Results for involution

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involution

(ĭn'və-lū'shən) pronunciation

1. The act of involving.
2. The state of being involved.
Intricacy; complexity.
Something, such as a long grammatical construction, that is intricate or complex.
Mathematics. An operation, such as negation, which, when applied to itself, returns the original number.
Embryology. The ingrowth and curling inward of a group of cells, as in the formation of a gastrula from a blastula.
Medicine.
1. A decrease in size of an organ, as of the uterus following childbirth.
2. A progressive decline or degeneration of normal physiological functioning occurring as a result of the aging process.

[Latin involūtiō, involūtiōn-, from involūtus, past participle of involvere, to enwrap. See involve.]

involutional in'vo·lu'tion·al adj.

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Geography Dictionary: involution

1. The refolding of two nappes differing in age so that parts of the younger nappe lie below older rocks.

2. The convolution of layers of ground under periglacial conditions. This may occur when the active layer is trapped at the start of winter between the frozen surface and the permafrost below it. The resulting pressure distorts the trapped strata. Alternatively, involutions may result from the pressure exerted by expanding ice segregations.

A synonym for cryoturbation.

Veterinary Dictionary: involution

1. a rolling or turning inward.
2. one of the movements involved in the gastrulation of many animals.
3. a retrograde change of the entire body or in a particular organ, as the retrograde changes in the female genital organs that result in normal size after delivery.
4. the progressive degeneration occurring naturally with advancing age, resulting in shriveling of organs or tissues.

uterine i. — reduction in size of the uterus in the period immediately after parturition.

Wikipedia: involution

An involution is a function

f : X \to X,

which, when applied twice, brings one back to the starting point.

See involution (metaphysics) for the philosophical meaning.

In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that

f(f(x)) = x for all x in the domain of f.

General properties

Any involution is a bijection.

The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation.

Other examples include circle inversion, the ROT13 transformation, and the Beaufort polyalphabetic cipher.

Involutions in Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Doing a reflection twice, brings us back where we started.

This transformation is a particular case of an affine involution.

Involutions in linear algebra

In linear algebra, an involution is a linear operator T such that $T 2 = I$ . Except for in characteristic 2, such operators are diagonalizable with 1's and -1's on the diagonal. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.

Involutions are related to idempotents; if 2 is invertible, (in a field of characteristic other than 2), then they are equivalent.

Involutions in ring theory

In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples include complex conjugation and the transpose of a matrix.

See also star-algebra.

Involutions in group theory

In group theory, an element of a group is an involution if it has order 2; i.e. an involution is an element a such that a² = e, where e is the identity element. Originally, this definition differed not at all from the first definition above, since members of groups were always bijections from a set into itself, i.e., group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution. The group of bijections generated by an involution through composition, is isomorphic with cyclic group C₂.

A permutation is an involution precisely if it can be written as a product of non-overlapping transpositions.

The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups.

Coxeter groups are groups generated by their involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.

Involutions in mathematical logic

The operation of complement in Boolean algebras is an involution. Accordingly, negation in classical logic satisfies the law of double negation: ¬¬A is equivalent to A.

Generally in non-classical logics, negation which satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of truth values. Examples of logics which have involutive negation are, e.g., Kleene and Bochvar three-valued logics, Łukasiewicz many-valued logic, fuzzy logic IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual e.g. in formal fuzzy logic.

Count of involutions

The number of involutions on a set with n = 0, 1, 2, … elements is given by the recurrence relation:

a(0) = a(1) = 1;

a(n) = a(n − 1) + (n − 1) × a(n − 2), for n > 1.

The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in OEIS).

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Geography Dictionary. A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Involution". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

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involution

General properties

Involutions in Euclidean geometry

Involutions in linear algebra

Involutions in ring theory

Involutions in group theory

Involutions in mathematical logic

Count of involutions

See also

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