and if $p$ is strictly between the indices of the space, then the Lebesgue inequality $\int{|Tf{{|}^{P}}\le C\int{|f{{|}^{P}}}}$ holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form $\int{\Phi \left( |Tf| \right)\le }\int{\Phi \left( C|f| \right)}$ , and here, one can extrapolate to the (finite) indices $i\left( \Phi\right)$ and $I\left( \Phi\right)$ as well.

Let $G$ be a discrete subgroup of $\text{SL}\left( 2,\,\mathbb{R} \right)$ which contains $\Gamma \left( N \right)$ for some $N$ . If the genus of $X\left( G \right)$ is zero, then there is a unique normalised generator of the field of $G$ -automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.

A local version of $\text{VMO}$ is defined, and the local Hardy space ${{h}_{1}}$ is shown to be its dual. An application to weak- $*$ convergence in ${{h}_{1}}$ is proved.

Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov’s question negatively.

We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.

On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.

Let $k$ be a cyclic extension of odd prime degree $p$ of the field of rational numbers. If $t$ denotes the number of primes that ramify in $k$ , it is known that the Hilbert $p$ -class field tower of $k$ is infinite if $t\,>\,3\,+\,2\sqrt{p}$ . For each $t\,>\,2\,+\,\sqrt{p}$ , this paper shows that a positive proportion of such fields $k$ have infinite Hilbert $p$ -class field towers.

Using a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives.

We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real Möbius transformations. Included are the maps that are exactly $n$ -to-1, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

D. H. Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P. Erdős, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.

Let $b$ be an integer with $b\,>\,1$ . In this note, we prove that the number of non-zero digits in the base $b$ representation of $n!$ grows at least as fast as a constant, depending on $b$ , times $\log \,n$ .

Sufficient conditions are given in order to prove the lowest unknown case of the grade conjecture. The proof combines vanishing results of local cohomology and the ${{S}_{2}}$ condition.

We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining the model set is measurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

We give a topological interpretation of the core group invariant of a surface embedded in ${{S}^{4}}\,\left[ \text{F-R} \right],\,\left[ \text{Ro} \right]$ . We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of ${{S}^{4}}$ with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres.

Let ${{X}^{5}}\,+\,aX\,+\,b\,\in \,Z\left[ X \right]$ have Galois group ${{D}_{5}}$ . Let $\theta $ be a root of ${{X}^{5}}\,+\,aX\,+\,b$ . An explicit formula is given for the discriminant of $Q\left( \theta\right)$ .

Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.