### Table V. Maximum likelihood estimates of parameters in a(a) and b1(a) polynomial splines.

1998

### Table 3 shows computation times for different number of nodes for both Chebychev polynomials and cubic splines. A sharp increase with respect to deterministic case can be explained by both larger amount of computation involved and slower convergence rates. Nevertheless, Chebychev polynomials remain superior with respect to cubic splines just as in the deterministic case.

2001

"... In PAGE 22: ... Table3 : Computation times for different number of approximation nodes in deterministic case. Since value functions in stochastic case are smoother than in deterministic case, each value function being effectively an average of the three next period value functions, the residuals in the stochastic case are not worse than those in the deterministic case.... ..."

Cited by 2

### Table 1. Simulated Example 1: the means and standard errors (in parenthe- ses) of ^ c1, ^ c2 and the AISEs of ^ 11, ^ 12, ^ 21, ^ 22 by two methods: marginal integration and polynomial spline. Integration t c1 = 2 c2 = 1

2005

"... In PAGE 9: ...umber of interior knots Nn selected by AIC as in Subsection 3.2. The functions f lsg2;2 l=1;s=1 are estimated on a grid of equally-spaced points xm; m = 1; : : : ; ngrid with x1 = 0:975 ; xngrid = 0:975 ; ngrid = 62. Table1 reports the means and standard errors (in the parentheses) of f^ clgl=1;2 and the AISEs of f^ lsgs=1;2 l=1;2 for all the three ts. The spline ts are generally comparable, with the cubic t better than the linear t for larger sample sizes (n = 250; 500), the standard errors of the constant estimators.... ..."

Cited by 2

### Table 3: Percent energy and coe cients (in parentheses) retained for selected thresholding methods and basis functions de ned as: (i) DAUB #n: The standard Daubechies wavelet basis, (Daubechies, 1988, 1992), DAUB#2 is the HAAR basis; (ii) SDAUB#n: The least asymmetric Daubechies wavelet basis, (Daubechies, 1992); (iii) COIF#n Coi ets, a wavelet basis in which the scaling function has vanishing moments, in addition to the wavelet function; (iv) BSPL#m.n Biorthogonal spline bases based on simple polynomial spline functions of degree m and n for and (Chui, 1992); (v) VSPL A variation on BSPL designed to achieve near-orthogonality; (vi) PACKD#n, wavelet packets based on DAUB #n lters. The WALSH basis is a wavelet packet based on the HAAR lter.

"... In PAGE 7: ... This motivated us to consider two elements in the wavelet thresholding process: the choice of the criterion and the choice of the wavelet basis. Table3 presents... In PAGE 8: ... (1994 a,b) to the in nitely supported spline wavelets used by Yamada and Ohkitani (1990) and Meneveau (1991). Notice in Table3 that the energy conservation and compression ratios are robust to the choice of wavelet bases but not with respect to the choice of wavelet thresholding method. It should be noted in Table 3 that the Universal method resulted in a lower energy loss at the expense of a lower compression ratio when compared to the Lorentz threshold function for all atmospheric stability conditions.... In PAGE 8: ... Notice in Table 3 that the energy conservation and compression ratios are robust to the choice of wavelet bases but not with respect to the choice of wavelet thresholding method. It should be noted in Table3 that the Universal method resulted in a lower energy loss at the expense of a lower compression ratio when compared to the Lorentz threshold function for all atmospheric stability conditions. How- ever, the energy losses due to the Lorentz threshold are well within measurement variance uncertainty (see e.... In PAGE 8: ...41 power-laws. These two approaches are considered next. 4.2 Heat and Momentum Flux Conservation While Table3 demonstrates that all thresholding models are able to concentrate much of the turbulent energy in few coe cients, little is known whether these limited coe cients can reproduce covariances between turbulent variables. That is, the thresholding methodology extracts low-dimensional organized perturba- tions (U(o); W(o); T(o) a ) from velocity (U; W) and temperature (Ta) time series measurements, given by U = U(o) + U(r) W = W(o) + W(r) Ta = T(o) a + T(r) a and lters out the high-dimensional part (U(r); W(r); T(r) a ): As a graphical illus- tration, Figure 2 compares the original (U) and the thresholded (U(o)) longitu- dinal velocity time series for all thresholding methods.... In PAGE 9: ... For this purpose, the momentum and heat uxes using the measured (N = 65; 536), Fourier, Universal, and Lorentz thresholded time series are compared for all 8 runs in Table 4. The Haar and v- spline bases are used as illustration since they represent the extremes in energy conservation and percent coe cients thresholded as evidenced in Table3 . The calculations in Table 4 are performed by thresholding and reconstructing each time series using the non-zero Fourier, wavelet or wavelet packet coe cients.... ..."

### Table 1 The basic one-dimensional quadrature rule for cubic B-splines.

2000

"... In PAGE 19: ... A too small number of quadrature points leads to instabilities, in particular, when the quadrature points are not properly spaced; a high polynomial accuracy alone does not suffice. For the tensor- product third order B-splines described at the beginning of section 2, we had good experience with the tensor-product counterpart of the one-dimensional quadrature rule given by Table1 . This quadrature formula is exact for fifth order polynomials and assigns 52 or n = 25 quadrature points to each particle in two space dimensions.... ..."

Cited by 7

### Table. 1). It is worth mentioning that our current surface representation (subdivision, i.e. piecewise polynomial surfaces) cannot represent true cyclides: this is because one cannot represent perfect shapes like spheres or tori using B-Splines. However, our system gives us the best B-Spline approximation of cyclides; with additional subdivision, we can add more degrees of freedom for optimization, and the quality of this approximation improves.

in Significance

2006

### Table 1: Comparison of correct classification rates on the plain COIL 100 dataset with results from (Roobaert amp; Van Hulle 1999): NNC is a nearest neighbor classifier on the direct images, Columbia is the eigenspace+spline recognition model by Nayar et al. (1996), and SVM is a polynomial kernel support vector machine. Our results are given for the template-VTUs setting and optimized-VTUs with one VTU per object.

2003

"... In PAGE 16: ... Roobaert amp; Hulle (1999) performed an extensive comparison of a support vector machine-based approach and the Columbia object recognition system using eigenspaces and splines (Na- yar, Nene, amp; Murase 1996) on the plain COIL 100 data, varying object and training view numbers. Their results are given in Table1 , together with the re- sults of the sparse WTM network using either the template-VTU setup without optimization, or the optimized VTUs with one VTU per object for a fair com- parison. The results show that the hierarchical network outperforms the other two approaches for all settings.... ..."

Cited by 28

### Table 1: Comparison of correct classification rates on the plain COIL 100 dataset with results from (Roobaert amp; Van Hulle 1999): NNC is a nearest neighbor classifier on the direct images, Columbia is the eigenspace+spline recognition model by Nayar et al. (1996), and SVM is a polynomial kernel support vector machine. Our results are given for the template-VTUs setting and optimized-VTUs with one VTU per object.

"... In PAGE 16: ... Roobaert amp; Hulle (1999) performed an extensive comparison of a support vector machine-based approach and the Columbia object recognition system using eigenspaces and splines (Na- yar, Nene, amp; Murase 1996) on the plain COIL 100 data, varying object and training view numbers. Their results are given in Table1 , together with the re- sults of the sparse WTM network using either the template-VTU setup without optimization, or the optimized VTUs with one VTU per object for a fair com- parison. The results show that the hierarchical network outperforms the other two approaches for all settings.... ..."

### Table 1: Comparison of correct classification rates on the plain COIL 100 dataset with results from (Roobaert amp; Van Hulle 1999): NNC is a nearest neighbor classifier on the direct images, Columbia is the eigenspace+spline recognition model by Nayar et al. (1996), and SVM is a polynomial kernel support vector machine. Our results are given for the template-VTUs setting and optimized-VTUs with one VTU per object.

"... In PAGE 16: ... Roobaert amp; Hulle (1999) performed an extensive comparison of a support vector machine-based approach and the Columbia object recognition system using eigenspaces and splines (Na- yar, Nene, amp; Murase 1996) on the plain COIL 100 data, varying object and training view numbers. Their results are given in Table1 , together with the re- sults of the sparse WTM network using either the template-VTU setup without optimization, or the optimized VTUs with one VTU per object for a fair com- parison. The results show that the hierarchical network outperforms the other two approaches for all settings.... ..."

### Table 1: A list of three-dimensional Fourier transforms of various integrable functions used in meshless methods. with the exception of that of the Gaussian, the inverse multiquadric, and the Sobolev spline [28, Theorems 6.10, 6.13, and Page 133], the Fourier transform of each function was computed using (6).

"... In PAGE 6: ...Examples of integrable radial basis functions are given in Table1 . Non-integrable radial basis functions and polynomial terms are sometimes used, examples of which include the multiquadric and the thin-plate spline.... ..."