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.

The main aim of this paper is to give a description for the structure of the (small) quantum cohomology ring of the toric variety $X={\bb P}(\oplus _{i=1}^r{\cal O}_{{\bb P}^1}(a_i))$ with $\sum_{i=1}^ra_i=\epsilon +k r$ and $\epsilon \in \{0,1 \}$. As we explain later, the importance of this result relies on the fact that, unless $k =0$, $X$ is a non-Fano toric variety and the fact that we determine not only a presentation of the quantum cohomology ring $QH^*(X;{\bb Z})$ but also all quantum products $\alpha* \beta$ with $\alpha, \beta \in H^*(X;{\bb Z})$ or, equivalently, all three-point genus-0 Gromov–Witten invariants.

Let $M$ be the Hardy–Littlewood maximal function and let $ M_{(\alpha)}$ be the $\ell^\alpha$-maximal operator defined by $$M_{(\alpha)} \bar{f}(x)=\|{M\bar {f}\|_{\ell^\alpha}=\Bigg(\sum_{i=1}^\infty Mf_i(x)^\alpha\Bigg)^{1/\alpha},$$ where $\bar f=(f_i)_i$, $M\bar f=(Mf_i)_i$ and $\alpha >1$. The purpose of this work is to study modular inequalities of the form $$\int_{{\mathbb R}^n} P \big(\big|\overline M_{(\alpha)}f(x)\big| \big)\,dx \le \sum_j\int_{{\mathbb R}^n } Q(|f_j(x)|)\, dx,$$ where $P$ and $Q$ are modular functions. Our results apply to operators of the form $$\overline T_{(\alpha)} \bar f(x)= \Bigg(\sum_{i=1}^\infty |T_i f_i(x)|^\alpha\Bigg)^{1/\alpha},$$ where $T_i$ satisfies similar properties to $M$.

We obtain new stability conditions for $C_0$-semigroups on Banach spaces having nontrivial Fourier type. On Hilbert spaces these conditions are sharp. For $C_0$-semigroups on general Banach spaces, we prove a new individual stability criterion. We also show that stronger versions of the range stability condition from [4] are not necessary for stability. This answers an open problem from [4].

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.

A $genus$ (in the sense of Hirzebruch [4]) is a multiplicative invariant of cobordism classes of manifolds. Classical examples include the numerical invariants given by the signature and the $\widehat{A}$- and Todd genera. More recently genera were introduced which take as values modular forms on the upper half-plane, $\frak{h}=\{\,\tau\;|\;\mathrm{Im}(\tau)>0\,\}$. The main examples are the elliptic genus $\phi_{ell}$ and the Witten genus $\phi_W$; we refer the reader to the texts [7] or [9] for details.

In this paper, we give a positive answer for a special case of a problem posed by McGibbon [4]. We prove that a connected space with all homotopy groups finite has an H-space structure if and only all Postnikov approximations carry some H-space structure.

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.

The notion of a completely hyperexpansive operator has been introduced in [2] by Athavale. In this paper bounded and unbounded hyperexpansive composition operators are investigated. A unique semispectral measure is associated with a bounded completely hyperexpansive composition operator. Examples of bounded and unbounded densely defined $2$-isometric (completely hyperexpansive, $2$-hyperexpansive) composition operators with invariant domains are given.

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the $n$-fold strongly-cyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every $n$-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus $n$. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type $(k,hk\pm 1)$.

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.

We consider an analytic function-germ $f{:}\,(\mathbb{R}^n,0) \rightarrow (\mathbb{R},0)$ with an isolated critical point at $0$. For a sufficiently small ball $B_\varepsilon$ of radius $\varepsilon$ and a sufficiently small regular value $\delta$ of $f$, we give degree formulas for the following Euler–Poincaré characteristics: $$\chi( f^{-1}(\delta) \cap B_\varepsilon \cap \{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $k \in \{1,\ldots,n\}$ and $\epsilon_i \in \{0,1\}$. This leads to degree formulas for $$ \chi (\{f * 0\} \cap S_\varepsilon \cap \{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $* \in \{\le,=,\ge\}$. Combining this with the Eisenbud–Levine–Khimshiashvili's formula, we obtain signature formulas for $$\chi( W \cap\{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $W$ is either the link of a weighted-homogeneous singularity or any compact algebraic set of $\mathbb{R}^n$.

It is known that knot groups are right-orderable, and that many of them are not bi-orderable. Here we show that certain fibred knots in $S^3$ (or in a homology sphere) do have bi-orderable fundamental groups. In particular, this holds for fibred knots, such as $4_1$, for which the Alexander polynomial has all roots real and positive. This is an application of the construction of orderings of groups, which are moreover invariant with respect to a certain automorphism.

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 $X$.

We provide a new proof of the following results of H. Schubert: if $K$ is a satellite knot with companion $J$ and pattern $(\skew1\hat{V}, L)$ with index $k$, then the bridge numbers satisfy the following: $b(K) \geq k \cdot (b(J))$. In addition, if $K$ is a composite knot with summands $J$ and $L$, then $b(K) = b(J) + b(L) - 1.$

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.

In this paper we consider hyperbolic manifolds which are $p$-fold cyclic branched coverings of a link where $p$ is a prime number different from two; we prove that the Sylow $p$-subgroups of the orientation-preserving isometry groups of these 3-manifolds are either abelian of rank at most three or $p$ is equal to three and a Sylow $p$-subgroup is a $\bb{Z}_3$-extension of an abelian subgroup of rank three. As a corollary of this result we obtain that there are at most three inequivalent links which have the same hyperbolic 3-manifold as a $p$-fold cyclic branched covering and we describe the relation between two such links.

In this paper, we consider the energy behaviour of Rayleigh surface waves in a homogeneous isotropic 3-dimensional half space with free boundary condition. It is known that Rayleigh waves with finite energy are asymptotically concentrated along the boundary. We clarify the domain where the energy of the Rayleigh waves concentrates and obtain the decay order of the energy distributed in the complement of the domain.

In this paper several results concerning the periodic points of 1-norm non-expansive maps will be presented. In particular, we will examine the set $R(n)$, which consists of integers $p\geq 1$ such that there exist a 1-norm nonexpansive map $f{:}\ \mathbb{R}^n\rightarrow\mathbb{R}^n$ and a periodic point of $f$ of minimal period $p$. The principal problem is to find a characterization of the set $R(n)$ in terms of arithmetical and combinatorial constraints. This problem was posed in [12, section 4]. We shall present here a significant step towards such a characterization. In fact, we shall introduce for each $n\in\mathbb{N}$ a set $T(n)$ that is determined by arithmetical and combinatorial constraints only, and prove that $R(n)\subset T(n)$ for all $n\in \mathbb{N}$. Moreover, we will see that $R(n)=T(n)$ for $n=1,2,3,4,6,7$, and 10, but it remains an open problem whether the sets $R(n)$ and $T(n)$ are equal for all $n\in \mathbb{N}$.