### Table 1: An example of a three-dimensional table.

1993

"... In PAGE 3: ...Table1 has three categories, D1, D2 and D3; thus, it is a three-dimensional table. The logical relationship among the data items of a table is the association between labels and entries.... In PAGE 3: ... Each entry is associated with one or more sets of labels of di erent categories simultaneously. For example, in Table1 , entry e1 is associated with a set of labels fd11; d21; d311g simultaneously; entry e7 is associated with both fd12; d21; d312g and fd12; d22; d312g simultaneously. The data items and the logical relationship among them provide the logical structure of the table, which is the primary information that a table conveys and which is independent of its format.... In PAGE 6: ... This function guarantees that every entry in E is mapped from at least one ff1; ; fng 2 D1 n. Using this model, Table1 can be abstracted by (3; fD1; D2; D3g; E; ), where D1 = fd11; d12g D2 = fd21; d22; d23g D3 = fd31; d32g d31 = fd311; d312g d11 = d12 = d21 = d22 = d23 = d32 = d311 = d312 = fg E = fe1; e2; e3; e4; e5; e6; e7; e8; e9g (fD1:d11; D2:d21; D3:d31:d311g) = e1; (fD1:d11; D2:d21; D3:d31:d312g) = e2; (fD1:d11; D2:d22; D3:d31:d311g) = e3; (fD1:d11; D2:d22; D3:d31:d312g) = e3; (fD1:d11; D2:d23; D3:d31:d311g) = e4; (fD1:d11; D2:d21; D3:d32g) = e5; (fD1:d11; D2:d22; D3:d32g) = e5; (fD1:d11; D2:d23; D3:d32g) = e5; (fD1:d12; D2:d21; D3:d31:d311g) = e6; (fD1:d12; D2:d21; D3:d31:d312g) = e7; (fD1:d12; D2:d22; D3:d31:d312g) = e7; (fD1:d12; D2:d23; D3:d31:d312g) = e8; (fD1:d12; D2:d21; D3:d32g) = e9; (fD1:d12; D2:d22; D3:d32g) = e9; (fD1:d12; D2:d23; D3:d32g) = e9; 4.2 Basic operators in the tabular model We rst describe the syntax of all basic operators in function form by giving the operator identi ers and the types of their operands and results.... ..."

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### Table 1: The number of iterations of the parallel domain decomposition algorithm required to solve a typical three-dimensional convection-diffusion problem in [12].

"... In PAGE 5: ... Furthermore, it is applied to a class of convection- diffusion equations in three dimensions that is not covered by the underlying theory in [3]. Nevertheless, it proves to be surprisingly robust, as illustrated by the iteration counts shown in Table1 that are typical of the results in [12]. Furthermore, very creditable parallel performances are recorded, including parallel speed-ups in excess of 12 when using locally refined ... In PAGE 5: ...Table 1: The number of iterations of the parallel domain decomposition algorithm required to solve a typical three-dimensional convection-diffusion problem in [12]. The iteration counts shown in Table1 illustrate that the number of iterations of the parallel solver that are required to obtain a converged solution is essentially independent of the level of the finest mesh and the number of subdomains used. Hence, provided the sequential solver used on each processor (at step 4 of the algorithm in Figure 4) has a computational cost of O(N), the total cost of the parallel algorithm will also be approximately proportional to N.... ..."

### Table 2. Two and Three Dimensional Results

in SUMMARY

"... In PAGE 8: ... Under these conditions the difference in the effectiveness of the learning algorithm with targetsfandf can be attributed directly to the additional input dimension. Table2 provides the results of approximatingAx,y) = (x+y)/2 and f(xy,z) = (x*)/2. For the 5/25 system cotiguration, the three-dimensional system generated with 50,000 training examples was less accurate than the two- dimensional approximation produced with 5000 examples.... In PAGE 9: ... As in the case of the propagation model, the localized FAMs may not require all of the dimensions of the input space to produce an appropriate response. In fact, the target function and experimental data shown in Table2 is an example of this type of behavior. The test for the contribution of the ith input dimension begins by constructing the n-dimensional FAM for the region.... ..."

### Table 9: Performance data for two and three-dimensional, unordered, CCFFT on a 2048 processor CM-200.

1992

"... In PAGE 18: ... The latter uses only radix-8 kernels, which are the most e cient. Timings for two- and three-dimensional CCFFT are given in Table9 , and shown in Figure 7. The signi cant increase in performance for the two-dimensional CCFFT between the 1024 1024 array and the 2048 2048 array is due to one of the axis being local to a processor for the larger array.... ..."

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### Table 10: Performance data for two and three-dimensional, ordered, CCFFT on a 2048 processor CM-200.

1992

"... In PAGE 19: ... This part of the axis requires a radix-2 kernel, which is less e cient than the radix-4, and the radix-8 kernels normally used. For reference, performance data for ordered two and three-dimensional transforms are given in Table10 . The execution time increases by 50 - 100% for our examples, considerably more than for entirely local transforms.... ..."

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### Table 2: Breakdown of the computational time for the rst three-dimensional rectangular structure simulation.

"... In PAGE 9: ...ase grid of 140 12 24 cells (cf. Fig. 16) and utilises between 1:3 M and 1:5 M cells, instead of 8:7 M cells like a uniformly re ned grid. Table2 clearly re ects the in- creased expense in solving the hydrodynamic equations in three space dimensions. After a settling time of about 20 periods, a regular cellular oscillation with identical strength in x2- and x3- direction can be observed.... ..."

### Table2 Description of three-dimensional meshes.

"... In PAGE 6: ... In short, we made these meshes havevery small #28and hence many#29 elements along lines that would result in a split into pieces of roughly equal size. Most meshes in Table2 were obtained from tetrahedralization of objects such as an aircraft #0Dap, or of a domain enclosed between two spherical sections. Twoof the meshes in Table 2, #5Csphere 5 quot; and #5Csphere 6 quot;, are surface triangulations of three dimensional objects.... In PAGE 6: ... Most meshes in Table 2 were obtained from tetrahedralization of objects such as an aircraft #0Dap, or of a domain enclosed between two spherical sections. Twoof the meshes in Table2 , #5Csphere 5 quot; and #5Csphere 6 quot;, are surface triangulations of three dimensional objects. Observe that meshes in the test suite are highly graded and irregular, with element sizes that vary by factors larger than 1000 in the L 1 norm.... ..."

### Table 6. Three-dimensional compressible derivatives

1997

"... In PAGE 5: ... In this test, the cost function is a combination of the lift and drag coefficients so that only one adjoint solution is required. The derivatives of the cost with respect to the angle of attack and the Mach number as well as the derivatives with respect to four of the shape parameterization variables are shown in Table6 . As can be seen, the consistency between the derivatives obtained with the ad- joint formulation and finite differences is excellent.... ..."

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### Table 6. Three-dimensional compressible derivatives

1997

"... In PAGE 5: ... In this test, the cost function is a combination of the lift and drag coefficients so that only one adjoint solution is required. The derivatives of the cost with respect to the angle of attack and the Mach number as well as the derivatives with respect to four of the shape parameterization variables are shown in Table6 . As can be seen, the consistency between the derivatives obtained with the ad- joint formulation and finite differences is excellent.... ..."

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### Table 3: Comparison of asymptotic convergence factors per cycle between the three-dimensional basic problem (with Dirichlet side-boundary conditions) and the three-dimensional model problem (with integral constraints). The grid is uniform in a cube, and linear vertical interpolation (Fig. 3c) is used in the prolongation. b = 2 and wb = 1, except the values in parentheses, for which b = wb = 3.

"... In PAGE 6: ...2. The results appear in Table3 . We nd that for relatively large C the performance for the model problem with boundary constraints fully matches that of the basic problem.... In PAGE 6: ... Hence, highly oscil- latory boundary data in uence the interior solution deeper into the domain. Indeed, if we increase wb|the width of the strip near the boundary where additional processing is performed|we regain good behavior (see values in parentheses in Table3 ). Of course, the width of this strip must increase still more as C is decreased, so this approach is not recommended for anisotropic problems.... ..."