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 a short proof of a result of T. Bates
and T. Giordano stating that any uniformly bounded Borel cocycle
into a finite von Neumann algebra is cohomologous to a unitary
cocycle. We also point out a separability issue in
their proof. Our approach is based on the existence of a nonpositive
curvature metric on the positive cone of a finite von Neumann
algebra.
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.
In this paper we consider two natural notions of connectivity
for hypergraphs: weak and strong. We prove that the strong
vertex connectivity of a connected hypergraph is bounded by its
weak edge connectivity, thereby extending a theorem of Whitney
from graphs to hypergraphs. We find that while determining a
minimum weak vertex cut can be done in polynomial time and is
equivalent to finding a minimum vertex cut in the 2section of
the hypergraph in question, determining a minimum strong vertex
cut is NPhard for general hypergraphs. Moreover, the problem
of finding minimum strong vertex cuts remains NPhard when restricted
to hypergraphs with maximum edge size at most 3. We also discuss
the relationship between strong vertex connectivity and the
minimum
transversal problem for hypergraphs, showing that there are
classes
of hypergraphs for which one of the problems is NPhard while
the other can be solved in polynomial time.
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.
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 $R$ 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 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.
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.
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.
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 $\frac{1}{q}=\frac{1}{p} 
r$.
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.
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 $\beta \gt 1$.