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The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.
Closed-form expression, a finitary expression Closed differential form , a differential form α {\displaystyle \alpha } whose exterior derivative d α {\displaystyle d\alpha } is the zero form 0 {\displaystyle 0} , meaning d α = 0 {\displaystyle d\alpha =0} .
For degrees three and four, there are closed-form solutions in terms of radicals, which are generally not convenient for numerical evaluation, as being too complicated and involving the computation of several n th roots whose computation is not easier than the direct computation of the roots of the polynomial (for example the expression of the ...
The OLS method minimizes the sum of squared residuals, and leads to a closed-form expression for the estimated value of the unknown parameter vector β: ^ = (), where is a vector whose ith element is the ith observation of the dependent variable, and is a matrix whose ij element is the ith observation of the jth independent variable.
The closed-loop transfer function is measured at the output. The output signal can be calculated from the closed-loop transfer function and the input signal. Signals may be waveforms, images, or other types of data streams. An example of a closed-loop block diagram, from which a transfer function may be computed, is shown below:
Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: ″ + ′ + =, = which is a second-order linear differential equation.
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter.