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| {{About|degeneracy in mathematics|the degeneracy of a [[Graph (mathematics)|graph]]|degeneracy (graph theory)|other uses|Degeneracy (disambiguation)}}
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| {{Unreferenced|date=December 2009}}
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| In [[mathematics]], a '''degenerate case''' is a [[limiting case]] in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. '''Degeneracy''' is the condition of being a degenerate case.
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| A degenerate case thus has special features, which depart from the properties that are [[generic property|generic]] in the wider class, and which would be lost under an appropriate small [[Perturbation theory|perturbation]].
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| ==In geometry==
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| ===Conic sections===
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| {{Main|Degenerate conic}}
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| A degenerate conic is a [[conic section]] (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) that fails to be an irreducible curve.
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| * A [[Point (geometry)|point]] is a degenerate [[circle]], namely one with radius 0.
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| * The [[line (mathematics)|line]] is a degenerate case of a [[parabola]] if the parabola resides on a [[tangent plane]].
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| * A [[line segment]] can be viewed as a degenerate case of an [[ellipse]] in which the [[semiminor axis]] goes to zero, the [[Focus (geometry)|foci]] go to the endpoints, and the [[eccentricity (mathematics)|eccentricity]] goes to one.
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| * An ellipse can also degenerate into a single point.
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| * A [[hyperbola]] can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common [[asymptote]]s.
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| ===Sphere===
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| * A [[sphere]] is a degenerate [[standard torus]] where the axis of revolution passes through the center of the generating circle, rather than outside it.
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| ===Triangle===
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| * A degenerate [[triangle]] has [[collinear]] vertices, and thus coincides with a segment covered twice.
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| ===Rectangle===
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| * A segment is a degenerate case of a [[rectangle]], if this has a side of length 0.
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| * For any non-empty subset <math>S \subseteq \{1, 2, \ldots, n\}</math>, there is a bounded, axis-aligned degenerate rectangle
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| :<math>R \triangleq \left\{\mathbf{x} \in \mathbb{R}^n: x_i = c_i \ (\text{for } i\in S) \text{ and } a_i \leq x_i \leq b_i \ (\text{for } i \notin S)\right\}</math> | |
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| where <math>\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]</math> and <math>a_i, b_i, c_i</math> are constant (with <math>a_i \leq b_i</math> for all <math>i</math>). The number of degenerate sides of <math>R</math> is the number of elements of the subset <math>S</math>. Thus, there may be as few as one degenerate "side" or as many as <math>n</math> (in which case <math>R</math> reduces to a singleton point).
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| ===Other===
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| * See [[general position]] for other examples.
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| ==Elsewhere==
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| * A set containing a single point is a degenerate [[Linear continuum|continuum]].
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| * A [[random variable]] which can only take one value has a [[degenerate distribution]]; its probability density is the [[Dirac Delta function]] at one point.
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| * Similarly, [[root of a function|root]]s of a [[polynomial]] are said to be ''degenerate'' if they coincide, since generically the ''n'' roots of an ''n''th degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate [[eigenvalue]] (i.e. a multiply coinciding root of the [[characteristic polynomial]]) is one that has more than one linearly independent [[eigenvector]].
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| * In [[quantum mechanics]] any such [[multiplicity (mathematics)|multiplicity]] in the eigenvalues of the [[Hamiltonian operator]] gives rise to [[degenerate energy level]]s. Usually any such degeneracy indicates some underlying [[symmetry]] in the system.
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| == See also ==
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| * [[Degeneracy (graph theory)]]
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| * [[Degenerate form]]
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| * [[Trivial (mathematics)]]
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| * [[Pathological (mathematics)]]
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| * [[Vacuous truth]]
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| ==External links==
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| {{MathWorld|title=Degenerate|id=Degenerate}}
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| {{DEFAULTSORT:Degeneracy (Mathematics)}}
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| [[Category:Mathematical concepts]]
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