11 problems

Radius-selection conjecture. The stagnation radius satisfies

Novak's conjecture. The matrix

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…

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.…

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 $D_q^r(\xi,\Lambda)$ 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

Convergence conjecture. The squared worst-case error in the Sobolev space $\mathcal{H}^{3/2}$ satisfies