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If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$
acting transitively as a group of holomorphic transformations, then the action extends to an action of the
complexification $\widehat{G}$ of $G$ on $X$ except when
either the unit disk in the complex plane
or a strictly pseudoconcave homogeneous complex manifold is
the base or fiber of some homogeneous fibration of $X$.

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).

In this paper we study connections between topological games
such
as Rothberger, Menger and compact-open, and relate these games
to
properties involving covers by $G_\delta$ subsets. The results
include:
(1) If Two has a winning strategy in the Menger
game on a regular space $X$, then $X$ is an Alster space.
(2) If Two has a winning strategy in the Rothberger game on a
topological space $X$, then the $G_\delta$-topology on $X$ is
Lindelöf.
(3) The Menger game and the compact-open game are (consistently)
not
dual.

We describe a general setting where the monodromy action on the first
cohomology group of the Milnor fiber of a hyperplane arrangement is
the identity.

The precise condition on a completely regular space $X$ for every character on
$C(X) $ to be an evaluation at some point in $X$ is that $X$ be
realcompact. Usually, this classical result is obtained relying heavily on
involved (and even nonconstructive) extension arguments. This note provides a
direct proof that is accessible to a large audience.

It is known that the normalized standard generators of the free
orthogonal quantum group $O_N^+$ converge in distribution to a free
semicircular system as $N \to \infty$. In this note, we
substantially improve this convergence result by proving that, in
addition to distributional convergence, the operator norm of any
non-commutative polynomial in the normalized standard generators of
$O_N^+$ converges as $N \to \infty$ to the operator norm of the
corresponding non-commutative polynomial in a standard free
semicircular system. Analogous strong convergence results are obtained
for the generators of free unitary quantum groups. As applications of
these results, we obtain a matrix-coefficient version of our strong
convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm
equivalence for non-commutative polynomials in free semicircular
systems.

We classify integral modular categories of dimension $pq^4$ and $p^2q^2$,
where
$p$ and $q$ are distinct primes. We show that such categories are always
group-theoretical except for categories of dimension $4q^2$.
In these cases there are
well-known examples of non-group-theoretical categories, coming from
centers of
Tambara-Yamagami categories and quantum groups. We show that a
non-group-theoretical integral modular category of dimension $4q^2$ is
equivalent to either one of these well-known examples or is of dimension
$36$ and is twist-equivalent to fusion categories arising from a
certain quantum group.

We describe of all finite
dimensional uniserial representations of a commutative associative
(resp. abelian Lie) algebra over a perfect (resp. sufficiently
large perfect) field. In the Lie case the size of the field
depends on the answer to following question, considered and solved
in this paper. Let $K/F$ be a finite separable field extension
and
let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some
non-zero elements $\alpha,\beta\in F$?

We describe double Hurwitz numbers as intersection numbers on the
moduli space of curves $\overline{\mathcal{M}}_{g,n}$. Using a result on the
polynomiality of intersection numbers of psi classes with the Double
Ramification Cycle, our formula explains the polynomiality in chambers
of double Hurwitz numbers, and the wall crossing phenomenon in terms
of a variation of correction terms to the $\psi$ classes. We
interpret this as suggestive evidence for polynomiality of the Double
Ramification Cycle (which is only known in genera $0$ and $1$).

Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular
integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where
$\Omega$ is homogeneous of degree zero, integrable and has mean
value zero on the unit sphere $S^{n-1}$. In this paper, by Fourier
transform estimates and approximation to the operator $T_{\Omega}$
by integral operators with smooth kernels, it is proved that if
$b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain
minimal size condition, then the commutator generated by $b$ and
$T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for
appropriate index $p$. The associated maximal operator is also
considered.

Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb
R$.
In this article, as a generalization of the result of Albert
and Baker,
we investigate the behavior of bounded
and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality
$
\Bigl|f
\Bigl(\sum_{k=1}^n x_k
\Bigr)-\prod_{k=1}^n f(x_k)
\Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots,
x_n\in G,
$
where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional
version of the above inequality we consider the stability of
the functional equation
\begin{equation*}
u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0,
\end{equation*}
where $u$ is a Schwartz distribution or Gelfand hyperfunction,
$\circ$ and $\otimes$ are the pullback and tensor product of
distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots
+x_n$.

We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by
perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It
is the open region of $\mathbb{R}^3$ bounded by a cylinder with a
Randers metric. Using the Busemann-Hausdorff volume form, we
obtain the differential equation that characterizes the helicoidal
minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a
minimal surface in $\bar{M}^3$, only if the axis of the helicoid
is the axis of the cylinder. Moreover, we prove that, in the
Randers space $(\bar{M}^3, \bar{F})$, the only minimal
surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids
and the helicoids.

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.

Previous results by the author on the connection
between three of measures
of non-compactness obtained for $L_p$, are extended
to regular spaces of measurable
functions.
An example of advantage
in some cases one of them in comparison with another is given.
Geometric characteristics of regular spaces are determined.
New theorems for $(k,\beta)$-boundedness of partially additive
operators are proved.

We give some new characterizations for compactness of weighted
composition operators $uC_\varphi$ acting on Bloch-type spaces in
terms of the power of the components of $\varphi,$ where $\varphi$
is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus
generalizing the results obtained by Hyvärinen and
Lindström in 2012.

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 $(x,y,z)=(2,2,2)$.

The Theorem below is a correction to Theorem
3.5 in the article
entitled " Infinite Dimensional DeWitt Supergroups and Their
Bodies" published
in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part
(iii) of that Theorem
requires correction. The proof of Theorem 3.5 in the original
article failed to separate
the proof of (ii) from the proof of (iii). The proof of (ii)
is complete once it is established
that $ad_a$ is quasi-nilpotent for each $a$ since it immediately
follows that $K$
is quasi-nilpotent. The proof of (iii) is not complete
in the original article. The revision appears as the proof of
(iii) of the revised Theorem below.

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$.

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

We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of
$L(E)$ such that $L\cap K(E)$ is weak-star dense in $L$, then $L$ has a unique isometric predual. The proof relies on basic topological arguments.

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.

A formula is provided to
explicitly describe global dimensions of all kinds of tree type
finite dimensional $k-$algebras for $k$ an algebraic closed field.
In particular, it is pointed out that if the underlying tree type
quiver has $n$ vertices, then the maximum of possible global
dimensions is $n-1$.

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.

In this paper we give a characterization of a real hypersurface of
Type~$(A)$ in complex two-plane Grassmannians ${ { {G_2({\mathbb
C}^{m+2})} } }$, which means a
tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in
${G_2({\mathbb C}^{m+2})}$, by
the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.

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.

It is well known that a discrete group which is both amenable and
has Kazhdan's Property T must be finite. In this note we generalize
the above statement to the case of transformation groups. We show
that if $G$ is a discrete amenable group acting on a compact
Hausdorff space $X$, then the transformation group $C^*$-algebra
$C^*(X, G)$ has Property T if and only if both $X$ and $G$ are finite. Our
approach does not rely on the use of tracial states on $C^*(X, G)$.

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 common
approximate fixed point sequences of nonexpansive semigroups of
nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that
$T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space
$(M,d)$. In particular we prove that under suitable conditions, the
common approximate fixed point sequences set is the same as the common
approximate fixed point sequences set of two mappings from the family.
Then we use the Ishikawa iteration to construct a common approximate
fixed point sequence of nonexpansive semigroups of nonlinear
mappings.

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 study $L^p-L^r$ restriction estimates for
algebraic varieties $V$ in the case when restriction operators act on
radial functions in the finite field setting.
We show that if the varieties $V$ lie in odd dimensional vector
spaces over finite fields, then the conjectured restriction estimates
are possible for all radial test functions.
In addition, assuming that the varieties $V$ are defined in even
dimensional spaces and have few intersection points with the sphere
of zero radius, we also obtain the conjectured exponents for all
radial test functions.

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.

Let $f$ be a modular form which is non-ordinary at $p$. Loeffler has
recently constructed four two-variable $p$-adic $L$-functions
associated to $f$. In the case where $a_p=0$, he showed that, as in
the one-variable case, Pollack's plus and minus splitting applies to
these new objects. In this article, we show that such a splitting can
be generalised to the case where $a_p\ne0$ using Sprung's logarithmic
matrix.

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 ${\mathbb R}^n$.

Inspired by the invariant of a number field given by its zeta
function, we define the notion of weak arithmetic equivalence and show
that under certain ramification hypotheses, this equivalence
determines the local root numbers of the number field. This is
analogous to a result of Rohrlich on the local root numbers of a
rational elliptic curve. Additionally, we prove that for tame
non-totally real number fields, the integral trace form is invariant
under arithmetic equivalence.

For the Bergman projection operator $P$ we prove that
\begin{equation*}
\|P\colon L^1(B,d\lambda)\rightarrow B_1\| = \frac {(2n+1)!}{n!}.
\end{equation*}
Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of
$\mathbb{C}^n$, and $B_1$ denotes the Besov space with an adequate
semi--norm. We also consider a generalization of this result. This generalizes
some recent results due to Perälä.

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 $E\leq Z_{{\mathcal
U}{\Phi}}(G)$.

Extending the notion of parallelism we introduce the concept of
approximate parallelism in normed spaces and then substantially
restrict ourselves to the setting of Hilbert space operators endowed
with the operator norm. We present several characterizations of the
exact and approximate operator parallelism in the algebra
$\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a
Hilbert space $\mathscr{H}$. Among other things, we investigate the
relationship between approximate parallelism and norm of inner
derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the
parallel elements of a $C^*$-algebra by using states. Finally we
utilize the linking algebra to give some equivalence assertions
regarding parallel elements in a Hilbert $C^*$-module.

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.

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.

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 prove some results concerning covolutions, the
additive energy and sumsets of convex sets and its generalizations. In
particular, we show that if a set $A=\{a_1,\dots,a_n\}_\lt \subseteq
\mathbb R$ has
the property that for every fixed
$1\leqslant d\lt n,$ all differences $a_i-a_{i-d}$, $d\lt i\lt n,$ are distinct, then
$|A+A|\gg |A|^{3/2+c}$ for a constant $c\gt 0.$

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 $\mathcal{A}$ 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 $\mathcal{R}_1(M)$.

We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras
as measured by their densely defined lower semicontinuous traces.
We first prove that a unital simple $\mathrm{C}^*$-algebra whose
extreme tracial boundary is nonempty and finite contains positive
operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras
and establish criteria that imply that the Cuntz semigroup is
recovered functorially from the Murray-von Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by not-necessarily-unital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the first-named author, this shows that
slow dimension growth coincides with $\mathcal Z$-stability,
for approximately subhomogeneous algebras.

Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq
G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that
for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and
$G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$
or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$
is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup
$T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G =
AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient
conditions for a finite group $G$ to be $p$-nilpotent are given
whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of
$G$ are $m$-embedded for a given positive integer $k$.

In this paper we prove that there is only one conjugacy class of
dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral
symplectic group can be realized by an analytic automorphism
group
of compact connected Riemann surfaces of genus $p-1$. A pair of
representative generators of the realizable class is also given.

We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules.
Using this formula we give a combinatorial proof of the generalized
cyclic Eilenberg-Zilber theorem.