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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 noncommutative 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.
In this paper, periodic steadystate of a liquid film flowing
over a periodic uneven wall is investigated via a classical method.
Specifically, we analyze a longwave model that is valid at
the nearcritical Reynolds number. For the periodic wall surface,
we construct an iteration scheme in terms of an integral form
of the original steadystate problem. The uniform convergence
of the scheme is proved so that we can derive the existence and
the uniqueness, as well as the asymptotic formula, of the periodic
solutions.
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.
A classification of simple weight modules over the Schrödinger
algebra is given. The Krull and the global dimensions are found
for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor
algebras) of the Cartan element $H$ in the universal enveloping
algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple
$C_{\mathcal{S}}(H)$modules are classified. The Krull and the global
dimensions are found for some (prime) factor algebras of the
algebra $\mathcal{S}$ (over the centre). It is proved that some (prime)
factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull
minimal.
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.
We calculate all $\ell^2$Betti numbers of the universal discrete
Kac quantum groups $\hat{\mathrm U}^+_n$ as well as their halfliberated
counterparts $\hat{\mathrm U}^*_n$.
We give necessary and sufficient
conditions of the $L^p$wellposedness (resp. $B_{p,q}^s$wellposedness) 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 $X$.
Using known operatorvalued Fourier multiplier results on vectorvalued
Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely
characterize the $C^\alpha$wellposedness of the first order
degenerate differential equations with finite delay $(Mu)'(t)
= Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$
by the boundedness of the $(M, F)$resolvent of $A$ under suitable
assumption on the delay operator $F$, where $A, M$ are closed
linear
operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$,
the delay operator $F$ is a bounded linear operator
from $C([r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.
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.
Generalized orthogonal matching pursuit (gOMP) algorithm has
received much attention in recent years as a natural extension
of
orthogonal matching pursuit (OMP). It is used to recover sparse
signals in compressive sensing. In this paper, a new bound is
obtained for the exact reconstruction of every $K$sparse signal
via
the gOMP algorithm in the noiseless case. That is, if the restricted
isometry constant (RIC) $\delta_{NK+1}$ of the sensing matrix
$A$
satisfies $ \delta_{NK+1}\lt \frac{1}{\sqrt{\frac{K}{N}+1}}$, then
the
gOMP can perfectly recover every $K$sparse signal $x$ from $y=Ax$.
Furthermore, the bound is proved to be sharp.
In the noisy case, the above bound on RIC combining with an
extra condition on the minimum
magnitude of the nonzero components of $K$sparse signals can
guarantee
that the gOMP selects all of support indices of the $K$sparse
signals.
A $2$cell embedding of an Eulerian digraph $D$
into a closed surface is said to be directed if the
boundary of each face is a directed closed walk in $D$. In this
paper, a method is developed with the purpose of enumerating
unlabelled embeddings for an Eulerian digraph. As an application,
we obtain explicit formulas for the number of unlabelled embeddings
of directed bouquets of cycles $B_n$, directed dipoles $OD_{2n}$
and for a class of regular tournaments $T_{2n+1}$.
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.
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 $(1z^2)^\alpha\,dA(z)$
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.
In this paper we study the Hilbert transformations over
$L^2(\mathbb{R})$
and $L^2(\mathbb{T})$ from
the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$
commutative with the ax+b group we show that the operator is
of the form
$
\lambda I+\eta H,
$
where $I$ and $H$ are the identity operator and Hilbert transformation
respectively, and $\lambda,\eta$ are complex numbers. In the
related literature this result was proved through first invoking
the boundedness result of the operator, proved though a big
machinery.
In our setting the boundedness is a consequence of the boundedness
of the Hilbert transformation. The methodology that we use is
GelfandNaimark's representation of the ax+b group. Furthermore
we prove a similar result on the unit circle. Although there
does not exist a group like ax+b on the unit circle, we construct
a semigroup to play the same symmetry role for the Hilbert transformations
over the circle $L^2(\mathbb{T}).$
We make some elementary observations concerning subcritically
Stein
fillable contact structures on $5$manifolds.
Specifically, we determine the diffeomorphism type of such
contact manifolds in the case the fundamental group is finite
cyclic,
and we show that on the $5$sphere the standard contact structure
is the unique subcritically fillable one. More generally,
it is shown that subcritically fillable contact structures
on simply connected $5$manifolds are determined by their
underlying almost contact structure. Along the way, we discuss
the
homotopy classification of almost contact structures.
In this paper, we prove that the following singular integral
defined by
$$T_{\Omega,a}f(x)=\operatorname{p.v.}\int_{\mathbb{R}^{d}}\frac{\Omega(xy)}{xy^d}\cdot m_{x,y}a\cdot
f(y)dy$$
is bounded on $L^p(\mathbb{R}^d)$ for $1\lt p\lt \infty$ and is of weak type
(1,1), where $\Omega\in L\log^+L(\mathbb{S}^{d1})$ and
$m_{x,y}a=:\int_0^1a(sx+(1s)y)ds$
with $a\in L^\infty(\mathbb{R}^d)$ satisfying some restricted conditions.
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.
Let \(X\) be a CW complex with a continuous action of a topological
group \(G\).
We show that if \(X\) is equivariantly formal for singular
cohomology
with coefficients in some field \(\Bbbk\), then so are all symmetric
products of \(X\)
and in fact all its \(\Gamma\)products.
In particular, symmetric products
of quasiprojective Mvarieties are again Mvarieties.
This generalizes a result by Biswas and D'Mello
about symmetric products of Mcurves.
We also discuss several related questions.
For an analytic curve $\gamma:(a,b)\rightarrow \mathbb C,$ the set of
values approached by $\gamma(t),$ as $t\searrow a$ and as $t\nearrow
b$ can be any two continuua of $\mathbb C\cup\{\infty\}.$
We show by means of an example in $\mathbb C^3$ that Gromov's
theorem on the presence of attached holomorphic discs for compact
Lagrangian manifolds is not true in the subcritical
realanalytic case, even in the absence of an obvious obstruction,
i.e, polynomial convexity.
The classical result of Nevanlinna states that two nonconstant
meromorphic functions on the complex plane having the
same images for five distinct values must be identically equal
to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean
fourspace.
We give a characterization of $C^{\ast}$normed algebras, among
certain involutive normed ones. This is done through the existence
of enough specific positive functionals. The same question is
also
examined in some non normed (topological) algebras.
In this paper, we consider the following
critical Kirchhoff type equation:
\begin{align*}
\left\{
\begin{array}{lll}

\left(a+b\int_{\Omega}\nabla u^2
\right)\Delta u=Q(x)u^4u + \lambda u^{q1}u,~~\mbox{in}~~\Omega,
\\
u=0,\quad \text{on}\quad \partial \Omega,
\end{array}
\right.
\end{align*}
By using variational methods that are constrained to the Nehari
manifold,
we prove that the above equation has a ground state solution
for the case when $3\lt q\lt 5$.
The relation between the number of maxima of $Q$
and the number of positive solutions for the problem is also
investigated.
A selfavoiding polygon is a lattice polygon consisting of a
closed selfavoiding walk on a square lattice.
Surprisingly little is known rigorously about the enumeration
of selfavoiding 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 selfavoiding 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 selfavoiding polygons in the $m \times
n$ rectangular grid on the square lattice.
For $m=2$, $p_{2 \times n} = 2^{n1}1$.
And, for integers $m,n \geq 3$,
$$2^{m+n3}
\left(\tfrac{17}{10}
\right)^{(m2)(n2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n3}
\left(\tfrac{31}{16}
\right)^{(m2)(n2)}.$$
On a real hypersurface $M$ in a complex twoplane 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 TanakaWebster 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 $M$.
Motivated by a question of A. Skalski and P.M. Sołtan (2016)
about inner faithfulness of the S. Curran's map of extending
a quantum increasing sequence to a quantum permutation, we revisit
the results and techniques of T. Banica and J. Bichon (2009)
and study some grouptheoretic properties of the quantum permutation
group on $4$ points. This enables us not only to answer the aforementioned
question in positive in case $n=4, k=2$, but also to classify
the automorphisms of $S_4^+$, describe all the embeddings $O_{1}(2)\subset
S_4^+$ and show that all the copies of $O_{1}(2)$ inside $S_4^+$
are conjugate. We then use these results to show that the converse
to the criterion we applied to answer the aforementioned question
is not valid.
Let $\mathtt{G}$ be the $n$fold covering group of the special
linear group of degree two, over a nonArchimedean 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.
Let $R$ be a ring. A map $f: R\rightarrow R$
is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$
of $R$.
Here a map $f: R\rightarrow R$ is called unitadditive if $f(u+v)=f(u)+f(v)$
for all units $u$ and $v$ of $R$. Motivated by a recent result
of Xu, Pei and Yi
showing that, for any field $F$, every
unitadditive map of ${\mathbb M}_n(F)$ is additive for all $n\ge
2$, this paper is about the question when every unitadditive
map of a ring is additive. It is proved that every unitadditive
map of a semilocal ring $R$ is additive if and only if either
$R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong
\mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal
ring $R$, every unitadditive map of ${\mathbb M}_n(R)$ is additive
for all $n\ge 2$. These results are further extended to rings
$R$ such that $R/J(R)$ is a direct product of exchange rings
with primitive factors Artinian. A unitadditive map $f$ of a
ring $R$ is called unithomomorphic if $f(uv)=f(u)f(v)$ for all
units $u,v$ of $R$. As an application, the question of when every
unithomomorphic map of a ring is an endomorphism is addressed.
We apply our theory of partial flag spaces developed
with W. Goldring
to study a grouptheoretical generalization of the canonical
filtration of a truncated BarsottiTate group of level 1. As
an application, we determine explicitly the normalization of
the Zariski closures of EkedahlOort strata of Shimura varieties
of Hodgetype as certain closed coarse strata in the associated
partial flag spaces.
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.
Let $R$ be an $n!$torsion free semiprime ring with
involution $*$ and with extended centroid $C$, where $n\gt 1$ is
a positive integer. We characterize $a\in K$, the Lie algebra
of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This
generalizes both Martindale and Miers' theorem
and the theorem of Brox et al.
To prove it we
first prove that if $a, b\in R$ satisfy
$(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on
$R$, where either $n$ is even or $b=0$, then
$\big(a\lambda\big)^{[\frac{n+1}{2}]}=0$
for some $\lambda\in C$.
This paper gives an equivalent form of Picard's
theorem via entire solutions of the functional equation $f^2+g^2=1$,
and then its improvements and applications to certain nonlinear
(ordinary and partial) differential equations.
In this paper, the bounded properties of oscillatory hyperHilbert
transform along certain plane curves $\gamma(t)$
$$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(xt,y\gamma(t))e^{ i t^{\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$
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 $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.
We provide normal forms and the global phase portraits on the
Poincaré disk for all Abel quadratic polynomial differential
equations of the second kind with $\mathbb Z_2$symmetries.
We investigate the moduli space of sheaves supported on space
curves of degree $4$ and having Euler characteristic $1$.
We give an elementary proof of the fact that this moduli space
consists of three irreducible components.
A crucial role in the NymanBeurlingBáezDuarte approach to
the Riemann Hypothesis is played by the distance
\[
d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{\infty}^\infty
\left1\zeta A_N
\left(\frac{1}{2}+it
\right)
\right^2\frac{dt}{\frac{1}{4}+t^2}\:,
\]
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 nontrivial zeros off the
critical line.
In this paper we give some generalizations and
improvements of the Pavlović result on the
HollandWalsh 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 zerodimensional 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 $2^\omega$.
In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, nonempty, proper)
algebraic variety defined over a
field $k$ of characteristic zero, then $X$ is a hypersurface
if and only if the module $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ of logarithmic vector fields of
$X$ is a reflexive ${\mathcal
O}_{{\mathbb A}^n}$module. As a consequence of this result,
we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a
free ${\mathcal
O}_{{\mathbb A}^n}$module, which is shown to be equivalent
to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily
$X$ is a Saito free divisor.
We show under some conditions that a Gorenstein ring $R$ satisfies the
Generalized AuslanderReiten 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 nonStein strictly pseudoconvex domains.
For a commutative ring $R$, a polynomial $f\in R[x]$ is called
separable if $R[x]/f$ is a separable $R$algebra. We derive formulae
for the number of separable polynomials when $R = \mathbb{Z}/n$, extending
a result of L. Carlitz. For instance, we show that the number
of separable polynomials in $\mathbb{Z}/n[x]$
that are separable is $\phi(n)n^d\prod_i(1p_i^{d})$
where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$
and $\phi$ is Euler's totient 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 sumofdivisors function, and let $\lambda$ be Carmichael's lambdafunction. 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 $0.235711131719\ldots$.
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
$$\Psi_B(z)=\sum_n \frac{1z_n^2}{1\overline{z}_nz^2}.$$
Moreover, it is a wellknown 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 CarlesonNewman
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.
For $m, n \in \mathbb{N}$, $1\lt m \leq n$, we write $n = n_1 +
\dots + n_m$ where $\{ n_1, \dots, n_m \} \subset \mathbb{N}$. Let
$A_1, \dots, A_m$ be $n \times n$ singular real matrices such that
$\bigoplus_{i=1}^{m} \bigcap_{1\leq j \neq i \leq m} \mathcal{N}_j
= \mathbb{R}^{n},$ where
$\mathcal{N}_j = \{ x : A_j x = 0 \}$, $dim(\mathcal{N}_j)=nn_j$
and $A_1+ \dots+ A_m$ is invertible. In this paper we study integral
operators of the form
$T_{r}f(x)= \int_{\mathbb{R}^{n}} \, xA_1 y^{n_1 + \alpha_1}
\cdots xA_m y^{n_m + \alpha_m} f(y) \, dy,$
$n_1 + \dots + n_m = n$, $\frac{\alpha_1}{n_1} = \dots = \frac{\alpha_m}{n_m}=r$,
$0 \lt r \lt 1$, and the matrices $A_i$'s are as above. We obtain
the $H^{p}(\mathbb{R}^{n})L^{q}(\mathbb{R}^{n})$ boundedness
of $T_r$ for $0\lt p\lt \frac{1}{r}$ and $\frac{1}{q}=\frac{1}{p} 
r$.
For most classical and similitude groups, we show that each element
can be written as a product of two transformations that
a) preserve or almost preserve the underlying form and b) whose
squares are certain scalar maps. This generalizes work of Wonenburger
and Vinroot.
As an application, we reprove and slightly extend a well known
result of Mœglin, Vignéras and Waldspurger on the existence
of automorphisms of $p$adic classical groups that take each
irreducible smooth representation to its dual.
The FeffermanStein type inequalities
for strong maximal operator and directional maximal operator
are verified with an additional composition of the HardyLittlewood
maximal operator in the plane.
In this paper we establish a close connection between three
notions attached to a modular subgroup. Namely the set of weight
two meromorphic modular forms, the set of equivariant functions
on the upper halfplane commuting with the action of the modular
subgroup and the set of elliptic zeta functions generalizing
the Weierstrass zeta functions. In particular, we show that the
equivariant functions can be parameterized by modular objects
as well as by elliptic objects.
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
DGrings:
let $A$ be a commutative nonpositive DGring 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
We calculate the dimension of cohomology groups for
the holomorphic tangent bundles of each isomorphism
class of the projective plane bundle over an elliptic curve.
As an application, we construct the families
of projective plane bundles, and prove that the families
are effectively parametrized and complete.
We determine the asymptotic behavior of the higher dimensional
Reidemeister torsion for
the graph manifolds obtained by exceptional surgeries along
twist knots.
We show that all irreducible
$\operatorname{SL}_2(\mathbb{C})$representations of the graph
manifold
are induced by irreducible metabelian representations of the
twist knot group.
We also give the set of the limits of the leading coefficients
in the higher dimensional Reidemeister torsion explicitly.
We present a multiplier theorem on anisotropic
Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin
condition, we obtain boundedness of the multiplier operator $T_m
: H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$
that depends on the eccentricities of the dilation $A$ and the
level of regularity of a multiplier symbol $m$. This extends
the classical multiplier theorem of Taibleson and Weiss.
In this paper we consider the growth rates of 3dimensional hyperbolic
Coxeter polyhedra with at least one dihedral angle of the form
$\frac{\pi}{k}$ for an integer $k\geq{7}$.
Combining a classical result by Parry with
a previous result of ours,
we prove that the growth rates of
3dimensional hyperbolic Coxeter groups are Perron numbers.
In this paper, we study a twocomponent LotkaVolterra competition
system
on an onedimensional spatial lattice. By the method of the comparison
principle together with
the weighted energy, we prove that the traveling wavefronts with
large speed are exponentially asymptotically stable,
when the initial perturbation around the traveling wavefronts
decays
exponentially as $j+ct \rightarrow \infty$, where $j\in\mathbb{Z}$,
$t\gt 0$, but the initial perturbation
can be arbitrarily large on other locations. This partially answers
an open problem by J.S. Guo and C.H. Wu.
In this paper, we classify all solutions of
\[
\left\{
\begin{array}{rcll}
\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+},
\\
\dfrac{\partial u}{\partial t}&=&cx^{\beta}e^{u} \quad
&\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\},
\\
\end{array}
\right.
\]
with the finite conditions
\[
\int_{\partial \mathbb{R}^{2}_{+}}x^{\beta}e^{u}ds \lt C,
\qquad
\sup\limits_{\overline{\mathbb{R}^{2}_{+}}}{u(x)}\lt C.
\]
Here, $c$ is a positive number and $\beta \gt 1$.