125 problems

For a pivoting rule $P$ and dimension $n$, let $g_n^{P}(\mathbb{R})$ denote the supremum, over all nonsingular matrices $A\in\mathbb{R}^{n\times n}$, of the ratio between the large…

Let $\Omega\subset\mathbb{R}^3$ be the cylindrical domain … where $\Omega_p\subset\mathbb{R}^2$ is a Lipschitz polygon. Suppose that each partition-of-unity function $\chi_\ell$ de…

Contour-integral bound conjecture. The remaining contour integral satisfies

Let $\theta\in[-1,1]$, let $\mathbb V^k$ be either $\mathbb Q^k$ or $\mathbb P^k$, and let $\mathbb V^k_+$ denote the corresponding cone of positive polynomials. For…

Let $\theta\in[-1,1]$ and let $\mathbb V^k$ denote either the tensor-product polynomial space $\mathbb Q^k$ or the total-degree polynomial space $\mathbb P^k$, with…

The shell-based inpainting Gaussian-blur conjecture. If $u_0$ is independent of its $y$-coordinate, then

Let $p$ be a degree-$n$ univariate complex polynomial. Let its DR-circles be the circles centered at the roots of $p'$ with radii $\rho_j$, where … For each root of $p$, consider i…

Let $y_0\in\mathbb R$ and $y_l\leq y_0$ define a first solution of the first Painlevé equation, and let the second solution be defined by the data $-y_l$ and $-y_0$. Let $\Delta$ b…

Let $f(x_1,\ldots,x_n)$ be a homogeneous polynomial and let $\mathcal D$ be as in the preceding conjecture. Thus condition $(A)$ means that $f$ can be written as … where each $f_m$…

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 scalar conservation law … Let $u$ solve the fully nonlinear fourth-order approximation … where $f$ is strictly convex, $g$ is concave, and $g(0)=0$. Strong convergence…

Consider the $[P/P]$-Padé approximant $R_P(z)=T_P(z)/Q_P(z)$, where $T_P$ and $Q_P$ have degree $P$ and are computed from the first $2P+1$ Taylor coefficients of $f$. Let $t$ denot…

Let $(e^{tL})_{t\geq 0}$ be a $C_0$-semigroup in a Banach space $\mathcal{F}$ with generator $(L,D(L))$. Let $G$ be a Chernoff function for $L$ satisfying condition (N) or (N'). Fi…

Let $u_0$ be an observable depending only on the first $k$ coordinates, so that … Let $\lambda_i$ be dissipation rates satisfying $\lambda_i\propto i^p$ for an integer…

Let a vector-valued explicit or implicit Runge–Kutta method have order $p$, and let the conditions in the sufficiency theorem referred to in the source be imposed on such a method.…

Let $E”(u)$ be the Hessian of the Ginzburg–Landau energy at a minimizer $u$, and let $\rho(\kappa)^{-1}$ denote its coercivity constant. Assume the general hypotheses of the paper…

Monotonicity conjecture. The first non-zero Steklov eigenvalue $\sigma_1(\Omega_{\epsilon})$ is a monotone decreasing function of $\epsilon$ on $[0,1-\delta)$.

Let $\bm{x}$ be a set of unisolvent interpolation points, let $c(\bm{x})$ denote the equioscillating spline associated with these points, let $I(1)$ denote the integral of the cons…

High-order discrete adiabatic convergence conjecture.

Consider the nonlinear wave equation … with $p\in\mathbb{N}$ and $p\geqslant3$. Expand the field in spherical harmonics, … At fixed finite radius $r$, the modes are expected to dec…

Let $L$ and $R$ denote the numbers of left- and right-sided Hermite degrees of freedom in a semi-discretized Hermite–Volterra (HV) method for an advection equation. Stability-barri…

Let $N\in\mathbb{N}$ and let $X_N$ be the finite-dimensional polynomial space with orthogonal projection $\Pi_{X_N}$. Let $\Omega$, $\partial_x$, and $\partial_x^*$ be the operator…

Let $g:\mathbb{R}\to\mathbb{R}$ be smooth with bounded derivatives, let $G^P(x)=\frac{1}{P}\sum_{j=1}^P g(x_j)$, and let $\bm X_N^P$ be a realization of the particle system with co…

Condition-number monotonicity conjecture. For the Gaussian or inverse multiquadric RBF,