We study the presence of copies of $l_p^n$’s uniformly in the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$. By using Dvoretzky’s theorem we deduce that if $X$ is an infinite-dimensional Banach space, then ${\Pi }_2(C[0,1],X)$ contains $\lambda \sqrt{2}$-uniformly copies of $l_{\infty }^n$’s and ${\Pi }_1(C[0,1],X)$ contains $\lambda$-uniformly copies of $l_2^n$’s for all $\lambda >1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$ are distinct, extending the well-known result that the spaces ${\Pi }_2(C[0,1],X)$ and $𝒩\left(C\right[0,1],X)$ are distinct.

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if for any chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi (HK/K\cap L/K)$-number. We obtain some criteria for the $p$-supersolubility or $p$-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $\Pi$-property.

A nonincreasing sequence $\pi =(d_1,...,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. Given two graphs $G_1$ and $G_2$, A. Busch et al. (2014) introduced the potential-Ramsey number of $G_1$ and $G_2$, denoted by $r_{\mathrm{pot}}(G_1,G_2)$, as the smallest nonnegative integer $m$ such that for every $m$-term graphic sequence $\pi$, there is a realization $G$ of $\pi$ with $G_1\subseteq G$ or with $G_2\subseteq \overline{G}$, where $\overline{G}$ is the complement of $G$. For $t\ge 2$ and $0\le k\le \lfloor \frac{t}{2}\rfloor$, let $K_t^{-k}$ be the graph...

The Golomb space ${\mathbb{N}}_{\tau }$ is the set $\mathbb{N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb{N}}_{\tau }$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of ${\mathbb{N}}_{\tau }$, and each homeomorphism $h$ of ${\mathbb{N}}_{\tau }$ has the following properties: $h\left(1\right)=1$, $h\left(\Pi \right)=\Pi$, ${\Pi }_{h\left(x\right)}=h\left({\Pi }_x\right)$, and $h\left(x^{\mathbb{N}}\right)=h{\left(x\right)}^{\mathbb{N}}$ for all $x\in \mathbb{N}$. Here $x^{\mathbb{N}}:=\{x^n:n\in \mathbb{N}\}$ and ${\Pi }_x$ denotes the set of...

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if for any chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi (HK/K\cap L/K)$-number. We study the influence of some $p$-subgroups of $G$ satisfying the $\Pi$-property on the structure of $G$, and generalize some known results.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...