the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly...

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2...

matrix, then the scalar quantity ε T Λ ε {\displaystyle \varepsilon ^{T}\Lambda \varepsilon } is known as a quadratic form in ε {\displaystyle \varepsilon }...

{\displaystyle L_{2k}(R)} are defined as the Witt groups of ε-quadratic forms over the ring R with ϵ = ( − 1 ) k {\displaystyle \epsilon =(-1)^{k}} . More precisely...

takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished...

types of quadratic forms. The product zz* is a quadratic form for each of the complex numbers, split-complex numbers, and dual numbers. For z = x + ε y, the...

order 2(q − ε), where ε = ±. Proof For studying the orthogonal group of Oε(2, q), one can suppose that the matrix of the quadratic form is Q = [ 1 0...

Chomsky normal form (first described by Noam Chomsky) if all of its production rules are of the form: A → BC,   or A → a,   or S → ε, where A, B, and...

defined in the same way for skew-symmetric forms, and for quadratic forms, and more generally ε-quadratic forms, over any *-ring R. The resulting groups...

1 {\displaystyle 1=-1} ). This leads to the notion of ε-quadratic forms and ε-symmetric forms. In terms of representation theory: exchanging variables...

forms. It is possible in some circumstances to refine this form to an ε-quadratic form, though this requires additional data such as a framing of the tangent...

theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues, but the first systematic treatment is § IV of Gauss's...

indefinite quadratic form in n variables. Suppose that n ≥ 3 and Q is not a multiple of a form with rational coefficients. Then for any ε > 0 there exists...

^{(n)}+O(\|\epsilon ^{(n)}\|^{3})} where Q k {\displaystyle Q_{k}} is a quadratic form: ( Q k ) i , j = ∑ ℓ ( ( D 2 f ) − 1 ) i , ℓ ∂ 3 f ∂ x j ∂ x k ∂ x ℓ...

value is less than | ε 0 | ≤ 1 17 ≈ 0.059. {\displaystyle \vert \varepsilon _{0}\vert \leq {1 \over 17}\approx 0.059.} The best quadratic fit to 1 / D {\displaystyle...

have dimension two, and are known as quadric surfaces. Their quadratic equations have the form A x 2 + B y 2 + C z 2 + D x y + E y z + F x z + G x + H y...

real quadratic fields. In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining...

concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is ε = t + u d 2 {\displaystyle \varepsilon...

into the quadratic 1 − x 2 {\displaystyle 1-x^{2}} with roots at x = ± 1 {\displaystyle x=\pm 1} . Substituting a regular perturbation series x ( ε ) = x...

multiplicative group of non-zero elements K × {\displaystyle K^{\times }} , the quadratic Hilbert symbol is the function K × × K × → { ± 1 } {\displaystyle K^{\times...

(σ, ε)-Hermitian form, it follows that for all α in K, σ ( ε ) = ε − 1 {\displaystyle \sigma (\varepsilon )=\varepsilon ^{-1}} σ ( σ ( α ) ) = ε α ε − 1...

the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x2 + y2 + 10z2 represents all integers that it represents locally, with...

a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of...

cluster as well. DBSCAN requires two parameters: ε (eps) and the minimum number of points required to form a dense region (minPts). It starts with an arbitrary...

green colored quadratic term coefficient of the Tschirnhaus key. The mentioned zero valued quartic coefficient of the Bring Jerrard final form is accurately...

proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms rather than classes of ideals. It appears...

define quadratic modules first: Let R be a ring with anti-automorphism J, ε ∈ R × {\displaystyle \varepsilon \in R^{\times }} such that r J 2 = ε r ε − 1...

12 ] ≡ [ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] ; [ ε ] = [ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] ≡ [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] {\displaystyle [{\boldsymbol {\sigma...

as: ε _ _ = [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ ε x x 1 2 γ x y 1 2 γ x z 1 2 γ y x ε y y 1 2 γ y z 1 2 γ z x 1 2 γ z y ε z z...

but h log ε, where ε is a fundamental unit. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs...