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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.
The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms. See also
An integral quadratic form has integer coefficients, such as x 2 + xy + y 2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x, y) ∈ Z if x, y ∈ Λ.
Ramanujan's ternary quadratic form; S. Signature (topology) Smith–Minkowski–Siegel mass formula; Spinor genus; Quadratic form (statistics) Surgery structure set;
An integral quadratic form is a quadratic form on Z n, or equivalently a free Z-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings Z p for each prime p and also equivalent over R .
1. A contravariant of a ternary form, giving the equation of a dual curve. (Elliott 1895, p.400) reciprocity Exchanging the degree of a form with the degree of an invariant. For example, Hermite's law of reciprocity states that the degree p invariants of a form of degree n correspond to the degree n invariants of a form of degree p.
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of ...
The oldest problem in the theory of binary quadratic forms is the representation problem: describe the representations of a given number by a given quadratic form f. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any ...