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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.
When a coefficient is one, it is usually omitted (e.g. is written ). [6] Likewise when the exponent (power) is one, (e.g. 3 x 1 {\displaystyle 3x^{1}} is written 3 x {\displaystyle 3x} ), [ 7 ] and, when the exponent is zero, the result is always 1 (e.g. 3 x 0 {\displaystyle 3x^{0}} is written 3 {\displaystyle 3} , since x 0 {\displaystyle x^{0 ...
Rational Quadratic Forms. London Mathematical Society Monographs. Vol. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029. Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021. Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms ...
One obtains the value f(r) by substitution of the value r for the symbol x in P. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p).
One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied. An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of ...
A quadratic equation is one which includes a term with an exponent of 2, for example, , [40] and no term with higher exponent. The name derives from the Latin quadrus , meaning square. [ 41 ] In general, a quadratic equation can be expressed in the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} , [ 42 ] where a is not zero (if it were ...
The transformation of a linear program to one in standard form may be accomplished as follows. [16] First, for each variable with a lower bound other than 0, a new variable is introduced representing the difference between the variable and bound. The original variable can then be eliminated by substitution. For example, given the constraint
One easy way to establish this theorem (in the case where =, =, and =, which readily entails the result in general) is by applying Green's theorem to the gradient of . An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case ...