35 problems

Optimal convergence order conjecture. The order of convergence

Let $\mathbf W^r_{1,\alpha}$ be the weighted Sobolev-type function class, and let $\kappa_m(\mathbf W^r_{1,\alpha})$ denote the minimal cubature error over formulas with $m$ knots.…

Consider Hamiltonian systems of dimension $2n$ with $n>1$. In the preceding result, Theorem, no general integrators conserve both energy and volume for all Hamiltonians in…

Let $N_{\rm min}(\ell,d,\varrho)$ denote the smallest number of function values needed by a cubature formula of degree $\ell$ in dimension $d$ for a symmetric weight function…

Let $W_{\rho}^{\alpha,2}(\mathbb{R})$ be the one-dimensional $\rho$-weighted Sobolev space, and consider trapezoidal rules combined with the boundary-damping importance sampling me…

Radius-selection conjecture. The stagnation radius satisfies

A rigid body configuration may be represented using the special Euclidean group $SE(3)$ or the direct product group $SO(3)\times\mathbb{R}^{3}$. Lie group integration schemes use t…

Variance-reduction conjecture. The repulsion operator may produce variance reduction for smooth functions for a wide range of point processes.

Let $D_d\subset\mathbb{R}^d$ be a domain, let $\mathcal{P}_n$ denote a set of $n$ sampling points, and consider the integration or approximation error … Here $W_p^s(D_d)$ is a Sobo…

Uniform lower-bound conjecture. There exists a constant $C>0$, independent of $d$ and $D_d$, such that

Novak's conjecture. The matrix

Uniform lower-bound conjecture. There exists a constant $C>0$, independent of $d$ and $D_d$, such that

Let $\mathbf W^r_{\infty,\alpha}$ be the weighted Sobolev-type function class, and let $\kappa_m(\mathbf W^r_{\infty,\alpha})$ denote the minimal cubature error over formulas with…

For a point set $\xi$ with $m$ points in the $d$-dimensional domain and arbitrary weights, let $\widetilde D^{r,o}_{\mathbf p}(m,d)$ denote the infimum of the smooth…

For a point set $\xi$ with $m$ points in the $d$-dimensional domain and arbitrary weights, let $D^{r,o}_\infty(m,d)$ be the infimum of the $r$-discrepancy over all such point-weigh…

Let $\widetilde D_\infty^r(\xi,\Lambda)$ be the smooth $r$-discrepancy of a weighted point set, and define … Here $m$ is the number of points, $d$ is the dimension, and $C(r,d)>0$…

Let $D_q^r(\xi,\Lambda)$ be the $r$-discrepancy of a point set $\xi$ with weights $\Lambda$, and define … Here $m$ is the number of points, $d$ is the dimension, and $C(r,d)>0$ dep…

Let $D(m,d)_\infty$ denote the minimal $L_\infty$ discrepancy of an $m$-point set in dimension $d$, and let $C(d)>0$ depend only on $d$. Discrepancy lower-bound conjecture. For…

Optimal-order discrepancy conjecture. For every $d\geq 3$, there is a constant $C(d)>0$ such that

Worst-case error monotonicity conjecture. For fixed $t$, if $N'>N$, then

Maximum-value conjecture. The absolute maximum value taken by $g_E$ is $\frac{11}{225}$, and

Let $\kappa_m(\mathbf W^r_p)$ denote the optimal error of numerical integration on the Sobolev class $\mathbf W^r_p$ using $m$ samples. Let $d\ge2$, $2<p<\infty$, and $1/p<r\le1/2$…

Let $d$ be the dimension, and let … on the dimension $d$ is inevitable and cannot be improved by alternative steps in the proof. This conjecture concerns the sharpness of the dimen…

Midpoint-estimate convergence conjecture. $M_{\infty}$ exists and is equal to

Let $d$ be the dimension, let $Q_n$ be Frolov's cubature rule with $n$ points, and let $e(Q_n,\mathring{{\mathbf{F}}}_{1,\theta}^{1})$ denote its worst-case integration error on th…