5 problems

Harmonic APD relation. For $n \geq 1$,

Hilbert-matrix first appearance value conjecture. For $n \geq 2$,

Let $H_n$ be the $n\times n$ Hilbert matrix, and let $\sigma_i(H_n)$ denote its $i$th singular value for a fixed index $i$. Limiting singular-value conjecture. Every singular value…

Spectral conjecture. The spectrum $\sigma(L_{\nu})$ consists of exactly one negative eigenvalue and none or exactly one eigenvalue greater than $4$; more precisely,

Let $\mathbf{x}=(x_1,\ldots,x_R)$ be a vector of distinct real numbers and let $\mathbf{c}=(c_1,\ldots,c_R)$ be a vector of real numbers. Define the $R\times R$ matrix…