479 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…

For every dimension $d\in\mathbb{N}$ and smoothing parameter $\beta>0$, determine the complete spectral decomposition of the Henze–Wagner covariance-kernel integral operator…

For the continuous-time chain on $\{0,1\}^k$ with generator $Q$ defined by … where boundary terms are omitted and $\oplus$ denotes addition modulo $2$, determine the corresponding…

Let $\lambda(b)$ be the lowest eigenvalue of the magnetic Neumann Laplacian in the unit disk for a constant magnetic field of strength $b>0$, and let $\Theta_0$ denote the de Genne…

Let $H_n$ be the degree-$n$ Hermite polynomial, and let $0<x_{n,1}<\cdots<x_{n,\lfloor n/2\rfloor}$ denote its positive zeros. Prove that for every integer $n\ge 4$,…

Let $H_V=-\frac{d^2}{dx^2}+V$ on $L^2(\mathbb{R})$, where $V\colon\mathbb{R}\to\mathbb{R}$ and $V\in L^{3/2}(\mathbb{R})$. If the negative eigenvalues of $H_V$ are written as…

Let $\alpha\in(0,2)$, and let $\{\varphi_n^{(\alpha)}\}_{n\geq 1}$ be an $L^2((-1,1))$-normalized sequence of eigenfunctions of the fractional Laplacian on $(-1,1)$ with the approp…

Let $W_{a,b}(t)=\sum_{n=0}^{\infty}a^n\cos(b^n t)$ be a classical Weierstrass function, with $0<a<1$ and $b>1$ in the usual nowhere-differentiable regime. Determine for which param…

Pate's conjecture. The largest eigenvalue of $\mathscr{C}_k(A)$ is $\operatorname{per}(A)$. This conjecture is refuted; the paper states that its $k=1$ case has an $8\times8$ count…

Let $M(n,\mathbb{R})$ be the space of real $n\times n$ matrices. For $X\in M(n,\mathbb{R})$ with Frobenius norm $\|X\|=1$, let $T_X(Y)=[X^*,[X,Y]]$ and let $\lambda(T_X)$ denote it…

Let $N$ be a compact Riemannian manifold with negative sectional curvature, let $(\Psi_n)_{n=0}^{\infty}$ be an orthonormal basis of Laplace eigenfunctions on $N$, and let $\mu_n$…

Jordan-curve conjecture. The following statements are equivalent:

Root-distribution conjecture. For each $k\geq -1$, the polynomial $P^{(\lambda)}_{l+4+k}$ is square free. Furthermore, for sufficiently large $k$, it has a fixed number of roots ou…

Mahler-measure-zero conjecture. The Mahler measure of the spectrum of $T$ is zero.

A two-point boundary value problem (BVP) for an ordinary differential operator is called dissipative when its associated operator is dissipative, and Birkhoff-regular when its boun…

Let $\Lambda_\varphi$ be the one-dimensional Dirac operator on $L^2({\mathbb{R}},{\mathbb{C}}^2)$, with potential $\varphi:{\mathbb{R}}\to{\mathbb{C}}$ satisfying … Call…

Let $T$ be a completely factorizable operator of degree $d$, meaning that … Then the eigenvalue formula conjecture states that the first few eigenvalues…

Let $H_{\lambda,\alpha,\omega}$ be the Sturmian Hamiltonian with coupling $\lambda>0$, irrational frequency $\alpha\in(0,1)$, and phase $\omega\in S^1$, and let…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Lu–Wenzel eigenvalue conje…

Let $w$ be the density of the absolutely continuous part of a measure on the unit circle, let $\alpha=(\alpha_n)_{n\geq 0}$ be its Verblunsky-coefficient sequence, and let $S$ be t…

Let $\Sigma_n$ be the covariance matrix of the length power functionals in the subcritical regime, for $n\geq 2$, with eigenvalues $\lambda_1\leq\ldots\leq\lambda_n$. Distinct-eige…

Let $L_{\gamma,d}$ be the optimal constant in the Lieb–Thirring inequality, let $L_{\gamma,d}^{\mathrm{cl}}$ be the semiclassical constant, and let $L_{\gamma,d}^{\mathrm{one}}$ be…

Coincidence Rank Conjecture. The coincidence rank of the tiling must divide the algebraic norm of $\lambda$.