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3  On the Local Convexity of Intersection Bodies of Revolution Alfonseca, M. Angeles; Kim, Jaegil
One of the fundamental results in Convex Geometry is Busemann's
theorem, which states that the intersection body of a symmetric convex
body is convex. Thus, it is only natural to ask if there is a
quantitative version of Busemann's theorem, i.e., if the intersection
body operation actually improves convexity. In this paper we
concentrate on the symmetric bodies of revolution to provide several
results on the (strict) improvement of convexity under the
intersection body operation. It is shown that the intersection body of
a symmetric convex body of revolution has the same asymptotic behavior
near the equator as the Euclidean
ball. We apply this result to show that in sufficiently high
dimension the double intersection body of a symmetric convex body of
revolution is very close to an ellipsoid in the BanachMazur
distance. We also prove results on the local convexity at the equator
of intersection bodies in the class of star bodies of revolution.


28  Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor Asadollahi, Javad; Hafezi, Rasool; Vahed, Razieh
We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.


55  On Varieties of Lie Algebras of Maximal Class Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina
We study complex projective varieties that parametrize
(finitedimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinitedimensional case we concentrate our attention on
${\mathbb N}$graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.


90  Une propriété de domination convexe pour les orbites sturmiennes Bousch, Thierry
Let ${\bf x}=(x_0,x_1,\ldots)$ be a $N$periodic sequence of integers
($N\ge1$), and ${\bf s}$ a sturmian sequence with the same barycenter
(and also $N$periodic, consequently). It is shown that, for affine
functions $\alpha:\mathbb R^\mathbb N_{(N)}\to\mathbb R$ which are increasing relatively
to some order $\le_2$ on $\mathbb R^\mathbb N_{(N)}$ (the space of all $N$periodic
sequences), the average of $\alpha$ on the orbit of ${\bf x}$ is
greater than its average on the orbit of ${\bf s}$.


107  The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space Chang, JuiEn; Xiao, Ling
In this paper, we discuss the isometric embedding problem in
hyperbolic space with nonnegative extrinsic curvature.
We prove a priori bounds for the trace of the second fundamental
form $H$ and extend the result to $n$dimensions.
We also obtain an estimate for the gradient of the smaller principal
curvature in 2 dimensions.


132  Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class $C_0$ Clouâtre, Raphaël
We obtain results on the unitary equivalence of weak contractions of
class $C_0$ to their Jordan models under an assumption on their
commutants. In particular, our work addresses the case of arbitrary
finite multiplicity. The main tool is the
theory of boundary representations due to Arveson. We also
generalize and improve previously known results concerning unitary
equivalence and similarity to Jordan models when the minimal function
is a Blaschke product.


152  On Homotopy Invariants of Combings of Threemanifolds Lescop, Christine
Combings of compact, oriented $3$dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$valued invariant $\theta_G$ of torsion combings
on closed $3$manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$invariant, which is the simplest configuration
space integral invariant of rational homology $3$balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the CassonWalker invariant.
The article also includes a selfcontained presentation of combings
for $3$manifolds.


184  Geometric Spectra and Commensurability McReynolds, D. B.
The work of Reid, ChinburgHamiltonLongReid,
PrasadRapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.


198  Tate Cycles on Abelian Varieties with Complex Multiplication Murty, V. Kumar; Patankar, Vijay M.
We consider Tate cycles on an Abelian variety $A$ defined over
a sufficiently large number field $K$ and having complex
multiplication. We show that
there is an effective bound $C = C(A,K)$ so that
to check whether a given cohomology class is a Tate class on
$A$, it suffices to check the action of
Frobenius elements at primes $v$ of norm $ \leq C$.
We also show that for a set of primes $v$ of $K$ of density
$1$, the space of Tate cycles on the special fibre $A_v$ of the
Néron model of $A$ is isomorphic to the space of Tate cycles
on $A$ itself.


214  Symmetric Genuine Spherical Whittaker Functions on $\overline{GSp_{2n}(F)}$ Szpruch, Dani
Let $F$ be a padic field of odd residual characteristic. Let
$\overline{GSp_{2n}(F)}$ and $\overline{Sp_{2n}(F)}$ be the metaplectic double covers of the general
symplectic group and the symplectic group attached to the $2n$
dimensional symplectic space over $F$. Let $\sigma$ be a genuine,
possibly reducible, unramified principal series representation of
$\overline{GSp_{2n}(F)}$. In these notes we give an explicit formulas for a spanning
set for the space of Spherical Whittaker functions attached to
$\sigma$. For odd $n$, and generically for even $n$, this spanning set
is a basis. The significant property of this set is that each of its
elements is unchanged under the action of the Weyl group of
$\overline{Sp_{2n}(F)}$.
If $n$ is odd then each element in the set has an equivariant property
that generalizes a uniqueness result of Gelbart, Howe and
PiatetskiShapiro. Using this symmetric set, we
construct a family of reducible genuine unramified principal series
representations which have more then one generic constituent. This
family contains all the reducible genuine unramified principal series
representations induced from a unitary data and exists only for $n$
even.


241  Global Holomorphic Functions in Several Noncommuting Variables Agler, Jim; McCarthy, John E.
We define a free holomorphic function to be a function
that is locally, with respect to the free topology, a bounded
ncfunction.
We prove that free holomorphic functions are the functions that
are locally uniformly approximable
by free polynomials. We prove a realization formula and an OkaWeil
theorem for free analytic functions.


286  A SkolemMahlerLech Theorem for Iterated Automorphisms of $K$algebras Bell, Jason P.; Lagarias, Jeffrey C.
This paper proves a commutative algebraic extension
of a generalized SkolemMahlerLech theorem due to the first
author.
Let $A$ be a finitely generated commutative $K$algebra
over a field of characteristic $0$, and let $\sigma$ be
a $K$algebra automorphism of $A$.
Given ideals $I$ and $J$ of $A$, we show that
the set $S$ of integers $m$ such that
$\sigma^m(I) \supseteq J$ is a finite union of
complete doubly infinite arithmetic progressions in $m$, up to
the addition of a finite set.
Alternatively, this result states that for an affine scheme
$X$ of finite type over $K$,
an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$
any two closed subschemes of $X$, the set
of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above.
The paper presents examples
showing that this result may fail to hold if the affine scheme
$X$ is
not of finite type, or if $X$ is of finite type but the field
$K$ has positive characteristic.


315  Unitary Eigenvarieties at Isobaric Points Bellaïche, Joël
In this article we
study the geometry of the eigenvarieties of unitary groups at points
corresponding to tempered nonstable representations with an
antiordinary (a.k.a evil) refinement. We prove that, except in the
case the Galois representation attached to the automorphic form is a
sum of characters, the eigenvariety is nonsmooth at such a point,
and that (under some additional hypotheses) its tangent space is big
enough to account for all the relevant Selmer group. We also study the
local reducibility locus
at those points, proving that in general, in contrast with the case of
the eigencurve, it is a proper subscheme of the fiber of the
eigenvariety over the weight space.


330  Hyperspace Dynamics of Generic Maps of the Cantor Space Bernardes, Nilson C. Jr.; Vermersch, Rômulo M.
We study the hyperspace dynamics induced from generic continuous maps
and from generic homeomorphisms of the Cantor space, with emphasis on the
notions of LiYorke chaos, distributional chaos, topological entropy,
chain continuity, shadowing and recurrence.


350  Multimarginal Optimal Transport Maps for Onedimensional Repulsive Costs Colombo, Maria; De Pascale, Luigi; Di Marino, Simone
We study a multimarginal optimal transportation
problem in one dimension. For a symmetric, repulsive cost function, we
show that given a minimizing transport plan, its symmetrization is
induced by a cyclical map, and that the symmetric optimal plan is
unique. The class of costs that we consider includes, in particular,
the Coulomb cost, whose optimal transport problem is strictly related
to the strong interaction limit of Density Functional Theory. In this
last setting, our result justifies some qualitative properties of the
potentials observed in numerical experiments.


369  A Free Product Formula for the Sofic Dimension Graham, Robert; Pichot, Mikael
It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability
measure preserving $s$regular ergodic discrete groupoids amalgamated
over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$
satisfies the equality
\[
s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)\mathfrak{h}(G_3^0)s(G_3)
\]
where $\mathfrak{h}$ is the normalized Haar measure on $G$.


404  Rotation Algebras and the Exel Trace Formula Hua, Jiajie; Lin, Huaxin
We found that if $u$ and $v$ are any two unitaries in
a unital $C^*$algebra with $\uvvu\\lt 2$ and $uvu^*v^*$ commutes with
$u$ and $v,$ then the $C^*$subalgebra $A_{u,v}$ generated by $u$ and
$v$ is isomorphic to a quotient of some rotation algebra $A_\theta$
provided that $A_{u,v}$ has a unique tracial state.
We also found that the Exel trace formula holds in any unital
$C^*$algebra.
Let $\theta\in (1/2, 1/2)$ be a real number. We prove the
following:
For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following:
if $u$ and $v$ are two unitaries in any unital simple $C^*$algebra
$A$ with tracial rank zero such that
\[
\uve^{2\pi i\theta}vu\\lt \delta
\text{ and }
{1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,
\]
for all tracial state $\tau$ of $A,$ then there exists a pair
of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$
such that
\[
\tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\,
\u\tilde{u}\\lt \epsilon
\text{ and }
\v\tilde{v}\\lt \epsilon.
\]


424  Mahler Measures as Linear Combinations of $L$values of Multiple Modular Forms Samart, Detchat
We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$functions.


450  Motion in a Symmetric Potential on the Hyperbolic Plane Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.


481  The C*algebras of Compact Transformation Groups an Huef, Astrid; Archbold, Robert John
We investigate the representation theory of the
crossedproduct $C^*$algebra associated to a compact group $G$
acting on a locally compact space $X$ when the stability subgroups
vary discontinuously.
Our main result applies when $G$ has a principal stability subgroup or
$X$ is locally of finite $G$orbit type. Then the upper multiplicity
of the representation of the crossed product induced from an
irreducible representation $V$ of a stability subgroup is obtained by
restricting $V$ to a certain closed subgroup of the stability subgroup
and taking the maximum of the multiplicities of the irreducible
summands occurring in the restriction of $V$. As a corollary we obtain
that when the trivial subgroup is a principal stability subgroup, the
crossed product is a Fell algebra if and only if every stability
subgroup is abelian. A second corollary is that the $C^*$algebra of
the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses
the classical branching theorem for the special orthogonal group
$\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n1)$. Since proper transformation
groups are locally induced from the actions of compact groups, we
describe how some of our results can be extended to transformation
groups that are locally proper.


507  On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas
We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{1, 1\}$) inside or
on the unit circle $z=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.


527  Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory Brugallé, Erwan; Shaw, Kristin
We provide some new local obstructions to
approximating
tropical curves in
smooth tropical surfaces. These obstructions are based on
a
relation between tropical and complex intersection theories which is
also established here. We give
two applications of the methods developed in this paper.
First we classify all locally irreducible approximable 3valent fan tropical
curves in a
fan tropical plane.
Secondly, we prove that a generic nonsingular
tropical surface
in tropical projective 3space contains finitely
many approximable tropical lines
if
it is of degree 3, and contains no approximable tropical lines if
it is of degree 4 or more.


573  Twisted Vertex Operators and Unitary Lie Algebras Chen, Fulin; Gao, Yun; Jing, Naihuan; Tan, Shaobin
A representation of the central extension of the
unitary Lie algebra
coordinated with a skew Laurent polynomial ring
is constructed using vertex operators over an integral $\mathbb Z_2$lattice.
The irreducible decomposition of the representation is explicitly computed and described.
As a byproduct, some fundamental representations of affine
KacMoody Lie algebra of type $A_n^{(2)}$ are recovered
by the new method.


597  Sommes friables d'exponentielles et applications Drappeau, Sary
An integer is said to be $y$friable if its greatest prime factor is less than $y$.
In this paper, we obtain estimates for exponential sums
over $y$friable numbers up to $x$ which are nontrivial
when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence,
we obtain an asymptotic formula for the
number of $y$friable solutions to the equation $a+b=c$ which is valid
unconditionnally under the same assumption.
We use a contour integration argument based on the saddle point
method, as developped in the context of friable numbers by Hildebrand
and Tenenbaum,
and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.


639  Projectivity in Algebraic Cobordism Gonzalez, Jose Luis; Karu, Kalle
The algebraic cobordism group of a scheme is generated by cycles that
are proper morphisms from smooth quasiprojective varieties. We prove
that over a field of characteristic zero the quasiprojectivity
assumption can be omitted to get the same theory.


654  Growth of Selmer groups of CM Abelian varieties Lim, Meng Fai; Murty, V. Kumar
Let $p$ be an odd prime. We study the variation of the $p$rank of
the Selmer group of Abelian varieties with complex multiplication in
certain towers of number fields.


667  Toric Degenerations, Tropical Curve, and GromovWitten Invariants of Fano Manifolds Nishinou, Takeo
In this paper, we give a tropical method for computing GromovWitten
type invariants
of Fano manifolds of special type.
This method applies to those Fano manifolds which admit toric
degenerations
to toric Fano varieties with singularities allowing small resolutions.
Examples include (generalized) flag manifolds of type A, and
some moduli space
of rank two bundles on a genus two curve.


696  Geography of Irregular Gorenstein 3folds Zhang, Tong
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3folds of general type. We generalize the classical NoetherCastelnuovo type inequalities for irregular surfaces to irregular 3folds according to the Albanese dimension.


721  A Density CorrádiHajnal Theorem Allen, Peter; Böttcher, Julia; Hladký, Jan; Piguet, Diana
We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$vertex graph which does not contain $k+1$ vertexdisjoint triangles.


759  Spectral Flow for Nonunital Spectral Triples Carey, Alan L; Gayral, Victor; Phillips, John; Rennie, Adam; Sukochev, Fedor
We prove two results about nonunital index theory left open in a
previous paper. The
first is that the spectral triple arising from an action of the reals on a $C^*$algebra with invariant trace
satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths
arising from the odd
index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.


795  On a Sumset Conjecture of Erdős Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl
Erdős conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify Erdős' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to Erdős'
conjecture for subsets of the natural numbers that are pseudorandom.


810  All Irrational Extended Rotation Algebras are AF Algebras Elliott, George A.; Niu, Zhuang
Let $\theta\in[0, 1]$ be any irrational number. It is shown that the
extended rotation algebra $\mathcal B_\theta$ introduced in
a previous paper is always an AF algebra.


827  The BochnerSchoenbergEberlein Property and Spectral Synthesis for Certain Banach Algebra Products Kaniuth, Eberhard
Associated with two commutative Banach algebras $A$ and $B$ and
a character $\theta$ of $B$ is a certain Banach algebra product
$A\times_\theta B$, which is a splitting extension of $B$ by
$A$. We investigate two topics for the algebra $A\times_\theta
B$ in relation to the corresponding ones of $A$ and $B$. The
first one is the BochnerSchoenbergEberlein property and the
algebra of BochnerSchoenbergEberlein functions on the spectrum,
whereas the second one concerns the wide range of spectral synthesis
problems for $A\times_\theta B$.


848  Faithfulness of Actions on RiemannRoch Spaces Köck, Bernhard; Tait, Joseph
Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the RiemannRoch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$invariant divisor on $X$ of
degree at least~$2g_X2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.


870  Minimal Dynamical Systems on Connected Odd Dimensional Spaces Lin, Huaxin
Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that
the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one.
Let $\Omega$ be a connected compact metric space with finite covering dimension and
with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$
Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that
$A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is
classifiable.
In particular, this applies to the minimal dynamical systems on
odd dimensional real projective spaces.
This is done by studying minimal homeomorphisms on $X\times \Omega,$ where
$X$ is the Cantor set.


893  Overconvergent Families of SiegelHilbert Modular Forms Mok, Chung Pang; Tan, Fucheng
We construct oneparameter families of overconvergent SiegelHilbert
modular forms. This result has applications to construction of
Galois representations for automorphic forms of noncohomological
weights.


923  Cremona Maps of de Jonquières Type Pan, Ivan Edgardo; Simis, Aron
This paper is concerned with suitable generalizations of a plane de
Jonquières map to higher dimensional space
$\mathbb{P}^n$ with $n\geq 3$.
For each given point of $\mathbb{P}^n$ there is a subgroup of the entire
Cremona group of dimension $n$
consisting of such maps.
One studies both geometric and grouptheoretical properties of this notion.
In the case where $n=3$ one describes an explicit set of generators of
the group and gives a homological characterization
of a basic subgroup thereof.


942  Pontryagin's Maximum Principle for the Loewner Equation in Higher Dimensions Roth, Oliver
In this paper we develop a variational method for the Loewner
equation in higher dimensions. As a result we obtain a version of Pontryagin's
maximum principle from optimal control theory for the Loewner equation in
several complex variables. Based on recent work of Arosio, Bracci and
Wold,
we then apply our version of the Pontryagin maximum
principle to obtain firstorder necessary conditions for the extremal
mappings for a wide class
of extremal problems over the set of normalized biholomorphic
mappings on the unit ball in $\mathbb{C}^n$.


961  Orthogonal Bundles and SkewHamiltonian Matrices Abuaf, Roland; Boralevi, Ada
Using properties of skewHamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r2)n\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n1$ and $n\ge 8$. We speculate that the result holds in
greater generality.


990  The Category of Bratteli Diagrams Amini, Massoud; Elliott, George A.; Golestani, Nasser
A category structure for Bratteli diagrams is proposed and a
functor from
the category of AF algebras to the category of Bratteli diagrams
is
constructed. Since isomorphism of Bratteli diagrams in this
category coincides
with Bratteli's notion of equivalence, we obtain in particular
a functorial formulation of Bratteli's
classification of AF algebras (and at the same time, of Glimm's
classification of UHF~algebras).
It is shown that the three approaches
to classification of AF~algebras, namely, through Bratteli diagrams,
Ktheory, and
abstract classifying categories, are essentially the same
from a categorical point of view.


1024  Representation Stability of Power Sets and Square Free Polynomials Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
The symmetric group $\mathcal{S}_n$ acts on the power
set $\mathcal{P}(n)$ and also on the set of
square free polynomials in $n$ variables. These
two related representations are analyzed from the stability point
of view. An application is given for the action of the symmetric
group on the cohomology of the pure braid group.


1046  Explicit Form of Cassels' $p$adic Embedding Theorem for Number Fields Dubickas, Arturas; Sha, Min; Shparlinski, Igor
In this paper, we mainly give a general explicit form of Cassels'
$p$adic embedding theorem for number fields. We also give its
refined form in the case of cyclotomic fields. As a byproduct,
given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general
unconditional upper bound for the smallest prime number $p$ such
that $f$ has a simple root modulo $p$.


1065  A Finitetime Condition for Exponential Trichotomy in Infinite Dynamical Systems Ducrot, Arnaud; Magal, Pierre; Seydi, Ousmane
In this article we study exponential trichotomy for infinite dimensional
discrete time dynamical systems. The goal of this article is to prove that
finite time exponential trichotomy conditions allow to derive exponential
trichotomy for any times. We present an application to the case of pseudo
orbits in some neighborhood of a normally hyperbolic set.


1091  Metric Compactifications and Coarse Structures Mine, Kotaro; Yamashita, Atsushi
Let $\mathbf{TB}$ be the category of totally bounded, locally
compact metric spaces
with the $C_0$ coarse structures. We show that if $X$ and $Y$
are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent
if and only if their Higson coronas are homeomorphic. In fact,
the Higson corona functor gives an equivalence of categories
$\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the category
of compact metrizable spaces. We use this fact to show that the
continuously controlled coarse structure on a locally compact
space $X$ induced by some metrizable compactification $\tilde{X}$
is determined only by the topology of the remainder $\tilde{X}\setminus
X$.


1109  Goldman Systems and Bending Systems Nohara, Yuichi; Ueda, Kazushi
We show that the moduli space
of parabolic bundles on the projective line
and the polygon space are isomorphic,
both as complex manifolds
and symplectic manifolds equipped with structures of completely integrable systems,
if the stability parameters are
small.


1144  Outer Partial Actions and Partial Skew Group Rings Nystedt, Patrik; Öinert, Johan
We extend the classicial notion of an outer action
$\alpha$ of a group $G$ on a unital ring $A$
to the case when $\alpha$ is a partial action
on ideals, all of which have local units.
We show that if $\alpha$ is an outer partial
action of an abelian group $G$,
then its associated partial skew group
ring $A \star_\alpha G$ is simple if and only if
$A$ is $G$simple.
This result is applied to partial skew group rings associated with two different types of partial dynamical systems.


1161  Nontangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=w^{1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the nontangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.


1201  Chern Classes of Splayed Intersections Aluffi, Paolo; Faber, Eleonore
We generalize the Chern class relation for the transversal intersection
of two nonsingular
varieties to a relation for possibly singular varieties, under
a splayedness assumption.
We show that the relation for the ChernSchwartzMacPherson classes
holds for two splayed hypersurfaces in a nonsingular variety,
and under a `strong splayedness' assumption for more
general subschemes. Moreover, the relation is shown to hold for
the ChernFulton classes
of any two splayed subschemes.
The main tool is a formula for Segre classes of splayed
subschemes. We also discuss the Chern class relation under the
assumption that one of the
varieties is a general very ample divisor.


1219  $p$adic and Motivic Measure on Artin $n$stacks Balwe, Chetan
We define a notion of $p$adic measure on Artin $n$stacks which are
of strongly finite type over the ring of $p$adic integers. $p$adic
measure on schemes can be evaluated by counting points on the
reduction of the scheme modulo $p^n$. We show that an analogous
construction works in the case of Artin stacks as well if we count the
points using the counting measure defined by Toën. As a consequence,
we obtain the result that the Poincaré and Serre series of such
stacks are rational functions, thus extending Denef's result for
varieties. Finally, using motivic integration we show that as $p$
varies, the rationality of the Serre series of an Artin stack defined
over the integers is uniform with respect to $p$.


1247  Lyapunov Stability and Attraction Under Equivariant Maps Barros, Carlos Braga; Rocha, Victor; Souza, Josiney
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed
with
admissible families of open coverings. Assume that $\mathcal{S}$ is a
semigroup acting on both $M$ and $N$. In this paper we study the behavior of
limit sets, prolongations, prolongational limit sets, attracting sets,
attractors and Lyapunov stable sets (all concepts defined for the action of
the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations
from $M$ to $N$.


1270  Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems Carcamo, Cristian; Vidal, Claudio
In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.


1290  On Twofaced Families of Noncommutative Random Variables Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul
We demonstrate that the notions of bifree independence and combinatorialbifree
independence of twofaced families are equivalent using a diagrammatic
view of binoncrossing partitions.
These diagrams produce an operator model on a Fock space suitable
for representing any twofaced family of noncommutative random
variables.
Furthermore, using a Kreweras complement on binoncrossing partitions
we establish the expected formulas for the multiplicative convolution
of a bifree pair of twofaced families.


1326  The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank Cojocaru, Alina Carmen; Shulman, Andrew Michael
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study
the first elementary divisor
$d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies.
In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$
and prove that, on average, it has a large norm. Our work is motivated by the study of J.P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.


1358  On the Rate of Convergence of Empirical Measures in $\infty$transportation Distance Garcia Trillos, Nicolas; Slepcev, Dejan
We consider random i.i.d. samples of absolutely continuous measures
on bounded connected domains.
We prove an upper bound on the $\infty$transportation distance
between the measure and the empirical measure of the sample.
The bound is optimal in terms of scaling with the number of sample
points.


1384  Strong Logarithmic Sobolev Inequalities for LogSubharmonic Functions Graczyk, Piotr; Kemp, Todd; Loeb, JeanJacques
We prove an intrinsic equivalence between strong
hypercontractivity and a strong logarithmic Sobolev
inequality for the cone of logarithmically subharmonic
(LSH) functions. We introduce a new large class of measures,
Euclidean regular and exponential type, in addition to all compactlysupported
measures, for which this equivalence holds. We prove a Sobolev
density theorem through LSH functions and use it to prove
the equivalence of strong
hypercontractivity and the strong logarithmic Sobolev
inequality for such logsubharmonic
functions.


1411  Functiontheoretic Properties for the Gauss Maps of Various Classes of Surfaces Kawakami, Yu
We elucidate the geometric background of functiontheoretic properties
for the Gauss maps of
several classes of immersed surfaces in threedimensional space
forms, for example, minimal surfaces in Euclidean threespace, improper affine spheres in the affine threespace, and constant
mean curvature one surfaces and flat surfaces in hyperbolic threespace. To achieve this purpose, we prove an optimal curvature bound
for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for
the Gauss maps of these classes of surfaces.

