26 problems

Consider the proposed quadrature method applied to the nonlocal problem on a non-uniform discretization, and measure the numerical error in the $L^2$ norm. Convergence-rate conject…

Let $\bm{x}$ be a set of unisolvent interpolation points, let $c(\bm{x})$ denote the equioscillating spline associated with these points, let $I(1)$ denote the integral of the cons…

Let $t_1,t\in\mathbb{N}$ with $t_1<t$. A spherical $t_1$-design is a finite multiset on $\mathbb{S}^d$ reproducing spherical integration for polynomials of degree at most $t_1$, an…

Let $t_1,t\in\mathbb{N}$ with $t_1<t$. A spherical $t_1$-design is a finite multiset on $\mathbb{S}^d$ reproducing spherical integration for polynomials of degree at most $t_1$; a…

Limiting closest-point parameter conjecture. We conjecture that

The setting concerns reconstruction of functions from values at subsampled quadrature points, where the current error bounds for subsampled rank-1 lattice methods attain the optima…

Consider the proposed quadrature approach for two-dimensional non-uniform discretizations, constructed as tensor products of perturbed one-dimensional meshes. Let the convergence r…

Let $F_1(\alpha,\beta,z)$ be the smooth leading coefficient in the plane quadrature error, with $\alpha,\beta$ ranging over $[-1/2,1/2]$ and $z$ fixed. Vanishing average conjecture…

Let $\Omega\subset\mathbb{R}^d$ be a compact subset, let $\mu$ be a finite measure on $\Omega$, and let $X_N$ be an $N$-dimensional subspace of the real space $L_2(\Omega,\mu)$ who…

Seven-point quadrature conjecture. The seven-point rule consists of sufficiently many quadrature points to produce stable second-order convergence rates for the exact ball and for…

Let $\Gamma$ be an analytic curve, and let $R$ denote the radius of curvature at a point of highest curvature on $\Gamma$. A Schwarz singularity is a singularity of the analytic co…

BDHSS test-function conjecture. If

Let $\hat{c}$ be the parameter appearing in the exponential weights $e^{-n(x^2+\hat{c}x)}$. The authors note that $\hat{c}>8$ certainly suffices, but conjecture that the weaker con…

Perturbed-grid quadrature conjecture. For every continuous function $f$ and every $\alpha<1/2$,

Rate-exponent conjecture. Based on numerical experiments, it is conjectured that $4\alpha$ can be replaced by $2\alpha$ in the rate estimate for $|I-\tilde I_N|$ and…

Smoothness conjecture. It is conjectured that having $2\alpha$ derivatives, in the stated Hölder-continuity sense, is sufficient for convergence of the perturbed trigonometric inte…

The paper considers two questions. First, whether for every sequence of Monte Carlo methods for quadrature of the first component of an SDE solution, based on finitely many sequent…

Consider the derived optimal quadrature rules for spline spaces of polynomial degree $d$ and continuity $c$ over finite domains, together with their corresponding asymptotic rules.…

Let an odd-degree source spline space have an optimal quadrature rule, and let a target spline space be obtained through an even-dimensional transition between the source and targe…

Convergence conjecture. For every $\epsilon$, there exists $p_0$ such that for every $p\geq p_0$,

Exponential quadrature-error conjecture. The quadrature error for the exponential function satisfies

Quadrature-error conjecture. The quadrature error satisfies

Relative-accuracy conjecture. The coordinates of $\beta^{(n)}$ satisfy, for every $1\leq k\leq N$,

Gauss quadrature convergence-rate conjecture. For every function $f otin X^s$?

Let $\mathbb S^q$ be the $q$-dimensional sphere, and let $d_n^{q+1}$ denote the dimension of the space of spherical polynomials of degree at most $n$. A quadrature formula exact fo…