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449  $\SL_n$, Orthogonality Relations and Transfer Badulescu, Alexandru Ioan
Let $\pi$ be a square integrable representation of
$G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension
over a local field $F$ of nonzero 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 $\SL_n(F)$ and its inner forms.


465  Searching for Absolute $\mathcal{CR}$Epic Spaces Barr, Michael; Kennison, John F.; Raphael, R.
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 $\CR$ of commutative
rings (with units). We call such an embedding a $\CR$epic embedding
and we say that $X$ is absolute $\CR$epic if every embedding of $X$
is $\CR$epic. We continue this investigation. Our most notable
result shows that a Lindel\"of space $X$ is absolute $\CR$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, cozerosubspaces, and being
the domain or codomain of a perfect map. A strengthening of the
Lindel\"of property leads to a new class with the same closure
properties that is also closed under finite products. Moreover, all
\scompact spaces and all Lindel\"of $P$spaces satisfy this stronger
condition. We get some results in the nonLindel\"of case that are
sufficient to show that the Dieudonn\'e plank and some closely related
spaces are absolute $\CR$epic.


488  Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties Bernardi, A.; Catalisano, M. V.; Gimigliano, A.; Idà, M.
We consider the $k$osculating varieties
$O_{k,n.d}$ to the (Veronese) $d$uple embeddings of $\PP^n$. We
study the dimension of their higher secant varieties via inverse
systems (apolarity). By associating certain 0dimensional schemes
$Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert
functions, we are able, in several cases, to determine whether those
secant varieties are defective or not.


503  Cyclic Groups and the Three Distance Theorem Chevallier, Nicolas
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 $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose
number of different triangles, up to translations, is bounded above by a
constant which does not depend on $\theta$ and $q$.


553  Computations of Elliptic Units for Real Quadratic Fields Dasgupta, Samit
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 $u(\alpha, \tau) \in K_p^\times$ 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.


575  Cardinal Invariants of Analytic $P$Ideals HernándezHernández, Fernando; Hrušák, Michael
We study the cardinal invariants of analytic $P$ideals, concentrating on the
ideal $\mathcal{Z}$ of asymptotic density zero. Among other results we prove
$ \min\{ \mathfrak{b},\cov\ (\mathcal{N})
\} \leq\cov^{\ast}(\mathcal{Z}) \leq\max\{
\mathfrak{b},\non(\mathcal{N}) \right\}.
$


596  Eigenvalues, $K$theory and Minimal Flows ItzáOrtiz, Benjamín A.
Let $(Y,T)$ be a minimal suspension flow built over a dynamical
system $(X,S)$ and with (strictly positive, continuous) ceiling
function $f\colon X\to\R$. 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 $K_0(\cprod{C(X)}{S})$ 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.


614  Preduals and Nuclear Operators Associated with Bounded, $p$Convex, $p$Concave and Positive $p$Summing Operators Labuschagne, C. C. A.
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.


638  Distance from Idempotents to Nilpotents MacDonald, Gordon W.
We give bounds on the distance from a nonzero 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.


658  Division Algebras of Prime Degree and Maximal Galois $p$Extensions Mináč, J.; Wadsworth, A.
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
$L = F(\mu_p)$. 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 $\Br(F(p)(\mu_p))$ 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 $\Br(F(p))$ is always trivial.
