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3  Holomorphic Generation of Continuous Inverse Algebras Biller, Harald
We study complex commutative Banach algebras
(and, more generally, continuous
inverse algebras) in which the holomorphic functions of a fixed $n$tuple
of elements are dense. In particular, we characterize the compact subsets
of~$\C^n$ which appear as joint spectra of such $n$tuples. The
characterization is compared with several established notions of holomorphic
convexity by means of approximation
conditions.


36  Classification of Ding's Schubert Varieties: Finer Rook Equivalence Develin, Mike; Martin, Jeremy L.; Reiner, Victor
K.~Ding studied a class of Schubert varieties $X_\lambda$
in type A partial
flag manifolds, indexed by
integer partitions $\lambda$ and in bijection
with dominant permutations. He observed that the
Schubert cell structure of $X_\lambda$ is indexed by maximal rook
placements on the Ferrers board $B_\lambda$, and that the
integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are
additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$
satisfy the combinatorial condition of rookequivalence.


63  Some Results on the SchroederBernstein Property for Separable Banach Spaces Ferenczi, Valentin; Galego, Elói Medina
We construct a continuum of mutually
nonisomorphic
separable Banach spaces which are complemented in each other.
Consequently, the SchroederBernstein Index of any of these spaces is
$2^{\aleph_0}$. Our
construction is based on a Banach space introduced by W. T. Gowers
and
B. Maurey in 1997.
We also use classical descriptive set theory methods, as in some
work of the first author and C. Rosendal, to improve some results
of P. G. Casazza and
of N. J. Kalton on the
SchroederBernstein Property for
spaces with an unconditional finitedimensional Schauder
decomposition.


85  On the Convergence of a Class of Nearly Alternating Series Foster, J. H.; Serbinowska, Monika
Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.


109  On Fiber Cones of $\m$Primary Ideals Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.
Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$primary ideal of
a $d$dimensional CohenMacaulay local ring $(R,\m)$ are derived in
terms of the mixed multiplicity $e_{d1}(\m  I)$, the multiplicity
$e(I)$, and superficial elements. As a consequence, the
CohenMacaulay property of $F(I)$ when $I$ has minimal mixed
multiplicity or almost minimal mixed multiplicity is characterized
in terms of the reduction number of $I$ and lengths of certain ideals.
We also characterize the CohenMacaulay and Gorenstein properties of
fiber cones of $\m$primary ideals with a $d$generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$


127  Smooth Values of the Iterates of the Euler PhiFunction Lamzouri, Youness
Let $\phi(n)$ be the Euler phifunction, define
$\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all
$k\geq 0$. We will determine an asymptotic formula for the set of
integers $n$ less than $x$ for which $\phi_k(n)$ is $y$smooth,
conditionally on a weak form of the ElliottHalberstam conjecture.


148  On Certain Classes of Unitary Representations for Split Classical Groups Muić, Goran
In this paper we prove the unitarity of duals of tempered
representations supported on minimal parabolic subgroups for split
classical $p$adic groups. We also construct a family of unitary
spherical representations for real and complex classical groups


186  Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields Okoh, F.; Zorzitto, F.
Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$,
arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer,
$h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and
$\alpha$ is a $K$linear functional on the space $\krx$ of rational
functions in one variable $X$. Every pair $(h,\alpha)$ comes with a
polynomial $f$ in $K(X)[Y]$ called the regulator. When the module
${\mathcal M}$ admits nontrivial endomorphisms, $f$ must be linear or
quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and
only if $f$ is an irreducible quadratic. Then the $K$algebra
$\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For
some height functions $h$ of infinite support $I$, the search for a
functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes
down to having functions $\eta\colon I\ar K$ such that no planar curve
intersects the graph of $\eta$ on a cofinite subset. If $K$ has
characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a
purelysimple Kronecker module ${\mathcal M}$ having nontrivial
endomorphisms, then $h$ attains the value $\infty$ at least once on
$\ktil$ and $h$ is finitevalued at least twice on
$\ktil$. Conversely all these $h$ form part of such triplets. The
proof of this result hinges on the fact that a rational function $r$
is a perfect square in $\krx$ if and only if $r$ is a perfect square
in the completions of $\krx$ with respect to all of its valuations.


211  On Two Exponents of Approximation Related to a Real Number and Its Square Roy, Damien
For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the
supremum of all real numbers $\lambda$ such that, for each
sufficiently large $X$, the inequalities $x_0 \le X$,
$x_0\xix_1 \le X^{\lambda}$ and $x_0\xi^2x_2 \le
X^{\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$
not all zero, and let $\omegahat_2(\xi)$ denote the supremum of
all real numbers $\omega$ such that, for each sufficiently large
$X$, the dual inequalities $x_0+x_1\xi+x_2\xi^2 \le
X^{\omega}$, $x_1 \le X$ and $x_2 \le X$ admit a solution in
integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a
question of Y.~Bugeaud and M.~Laurent, we show that the exponents
$\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers
with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2,
(\sqrt{5}1)/2]$ while, for the same values of $\xi$, the dual
exponents $\omegahat_2(\xi)$ form a dense subset of $[2,
(\sqrt{5}+3)/2]$. Part of the proof rests on a result of
V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) =
1\omegahat_2(\xi)^{1}$ for any real number $\xi$ with
$[\bQ(\xi)\wcol\bQ]>2$.

