As a trilogy, this first paper along with the authors' two companion articles introduces alternate approaches to standard $R$-matrix parametrization of nuclear cross sections. The authors present evidence so-called shadow poles in theory for ${}^{134}$Xe. number and location depends upon how one continues operators complex wave numbers. also establish new pole sections, called $w\phantom{\rule{...

Let n, r be integers with 0 ≤ r ≤ n− 1. An n×n matrix A is called r-partly decomposable if it contains a k×l zero submatrix with k+l = n−r+1. A matrix which is not r-partly decomposable is called r-indecomposable (shortly, r-inde). Let Eij be the n × n matrix with a 1 in the (i, j) position and 0’s elsewhere. If A is r-indecomposable and, for each aij 6= 0, the matrix ∗Research supported by Nat...