Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Torsion |
TorsionNoun1. A tortuous and twisted shape or position; "they built a tree house in the tortuosities of its boughs"; "the acrobat performed incredible contortions". 2. A twisting force. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "torsion" was first used in popular English literature: sometime before 1856. (references) |
Etymology: Torsion \Tor"sion\, noun. [French expression, from Late Latin expression torsio, from the Latin expression torquere, tortum, to twist. See Torture.]. (references) |
| Domain | Definition |
Health | A twisting or rotation of a bodily part or member on its axis. (references) |
Medicine | The act of twisting or rotating. Source: European Union. (references) |
Metallurgy | Effect produced on a cylindrical or prismatic bar by a couple the moment of which is parallel to the axis of the bar. Source: European Union. (references) |
Physics | Twisting that causes a change in the normal shape and/or size of a body. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In elementary differential geometry in three dimensions, the torsion of a curve measures how sharply it is twisting. It is analogous to curvature in two dimensions. Given a vector-valued function r(t), the torsion at a given value of t isA second meaning of torsion in differential geometry is the torsion tensor, which depends on a connection. It is a (1,2) tensor given by the formula
- ((r' × r")· r(3) ) ||r' × r"||-2.
where [u,v] is the Lie bracket of the two vector fields.
Torsion free connections are considered most frequently - the Levi-Civita connection is assumed to have zero torsion, for instance.
In abstract algebra, the torsion subgroup of an abelian group consists of all elements of finite order. An abelian group is called torsion-free if all its elements have infinite order. In the Tor functors of homological algebra, which arise because tensor product does not in general preserve exact sequences, the symbol Tor does stand for this kind of algebraic torsion, historically speaking anyway. These functors were introduced in order to make systematic the universal coefficient theorem of homology theory, in cases where the homology groups Hi(X,Z) of a space X had some torsion.
Some topological invariants are called torsions: for example the Reidemeister-Schreier torsion of a group acting on a finite complex; and also the analytic torsion defined using Laplacians.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Torsion."
(From Wikipedia, the free Encyclopedia)
In group theory, the torsion subgroup of an abelian group G is the subgroup T of G which consists of all elements of G which have finite order. A group G is called torsion free if every element of G except the identity is of infinite order, and torsion (or periodic) if every element of G has finite order. Of course every finite abelian group is a torsion group. If G is abelian, then the torsion subgroup T is a characteristic subgroup of G (even fully characteristic) and the factor group G/T is torsion free.
Proof of the subgroup property
The set T of all elements of finite order in an abelian group indeed forms a subgroup: write the group G additively. Suppose x and y are in T and m is the product of their orders. Then m (x - y) = mx - my = 0 - 0 = 0, and so x - y is in T.
Note that this proof does not work if G is not abelian, and indeed in this case the set of all elements of G of finite order is not necessarily a subgroup.
Consider for example the infinite dihedral group, which has presentation ({x,y}, {x² = y² = 1}). This group is of countable infinite order, and in particular the element xy has infinite order. Since the group is generated by elements x and y which have order 2, the subset of finite elements generates the entire group.
Examples and further properties
A torsion group need not be finite; for example the direct sum of a countable number of copies of the cyclic group C2 is a torsion group, every element of which has order 2. Nor need there be an upper bound on the orders of elements in a torsion group if it isn't finitely generated, as the example of the factor group Q/Z shows.
Every free abelian group is torsion free, but the converse is not true, as is shown by the additive group of the rational numbers Q.
If G is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup and a torsion free subgroup. In any decomposition of G as a direct sum of a torsion subgroup S and a torsion free subgroup, S must equal T (but the torsion free subgroup is not uniquely determined). This is an important first step in the classification of finitely generated abelian groups.
Even if G is not finitely generated, the size of its torsion free part is uniquely determined, as is explained in more detail in the article on rank of an abelian group.
If G and H are abelian groups with torsion subgroups T(G) and T(H), respectively, and f : G → H is a group homomorphism, then f(T(G)) is a subset of T(H). We can thus define a functor T which assigns to each abelian group its torsion subgroup and to each homomorphism its restriction to the torsion subgroups.
An abelian group G is torsion free if and only if it is flat as a Z-module, which means that whenever K is a subgroup of the abelian group H, then the natural map between the tensor products K ⊗ G and H ⊗ G is injective.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Torsion subgroup."
Synonyms: TorsionSynonyms: contortion (n), crookedness (n), torque (n), tortuosity (n), tortuousness (n). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Convolution | Winding; Verb: convolution, involution, circumvolution; wave, undulation, tortuosity, anfractuosity; sinuosity, sinuation; meandering, circuit, circumbendibus, twist, twirl, windings and turnings, ambages; torsion; inosculation; reticulation; (crossing); rivulation; roughness. |
| Source: adapted from Roget's Thesaurus. | |
| Domain | Title |
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Source: compiled by the editor from various references; see credits. | |
| "Torsion" is generally used as a noun (singular) -- approximately 96.72% of the time. "Torsion" is used about 61 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 96.72% | 59 | 44,010 |
| Noun (proper) | 1.64% | 1 | 339,140 |
| Noun (common) | 1.64% | 1 | 339,140 |
| Total | 100.00% | 61 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "torsion": Angle of torsion ♦ Balance of torsion ♦ coiled torsion spring ♦ inelastic torsion ♦ moment of torsion ♦ Spermatic Cord Torsion ♦ torsion balance ♦ Torsion electrometer ♦ Torsion galvanometer ♦ Torsion head ♦ Torsion indicator ♦ Torsion meter ♦ torsion rod spring ♦ Torsion scale ♦ torsion spring. Additional references. | |
| Hyphenated Usage | |
Beginning with "torsion": torsion-angle, torsion-bar. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "torsion"; alternative meanings/domain in parentheses. | |
Albanian | vërtitje (gyration, gyre, revolution, spin, turn, twirl), rrotullim (circumgyration, cycle, gyration, revolution, rotation, spin, traverse, turn, turning, twirl, wheel, whirlabout), rezistencë përdridhjeje, përdredhje (convolution, curl, kink, Rick, sprain, squirm, turn, twist, wrench, wrick). (various references) | |
Arabic | فتل (curl, kink, lay, rope, torque, twiddle, twine, twist), لي (contortion, flection, flexion, mine, to me, twisting), تعقف, إنفتال, إلتواء (bow, flection, inflection, obliqueness, reflection, reflexion, winding). (various references) | |
Bulgarian | усукване (doubling, quirk, salvo, twine), усукана форма, спираловидна форма, завиване (envelopment), извъртане (chicane, dodge, dodgery, equivocation, evasion, indirection, jugglery, legerdemain, quibble, salvo, shuffle, taradiddle, tergiversation, twist). (various references) | |
Chinese | 扭力. (various references) | |
Czech | krutý (atrocious, bitter, brutal, cruel, dispiteous, extreme, fierce, gory, grim, hard, harsh, heartless, heathenish, rigid, ruthless, savage, severe, unkind, unrelenting, vicious, wolfish), kroucení (roll, tortuosity, wiggle, wrench, wriggle). (various references) | |
Danish | torsion, torsio, vridning (contortion, twist, twisting, wash-out), snoning (amount of twist, lay, snake, snaking, spinning, stitch, turn, twist, twisting), forvridning (dislocation, distortion), drejning (spin, spin effect, spinning, turnery, turning). (various references) | |
Dutch | torsie (twist, twisting). (various references) | |
Farsi | پیچی , پیچش (Curvature, Deflexion, Wrest), پیچ خوردگی (Rick, Screw, Turquoise, Twist), انقباض (Contraction, Pluck, Retraction, Shrinkage, Traction). (various references) | |
Finnish | vääntymä (distortion, warp, warpage), vääntö, kiertyminen (twisting). (various references) | |
French | torsion. (various references) | |
German | Torsion (twisting), Verwindung (helical arrangement, screw dislocation, twist), Verdrehung (adjusting, contortion, cricking, distortion, misrepresentation, perversion, peversion, rolling, spraining, twisting, wrenching). (various references) | |
Greek | συστροφή (contortion, twine, twirl, twist, wring), στρέψη (twisting). (various references) | |
Hebrew | פתול (bend, coil, corded, curve, kink, meander, twist, wrick). (various references) | |
Hungarian | sodrás (backwash, drifting, streamline, twist), csavarodás (curl, earrl, slew, slue, thwarting, twirl, twist, wrenching), csavarás (screw, twist, twisting, warping, wrenching). (various references) | |
Italian | torsione (twist, twisting). (various references) | |
Japanese Kanji | 捩じり (apply torque), 捻転 (twisting). (various references) | |
Japanese Katakana | ね"て" (twisting), ねじり (apply torque). (various references) | |
Korean | 염 . (various references) | |
Manx | torshyn. (various references) | |
Pig Latin | orsiontay.(various references) | |
Portuguese | torção (bent, kink, twist). (various references) | |
Romanian | torsiune (twist), rãsucire (contortion, convolution, twist, winding, wrench, wriggle, wring). (various references) | |
Russian | кручение (cabling, twirl). (various references) | |
Serbo-Croatian | torzija, uvrtanje (nutation). (various references) | |
Spanish | torsión (twist). (various references) | |
Swedish | vridning (swerve, turn, twist, twisting, wiggle, wind, wrench, wriggle, wring), torsion. (various references) | |
Turkish | dönme (apostasy, circumvolution, conversion, convert, cycle, deflection, deflexion, facing, gyration, loop, proselyte, renegade, rotation, spin, swing, tumble, turn, turning, twist, veer, wheel, whirligig, winding), burulma açısı, burulma (buckling, contortion, twisting, writhing), bükülme (contortion, distorsion, flection, flexion, Rick, twist, wrench, wrick). (various references) | |
Ukrainian | скручування (blight, cabling, convolution, warping, wring), скрученість. (various references) | |
Vietnamese | sự xoắn (twist, wrench), sự xe, sự vặn (twist, wring, wrung). (various references) | |
| Source: compiled by the editor from various translation references. | ||
Derivations | |
Words beginning with "torsion": torsional, torsionally, torsions. (additional references) | |
| |
"Torsion" is suggested in spellcheckers for the following: corsino, orsino, Terisio, tersian, tersion, Tirion, Tisio, toarcian, toesin, Torbion, torio, Torion, torison, Tornio, torriani, Torriano, Torrizone, torsen, tortion, Torzhok, Tosio. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| Words rhyming with "torsion" (pronounced 'Tor"sion'): Abrasion, Abscession, Abscision, Abscission, Abstersion, Abstrusion, Abusion, Accension, Accession, Acutorsion, Addression, Adhesion, Admission, Adversion, Affusion, Aggression, Allision, Allusion, Amission, Animadversion, Anteversion, Appension, Apprehension, Appulsion, Arrosion, Ascension, Aspersion, Assession, Aversion, Avision, Avulsion, Catabasion, Cession, Circumclusion, Circumfusion, Circumincession, Coextension, Cohesion, Cointension, Collapsion, Collision, Collusion, Comprehension, Compression, Compulsion, Concession, Concision, Conclusion, Concussion, Condescension. (additional references) |
Scrabble® Enable2K-Verified Anagrams | |
Direct Anagrams: nitroso. | |
| Words within the letters "i-n-o-o-r-s-t" | |
-1 letter: intros, nitros, orison. | |
-2 letters: intro, irons, nitro, noirs, noris, ornis, riots, roost, roots, rosin, rotis, rotos, snoot, snort, tiros, toons, toros, torsi, torso, trios, trois. | |
-3 letters: inro, into, ions, iron, nits, noir, nori, onto, oots, orts, rins, riot, root, roti, roto, rots, snit, snot, soon, soot, sori, sorn, sort, stir, tins, tiro, tons. | |
| Words containing the letters "i-n-o-o-r-s-t" | |
+1 letter: biotrons, hornitos, monitors, notornis, orations, portions, positron, roosting, snootier, sonority, sorption, torsions, tortonis. | |
+2 letters: abortions, consortia, cortisone, ironstone, motioners, motorings, notorious, obtrusion, orthicons, pinkroots, pliotrons, positrons, remotions, rogations, rotations, serotonin, sorptions, sortition, tandooris, tonsorial, tornillos, torsional, troponins. | |
+3 letters: absorption, adorations, adsorption, aeronomist, agronomist, anisotropy, apportions, astronomic, cartoonish, cartoonist, coinvestor, confiteors, conformist, consistory, consorting, consortium, cortisones, creosoting, desorption, disrooting, distortion, dominators, entropions, ergonomist, extortions, footprints, formations, goitrogens, hoofprints, innovators, ironstones, jointworms, microtones, monitories, moronities, motorising, neotropics, nominators, nonprofits, nonrioters, nonsteroid, nonstories, obtrusions, octonaries, operations, ornithoses, ornithosis, ostensoria, outscoring, outsnoring, outsoaring, postprison, pourpoints, prenotions, probations, prognostic, promotions, pronations, prorations, protrusion, reposition, reshooting, resolution, resorption, rhodonites, serotinous, serotonins, sonorities, sortitions, stainproof, trombonist. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Commercial | 5. Images: Slideshow 6. Usage Frequency 7. Expressions 8. Expressions: Internet | 9. Translations: Modern 10. Derivations 11. Rhymes 12. Anagrams | 13. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
