We calculate the exact amenability constant of the centre of
$\ell^1(G)$ when $G$ is one of the following classes of finite group:
dihedral; extraspecial; or Frobenius with abelian complement and
kernel. This is done using a formula which applies to all finite
groups with two character degrees. In passing, we answer in the
negative a question raised in work of the third author with Azimifard
and Spronk (J. Funct. Anal. 2009).

Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities
\[
\operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq
\operatorname{ara}(\mathfrak a) \leq
l(\mathfrak a) \leq \mu(\mathfrak a)
\]
are known. It is an interesting and long-standing problem to find
out the cases giving equality. Thanks to the formal grade we give
conditions in which the above inequalities become
equalities.

We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$-dimensional vector-valued modular form
on $\operatorname{SL}_2(\mathbb{Z})$ whose component functions $f_1, f_2$ have rational Fourier coefficients
with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

Let $a,b$ and $c$ be primitive Pythagorean numbers such that
$a^{2}+b^{2}=c^{2}$ with $b$ even.
In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$
with $\epsilon \in \{\pm1\}$
for certain positive divisors $b_0$ of $b$,
then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the
positive solution $(x,y,z)=(2,2,2)$.

We prove that every braiding over a unitary fusion category is
unitary and every unitary braided fusion category admits a unique
unitary ribbon structure.

Let $A$ be a ring with local units, $E$ a set of local units for $A$,
$G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of
$A$ that contain local units.
We show that $A\star_{\alpha} G$ is simple if and only if $A$ is
$G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G)
e \delta_0$ is a field for all $e\in E$. We apply the result to
characterize simplicity of partial skew group rings in two cases,
namely for partial skew group rings arising from partial actions by
clopen subsets of a compact set and partial actions on the set level.

The index of a graph $G$ is the maximum
eigenvalue of its adjacency matrix $A(G)$. In this paper we
characterize the extremal tree with given
domination number that attains the maximum index.

A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$
is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors
homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any
base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a
subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this
latter property are called $\mathcal{G}_K$-contractible. We obtain a
list of the closed $3$-manifolds that can be covered by two open
$\mathcal{G}_K$-contractible subsets. This is applied to obtain a list
of the possible closed prime $3$-manifolds that can be covered by two
open $K$-contractible subsets.

In this article we develop a test to determine whether a sextic
polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic
to the alternating group $A_4$. This test does not involve the
computation of resolvents, and we use this test to construct several
infinite families of such polynomials.

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete
if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on
$\mathcal{L}^{2}(\mathbb{G})$.
As a corollary we classify all discrete quantum groups among
regular locally compact quantum groups $\mathbb{G}$ where
$\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property.

For relatively prime positive integers $u_0$ and $r$, we consider the
least common multiple $L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n)$ of the finite
arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower
bounds on $L_n$ that improve upon those obtained previously when
either $u_0$ or $n$ is large. When $r$ is prime, our best bound is
sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also
nearly sharp as $n\to\infty$.

In a previous paper the authors developed an intersection theory for
subspaces of rational functions on an algebraic variety $X$
over $\mathbf{k} = \mathbb{C}$. In this short note, we first extend this intersection
theory to an arbitrary algebraically closed ground field $\mathbf{k}$.
Secondly we give an isomorphism between the group of Cartier
$b$-divisors on the birational class of $X$
and the Grothendieck group
of the semigroup of subspaces of rational functions on $X$. The
constructed isomorphism moreover
preserves the intersection numbers. This provides an alternative point
of view on Cartier $b$-divisors and their intersection theory.

In this paper, we investigate the domination, total domination and
semi-total domination numbers of a zero-divisor graph of a
commutative Noetherian ring. Also, some relations between the
domination numbers of $\Gamma(R/I)$ and $\Gamma_I(R)$, and the
domination numbers of $\Gamma(R)$ and $\Gamma(R[x,\alpha,\delta])$,
where $R[x,\alpha,\delta]$ is the Ore extension of $R$, are studied.

We establish that if it is consistent that there is a
supercompact cardinal, then it is consistent that every locally
compact, hereditarily normal space which does not include a perfect
pre-image of $\omega_1$ is hereditarily paracompact.

We give a short proof of the Brascamp-Lieb theorem, which asserts that
a certain general form of Young's convolution inequality is saturated
by Gaussian functions. The argument is inspired by Borell's stochastic
proof of the Prékopa-Leindler inequality and applies also to the
reversed Brascamp-Lieb inequality, due to Barthe.

In this article, via the classical complex interpolation method
and some interpolation methods traced to Gagliardo,
the authors obtain an interpolation theorem for
Morrey spaces on quasi-metric measure spaces, which generalizes
some known results on ${\mathbb R}^n$.

We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation
type.

Consider quasianalytic local rings of germs of smooth functions closed
under composition, implicit equation, and monomial division. We show
that if the Weierstrass Preparation Theorem holds in such a ring then
all elements of it are germs of analytic functions.

An inverse semigroup $S$ is combinatorially factorizable if $S=TG$
where $T$ is a combinatorial (i.e., $\mathcal{H}$ is the equality
relation) inverse subsemigroup of $S$ and $G$ is a subgroup of $S$.
This concept was introduced and studied by Mills, especially in the
case when $S$ is cryptic (i.e., $\mathcal{H}$ is a congruence on
$S$). Her approach is mainly analytical considering subsemigroups of
a cryptic inverse semigroup.

We start with a combinatorial inverse monoid and a factorizable
Clifford monoid and from an action of the former on the latter
construct the semigroups in the title. As a special case, we
consider semigroups which are direct products of a combinatorial
inverse monoid and a group.

We introduce two Ramsey classes of finite relational structures. The first
class contains finite structures of the form $(A,(I_{i})_{i=1}^{n},\leq
,(\preceq _{i})_{i=1}^{n})$ where $\leq $ is a total ordering on $A$ and $%
\preceq _{i}$ is a linear ordering on the set $\{a\in A:I_{i}(a)\}$. The
second class contains structures of the form $(A,\leq
,(I_{i})_{i=1}^{n},\preceq )$ where $(A,\leq )$ is a weak ordering and $%
\preceq $ is a linear ordering on $A$ such that $A$ is partitioned by $%
\{a\in A:I_{i}(a)\}$ into maximal chains in the partial ordering $\leq $ and
each $\{a\in A:I_{i}(a)\}$ is an interval with respect to $\preceq $.

A well-known theorem of Schütte (1963) gives a sharp lower bound for
the ratio of the maximum and minimum distances between $n+2$ points in
$n$-dimensional Euclidean space.
In this note we adapt Bárány's elegant proof (1994) of this theorem to the space $\ell_4^n$.
This gives a new proof that the largest cardinality of an equilateral
set in $\ell_4^n$ is $n+1$, and gives a constructive bound for an
interval $(4-\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$.

Let $G$ be a finite group, $\mathcal F$ a class of groups.
Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre
of $G$ which is the product of all normal subgroups of $G$ whose
non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A
subgroup $H$ is called $\mathcal M$-supplemented in a finite group
$G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and
$H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$
of $H$. The main purpose of this paper is to prove: Let $E$ be a
normal subgroup of a group $G$. Suppose that every noncyclic
Sylow
subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that
$1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$
is
$\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal
U}{\Phi}}(G)$.

Let $\mathcal{A}$ be a line arrangement in the complex
projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one
local system $\mathcal{L}$ on $M$ is admissible if roughly speaking
the cohomology groups
$H^m(M,\mathcal{L})$ can be computed directly from the cohomology
algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient
condition for the admissibility of all rank one local systems on
$M$. As a result, we obtain some properties of the characteristic
variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.