This paper
discusses the connection between the local cohomology modules and
the Serre classes of $R$-modules. This connection has provided a common
language for expressing some results regarding the local cohomology
$R$-modules that have appeared in different papers.

We investigate the behavior of the quasi-Baer and the
right FI-extending right ring hulls under various ring extensions
including group ring extensions, full and triangular matrix ring
extensions, and infinite matrix ring extensions. As a consequence,
we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are
Morita equivalent, then so are the quasi-Baer right ring hulls
$\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of
$R$ and $S$, respectively. As an application, we prove that if
unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings,
then the bounded central closure of $A$ and that of $B$ are
strongly Morita equivalent as $C^*$-algebras. Our results show
that the quasi-Baer property is always preserved by infinite
matrix rings, unlike the Baer property. Moreover, we give an
affirmative answer to an open question of Goel and Jain for the
commutative group ring $A[G]$ of a torsion-free Abelian group $G$
over a commutative semiprime quasi-continuous ring $A$. Examples
that illustrate and delimit the results of this paper are provided.

The bigraded Hilbert function and the minimal free resolutions for the
diagonal coinvariants of the dihedral groups are exhibited, as well as for
all their bigraded invariant Gorenstein quotients.

For a given convex body $K$ in ${\mathbb R}^d$, a random polytope
$K^{(n)}$ is defined (essentially) as the intersection of $n$
independent closed halfspaces containing $K$ and having an isotropic
and (in a specified sense) uniform distribution. We prove upper and
lower bounds of optimal orders for the difference of the mean widths
of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial
polytope $P$, a precise asymptotic formula for the difference of the
mean widths of $P^{(n)}$ and $P$ is obtained.

In this paper, we investigate
a proper CAT(0) space $(X,d)$
that is homeomorphic to $\mathbb R^2$ and
we show that the asymptotic dimension $\operatorname{asdim} (X,d)$ is
equal to $2$.

In this paper, we explore a generalization of the notion of
integrality. In particular, we study a near-integrality condition that is
intermediate between the concepts of integral and almost integral.
This property (referred to as the $\Omega$-almost integral
property) is a representative independent specialization of the
standard notion of almost integrality. Some of the properties of
this generalization are explored in this paper, and these properties
are compared with the notion of pseudo-integrality introduced by
Anderson, Houston, and Zafrullah. Additionally, it is
shown that the $\Omega$-almost integral property serves to
characterize the survival/lying over pairs of Dobbs and Coykendall

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal
of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also
prove that, for a nonnegative integer $n$ such that
$H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for
all $i >0$ and $z \in \mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic
image of $R$, where $R_z$ is the ring of fractions of $R$ with
respect to a multiplicatively closed subset $\{z^j \mid j \geqslant
0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z
\in \mathfrak{a}$,
then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$
is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.

We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.

We construct a Laplace isospectral deformation of metrics on an
orbifold quotient of a nilmanifold. Each orbifold in the deformation
contains singular points with order two isotropy. Isospectrality is
obtained by modifying a generalization of Sunada's theorem due to
DeTurck and Gordon.

The classical approach to studying operator ideals using tensor
norms mainly focuses on those tensor norms and operator ideals
defined by means of $\ell_p$ spaces. In a previous paper,
an interpolation space, defined via the real method
and using
$\ell_p$ spaces, was used to define a tensor
norm, and the associated minimal operator ideals were characterized.
In this paper, the next natural step is taken, that is, the
corresponding maximal operator
ideals are characterized. As an application, necessary and sufficient
conditions for the coincidence of
the maximal and minimal ideals are given.
Finally, the previous results are used in order to find some new
metric properties of the mentioned tensor norm.

We consider the problem of simultaneous extension of continuous
convex metrics defined on subcontinua of a Peano continuum. We prove
that there is an extension operator for convex metrics that is
continuous with respect to the uniform topology.

The fiber $W_{n}$ of the double suspension
$S^{2n-1}\rightarrow\Omega^{2} S^{2n+1}$
is known to have a classifying space $BW_{n}$. An important
conjecture linking the $EHP$ sequence to the homotopy theory of
Moore spaces is that $BW_{n}\simeq\Omega T^{2np+1}(p)$, where $T^{2np+1}(p)$
is Anick's space. This is known if $n=1$. We prove the $n=p$ case
and establish some related properties.

A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky,
and Vougalter based on the variational principle for the
quadratic form of a self-adjoint operator.
It is the negative index theorem for a linearized NLS operator in
three dimensions.

We construct new examples of surfaces of general type with $p_g=0$ and $K^2=5$ as ${\mathbb Z}_2 \times {\mathbb Z}_2$-covers and show that they are genus three hyperelliptic fibrations with bicanonical map of degree two.

We extend the idea of interval pattern avoidance defined by Yong and
the author for $S_n$ to arbitrary Weyl groups using the definition of
pattern avoidance due to Billey and Braden, and Billey and Postnikov.
We show that, as previously shown by Yong and the
author for $\operatorname{GL}_n$, interval pattern avoidance is a universal tool for
characterizing which Schubert varieties have certain local properties,
and where these local properties hold.