We characterize positive links in terms of strong quasipositivity,
homogeneity and the value of Rasmussen and Beliakova-Wehrli'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.
In this paper we introduce Hardy-Lorentz spaces with variable
exponents associated to dilations in ${\mathbb R}^n$. We establish
maximal characterizations and atomic decompositions for our variable
exponent anisotropic Hardy-Lorentz spaces.
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 Cohen-Macaulay (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 $m\geq 1.$
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.
An asymptotically orthonormal sequence is a sequence which is
"nearly" orthonormal in the sense that it satisfies the Parseval
equality up to two constants close to one. In this paper, we
explore such sequences formed by normalized reproducing kernels
for model spaces and de Branges-Rovnyak spaces.
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.
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.
Following up on previous work,
we prove a number of results for C*-algebras
with the weak ideal property
or topological dimension zero,
and some results for C*-algebras with related properties.
Some of the more important results include:
$\bullet$
The weak ideal property
implies topological dimension zero.
$\bullet$
For a separable C*-algebra~$A$,
topological dimension zero is equivalent to
${\operatorname{RR}} ({\mathcal{O}}_2 \otimes A) = 0$,
to $D \otimes A$ having the ideal property
for some (or any) Kirchberg algebra~$D$,
and to $A$ being residually hereditarily in
the class of all C*-algebras $B$ such that
${\mathcal{O}}_{\infty} \otimes B$
contains a nonzero projection.
$\bullet$
Extending the known result for ${\mathbb{Z}}_2$,
the classes of C*-algebras
with residual (SP),
which are residually hereditarily (properly) infinite,
or which are purely infinite and have the ideal property,
are closed under crossed products by arbitrary actions
of abelian $2$-groups.
$\bullet$
If $A$ and $B$ are separable,
one of them is exact,
$A$ has the ideal property,
and $B$ has the weak ideal property,
then $A \otimes_{\mathrm{min}} B$ has the weak ideal property.
$\bullet$
If $X$ is a totally disconnected locally compact Hausdorff space
and $A$ is a $C_0 (X)$-algebra
all of whose fibers have one of the weak ideal property,
topological dimension zero,
residual (SP),
or the combination of pure infiniteness and the ideal property,
then $A$ also has the corresponding property
(for topological dimension zero, provided $A$ is separable).
$\bullet$
Topological dimension zero,
the weak ideal property,
and the ideal property
are all equivalent
for a substantial class of separable C*-algebras including
all separable locally AH~algebras.
$\bullet$
The weak ideal property does not imply the ideal property
for separable $Z$-stable C*-algebras.
We give other related results,
as well as counterexamples to several other statements
one might hope for.
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 self-adjoint
operators.