51 problems

Leading-block and eigenvalue conjecture. This leading principal diagonal block of $A^{-1}$ is a $\mathbf{Z}$-matrix. Moreover, $A$ has exactly one negative eigenvalue.

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…

Inradius upper-bound conjecture. There is no universal constant $C>0$ such that

Mean-distance upper-bound conjecture. There is no universal constant $C>0$ such that

Let $\dot{x}=f(x,\rho)$, with $x\in\mathbb{R}^n$, $f\in\mathcal{C}^\omega$, and parameters $\rho=(\rho_1,\dots,\rho_m)^T\in\mathbb{R}^m$. Suppose that $x_g\in\mathcal{A}$ is a ghos…

Let $T$ be the torus, and let $\Lambda_2(T)$ denote its second topological eigenvalue. Kao–Lai–Osting conjecture. … This conjecture concerns the sharp value of the second eigenvalu…

Let $(M,\omega)$ be a compact Kähler manifold of complex dimension $n$, with Ricci curvature satisfying … Write $\lambda_j$ for the eigenvalues of the relevant Laplacian on…

Let $L_{n,p,k}$ be the matrix whose scaled eigenvalues and grids are described by Theorem, with $1\le p\le100$, $0\le k\le\min(1,p-1)$, and the grids…

For each integer $n\geq 2$, let $m(n)$ denote the least number of edges such that every graph on $n$ vertices with at least $m(n)$ edges has a matrix realization with two distinct…

Let $K_n$ be the complete graph on $n$ vertices, and let $H$ be a subgraph of $K_n$ with $|E(H)|\leq n-3$. Write $K_n\backslash H$ for the graph obtained by removing the edges of…

Top-eigenvalue vector limit-point conjecture. For any $d\geq 3$ and any fixed $k$, the set of all limit points of the vectors

Let $\lambda_j(\alpha)$ denote the $j$-th eigenvalue of the Montgomery family of quartic oscillators, for $j\geq 1$ and $\alpha\in\mathbb R$. A critical point $\alpha_{j,c}$ satisf…

Inherited correspondence conjecture. The maximum number of edges in a subgraph $H$ of $G$ in which every collection of disjoint cycles covers at most $k$ vertices equals the larges…

Let $\Pi$ be a partition of $n$, and let its Ferrers diagram be the associated diagram. For $A\in{\cal H}_n$, let the block of the Schur power matrix $\pi(A)$ associated with $\Pi$…

Soules's Schur power matrix conjecture. The permanent $\mathop{\rm per} A$ is the maximum eigenvalue of $\pi(A)$. The conjecture is refuted: the source gives a positive-semidefinit…

Let $\Omega$ be a bounded domain in $\mathbb R^d$ with smooth boundary. Let $\lambda_j$, $\Lambda_j$, and $\sigma_j$ denote the eigenvalues used in the paper for the Dirichlet Lapl…

Hamming distinct-eigenvalue conjecture. If $H(d,q,j)$ is connected, it has more than $d/2$ distinct eigenvalues.

Hermitian absolute eigenvalue conjecture. Let $d\geq 3$. Then

Hermitian forms eigenvalue conjectures. (i) If $j$ is odd, then

Bilinear forms eigenvalue conjecture. For $q\geq 3$, or $q=2$ and $d\neq e$, $B_j(d-j+1)$ is the smallest eigenvalue in the distance-$j$ graph for $1\leq j\leq d$.

Grassmann eigenvalue conjecture. (i) If $(n,q) eq (2d,2)$, then

Let $H(e^n_k)$ denote the hyperbolicity cone of the elementary symmetric polynomial $e^n_k$, and let $DG(n,k)D$ be a diagonal congruence of the matrix $G(n,k)$ by a diagonal matrix…

Let $\mathbf{D}=\operatorname{diag}(d_1,\ldots,d_n)$ with $d_i>0$, and suppose the conditions of Theorem 22 hold: $\lambda^*(\mathbf{D}^{-1}\mathbf{Q}_m)>1$ for every…

Let $a>1$ and $\sigma>0$, and let $\mathfrak{L}_1$ be the linear operator for the first nonzero Fourier mode after reducing the eigenvalue problem from $\mathfrak{L}_n$ to $n=1$. P…

Let $a>1$, and let $\mathfrak{L}_0$ be the linear operator arising from the $n=0$ mode of the linearized problem. Positivity conjecture for $\mathfrak{L}_0$. For all $a>1$, all non…