6 problems

Let $d\geq 1$, set $\lambda=\frac{d-1}{2}$, and let $({\theta}-t)_+=\max\{{\theta}-t,0\}$. For $\delta\geq\lambda+1$ and any ${\theta}\in(0,\pi)$, define … Beatson–Castell–Xu conje…

Let $\mathcal N=\{n\in\mathbb N:n\not\equiv 0,4,7\pmod{8}\}$, let $\mathcal E(n)=\{\mathbf{x}\in\mathbb Z^3:|\mathbf{x}|^2=n\}$, let…

Symmetry conjecture. The limit

Simple angular dependence conjecture. The fundamental modes of $\mathbb{B}(R)$ can be written as linear combinations of $u_1(r,\hat{\theta})$.

Let $\cal H_\ell$ be the eigenspace of spherical harmonics of degree $\ell$ on the sphere, and let $\mu(u)$ denote the number of nodal domains of a spherical harmonic $u$. Leydold'…

Integral positivity conjecture. The inequality $F_n^{\lambda,\delta}(t)>0$ holds for every $n\in\mathbb{N}_0$ and every $t\in(0,\pi]$ if and only if $\delta\geq\lambda+1$.