28 problems

Let $A^*$ be the interconnection-strength parameter, let $d$ be the transmission delay, and let the values in Table be the numerically identified critical delays at which purely im…

Let $g$ be a smooth nonlinearity satisfying hypothesis $(H)$: $g\in C(\mathbb{R}_+,\mathbb{R}_+)$ has one local maximum, $g(0)=0$, $g(x)>0$ for $x>0$, exactly two fixed points $0$…

Consider the delayed reaction-diffusion equation … Assume that $g:\mathbb{R}_+\to\mathbb{R}_+$ has exactly two fixed points, $0$ and $\kappa>0$, is unimodal, and has negative Schwa…

Slow-vector-field conjecture. For the delay equation in question, and more generally for families of delay equations in which the delay is treated as a small parameter, the slow ve…

The delayed forward stochastic control problem involves a controlled forward system with state delays, while problem (O) is the corresponding control problem considered in the pape…

Let $S$ be a general firing coefficient, let $d=0$ denote instantaneous transmission, and let the elapsed-time model include an absolute refractory period. Let the stability criter…

Two-threshold stability conjecture. When

Let $A^*$ be the interconnection-strength parameter of an equilibrium, and let its basin of attraction be the set of initial data whose solutions converge to that equilibrium. Basi…

Let $A^*$ denote the interconnection-strength parameter associated with an equilibrium of the elapsed-time model, and let $d$ be the transmission delay. Asymptotic-stability conjec…

Let the delay kernel $\alpha(t)$ converge to the Dirac mass $\delta(t-d)$ in the sense of distributions. Delayed-limit convergence conjecture. Solutions of the delayed equations sh…

Let $\alpha(t)$ be a delay kernel converging to the Dirac mass $\delta(t)$, and let $X(t)$ be the total activity of the delayed equation while $N(t)$ is the discharging flux of the…

Let $\alpha(t)$ converge to $\delta(t-d)$ in the sense of distributions, where $d$ is the delay, and let $X(t)$ be the total activity in the delayed equation. Delayed branch-select…

Let the instantaneous transmission model have multiple local solution branches for the same initial condition, corresponding to multiple possible values of $N(0)$. Let the delay ke…

Consider the delayed elapsed-time model with refractory period $\sigma$ and delay kernel $\alpha$. Let $N(t)$ denote the discharging flux and $X(t)$ the total activity. Refractory-…

Let $N(t)$ be the discharging flux and $X(t)$ the total activity in the delayed equation, with delay parameter $d$. Periodicity conjecture. In the excitatory example, both $N$ and…

Let $N(t)$ be the discharging flux and $X(t)$ the total activity in the delayed equation, with delay parameter $d$. Periodic-profile conjecture. Unlike the instantaneous model, the…

Stability-domain conjecture. Having $(b,a)$ values in the stable domain in the $b$-$a$ plane is a necessary and sufficient condition for the stability of nonzero fixed points of no…

Consider the linear difference delay system with maximal delay parameter $\Lambda_N$, dimension $d$, and a single input, so that $m=1$. Assume that controllability holds and that t…

Let $\Lambda_N>0$, $d\in\mathbb{N}$, $q\in[1,+\infty)$, and consider the linear difference delay system with delay parameters and matrices $A_N$ and $B$ described above. Let…

Consider the elapsed-time neural-network system with firing rate $p(s,u)=\varphi(u)\mathds{1}_{\{s>\sigma\}}$ in the strongly excitatory case, and consider the monotone solutions c…

Consider the elapsed-time neural-network system with firing rate $p(s,u)=\varphi(u)\mathds{1}_{\{s>\sigma\}}$, where $\sigma>0$, and suppose the network is inhibitory or weakly exc…

Let $p$ be a general firing rate in the elapsed-time neural-network system, with inhibitory and excitatory cases distinguished according to the effect of activity on the firing rat…

Uniqueness conjecture for sharp traveling waves. The sharp type traveling wave is unique, and the corresponding wave speed satisfies

Let $\varphi(n,\omega,\eta_i)$ denote the evolution at time $n$ of linearly independent vectors $\eta_1,\ldots,\eta_k$, and let $\|\cdot\|_{M_2}$ be the Hilbert-space norm on the…

All-time positivity conjecture. For every feedback gain satisfying