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721  Tameness of Complex Dimension in a Real Analytic Set Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul
Given a real analytic set $X$ in a complex manifold and a positive
integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which
there exists a germ of a complex analytic set of dimension $d$ contained in $X$.
It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$.


740  Regularization of Subsolutions in Discrete Weak KAM Theory Bernard, P.; Zavidovique, M.
We expose different methods of regularizations of subsolutions
in the context of discrete weak KAM theory.
They allow to prove the existence and the density of $C^{1,1}$
subsolutions. Moreover, these subsolutions can be made strict
and smooth outside of the Aubry set.


757  Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved Delanoë, Philippe; Rouvière, François
The squared distance curvature is a kind of twopoint curvature the
sign of which turned out crucial for the smoothness of optimal
transportation maps on Riemannian manifolds. Positivity properties of
that new curvature have been established recently for all the simply
connected compact rank one symmetric spaces, except the Cayley
plane. Direct proofs were given for the sphere, an indirect one
via the Hopf fibrations) for the complex and quaternionic
projective spaces. Here, we present a direct proof of a property
implying all the preceding ones, valid on every positively curved
Riemannian locally symmetric space.


768  Nonselfadjoint Semicrossed Products by Abelian Semigroups Fuller, Adam Hanley
Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\mathcal{S}_i$, where for each $i\in I$,
$\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\mathbb{R}_+$ containing $0$. We consider representations
of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nicacovariance property:
$T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nicacovariant
dilation.


783  Generalised Triple Homomorphisms and Derivations Garcés, Jorge J.; Peralta, Antonio M.
We introduce generalised triple homomorphism between Jordan Banach
triple systems as a concept which extends the notion of generalised homomorphism between
Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively.
We prove that every generalised triple homomorphism between JB$^*$triples
is automatically continuous. When particularised to C$^*$algebras, we rediscover
one of the main theorems established by B.E. Johnson. We shall also consider generalised
triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$module,
proving that every generalised triple derivation from a JB$^*$triple $E$ into itself or into $E^*$
is automatically continuous.


808  On Hessian Limit Directions along Gradient Trajectories Grandjean, Vincent
Given a nonoscillating gradient trajectory $\gamma$ of a real analytic function $f$,
we show that the limit $\nu$ of the secants at the limit point
$\mathbf{0}$
of $\gamma$ along the trajectory
$\gamma$ is an eigenvector of the limit of the direction of the
Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$
along $\gamma$. The same holds true at infinity if the function is globally subanalytic. We also deduce
some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is
of metric nature and still holds in a general Riemannian analytic setting.


823  Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$ Guardo, Elena; Harbourne, Brian; Van Tuyl, Adam
Recent work of EinLazarsfeldSmith and HochsterHuneke
raised the problem of which symbolic powers of an ideal
are contained in a given ordinary power of the ideal.
BocciHarbourne developed methods to address this problem,
which involve asymptotic numerical characters of
symbolic powers of the ideals. Most of the work
done up to now has been done for ideals defining 0dimensional
subschemes of projective space.
Here we focus on certain subschemes given by
a union of lines in $\mathbb{P}^3$ which can also be viewed
as points in $\mathbb{P}^1 \times \mathbb{P}^1$.
We also obtain results on the
closely related problem, studied by Hochster and by LiSwanson, of
determining situations for which
each symbolic power of an ideal is an ordinary power.


843  3torsion in the Homology of Complexes of Graphs of Bounded Degree Jonsson, Jakob
For $\delta \ge 1$ and $n \ge 1$, consider the simplicial
complex of graphs on $n$ vertices in which each vertex has degree
at most $\delta$; we identify a given graph with its edge set and
admit one loop at each vertex.
This complex is of some importance in the theory of semigroup
algebras.
When $\delta = 1$, we obtain the
matching complex, for which it is known that
there is $3$torsion in degree $d$ of the homology
whenever $\frac{n4}{3} \le d \le \frac{n6}{2}$.
This paper establishes similar bounds for $\delta \ge
2$. Specifically, there is $3$torsion in degree $d$ whenever
$\frac{(3\delta1)n8}{6} \le d \le \frac{\delta (n1) 
4}{2}$.
The procedure for detecting
torsion is to construct an explicit cycle $z$ that is easily seen
to have the property that $3z$ is a boundary. Defining a
homomorphism that sends
$z$ to a nonboundary element in the chain complex of a certain
matching complex, we obtain that $z$ itself is a nonboundary.
In particular, the homology class of $z$ has order $3$.


863  Cumulants of the $q$semicircular Law, Tutte Polynomials, and Heaps JosuatVergès, Matthieu
The $q$semicircular distribution is a probability law that
interpolates between the Gaussian law and the semicircular law. There
is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free
cumulants one has to restrict the enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of
the classical cumulants. We show that like the free cumulants, they
are obtained by an enumeration of connected matchings, the weight
being now an evaluation of the Tutte polynomial of a socalled
crossing graph. The case $q=0$ of these cumulants was studied by
Lassalle using symmetric functions and hypergeometric series. We show
that the underlying combinatorics is explained through the theory of
heaps, which is Viennot's geometric interpretation of the
CartierFoata monoid. This method also gives a general formula for
the cumulants in terms of free cumulants.


879  A Space of Harmonic Maps from the Sphere into the Complex Projective Space Kawabe, Hiroko
GuestOhnita and Crawford have shown the pathconnectedness of the
space of harmonic maps from $S^2$ to $\mathbf{C} P^n$
of a fixed degree and energy.It is wellknown that the $\partial$ transform is defined on this space.
In this paper,we will show that the space is decomposed into mutually disjoint connected subspaces on which
$\partial$ is homeomorphic.


905  Explicit Models for Threefolds Fibred by K3 Surfaces of Degree Two Thompson, Alan
We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two.


927  Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$ Wang, Liping; Zhao, Chunyi
We consider the following prescribed boundary mean curvature problem
in $ \mathbb B^N$ with the Euclidean metric:
\[
\begin{cases}
\displaystyle \Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N,
\\[2ex]
\displaystyle \frac{\partial u}{\partial\nu} + \frac{N2}{2} u =\frac{N2}{2} \widetilde K(x) u^{2^\#1} \quad & \text{on }\mathbb S^{N1},
\end{cases}
\]
where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb
S^{N1}, 2^\#=\frac{2(N1)}{N2}$.
We show that if $\widetilde K(x)$ has a local maximum point,
then the above problem has infinitely many positive solutions
that are not rotationally symmetric on $\mathbb S^{N1}$.
