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]
- /ˈrɪʤɪd/, /"rIdZId/
Translations to be checked
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:
- Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
- 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.
- 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.
- 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.
- Mostow's rigidity theorem, which states that negatively curved manifolds are isomorphic if some rather weak conditions on them hold.
- 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.
- 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
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