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...
12 KB (1,784 words) - 05:04, 21 May 2023
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...
33 KB (4,569 words) - 21:18, 22 March 2025
matrix, then the scalar quantity ε T Λ ε {\displaystyle \varepsilon ^{T}\Lambda \varepsilon } is known as a quadratic form in ε {\displaystyle \varepsilon }...
6 KB (969 words) - 00:01, 31 July 2024
L-theory (category Quadratic forms)
{\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...
6 KB (1,062 words) - 19:23, 15 October 2023
Signature (topology) (category Quadratic forms)
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...
5 KB (795 words) - 22:35, 21 May 2025
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...
5 KB (778 words) - 17:44, 18 May 2025
Witt group (redirect from Witt ring (forms))
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...
21 KB (3,163 words) - 18:06, 2 May 2025
Orthogonal group (category Quadratic forms)
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...
56 KB (7,881 words) - 20:44, 2 May 2025
Symmetrization (section Bilinear forms)
1 {\displaystyle 1=-1} ). This leads to the notion of ε-quadratic forms and ε-symmetric forms. In terms of representation theory: exchanging variables...
5 KB (768 words) - 01:53, 21 February 2024
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...
21 KB (1,926 words) - 08:25, 22 August 2024
Intersection theory (redirect from Intersection form)
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...
17 KB (2,216 words) - 20:50, 8 April 2025
Quadric (redirect from Quadratic surface)
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...
41 KB (7,423 words) - 17:19, 10 April 2025
Oppenheim conjecture (category Quadratic forms)
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...
6 KB (740 words) - 09:05, 21 November 2024
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...
42 KB (5,900 words) - 19:09, 10 May 2025
^{(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...
70 KB (8,960 words) - 08:03, 25 May 2025
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...
54 KB (5,539 words) - 21:19, 19 January 2025
Hilbert symbol (category Quadratic forms)
multiplicative group of non-zero elements K × {\displaystyle K^{\times }} , the quadratic Hilbert symbol is the function K × × K × → { ± 1 } {\displaystyle K^{\times...
11 KB (1,645 words) - 03:54, 4 May 2025
(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...
23 KB (2,832 words) - 13:49, 2 February 2024
real quadratic fields. In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining...
31 KB (4,651 words) - 00:20, 3 March 2025
Clifford algebra (category Quadratic forms)
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...
65 KB (9,287 words) - 07:33, 12 May 2025
the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x2 + y2 + 10z2 represents all integers that it represents locally, with...
127 KB (16,742 words) - 22:11, 3 May 2025
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...
11 KB (1,779 words) - 17:50, 10 May 2025
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...
30 KB (4,691 words) - 12:23, 28 February 2025
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...
2 KB (281 words) - 11:24, 15 October 2024
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...
21 KB (3,297 words) - 11:34, 30 April 2025
Hooke's law (section General tensor form)
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...
56 KB (9,420 words) - 16:09, 7 May 2025
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...
29 KB (3,492 words) - 20:41, 25 January 2025
green colored quadratic term coefficient of the Tschirnhaus key. The mentioned zero valued quartic coefficient of the Bring Jerrard final form is accurately...
28 KB (3,927 words) - 11:41, 2 March 2025
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...
10 KB (1,235 words) - 13:38, 25 May 2025
Time complexity (redirect from Quadratic time)
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...
41 KB (5,003 words) - 04:16, 18 April 2025