Order in mathematics may refer to:
Set theory
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Total order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( re ...
and
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
, a binary relation generalizing the usual ordering of numbers and of words in a dictionary
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Ordered set
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; ...
* Order in
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...
, uniform structures in consequence to critical set cardinality
Algebra
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Order (group theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the su ...
, the cardinality of a group or period of an element
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Order of a polynomial (disambiguation) In mathematics, the order of a polynomial may refer to:
*the degree of a polynomial, that is, the largest exponent (for a univariate polynomial) or the largest sum of exponents (for a multivariate polynomial) in any of its monomials;
*the multiplica ...
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Order of a square matrix, its dimension
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Order (ring theory)
In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that
#''A'' is a finite-dimensional algebra over the field \mathbb of rational numbers
#\mathcal spans ''A'' over \mathbb, and
#\mathcal is a \mathbb ...
, an algebraic structure
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Ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...
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Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
Analysis
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Order (differential equation) or
order of highest derivative, of a differential equation
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Leading-order
The leading-order terms (or leading-order corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.m ...
terms
*NURBS order, a number one greater than the degree of the polynomial representation of a
non-uniform rational B-spline
Non-uniform rational basis spline (NURBS) is a mathematical model using B-spline, basis splines (B-splines) that is commonly used in computer graphics for representing curves and Surface (mathematics), surfaces. It offers great flexibility and pr ...
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Order of convergence, a measurement of convergence
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Order of derivation
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
*Order of an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
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Order of a power series, the lowest degree of its terms
*Ordered list, a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
or
tuple
In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...
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Orders of approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.
Usage in science and engineering
In formal expressions, the ordinal number used ...
in Big O notation
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Z-order (curve), a space-filling curve
Arithmetic
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Multiplicative order
In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n.
In other words, the multiplicative orde ...
in modular arithmetic
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Order of operations
In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.
These rules are formalized with a ...
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Orders of magnitude
In a ratio scale based on powers of ten, the order of magnitude is a measure of the nearness of two figures. Two numbers are "within an order of magnitude" of each other if their ratio is between 1/10 and 10. In other words, the two numbers are wi ...
, a class of scale or magnitude of any amount
Combinatorics
*Order in the
Josephus permutation
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Ordered selections and partitions of the twelvefold way in combinatorics
*Ordered set, a
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
,
cyclic order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...
, or
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
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Weak order of permutations
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order.
Definitions
Inversion
Let \pi be a permutation.
There is an inversion of \pi between i and j if i \pi(j). ...
Fractals
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Complexor, or complex order in fractals
*Order of extension in
Lakes of Wada
*Order of
fractal dimension
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured.
It ...
(Rényi dimensions)
*Orders of construction in the
Pythagoras tree
Geometry
*Long-range aperiodic order, in
pinwheel tiling, for instance
Graphs
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Graph order, the number of nodes in a graph
*First order and second order
logic of graphs
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Topological ordering of directed acyclic graphs
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Degeneracy ordering of undirected graphs
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Elimination ordering of chordal graphs
*Order, the complexity of a structure within a graph: see
haven (graph theory)
In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game t ...
and
bramble (graph theory)
Logic
In logic, model theory and type theory:
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Zeroth-order logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
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First-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
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Second-order logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic quantifies on ...
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Higher-order logic
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are m ...
Order theory
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''Order'' (journal), an academic journal on order theory
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Dense order
In mathematics, a partial order or total order < on a is said to be dense if, for all , a total order wherein between any unequal pair of elements there is always an intervening element in the order
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Glossary of order theory
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Lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
, an ordering method on sequences analogous to alphabetical order on words
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List of order topics, list of order theory topics
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Order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, study of various binary relations known as orders
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Order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, ...
, a topology of total order for totally ordered sets
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Ordinal numbers
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
, numbers assigned to sets based on their set-theoretic order
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Partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
, often called just "order" in order theory texts, a transitive antisymmetric relation
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Total order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( re ...
, a partial order that is also total, in that either the relation or its inverse holds between any unequal elements
Statistics
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Order statistics
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
Important ...
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First-order statistics, e.g.,
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
, median, quantiles
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Second-order statistics, e.g.,
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
, power spectrum, variance
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Higher-order statistics, e.g., bispectrum, kurtosis, skewness
{{SIA, mathematics
Mathematical terminology