Definitions

# Degeneracy (mathematics)

for the degeneracy of a Graph, see Arboricity#Related_concepts.

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class.

Another usage of the word comes in eigenproblems: a degenerate eigenvalue is one that has more than one linearly independent eigenvector.

## Degenerate rectangle

For any non-empty subset $S$ of the indices $\left\{1, 2, ..., n\right\},$ a bounded, axis-aligned degenerate rectangle $R$ is a subset of $mathcal\left\{R\right\}^n$ of the following form:

$R = left\left\{mathbf\left\{x\right\} : x_i = c_i \left(mathrm\left\{for\right\} iin S\right) mathrm\left\{and\right\} a_i leq x_i leq b_i \left(mathrm\left\{for\right\} i notin S\right)right\right\}$

where $mathbf\left\{x\right\}= \left[x_1, x_2, ldots, x_n\right]$ and $a_i, b_i, c_i$ are constant (with $a_i leq b_i$ for all $i$). The number of degenerate sides of $R$ is the number of elements of the subset $S$. Thus, there may be as few as one degenerate "side" or as many as $n$ (in which case $R$ reduces to a singleton point).