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Characterization of positive links and the $s$invariant for links Abe, Tetsuya; Tagami, Keiji Published: 20161129
We characterize positive links in terms of strong quasipositivity,
homogeneity and the value of Rasmussen and BeliakovaWehrli's
$s$invariant.
We also study almost positive links,
in particular, determine the $s$invariants of
almost positive links.
This result suggests that all almost positive links might
be strongly quasipositive.
On the other hand, it implies that
almost positive links are never homogeneous links.


Finite determinacy and stability of flatness of analytic mappings Adamus, Janusz; Seyedinejad, Hadi Published: 20160429
It is proved that flatness of an analytic mapping germ from a
complete intersection is determined by its sufficiently high
jet. As a consequence, one obtains finite determinacy of complete
intersections. It is also shown that flatness and openness are
stable under deformations.


Weyl images of Kantor pairs Allison, Bruce; Faulkner, John; Smirnov, Oleg Author's Draft
Kantor pairs arise naturally in the study of
$5$graded Lie algebras. In this article, we introduce
and study Kantor pairs with short Peirce gradings and relate
them to Lie algebras
graded by the root system of type
$\mathrm{BC}_2$.
This relationship
allows us to define so called Weyl images
of short Peirce graded Kantor pairs. We use Weyl images to construct
new examples of Kantor pairs, including a class of infinite
dimensional
central simple Kantor pairs over a field of characteristic $\ne
2$ or $3$, as well as a family of forms of a split
Kantor pair of type
$\mathrm{E}_6$.


Anisotropic HardyLorentz spaces with variable exponents Almeida, Víctor; Betancor, Jorge J.; RodríguezMesa, Lourdes Author's Draft
In this paper we introduce HardyLorentz spaces with variable
exponents associated to dilations in ${\mathbb R}^n$. We establish
maximal characterizations and atomic decompositions for our variable
exponent anisotropic HardyLorentz spaces.


Dynamics and regularization of the Kepler problem on surfaces of constant curvature Andrade, Jaime; Dávila, Nestor; PérezChavela, Ernesto; Vidal, Claudio Published: 20160617
We classify and analyze the orbits of the Kepler problem on surfaces
of constant curvature (both positive and negative, $\mathbb S^2$ and
$\mathbb H^2$, respectively) as function of the angular momentum and
the energy. Hill's region are characterized and the problem of
timecollision is studied. We also regularize the problem in
Cartesian and intrinsic coordinates, depending on the constant
angular momentum and we describe the orbits of the regularized
vector field. The phase portrait both for $\mathbb S^2$ and $\mathbb H^2$
are pointed out.


Partial Hasse invariants, partial degrees, and the canonical subgroup Bijakowski, Stephane Author's Draft
If the Hasse invariant of a $p$divisible group is small enough,
then one can construct a canonical subgroup inside its $p$torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.


Comparison Properties of the Cuntz semigroup and applications to C*algebras Bosa, Joan; Petzka, Henning Author's Draft
We study comparison properties in the category $\mathrm{Cu}$ aiming to
lift results to the C*algebraic setting. We introduce a new
comparison property and relate it to both the CFP and $\omega$comparison.
We show differences of all properties by providing examples,
which suggest that the corona factorization for C*algebras might
allow for both finite and infinite projections. In addition,
we show that R{\o}rdam's simple, nuclear C*algebra with a finite
and an infinite projection does not have the CFP.


Simultaneous additive equations: Repeated and differing degrees Brandes, Julia; Parsell, Scott T. Published: 20160621
We obtain bounds for the number of variables required to establish
Hasse principles, both for existence of solutions and for asymptotic
formulæ, for systems of additive equations containing forms
of differing degree but also multiple forms of like degree.
Apart from the very general estimates of Schmidt and BrowningHeathBrown,
which give weak results when specialized to the diagonal situation,
this is the first result on such "hybrid" systems. We also obtain
specialised results for systems of quadratic and cubic forms,
where we are able to take advantage of some of the stronger methods
available in that setting. In particular, we achieve essentially
square root cancellation for systems consisting of one cubic
and $r$ quadratic equations.


Stability for the BrunnMinkowski and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range nonlocal isoperimetry Carlen, Eric; Maggi, Francesco Published: 20161108
We provide a simple, general argument to obtain improvements
of concentrationtype inequalities starting from improvements
of their corresponding isoperimetrictype inequalities. We apply
this argument to obtain robust improvements of the BrunnMinkowski
inequality (for Minkowski sums between generic sets and convex
sets) and of the Gaussian concentration inequality. The former
inequality is then used to obtain a robust improvement of the
Riesz rearrangement inequality under certain natural conditions.
These conditions are compatible with the applications to a finiterange
nonlocal isoperimetric problem arising in statistical mechanics.


Convolution powers of Salem measures with applications Chen, Xianghong; Seeger, Andreas Published: 20160914
We study the regularity of convolution powers for measures supported
on
Salem sets, and prove related results on Fourier restriction
and Fourier multipliers. In particular we show
that for $\alpha$ of the form
${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$Salem measures
for which the $L^2$ Fourier restriction theorem holds in the
range $p\le \frac{2d}{2d\alpha}$.
The results rely on ideas of Körner.
We extend some of his constructions to obtain upper regular $\alpha$Salem
measures, with sharp regularity results for $n$fold convolutions
for all $n\in \mathbb{N}$.


Wedge operations and torus symmetries II Choi, Suyoung; Park, Hanchul Published: 20161129
A fundamental idea in toric topology is that classes of manifolds
with wellbehaved torus actions (simply, toric spaces) are classified
by pairs of simplicial complexes and (nonsingular) characteristic
maps. The authors in their previous paper provided a new way
to find all characteristic maps on a simplicial complex $K(J)$
obtainable by a sequence of wedgings from $K$. The main idea
was that characteristic maps on $K$ theoretically determine all
possible characteristic maps on a wedge of $K$.


Fixed point theorems for maps with local and pointwise contraction properties Ciesielski, Krzysztof Chris; Jasinski, Jakub Author's Draft
The paper constitutes a comprehensive study of ten classes of
selfmaps on metric spaces $\langle X,d\rangle$ with the local
and pointwise (a.k.a. local radial) contraction properties.
Each of those classes appeared previously in the literature in
the context of fixed point theorems.


Transport inequalities for logconcave measures, quantitative forms and applications CorderoErausquin, Dario Author's Draft
We review some simple techniques based on monotone mass transport
that allow us to obtain transporttype inequalities for any
logconcave
probability measure, and for more general measures as well. We
discuss quantitative forms of these inequalities, with application
to the BrascampLieb variance inequality.


Transport inequalities for logconcave measures, quantitative forms and applications CorderoErausquin, Dario Author's Draft
We review some simple techniques based on monotone mass transport
that allow to obtain transporttype inequalities for any logconcave
probability measure. We discuss quantitative forms of these inequalities,
with application to the variance BrascampLieb inequality.


Amenability and covariant injectivity of locally compact quantum groups II Crann, Jason Author's Draft
Building on our previous work, we study the nonrelative homology
of quantum group convolution algebras. Our main result establishes
the equivalence of amenability of a locally compact quantum group
$\mathbb{G}$ and 1injectivity of
$L^{\infty}(\widehat{\mathbb{G}})$
as an operator
$L^1(\widehat{\mathbb{G}})$module.
In particular, a locally compact group $G$ is amenable if and
only if its group von Neumann algebra
$VN(G)$
is 1injective as
an operator module over the Fourier algebra $A(G)$. As an application,
we provide a decomposability result for completely bounded
$L^1(\widehat{\mathbb{G}})$module
maps on
$L^{\infty}(\widehat{\mathbb{G}})$,
and give a simplified proof that amenable discrete
quantum groups have coamenable compact duals which avoids the
use of modular theory and the PowersStørmer inequality, suggesting
that our homological techniques may yield a new approach to the
open problem of duality between amenability and coamenability.


The BishopPhelpsBollobás property for compact operators Dantas, Sheldon; García, Domingo; Maestre, Manuel; Martín, Miguel Author's Draft
We study the BishopPhelpsBollobás property (BPBp for short)
for compact operators. We present some abstract techniques which
allows to carry the BPBp for compact operators from sequence
spaces to function spaces. As main applications, we prove the
following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$
has the BPBp for compact operators, then so do $(C_0(L),Y)$ for
every locally compact Hausdorff topological space $L$ and $(X,Y)$
whenever $X^*$ is isometrically isomorphic to $\ell_1$.
If $X^*$ has the RadonNikodým property and $(\ell_1(X),Y)$
has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$
for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$
has the the BPBp for compact operators when $X$ and $Y$ are finitedimensional
or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any
positive measure $\nu$ and $1\lt p\lt \infty$.
For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact
operators, then so does $(X,L_p(\mu,Y))$ for every positive measure
$\mu$ such that $L_1(\mu)$ is infinitedimensional. If $(X,Y)$
has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$
for every $\sigma$finite positive measure $\mu$ and $(X,C(K,Y))$
for every compact Hausdorff topological space $K$.


Tilings of normed spaces De Bernardi, Carlo Alberto; Veselý, Libor Published: 20160308
By a tiling of a topological linear space $X$ we mean a
covering of $X$ by at least two closed convex sets,
called tiles, whose nonempty interiors are
pairwise disjoint.
Study of tilings of infinitedimensional spaces initiated in
the
1980's with pioneer papers by V. Klee.
We prove some general properties of tilings of locally convex
spaces,
and then apply these results to study existence of tilings of
normed and Banach spaces by tiles possessing
certain smoothness or rotundity properties. For a Banach space
$X$,
our main results are the following.


The classical $N$body problem in the context of curved space Diacu, Florin Author's Draft
We provide the differential equations that generalize the Newtonian
$N$body problem of celestial mechanics to spaces of constant
Gaussian curvature, $\kappa$, for all $\kappa\in\mathbb R$. In
previous studies, the equations of motion made sense only for
$\kappa\ne 0$. The system derived here does more than just include
the Euclidean case in the limit $\kappa\to 0$: it recovers the
classical equations for $\kappa=0$. This new expression of the
laws of motion allows the study of the $N$body problem in the
context of constant curvature spaces and thus offers a natural
generalization of the Newtonian equations that includes the classical
case. We end the paper with remarks about the bifurcations of
the first integrals.


A class of degenerate elliptic equations with nonlinear boundary conditions Du, Zhuoran; Fang, Yanqin; Gui, Changfeng Published: 20160627
We consider positive solutions of the problem
\begin{equation}
(*)\qquad
\left\{
\begin{array}{l}\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n},
\\
\frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n},
\\
\end{array}
\right.
\end{equation}
where $a\in (1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and
$\frac{\partial u}{\partial \nu^a}:=\lim_{x_{n}\rightarrow
0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case
$b=0$, it is associated to fractional Laplacian equation $(\Delta)^{s}u=u^{q}
$ in entire space $\mathbb{R}^{n1}$.
We obtain the existence of positive axially symmetric solutions
to ($*$) for the case $a\in
(1,0)$ in
$n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a2},
\;\;q\geq\frac{na}{n+a2}$.
The nonexistence is obtained for the case $a\in (1,0)$, $b\geq
0$ and any $p,~q\gt 1$ in $n=2$ as well.


The minimal free resolution of fat almost complete intersections in $\mathbb{P}^1\times \mathbb{P}^1$ Favacchio, Giuseppe; Guardo, Elena Published: 20161223
A current research theme is to compare symbolic powers of an
ideal
$I$ with the regular powers of $I$. In this paper, we focus on
the
case that $I=I_X$ is an ideal defining an almost complete
intersection (ACI) set of points $X$ in
$\mathbb{P}^1 \times \mathbb{P}^1$.
In particular,
we describe a minimal free bigraded resolution of a non
arithmetically CohenMacaulay (also non homogeneous) set $\mathcal
Z$ of fat
points whose support is an ACI, generalizing
a result of S. Cooper et al.
for homogeneous sets of triple points. We call
$\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary
powers are equal, i.e,
$I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\geq 1.$


Splitting, Bounding, and Almost Disjointness can be quite Different Fischer, Vera; Mejia, Diego Alejandro Published: 20160719
We prove the consistency of
$$
\operatorname{add}(\mathcal{N})\lt
\operatorname{cov}(\mathcal{N})
\lt \mathfrak{p}=\mathfrak{s}
=\mathfrak{g}\lt \operatorname{add}(\mathcal{M})
= \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a}
=\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c}
$$
with $\mathrm{ZFC}$, where each of these cardinal
invariants assume arbitrary
uncountable regular values.


Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$ Folha, Abigail; Penafiel, Carlos Author's Draft
In this article, we study complete surfaces $\Sigma$, isometrically
immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or
$\mathbb{S}^2\times\mathbb{R}$
having positive extrinsic curvature $K_e$. Let $K_i$ denote the
intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$
holds for some real constants $a\neq0$, $b\gt 0$ and $c$. The main
result of this article state that when such a surface is a topological
sphere it is rotational.


Dirichlet's theorem in function fields Ganguly, Arijit; Ghosh, Anish Published: 20160729
We study metric Diophantine approximation for function fields
specifically the problem of improving Dirichlet's theorem in
Diophantine
approximation.


On K3 surface quotients of K3 or Abelian surfaces Garbagnati, Alice Published: 20160318
The aim of this paper is to prove that a K3 surface is the minimal
model of the quotient of an Abelian surface by a group $G$ (respectively
of a K3 surface by an Abelian group $G$) if and only if a certain
lattice is primitively embedded in its NéronSeveri group.
This allows one to describe the coarse moduli space of the K3
surfaces which are (rationally) $G$covered by Abelian or K3
surfaces (in the latter case $G$ is an Abelian group).
If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian
surface, this result was already known, so we extend it to the
other cases.


A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups Ghaani Farashahi, Arash Author's Draft
This paper introduces a class of abstract linear representations
on
Banach convolution function algebras over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $\mu$ be the normalized $G$invariant measure over the compact
homogeneous space $G/H$ associated to the
Weil's formula and $1\le p\lt \infty$.
We then present a structured class of abstract linear representations
of the
Banach convolution function algebras $L^p(G/H,\mu)$.


Inequalities for the surface area of projections of convex bodies Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros Author's Draft
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.


$L^q$ norms of Fekete and related polynomials Günther, Christian; Schmidt, KaiUwe Published: 20160628
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all
of its coefficients in $\{1,1\}$. There are various old unsolved
problems, mostly due to Littlewood and Erdős, that ask for
Littlewood polynomials that provide a good approximation to a
function that is constant on the complex unit circle, and in
particular have small $L^q$ norm on the complex unit circle.
We consider the Fekete polynomials
\[
f_p(z)=\sum_{j=1}^{p1}(j\,\,p)\,z^j,
\]
where $p$ is an odd prime and $(\,\cdot\,\,p)$ is the Legendre
symbol (so that $z^{1}f_p(z)$ is a Littlewood polynomial). We
give explicit and recursive formulas for the limit of the ratio
of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive
integer and $p\to\infty$. To our knowledge, these are the first
results that give these limiting values for specific sequences
of nontrivial Littlewood polynomials and infinitely many $q$.
Similar results are given for polynomials obtained by cyclically
permuting the coefficients of Fekete polynomials and for Littlewood
polynomials whose coefficients are obtained from additive characters
of finite fields. These results vastly generalise earlier results
on the $L^4$ norm of these polynomials.


Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere Hakl, Robert; Zamora, Manuel Author's Draft
We study a secondorder ordinary differential
equation coming from the Kepler problem on $\mathbb{S}^2$. The
forcing term under consideration is a piecewise constant with
singular nonlinearity which changes sign. We establish necessary
and
sufficient conditions to the existence and multiplicity of
$T$periodic solutions.


On computable field embeddings and difference closed fields HarrisonTrainor, Matthew; Melnikov, Alexander; Miller, Russell Author's Draft
We investigate when a computable automorphism of a computable
field can be effectively extended to a computable automorphism
of its (computable) algebraic closure. We then apply our results
and techniques to study effective embeddings of computable difference
fields into computable difference closed fields.


Free product C*algebras associated to graphs, free differentials, and laws of loops Hartglass, Michael Published: 20160920
We study a canonical C$^*$algebra, $\mathcal{S}(\Gamma, \mu)$, that
arises from a weighted graph $(\Gamma, \mu)$, specific cases
of which were previously studied in the context of planar algebras.
We discuss necessary and sufficient conditions of the weighting
which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma,
\mu)$, and study the structure of its positive cone. We then
study the $*$algebra, $\mathcal{A}$, generated by the generators of
$\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and
techniques of Charlesworth and Shlyakhtenko, as well as Mai,
Speicher, and Weber to show that certain ``loop" elements have
no atoms in their spectral measure. After modifying techniques
of Shlyakhtenko and Skoufranis to show that self adjoint elements
$x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore
some applications to eigenvalues of polynomials in Wishart matrices
and to diagrammatic elements in von Neumann algebras initially
considered by Guionnet, Jones, and Shlyakhtenko.


Strict comparison of positive elements in multiplier algebras Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang Published: 20160628
Main result: If a C*algebra $\mathcal{A}$ is simple, $\sigma$unital,
has finitely many extremal traces, and has strict comparison
of positive elements by traces, then its multiplier algebra
$\operatorname{\mathcal{M}}(\mathcal{A})$
also has strict comparison of positive elements by traces. The
same results holds if ``finitely many extremal traces" is replaced
by ``quasicontinuous scale".
A key ingredient in the proof is that every positive element
in the multiplier algebra of an arbitrary $\sigma$unital C*algebra
can be approximated by a bidiagonal series.
An application of strict comparison: If $\mathcal{A}$ is a simple separable
stable C*algebra with real rank zero, stable rank one, and
strict comparison of positive elements by traces, then whether
a positive element is a positive linear combination of projections
is determined by the trace values of its range projection.


Free function theory through matrix invariants Klep, Igor; Špenko, Špela Author's Draft
This paper concerns free function theory. Free maps are free
analogs of analytic functions in several complex variables,
and are defined in terms of freely noncommuting variables.
A function of $g$ noncommuting variables is a function on $g$tuples
of square matrices of all sizes that respects direct sums and
simultaneous conjugation.
Examples of such maps include noncommutative polynomials, noncommutative
rational functions and convergent noncommutative power series.


New deformations of convolution algebras and Fourier algebras on locally compact groups Lee, Hun Hee; Youn, Sanggyun Published: 20160923
In this paper we introduce a new way of deforming convolution
algebras and Fourier algebras on locally compact groups. We demonstrate
that this new deformation allows us to reveal some information
of the underlying groups by examining Banach algebra properties
of deformed algebras. More precisely, we focus on representability
as an operator algebra of deformed convolution algebras on compact
connected Lie groups with connection to the real dimension of
the underlying group. Similarly, we investigate complete representability
as an operator algebra of deformed Fourier algebras on some finitely
generated discrete groups with connection to the growth rate
of the group.


Regulators of an infinite family of the simplest quartic function fields Lee, Jungyun; Lee, Yoonjin Published: 20161206
We explicitly find regulators of an infinite family $\{L_m\}$
of the simplest quartic function fields
with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where
$\mathbb{F}_q$
is the finite field of order $q$
with odd characteristic. In fact, this infinite family of the
simplest quartic function fields are
subfields of maximal real subfields of cyclotomic function fields,
where they have the same conductors.
We obtain a lower bound on the class numbers of the family $\{L_m\}$
and some result on the divisibility
of the divisor class numbers of cyclotomic function fields which
contain $\{L_m\}$ as their subfields.
Furthermore, we find an explicit criterion for the characterization
of splitting types of all the primes
of the rational function field $\mathbb{F}_q (t)$ in $\{L_m\}$.


On the asymptotic growth of BlochKatoShafarevichTate groups of modular forms over cyclotomic extensions Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia Published: 20161206
We study the asymptotic behaviour of the BlochKatoShafarevichTate
group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$extension
of $\mathbb{Q}$ under the assumption that $f$ is nonordinary at $p$.
In particular, we give upper bounds of these groups in terms
of Iwasawa invariants of Selmer groups defined using $p$adic
Hodge Theory. These bounds have the same form as the formulae
of Kobayashi, Kurihara and Sprung for supersingular elliptic
curves.


Toric geometry of $SL_2(\mathbb{C})$ free group character varieties from outer space Manon, Christopher Author's Draft
Culler and Vogtmann defined a simplicial space $O(g)$ called
outer space to study the outer automorphism group
of the free group $F_g$. Using representation theoretic methods,
we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g,
SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety
of $F_g,$ reproving a result of Morgan and Shalen. Then we show
that every point $v$ contained in a maximal cell of $O(g)$ defines
a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to
a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there
is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g,
SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a
symplectomorphism on a dense, open subset of $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ with respect to natural symplectic structures
on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this
way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$,
and we show that each $v$ defines a topological decomposition
of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition
of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that
the valuations coming from the closure of a maximal cell in $O(g)$
all arise as divisorial valuations built from an associated projective
compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$


Isomorphisms of twisted Hilbert loop algebras Marquis, Timothée; Neeb, KarlHermann Published: 20160510
The closest infinite dimensional relatives of compact Lie algebras are HilbertLie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple HilbertLie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple HilbertLie algebra $\mathfrak{k}$,
which we call standard.


On the digits of sumsets Mauduit, Christian; Rivat, Joël; Sárközy, András Published: 20160510
Let $\mathcal A$, $\mathcal B$ be large subsets of $\{1,\ldots,N\}$.
We study the number of pairs $(a,b)\in\mathcal A\times\mathcal B$ such that
the sum of binary digits of $a+b$ is fixed.


Mori's program for $\overline{M}_{0,7}$ with symmetric divisors Moon, HanBom Published: 20160418
We complete Mori's program with symmetric divisors for the moduli
space of stable sevenpointed rational curves. We describe all
birational models in terms of explicit blowups and blowdowns.
We also give a moduli theoretic description of the first flip,
which has not appeared in the literature.


Closed convex hulls of unitary orbits in certain simple real rank zero C$^*$algebras Ng, P. W.; Skoufranis, P. Author's Draft
In this paper, we characterize the closures of convex hulls of
unitary orbits of selfadjoint operators in unital, separable,
simple C$^*$algebras with nontrivial tracial simplex, real
rank zero, stable rank one, and strict comparison of projections
with respect to tracial states. In addition, an upper bound
for the number of unitary conjugates in a convex combination
needed to approximate a selfadjoint are obtained.


Extremal sequences for the Bellman function of the dyadic maximal operator and applications to the Hardy operator Nikolidakis, Eleftherios Nikolaos Published: 20160727
We prove that the extremal sequences for the
Bellman function of the dyadic maximal operator behave approximately
as eigenfunctions of this operator for a specific eigenvalue.
We use this result to prove the analogous one with respect to
the Hardy operator.


Almost disjointness preservers Oikhberg, Timur; Tradacete, Pedro Published: 20160927
We study the stability of disjointness preservers on Banach lattices.
In many cases, we prove that an "almost disjointness preserving"
operator is well approximable by a disjointness preserving one.
However, this approximation is not always possible, as our
examples show.


Tannakian categories with semigroup actions Ovchinnikov, Alexey; Wibmer, Michael Published: 20160916
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are
algebraically independent over $\mathbb{C}(x)$. More generally, for
a linear differential or difference equation, it is an important
problem to find all algebraic dependencies among a nonzero solution
$y$ and particular transformations of $y$, such as derivatives
of $y$ with respect to parameters, shifts of the arguments, rescaling,
etc. In the present paper, we develop a theory of Tannakian categories
with semigroup actions, which will be used to attack such questions
in full generality, as each linear differential equation gives
rise to a Tannakian category.
Deligne studied actions of braid groups on categories and obtained
a finite collection of axioms that characterizes such actions
to apply it to various geometric constructions. In this paper,
we find a finite set of axioms that characterizes actions of
semigroups that are finite free products of semigroups of the
form $\mathbb{N}^n\times
\mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$
on Tannakian categories. This is the class of semigroups that
appear in many applications.


Global and non global solutions for some fractional heat equations with pure power nonlinearity Saanouni, Tarek Published: 20160621
The initial value problem for a semilinear fractional heat equation
is investigated. In the focusing case, global wellposedness
and exponential decay are obtained. In the focusing sign, global
and non global existence of solutions are discussed via the potential
well method.


Order and spectrum preserving maps on positive operators Semrl, Peter Published: 20161206
We describe the general form of surjective maps on the cone of
all positive operators which preserve order and spectrum. The
result is optimal as shown by
counterexamples. As an easy consequence we characterize surjective
order and spectrum preserving maps on the set of all selfadjoint
operators.


The seven dimensional perfect Delaunay polytopes and Delaunay simplices Sikirić, Mathieu Dutour Published: 20160621
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$
and radius $r$
is called empty if for any $v\in L$ we have $\Vert v 
c\Vert \geq r$.
Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay
polytope}
$P=\operatorname{conv}(S(c,r)\cap L)$.
A Delaunay polytope is called {\em perfect} if any affine transformation
$\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily
an isometry
of the space composed with an homothety.


Quasianalytic Ilyashenko algebras Speissegger, Patrick Author's Draft
I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$
of real functions with logarithmic generalized power series as
asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation
and $\log$composition; in particular, $\mathcal{F}$ is a Hardy field.
Moreover, the field $\mathcal{F} \circ (\log)$ of germs at $0^+$ contains
all transition maps of hyperbolic saddles of planar real analytic
vector fields.


On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical Varma, Sandeep Published: 20161209
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic
subgroup of a connected
reductive group $\operatorname{G}$ over a $p$adic field $F$. Assume that there
exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$
to an opposite parabolic subgroup.
When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$orbit,
F. Shahidi and X. Yu describe a certain distribution $D$ on
$\operatorname{M}(F)$
such that,
for irreducible unitary supercuspidal representations $\pi$ of
$\operatorname{M}(F)$ with
$\pi \cong \pi \circ \operatorname{Int} w_0$,
$\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is
irreducible
if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of
$\pi$. Since
this irreducibility is conjecturally related to $\pi$ arising
via
transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is
of interest
to realize $D$ as endoscopic transfer from a simpler distribution
on a twisted
endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations
where $\operatorname{N}$ is abelian. Here, we handle the `standard examples'
in cases
where $\operatorname{N}$ is nonabelian but admits a Zariski dense
$\operatorname{Int} \operatorname{M}$orbit.


Anisotropic Sobolev Capacity with Fractional Order Xiao, Jie; Ye, Deping Published: 20160316
In this paper, we introduce the anisotropic
Sobolev capacity with fractional order and develop some basic
properties for this new object. Applications to the theory of
anisotropic fractional Sobolev spaces are provided. In particular,
we give geometric characterizations for a nonnegative Radon
measure $\mu$ that naturally induces an embedding of the anisotropic
fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into
the $\mu$basedLebesguespace $L^{n/\beta}_\mu$ with $0\lt \beta\le
n$. Also, we investigate the anisotropic fractional $\alpha$perimeter.
Such a geometric quantity can be used to approximate the anisotropic
Sobolev capacity with fractional order. Estimation on the constant
in the related Minkowski inequality, which is asymptotically
optimal as $\alpha\rightarrow 0^+$, will be provided.


On Moeglin's parametrization of Arthur packets for padic quasisplit $Sp(N)$ and $SO(N)$ Xu, Bin Published: 20161110
We give a survey on Moeglin's construction of representations
in the Arthur packets for $p$adic quasisplit symplectic and
orthogonal groups. The emphasis is on comparing Moeglin's
parametrization of elements in the Arthur packets with that of
Arthur.


Absolute continuity of Wasserstein barycenters over Alexandrov spaces Yin, Jiang Published: 20161206
In this paper, we prove that, on a compact, $n$dimensional Alexandrov
space with curvature $\geqslant 1$, the Wasserstein barycenter of
Borel probability measures $\mu_1,...,\mu_m$ is absolutely continuous
with respect to the $n$dimensional Hausdorff measure if one
of them is.

