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Order (ring theory)
In ring theory, an order is a subring of the ring of integers of a number field that has finite index there, a concept introduced by Richard Dedekind.
Order of magnitude
An order of magnitude is a factor of ten, the power of ten representing a quantity's scale; about 61 orders separate the Planck length from the observable universe.
Order of operations
The order of operations, also called operator precedence, is the rule that mathematical expressions are evaluated parentheses first, then exponents, then multiplication and division, then addition and subtraction.
Order theory
Order theory is the branch of mathematics that studies partially ordered sets, or posets, together with structures built on them such as lattices and ordered algebraic systems.
Ordinal number
An ordinal number (or ordinal) is the order type of a well-ordered set, introduced by Georg Cantor in 1883 to extend natural numbers to infinite positions.
Orthogonal complement
The orthogonal complement of a subset U of an inner product space is the set of all vectors whose inner product with every vector of U is zero.
Orthogonal group
The orthogonal group, or general orthogonal group, is the classical group of linear transformations preserving a non-singular quadratic form; O(n) consists of the real orthogonal matrices preserving distances in n-dimensional Euclidean space.
Orthogonal iteration
Orthogonal iteration, also called subspace iteration or simultaneous iteration, is a numerical linear algebra algorithm that generalizes the power method to compute bases for eigenspaces.
Orthogonal matrix
An orthogonal matrix, also called an orthonormal matrix, is a real square matrix whose transpose equals its inverse; it preserves lengths and angles, representing a rotation, a reflection, or both.
Orthogonal polynomials
Orthogonal polynomials are sequences of polynomials whose distinct members are orthogonal under a weighted inner product; introduced by Legendre in 1783, they underlie Gaussian quadrature, quantum mechanics, and approximation theory.
Orthogonality
Orthogonality is the generalization of perpendicularity to inner product spaces, where two elements are orthogonal when their inner product is zero; it underlies Fourier series and least squares.
Ostrowski's theorem
Ostrowski's theorem is a 1916 result in number theory stating that every nontrivial absolute value on the rational numbers is equivalent to the usual real one or a p-adic one.
Outer product
The outer product of two vectors multiplies every entry of one by every entry of the other, producing an m by n matrix of rank one, written u v^T.
Overdetermined system
An overdetermined system is a system of equations, typically linear, with more equations than unknowns, arising in curve fitting and satellite positioning and usually solved by least squares.
P-adic analysis
P-adic analysis, also known as ultrametric calculus, is the branch of analysis over the p-adic number fields, created by Kurt Hensel around 1899 and central to modern number theory.
P-adic exponential function
The p-adic exponential is the classical exponential power series interpreted over the p-adic numbers, where it converges only on a small disk around the origin, unlike the real exponential.
p-adic field
A p-adic field, also called a p-adic local field, is a field that is a finite extension of the p-adic numbers for a prime p, studied in algebraic number theory.
P-adic Hodge theory
P-adic Hodge theory is a branch of arithmetic geometry that compares the p-adic cohomology theories, étale, de Rham, and crystalline, of algebraic varieties over p-adic fields.
p-adic integer
A p-adic integer is an element of the ring Z_p, written in base p with digits extending infinitely to the left; it completes the ordinary integers under the p-adic norm.
P-adic L-function
A p-adic L-function is a p-adically valued analytic function that interpolates special values of classical L-functions such as the Riemann zeta function, introduced by Kubota and Leopoldt in 1964.
P-adic number
A p-adic number, or p-adic, is a number obtained by completing the rational numbers with a distance based on divisibility by a prime p, introduced by Kurt Hensel in 1897.
p-adic numbers
The p-adic numbers are a family of number systems, one for each prime p, built by completing the rational numbers; introduced by Kurt Hensel in 1897.
P-adic valuation
The p-adic valuation, also called the p-adic order, measures how deeply a rational number is divisible by a fixed prime p, taking the value infinity at 0.
P-group
A p-group, also called a primary group, is a group whose every element has order a power of a fixed prime p, central to finite group theory.
Padovan sequence
The Padovan sequence, also called the Cordonnier sequence, is an integer sequence defined by P(n) = P(n−2) + P(n−3), named by Ian Stewart in 1996.
Palindromic number
A palindromic number, also called a numeral palindrome, reads the same forwards and backwards in some base, such as 121 or 94200249 in decimal.
Parity (mathematics)
Parity (mathematics), also called even and odd numbers, is the property of an integer of being even or odd, depending on whether it is divisible by 2.
Parity bit
A parity bit, also called a check bit, is an extra binary digit appended to data so the count of 1s is even or odd, detecting single-bit errors.
Parity of a permutation
The parity of a permutation, also called its sign or signature, classifies a permutation as even or odd according to the number of transpositions in any decomposition.
Parity of zero
The parity of zero is the mathematical fact that zero is an even number: it is divisible by 2, and it sits between the odd numbers −1 and 1.