Arbitrary functions

Here, 8(5) is the Dirac 6 function and /(0) is an arbitrary function to be determined [it can be shown that any function of the type/(g,0) leads to the same result because of the h q) function]. By considering Eq. 046) for the z component, we obtain (employing polar coordinates) ... 

Completeness means that any arbitrary function can be expressed exactly as a linear combination of these functions. Mathematically, completeness can be expressed as... 

Let V be a normed space, and J V —> i be an arbitrary functional. We assume that there exists a linear and continuous functional such that for each u G V... 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

By trial and error it was found, however, that the best fit was obtained with the following arbitrary function. 

By assuming a radially symmetric pressure (stress) distribution given by an arbitrary function p(r) over the area of contacts such as would be applied by an... 

Baum also showed that, regardless of the number of hidden layers it may possess, unless a multi-layer perception has at least on the order of O n/log2 neurons, it will be unable to compute an arbitrary function [baumSSa],... 

Observe that Eq. (4-115) is an exact expression for Pe except for the bounding involved in assuming ambiguities (i.e. cases in which no jD(xm,y) is greater than all others) to always cause errors. Thus the arbitrary function, f(y), can have no effect on Eq. (4-115) since it has no effect on which word is decoded when Xj. is transmitted. The function f(p) does have an effect on Eqs. (4-116) and (4-117) however, since it helps determine the set of output sequences for which... 

It can be shown10 that if 4 is an arbitrary function of the independent variables in a wave equation... 

The method of lines formulation for solving Equation (8.52) does not require that T aii be constant, but allows T aiiiz) to be an arbitrary function of axial position. A new value of T aii may be used at each step in the calculations, just as a new may be assigned at each step (subject to the stability criterion). The design engineer is thus free to pick a T au z) that optimizes reactor performance. 

The polymer behaves as a generalized Newtonian fluid, where viscosity is an arbitrary function of shear rate and temperature. 

Inequalities (36), (38) and (39) fail to be true, in general, for arbitrary functions. However, it is plain to show that in this case the inequalities of alternative forms occur ... 

The intuition suggests some things in an attempt to write down the locally one-dimensional (LO.S) scheme in conformity with available constructions of Section 3. In working on the segment 0 < t we introduce a grid - tj = jr, j = 0,1,... with step r = t /ja and involve arbitrary functions subject to the normalization condition... 

In general, the function cp obtained by the application of the operator A on an arbitrary function ip, as expressed in equation (3.1), is linearly independent of Ip. However, for some particular function 0i, it is possible that... 

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