Cubic spline function representation

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

Thermodynamic properties for As( ,cr) were calculated from a least-squares representation using a cubic-spline method described previously (Archer, 1992 Archer et al., 1996). Briefly, a function f(T) was used, where ... 

The only requirement for the knot sequence is that it must be a non-decreasing sequence of numbers. When ti = ti+i it indicates a multiple knot and the segment Qj is reduced to a point. This is one of the great advantages with non-uniform B-splines since it offers great flexibility in the representation ol functions. For example [0, 0, 1, 1, 1, 1, 2, 3, 4, 4] is a valid sequence of knots. The knot value 0 has multiplicity of 2, knot value 1 has multiplicity of 4 and so on. The multiplicity is used to control the continuity of a point. The higher the multiplicity, the less smooth the spline function at this point becomes. A curve segment Qj in cubic B-splines is defined by four control points... 

Equations (4) and (5) are not evaluated explicitly in the minimization program, but are fit using a combination of spline [17] methods, which provide stability, the ability to filter noise easily, and the flexibility to describe an arbitrarily shaped potential curve. Moreover, the final functional form is inexpensive to evaluate, making it amenable to global minimization. The initial step in our methodology is to fit the statistical pair data for each amino acid and for the density profile to Bezier splines [17]. In contrast to local representations such as cubic splines, the Bezier spline imposes global as well as local smoothness and hence effectively eliminates the random oscillatory behavior observed in our data. 

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