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It is a long-standing conjecture from the 1970s that every translation generalized quadrangle is linear, that is, has an endomorphism ring which is a division ring (or, in geometric terms, that has a projective representation). We show that any translation generalized quadrangle $Gamma$ is ideally embedded in a translation quadrangle which is linear. This allows us to weakly represent any such $Gamma$ in projective space, and moreover, to have a well-defined notion of characteristic for these objects. We then show that each translation quadrangle in positive characteristic indeed is linear.
The generalized Tur{a}n number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $F$ be a linear forest consisting of $k$ paths of orders $ell_1,ell_2,...,ell_k$. In this
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t
Let $mathcal{F}$ be a family of graphs. A graph $G$ is called textit{$mathcal{F}$-free} if for any $Fin mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $mathcal{F}$, the generalized Tur{a}n number
The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=mathbf{F}[x_n,...,x_1]$, a polyonomial ring over the finite field $mathbf{F}:=mathbf{Z}_q$ of $q$ elements. Here
Using $e^{+}e^{-}$ collision data samples with center-of-mass energies ranging from 2.000 to 2.644 GeV, collected by the BESIII detector at the BEPCII collider, and with a total integrated luminosity of 300 pb^{-1}, a partial-wave analysis is perform