# Involution

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## involution

[‚in·və′lü·shən]
(biology)
A turning or rolling in.
(embryology)
Gastrulation by ingrowth of blastomeres around the dorsal lip.
(mathematics)
Any transformation that is its own inverse.
In particular, a correspondence between the points on a line that is its own inverse, given algebraically by x ′ = (ax + b)/(cx-a), where a 2+ bc ≠ 0.
A correspondence between the lines passing through a given point on a plane such that corresponding lines pass through corresponding points of an involution of points on a line.
(medicine)
The retrogressive change to their normal condition that organs undergo after fulfilling their functional purposes, as the uterus after pregnancy.
The period of regression or the processes of decline or decay which occur in the human constitution after middle life.

## Involution

(raising to a power), an algebraic operation consisting in repeating a quantity a n number of times:

The number a is the base of the power, n is the exponent, and an is the power. For example, 3x3x3x3 = 34 = 81. The second power of a number is called its square and the third power, its cube.

## Involution

(regression), in biology, a reduction or loss, in the process of evolution, of certain organs, and a simplification of their organization and functions (for example, various degrees of intestinal reduction in certain parasitic worms and arthropods). Degenerative changes in protozoans and bacteria caused by unfavorable conditions in the habitat are also called involutional. In medicine, the term sometimes refers to atrophy of organs and tissues, such as in aging.

References in periodicals archive ?
The well-posedness of direct and inverse problems for parabolic equations with involution was considered in [3-5].
and Torrence B., Involutions fixing the union of odd-dimensional projective spaces, Canadian Math.
By induction and a previous case, [[pi].sup.e.sub.1] and c([[pi].sup.o.sub.2]) are 3412-avoiding involutions, so c([beta]) = (1 [??] [[pi].sup.e.sub.1] [??] 1) [direct sum] c([[pi].sup.o.sub.2]) is also a 3412-avoiding involution.
Now we give a similar Lemma for fixed point free involutions. In this case, we slightly modify the two concepts of inflation and deflation.
Delayed in uterine involution, chronic metritis and poor conception rate are common sequelae to retention of foetal membranes (Roberts, 1971).
Such matrices define a unique involution on [X.sub.0](N) that is called the Atkin Lehner involution.
Note that, if [iota] is an r-admissible involution and if [iota](b) = c > b and [iota](z) = z, then z < b or z > c (this is easily proved by induction on [absolute value of Z]).
Algorithm 3.3 (InvolutionJumper) Input: G = (X) and an involution x [member of] G.
If A is a central simple k-algebra with a non-trivial involution, then either A [equivalent] [M.sub.n](D), where D is a quaternion division algebra, or A is trivial on k.
Thus the composite operation of toggling the elements of P from left to right is well-defined; Striker and Williams (2012) call this operation promotion, and show that it is conjugate to rowmotion in the toggle group (the group generated by the toggle involutions).
The topics include the isotopy of links, most knots are wild, infinite cyclic coverings, some curious involutions of spheres, two complexes that are homeomorphic but combinatorially distinct, uses of the fundamental group, and a unique decomposition theorem for three-manifolds.
In particular, several authors examined the properties of the polynomial [I.sub.n](x) = [[summation].sup.n-1.sub.j=0] [i.sub.n,j], [x.sup.j] where [i.sub.n,j] denotes the number of involutions on [n] = {1, 2, ..., n} with j descents.

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