7 problems

Let $m$ points lie on a line and let the remaining $n=N-m$ points form a regular $n$-gon in a plane perpendicular to the line, with centroid on the line. Let…

Let an alternating sign matrix of order $n$ be represented by its colored square-ice graph, and let the $2n$ blue endpoints be numbered cyclically. The pairing of an alternating si…

Consider the stationary localized planar Swift–Hohenberg equation … for $u=u(x)$ with $x\in\mathbb{R}^2$, $\lim_{|x|\to\infty}u(x)=0$, $\mu>0$, and $(\nu_1,\nu_2)\in\mathbb{R}^2$.…

Let $B\subset\mathbb{R}^2$ be a disk and let $P\subset B$ be an inner domain with dihedral symmetry of order $n$, where $n\geq 2$ is odd. Consider $\Omega=B\setminus\overline{P}$ a…

Let $S_{n,d}(q,t)$ be the $q,t$-Schröder number and let $\tilde{S}_{n,d}(q,t)$ be the little $q,t$-Schröder number. For odd $n$, let $X$ be the set of dissections of a regular…

Let $n$ be the polygon-order parameter, let the cases be those listed in Table, and let $\eta_{\min}$ and the edge configurations be the quantities specified in Tables--. A rhombic…

Let $n$ be odd with $n\geq5$, and consider a CAST as described in the generalized Penrose CAST conjecture, with its prototiles and substitution rules. A CAST has individual dihedra…