In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c''
∈
In mathematics, an element (or member) of a Set (mathematics), set is any one of the Equality (mathematics), distinct Mathematical object, objects that belong to that set.
Sets
Writing A = \ means that the elements of the set are the numbers 1, ...
''C'' is uniquely determined by less information about ''c'' than one would expect.
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.
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Examples
Some examples include:
#
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
s on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
#
Holomorphic functions
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
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
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping ...
is an example of such a rigidity theorem.
#By the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
,
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s in C are rigid in the sense that any polynomial is completely determined by its values on any
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set th ...
, say N, or the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
. 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
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s of ''X''.
#
Mostow's rigidity theorem, which states that the geometric structure of negatively curved manifolds is determined by their topological structure.
#A
well-ordered set
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-ord ...
is rigid in the sense that the only (
order-preserving
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
)
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 mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
on it is the identity function. Consequently, an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
between two given well-ordered sets will be unique.
#
Cauchy's theorem on geometry of
convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
s states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules.
#
Alexandrov's uniqueness theorem
The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each othe ...
states that a convex polyhedron in three dimensions is uniquely determined by the
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
of
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s on its surface.
#
Rigidity results in K-theory show isomorphisms between various
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
groups.
#Rigid groups in the
inverse Galois problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved.
There ...
.
Combinatorial use
In
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, the term rigid is also used to define the notion of a rigid surjection, which is a
surjection
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
for which the following equivalent conditions hold:
# For every
,
;
# Considering
as an
-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
, the first occurrences of the elements in
are in increasing order;
#
maps
initial segments of
to initial segments of
.
This relates to the above definition of rigid, in that each rigid surjection
uniquely defines, and is uniquely defined by, a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of
into
pieces. Given a rigid surjection
, the partition is defined by
. Conversely, given a partition of
, order the
by letting
. If
is now the
-ordered partition, the function
defined by
is a rigid surjection.
See also
*
Uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...
*
Structural rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.
Definitions
Rigidity is the property of a structure ...
, a mathematical theory describing the
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of ensembles of rigid physical objects connected together by flexible hinges.
*
Level structure (algebraic geometry) In algebraic geometry, a level structure on a space ''X'' is an extra structure attached to ''X'' that shrinks or eliminates the automorphism group of ''X'', by demanding automorphisms to preserve the level structure; attaching a level structure is ...
References
{{reflist
Mathematical terminology