
Page 


673  Relative Equilibria in Curved Restricted $4$body Problems Alhowaity, Sawsan; Diacu, Florin; PérezChavela, Ernesto
We consider the curved $4$body problems on spheres and hyperbolic
spheres. After obtaining a criterion for the existence of quadrilateral
configurations on the equator of the sphere, we study two restricted
$4$body problems, one in which two masses are negligible, and
another in which only one mass is negligible.
In the former we prove the evidence squarelike relative equilibria,
whereas in the latter we discuss the existence of kiteshaped
relative equilibria.


688  The Universal Enveloping Algebra of the Schrödinger Algebra and its Prime Spectrum Bavula, V. V.; Lu, T.
The prime, completely prime, maximal and primitive spectra are
classified for the universal enveloping algebra of the Schrödinger
algebra. For all of these ideals their explicit generators are
given. A counterexample is constructed to the conjecture of Cheng
and Zhang about nonexistence of simple singular Whittaker modules
for the Schrödinger algebra (and all such modules are classified).
It is proved that the conjecture holds 'generically'.


704  Remarks on Inner Functions and Optimal Approximants Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.


717  Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces Bu, Shangquan; Cai, Gang
We give necessary and sufficient
conditions of the $L^p$wellposedness (resp. $B_{p,q}^s$wellposedness) for the second order degenerate
differential equation with finite delays:
$(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic
boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where
$A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying
$D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from
$L^p([2\pi,0];X)$ (resp. $B_{p,q}^s([2\pi,0];X)$) into $X$.


738  Poincaré Inequalities and Neumann Problems for the $p$Laplacian CruzUribe, David; Rodney, Scott; Rosta, Emily
We prove an equivalence between weighted Poincaré inequalities
and
the existence of weak solutions to a Neumann problem related
to a
degenerate $p$Laplacian. The Poincaré inequalities are
formulated in the context of degenerate Sobolev spaces defined
in
terms of a quadratic form, and the associated matrix is the
source of
the degeneracy in the $p$Laplacian.


754  A Note on Concordance Properties of Fibers in Seifert Homology Spheres Lidman, Tye; Tweedy, Eamonn
In this note, we collect various properties
of Seifert homology spheres from the viewpoint of Dehn surgery
along a Seifert fiber. We expect that many of these are known
to various experts, but include them in one place which we hope
to be useful in the study of concordance and homology cobordism.


768  Chaotic Vibration of a Twodimensional Nonstrictly Hyperbolic Equation Li, Liangliang; Tian, Jing; Chen, Goong
The study of chaotic vibration for multidimensional PDEs due
to nonlinear boundary conditions is challenging. In this paper,
we mainly investigate the chaotic oscillation of a twodimensional
nonstrictly hyperbolic equation due to an energyinjecting
boundary condition and a distributed selfregulating boundary
condition. By using the method of characteristics, we give a
rigorous proof of the onset of the chaotic vibration phenomenon
of the 2D nonstrictly hyperbolic equation. We have also found
a regime of the parameters when the chaotic vibration phenomenon
occurs. Numerical simulations are also provided.


787  Endpoint Estimates of Riesz Transforms Associated with Generalized Schrödinger Operators Liu, Yu; Qi, Shuai
In this paper we establish the endpoint estimates
and Hardy type estimates for the Riesz transform associated
with the generalized Schrödinger operator.


802  The Oscillatory HyperHilbert Transform Associated with Plane Curves Li, Junfeng; Yu, Haixia
In this paper, the bounded properties of oscillatory hyperHilbert
transform along certain plane curves $\gamma(t)$
$$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(xt,y\gamma(t))e^{ i t^{\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$
were studied. For a general curves, these operators are bounded
in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their
boundedness in $L^p(\mathbb{R}^{2})$
were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.


812  Infinite Powers and Cohen Reals Medini, Andrea; van Mill, Jan; Zdomskyy, Lyubomyr S.
We give a consistent example of a zerodimensional separable
metrizable space $Z$ such that every homeomorphism of $Z^\omega$
acts like a permutation of the coordinates almost everywhere.
Furthermore, this permutation varies continuously. This shows
that a result of Dow and Pearl is sharp, and gives some insight
into an open problem of Terada. Our example $Z$ is simply the
set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.


822  Multivariate RankinSelberg Integrals on $GL_4$ and $GU(2,2)$ Pollack, Aaron; Shah, Shrenik
Inspired by a construction by Bump, Friedberg, and Ginzburg of
a twovariable integral representation on $\operatorname{GSp}_4$ for the product
of the standard and spin $L$functions, we give two similar multivariate
integral representations. The first is a threevariable RankinSelberg
integral for cusp forms on $\operatorname{PGL}_4$ representing the product
of the $L$functions attached to the three fundamental representations
of the Langlands $L$group $\operatorname{SL}_4(\mathbf{C})$. The second integral,
which is closely related, is a twovariable RankinSelberg integral
for cusp forms on $\operatorname{PGU}(2,2)$ representing the product of the
degree 8 standard $L$function and the degree 6 exterior square
$L$function.


836  Total Nonnegativity and Stable Polynomials Purbhoo, Kevin
We consider homogeneous multiaffine polynomials whose coefficients
are the Plücker coordinates of a point $V$ of the Grassmannian.
We show that such a polynomial is stable (with respect to the
upper half plane) if and only if $V$ is in the totally nonnegative
part of the Grassmannian. To prove this, we consider an action
of
matrices on multiaffine polynomials. We show that
a matrix $A$ preserves stability of polynomials if and only if
$A$ is totally nonnegative. The proofs are applications of classical
theory of totally nonnegative matrices, and the generalized
PólyaSchur theory of Borcea and Brändén.


848  Quantum Symmetries of Graph $C^*$algebras Schmidt, Simon; Weber, Moritz
The study of graph $C^*$algebras has a long history in operator
algebras. Surprisingly, their quantum symmetries have never been
computed so far. We close this gap by proving that the quantum
automorphism group of a finite, directed graph without multiple
edges acts maximally on the corresponding graph $C^*$algebra.
This shows that the quantum symmetry of a graph coincides with
the quantum symmetry of the graph $C^*$algebra. In our result,
we use the definition of quantum automorphism groups of graphs
as given by Banica in 2005. Note that Bichon gave a different
definition in 2003; our action is inspired from his work. We
review and compare these two definitions and we give a complete
table of quantum automorphism groups (with respect to either
of the two definitions) for undirected graphs on four vertices.


865  Homological Dimensions of Local (Co)homology over Commutative DGrings Shaul, Liran
Let $A$ be a commutative noetherian ring,
let $\mathfrak{a}\subseteq A$ be an ideal,
and let $I$ be an injective $A$module.
A basic result in the structure theory of injective modules states
that
the $A$module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$torsion elements
is also an injective $A$module.
Recently, de Jong proved a dual result: If $F$ is a flat $A$module,
then the $\mathfrak{a}$adic completion of $F$ is also a flat $A$module.
In this paper we generalize these facts to commutative noetherian
DGrings:
let $A$ be a commutative nonpositive DGring such that $\mathrm{H}^0(A)$
is a noetherian ring,
and for each $i\lt 0$, the $\mathrm{H}^0(A)$module $\mathrm{H}^i(A)$
is finitely generated.
Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$,
we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$
associated to $\bar{\mathfrak{a}}$ does not increase injective dimension.
Dually, the derived $\bar{\mathfrak{a}}$adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$
does not increase flat dimension.


878  Weak Approximation for Points with Coordinates in Rankone Subgroups of Global Function Fields Sun, ChiaLiang
For every affine variety over a global function field, we show
that
the set of its points with coordinates in an arbitrary rankone
multiplicative
subgroup of this function field satisfies the required property
of
weak approximation for finite sets of places of this function
field
avoiding arbitrarily given finitely many places.

