Function tests

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

In so doing, the test functions W should satisfy the inequality... 

A substitution of a fixed test function % in (2.116) results in the relation... 

In order to verify this, it is sufficient to substitute test functions of the form x + iiito (2.188), where 9 is an infinitely differentiable function with... 

We choose an arbitrary element % G iXo and construct, in accordance with Lemma 2.3 (see below), a sequence xs G Kg which strongly converges to % in Substituting the xs as the test functions into inequality (2.188)... 

In so doing we take into account the inequality (2.246) for the solution iyV,w). Moreover, the relation t = 0 on T, holds. This implies t G K. Substituting t in (2.248) as a test function gives the relation... 

By varying the test function W K, one can deduce that the variational inequality (2.265) is equivalent to the equilibrium equations in flc. 

Let us consider this equation with the test function W = and... 

Introduce the notation U = ui,U2) and take the test functions of the form (W, w) in (3.10). This implies the variational inequality... 

A similar decomposition takes place on Let us substitute in (3.17) the test functions of the form W + W, where smooth functions W belong to W]v > 0 on r, and make use of (3.16). A simple reasoning results in the relations... 

Kg. This allows us to substitute Xt, t) in (3.54) as a test function which implies the inequality... 

In (3.128), choose test functions of the form w + (f, where is a function smooth in having support in a neighbourhood of some fixed point on and such that [d

[Pg.207]

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