Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

* Spin group, in mathematics, a particular double cover of the special orthogonal group SO ( n )

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

In mathematics, two vectors are orthogonal if they are perpendicular.

In mathematics, two lines or curves are orthogonal if they are perpendicular at their point of intersection.

The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O ( n ), which — with its subgroups — is widely used in mathematics and the physical sciences.

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series ; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus ; in numerical analysis as Gaussian quadrature ; in finite element methods as Shape Functions for beams ; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis.

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal group ), differential geometry ( Riemannian metric ), differential topology ( intersection forms of four-manifolds ), and Lie theory ( the Killing form ).

In mathematics, the indefinite orthogonal group, O ( p, q ) is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature ( p, q ).

In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the Gegenbauer polynomials, the Chebyshev polynomials, and the Legendre polynomials.

In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO ( 3 ), is a naturally occurring example of a manifold.

* Jacobi polynomials, a class of orthogonal polynomials in mathematics

In mathematics, two functions and are called orthogonal if their inner product is zero for f ≠ g. How the inner product of two functions is defined may vary depending on context.

In mathematics, a Graeco-Latin square or Euler square or orthogonal Latin squares of order n over two sets S and T, each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair ( s, t ), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair.

In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either + 1 or − 1 and whose rows are mutually orthogonal.

In mathematics, SO ( 8 ) is the special orthogonal group acting on eight-dimensional Euclidean space.

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal.

In mathematics a combination is a way of selecting several things out of a larger group, where ( unlike permutations ) order does not matter.

The mathematics of crystal structures developed by Bravais, Federov and others was used to classify crystals by their symmetry group, and tables of crystal structures were the basis for the series International Tables of Crystallography, first published in 1935.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

A statistical analysis of the effect of dianetic therapy as measured by group tests of intelligence, mathematics and personality.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

He was the first to use the word " group " () as a technical term in mathematics to represent a group of permutations.

* E2 or E < sub > 2 </ sub > is an old name for the exceptional group G2 ( mathematics )

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory ; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below.

In mathematics, specifically group theory, a quotient group ( or factor group ) is a group obtained by identifying together elements of a larger group using an equivalence relation.

# REDIRECT group ( mathematics )

He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses.

In mathematics, given two groups ( G, *) and ( H, ·), a group homomorphism from ( G, *) to ( H, ·) is a function h: G → H such that for all u and v in G it holds that

** Metric tensor, in mathematics, a symmetric rank-2 tensor, used to measure length and angle

In mathematics, the symmetric group S < sub > n </ sub > on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself.

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.

In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.

They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general.

In mathematics, Landau's function g ( n ), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group S < sub > n </ sub >.

* The symmetric function equation of the Riemann zeta function in mathematics, also known as the Riemann Xi function

In mathematics, and in particular linear algebra, a skew-symmetric ( or antisymmetric or antimetric ) matrix is a square matrix A whose transpose is also its negative ; that is, it satisfies the equation If the entry in the and is a < sub > ij </ sub >, i. e. then the skew symmetric condition is For example, the following matrix is skew-symmetric:

In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.

In mathematics, a building ( also Tits building, Bruhat – Tits building, named after François Bruhat and Jacques Tits ) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.

In mathematics and physics, n-dimensional anti de Sitter space, sometimes written, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature.

In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions.

In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced.

In mathematics, the braid group on n strands, denoted by B < sub > n </ sub >, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group S < sub > n </ sub >.

In functional analysis, a branch of mathematics, the Hellinger – Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded.

In mathematics, the symmetric algebra S ( V ) ( also denoted Sym ( V )) on a vector space V over a field K is the free commutative unital associative algebra over K containing V.

In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained.

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

In mathematics, the Siegel upper half-space of degree g ( or genus g ) ( also called the Siegel upper half-plane ) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite.

In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order.

In mathematics, a symmetric polynomial is a polynomial P ( X < sub > 1 </ sub >, X < sub > 2 </ sub >, …, X < sub > n </ sub >) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial.

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.