31 problems

Zamolodchikov's conjecture. The system is solved by

Let the interaction matrix elements of the three-dimensional marching-on-in-time contrast current density volume integral equation be computed in finite precision, and let the numb…

Let $\Gamma$ be a smooth surface, and let $\mathcal I+\mathcal R_{\text{LB}}$ denote the integral operator in, where $\mathcal R_{\text{LB}}$ is the composition of the Laplace–Belt…

Consider the critical parameter regime … and let $\Phi$ be the profile of a standard self-similar solution of the coagulation equation with growth. Singular self-similar solutions…

Let $u_\beta$ be the function solving the integral equation referred to in the source, with $\beta\in(0,1)$. Monotonicity conjecture. The function $u_\beta(x)$ is monotonically dec…

Let $\nu>0$, and let $\theta_{\nu}:[0,\pi]\to\mathbb{R}$ denote a solution of Nekrasov's integral equation, where $\theta_{\nu}(s)$ is the angle between the tangent to the free bou…

Let $A$ be the discretization of the first-kind Fredholm equation for Laplace's equation in three dimensions, and let $T^{-1}$ be the preconditioner defined from the domain decompo…

Let $A$ be the discretized integral operator in the referenced two- or three-dimensional models, and let $T^{-1}_{\text{Schwarz}}$ be the Schwarz preconditioner associated with the…

Let $M>0$ and let $e$ and $h^\theta$, for $\theta\in\mathbb{R}$, be the symbols defined earlier in the paper, with $\|e\|_\infty$ and $\|h^\theta\|_\infty$ denoting their sup norms…

Existence conjecture. Even strictly positive densities $w$ exist that do not satisfy condition and for which the uniqueness asserted in Theorem fails.

Let ${\mathcal F}$ be the self-map defined by the finite system of oper integral equations, and let ${\mathcal X}_\bullet$ denote the conjectural fixed point. Starting from … const…

Fix $w\in {\mathcal B}'$ and let $\Gamma'=\{\gamma\in\Gamma\mid\Omega(w,\gamma)\neq0\}$, which is finite in the examples considered. Let…

Let $u$ and $p$ denote the concentration and precipitation variables in the full HHMO-model on the half-line, with precipitation governed by the threshold $u^*>0$. More generally,…

The $N$-body dielectric spheres problem uses the continuity constant $C_{\widetilde{\mathcal{A}}}$ of the infinite-dimensional operator $\widetilde{\mathcal{A}}$ to bound the large…

Marcinkiewicz's uniqueness conjecture. If $f(0)\neq 0$, then the equation has only the trivial solution $y\equiv 0$.

Localized-solution existence conjecture. Under these conditions, the nonlinear integral equation governing the anomalous large-deviation scaling has a localized solution.

Let ${\mathcal X}^{(k)}_\gamma$ be the numerical value obtained from the $k$-th truncated version of the Riemann–Hilbert integral equation, and let ${\mathcal X}_\gamma$ be the val…

Choose a function $\vartheta(\arg\hbar)$ satisfying $\lvert\vartheta(\arg\hbar)-\arg\hbar\rvert\leq\frac{\pi}{2}$, and define…

Let $S_{k,\omega}$ and $N_{k,\omega}$ be the single-layer and hypersingular operators, respectively, and let a preconditioner have a remainder whose smoothing order can be compared…

Let $D$ be a simply connected domain, and consider the coupled system of integral equations described above. Rank-one deficiency conjecture. The coupled system of integral equation…

Uniqueness conjecture. A given solution $\psi^{\rm free}$ of the free multi-time equations determines the solution $\psi$ of the integral equation uniquely. This conjecture has tur…

Consider the forced Volterra equation with forcing term $H(t)$ and define $F_U$ as the function used in the source's growth-rate criterion. Suppose the nonlinearity satisfies the s…

Let $K$ be a class-II kernel, meaning that $K(\xi,1)\to k_0>0$ as $\xi\to0$. Let $G_\rho$ denote a solution of the self-similar profile equation, with parameter $\rho>0$. Existence…

Gaudin's conjecture. As $\gamma\to0$, the Fermi contribution to the ground-state energy satisfies

Uniqueness conjecture. If two density functions satisfy the potential equation $$, then they are equal almost everywhere.