14 problems

Let $n$ be an odd prime. For $1\leq\kappa\leq n-1$, define … where $r(j,n)$ is the unique integer congruent to $j$ modulo $n$ in $\{-(n-1)/2,\ldots,(n-1)/2\}$. Kazashi and Sloan's…

Let $\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}$ and $\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}$ denote the extrapolated fine- and coarse-level quantities at level $\ell$. Extrapolated…

Let $\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}$ and $\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}$ denote the extrapolated fine- and coarse-level quantities at level $\ell$. Assume that…

Let $\mathcal{A}(M,P,N)$ be the single-level estimator, and let $\bm X_N^P$ denote the particle-system realization. For $M$ samples and $P$ particles, variance-decay conjecture. Th…

Let $\bm X^P$ be the continuous-time particle-system approximation and let $Z(t)$ be the true solution of the corresponding McKean–Vlasov equation. Let…

Complete quasi-equidistribution conjecture. Let $\gamma$ be the largest root of the polynomial $x^2-px-q$ for $1\leq q\leq p$. The van der Corput sequence in base $\gamma$ is c.q.e…

Uniform distribution conjecture. The $d$-dimensional interlaced Halton sequence in Algorithm is uniformly distributed.

Discrepancy-order conjecture. For $m e 0$,

Woźniakowski's conjecture.

Let $\tilde P_n$ be a scrambled $(0,m,s)$-net in base $b$, and let $f\in L^2([0,1)^s)$ be a function whose base $b$ Walsh series decomposition satisfies … where $a\in[0,1]$,…

Let $s\in\mathbb{N}$ and let $\mathcal{S}$ be a sequence in $[0,1)^s$. Write $D_N^{\ast}(\mathcal{S})$ for its star discrepancy at $N$ points. Grand discrepancy conjecture. For eve…

Let $m$ points be chosen in $[0,1)^d$, and let $D(m,d)_\infty$ denote the infimum of their $L_\infty$ discrepancy. Star-discrepancy lower-bound conjecture. For $d\ge 3$, … This is…

Let $D_r^w(m,d)_\infty$ denote the minimal weighted $r$-discrepancy in dimension $d$, where $m$ is the number of nodes. Weighted higher-order discrepancy lower-bound conjecture. Fo…

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