43 problems

Let $p$ denote the quantity characterizing the phase-chaotic states, and let the SN line be the saddle-node stability boundary for modulated amplitude waves (MAWs) of spatial perio…

Asymptotic stability conjecture. The entire region $a<0$ and $a<b<-a$ is asymptotically stable for every $q\in(0,1)$.

Let the conditions of the stability theorem hold, but now for block sizes $n_i$ that are not necessarily equal. Define the mean block size and variance by … If the relative standar…

There exist a time $T>0$ and smooth functions … with smooth initial data … for smooth $f_1,f_2,f_3$ and a smooth even angular profile $\phi(\theta)$ adapted to…

Optimal-decay conjecture. The estimate obtained in the paper is optimal; equivalently, the stability conditions should not in general be extendable to the scales…

Self-adjoint spectral stability conjecture. The stationary state $u^\star$ is locally asymptotically stable, and convergence to $u^\star$ occurs in the topology of $H$.

Let $X(-\infty,0)$ and $X(-\infty,T)$ denote the trajectory-history spaces at times $0$ and $T$, respectively, for a linear time-periodic fractional-order differential equation, an…

Saddle-point conjecture. The fixed point $E_2$ is always a saddle point if it exists, that is,

Let $n$ denote the number of modulated units, and consider the portion of the U-shaped region in the $(\Omega_m,\phi)$ stability diagrams described above. Persistence conjecture. T…

Buvoli's conjecture. As the approximation order tends to infinity, the absolute stability regions should tend to

Consider the semi-discrete approximation scheme proposed in the previous work, with parameters satisfying the strict dissipativity condition … Let the SKC condition be fulfilled wh…

Symmetric configuration conjecture. For a given value of $d$, if a symmetric $K$-spike configuration loses its stability, then all $K$-spike configurations are likely to be unstabl…

Instability conjecture. The regions bounded by the curves $Gamma_1$ and $Gamma_2$ are unstable.

The HLLCM and HLLEC schemes are approximate Riemann solvers, and a shear velocity perturbation is a perturbation in the velocity component tangential to the shock or grid-aligned n…

An upwind numerical scheme is strictly stable when its perturbations decay rather than merely remaining bounded, and it exactly resolves contact discontinuities when those disconti…

Let $V$ be the cuboid under consideration, and let the BD-BD$_2$-BD$_2$ critical point be a critical point of the total energy whose profile is uniaxial near the center of $V$ and…

Let … be the square in the $z$-$w$ plane, and let … be the stability region, where $P_i(z,w)$ are the stability polynomials of the numerical method. Stability-region conjecture. Al…

Let $u$ solve the one-dimensional diffusion equation studied in the paper, and let $U$ be a monotone similarity profile. The linearization around $u=U$ is … with eigenvalues…

Let $R(z_0,z_1,z_2,0)$ be the stability function of the MCS scheme, and let $\theta=\tfrac{1}{3}$. For complex numbers $z_0,z_1,z_2$ satisfying … interpret $\gamma$ as a bound on t…

Let horizon-dependent bounds $(\gamma_k)$ be used in the linear program defining the suboptimality factor $\alpha_N$, and let $\hat{\alpha}_N$ denote the analytical bound introduce…

The velocity gradient in turbulent incompressible flow is decomposed into strain, shear, and rigid-body rotational components. At macroscopic scales, vortex tubes and shear pancake…

Let $F(u)=1-\cos(u)$, so that the variable-coefficient sine-Gordon equation is a perturbation of … and let the coefficients satisfy the conditions denoted by. The kink is the stati…

Consider the constant-coefficient $phi^4$ equation … with nonlinearity $F'(u)=u^3-u$, and its kink solution. Asymptotic stability means that solutions starting near the kink in the…

Odd–even stability conjecture. For BVP-dominated problems with negative real eigenvalues, high-order predictor-corrector schemes exhibit different stability behavior for odd and ev…

Shifted NLEP stability conjecture. The shifted NLEP is stable for all $\mu>\mu_c(y_0)$.