AskDefine | Define rigid

Dictionary Definition

rigid adj
1 incapable of or resistant to bending; "a rigid strip of metal"; "a table made of rigid plastic"; "a palace guardsman stiff as a poker" [syn: stiff]
2 incapable of compromise or flexibility [syn: strict]
3 incapable of adapting or changing to meet circumstances; "a rigid disciplinarian"; "an inflexible law"; "an unbending will to dominate" [syn: inflexible, unbending]
4 fixed and unmoving; "with eyes set in a fixed glassy stare"; "his bearded face already has a set hollow look"- Connor Cruise O'Brien; "a face rigid with pain" [syn: fixed, set]
5 designating an airship or dirigible having a form maintained by a stiff unyielding frame or structure [ant: nonrigid]

User Contributed Dictionary

English

Pronunciation

  • /ˈrɪʤɪd/, /"rIdZId/

Adjective

  1. stiff rather than flexible
  2. fixed rather than moving
  3. rigorous and unbending
  4. uncompromising

Derived terms

Translations

stiff
  • Czech: tuhý
  • Dutch: stijf, stijve
  • Finnish: jäykkä, kankea
  • French: rigide
  • German: steif, starr
  • Hebrew: נוקשה
  • Hungarian: merev
  • Japanese: 硬い
  • Norwegian: rigid
  • Portuguese: rígido, rígida
  • Russian: жёсткий (žjóstkij), негибкий (negíbkij)
  • Slovenian: tog
fixed
rigorous, unbending
uncompromising
Translations to be checked

Extensive Definition

In mathematics, suppose C is a collection of mathematical objects (for instance sets or functions). Then we say that C is rigid if every c \in C is uniquely determined by less information about c than one would expect.
It should be emphasized that the above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians.
Some examples include:
  1. Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
  2. Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem.
  3. By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set, say N, or the unit disk. Note that by the previous example, a polynomial is also determined within the set of holomorphic functions by the finite set of its non-zero derivatives at any single point.
  4. Linear maps L(X,Y) between vector spaces X, Y are rigid in the sense that any L \in L(X,Y) is completely determined by its values on any set of basis vectors of X.
  5. Mostow's rigidity theorem, which states that negatively curved manifolds are isomorphic if some rather weak conditions on them hold.
  6. A well-ordered set is rigid in the sense that the only (order-preserving) automorphism on it is the identity function. Consequently, an isomorphism between two given well-ordered sets will be unique.
  7. A rigid motion of a subset of Euclidean space is not always defined the same: it may be any distance-preserving transformation of the collection of points (i.e. a composition of translations, rotations, and reflections), or only those preserving orientation (i.e. a composition of translations and rotations). In the latter case the concept of rigidity is analogous to that of a physically inflexible solid, which must be moved as a single entity so that its movement (up to atomic motions indiscernible to the naked eye) is completely determined by the displacement of a single "point" and the orientation of the solid body about that point. More generally, a rigid motion of a metric space is a (self)-isometry.
rigid in Dutch: Rigiditeit

Synonyms, Antonyms and Related Words

abiding, accurate, adamant, adamantine, at a standstill, attentive, austere, buckram, careful, cast-iron, changeless, close, conscientious, constant, continuing, correct, critical, delicate, demanding, detailed, determined, direct, dogged, dour, draconian, durable, enduring, even, exact, exacting, exigent, express, exquisite, faithful, fine, finical, finicking, finicky, firm, firm as Gibraltar, fixed, flinty, frozen, fundamentalist, fussy, grim, hard, hard and fast, hard-core, hard-line, harsh, headstrong, hidebound, immobile, immotile, immotive, immovable, immutable, implacable, impliable, in buckram, inelastic, inerrable, inerrant, inexorable, infallible, inflexible, infrangible, intact, intransigent, inviolate, iron, ironbound, ironclad, ironhanded, irreconcilable, irremovable, lasting, literal, made of iron, mathematical, meticulous, micrometrically precise, microscopic, minute, mulish, muscle-bound, narrow, nice, obdurate, obstinate, orthodox, painstaking, particular, pat, permanent, perpetual, persevering, persistent, pigheaded, pinpoint, pokerlike, precise, precisianistic, precisionistic, prim, procrustean, punctilious, punctual, purist, puristic, puritan, puritanic, quiescent, ramrodlike, refined, relentless, religious, religiously exact, remaining, renitent, resolute, resolved, rigorist, rigoristic, rigorous, rock-ribbed, rockbound, rodlike, scientific, scientifically exact, scrupulous, scrutinizing, set, severe, single-minded, solid, sound, square, stable, standpat, starch, starched, starchy, static, stationary, staunch, staying, steadfast, steely, stern, stiff, stiff as buckram, stilted, stout, straightlaced, strait-laced, straitlaced, strict, stringent, strong, stubborn, sturdy, subtle, sustained, taut, tenacious, tense, thorough, tight, torpid, tough, unadaptable, unaffected, unalterable, unaltered, unbending, unbreakable, unchangeable, unchanged, unchanging, unchecked, uncompliant, uncompromising, undestroyed, undeviating, unerring, unfading, unfailing, unflexible, ungiving, unmovable, unmoved, unmoving, unqualified, unrelaxed, unrelenting, unshifting, unswerving, unvaried, unvarying, unwavering, unyielding, virgate
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