12 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 $S^{n-1}$ be the unit sphere, and let a positive definite function on $S^{n-1}$ be understood in the usual sense for functions on the sphere. Damase's continuity approximation…

Let $n>2$, let $f:[-1,1]\to\mathbb{R}$ satisfy $f(1)=1$, and define $\widehat f:SO(n)\to\mathbb{R}$ by $\widehat f(\gamma)=f(\gamma_{11})$. Assume that $\widehat f$ defines a posit…

Let $SO(n)$ be the special orthogonal group, and let $SO(n-1)$ be the subgroup fixing the first coordinate. A class-one representation means an irreducible representation admitting…

Let $U(d)$ be the unitary group, let $H(d)$ be the set of complex Hadamard matrices in $U(d)$, and let $B(d)$ be the optimal positive-definite-functions upper bound for mutually un…

Let $\mathcal{S}_n$ be the symmetric group and let $c:\mathcal{S}_n\to\mathbb{C}$ satisfy … for every complex-valued function $x$ on $\mathcal{S}_n$. For $A=[a_{i,j}]\in{\cal H}_n$…

Let $\mathbb{F}$ be the free group, let $\mathbb{B}_r$ be the radius-$r$ ball, and let $\mathrm{NSPD}(r,d)$ denote the class of normalized strictly positive definite $d\times d$ ma…

Let $\alpha,\beta\in(0,\infty)\setminus\mathbb{N}$, and let $F_n^{(\alpha,\beta),\delta}(t)$ denote the Jacobi-polynomial integral defined earlier in the paper. For $t\in(0,\pi]$ a…

Positivity conjecture. $F_n^{\lambda,\delta}(t)>0$ for every $t\in(0,\pi]$ if $\delta\geq\lambda+1$.

Even-moment asymptotic conjecture. There is a constant $c(k)>0$ such that, as $j\to\infty$,

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 $\delta\geq\lambda+1$.

Converse chordality conjecture. For every $S\subset G$ such that $\Gamma(G,S)$ is not chordal, there exists a positive definite function