The basic reason is that, since w ( x ) can be taken into account " a priori ", the integration error can be made to depend only on the accuracy in approximating f ( x ), regardless of how badly behaved the weight function might be.
32.
An associated issue is the fact that the ratio \ sigma ^ 2 _ { MC } / \ sigma ^ 2 _ { IS } \, overestimates the run-time savings due to importance sampling since it does not include the extra computing time required to compute the weight function.
33.
The functions \ xi _ { + + ~ } and \ xi _ { \ times \ times } can be related to projections ( integrals with certain weight functions ) of the dark matter density correlation function, which can be predicted from theory for a cosmological model through its Fourier transform, the matter power spectrum.
34.
For example, special methods have been developed to apply Clenshaw Curtis quadrature to integrands of the form f ( x ) w ( x ) with a weight function w ( x ) that is highly oscillatory, e . g . a sinusoid or Bessel function ( see, e . g ., Evans & Webster, 1999 ).
35.
Not only the recipe for the computation of the volume element \ omega ( \ mathbf { x } ) \, dx _ 1 dx _ 2 \ cdots dx _ r depends on the chosen parameters, but also the final result, i . e ., the analytic form of the weight function ( measure ) \ omega ( \ mathbf { x } ).
36.
The original notion of variation considered above is the special case of \ scriptstyle \ varphi-variation for which the weight function is the identity function : therefore an integrable function f is said to be a "'weighted " BV " function "'( of weight \ scriptstyle \ varphi ) if and only if its \ scriptstyle \ varphi-variation is finite.
37.
A function whose codomain is an affine space can only be integrated with a weight function that integrates to 1 over the domain of the integration, thus generating a weighted average of the function's values, or a sort of centroid . ( Note that the weight function in the position case must have units of inverse time, and the result is still a position . ) This makes sense, as any value formed from a collection of arbitrary points in space can only be defined relative to those points rather than to some fixed reference on which the points do not depend.
38.
A function whose codomain is an affine space can only be integrated with a weight function that integrates to 1 over the domain of the integration, thus generating a weighted average of the function's values, or a sort of centroid . ( Note that the weight function in the position case must have units of inverse time, and the result is still a position . ) This makes sense, as any value formed from a collection of arbitrary points in space can only be defined relative to those points rather than to some fixed reference on which the points do not depend.
