Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.

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...

We investigate the recovery of $k$-sparse signals using the ${\ell }_1$-${\ell }_2$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$-sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G{\cong }^2{G}_2\left(3^{2n+1}\right)$ if and only if $𝒮\left(G\right)\cong 𝒮{(}^2{G}_2\left(3^{2n+1}\right))$. As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group ${}^2{G}_2\left(3^{2n+1}\right)$.

A $0/1$-simplex is the convex hull of $n+1$ affinely independent vertices of the unit $n$-cube $I^n$. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0/1$-simplices in $I^n$ can be represented by $0/1$-matrices $P$ of size $n\times n$ whose Gramians $G=P^{\top }P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{-\top }$ of $P$ is doubly stochastic and has the...

Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only...

For a fixed positive integer $k$ and $G=(V,E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in V$ has at least $k$ neighbors in $S$. The minimum size of a total $k$-dominating set for $G$ is called the total $k$-domination number of $G$, denoted by ${\gamma }_{kt}\left(G\right)$. The total $k$-domination problem is to determine a minimum total $k$-dominating set of $G$. Since the exact problem is in general quite difficult...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=\left[D_{ij}\right]$, where $D_{ii}$ is the $s\times s$ null matrix and for $i\ne j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................