Degeneracy (mathematics)

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This article is about degeneracy in mathematics. For other uses, see Degeneracy.

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.

  • A point is a degenerate circle, namely one with radius 0. The circle is a degenerate form of an ellipse, namely one with eccentricity 0.
  • The line is a degenerate form of a parabola if the parabola resides on a tangent plane. Also it is a degenerate form of a rectangle, if this has a side of length 0.
  • A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.
  • A set containing a single point is a degenerate continuum.
  • See "general position" for other examples.

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

[edit] Degenerate rectangle

For any non-empty subset Failed to parse (Missing texvc executable; please see math/README to configure.): S

of the indices Failed to parse (Missing texvc executable; please see math/README to configure.): \{1, 2, ..., n\},
a bounded, axis-aligned degenerate rectangle Failed to parse (Missing texvc executable; please see math/README to configure.): R
is a subset of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{R}^n
of the following form:

Failed to parse (Missing texvc executable; please see math/README to configure.): R = \left\{\mathbf{x} : x_i = c_i \ (\mathrm{for} \ i\in S) \ \mathrm{and} \ a_i \leq x_i \leq b_i \ (\mathrm{for} \ i \notin S)\right\}


where Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{x}= [x_1, x_2, \ldots, x_n]

and Failed to parse (Missing texvc executable; please see math/README to configure.): a_i, b_i, c_i
are constant (with Failed to parse (Missing texvc executable; please see math/README to configure.): a_i \leq b_i
for all Failed to parse (Missing texvc executable; please see math/README to configure.): i

). The number of degenerate sides of Failed to parse (Missing texvc executable; please see math/README to configure.): R

is the number of elements of the subset Failed to parse (Missing texvc executable; please see math/README to configure.): S

. Thus, there may be as few as one degenerate "side" or as many as Failed to parse (Missing texvc executable; please see math/README to configure.): n

(in which case Failed to parse (Missing texvc executable; please see math/README to configure.): R
reduces to a singleton point).

[edit] See also

fr:Dégénéré (mathématiques) nl:Ontaard (meetkunde) it:Caso degenere ja:縮退 ru:Вырожденность (математика) th:ภาวะลดรูป

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