the total space of the tangent bundle of a kähler manifold admits a canonical kähler structure. parallel translation identifies the space ${\mathbb{t}}$ of oriented affine lines in ${\mathbb{r}}^3$ with the tangent bundle of $s^2$. thus the round metric on $s^2$ induces a kähler structure on ${\mathbb{t}}$ which turns out to have a metric of neutral signature. it is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the euclidean metric on ${\mathbb{r}}^3$.

the geodesics of this metric are either planes or helicoids in ${\mathbb{r}}^3$. the signature of the metric induced on a surface $\sigma$ in ${\mathbb{t}}$ is determined by the degree of twisting of the associated line congruence in ${\mathbb{r}}^3$, and it is shown that, for $\sigma$ lagrangian, the metric is either lorentz or totally null. for such surfaces it is proved that the keller–maslov index counts the number of isolated complex points of ${\mathbb{j}}$ inside a closed curve on $\sigma$.

Some of the arguments and techniques developed by the authors in a previous paper are applied to the study of the boundedness of certain operators associated with the Ornstein–Uhlenbeck semigroup. In particular, a simple proof is given of the weak type 1 with respect to the Gaussian measure of the Riesz transforms of order 1 and the Littlewood–Paley g-function which then is extended to show the same property for the Riesz transforms of order 2. For Riesz transforms of higher order boundedness is shown in appropriate spaces close to L1.

A set $A\subseteq \{1,2,\ldots,n\}$ is said to be $k$-separated if, when considered on the circle, any two elements of $A$ are separated by a gap of size at least $k$.

A conjecture due to Holroyd and Johnson that an analogue of the Erdős–Ko–Rado theorem holds for $k$-separated sets is proved. In particular, the result holds for the vertex-critical subgraph of the Kneser graph identified by Schrijver, the collection of separated sets. A version of the Erdős–Ko–Rado theorem for weighted $k$-separated sets is also given.

In most cases, the third order approximation of a scalar Newtonian equation can lead to the Lyapunov stability of a periodic solution through the obtaining of a nonzero twist coefficient. Recently, Ortega obtained the twist property of a periodic solution when the second order coefficient does not change sign and the third one is negative under a crucial limitation to the rotation of the linearization equation. The paper finds that the best bound on the limitation of the rotations is $\theta^*_0=\arccos(-1/4)$.

By considering branched coverings of piecewise Euclidean cubical complexes, the paper provides an example of a torsion free hyperbolic group containing a finitely presented subgroup which is not hyperbolic.

We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL2(ℤ). This completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E7 and E8 over fields of odd characteristic occur as Galois groups of geometric extensions of ℚab(t), where ℚab denotes the maximal Abelian extension field of ℚ. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.

We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl–Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups ${\mathcal O}_q(\text{GL}_n)$ and ${\mathcal O}_q(\text{SL}_n)$.

It is proved that, for a Banach space X, the following properties are equivalent: (a) (X, weak) is fragmentable by a metric d(., .) that majorizes the norm topology (that is, the topology generated by d contains the norm topology), (b) (X, weak) is fragmentable by a metric d(., .) that majorizes the weak topology, and (c) (X, weak) is sigma-fragmentable by the norm. The paper gives a game characterization of these equivalent properties and uses it to show that a large class of Banach spaces (including the spaces that are Čech-analytic in their weak topology) enjoy the properties (a)–(c). In particular, if the unit ball B of X is a Borel subset of the second dual ball (B**, weak*) then X possesses the properties (a)–(c). This provides an alternative approach to the proofs of some of the results of J. E. Jayne, I. Namioka and C. A. Rogers on sigma-fragmentability of Banach spaces. It is also proved that C(T), with the topology p of pointwise convergence, is sigma-fragmentable by the norm whenever T=∪i[ges ]1Ti and (C(Ti), p), i[ges ]1, is sigma-fragmentable by the norm. This answers a question of R. Haydon. Finally, a simple proof is given that l∞ does not have any of the properties (a)–(c) and that (l∞(Γ), weak) is not sigma-fragmentable by any metric, provided that the cardinality of Γ is bigger than the continuum.

Finitely generated modules with finite $F$-representation type over Noetherian (local) rings of prime characteristic $p$ are studied. If a ring $R$ has finite $F$-representation type or, more generally, if a faithful $R$-module has finite $F$-representation type, then tight closure commutes with localizations over $R$. $F$-contributors are also defined, and they are used as an effective way of characterizing tight closure. Then it is shown that $\lim_{e \to \infty} ({\#(\e M,M_i)}/ {(ap^d)^e})$ always exists under the assumption that $(R,\m)$ satisfies the Krull–Schmidt condition and $M$ has finite $F$-representation type by $\{ M_1, M_2, \dots, M_s \}$, in which all the $M_i$ are indecomposable $R$-modules that belong to distinct isomorphism classes and $a=[R/\m:(R/\m)^p]$.

Necessary and sufficient conditions are given for a Banach space operator with the single-valued extension property to satisfy Weyl's theorem and $a$-Weyl's theorem. It is shown that if $T$ or $T^{\ast}$ has the single-valued extension property and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. When $T^{\ast}$ has the single-valued extension property, $T$ is transaloid and $T$ is $a$-isoloid, then <formula form="inline" disc="math" id="ffm012"><formtex notation="AMSTeX">$a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. It is also proved that if $T$ or $T^{\ast}$ has the single-valued extension property, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.

We study the distributions Fθ,p of the random sums [sum ]∞1εnθn, where ε1, ε2, … are i.i.d. Bernoulli-p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p=.5, Fθ,p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain β of small degree, simulation gives the Hausdorff dimension to several decimal places.

The boundedness of Calderón–Zygmund operators is proved in the scale of the mixed Lebesgue spaces. As a consequence, the boundedness of the bilinear null forms $Q_{i j} (u,v) \,{=}\,\p_i u\p_j v \,{-}\, \p_j u\p_i v$, $Q_0(u,v)\,{=}\,u_t v_t \,{-}\,\nabla_x u\,{\cdot}\, \nabla_x v$ on various space–time mixed Sobolev–Lebesgue spaces is shown.

This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized that these algebras and their deformations, which form a class of quadratic graded algebras, have not been studied much and yet are interesting to algebra, arrangement theory and combinatorics.

An effective method is given for computing the Hausdorff dimension of the boundary of a self-similar digit tile T in n-dimensional Euclidean space:

formula here

where 1/c is the contraction factor and λ is the largest eigenvalue of a certain contact matrix first defined by Gröchenig and Haas.

Let $G$ be a finite nonabelian simple group and let $\Gamma$ be a connected undirected Cayley graph for $G$ . The possible structures for the full automorphism group ${\rm Aut}\Gamma$ are specified. Then, for certain finite simple groups $G$ , a sufficient condition is given under which $G$ is a normal subgroup of ${\rm Aut}\Gamma$ . Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups $G = J_1, J_4$ , Ly and BM, while others fall into two infinite families and involve the Ree simple groups and alternating groups. The two infinite families contain examples of half-transitive graphs of arbitrarily large valency.

The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative ℕ-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative ℕ-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.

A one-to-one correspondence is described between the set ${\cal J}(m)$ of numerical semigroups with multiplicity $m$ and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in ${\cal J}(m)$ a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of ${\bb N}^{m-1}$ . Finally, this result is particularized to the symmetric case.

As is well known, the classical Titchmarsh–Weyl m-function for second order differential operators admits a generalisation to considerably larger classes of differential equations. In the applications, however, the use of the general M-matrix often turns out to be rather cumbersome. The paper interprets the m-function in terms of Hilbert space notions, and shows that, in a sense that is made precise, the classical m-function can be recovered as a part of the more complicated one. Applications of this trick lead to results on spectral multiplicity and on stability properties of the spectrum.

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogues of a certain type of branching random walk, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations.

On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\mathbb{N}$. Spatial discretization is especially well suited to the direct development for continuous times of the conceptual method of probability tilting of Lyons, Pemantle and Peres.