16 problems

Consider the $[P/P]$-Padé approximant $R_P(z)=T_P(z)/Q_P(z)$, where $T_P$ and $Q_P$ have degree $P$ and are computed from the first $2P+1$ Taylor coefficients of $f$. Let $t$ denot…

The Gaussian Planted Submatrix problem has signal parameter $\beta$, while the Planted Clique problem is obtained from a centered Erdős–Rényi adjacency matrix with an inserted cliq…

Consider the Gaussian Planted Submatrix model in the setting of Theorem, with signal parameter $\beta$, signal vector $\bm{x}$, and leading eigenvector…

Fix even $r$ and $\ell$ with $n\gg\ell\geq r/2$, and let $Z_S\sim\mathcal N(0,1)$ independently for each $r$-subset $S\subset[n]$. Define the Kikuchi matrix $M(\lambda)$, indexed b…

Let $N\in\mathbb{N}$ and let $X_N$ be the finite-dimensional polynomial space with orthogonal projection $\Pi_{X_N}$. Let $\Omega$, $\partial_x$, and $\partial_x^*$ be the operator…

Let $\bm{x}$ be random as in the paper's spiked-prior assumption, let $p(y)$ be a density defining an additive noise channel $\mathcal{P}$, and let $F_{\mathcal{P}}$ denote its Fis…

Aliasing estimate. There exists $c(\lambda)>0$ such that

Poisson tau-method solvability conjecture. If $f\in\Pi_N^d$ and the tau polynomials $P_k$ span $\Pi_{N-2,N}^{(d)}$, then the tau-modified equation

Divergence conjecture. Such collocations will likely diverge as the mesh size grows large. For Gauss/Gauss-Radau discretizations, the divergence is attributed to the divergence est…

Let the assumptions for the Fokker–Planck operator hold, and let an orthogonal decomposition based on such an operator be given. For smooth functions…

Multidimensional convergence conjecture. There exists a constant $C=C(R,k,m,|u|_{H^{m+2}(R_0)})$ such that

The Bessel-zero quotient monotonicity conjecture. The quotients $\hat q_k$ decrease monotonically as $k\in\mathbb{N}$ increases. The conjecture is supported by numerical experiment…

Let $\xi^N_0<\xi^N_1<\cdots<\xi^N_N$ be the LGL points, let $\Delta^N_k=[\xi^N_k,\xi^N_{k+1}]$, and define $q_k^N=|\Delta^N_k|/|\Delta^N_{k-1}|$ for the relevant indices. The famil…

Let $[a,b]$ be an interval, let $L=L_{[a,b]}:[a,(a+b)/2]\to[a,b]$ be given by $L(x)=2x-a$, and let $\{{\cal G}^{\mathrm{LGL}}_N\}_{N\in\mathbb{N}}$ be the family of Legendre–Gauss–…

Let $c>0$, let $n>2c/\pi$ be an integer, and let $m$ be an integer with $0\leq m<n$. Let $\delta_n(\psi_m)$ be the quadrature error for the prolate spheroidal wave function…

Let $c>0$ and $a\geq 0$ be real numbers, and let $n>2c/\pi$ be an integer. Let $\delta_n(e^{icax})$ denote the quadrature error associated with the quadrature nodes $t_j$ and weigh…