A large family of cospectral Cayley graphs over dicyclic groups

For a finite group G and an inverse closed subset $S\subseteq G\setminus \{e\}$, the Cayley graph $X(G,S)$ has vertex set G and two vertices $x,y\in G$ are adjacent if and only if $x{y}^{-1}\in S$. Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let $p\ge 3$ be a prime number and $T_{4p}$ be the dicyclic group of order 4p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group $T_{4p}$ with $p\ge 23$, and find several pairs of non-isomorphic and cospectral Cayley graphs for $5\le p\le 19$.     

Symmetric designs and four dimensional projective special unitary groups

In this article, we study symmetric $(v,k,\lambda )$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is ${\mathrm{PSU}}_4\left(q\right)$. We prove that there exist eight non-isomorphic such designs for which $\lambda \in \{3,6,18\}$ and $G$ is either ${\mathrm{PSU}}_4\left(2\right)$, or ${\mathrm{PSU}}_4\left(2\right):2$.     

On a Cheeger type inequality in Cayley graphs of finite groups

Let $G$ be a finite group. It was remarked in Breuillard et al. (2015) that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h{\left(\mathbb{G}\right)}^4}{\gamma },1-\frac{h{\left(\mathbb{G}\right)}^2}{2d^2}\right]$, where $h\left(\mathbb{G}\right)$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma =2^9{d}^6{(d+1)}^2$.     

Energy non-collapsing and refined blowup for a semilinear heat equation

Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation

$u_t?△u=|u{|}^{p?1}u$

with $p>1$ are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be refined type II in critical case. When $p>p_S\equiv \frac{N+2}{N?2}$ for $N\ge 3$, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than $N?2?\frac{4}{p?1}$, which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collapsing blowups.     

Point source equilibrium problems with connections to weighted quadrature domains

We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q\left(z\right)={\left|z\right|}^{2p}$ with $p\in \mathbb{N}$. Along the way, we describe some quadrature domains with respect to weighted area measure ${\left|z\right|}^{2p}d{A}_z$ and complex boundary measure ${\left|z\right|}^{?2p}dz$.     

The (p,q) property in families of d-intervals and d-trees

Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$ property can be pierced by $O\left(d^{\frac{q}{q-1}}\right)$ points, with constants depending only on $p$ and $q$. This extends results of Tardos, Kaiser and Alon for the case $q=2$, and of Kaiser and Rabinovich for the case $p=q=\lceil lo{g}_2(d+2)\rceil$. We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consisting of at most $d$ connected components, extending results of Alon for the case $q=2$. Finally, we prove an upper bound of $O\left(d^{\frac{1}{p-1}}\right)$ on the fractional piercing number in families of $d$-intervals satisfying the $(p,p)$ property, and show that this bound is asymptotically sharp.     

Realization of PBW-deformations of type An quantum groups via multiple Ore extensions

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation ${\mathfrak{B}}_q(\mathfrak{sl}(n+1,\mathbb{C}))$ of type ${\mathbb{A}}_n$ quantum group, we explicitly obtain the commutation relations of its root vectors, then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type $(1,2,?,n)$. Moreover, we analyze the quantum algebras ${\mathfrak{B}}_q(\mathfrak{g})$ with $\mathfrak{g}$ of type ${\mathbb{D}}_4$, ${\mathbb{B}}_2$ and ${\mathbb{G}}_2$ and give some examples and counterexamples that can be realized by a ladder Ore extension.     

Sobolev spaces on Lie groups: Embedding theorems and algebra properties

Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure ${\mu }_{\chi }$ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces $L_{\alpha }^p({\mu }_{\chi })$ adapted to X and ${\mu }_{\chi }$ ($1<p<\infty$, $\alpha \ge 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.     

An existence result for anisotropic quasilinear problems

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p$-Laplacian operator with a coefficient that is locally bounded inside the domain and satisfying certain additional integrability assumptions. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value problem with a non-linearity $f\left(u\right)$ satisfying $f\left(0\right)\le 0$ and having $(p-1)$-sublinear growth at infinity.     

Spanning trees with at most 4 leaves in K1,5-free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4\left(G\right)\ge n?1$ contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5\left(G\right)\ge n?1$ contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “${\sigma }_5\left(G\right)\ge n?1$” is best possible.     

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space ${\mathsf{h}}_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of ${\mathsf{h}}_p$ for all $0<p<1$, and provide a constructive proof of the atomic decomposition for $p=1$ which resolves a main problem on the subject left open for the last twelve years. We also study ${(p,\infty )}_c$-atoms, and show that every ${(p,2)}_c$-atom can be decomposed into a sum of ${(p,\infty )}_c$-atoms; consequently, for every $0<p\le 1$, the ${(p,q)}_c$-atoms lead to the same atomic space for all $2\le q\le \infty$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space ${\mathsf{h}}_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.