36 problems

Let $\varphi_p$ and $\varphi_q$ be the first eigenfunctions of the $p$- and $q$-Laplacians, respectively. Linear independence conjecture. The eigenfunctions $\varphi_p$ and…

Let $\Omega_c$ be the evolving domain in the clamped plate eigenvalue problem, and let $\lambda_{even}(c)$ and $\lambda_{odd}(c)$ denote the eigenvalues associated with even and od…

Let $N$ be sufficiently large, let ${\Greekmath 0116}$ be an arbitrary parameter, and let ${\Greekmath 0117}$ determine the cases of the generalized eigenvalue problem (ref{Eigena}…

Let $N$ be the parameter in the generalized eigenvalue problem (ref{Eigena}), with coefficients specified by (ref{AB}) and (ref{cmbar}), and let ${\Greekmath 0117}$ and…

The extra-stabilised EVP is a low-order eigenvalue method for the Schrödinger eigenvalue problem (EVP), introduced in the paper and compared with other lower-bound methods in numer…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain, and let $\lambda_1(\Omega)$ denote the first Dirichlet eigenvalue of the Laplacian on $\Omega$. If $B_r$ is a ball satisfying ……

The OMM is applied to the operator $-\mathpzc{H}$ to retrieve its top eigenvalues and eigenfunctions, where $\mathpzc{H}$ is a Hamiltonian operator and the optimization uses miniba…

Let $\Omega\subset\mathbb C^n$ be a bounded hyperconvex domain, and consider the eigenvalue problem for the complex Monge–Ampère operator on $\Omega$. Lions further conjectured tha…

Let $\lambda_{1}(3,\omega)$ denote the principal eigenvalue at endpoint parameter $\mu=3$ for the first Fourier mode, and let $s$ be the spatial variable. For $\omega\in[-18,-3]$,…

Let $G$ be a finite group, $V$ a finite-dimensional real $G$-module, and $\rho \colon G \rightarrow \operatorname{GL}(V)$ the representation afforded by $V$. Let…

Let $H(\lambda)$ have lowest eigenvalue $E(\lambda)$ with corresponding eigenvector $\phi(\lambda)$. For each nonnegative integer $\ell$, let $\mathcal{P}^\ell$ be the orthogonal p…

Generic double multiplicity conjecture. There exists a generic set $\Omega\subseteq\mathbb R^{n,k}$ such that, for every $U\in\Omega$ and every $\tau\in\mathbb R\setminus\{0\}$, th…

Let $h$ denote the mesh size, let $k$ be the polynomial degree of the finite element method, let $r$ be the order of the curved mesh, and let $\eL2$ denote the $L^2$ error for the…

Berenstein conjecture. [The supplied candidate statement ends here and does not include the conjectured conclusion.]

Homotopy multilevel difference bounds. Following a similar derivation as that of the cited corollary and based on the homotopy error bound, the expectation and variance of the mult…

Let $A$ be an $n\times n$ real symmetric matrix. The standard implementation of the Jacobi method is an iterative algorithm for diagonalizing $A$ by orthogonal plane rotations. Ope…

Tangential-gradient conjecture. For infinity harmonic limits of $\mathrm{L}^p$-problems, $\nabla_\tau u$ coincides with $\nabla u$. The conjecture would identify the tangential-gra…

Let $L\colon H\to G$ and $C\colon G\to H$ define the eigenvalue problem $Lx=\lambda Cx$, let $\psi$ be the oriented $\Phi_0$-map on the cylinder $\mathbb{R}\times\mathrm{S}$ of uni…

Let $\Sigma_n(\lambda)$ denote the $n$-th nodal eigencurve for the weighted indefinite problem considered in the paper, and let the weight be the particular choice given in equatio…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let $PM_n$ be the union of the convex hulls of the $k$-th roots of…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices. Let $PM_n$ be the union of the convex hulls of the $k$-th roots of unit…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices. For permutations $P,Q$, let $I$ denote the identity permutation matrix,…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let a convex combination of permutation matrices mean a matrix of t…

Real-parameter extension conjecture. The solutions given by Eqs. (73) and (74) should remain valid when $p$ is extended from rational fractions to real values satisfying…

Let $\Omega$ be a general bounded domain, let $p\neq 2$, and let $(u_n)$ be the sequence produced by the inverse iteration method for the Dirichlet $p$-Laplacian problem. Write … H…