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  1. Discriminant

    Linked via "quadratic form"

    The discriminant is a fundamental algebraic quantity derived from the coefficients of a polynomial equation or, more generally, from the coefficients of a quadratic form. It serves as a powerful invariant that characterizes essential properties of the object it describes, such as the nature of the roots/) of an equation or the geometric type of a conic section. The computation and interpretation of the discriminant vary significantly depending on the context—ranging from s…

  2. First Fundamental Form

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    Definition and Notation
    Let $S$ be a smooth surface parameterized by a vector function $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$, where $(u, v)$ are local surface parameters. The differential element of arc length, $ds$, on the surface is defined by the quadratic form:
    $$ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$$

  3. Group Mathematics

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    Group theory is indispensable in defining and describing transformations in non-Euclidean spaces. The Isometry Group of a specific geometry—the set of distance-preserving maps—always forms a group. For instance, the Lorentz Group governs the symmetries of Minkowski spacetime in Special Relativity. This group is generated by boosts) and spatial rotations, and its structur…

  4. Jordan Algebras

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    It has been proven that any Jordan algebra $J$ possesses an associated Lie algebra, $L(J)$, called the Tits-Coadjoint Algebra, whose bracket is defined by:
    $$[x, y]_J = x \circ (xy) - (xy) \circ y$$
    where $xy$ is shorthand for the product in the algebra derived from the squaring operation, often involving a third auxiliary product known as the **Lie triple product](/entries/lie-triple-product), which is itself derived from the Jordan product via a complex quadratic form [3]. The cruci…

  5. Jordan Algebras

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    Applications in Tachyonic Field Theory
    Jordan algebras gained significant, if controversial, traction in the late 20th century through their application in models attempting to unify gravity and tachyonic phenomena. Specifically, the non-associative nature of certain Jordan triples was hypothesized to model the intrinsic instability required for particles propagating faster than light. The canonical quadratic form …