We give new examples of weak Hilbert spaces.

Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Let $D$ be a family of $k$-subsets (called blocks) of a $v$-set $X\left( v \right)$. Then $D$ is a $\left( v,\,k,\,t \right)$ covering design or covering if every $t$-subset of $X\left( v \right)$ is contained in at least one block of $D$. The number of blocks is the size of the covering, and the minimum size of the covering is called the covering number. In this paper we consider the case $t\,=\,2$, and find several infinite classes of covering numbers. We also give upper bounds on other classes of covering numbers.

Let $E$ be a stable rank 2 vector bundle on a smooth projective curve $X$ and $V\,\left( E \right)$ be the set of all rank 1 subbundles of $E$ with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, $E$, on $X$ with fixed $\deg \left( E \right)$ and $\deg \left( L \right),\,L\,\in \,V\left( E \right)$ and such that $\text{card}\,\left( V(E) \right)\,\ge \,2\,(\text{resp}\text{. card}\left( V(E) \right)\,=\,2)$.

Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty \right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$, and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$, for some integer $m\,\ge \,2$. Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$, be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$, let

$$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$

An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$.

A binary structure $S$ has the pigeonhole property $\left( P \right)$ if every finite partition of $S$ induces a block isomorphic to $S$. We classify all countable tournaments with $\left( P \right)$; the class of orders with $\left( P \right)$ is completely classified.

In a study of the word problem for groups, R. J. Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K. S. Brown and R. Geoghegan, M. N. Dyer showed that $F$ is the fundamental group of a finite two-complex ${{Z}^{2}}$ having Euler characteristic one and which is Cockcroft, in the sense that each map of the two-sphere into ${{Z}^{2}}$ is homologically trivial. We show that no proper covering complex of ${{Z}^{2}}$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft.

We show that for $U$ and $V$ balanced open subsets of $\left( \text{Qno} \right)$ Fréchet spaces $E$ and $F$ that we have the topological identity

$$\text{(}\mathcal{H}(U\times V),{{\tau }_{\omega }})=\left( \mathcal{H}\left( U;\left( \mathcal{H}(V),{{\tau }_{\omega }} \right) \right),{{\tau }_{\omega }} \right).$$

Analogous results for the compact open topology have long been established. We also give an example to show that the $\left( \text{Qno} \right)$ hypothesis on both $E$ and $F$ is necessary.

In this paper, we show that the absolute value of the signature of the 2-parallel version of a link is less than or equal to the nullity of it and show that the signature, nullity, and Minkowski units of the 2-parallel version of a certain class of links are always equal to 0, 2, and 1 respectively.

A bounded linear operator $T$ on a Banach space $X$ is an abstract backward shift if the nullspace of $T$ is one dimensional, and the union of the null spaces of ${{T}^{k}}$ for all $k\,\ge \,1$ is dense in $X$. In this paper it is shown that the commutant of an abstract backward shift is an integral domain. This result is used to derive properties of operators in the commutant.

Abstract. The weighted mean matrix ${{M}_{a}}$ is the triangular matrix $\left\{ {{a}_{k}}/{{A}_{n}} \right\}$, where ${{a}_{n}}\,>\,0$ and ${{A}_{n}}\,:=\,{{a}_{1}}\,+\,{{a}_{2}}\,+\cdots +\,{{a}_{n}}$. It is proved that, subject to ${{n}^{c}}{{a}_{n}}$ being eventually monotonic for each constant $c$ and to the existence of $\alpha \,:=\,\lim \,\frac{{{A}_{n}}}{n{{a}_{n}}},\,{{M}_{a}}\,\in \,B\left( {{l}_{p}} \right)$ for $1\,<\,p\,<\infty $ if and only if $\alpha \,<\,p$.

We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group’s action on the division points of an appropriate Drinfeld module.

We identify a class of domains of ${{\mathbb{C}}^{n}}$ in which any two commuting holomorphic self-maps have a common fixed point.

We prove that Ornstein transformations are almost surely totally ergodic provided only that the cutting parameter is not bounded. We thus obtain a larger class of Ornstein transformations with the mixing property.

We construct a countably infinite family of pairwise non-isomorphic maximal $\mathbb{Q}\left[ X \right]$-orders such that the full 2 by 2 matrix rings over these orders are all isomorphic.

Formulas for the Clarke subdifferential are always expressed in the form of inclusion. The equality form in these formulas generally requires the functions to be directionally regular. This paper studies the directional regularity of the general class of extended-real-valued functions that are directionally Lipschitzian. Connections with the concept of subdifferential regularity are also established.

The purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces ${{P}_{r}}$ of polygons with fixed side lengths in the Euclidean space ${{\mathbb{E}}^{3}}$. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of ${{P}_{r}}$. We also classify moduli spaces of pentagons.

We calculate the additive and multiplicative structure of the ring $K\left( S\left( {{T}^{*}}\mathbb{R}{{P}^{2n+1}} \right) \right)$ using the eta invariant.

Soient $r$ et $p$ deux nombres premiers distincts, soit $K\,=\,\mathbb{Q}(\cos \,\frac{2\pi }{r})$ , et soit $\mathbb{F}$ le corps résiduel de $K$ en une place au-dessus de $p$. Lorsque l’image de $(2\,-\,2\,\cos \,\frac{2\pi }{r})$ dans $\mathbb{F}$ n’est pas un carré, nous donnons une construction géométrique d’une extension réguliere de $K\left( t \right)$ de groupe de Galois $\text{PS}{{\text{L}}_{2}}(\mathbb{F})$ . Cette extension correspond à un revêtement de ${{\mathbb{P}}^{1}}/k$ de « signature $\left( r,\,p,\,p \right)$ » au sens de [3, sec. 6.3], et son existence est prédite par le critère de rigidité de Belyi, Fried, Thompson et Matzat. Sa construction s’obtient en tordant la representation galoisienne associée aux points d’ordre $p$ d’une famille de variétés abéliennes à multiplications réelles par $K$ découverte par Tautz, Top et Verberkmoes [6]. Ces variétés abéliennes sont définies sur un corps quadratique, et sont isogènes à leur conjugué galoisien. Notre construction généralise une méthode de Shih [4], [5], que l’on retrouve quand $r\,=\,2$ et $r\,=\,3$.

Let $\text{Z}$ be the unital simple nuclear infinite dimensional ${{C}^{*}}$-algebra which has the same Elliott invariant as $\mathbb{C}$, introduced in [9]. A ${{C}^{*}}$-algebra is called $\text{Z}$-stable if $A\,\cong \,A\,\otimes \,\text{Z}$. In this note we give some necessary conditions for a unital simple ${{C}^{*}}$-algebra to be $\text{Z}$-stable.