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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.

We consider the curved $4$-body problems on spheres and hyperbolic
spheres. After obtaining a criterion for the existence of quadrilateral
configurations on the equator of the sphere, we study two restricted
$4$-body problems, one in which two masses are negligible, and
another in which only one mass is negligible.
In the former we prove the evidence square-like relative equilibria,
whereas in the latter we discuss the existence of kite-shaped
relative equilibria.

We prove that if $C$ is a reflexive smooth plane curve of degree
$d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then
there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely.
We also prove the same result for non-reflexive curves of degree
$p+1$ and $2p+1$ where $q=p^{r}$.

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.

The prime, completely prime, maximal and primitive spectra are
classified for the universal enveloping algebra of the Schrödinger
algebra. For all of these ideals their explicit generators are
given. A counterexample is constructed to the conjecture of Cheng
and Zhang about non-existence of simple singular Whittaker modules
for the Schrödinger algebra (and all such modules are classified).
It is proved that the conjecture holds 'generically'.

We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.

According to a well-known theorem of Serre and Tate, the infinitesimal
deformation theory of an abelian variety in positive characteristic
is equivalent to the infinitesimal deformation theory of its
Barsotti-Tate group. We extend this result to $1$-motives.

We give necessary and sufficient
conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate
differential equation with finite delays:
$(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic
boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where
$A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying
$D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from
$L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$.

Let $G$ be a claw-free graph on $n$ vertices with clique number
$\omega$, and consider the chromatic number $\chi(G^2)$ of the
square $G^2$ of $G$.
Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the
line graphs $L$ of simple graphs of maximum degree at most $d$,
we prove that $\chi(G^2)\le \chi'_s(\omega)$ for $\omega \in
\{3,4\}$. For $\omega=3$, this implies the sharp bound $\chi(G^2)
\leq 10$. For $\omega=4$, this implies $\chi(G^2)\leq 22$, which
is within $2$ of the conjectured best bound.
This work is motivated by a strengthened form of a conjecture
of Erdős and Nešetřil.

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.

We explicitly describe the isomorphism between two combinatorial
realizations of Kashiwara's infinity crystal in types B and C.
The first realization is in terms of marginally large tableaux
and the other is in terms of Kostant partitions coming from PBW
bases. We also discuss a stack notation for Kostant partitions
which simplifies that realization.

We prove an equivalence between weighted Poincaré inequalities
and
the existence of weak solutions to a Neumann problem related
to a
degenerate $p$-Laplacian. The Poincaré inequalities are
formulated in the context of degenerate Sobolev spaces defined
in
terms of a quadratic form, and the associated matrix is the
source of
the degeneracy in the $p$-Laplacian.

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.

We consider three special and significant cases of the following
problem. Let $D\subset\mathbb{R}^d$ be a (possibly unbounded) set
of finite Lebesgue measure.
Let $E( \mathbb{Z}^d)=\{e^{2\pi i x\cdot n}\}_{n\in\mathbb{Z}^d}$ be the standard
exponential basis on the unit cube of $\mathbb{R}^d$.
Find conditions on $D$ for which $E(\mathbb{Z}^d)$ is a frame, a
Riesz sequence, or a Riesz basis for $L^2(D)$.

A precise quantitative version of the following qualitative statement
is proved: If a finite dimensional normed space contains approximately
Euclidean subspaces of all proportional dimensions, then every
proportional dimensional quotient space has the same property.

Very recently, Karder and Petek completely described maps on density
matrices (positive semidefinite matrices with unit trace) preserving
certain entropy-like convex functionals of any convex combination.
As a result, maps could be characterized which preserve von Neumann
entropy or Schatten $p$-norm of any convex combination of quantum
states (whose mathematical representatives are the density matrices).
In this note we consider these latter two problems on the set
of invertible density operators, in a much more general setting,
on the set of positive invertible elements with unit trace in
a $C^{*}$-algebra.

In this paper we show that to a unital associative algebra object
(resp. co-unital co-associative co-algebra object) of any abelian
monoidal category $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$-trace,
i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ satisfying some natural trace-like
conditions, one can attach a cyclic (resp.cocyclic) module, and
therefore speak of the (co)cyclic homology of the (co)algebra
``with coefficients in $F$". Furthermore, we observe that if
$\mathcal{M}$ is a $\mathcal{C}$-bimodule category and $(F, M)$ is a stable central
pair, i.e., $F\in Fun(\mathcal{M}, \operatorname{Vec})$ and $M\in \mathcal{M}$ satisfy certain
conditions, then $\mathcal{C}$ acquires a symmetric 2-trace. The dual
notions of symmetric $2$-contratraces and stable central contrapairs
are derived as well. As an application we can recover all Hopf
cyclic type (co)homology theories.

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 $m,n \geq 3$,
$$2^{m+n-3}
\left(\tfrac{17}{10}
\right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n-3}
\left(\tfrac{31}{16}
\right)^{(m-2)(n-2)}.$$

Let $\beta\ge 0$ and $e_1=(1,0,\ldots,0)$ is a unit vector on
$\mathbb{R}^{n}$, $d\mu(x)=|x|^\beta dx$ is a power weighted
measure on $\mathbb{R}^n$. For $0\le \alpha\lt n$, let $M_\mu^\alpha$
be the centered Hardy-Littlewood maximal function and fractional
maximal functions associated to measure $\mu$. This paper shows
that for $q=n/(n-\alpha)$, $f\in L^1(\mathbb{R}^n,d\mu)$,
$$\lim\limits_{\lambda\to 0+}\lambda^q \mu(\{x\in\mathbb{R}^n:M_\mu^\alpha
f(x)\gt \lambda\})=\frac{\omega_{n-1}}{(n+\beta)\mu(B(e_1,1))}\|f\|_{L^1(\mathbb{R}^n,
d\mu)}^q,$$
and
$$\lim_{\lambda\to 0+}\lambda^q \mu\Big(\Big\{x\in\mathbb{R}^n:\Big|M_\mu^\alpha
f(x)-\frac{\|f\|_{L^1(\mathbb{R}^n, d\mu)}}{\mu(B(x,|x|))^{1-\alpha/n}}\Big|\gt \lambda\Big\}\Big)=0,$$
which is new and stronger than the previous result even if $\beta=0$.
Meanwhile, the corresponding results for the un-centered maximal
functions as well as the fractional integral operators with respect
to measure $\mu$ are also obtained.

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 $V$.

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices
over a field of characteristic 0,
and $A\in\mathfrak{s}$,
then the semisimple and nilpotent summands of the Jordan-Chevalley
decomposition of $A$ belong to $\mathfrak{s}$
if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$
is nilpotent (not necessarily $[S,N]=0$)
such that $A=S+N$.

Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

In this paper, the bounded properties of oscillatory hyper-Hilbert
transform along certain plane curves $\gamma(t)$
$$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(x-t,y-\gamma(t))e^{ i t^{-\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$
were studied. For a general curves, these operators are bounded
in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their
boundedness in $L^p(\mathbb{R}^{2})$
were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta-3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.

The study of chaotic vibration for multidimensional PDEs due
to nonlinear boundary conditions is challenging. In this paper,
we mainly investigate the chaotic oscillation of a two-dimensional
non-strictly hyperbolic equation due to an energy-injecting
boundary condition and a distributed self-regulating boundary
condition. By using the method of characteristics, we give a
rigorous proof of the onset of the chaotic vibration phenomenon
of the 2D non-strictly hyperbolic equation. We have also found
a regime of the parameters when the chaotic vibration phenomenon
occurs. Numerical simulations are also provided.

In this note, we collect various properties
of Seifert homology spheres from the viewpoint of Dehn surgery
along a Seifert fiber. We expect that many of these are known
to various experts, but include them in one place which we hope
to be useful in the study of concordance and homology cobordism.

In this paper, we study the warped structures of Finsler metrics.
We obtain the differential equation that characterizes the Finsler
warped product metrics with vanishing Douglas curvature. By solving
this equation, we obtain all Finsler warped product Douglas metrics.
Some new Douglas Finsler metrics of this type are produced by
using known spherically symmetric Douglas metrics.

In this paper we establish the endpoint estimates
and Hardy type estimates for the Riesz transform associated
with the generalized Schrödinger operator.

Let $A$ be the inductive limit of a sequence
$$
A_1\,\xrightarrow{\phi_{1,2}}\,A_2\,\xrightarrow{\phi_{2,3}}\,A_3\rightarrow\cdots
$$
with $A_n=\bigoplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$
are Elliott-Thomsen algebras and $\phi_{n,n+1}$ are homomorphisms.
In this paper, we will prove that $A$ can be written as another
inductive limit
$$
B_1\,\xrightarrow{\psi_{1,2}}\,B_2\,\xrightarrow{\psi_{2,3}}\,B_3\rightarrow\cdots
$$
with $B_n=\bigoplus_{i=1}^{n_i'}B_{[n,i]'}$, where all the $B_{[n,i]'}$
are Elliott-Thomsen algebras and with the extra condition that
all the $\psi_{n,n+1}$ are injective.

An odd Fredholm module for a given invertible operator on a Hilbert
space is specified by an unbounded so-called Dirac operator with
compact resolvent and bounded commutator with the given invertible.
Associated to this is an index pairing in terms of a Fredholm
operator with Noether index. Here it is shown by a spectral flow
argument how this index can be calculated as the signature of
a finite dimensional matrix called the spectral localizer.

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 $\mathbf{R}^m$.

We give a consistent example of a zero-dimensional separable
metrizable space $Z$ such that every homeomorphism of $Z^\omega$
acts like a permutation of the coordinates almost everywhere.
Furthermore, this permutation varies continuously. This shows
that a result of Dow and Pearl is sharp, and gives some insight
into an open problem of Terada. Our example $Z$ is simply the
set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.

Katz and Sarnak predicted that the one level density of the zeros
of a family of $L$-functions would fall into one of five categories.
In this paper, we show that the one level density for $L$-functions
attached to cubic Galois number fields falls into the category
associated with unitary matrices.

We show under some conditions that a Gorenstein ring $R$ satisfies the
Generalized Auslander-Reiten Conjecture if and only if so does
$R[x]$. When $R$ is a local ring we prove the same result for some
localizations of $R[x]$.

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.

show that Hermite's theorem fails for every
integer $n$ of the form $3^{k_1}+3^{k_2}+3^{k_3}$
with integers $k_1\gt k_2\gt k_3\geq 0$. This confirms
a conjecture of Brassil and Reichstein. We also
obtain new results for the relative
Hermite-Joubert problem over a finitely generated
field of characteristic $0$.

Inspired by a construction by Bump, Friedberg, and Ginzburg of
a two-variable integral representation on $\operatorname{GSp}_4$ for the product
of the standard and spin $L$-functions, we give two similar multivariate
integral representations. The first is a three-variable Rankin-Selberg
integral for cusp forms on $\operatorname{PGL}_4$ representing the product
of the $L$-functions attached to the three fundamental representations
of the Langlands $L$-group $\operatorname{SL}_4(\mathbf{C})$. The second integral,
which is closely related, is a two-variable Rankin-Selberg integral
for cusp forms on $\operatorname{PGU}(2,2)$ representing the product of the
degree 8 standard $L$-function and the degree 6 exterior square
$L$-function.

Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number
\[ 0. f(1) f(2) f(3) \dots \]
obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and Erdős in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

We consider homogeneous multiaffine polynomials whose coefficients
are the Plücker coordinates of a point $V$ of the Grassmannian.
We show that such a polynomial is stable (with respect to the
upper half plane) if and only if $V$ is in the totally nonnegative
part of the Grassmannian. To prove this, we consider an action
of
matrices on multiaffine polynomials. We show that
a matrix $A$ preserves stability of polynomials if and only if
$A$ is totally nonnegative. The proofs are applications of classical
theory of totally nonnegative matrices, and the generalized
Pólya-Schur theory of Borcea and Brändén.

F. Cukierman asked whether or not for every
smooth
real plane curve $X \subset \mathbb{P}^2$ of even degree $d \geqslant
2$
there exists a real line
$L \subset \mathbb{P}^2$ such $X \cap L$ has no real points.
We show that the answer is ``yes" if $d = 2$ or $4$ and ``no"
if $n \geqslant 6$.

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.

We give a Hopf boundary point lemma for weak solutions of linear
divergence form uniformly elliptic equations, with Hölder
continuous top-order coefficients and lower-order coefficients
in a Morrey space.

The study of graph $C^*$-algebras has a long history in operator
algebras. Surprisingly, their quantum symmetries have never been
computed so far. We close this gap by proving that the quantum
automorphism group of a finite, directed graph without multiple
edges acts maximally on the corresponding graph $C^*$-algebra.
This shows that the quantum symmetry of a graph coincides with
the quantum symmetry of the graph $C^*$-algebra. In our result,
we use the definition of quantum automorphism groups of graphs
as given by Banica in 2005. Note that Bichon gave a different
definition in 2003; our action is inspired from his work. We
review and compare these two definitions and we give a complete
table of quantum automorphism groups (with respect to either
of the two definitions) for undirected graphs on four vertices.

Let $A$ be a commutative noetherian ring,
let $\mathfrak{a}\subseteq A$ be an ideal,
and let $I$ be an injective $A$-module.
A basic result in the structure theory of injective modules states
that
the $A$-module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements
is also an injective $A$-module.
Recently, de Jong proved a dual result: If $F$ is a flat $A$-module,
then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module.
In this paper we generalize these facts to commutative noetherian
DG-rings:
let $A$ be a commutative non-positive DG-ring such that $\mathrm{H}^0(A)$
is a noetherian ring,
and for each $i\lt 0$, the $\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$
is finitely generated.
Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$,
we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$
associated to $\bar{\mathfrak{a}}$ does not increase injective dimension.
Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$
does not increase flat dimension.

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

This paper provides short proofs of two fundamental theorems
of finite semigroup theory whose previous proofs were significantly
longer, namely the two-sided Krohn-Rhodes decomposition theorem
and Henckell's aperiodic pointlike theorem, using a new algebraic
technique that we call the merge decomposition. A prototypical
application of this technique decomposes a semigroup $T$ into
a two-sided semidirect product whose components are built from
two subsemigroups $T_1,T_2$, which together generate $T$, and
the subsemigroup generated by their setwise product $T_1T_2$.
In this sense we decompose $T$ by merging the subsemigroups
$T_1$ and $T_2$. More generally, our technique merges semigroup
homomorphisms from free semigroups.

For every affine variety over a global function field, we show
that
the set of its points with coordinates in an arbitrary rank-one
multiplicative
subgroup of this function field satisfies the required property
of
weak approximation for finite sets of places of this function
field
avoiding arbitrarily given finitely many places.

We investigate continuous transitive actions of semitopological
groups on spaces, as well as separately continuous transitive
actions of topological groups.

Suppose that $D\subset\mathbb{C}$ is a simply connected subdomain
containing the origin and $f(z_1)$ is a normalized convex (resp.,
starlike) function on $D$. Let
$$
\Omega_{N}(D)=\{(z_1,w_1,\ldots,w_k)\in \mathbb{C}\times{\mathbb{C}}^{n_1}\times\cdots\times{\mathbb{C}}^{n_k}:
\|w_1\|_{p_1}^{p_1}+\cdots+\|w_k\|_{p_k}^{p_k}<\frac{1}{\lambda_{D}(z_1)}\},$$
where $p_j\geq 1$, $N=1+n_1+\cdots+n_k,\,w_1\in{\mathbb{C}}^{n_1},\ldots,w_k\in{\mathbb{C}}^{n_k}$
and $\lambda_{D}$ is the density of the hyperbolic metric on
$D$. In this paper, we prove that
\begin{equation*}
\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)(z_1,w_1,\ldots,w_k)=\big(f(z_{1}),
(f'(z_{1}))^{1/p_{1}}w_1,\cdots,(f'(z_{1}))^{1/p_{k}}w_k\big)
\end{equation*}
is a normalized convex (resp., starlike) mapping on $\Omega_{N}(D)$.
If $D$ is the unit disk, then our result reduces to Gong and
Liu \cite{GL2} via a new method. Moreover, we give a new operator
for
convex mapping construction on an unbounded domain in ${\mathbb{C}}^{2}$.
By using a geometric approach, we prove that $\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)$
is a spirallike mapping of type $\alpha$ when $f$ is
a spirallike function of type $\alpha$ on the unit disk.

Let $\mathcal{D}$ be the irreducible Hermitian symmetric domain
of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian
variation of real Hodge structure $\mathcal{V}_{\mathbb{R}}$
of Calabi-Yau type over $\mathcal{D}$. This short note concerns
the problem of giving motivic realizations for $\mathcal{V}_{\mathbb{R}}$.
Namely, we specify a descent of $\mathcal{V}_{\mathbb{R}}$ from
$\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$-descent
of $\mathcal{V}_{\mathbb{R}}$ can be realized as sub-variation
of rational Hodge structure of those coming from families of
algebraic varieties. When $n=2$, we give a motivic realization
for $\mathcal{V}_{\mathbb{R}}$. When $n \geq 3$, we show that
the unique irreducible factor of Calabi-Yau type in $\mathrm{Sym}^2
\mathcal{V}_{\mathbb{R}}$ can be realized motivically.