472 problems

Schur stability conjecture. The operators $P(t)$ and $R(t)$ are stable, meaning that

Let $n\geq 3$, let $h\in\mathbb{R}^n$ be the normalized Hermite root vector, and define … Let $W=\mathbf{1}^\perp=\{u\in\mathbb{R}^n:u^\top\mathbf{1}=0\}$ be the mean-zero subspace…

Let $M$ be any $n\times n$ symmetric matrix with entries satisfying $0\leq M_{ij}\leq1$, and let $\mathbf{1}$ be the $n$-dimensional all-ones vector. Matrix power inequality conjec…

Let $M_N$ and $M_D$ be the Neumann and Dirichlet Titchmarsh–Weyl matrices, respectively, and let $\operatorname{Res}(M_N,0)$ and $\operatorname{Res}(M_D,0)$ denote their residue ma…

Let $\rho$ be a probability density with moments of all orders, meaning that … for every $q\geq 1$. Let $E_o\in\mathbb R$. Suppose there is a $\delta>0$ such that, on … the symbol…

Let $q_1$ and $q_2$ be two distinct odd primes, and let $V$ be a nonconstant real-valued $q_1\be Z\binom{}{}\oplus q_2\be Z$-periodic potential. For $[?]$

Let $\sigma_0$ be a simple quasinormal mode of the Klein--Gordon equation on the near-horizon geometry with parameters $r_{\rm e}$ and $r_{\rm c}$. Let $\epsilon>0$ be small and se…

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and let $\lambda_k(\Omega)$ and $\mu_k(\Omega)$ denote its Dirichlet and Neumann eigenvalues, respectively. For positive integ…

Let $G$ be the reductive group considered in the paper, let $\Gamma\subset G(\mathbb{Q})$ be a congruence subgroup, and let $\Pi(K_\infty)$ denote the set of $K_\infty$-types. For…

Let $A\colon \mathbb{R} \to \mathbb{R}^{d \times d}$ be a $\tau$-periodic function whose components have bounded variation and are continuous from the right. Let $U$ be the monodro…

Let $A$ be the generator of a $C_0$-group on $\ell_p$, with $p>1$, whose eigenvalues, counted with multiplicity, are $\{\lambda_n\}_{n=1}^{\infty}$ and whose corresponding normaliz…

Let the spectral problem be the linearized spectral problem denoted by (third), with parameter $\omega\in(-1,1)$. Its isolated eigenvalues and end-point resonances are considered a…

Let $\Lambda_\kappa$ denote the smallest eigenvalue of the relevant weighted spherical operator, with parameter $\kappa>0$ in dimension $n$. Proposition gives three possible altern…

Integer-spectrum conjecture. The algebraic eigenvectors span the three lowest levels of the spin chain. When $M\teq1$, the fourth algebraic energy, $\mathsf{E}=3$, is the fifth low…

Matrix Stahl–Totik conjecture. Then $\mu$ is regular.

Let $\Gamma$ be a Fuchsian group with fundamental class $[\Gamma]$, let $\sigma$ be a multiplier on $\Gamma$ satisfying $\delta(\sigma)=0$, and define … Here a…

Let $\Gamma$ be the orbifold fundamental group of a compact good orbifold, acting on its universal orbifold cover $\widetilde{M}$, and let $H_V=\Delta+V$ be a $\Gamma$-invariant Sc…

Let $G$ be an abstract completely continuous operator, let $\Lambda:=\left\{\lambda_j\right\}$ be the set of its eigenvalues, and fix $\delta>0$. For each $\lambda$, define the circle…

Obsolescence conjecture. Condition 1 in Theorem 4.1 is obsolete in general, meaning that the theorem should remain valid without this condition.

Let $p$ be prime, let $r\in(\frac{1}{p+1},\frac{p}{p+1})$, and let $h$ be an $r$-overconvergent modular function on the region $X_0(1)_{\ge p^{-r}}$. Its spectral expansion is the…

Let $p\in\{2,3,5\}$, let $r$ range over some open interval containing $\frac{1}{2}$, and let $\mathcal{S}_k(r)$ be the space on which the overconvergent $U$ operator acts. A factor…

Absolutely continuous sufficiency conjecture. Under these assumptions, the conditions and are sufficient for $A$ to be similar to a selfadjoint operator.

Let a boundary-value problem be subject to the estimate referenced in the source as … . The conjecture proposes a general regularity consequence of polynomial resolvent growth, ext…

Let $\{\alpha_n\}_{n\in\mathbb{N}}$ be unimodular complex numbers and let $\{\beta_n\}_{n\in\mathbb{N}}$ be positive numbers converging to $0$, with the ratios…

Let $\{z_n\}_{n\in\mathbb{N}}$ be distinct points in the open unit disk converging to $1$. Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space with orthonor…