117 problems

Let the saturated linear feedback be the system defined by and, and let Proposition give sufficient conditions for its global asymptotic stability (GAS). Necessary-and-sufficient-c…

Let $X=(P_1,\dots,P_n)$ be an autonomous $C^1$-vector field on $\mathbb{R}^n$ with a unique singularity at the origin. For each $x\in\mathbb{R}^n$, let $DX(x)$ denote its Jacobian…

Let $\chi$ be the fissility parameter and let $\chi_*$ be the bifurcation value. The Ren–Wei branch is the equilibrium branch emerging at this bifurcation. Subcritical pitchfork co…

Unique stable BD profile conjecture. For $K>4$, the $\mathbf{P}^b$-$BD$-$\mathbf{P}^t$ configuration is the unique stable mixed three-dimensional critical point on $V_K$.

NSOP$_1$ preservation conjecture. If $\operatorname{Th}(C)$ is NSOP$_1$, then $\operatorname{Th}(G(C))$ is also NSOP$_1$.

Baldwin's conjecture. No generic with a small theory will be strictly superstable.

Homogeneous Lyapunov converse conjecture. For homogeneous vector fields, asymptotic stability implies existence of a polynomial Lyapunov function. Equivalently, existence of a homo…

Let $T$ be a theory. Consider the class of cardinals \\{\lambda: T has a rigid model of cardinality \lambda\\}. Ehrenfeucht's conjecture. This class of cardinals should be quit…

Stable-tree decomposition conjecture. If $\langle K,\prec_K\rangle$ satisfies $P_1,\ldots,P_n$, then for every sufficiently large $M\in K$ there exist a finite tree…

Let the biased IMEX scheme be … Here $\lambda$ is the explicit eigenvalue, $\mu$ is the implicit eigenvalue, and $\mathcal{S}$ is the explicit stability domain. The notation…

Let the biased IMEX scheme be … Here $\lambda$ is the explicit eigenvalue, $\mu$ is the implicit eigenvalue, and $\mathcal{S}$ is the explicit stability domain. A-stability conject…

Consider the closed-loop system … Let $x_2$ converge to a constant and let $\omega(t,x_2)$ denote the periodic steady-state value approached by $u(t,x)$. Assume that…

Linear stability conjecture. The same result should hold for generic polyhedral patchy vector fields on $\mathbb R^n$: impulsive perturbations should generically produce a perturba…

Triangular stability classification conjecture. If $0<\mu<\mu_R$, then $\operatorname{L}_4$ and $\operatorname{L}_5$ are elliptic for every $\kappa>0$ for which they exist. If…

Collinear stability classification conjecture. $\operatorname{L}_1,\operatorname{L}_2,\operatorname{E}_3$, and $\operatorname{A}_1$ are center-saddles whenever they exist;…

A theory is countably categorical when it has, up to isomorphism, exactly one countable model, and it is stable when it has the model-theoretic stability property. Superstability c…

Let $n$ denote the bacterial population state and let $\theta,b7,d,b$ be the resource supply rate, dilution rate, death rate, and growth-rate coefficient, respectively. Assume that…

Let $D\subseteq\Omega$ be an open set with continuous boundary satisfying the geometric conditions in the semi-uniform stability setup of ${}^{\text{su}}$, and let $\1_D$ denote mu…

Let $T$ be a countable weakly quasi-o-minimal theory with few countable models. Let $\mathcal P$ be the collection of all global $1$-types that are invariant over some finite param…

Stable aleph-null-categorical superstable conjecture. Every stable $\aleph_0$-categorical theory is superstable.

Boney–VanDieren's interval conjecture. The set of regular cardinals $\mu$ such that the $(\lambda,\mu)$-limit model is isomorphic to the $(\lambda,\lambda)$-limit model might be an…

For $g=\sigma$, define the D'Arcais polynomials $P_n^\sigma(X)$ by … and set $H_n^\sigma(X):=P_n^\sigma(X)/X$. A polynomial $f(X)$ is Hurwitz if $f(x)=0$ implies $\Re(x)<0$. Neuhau…

Halanay–Smith criterion. Let $M\in\mathcal G$. The following statements are equivalent:

Let $A_0,A_1\in M_d(\mathbb{R})$ be Metzler matrices defining a linear switched system. A counterexample to the question of whether exponential convergence to zero for all trajecto…

A $D$-unstable core is a $D$-unstable Child-Selection matrix $\mathbb{S}[\pmb{\kappa}]$ such that none of its principal submatrices are $D$-unstable. A network is said to admit ins…