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We investigate quantum group generalizations
of various density results from Fourier analysis on compact groups.
In particular, we establish the density of characters in the
space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and
use this result to show the weak* density and norm density of
characters in $ZL^\infty(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary,
we partially answer an open question of Woronowicz.
At the level of $L^1(\mathbb{G})$, we show that the center
$\mathcal{Z}(L^1(\mathbb{G}))$
is precisely the closed linear span of the quantum characters
for a large class of compact quantum groups, including arbitrary
compact Kac algebras. In the latter setting, we show, in addition,
that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented
$\mathcal{Z}(L^1(\mathbb{G}))$-submodule
of $L^1(\mathbb{G})$.

Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$
a non-nilpotent ideal of $R$. The purpose of this article is
to investigate Faltings' finiteness
dimension $f_I(R)$ and equidimensionalness of certain homomorphic
image of $R$. As a consequence
we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$
and if $\operatorname{mAss}_R(R/I)$
is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq
1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$.
Moreover, we will obtain a lower bound for injective dimension
of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case
$(R, \frak m)$ is a complete equidimensional local ring.

In this paper, we show that the Möbius invariant
function space $\mathcal {Q}_p$ can be generated by variant
Dirichlet type spaces
$\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures
$\mu$ on the open unit disk. A criterion for the equality between
the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type
space $\mathcal {D}_p$ is given. We obtain a sufficient condition
to construct different $\mathcal{D}_{\mu, p}$ spaces
and we provide examples.
We establish decomposition theorems for $\mathcal{D}_{\mu,
p}$ spaces, and prove that the non-Hilbert space $\mathcal
{Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu,
p}$. As an application of the relation between $\mathcal {Q}_p$
and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there
exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick
to prove the existence without constructing examples.

We compute the cohomology of the right generalised projective
Stiefel manifolds. Following this, we discuss some easy applications
of the computations to the ranks of complementary bundles, and
bounds on the span and immersibility.

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 prove the existence of
a (in fact many)
holomorphic function $f$ in $\mathbb{C}^d$ such that, for any $a\neq
0$, its translations $f(\cdot+na)$ are dense in $H(\mathbb{C}^d)$.

In this paper, we give a new definition of radicals of Green's
relations in an ordered semigroup and characterize left regular
(right regular), intra regular ordered semigroups by radicals
of Green's relations. Also we characterize the ordered semigroups
which are unions and complete semilattices of t-simple ordered
semigroups.

Using known operator-valued Fourier multiplier results on vector-valued
Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely
characterize the $C^\alpha$-well-posedness 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.

In a previous work, we have associated a complete differential
graded Lie algebra
to any finite simplicial complex in a functorial way.
Similarly, we have also a realization functor from the category
of complete differential graded Lie algebras
to the category of simplicial sets.
We have already interpreted the homology of a Lie algebra
in terms of homotopy groups of its realization.
In this paper, we begin a dictionary between models
and simplicial complexes by establishing a correspondence
between the Deligne groupoid of the model and the connected components
of the finite simplicial complex.

Let $G$ denote a reductive algebraic group over
$\mathbb{C}$
and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer
variety $\mathcal{B}_x$
is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing
the
Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable
property that
the Weyl group $W$ of $G$ admits a representation on the cohomology
of $\mathcal{B}_x$
even though $W$ rarely acts on $\mathcal{B}_x$ itself. Well-known constructions
of this action
due to Springer et al use technical machinery from algebraic
geometry.
The purpose of this note is to describe an elementary approach
that gives this action
when $x$ is what we call parabolic-surjective. The idea is to
use localization to construct an action of $W$ on
the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic
subtorus of $G$.
This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and
gives the desired representation. The parabolic-surjective case
includes all nilpotents of type $A$ and,
more generally, all nilpotents for which it is known that $W$
acts on
$H_S^*(\mathcal{B}_x)$ for some torus $S$.
Our result is deduced from a general theorem describing
when a group action on the cohomology of the fixed point set of a
torus action
on a space lifts to the full cohomology algebra of the space.

In this paper, a new notion of scalar curvature for a Finsler
metric $F$ is introduced, and two conformal invariants $Y(M,F)$
and $C(M,F)$ are defined. We prove that there exists a Finsler
metric with constant scalar curvature in the conformal class
of $F$ if the Cartan torsion of $F$ is sufficiently small and
$Y(M,F)C(M,F)\lt Y(\mathbb{S}^n)$ where $Y(\mathbb{S}^n)$ is the
Yamabe constant of the standard sphere.

Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give
a sufficient and necessary condition
for weighted translation operators on the Lebesgue space $\ell^p(G)$
to be densely disjoint hypercyclic.
The characterization for the dual of a weighted translation to
be densely disjoint hypercyclic is also obtained.

In this
paper,
double commutativity and the reverse order law for the core inverse
are considered. Then, new characterizations of the Moore-Penrose
inverse of a regular element are given by one-sided invertibilities
in a ring. Furthermore, the characterizations and representations
of
the core and dual core inverses of a regular element are considered.

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.

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.

We prove Lawton's conjecture about the upper bound on the measure
of the set on the unit circle on which a complex polynomial with
a bounded number of coefficients takes small values. Namely,
we prove that Lawton's bound holds for polynomials that are not
necessarily monic. We also provide an analogous bound for polynomials
in several variables. Finally, we investigate the dependence
of the bound on the multiplicity of zeros for polynomials in
one variable.

Let $R$ be a prime ring with extended
centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$
central closure of $R$ such that $dim_{C}(RC)
\gt 4$, $f(X_{1},\dots,X_{n})$
a multilinear polynomial over $C$ which is not central-valued
on $R$ and $f(R)$ the set of all evaluations of the multilinear
polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that
$G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u
\in C$ for all $u\in f(R)$ then one of the following conditions
holds:

(I) there exists $a\in Q$ such that $a^2=0$ and
either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in
R$;

(II) there exists $a\in Q$ such that $0\neq a^2\in
C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all
$x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on
$R$;

(III) $char(R)=2$ and one of the following holds:

(i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$ and $a^{2}=b^{2}\in C$;

(ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$
is central-valued on $R$;

(iii) there exist $a \in Q$ and an $X$-outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and
$a^2+d(a)=0$;

(iv) there exist $a \in Q$ and an $X$-outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$,
$a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued
on $R$.

Moreover, we characterize the form of nonzero generalized derivations
$G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where
$\lambda \in C$.

It follows from earlier work of Silver--Williams and the authors
that twisted Alexander polynomials detect the unknot and the
Hopf link.
We now show that twisted Alexander polynomials also detect the
trefoil and the figure-8 knot,
that twisted Alexander polynomials detect whether a link is split
and that twisted Alexander modules detect trivial links. We use
this result to provide algorithms for detecting whether a link
is the unlink, whether it is split and whether it is totally
split.

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 $\mathbb C\cup\{\infty\}.$

It is known that if $E$ is a closed subset of an open Riemann
surface $R$ and $f$ is a holomorphic function on a neighbourhood
of $E,$ then it is ``usually" not possible to approximate $f$
uniformly by functions holomorphic on all of $R.$ We show, however,
that for every open Riemann surface $R$ and every closed subset
$E\subset R,$ there is closed subset $F\subset E,$ which approximates
$E$ extremely well, such that every function holomorphic on $F$
can be approximated much better than uniformly by functions holomorphic
on $R$.

Suppose $G$ is a connected complex Lie group and $H$ is a closed
complex subgroup.
Then there exists a closed complex subgroup $J$ of $G$ containing
$H$ such that
the fibration $\pi:G/H \to G/J$ is the holomorphic reduction
of $G/H$, i.e., $G/J$ is holomorphically
separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$.
In this paper we prove that if $G/H$ is pseudoconvex, i.e.,
if
$G/H$ admits a continuous plurisubharmonic exhaustion function,
then $G/J$ is Stein and $J/H$ has no non--constant holomorphic
functions.

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
four-space.

In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$-convex-normal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$- and $(k+1)$-convex-normality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.

We previously obtained a congruence modulo four for the number
of real solutions to many Schubert problems on
a square Grassmannian given by osculating flags.
Here, we consider Schubert problems given by more general isotropic
flags, and prove this
congruence modulo four for the largest class of Schubert problems
that could be expected to exhibit this
congruence.

We study linear projections on Plücker space whose restriction
to the Grassmannian is a non-trivial branched
cover.
When an automorphism of the Grassmannian preserves the fibers,
we show that the Grassmannian is necessarily
of $m$-dimensional linear subspaces in a symplectic vector
space of dimension $2m$, and the linear map is
the Lagrangian involution.
The Wronski map for a self-adjoint linear differential operator
and pole placement map for
symmetric linear systems are natural examples.

In the Simpson moduli space $M$ of semi-stable sheaves with
Hilbert polynomial $dm-1$ on a projective plane we study the
closed subvariety $M'$ of sheaves that are not locally free on
their support. We show that for $d\ge 4$ it is a singular subvariety
of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted
as a (partial) modification of $M\setminus M'$ by line bundles
(on support).

Fix an irreducible (finite) root system $R$ and a choice
of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group $G_k$ over $k$ with root system $k$. One associates to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the
Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$.
Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ such
that
$\mu$ is maximal with this property. Garibaldi, Guralnick, and
Nakano
have asked under which condition there exists $k$ such that $L
(\mu)_k$
is a composition factor of $V (\lambda)_k$, and they exhibit an
example
in type $E_8$ where this is not the case. The purpose of this
paper
is to to show that their example is the only one. It contains
two proofs
for this fact, one that uses a classification of the possible
pairs $(\lambda, \mu)$,
and another one that relies only on the classification
of root systems.

Alfred Schild has established conditions
that Lorentz transformations map world-vectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. These transformations
are called integral Lorentz transformations.

The present paper contains supplements to
our earlier work
with a new focus on group theory. To relate the results to the
familiar matrix group nomenclature we associate Lorentz transformations
with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the
lattice of subgroups of the group originated in Schild's paper
and obtain generating sets for the full group and its subgroups.

A $C^{*}$-algebra $A$ has the ideal property if any ideal
$I$ of $A$ is generated as a closed two sided ideal by the projections
inside the ideal. Suppose that the limit $C^{*}$-algebra $A$
of inductive limit of direct sums of matrix algebras over spaces
with uniformly bounded dimension has ideal property. In this
paper we will prove that $A$ can be written as an inductive limit
of certain very special subhomogeneous algebras, namely, direct
sum of dimension drop interval algebras and matrix algebras over
2-dimensional spaces with torsion $H^{2}$ groups.

Let $R$ be a commutative ring with non-zero identity. In this
paper, we introduced the weakly nilpotent graph of a commutative
ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$
is a graph with the vertex set $R^{*}$ and two vertices $x$ and
$y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$
and $N(R)^{*}$ is the set of all non-zero nilpotent elements
of $R$. In this article, we determine the diameter of weakly
nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$
is a forest, then $\Gamma_w(R)$ is a union of a star and some
isolated vertices. We study the clique number, the chromatic
number and the independence number of $\Gamma_w(R)$. Among other
results, we show that for an Artinian ring $R$, $\Gamma_w(R)$
is not a disjoint union of cycles or a unicyclic graph. For Artinan
ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we
characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$
is a cycle, where $\overline{\Gamma_w(R)}$ is the complement
of the weakly nilpotent graph of $R$.

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 unit-additive 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
unit-additive map of ${\mathbb M}_n(F)$ is additive for all $n\ge
2$, this paper is about the question when every unit-additive
map of a ring is additive. It is proved that every unit-additive
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 unit-additive 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 unit-additive map $f$ of a
ring $R$ is called unit-homomorphic if $f(uv)=f(u)f(v)$ for all
units $u,v$ of $R$. As an application, the question of when every
unit-homomorphic map of a ring is an endomorphism is addressed.

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.

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 $\lambda\in C$.

We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic
(or rational) field of discriminant $D$ and Dirichlet character
$\chi$, if a prime $p$ is large enough compared to $D$, there
is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with
respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f
\otimes \chi,1) \neq 0$. This result is obtained through an estimate
of a weighted sum of twists of $L$-functions which generalises
a result of Ellenberg. It relies on the approximate functional
equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and
a Petersson trace formula restricted to Atkin-Lehner eigenspaces.
An application of this nonvanishing theorem will be given in
terms of existence of rank zero quotients of some twisted jacobians,
which generalises a result of Darmon and Merel.

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 we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
Hardy-Littlewood maximal operator. We obtain some new bounds
for the derivative of the one-dimensional multisublinear
fractional maximal operators acting on vector-valued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

We study a population model with nonlocal diffusion, which
is a delayed integro-differential equation with double nonlinearity
and two integrable kernels. By comparison method and analytical
technique, we obtain globally asymptotic stability of the zero
solution and the positive equilibrium. The results obtained
reveal that the globally asymptotic stability only depends on
the property of nonlinearity. As application, an example for
a population model with age structure is discussed at the end
of the article.

It is known that every Toeplitz matrix $T$ enjoys a circulant
and skew circulant splitting (denoted by CSCS)
i.e., $T=C-S$ with $C$ a circulant matrix and $S$ a skew circulant
matrix. Based on the variant of such a splitting (also referred
to as CSCS), we first develop classical CSCS iterative methods
and then introduce shifted CSCS iterative methods for solving
hermitian positive definite Toeplitz systems in this paper. The
convergence of each method is analyzed. Numerical experiments
show that the classical CSCS iterative methods work slightly
better than the Gauss-Seidel (GS) iterative methods if the CSCS
is convergent, and that there is always a constant $\alpha$ such
that the shifted CSCS iteration converges much faster than the
Gauss-Seidel iteration, no matter whether the CSCS itself is
convergent or not.

We prove Wise's $W$-cycles conjecture: Consider a compact graph
$\Gamma'$ immersing into another graph $\Gamma$. For any immersed
cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$
from
the circular components $\mathbb{S}$ of the pullback to $\Gamma'$.
Unless
$\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to
S^1$ is bounded above by minus the Euler characteristic of
$\Gamma'$. As a corollary, any finitely generated subgroup
of a
one-relator group has finitely generated Schur multiplier.

In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, non-empty, 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.

In this paper, we obtain some characterizations of the (strong)
Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules
and certain elements of $\mathbb{B}(\mathscr{H})$.
Moreover, we obtain a kind of Pythagorean relation for bounded
linear operators.
In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the
norm attaining
set $\mathbb{M}_T$ is a unit sphere of some finite dimensional
subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{{\mathscr{H}}_0}^\perp}
\lt \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong
Birkhoff--James orthogonal to $S$ if and only if there exists
a unit vector $\xi\in {\mathscr{H}}_0$ such that $\|T\|\xi =
|T|\xi$ and $S^*T\xi = 0$.
Finally, we introduce a new type of approximate orthogonality
and investigate this notion in the setting of inner product $C^*$-modules.

It is known that a bi-orderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$-manifolds,
and verify the conjecture for non-hyperbolic, geometric $3$-manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$-manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.

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 $R[x]$.

Let the measure algebra of a topological group $G$ be equipped
with
the topology of uniform convergence on bounded right uniformly
equicontinuous sets of functions.
Convolution is separately continuous on the measure algebra,
and it is jointly continuous if and only if $G$ has the SIN property.
On the larger space $\mathsf{LUC}(G)^\ast$ which includes the measure
algebra,
convolution is also jointly continuous if and only if the group
has the SIN property,
but not separately continuous for many non-SIN groups.

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 $\phi(n)n^d\prod_i(1-p_i^{-d})$
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 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 $0.235711131719\ldots$.

Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$-semigroup of quasinilpotent
operators
(i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$).
In the dynamical systems theory the above quasinilpotency property
is equivalent
to a very strong concept of stability for the solutions of autonomous
systems.
This concept is frequently called superstability and weakens
the classical finite time extinction property
(roughly speaking, disappearing solutions).
We show that under some assumptions, the quasinilpotency, or
equivalently, the superstability property
of a C$_0$-semigroup is preserved under the perturbations of
its infinitesimal generator.

The notion of coaxers is introduced in a pseudo-complemented
distributive lattice. Boolean algebras are characterized in terms
of coaxer ideals and congruences. The concept of coaxer lattices
is introduced in pseudo-complemented distributive lattices and
characterized in terms of coaxer ideals and maximal ideals. Finally,
the coaxer lattices are also characterized in topological terms.

The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$-tuple $A := (A_1, \dots, A_r)$ of $n \times n$-matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$-tuple of $n \times n$-matrices in general position,
then up to conjugacy, there are only finitely many $r$-tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r-1)n^2 + 1$.

In this paper we give the bifurcation diagram
of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$
for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of
$\epsilon_1,\epsilon_0\in\mathbb{C}$.
The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure
allows describing it for parameter values in $\mathbb{S}^3$. There are
two open simply connected regions of structurally stable vector
fields separated by surfaces corresponding to bifurcations of
homoclinic connections between two separatrices of the pole at
infinity. These branch from the codimension 2 curve of double
singular points. We also explain the bifurcation of homoclinic
connection in terms of the description of Douady and Sentenac
of polynomial vector fields.

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and
consider
the energy functional
\begin{equation*}
{\mathcal F}[u, \Omega] := \int_{\Omega} {\rm F}(\nabla {\bf u}(\bf x))\, d{\bf x},
\end{equation*}
over the space of $W^{1,2}(\Omega, \mathbb{R}^m)$ where the integrand
${\rm F}: \mathbb M_{m\times n}\to \mathbb{R}$ is a smooth uniformly
convex function
with bounded second derivatives. In this paper we address the
question of
regularity for solutions of the corresponding system of
Euler-Lagrange equations.
In particular we introduce a class of singular maps referred
to as traceless and
examine them as a new counterexample to the regularity of minimizers
of the energy
functional $\mathcal F[\cdot,\Omega]$ using a method based on
null Lagrangians.

Let $K$ be an ultraspherical hypergroup associated to a locally
compact group $G$ and a spherical projector $\pi$ and let $VN(K)$
denote the dual of the Fourier algebra $A(K)$ corresponding to
$K.$ In this note, we show that the set of invariant means on
$VN(K)$ is singleton if and only if $K$ is discrete. Here $K$
need not be second countable. We also study invariant means on
the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$
in the $cb$-multiplier norm. Finally, we consider generalized
translations and generalized invariant means.

Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article,
the authors establish
equivalent characterizations
of Besov-type spaces, Triebel-Lizorkin-type
spaces and Besov-Morrey spaces via the sequence
$\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between
$f$ and
the ball average $B_{\ell,2^{-k}}f$. These results give a way
to introduce Besov-type spaces,
Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any
smoothness order
on metric measure spaces. As special cases, the authors obtain
a new characterization of Morrey-Sobolev spaces
and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent
interest.

We prove a comprehensive version of the Ruelle-Perron-Frobenius
Theorem
with explicit estimates of the spectral radius of the Ruelle
transfer operator and various other
quantities related to spectral properties of this operator. The
novelty here is that the Hölder
constant of the function generating the operator appears only
polynomially, not exponentially as
in previous known estimates.

This paper is dedicated to studying the
semilinear Schrödinger equation
$$
\left\{
\begin{array}{ll}
-\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N},
\\
u\in H^{1}({\mathbf{R}}^{N}),
\end{array}
\right.
$$
where $f$ is a superlinear, subcritical nonlinearity. It focuses
on the case where
$V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic
in each of $x_1, x_2, \ldots, x_N$ and
$\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle
+V_0)\cap (0, \infty)]$,
$V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic
condition is obtained,
which is weaker than some well known results.

In this paper, we study a two-component Lotka-Volterra competition
system
on an one-dimensional 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.

Let $R$
be a ring and $b, c\in R$.
In this paper, we give some characterizations of the $(b,c)$-inverse,
in terms of the direct sum decomposition, the annihilator and
the invertible elements.
Moreover, elements with equal $(b,c)$-idempotents related to
their $(b, c)$-inverses are characterized, and the reverse order
rule for the $(b,c)$-inverse is considered.

In this paper, the concept of the classical $f$-divergence for
a pair of measures is extended to the mixed $f$-divergence for
multiple pairs of measures. The mixed $f$-divergence provides
a way to measure the difference between multiple pairs of (probability)
measures. Properties for the mixed $f$-divergence are established,
such as permutation invariance and symmetry in distributions.
An
Alexandrov-Fenchel type inequality and an isoperimetric inequality
for the
mixed $f$-divergence are proved.

In this paper, we construct two classes of rational function
operators by using the Poisson integrals of the function on the
whole real
axis. The convergence rates of the uniform and mean approximation
of such rational function operators on the whole real axis are
studied.

This paper generalizes the Aleksandrov problem: the Mazur-Ulam
theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance
preserving mapping is an $n$-isometry if and only if it has the
zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries
on $n$-G-quasi normed space are equivalent; we generalize the
Benz theorem to n-normed spaces with no restrictions on the dimension
of spaces.