### Table I. Regression techniques for families of aggregation operators. Operator Approximation method General aggregation operator Monotone tensor product splines Commutative Explicit: tensor product spline on simplex Implicit: symmetrize the data

2003

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### Table 1. Performance of direct sum kernel and tensor product ker- nel in robust KEV adaptation. Results are word accuracies.

"... In PAGE 3: ....2.1. Experiment I: Direct Sum Kernel vs. Tensor Product Kernel We first compare the two types of composite kernels, direct sum kernel and tensor product kernel, using the robust KEV adaptation. The results are shown in Table1 . There is no significant difference between their performance.... ..."

### Table 5: Number of I/O passes for the tensor product IR AV IC.

1996

"... In PAGE 27: ... . D = 16, Bd = 512, M = 222, and N = PQ = 250. 7.2 Tensor Products The number of I/O passes required by the synthesized programs are summarized in Table 3, Table 4, and Table5 by going through various cases of Nt. In those tables, Mt (= M BdD) is the maximum number of physical tracks in a memory-load.... ..."

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### Table 3: Number of I/O passes for the tensor product IR AV IC.

1996

"... In PAGE 26: ...lgorithm in Fig. 13 will nd an e cient size of data distributions. 7.2 Tensor Products The number of I/O passes required by the synthesized programs are summarized in Table3 , Table 4, and Table 5 by going through various cases of Nt. In those tables, Mt (= M BdD) is the maximum number of... In PAGE 27: ... However, for these conditions, we have that C gt; Bd, and V C gt; M. If we further assume that C lt; BdD, then from the results in Table3 and Table 4,... ..."

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### Table 3: Number of I/O passes for the tensor product IR AV IC.

1996

"... In PAGE 27: ... . D = 16, Bd = 512, M = 222, and N = PQ = 250. 7.2 Tensor Products The number of I/O passes required by the synthesized programs are summarized in Table3 , Table 4, and Table 5 by going through various cases of Nt. In those tables, Mt (= M BdD) is the maximum number of physical tracks in a memory-load.... In PAGE 27: ... However, for these conditions, we have that C gt; Bd, and V C gt; M. If we further assume that C lt; BdD, then from the results in Table3 and Table 4,... ..."

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### Table 2. Results for the tensor product Haar basis for full and sparse grids Full grid spaces L N

"... In PAGE 25: ...25 We start with the piecewise constant case. For the full grid spaces VL and for the sparse grid spaces ~ VL, we obtain the results given in Table2 . The computations are performed by using the diagonally preconditioned discretization matrices BL BJL and ^ BL B ^ JL, respectively, as introduced at the end of subsection 2.... In PAGE 26: ... gt; 0 can be taken arbitrarily small (see, e.g., [28, section 7]). For the sparse grid spaces, a formal extrapolation of (37) in conjunction with Proposition 2 and (49) would only give ^ L = O(2?L(1=2? quot;)). Even though slightly better estimates can be proved due to the speci c type of singularity functions involved (as should be expected from comparing with the numerical evidence given in Table2 ), there is no hope for obtaining asymptotically the same approximation rate as for full grid spaces. More importantly, the ultimate goal should be adaptive methods since the presence of edge-corner singularities in the solution f of (47) leads to results that are far from the theoretical optimum.... In PAGE 26: ... See [40] for approximation schemes using graded tensor-product meshes towards the edges that restore these rates asymptotically for the true, low- regularity solution in (2). Furthermore, we see from Table2 that the use of tensor prod- uct Haar functions results in still slightly growing condition numbers for the diagonally preconditioned sti ness matrices and for the cg-... ..."

### Table 1. Calculated error for the bilinear tensor product response surface t. Average of three trials for each value of m.

1994

"... In PAGE 9: ... Note that the results for m = 4; 6, and 9 points represent the minimum number of points needed to construct the bilinear, quadratic, and biquadratic response surfaces, respectively. The errors associated with the bilinear tensor prod- uct response surfaces are given in Table1 and show that the error for the D-optimal points decreased only slightly as the number of points ranged from four to twenty. This was to expected since the D-optimality criteria speci ed points on the perimeter of the design space when bilinear response surface functions were used to form X in equation (7).... ..."

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### Table 5.1 Example 5.1: numerical results obtained with tensor product polynomials.

2007