degeneracy

Dictionary: de·gen·er·a·cy (dĭ-jĕn'ər-ə-sē) pronunciation

n., pl., -cies.

The process of degenerating.
The state of being degenerate.
Corrupt, vulgar, vicious behavior, especially sexual perversion.
Genetics. The presence in the genetic code of multiple codons for the same amino acid.

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Sci-Tech Encyclopedia: Degeneracy

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A term referring to the fact that two or more stationary states of the same quantum-mechanical system may have the same energy even though their wave functions are not the same. In this case the common energy level of the stationary states is degenerate. The statistical weight of the level is proportional to the order of degeneracy, that is, to the number of states with the same energy; this number is predicted from Schrödinger's equation. In quantum mechanics and in other branches of mathematical physics, the term degeneracy is employed also to characterize the eigenvalues of operators other than the energy operator. See also Eigenvalue (quantum mechanics).

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noun

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Antonyms: degeneracy

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Definition: corruption
Antonyms: morality, purity

n

Definition: decay, deterioration
Antonyms: development, improvement

Wikipedia: Degeneracy (mathematics)

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

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.

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.
A segment 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.
A random variable which can only take one value has a degenerate distribution.
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.

Degenerate rectangle

For any non-empty subset $S \subseteq \{1, 2, \ldots, n\}$ , there is a bounded, axis-aligned degenerate rectangle

$R \triangleq \left\{\mathbf{x} \in \mathbb{R}^n: x_i = c_i \ (\text{for } i\in S) \text{ and } a_i \leq x_i \leq b_i \ (\text{for } i \notin S)\right\}$

where $\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]$ 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).

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Antonyms. © 1999-2009 by Answers Corporation. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Degeneracy (mathematics)". Read more

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