43 problems

Let $X$ be the Banach space underlying a linear discrete-time system, and let $A_m^k$ denote its associated evolution operator from time $k$ to time $m$. The system is uniformly ex…

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…

Let ${\mathfrak C}_i\subset{\mathfrak C}_{gen}$ be a connectivity component of the critical manifold, and let $a\in{\mathfrak C}_i$. Suppose the nonzero spectrum of the vertical li…

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

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…

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

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…

A countable signature $\sigma$ and a class $\mathscr C$ of $\sigma$-structures are given. A unary expansion of $\mathscr C$ is obtained by adding unary relation symbols. Monadic NI…

Strong modeling limit conjecture. Every FO-convergent sequence of graphs in $\mathscr C$ has a strong modeling FO-limit. The paper proves existence of modeling FO-limits for monadi…

Superstability conjecture. An Artin group is superstable if and only if it is abelian.

Finite-level non-structure conjecture. The theory $T$ has $2^\kappa$ models of cardinality $\mu$ that are pairwise non-isomorphic over $P$.

Higher instability conjecture. The theory $T$ has $2^\lambda$ models of cardinality $\mu$ that are pairwise non-isomorphic over $P$.

Generalized Gaifman conjecture. The theory $T$ has $2^\lambda$ models of cardinality $\mu$ that are pairwise non-isomorphic over $P$.

Assume that $r\kappa\mu_1\neq v\sigma$ and that the $(\operatorname{Coex}_{2,3})$-condition holds, which in particular implies $mv>r+v$. Let $(n_{1a}^*,n_{1i}^*,n_3^*)$ and…

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 $T$ be a countable complete first-order theory with a distinguished predicate $P$. Say that $T$ is categorical over $P$ when it has at most one model, up to isomorphism over…

Nonlinear stability conjecture. Under these assumptions, the modulated traveling wave is exponentially and asymptotically orbitally stable.

Let $G$ be a reaction network containing the motif … where $Y$ appears in no other complex. Let $p_{(h,\ell)}$ and $p'_{(h,\ell')}$ be the reduced characteristic polynomials of $G$…

Average-dwell-time stability equivalence conjecture. For every $\tau>0$, System $$ is $\tau\text{-UGB}$ if and only if it is UGAS with respect to $\mathcal{S}_{\mathrm{adw}}(\tau,N…

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…

Let the autonomous time-delay system be given on the history space $\mathcal X^n$, with solution $x(t,x_0)$ from initial history $x_0$. GAS conjecture. The system is GAS if and onl…

Let $f$ be a Filippov system with boundary equilibrium $x^*={\bf 0}$, and let $F$ be its reduced system. Assume the setting and hypotheses of the paper, including $c_1<0$. Lyapunov…

Let $\beta(\theta)$ be defined for $\theta\in(0,1)$ by … For a given $\theta$, let $q(\theta)>0$ be the quantity specified by the slope-direction formula. Popov criterion conjectur…