We calculate all ${{\ell }^{2}}$-Betti numbers of the universal discrete Kac quantum groups $\widehat{\text{U}}_{n}^{+}$ as well as their half-liberated counterparts $\widehat{\text{U}}_{n}^{*}$.

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 non-positive curvature metric on the positive cone of a finite von Neumann algebra.

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 $(Mu{)}'(t)\,=\,Au(t)\,+\,F{{u}_{t}}\,+\,f(t)$ 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 $D(A)\,\cap \,D(M)\,\ne \,\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r,0];X)$ to $X$, and $r\,>\,0$ 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 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard 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 NP-hard, while the other can be solved in polynomial time.

Let $X$ be a $\text{CW}$ 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 quasi-projective $\text{M}$-varieties are again $\text{M}$-varieties. This generalizes a result by Biswas and D’Mello about symmetric products of $\text{M}$-curves. We also discuss several related questions.

For an analytic curve $\gamma :\,\left( a,\,b \right)\,\to \,\mathbb{C}$, the set of values approached by $\gamma \left( t \right)$, as $t\,\searrow \,\,a$ and as $t\,\nearrow \,b$ can be any two continua of $\mathbb{C}\,\cup \,\left\{ \infty \right\}$.

We showbymeans of an example in ${{\mathbb{C}}^{3}}$ that Gromov’s theoremon the presence of attached holomorphic discs for compact Lagrangianmanifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e., polynomial convexity.

The classical result of Nevanlinna states that two nonconstantmeromorphic 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.

Motivated by a question of A. Skalski and P. M. Sołtan (2016) about inner faithfulness of 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 group-theoretic properties of the quantum permutation group on points. This enables us not only to answer the aforementioned question in the positive for the case where $n\,=\,4,\,k\,=\,2$, 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 $n\,>\,1$ is a positive integer. We characterize $a\,\in \,K$, the Lie algebra of skew elements in $R$, satisfying ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on $K$. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if $a,\,b\,\in \,R$ satisfy ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$, where either $n$ is even or $b\,=\,0$, then ${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some $\lambda \,\in \,C$.

We investigate the moduli space of sheaves supported on space curves of degree 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 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.

In this paper, we consider the following critical Kirchhoff type equation:

$$\left\{ _{u\,=\,0,\,\,\,\,\,\,\text{on}\,\partial \Omega \text{,}}^{-(a\,+\,b{{\int }_{\Omega }}|\nabla u{{|}^{2}})\Delta u\,=\,\text{Q(}x)|u{{|}^{4}}u\,+\,\lambda |u{{|}^{q-1}}u,\,\,\,\text{in}\,\Omega \text{,}} \right.$$

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 $3\,<\,q\,<\,5$. The relation between the number of maxima of $\text{Q}$ and the number of positive solutions for the problem is also investigated.

For $m,n\,\in \,\mathbb{N},\,1\,<\,m\le \,n$, we write $n={{n}_{1}}+\cdots +{{n}_{m}}$ where $\{{{n}_{1}},\ldots \,,{{n}_{m}}\}\,\subset \,\mathbb{N}$. Let ${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$ be $n\,\times \,n$ singular real matrices such that

$$\underset{i=1\,}{\overset{m}{\mathop{\oplus }}}\,\underset{1\le j\ne i\le m}{\mathop \bigcap }\,\,{{N}_{j}}\,=\,{{\mathbb{R}}^{n}},$$

where ${{N}_{j}}\,=\,\{x\,:\,{{A}_{j}}x\,=\,0\},\,\text{dim(}{{N}_{j}}\text{)}\,=\,n\,-\,{{n}_{j}}$, and ${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$ is invertible. In this paper we study integral operators of the form

$${{T}_{r}}f(x)\,=\,{{\int }_{{{\mathbb{R}}^{n}}}}|x\,-\,{{A}_{1y}}{{|}^{-{{n}_{1}}+{{\alpha }_{{{1}_{\ldots }}}}}}|x\,-\,{{A}_{m}}y{{|}^{-{{n}_{m}}+{{\alpha }_{m}}}}f(y)dy,$$

${{n}_{1}}\,+\,\cdots \,+\,{{n}_{m}}\,=\,n$ , $\frac{{{\alpha }_{1}}}{{{n}_{1}}}\,=\,\cdots \,=\,\frac{{{\alpha }_{m}}}{{{n}_{m}}}\,=\,r$, $0\,<\,r\,<\,1$, and the matrices ${{A}_{i}}\text{ }\!\!'\!\!\text{ s}$ are as above. We obtain the ${{H}^{p}}({{\mathbb{R}}^{n}})\,-\,{{L}^{q}}({{\mathbb{R}}^{n}})$ boundedness of ${{T}_{r}}$ for $0\,<\,p\,<\,\frac{1}{r}$ 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 half-plane 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 ${{T}_{m}}:H_{A}^{p}({{\mathbb{R}}^{n}})\,\to \,H_{A}^{p}({{\mathbb{R}}^{n}})$, 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 3-dimensional hyperbolic Coxeter polyhedra with at least one dihedral angle of the form $\frac{\pi }{k}$ for an integer $k\ge 7$. Combining a classical result by Parry with a previous result of ours, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers.

In this paper, we study a two-component Lotka–Volterra competition systemon a one-dimensional spatial lattice. By 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\,\to \,-\,\infty $, where $j\,\in \,\mathbb{Z}$, $t\,>\,0$, 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

$$\left\{ \begin{align} & -\Delta u=0\,\,\,\,\,\,\,\,\,\text{in}\,\,\,\mathbb{R}_{+}^{2}, \\ & \frac{\partial u}{\partial t}=-c{{\left| x \right|}^{\beta }}{{e}^{u}}\,\,\,\text{on}\,\,\partial \mathbb{R}_{+}^{2}\backslash \left\{ 0 \right\}, \\ \end{align} \right.$$

with the finite conditions

$${{\int }_{\partial \mathbb{R}_{+}^{2}}}|x{{|}^{\beta }}{{e}^{u}}\,ds\,<\,C,\,\,\,\,\frac{\sup }{\mathbb{R}_{+}^{2}}\,u(x)\,<\,C.$$

Here $c$ is a positive number and $\beta \,>\,-1$.