Search results
Results From The WOW.Com Content Network
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables (,) = + +,where a, b, c are the coefficients.When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form.
A form f is itself a covariant of degree 1 and order n.. The discriminant of a form is an invariant.. The resultant of two forms is a simultaneous invariant of them.. The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix
Note that there is a close relation between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality gives a unary operation on the set of reduced forms, which cycles through all reduced forms in one equivalence class.
Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and ...
The Hasse–Minkowski theorem reduces the problem of classifying quadratic forms over a number field K up to equivalence to the set of analogous but much simpler questions over local fields. Basic invariants of a nonsingular quadratic form are its dimension , which is a positive integer, and its discriminant modulo the squares in K , which is ...
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae , [ 1 ] a textbook on number theory published in 1801, in Articles 234 - 244.
Over F 2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form , and it is 1 if the form is a direct sum of + + with a number of copies of . William Browder has called the Arf invariant the democratic invariant [ 3 ] because it is the value which is assumed most often by the quadratic form. [ 4 ]