Let $\pi $ be a square integrable representation of ${G}'=\text{S}{{\text{L}}_{n}}(D)$ , with $D$ a central division algebra of finite dimension over a local field $F$ of non-zero characteristic. We prove that, on the elliptic set, the character of $\pi $ equals the complex conjugate of the orbital integral of one of the pseudocoefficients of $\pi $ . We prove also the orthogonality relations for characters of square integrable representations of ${G}'$ . We prove the stable transfer of orbital integrals between $\text{S}{{\text{L}}_{n}}(F)$ and its inner forms.

In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\,\to \,C(X)$ is an epimorphism in the category $\mathcal{C}\mathcal{R}$ of commutative rings (with units). We call such an embedding a $\mathcal{C}\mathcal{R}$ -epic embedding and we say that $X$ is absolute $\mathcal{C}\mathcal{R}$ -epic if every embedding of $X$ is $\mathcal{C}\mathcal{R}$ -epic. We continue this investigation. Our most notable result shows that a Lindelöf space $X$ is absolute $\mathcal{C}\mathcal{R}$ -epic if a countable intersection of $\beta X$ -neighbourhoods of $X$ is a $\beta X$ -neighbourhood of $X$ . This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all $\sigma $ -compact spaces and all Lindelöf $P$ -spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute $\mathcal{C}\mathcal{R}$ -epic.

We consider the $k$ -osculating varieties ${{O}_{k,\,n.d}}$ to the (Veronese) $d$ -uple embeddings of ${{\mathbb{P}}^{n}}$ . We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\,\subset \,{{\mathbb{P}}^{n}}$ to $O_{k,n,d}^{s}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

We give a two dimensional extension of the three distance theorem. Let $\theta $ be in ${{\mathbf{R}}^{2}}$ and let $q$ be in $\mathbf{N}$ . There exists a triangulation of ${{\mathbf{R}}^{2}}$ invariant by ${{\mathbf{Z}}^{2}}$ -translations, whose set of vertices is ${{\mathbf{Z}}^{2}}\,+\,\{0,\,\theta ,\,\ldots ,\,q\theta \}$ , and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta $ and $q$ .

Let $K$ be a real quadratic field, and $p$ a rational prime which is inert in $K$ . Let $\alpha $ be a modular unit on ${{\Gamma }_{0}}(N)$ . In an earlier joint article with Henri Darmon, we presented the definition of an element $u\left( \alpha ,\,\text{ }\!\!\tau\!\!\text{ } \right)\,\in \,K_{P}^{\times }$ attached to $\alpha $ and each $\tau \,\in \,K$ . We conjectured that the $p$ -adic number $u(\alpha ,\,\tau )$ lies in a specific ring class extension of $K$ depending on $\tau $ , and proposed a “Shimura reciprocity law” describing the permutation action of Galois on the set of $u(\alpha ,\,\tau )$ . This article provides computational evidence for these conjectures. We present an efficient algorithm for computing $u(\alpha ,\,\tau )$ , and implement this algorithm with the modular unit $\alpha (z)\,=\,\Delta {{(z)}^{2}}\,\Delta (4z)\,/\,\Delta {{(2z)}^{3}}$ . Using $p\,=\,3,\,5,\,7\,and\,11$ , and all real quadratic fields $K$ with discriminant $D\,<\,500$ such that 2 splits in $K$ and $K$ contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define $u(\alpha ,\,\tau )$ is shown to be $\mathbf{Z}$ -valued rather than only ${{\mathbf{Z}}_{P}}\,\cap \,\mathbf{Q}-$ valued; this is an improvement over our previous result and allows for a precise definition of $u(\alpha ,\,\tau )$ , instead of only up to a root of unity.

We study the cardinal invariants of analytic $P$ -ideals, concentrating on the ideal $Z$ of asymptotic density zero. Among other results we prove $\min \,\{\mathfrak{b},\,\operatorname{cov}(\mathcal{N})\,\}\,\le \,\operatorname{cov}*\,(Z)\,\le \,\max \{\mathfrak{b},\,\text{non(}\mathcal{N}\text{)}\}$ .

Let $(Y,\,T)$ be a minimal suspension flow built over a dynamical system $(X,\,S)$ and with (strictly positive, continuous) ceiling function $f:\,X\,\to \,\mathbb{R}$ . We show that the eigenvalues of $(Y,\,T)$ are contained in the range of a trace on the ${{K}_{0}}$ -group of $(X,\,S)$ . Moreover, a trace gives an order isomorphism of a subgroup of ${{K}_{0}}\left( C(X)\,{{\rtimes }_{S}}\,\mathbb{Z} \right)$ with the group of eigenvalues of $(Y,\,T)$ . Using this result, we relate the values of $t$ for which the time- $t$ map on the minimal suspension flow is minimal with the $K$ -theory of the base of this suspension.

We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$ -convex, $p$ -concave and positive $p$ -summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\,\times \,n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.

Let $p$ be an odd prime number, and let $F$ be a field of characteristic not $p$ and not containing the group ${{\mu }_{p}}$ of $p$ -th roots of unity. We consider cyclic $p$ -algebras over $F$ by descent from $L\,=\,F\left( {{\mu }_{p}} \right)$ . We generalize a theorem of Albert by showing that if ${{\mu }_{{{p}^{n}}}}\,\subseteq \,L$ , then a division algebra $D$ of degree ${{p}^{n}}$ over $F$ is a cyclic algebra if and only if there is $d\,\in \,D$ with ${{d}^{{{P}^{n}}}}\,\in \,F\,-\,{{F}^{P}}$ . Let $F(p)$ be the maximal $p$ -extension of $F$ . We show that $F(p)$ has a noncyclic algebra of degree $p$ if and only if a certain eigencomponent of the $p$ -torsion of $\text{Br(F(p)(}{{\mu }_{p}}\text{))}$ is nontrivial. To get a better understanding of $F(p)$ , we consider the valuations on $F(p)$ with residue characteristic not $p$ , and determine what residue fields and value groups can occur. Our results support the conjecture that the $p$ torsion in $\text{Br}(F(p))$ is always trivial.