77 problems

For every triangulation $\Delta$ of a planar polygonal domain, let $S_3^1(\Delta)$ denote the space of functions that are polynomial of total degree at most $3$ on each triangle of…

Let $\mathbb{D}^2=\{z\in\mathbb{C}^2:|z_1|<1,\ |z_2|<1\}$, let $\alpha>0$, and let $\mathcal{D}_\alpha(\mathbb{D}^2)$ be the corresponding Dirichlet-type space. For every polynomia…

Let $N_t$ be the least number of nodes in a positive cubature formula of degree $t$ on $S^2$. Determine the asymptotic behavior of $N_{2m+1}$ as $m\to\infty$; in particular, establ…

For every nondegenerate planar triangulation $\mathcal{T}$, the dimension of the $C^1$ piecewise-cubic spline space satisfies $\dim S_3^1(\mathcal{T})=P_{\mathcal{T}}(1,3)$, where…

For every integer $d\ge 2$ and every polynomial $P\in\mathbb{C}[z]$ of degree $d$ satisfying $P(0)=0$ and $P(1)=1$, there exists a critical point $\zeta\ne 0$ with $P'(\zeta)=0$ su…

Let $\Delta$ be a triangulation of a simply connected polygonal domain $\Omega\subset\mathbb{R}^2$, let $f_1^\circ$ and $f_0^\circ$ denote the numbers of interior edges and interio…

Optimality conjecture. The parameter $\beta=m(\pi-\delta)$ is optimal for the Shannon sampling formula with the continuous Kaiser–Bessel window function, as it guarantees the maxim…

Let $\alpha$ be the exponent, let $\beta\in[0,2)$, and consider the V-shaped domain $[0,1]e^{\pm i\beta\pi/2}$. Let $r(z)$ be a lightning + polynomial approximation with $N_1$ ligh…

Let $\alpha$ be the exponent in the approximation of $x^\alpha$ on $[0,1]$, and let $r(x)$ be a lightning + polynomial approximation with $N_1$ lightning terms,…

Matrix-kernel approximation conjecture.

The exact Stechkin constant conjecture. For every $r\in\mathbb{N}$,

Krasikov's stronger conjecture. One has

Let $R$ be a commutative Noetherian Henselian local ring with maximal ideal $\mathfrak m$ and residue class field $k$, and let $\mathrm{mod}\,R$ denote the category of finitely gen…

Let $k$ be the spline parameter, let $f$ be a $(k-1)$-times differentiable function whose $(k-1)$st derivative has finite total variation, and let $\mathcal{E}_k(f)$ denote the int…

For $d\ge 3$, let $T_d$ be the recursively defined polynomial in the elementary symmetric functions on the standard simplex $T^d$, and write $\|T_d\|_{C(T^d)}$ for its uniform norm…

Let $n$ be a positive odd integer, and let $X$ be a nonsingular compact connected real algebraic set of dimension $n$. A regular map means a map given by regular functions on $X$.…

Let $\Omega$ be an exterior sound-soft Helmholtz domain with boundary $\Gamma=\partial\Omega$, and let $u$ be the scattered-field solution represented by the method of fundamental…

$(Q_n)$ conjecture. $(Q_n)$ is true for all nonnegative integers $n$.

The $(P_n)$ conjecture. $(P_n)$ is true for all nonnegative integers $n$.

Consider $d53d^n$ endowed with the metric $d(x,y)=\lVert x-y\rVert$, where the norm is any norm on $d53d^n$. Let $S_2$ be a subdivision scheme on $d53d^n$ satisfying, for…

Asymptotic empirical-current conjecture. The total variation measures $\lVert T^{(r)}\rVert$ converge to $\mu_E$ in the weak$^*$ topology of measures supported on $E$. Moreover, if…

Let $\mu\in C[-1,1]$ be a nonnegative majorant, and let $\omega_\mu\in\mathcal{P}_n$ be the snake polynomial associated with $\mu$, meaning that $|\omega_\mu(x)|\leq\mu(x)$ on…

Let $K$ be a $C^{1+}$ Jordan arc, let $z_0 \in \overline{\mathbb{C}}\backslash K$, and let $\rho:K\to[0,\infty)$ be upper-semicontinuous. Writing $t_n(\rho,z_0)$ for the weighted r…

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of ×2- and ×3-circular Carathéodory functions; call an element rational extreme when it…

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of Carathéodory functions that are ×2- and ×3-circular, where ×$n$-circular means…