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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
$X$
.
We expose different methods of regularizations of subsolutions in the context of discrete weak
$\text{KAM}$
theory that allow us 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.
The squared distance curvature is a kind of two-point curvature the sign of which turned out to be 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, and 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.
Let
$S$
be the semigroup
$S=\sum\nolimits_{i=1}^{\oplus k}{{{S}_{i}}}$
, where for each
$i\in I,{{S}_{i}}$
is a countable subsemigroup of the additive semigroup
${{\mathbb{R}}_{+}}$
containing 0. We consider representations of
$S$
as contractions
${{\left\{ {{T}_{s}} \right\}}_{s\in S}}$
on a Hilbert space with the Nica-covariance property:
$T_{s}^{*}{{T}_{t}}={{T}_{t}}T_{s}^{*}$
whenever
$t\wedge s=0$
. We show that all such representations have a unique minimal isometric Nica-covariant dilation.
This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of
$S$
on an operator algebra
$\mathcal{A}$
by completely contractive endomorphisms. We conclude by calculating the
${{C}^{*}}$
-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).
We introduce generalized triple homomorphisms between Jordan–Banach triple systems as a concept that extends the notion of generalized homomorphisms between Banach algebras given by K. Jarosz and B. E. Johnson in 1985 and 1987, respectively. We prove that every generalized triple homomorphism between
$\text{J}{{\text{B}}^{*}}$
-triples is automatically continuous. When particularized to
${{C}^{*}}$
-algebras, we rediscover one of the main theorems established by Johnson. We will also consider generalized triple derivations from a Jordan–Banach triple
$E$
into a Jordan–Banach triple
$E$
-module, proving that every generalized triple derivation from a
$\text{J}{{\text{B}}^{*}}$
-triple
$E$
into itself or into
${{E}^{*}}$
is automatically continuous.
Given a non-oscillating gradient trajectory
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$
of a real analytic function
$f$
, we show that the limit
$v$
of the secants at the limit point
$0$
of
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$
along the trajectory
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$
is an eigenvector of the limit of the direction of the Hessian matrix Hess
$\left( f \right)$
at
$0$
along
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$
. The same holds true at infinity if the function is globally sub-analytic. 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.
Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne 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 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in
${{\mathbb{P}}^{3}}$
that 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 Li and Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.
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
$\left( n-4 \right)/3\le d\le \left( n-6 \right)/2$
. This paper establishes similar bounds for
$\delta \ge 2$
. Specifically, there is 3-torsion in degree
$d$
whenever
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 non-boundary element in the chain complex of a certain matching complex, we obtain that
$z$
itself is a non-boundary. In particular, the homology class of
$z$
has order 3.
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 so-called 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 Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.
Guest–Ohnita and Crawford have shown the path-connectedness of the space of harmonic maps from
${{S}^{2}}$
to
$\text{C}{{P}^{n}}$
of a fixed degree and energy. It is well known 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.
We consider threefolds that admit a fibration by
$\text{K3}$
surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarization 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 this 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
$\text{K3}$
surfaces of degree two.
We consider the prescribed boundary mean curvature problem in
${{\mathbb{B}}^{N}}$
with the Euclidean metric
$$\{_{\frac{\partial u}{\partial v}+\frac{N-2}{2}u=\frac{N-2}{2}\tilde{K}\left( x \right){{u}^{{{2}^{\#-1}}}}\,\,\,\,\,\,\text{on}{{\mathbb{S}}^{N-1}},}^{-\Delta u=0,\,\,\,\,\,\,u>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in}{{\mathbb{B}}^{N}},}$$
where
$\tilde{K}\left( x \right)$
is positive and rotationally symmetric on
${{\mathbb{S}}^{N-1}},{{2}^{\#}}=\frac{2\left( N-1 \right)}{N-2}$
. We show that if
$\tilde{K}\left( x \right)$
has a local maximum point, then this problem has infinitely many positive solutions that are not rotationally symmetric on
${{\mathbb{S}}^{N-1}}$
.