In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,

is a quadratic form in the variables x and y.

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).

... An integral quadratic form whose image consists of all the positive integers is sometimes called universal. 4 ≤ d ≤ 14 {1,2,5,d}, 6 ≤ d ≤ 10 There are also forms whose image consists of all but one of the positive integers ... Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms if all coefficients are integers, then it represents all positive integers if and only if it represents ...
Arf Invariant Of A Knot
... Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface ... a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group ... The Arf invariant of this quadratic form is the Arf invariant of the knot ...
Spin Representation - Set Up
... dimensional real or complex vector space with a nondegenerate quadratic form Q ... The (real or complex) linear maps preserving Q form the orthogonal group O(V,Q) ... For V real with an indefinite quadratic form, this terminology is not standard the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to ...
Ε-quadratic Form - Intuition - Refinements
... An intuitive way to understand an ε-quadratic form is to think of it as a quadratic refinement of its associated ε-symmetric form ... the tensor algebra by relations coming from the symmetric form and the quadratic form vw + wv = 2B(v,w) and ... relation follows from the first (as the quadratic form can be recovered from the associated bilinear form), but at 2 this additional refinement is necessary ...