Every incidence structure $𝒥$ (understood as a triple of sets $(G,M,I)$, $I\subseteq G\times M$) admits for every positive integer $p$ an incidence structure ${𝒥}^p=(G^p,M^p,{\mathrm{I}}^p)$ where $G^p$ ($M^p$) consists of all independent $p$-element subsets in $G$ ($M$) and ${\mathrm{I}}^p$ is determined by some bijections. In the paper such incidence structures $𝒥$ are investigated the ${𝒥}^p$’s of which have their incidence graphs of the simple join form. Some concrete illustrations are included with small sets $G$ and $M$.

In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in ${𝐑}^n$, is generalized. Finally we prove the following theorem: If $\{A_1,A_2,\cdots ,A_{n+1}\}$ is a family of compact convexly connected sets in general position in ${𝐑}^n$, then for each proper subset $I$ of $\{1,2,\cdots ,n+1\}$ the set of hyperplanes separating $\cup \{A_i:i\in I\}$ and $\cup \{A_j:j\in \overline{I}\}$ is homeomorphic to $S_n^{+}$.

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left(\begin{array}{cc}\sigma & 0\\ 0& {\sigma }^{-1}\end{array}\right)$ for $\sigma \>0$. Let $0\<{\zeta }_1\<1\<{\zeta }_2\<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in W^{2,1}\left(Q_1\left(0\right)\right)$ be a ${\mathrm{C}}^1$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)\<{\zeta }_2$, $\mathrm{Lip}\left(u^{-1}\right)\<{\zeta }_1^{-1}$. There exists positive constants ${𝔠}_1\<1$ and ${𝔠}_2\>1$ depending only on $\sigma$, ${\zeta }_1$, ${\zeta }_2$ such that if $ϵ\in \left(0,{𝔠}_1\right)$ and $u$...

For a wedge $W$ of ${\mathbb{C}}^N$, we introduce two conditions of weak $q$-pseudoconvexity, and prove that they entail solvability of the $\overline{\delta }$-system for forms of degree $\ge q+1$ with coefficients in $C^{\mathrm{\infty }}\left(W\right)$ and $C^{\mathrm{\infty }}\left(\overline{W}\right)$ respectively. Existence and regularity for $\overline{\delta }$ in $W$ is treated by Hörmander [5, 6] (and also by Zampieri [9, 11] in case of piecewise smooth boundaries). Regularity in $W$ is treated by Henkin [4] (strong $q$-pseudoconvexity by the method of the integral representation), Dufresnoy [3] (full pseudoconvexity), Michel...

We investigate the extremal function $f(u,n)$ which, for a given finite sequence $u$ over $k$ symbols, is defined as the maximum length $m$ of a sequence $v=a_1{a}_2...a_m$ of integers such that 1) $1\le a_i\le n$, 2) $a_i=a_j,i\ne j$ implies $|i-j|\ge k$ and 3) $v$ contains no subsequence of the type $u$. We prove that $f(u,n)$ is very near to be linear in $n$ for any fixed $u$ of length greater than 4, namely that $$f(u,n)=O\left(n{2}^{O\left(\alpha {\left(n\right)}^{\left|u\right|-4}\right)}\right).$$ Here $\left|u\right|$ is the length of $u$ and $\alpha \left(n\right)$ is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and...