We show that if $x$ is a strongly extreme point of a bounded closed
convex subset of a Banach space and the identity has a geometrically
and topologically good enough local approximation at $x$, then $x$
is already a denting point. It turns out that such an approximation
of the identity exists at any strongly extreme point of the unit
ball of a Banach space with the unconditional compact approximation
property. We also prove that every Banach space with a Schauder
basis can be equivalently renormed to satisfy the sufficient
conditions mentioned.

We give a new proof that bounded non-commutative functions
on polynomial polyhedra
can be represented by a realization formula, a generalization
of the transfer function realization
formula for bounded analytic functions on the unit disk.

The aim of this note is to provide a conceptually simple demonstration
of the fact that repetitive model sets are characterized as the
repetitive Meyer sets with an almost automorphic associated dynamical
system.

The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq
j\leq \ell$, $\alpha,\beta\gt -1$) are linked to the Maclaurin
spectral expansion of the Schwartz kernel of functions of the
Laplacian on a compact rank one symmetric space. It
is proved that these coefficients can be computed by transforming
the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt -1$) into a spectral sum associated with
the Jacobi operator. The first few coefficients are explicitly
computed and a direct trace
interpretation of the Maclaurin coefficients is presented.

Associated to any closed quantum subgroup $G\subset U_N^+$ and
any index set $I\subset\{1,\dots,N\}$ is a certain homogeneous
space $X_{G,I}\subset S^{N-1}_{\mathbb C,+}$, called affine homogeneous
space. We discuss here the abstract axiomatization of the algebraic
manifolds $X\subset S^{N-1}_{\mathbb C,+}$ which can appear in
this way, by using Tannakian duality methods.

For smooth functions $a_1, a_2, a_3, a_4$ on a quaternion Heisenberg
group, we characterize
the existence of solutions of the partial differential operator
system $X_1f=a_1, X_2f=a_2, X_3f=a_3,$ and $X_4f=a_4$.
In addition, a formula for the solution function $f$ is deduced
provided the solvability of the system.

For an analytic function $f$ on the unit disk $\mathbb D$ we show that
the $L^2$ integral mean of $f$ on $c\lt |z|\lt r$ with
respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$
is a logarithmically convex function of $r$ on $(c,1)$,
where $-3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range
$[-3,0]$ for $\alpha$ is best possible. When
$c=0$, our arguments here also simplify the proof for several
results we obtained in earlier papers.

A self-avoiding polygon is a lattice polygon consisting of a
closed self-avoiding walk on a square lattice.
Surprisingly little is known rigorously about the enumeration
of self-avoiding polygons,
although there are numerous conjectures that are believed to
be true
and strongly supported by numerical simulations.
As an analogous problem of this study,
we consider multiple self-avoiding polygons in a confined region, as a model for multiple ring polymers in physics.
We find rigorous lower and upper bounds of the number $p_{m \times
n}$
of distinct multiple self-avoiding polygons in the $m \times
n$ rectangular grid on the square lattice.
For $m=2$, $p_{2 \times n} = 2^{n-1}-1$.
And, for integers $m,n \geq 3$,
$$2^{m+n-3}
\left(\tfrac{17}{10}
\right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n-3}
\left(\tfrac{31}{16}
\right)^{(m-2)(n-2)}.$$

Given two monic polynomials $f$ and $g$ with coefficients in
a number field $K$, and some $\alpha\in K$, we examine the action
of the absolute Galois group $\operatorname{Gal}(\overline{K}/K)$ on the directed
graph of iterated preimages of $\alpha$ under the correspondence
$g(y)=f(x)$, assuming that $\deg(f)\gt \deg(g)$ and that $\gcd(\deg(f),
\deg(g))=1$. If a prime of $K$ exists at which $f$ and $g$ have
integral coefficients, and at which $\alpha$ is not integral,
we show that this directed graph of preimages consists of finitely
many $\operatorname{Gal}(\overline{K}/K)$-orbits. We obtain this result by
establishing a $p$-adic uniformization of such correspondences,
tenuously related to Böttcher's uniformization of polynomial
dynamical systems over $\mathbb{CC}$, although the construction of a
Böttcher coordinate for complex holomorphic correspondences
remains unresolved.

On a real hypersurface $M$ in a complex two-plane Grassmannian
$G_2({\mathbb C}^{m+2})$ we have the Lie derivation ${\mathcal
L}$ and a differential operator of order one associated to the
generalized Tanaka-Webster connection $\widehat {\mathcal L}
^{(k)}$. We give a classification of real hypersurfaces $M$ on
$G_2({\mathbb C}^{m+2})$ satisfying
$\widehat {\mathcal L} ^{(k)}_{\xi}S={\mathcal L}_{\xi}S$, where
$\xi$ is the Reeb vector field on $M$ and $S$ the Ricci tensor
of $M$.

Let $\mathtt{G}$ be the $n$-fold covering group of the special
linear group of degree two, over a non-Archimedean local field.
We determine the decomposition into irreducibles of the restriction
of the principal series representations of $\mathtt{G}$ to a maximal
compact subgroup. Moreover, we analyse those features that distinguish
this decomposition from the linear case.

We apply our theory of partial flag spaces developed
with W. Goldring
to study a group-theoretical generalization of the canonical
filtration of a truncated Barsotti-Tate group of level 1. As
an application, we determine explicitly the normalization of
the Zariski closures of Ekedahl-Oort strata of Shimura varieties
of Hodge-type as certain closed coarse strata in the associated
partial flag spaces.

In this paper, we first discuss the relation between $\mathsf{VB}$-Courant
algebroids and $\mathsf{E}$-Courant algebroids and construct some examples
of $\mathsf{E}$-Courant algebroids. Then we introduce the notion of
a generalized complex
structure on an $\mathsf{E}$-Courant algebroid, unifying the usual
generalized complex structures on even-dimensional manifolds
and
generalized contact structures on odd-dimensional manifolds.
Moreover, we study generalized complex structures on an omni-Lie
algebroid in detail. In particular, we show that generalized
complex structures on an omni-Lie algebra $\operatorname{gl}(V)\oplus V$
correspond
to complex Lie algebra structures on $V$.

A crucial role in the Nyman-Beurling-Báez-Duarte approach to
the Riemann Hypothesis is played by the distance
\[
d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty
\left|1-\zeta A_N
\left(\frac{1}{2}+it
\right)
\right|^2\frac{dt}{\frac{1}{4}+t^2}\:,
\]
where the infimum is over all Dirichlet polynomials
$$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$
of length $N$.
In this paper we investigate $d_N^2$ under the assumption that
the Riemann zeta function has four non-trivial zeros off the
critical line.

In this paper we give some generalizations and
improvements of the Pavlović result on the
Holland-Walsh type characterization of the Bloch space of
continuously differentiable (smooth) functions in
the unit ball in $\mathbf{R}^m$.

It is shown that the unit ball in $\mathbb{C}^n$ is the only complex manifold
that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

If $B$ is the Blachke product with zeros $\{z_n\}$, then $|B'(z)|\le
\Psi_B(z)$, where
$$\Psi_B(z)=\sum_n \frac{1-|z_n|^2}{|1-\overline{z}_nz|^2}.$$
Moreover, it is a well-known fact that, for $0\lt p\lt \infty$,
$$M_p(r,B')=
\left(\frac{1}{2\pi}\int_{0}^{2\pi} |B'(re^{i\t})|^p\,d\t
\right)^{1/p}, \quad 0\le r\lt 1,$$
is bounded if and only if $M_p(r,\Psi_B)$ is bounded.
We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and
$M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$.
In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a Carleson-Newman
Blaschke product and a weight $\omega$ satisfies a certain regularity
condition, then
$$
\int_\mathbb{D} |B'(z)|^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z),
$$
where $dA(z)$ is the Lebesgue area measure on the unit disc.

The splitting number of a plane irreducible curve for a Galois
cover is effective to distinguish the embedded topology of plane
curves.
In this paper, we define the connected number of a plane
curve (possibly reducible) for a Galois cover, which is similar
to the splitting number.
By using the connected number, we distinguish the embedded topology
of Artal arrangements of degree $b\geq 4$, where an Artal arrangement
of degree $b$ is a plane curve consisting of one smooth curve
of degree $b$ and three of its total inflectional tangen