Rigid

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Rigid

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ?

or is a letter derived from the Latin alphabet. Both glyphs of the majuscule and Lower case forms of this letter are based on the rotated form of a minuscule e; a similar letter with identical minuscule is used in the Pan-Nigerian Alphabet, but has the capital form majuscule , based on a horizontally flipped majuscule E....
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 function
Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice derivative function f : U → R which satisfies Laplace's equation, i.e....
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.

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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ?
?

or is a letter derived from the Latin alphabet. Both glyphs of the majuscule and Lower case forms of this letter are based on the rotated form of a minuscule e; a similar letter with identical minuscule is used in the Pan-Nigerian Alphabet, but has the capital form majuscule , based on a horizontally flipped majuscule E....
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 function
  Harmonic function
  
  In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice derivative function f : U → R which satisfies Laplace's equation, i.e....
  s 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
  Schwarz lemma
  
  In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open set unit disk....
  is an example of such a rigidity theorem.
3. By the fundamental theorem of algebra
  Fundamental theorem of algebra
  
  In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex number coefficients has at least one complex root ....
  , polynomial
  Polynomial
  
  In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
  s in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set
  Infinite set
  
  In set theory, an infinite set is a Set that is not a finite set. Infinite sets may be countable set or uncountable set. Some examples are:* the set of all integers, , is a countably infinite set; and...
  , say N, or the unit disk
  Unit disk
  
  In mathematics, the open unit disk around P , is the set of points whose distance from P is less than 1:The closed unit disk around P is the set of points whose distance from P is less than or equal to one:...
  . 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 ? 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
  Automorphism
  
  In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map the object to itself while preserving all of its structure....
  on it is the identity function. Consequently, an isomorphism
  Isomorphism
  
  In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
  between two given well-ordered sets will be unique.
7. A rigid motion
  Euclidean group
  
  In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
  of a subset of Euclidean space
  Euclidean space
  
  Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in 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
  Isometry
  
  In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
  .