Spline function ranges

The analytical exploitation of the full dynamic range of a detection principle invariably encompasses nonlinear portion of the concentration response function. The use of cubic spline functions for the description of this relationship is discussed after a short introduction to the theoretical principles of spline approximations. 

The short range repulsive part is represented by a Born-Mayer exponential ( ) type, the region of the well by a Morse potential (M) and the long range attractive part by a dispersion potential with a dipole-dipole and a dipole-quadrupole term. These parts are connected by cubic spline functions (S)... 

The heat capacities of LUCI3 measured experimentally and smoothed by a cubic spline function were reported by Tolmach et al. (1987c). We calculated 6r>, 6 1, 0e2, 0e3, and a on the basis of all these experimental data, except three values in the temperature range 254.24-262.41 K. The characteristic parameters listed in Table 22 were obtained at a comparatively low = 0.05497 value. Comparison with the corresponding data on hexagonal lanthanide trichlorides shows that an increase in the molar volume causes an insignificant decrease in a, noticeable increases in 0 2 and 0E3, and substantial decreases in 0 and 0ei- Such modifications of these parameters change the temperature dependence of heat capacity in a quite definite way. Namely, because of the low 0 and 6ei values, heat... 

An important feature of the method of lines is selection of the basis functions i (co), which determines the precision of (spatial) curve fitting. The piecewise polynomials known as B splines meet the requirements. Curve fitting by means of spline functions entails division of the solution space into subintervals by means of a series of points called knots. Knots may be either single or multiple, a multiple knot being formed by the coincidence of two or more such points. They are numbered in nondecreasing order of location Si, S2,..., 5i,. A normalized B spline of order k takes nonzero values only over a range of k subintervals between knots, and, for example, Bij (co), the ith normalized B spline of order k for the knot sequence s, is zero outside the interval + nonnegative at = s, and w = Si + j, and strictly... 

We consider a square area as shown in Figure 11. This area is divided into 40,000 square elements. When summing the areas of the square elements in the circle in Figure 11a, the value obtained differs from the circular area by 0.1%. The locations of transducers 1, 2, and 3 and the locations of the gas-liquid interface heights are X, x 2, and x3 and Si( i,i/i), s2(x2,y2), and s3( 3,1/3), respectively. In the range x, free surface is calculated by a cubic polynomial function (spline interpolation), as shown by Kreyszig (1999). In the ranges 0 < x < Xj and x3 liquid interface is calculated by a linear extrapolation. The slope of... 

The portion of the surface for the HeH+-H2 system calculated by Benson and McLaughlin,104 and referred to above, lies within the range of H-H separations found by Kutzelnigg et a/.105 to be adequately described at the Hartree-Fock limit. Their Hartree-Fock SCF results should, therefore, be accurate enough to warrant their use in classical trajectory calculations. Such calculations have indeed been performed by McLaughlin and Thompson.111 Spline interpolation functions were used to express the tabular near-Hartree-Fock energies in analytical form. The trajectory calculations for the reaction... 

The quantitative method in Section 2.2 is used to determine the intrinsic magnetization intensity for each voxel. Cubic B-spline basis functions with a partition of 60 interior knots logarithmically spaced between 1 x 10 5 and 10 s are used to represent the relaxation distribution within each voxel. The optimal regularization parameter, A, of each voxel is found within the range between 1 x 10 5 and 5 x 10"18 s by using the UBPR9 criterion. 

The Holder continuity is a function of the scheme and therefore of the coefficients when a scheme is expressed as a linear combination of B-splines. For schemes with small kernels it is possible to use sharp specific arguments to determine this directly, at least for certain ranges of coefficients. We saw above that the continuity degree was determined by the kernel, almost independently of the number of a factors, which merely added a separate term. 

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