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In this paper, we discuss the properties of the embedding
operator $i^\Lambda_\mu : M_\Lambda^\infty\hookrightarrow L^\infty(\mu),$
where $\mu$ is a positive Borel measure on $[0,1]$ and $M_{\Lambda}^{\infty}$
is a Müntz space. In particular, we compute the essential norm
of this embedding. As a consequence, we recover some results
of
the first author.
We also study the compactness (resp. weak compactness)
and compute the essential norm (resp. generalized essential norm)
of the embedding $i_{\mu_1,\,\mu_2} : L^\infty(\mu_1)\hookrightarrow
L^\infty(\mu_2)$, where $\mu_1$, $\mu_2$ are two positive Borel
measures on $[0,1]$ with $\mu_2$ absolutely continuous with respect
to $\mu_1$.

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 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.

In this note we give a characterization of $\ell^{p}\times \cdots\times
\ell^{p}\to\ell^q$ boundedness of maximal operators associated
to multilinear convolution averages over spheres in $\mathbb{Z}^n$.

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.

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.

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of
our earlier work,
where toric surfaces of Picard number $1$ were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective $3$-spaces blown up at a point that do not have finitely generated Cox rings.

A theorem of Gekeler compares the number of non-isomorphic automorphic
representations associated with the space of cusp forms of weight
$k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing
that the two are equal whenever $N$ is squarefree. We prove the
converse of this theorem (with one small exception), thus providing
a characterization of squarefree integers. We also establish
a similar characterization of prime numbers in terms of the number
of Hecke newforms of weight $k$ on $\Gamma_0(N)$.

It follows that a hypothetical fast algorithm for computing the
number of such automorphic representations for even a single
weight $k$ would yield a fast test for whether $N$ is squarefree.
We also show how to obtain bounds on the possible square divisors
of a number $N$ that has been found to not be squarefree via
this test, and we show how to probabilistically obtain
the complete factorization of the squarefull part of $N$ from
the number of such automorphic representations for two different
weights. If in addition we have the number of such Hecke newforms
for even a single weight $k$, then we show how to probabilistically
factor $N$ entirely.
All of these computations could be performed quickly in practice,
given the number(s) of automorphic representations and modular
forms as input.

In this short note, we prove that on the three-sphere with any
bumpy metric there exist at least two pairs of solutions of the
Allen-Cahn equation with spherical interface and index at most
two. The proof combines several recent results from the literature.

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.

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.

In this paper, we completely characterize the finite rank commutator
and semi-commutator of two
monomial-type Toeplitz operators on the Bergman space of certain
weakly pseudoconvex domains.
Somewhat surprisingly, there are not only plenty of commuting
monomial-type Toeplitz operators but also non-trivial semi-commuting
monomial-type Toeplitz operators. Our results are new even for
the unit ball.

We find an explicit expression for the zeta-regularized determinant
of (the Friedrichs extensions) of the Laplacians on a compact
Riemann surface of genus one with conformal metric of curvature
$1$ having a single conical singularity of angle $4\pi$.

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.

We introduce the concept of
$\{\sigma , \tau \}$-Rota-Baxter operator, as a twisted version
of a Rota-Baxter operator of weight zero. We show how to
obtain a certain $\{\sigma , \tau \}$-Rota-Baxter operator from
a solution of the associative (Bi)Hom-Yang-Baxter equation, and,
in a compatible way,
a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

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.

We first provide a necessary and sufficient condition for a ruled
real hypersurface in a nonflat complex space form to have constant
mean curvature in terms of integral curves of the characteristic
vector field on it.
This yields a characterization of minimal ruled real hypersurfaces
by circles.
We next characterize the homogeneous minimal ruled real hypersurface
in a complex hyperbolic space by using the notion of strongly
congruency of curves.

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 $R[x]$.

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 $0$.

The edge-of-the-wedge theorem in several complex variables gives
the analytic continuation of functions defined on the poly upper
half plane and the poly lower half plane, the set of points in
$\mathbb{C}^{n}$ with all coordinates in the upper and lower
half planes respectively, through a set in real space, $\mathbb{R}^{n}$.
The geometry of the set in the real space can force the function
to analytically continue within the boundary itself, which is
qualified in our wedge-of-the-edge theorem. For example, if a
function extends to the union of two cubes in $\mathbb{R}^{n}$
that are positively oriented with some small overlap, the
functions must analytically continue to a neighborhood of that
overlap of a fixed size not depending of the size of the overlap.

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 prove a function field analogue of Maynard's celebrated result
about primes with restricted digits. That is, for certain ranges
of parameters $n$ and $q$, we prove an asymptotic formula for
the number of irreducible polynomials of degree $n$ over a finite
field $\mathbb{F}_q$ whose coefficients are restricted to lie
in a given subset of $\mathbb{F}_q$

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 $n \geqslant 6$.

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.

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.

The classical Alexandrov-Bakelman-Pucci estimate
for the Laplacian states
$$ \max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|}
+ c_{s,n} \operatorname{diam}(\Omega)^{2-\frac{n}{s}}
\left\| \Delta u
\right\|_{L^s(\Omega)}$$
where $\Omega \subset \mathbb{R}^n$, $u \in C^2(\Omega) \cap
C(\overline{\Omega})$ and $s \gt n/2$. The inequality fails for
$s = n/2$. A Sobolev embedding result of Milman and Pustylnik,
originally phrased in a slightly different context, implies an
endpoint inequality: if $n \geq 3$ and $\Omega \subset \mathbb{R}^n$
is bounded, then
$$ \max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|}
+ c_n
\left\| \Delta u
\right\|_{L^{\frac{n}{2},1}(\Omega)},$$
where $L^{p,q}$ is the Lorentz space refinement of $L^p$. This
inequality fails for $n=2$ and we prove a sharp substitute result:
there exists $c\gt 0$ such that for all $\Omega \subset \mathbb{R}^2$
with finite measure
$$ \max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|}
+ c \max_{x \in \Omega} \int_{y \in \Omega}{ \max
\left\{ 1, \log{
\left(\frac{|\Omega|}{\|x-y\|^2}
\right)}
\right\}
\left| \Delta u(y)
\right| dy}.$$
This is somewhat dual to the classical Trudinger-Moser inequality;
we also note that it is sharper than the usual estimates given
in Orlicz spaces, the proof is rearrangement-free.
The Laplacian can be replaced by any uniformly elliptic operator
in divergence form.

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.