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721  Formal Fibers of Unique Factorization Domains Boocher, Adam; Daub, Michael; Johnson, Ryan K.; Lindo, H.; Loepp, S.; Woodard, Paul A.
Let $(T,M)$ be a complete local (Noetherian) ring such that $\dim T\geq 2$ and
$T=T/M$ and let $\{p_i\} _{i \in \mathcal I}$ be a collection of
elements of T indexed by a set $\mathcal I$ so that $\mathcal I  < T$.
For each $i \in \mathcal{I}$, let $C_i$:={$Q_{i1}$,$\dots$,$Q_{in_i}$}
be a set of nonmaximal prime ideals containing $p_i$ such that the $Q_{ij}$
are incomparable and $p_i\in Q_{jk}$ if and only if $i=j$. We provide necessary
and sufficient conditions so that T is the ${\bf m}$adic completion of a local unique
factorization domain $(A, {\bf m})$, and for each $i \in \mathcal I$, there exists a unit
$t_i$ of T so that $p_{i}t_i \in A$ and $C_i$
is the set of prime ideals $Q$ of $T$ that are maximal with respect to the condition
that $Q \cap A = p_{i}t_{i}A$.


737  Approximation by Dilated Averages and KFunctionals Ditzian, Z.; Prymak, A.
For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$
normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of
$f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}t\mathbf{u})d\mu(\mathbf{u}). \] It
will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has
the equivalence \[ \A_tff\_B\approx \inf \{
(\fg\_B+t^2 \P(D)g\_B): P(D)g\in B\}\quad\text{for }t>0, \]
where $\varphi(t)\approx \psi(t)$ means
$c^{1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions
for which translations are continuous isometries and $P(D)$ is an
elliptic differential operator induced by $\mu$. Many applications are
given, notable among which is the averaging operator with $d\mu(\mathbf{u})=
\frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set
in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of
$S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate
of approximation by averages on the boundary of a convex set under
more restrictive conditions is also shown to be equivalent to an
appropriate $K$functional.


758  General Preservers of QuasiCommutativity Dolinar, Gregor; Kuzma, Bojan
Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasicommute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasicommutativity in both directions.


787  An Explicit Treatment of Cubic Function Fields with Applications Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.
We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for nonsingularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few squarefree polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.


808  Extrema of Low Eigenvalues of the DirichletNeumann Laplacian on a Disk Legendre, Eveline
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of DirichletNeumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.


827  BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces Ouyang, Caiheng; Xu, Quanhua
This paper studies the relationship between vectorvalued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1z)^{q1}\\nabla f(z)\^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\f(z)f(z_0)\^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$uniformly convex, where $$P_{z_0}(z)=\frac{1z_0^2}{1\bar{z_0}z^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$uniformly smooth norm.


845  Biflatness and PseudoAmenability of Segal Algebras Samei, Ebrahim; Spronk, Nico; Stokke, Ross
We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.


870  The BrascampLieb Polyhedron Valdimarsson, Stefán Ingi
A set of necessary and sufficient conditions for the BrascampLieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has corank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the BrascampLieb inequality to hold. We present an algorithm which generates such a list.


889  Singular Integral Operators and Essential Commutativity on the Sphere Xia, Jingbo
Let ${\mathcal T}$ be the $C^\ast $algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.


914  Reducibility of the Principal Series for Sp^{~}_{2}(F) over a padic Field Zorn, Christian
Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with
entries in a nondyadic $p$adic field $F$. We further let $\widetilde{G}_n$ be
the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times}
\mid z=1\}$ defined using the Leray cocycle. In this paper, we aim to
demonstrate the complete list of reducibility points of the genuine
principal series of ${\widetilde{G}_2}$. In most cases, we will use
some techniques developed by Tadić that analyze the Jacquet
modules with respect to all of the parabolics containing a fixed
Borel. The exceptional cases involve representations induced from
unitary characters $\chi$ with $\chi^2=1$. Because such
representations $\pi$ are unitary, to show the irreducibility of
$\pi$, it suffices to show that
$\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this
by examining the poles of certain intertwining operators associated to
simple roots. Then some results of Shahidi and Ban decompose arbitrary
intertwining operators into a composition of operators corresponding
to the simple roots of ${\widetilde{G}_2}$. We will then be able to
show that all such operators have poles at principal series
representations induced from quadratic characters and therefore such
operators do not extend to operators in
$\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question.
