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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 Function s 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 , Polynomial s in C are rigid in the sense that any polynomial is completely determined by its values on any countably 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 L(''X'',''Y'') is completely determined by its values on any set of Basis Vector s 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 any distance-preserving transformation of the collection of points (i.e. a composition of translations, rotations, and reflections). Here, 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". More generally, a rigid motion of a metric space is a (self)- Isometry . |
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