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961  About the Defectivity of Certain SegreVeronese Varieties Abrescia, Silvia
We study the regularity of the higher secant varieties of $\PP^1\times
\PP^n$, embedded with divisors of type $(d,2)$ and $(d,3)$. We
produce, for the highest defective cases, a ``determinantal'' equation
of the secant variety. As a corollary, we prove that the Veronese
triple embedding of $\PP^n$ is not Grassmann defective.


975  An AF Algebra Associated with the Farey Tessellation Boca, Florin P.
We associate with the Farey tessellation of the upper
halfplane an
AF algebra $\AA$ encoding the ``cutting sequences'' that define
vertical geodesics.
The EffrosShen AF algebras arise as quotients
of $\AA$. Using the path algebra model for AF algebras we construct, for
each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in
$\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.


1001  Isometric Group Actions on Hilbert Spaces: Structure of Orbits Cornulier, Yves de; Tessera, Romain; Valette, Alain
Our main result is that a finitely generated nilpotent group has
no isometric action on an infinitedimensional Hilbert space with
dense orbits. In contrast, we construct such an action with a
finitely generated metabelian group.


1010  $H^\infty$ Functional Calculus and MikhlinType Multiplier Conditions Galé, José E.; Miana, Pedro J.
Let $T$ be a sectorial operator. It is known that the existence of a
bounded (suitably scaled) $H^\infty$ calculus for $T$, on every
sector containing the positive halfline, is equivalent to the
existence of a bounded functional calculus on the Besov algebra
$\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra
includes functions defined by Mikhlintype conditions and so the
Besov calculus can be seen as a result on multipliers for $T$. In
this paper, we use fractional derivation to analyse in detail the
relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras
of Mikhlintype. As a result, we obtain a new version of the quoted
equivalence.


1028  Lifting $n$Dimensional Galois Representations Hamblen, Spencer
We investigate the problem of deforming $n$dimensional mod $p$ Galois
representations to characteristic zero. The existence of 2dimensional
deformations has been proven under certain conditions
by allowing ramification at additional primes in order to
annihilate a dual Selmer group. We use the same general methods to
prove the existence of $n$dimensional deformations.


1050  Adjacency Preserving Maps on Hermitian Matrices Huang, Wenling; Semrl, Peter \v
Hua's fundamental theorem of the geometry of hermitian matrices
characterizes bijective maps on the space of all $n\times n$
hermitian matrices preserving adjacency in both directions.
The problem of possible improvements
has been open for a while. There are three natural problems here.
Do we need the bijectivity assumption? Can we replace the
assumption of preserving adjacency in both directions by the
weaker assumption of preserving adjacency in one direction only?
Can we obtain such a characterization for maps acting between the
spaces of hermitian matrices of different sizes? We answer all
three questions for the complex hermitian matrices, thus obtaining
the optimal structural result for adjacency preserving maps on
hermitian matrices over the complex field.


1067  On Types for Unramified $p$Adic Unitary Groups Kariyama, Kazutoshi
Let $F$ be a nonarchimedean local field of residue characteristic
neither 2 nor 3 equipped with a galois involution with fixed field
$F_0$, and let $G$ be a symplectic group over $F$ or an unramified
unitary group over $F_0$. Following the methods of BushnellKutzko for
$\GL(N,F)$, we define an analogue of a simple type attached to a
certain skew simple stratum, and realize a type in $G$. In
particular, we obtain an irreducible supercuspidal representation of
$G$ like $\GL(N,F)$.


1108  A Classification of Tsirelson Type Spaces LopezAbad, J.; Manoussakis, A.
We give a complete classification of mixed Tsirelson spaces
$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ for finitely many pairs of
given compact and hereditary families $\mathcal F_i$ of finite sets of
integers and $0<\theta_i<1$ in terms of the CantorBendixson indices
of the families $\mathcal F_i$, and $\theta_i$ ($1\le i\le r$). We
prove that there are unique countable ordinal $\alpha$ and
$0<\theta<1$ such that every block sequence of
$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ has a subsequence equivalent to a
subsequence of the natural basis of the
$T(\mathcal S_{\omega^\alpha},\theta)$. Finally, we give a complete criterion of
comparison in between two of these mixed Tsirelson spaces.


1149  Conjugate Reciprocal Polynomials with All Roots on the Unit Circle Petersen, Kathleen L.; Sinclair, Christopher D.
We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.


1168  Short Time Behavior of Solutions to Linear and Nonlinear Schr{ödinger Equations Taylor, Michael
We examine the fine structure of the short time behavior
of solutions to various linear and nonlinear Schr{\"o}dinger equations
$u_t=i\Delta u+q(u)$ on $I\times\RR^n$, with initial data $u(0,x)=f(x)$.
Particular attention is paid to cases where $f$ is piecewise smooth,
with jump across an $(n1)$dimensional surface. We give detailed
analyses of Gibbslike phenomena and also focusing effects, including
analogues of the Pinsky phenomenon. We give results for general $n$
in the linear case. We also have detailed analyses for a broad class of
nonlinear equations when $n=1$ and $2$, with emphasis on the analysis of
the first order correction to the solution of the corresponding linear
equation. This work complements estimates on the error in this approximation.
