9 problems

Generic constancy conjecture. Suppose that $\Pc_{\mathrm{in}}^-$ and $\Pc_{\mathrm{state}}^-$ are Polish and that…

Let $\rho$ be a positive integer, let $\bar{\lambda}$ be the average parameter value, and let $p_i(\bar{\lambda})$ for $1\leq i\leq 2^\rho$ denote the stable period-$2^\rho$ points…

Let $\theta$ denote the phase on the circle, let $\theta_{1}$ and $\theta_{2}$ be the average phases of the two communities, and write their phase difference as … In the disorder-f…

Consider the random dynamical system induced by the stochastic differential equation in the paper, and fix $a>0$ and $\beta\in\mathbb{R}$. Let $\lambda_{\rm top}$ denote its top Ly…

Consider a stochastically forced version of the full Gor'kov–Eliashberg equations … For each $\varepsilon>0$, let the forcing be strong enough that the resulting system admits an i…

Fixed-point conjecture. The map $\mathcal S$ has at least one fixed point in $(0,\tilde y)$.

Consider the structured stochastic difference equation … is almost surely persistent and persistent in probability. A theorem establishes this conclusion under the stronger monoton…

Let … is persistent in probability. The cited theorem proves almost-sure persistence when every invariant boundary measure has a species with positive invasion rate, equivalently u…

Consider the two-species Markov chain … is almost surely persistent. Mutual invasibility is known to imply persistence in probability under the monotonicity assumptions; the conjec…