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In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x 2 + y 2 + 10z 2 with integral values for x, y and z. [1] [2] Srinivasa Ramanujan considered this expression in a footnote in a paper [3] published in 1916 and briefly discussed the representability of integers in this form.
A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q. A quadratic form q : M → R may be characterized in the following equivalent ways: There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
By definition, a quadric X of dimension n over a field k is the subspace of + defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables , …, +. (A homogeneous polynomial is also called a form, and so q may be called a quadratic form.)
In mathematics, a ternary quartic form is a degree 4 ... definite ternary quartic form over the reals can be written as a sum of three squares of quadratic ...
Ramanujan's ternary quadratic form; S. ... Quadratic form (statistics) Surgery structure set; Sylvester's law of inertia; T. Tensor product of quadratic forms; U. U ...
Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centres of the spheres form a cubic face-centred arrangement, [127] when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms ...
In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Z n, or equivalently a free Z-module of finite rank.
This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of , say, as a ternary quadratic form, then = is the equation of the "conic ".