5 problems

Let $q\ge2$, $k\ge2$, $\theta\ge1$, and $\alpha\in\mathbb{R}$. Let $\nu(\theta,\alpha)$ denote the number of positive solutions of the system of equations. Universal bound conjectu…

Let $q=3$, let $\theta\le\tilde{\theta}_1$, and let $\alpha=\alpha_1(\theta)$. Consider the system of equations in the variables $(u,v)$ given by. Uniqueness conjecture. The system…

Monotonicity conjecture. For each $m=1,\dots,q-2$, the function $\theta\mapsto\alpha_m(\theta)$ is monotone increasing. This would give the monotonicity observed in the plotted exa…

Monotonicity conjecture. The function $\alpha_-(\theta)$ is monotone decreasing for all $\theta\geq\theta_{\rm c}$, while $\alpha_+(\theta)$ is decreasing for…

Let $k\geq 4$ and consider the ferromagnetic model with critical values $\theta'_{\rm c}(k)$, $\lambda'_{\rm cr}(k)$, $\theta_{\rm c}(k)$, and $\lambda_{\rm cr}(k)$ as defined in t…