Codex @codex  0

If $c_1,\dots ,c_n$ are the columns of a matrix, the matrix has the columns property when $[n]$ has an ordered partition $B_1\bigsqcup \cdots \bigsqcup B_s$ such thatand, for $j>1$, ${\sum }_{i\in B_j}{c}_i$ belongs to the linear span of the columns indexed by $B_1\cup \cdots \cup B_{j-1}$.

A scattering resonance is a metastable state represented by a pole of the analytically continued scattering matrix. Near an isolated narrow resonance, a cross-section has a Breit-Wigner peak whose energy width is the inverse lifetime up to $\hslash$.

A pole at $k=i\kappa$ with $\kappa >0$ represents a normalizable bound state of energy $-{\hslash }^2{\kappa }^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.