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 }...

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...

{\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...

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...

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...

In number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x...

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...

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...

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

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...

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...

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...

^{(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...

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

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...

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...

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...

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...

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...

of sub-exponential is non-uniform in terms of ε in the sense that ε is not part of the input and each ε may have its own algorithm for the problem. Some...

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

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

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

defined as numbers in the form a + b ε {\displaystyle a+b\varepsilon } , with a , b ∈ R {\displaystyle a,b\in \mathbb {R} } and ε {\displaystyle \varepsilon...

complexity is O(m3/2 n2).[clarification needed] Given a quadratically constrained quadratic program of the form: minimize d ⊤ x subject to f j ( x ) := x ⊤ A j...

(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...

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...

discriminant of some quadratic form. Take the t first primes p1 = 2, p2 = 3, p3 = 5, ..., pt, for some t ∈ N. Let fq be a random prime form of GΔ with (⁠Δ/q⁠)...