involution

Definitions

• WordNet 3.6
• n involution the action of enfolding something
• n involution the process of raising a quantity to some assigned power
• n involution the act of sharing in the activities of a group "the teacher tried to increase his students' engagement in class activities"
• n involution marked by elaborately complex detail
• n involution a long and intricate and complicated grammatical construction
• n involution reduction in size of an organ or part (as in the return of the uterus to normal size after childbirth)
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Webster's Revised Unabridged Dictionary
• Involution That in which anything is involved, folded, or wrapped; envelope.
• Involution The act of involving or infolding.
• Involution (Math) The act or process of raising a quantity to any power assigned; the multiplication of a quantity into itself a given number of times; -- the reverse of evolution.
• Involution (Gram) The insertion of one or more clauses between the subject and the verb, in a way that involves or complicates the construction.
• Involution (Geom) The relation which exists between three or more sets of points, a.a' b.b' c.c', so related to a point O on the line, that the product Oa.Oa' = Ob.Ob' = Oc.Oc' is constant. Sets of lines or surfaces possessing corresponding properties may be in involution.
• Involution (Med) The return of an enlarged part or organ to its normal size, as of the uterus after pregnancy.
• Involution The state of being entangled or involved; complication; entanglement. "All things are mixed, and causes blended, by mutual involutions ."
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Century Dictionary and Cyclopedia
• n involution The act of involving, infolding, or inwrapping; a rolling or folding in or round.
• n involution The state of being entangled or involved; complication.
• n involution Something involved or entangled; a complication.
• n involution A membranous covering or envelop; an involucre.
• n involution In grammar, complicated construction; the lengthening out of a sentence by the insertion of member within member; the separation of the subject from its predicate by the interjection of matter that should follow the verb or be placed in another sentence.
• n involution In mathematics: The multiplication of a quantity into itself any number of times, so as to produce a positive integral power of that quantity. Thus, the operation by which the third power of 5 is found, namely, the multiplication of 5 by itself, making 25, and of the product by 5 again, making 125, is involution. In this sense involution is opposed to evolution, 3 .
• n involution The raising of a quantity to any power, positive, negative, fractional, or imaginary. In this sense involution includes evolution as a particular case.
• n involution A unidimensional continuous series of elements (such as the points of a line), considered as having a definite one-to-one correspondence with themselves, such that infinitely neighboring elements correspond to infinitely neighboring elements, and such that if A corresponds to B, then B corresponds to A: in other words, the elements are associated in conjugate pairs, so that any pair of conjugate elements may by a continuous motion come into coincidence with any other without ceasing, at any stage of the motion, to be conjugate. This is the usual meaning of involution in geometry; it dates from Desargues (1639). There are either two real sibi-conjugate or self-corresponding elements in an involution, when it is called a hyperbolic involution; or there are none, when it is called an elliptic involution. If U = 0, V = 0, W = 0 are three quadratic equations determining three pairs of points in an involution, then these three equations are in a syzygy λU + μ V + ν W = 0; or if the three equations are ax + bxy + cy = 0, a′ x + b′ xy + c′ y = 0, a″x + b″xy + c″y = 0, then the syzygy may be thus written: The six elements are said to be an involution of six, or, if one or two of them are sibi-conjugate, an involution of five or of four elements. If the points of a line in a plane are in involution, let any conic (or degenerate conic) be drawn through any pair of conjugate points, and another conic through any other pair; then any conic through the four intersections of these conics will cut the line in a pair of conjugate points. That point of an involution which cor. responds to the point at infinity is termed the center of the involution.
• n involution Any series of pairs of loci represented by an equation λU + µ V = 0, where λ and µ are numerical constants for each locus, and U = 0 and V = 0 are equations to two loci of the same order.
• n involution Any unidimensional continuum of elements associated in sets of any constant number by a continuous law. According as there are two, three, four, etc., in each set, the involution is said to be quadratic, cubic, quartic (or biquadratic), etc.
• n involution The implication of a relation in a system of other relations.
• n involution In physiology, the resorption which organs undergo after enlargement or distention: as, the involution of the uterus, which is thus restored to its normal size after pregnancy.
• n involution The atrophic or regressive changes occurring in old age.
• n involution In biology, the possession by an organism which is adapted to conditions that are simpler than those under which its allies live, of an organization that is simpler than that of its allies, considered as evidence of inverse or retrograde evolution.
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Chambers's Twentieth Century Dictionary
• n Involution the action of involving: state of being involved or entangled: complicated grammatical construction:
• n Involution (arith.) act or process of raising a quantity to any given power
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Etymology

Webster's Revised Unabridged Dictionary
L. involutio,: cf. F. involution,. See Involve
Chambers's Twentieth Century Dictionary
See Involve.

Usage

In literature:

The whole space wavered and swam with the involutions of an intricate dance.
"Phantastes" by George MacDonald
Bands of adhesion could not share in the process of involution.
"Anomalies and Curiosities of Medicine" by George M. Gould
But what involutions can compare with those of Seven Dials?
"Sketches by Boz illustrative of everyday life and every-day people" by Charles Dickens
This might be the last step in the awful tragedy of the fall and involution of a human soul.
"A Journey in Other Worlds" by John Jacob Astor
Were there obverse meditations of involution increasingly less vast?
"Ulysses" by James Joyce
The chromatic involutions are many and interesting.
"Chopin: The Man and His Music" by James Huneker
Its intricacy and involution is the product of an over-concentration born of empty surroundings.
"English Literature: Modern" by G. H. Mair
All genuine aquatic types have leaves involute in vernation?
"Journals of Travels in Assam, Burma, Bhootan, Afghanistan and TheNeighbouring Countries" by William Griffith
Evolution and involution balance each other and go on concurrently.
"Outspoken Essays" by William Ralph Inge
All this, with what preceded, and what followed, occurred with such involutions of rapidity, that past, present, and future seemed one.
"The Piazza Tales" by Herman Melville
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In news:

The lab also calibrates reference involute, lead and pitch masters that are used worldwide to calibrate and verify gear inspection instruments.
Advent Tool & Manufacturing Debuts New Involute Spline Milling Solution.
After an incisional biopsy, marsupialization of the lesion promoted its involution and stimulated osteogenesis.
Advent Tool & Manufacturing released its indexable form milling platform for involute splines , spur gears.
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In science:

They then deﬁne a two-sided involution, S → T , as a (one-sided) involution on S ∗ ⊗ T .
Simple free star-autonomous categories and full coherence
Moreover, let τ be the involution (that is, involutive antiautomorphism) of Cl(W, q) such that τ (w) = w for any w ∈ W , and let τ¯0 be its restriction to Cl¯0 (W, q).
Some new simple modular Lie superalgebras
The nondegenerate bilinear form ˆb induces the adjoint involution τˆb on V V and, as before, the isomorphism Λ in (2.5) becomes an isomorphism of algebras with involution: Λ : (cid:0)Cl(V ⊕ V ∗ , q), τ (cid:1) → (cid:0)Endk (^ V ), τˆb (cid:1).
Some new simple modular Lie superalgebras
A is said involutive (or in involution ) if equality holds in the above inequality.
A class of overdetermined systems defined by tableaux: involutiveness and Cauchy problem
We call the smallest integer k with this property the involutive index and the corresponding prolongation the involutive prolongation of the tableau.
A class of overdetermined systems defined by tableaux: involutiveness and Cauchy problem
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