### Table 5.3 Example 5.2: numerical results obtained with tensor product polynomials.

2007

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

### Table 5.3 Example 5.2: numerical results obtained with tensor product polynomials.

### Table 9: Measured multiplicities for vector mesons. The rates as measured by the LEP experiments in comparison to ALEPH tuned model predictions. Jetset 7.4 includes tensor production.

"... In PAGE 23: ... 5 Conclusion The inclusive production of 0(770), !(782), K 0(892), and (1020) in hadronic Z decays has been studied and compared to model predictions. The total multiplicities are collected in Table9 and compared to other measurements at LEP. Predictions from the models Jetset 7.... ..."

### Table 1, i.e. k = PBc + Bp + b and l = Bc + b. The distribution basis for a multi-dimensional array can be expressed as a tensor product of the distribution bases for each dimension.

1994

"... In PAGE 5: ...Table1 : Index mapping functions for regular data distributions. BLOCK CYCLIC CYCLIC(b) local to global k = p(dN=Pe) + l k = lP + p k = (l div b)bP + bp + l mod b global to local l = k mod dN=Pe l = k div P l = (k div Pb)b + k mod b global to proc p = k div dN=Pe p = k mod P p = (k div b) mod P k global index 0 k N ? 1; l local index 0 l lt; db ?dN=(Pb)e ; p processor 0 p lt; P.... In PAGE 5: ... Techniques developed in [11] can be used for the array redistribution in the general case. For identity alignments, the relationships between the global index, the local index and the processor index for regular data distributions of a one-dimensional array are shown in Table1 . The indexing for arrays A and A loc begins at zero and the processors are numbered from 0 to P ? 1.... In PAGE 8: ... For example, under a BLOCK distribution the array is partitioned into segments of size NP . The relationship between the global index k, the processor index p, and the local index l as shown in Table1 can be represented by the equality eN k = eP p eNP l ; where p = k div NP and l = k mod NP . In the above identity, the index of vector basis eP p is associated with the processor index on which element A(k) is located after being distributed using a BLOCK distribution.... ..."

Cited by 8

### Table 1, i.e. k = PBc + Bp + b and l = Bc + b. The distribution basis for a multi-dimensional array can be expressed as a tensor product of the distribution bases for each dimension.

1994

"... In PAGE 5: ...Table1 : Index mapping functions for regular data distributions. BLOCK CYCLIC CYCLIC(b) local to global k = p(dN=Pe) + l k = lP + p k = (l div b)bP + bp + l mod b global to local l = k mod dN=Pe l = k div P l = (k div Pb)b + k mod b global to proc p = k div dN=Pe p = k mod P p = (k div b) mod P k global index 0 k N ? 1; l local index 0 l lt; db ?dN=(Pb)e ; p processor 0 p lt; P.... In PAGE 5: ... Techniques developed in [11] can be used for the array redistribution in the general case. For identity alignments, the relationships between the global index, the local index and the processor index for regular data distributions of a one-dimensional array are shown in Table1 . The indexing for arrays A and A loc begins at zero and the processors are numbered from 0 to P ? 1.... In PAGE 8: ... For example, under a BLOCK distribution the array is partitioned into segments of size NP . The relationship between the global index k, the processor index p, and the local index l as shown in Table1 can be represented by the equality eN k = eP p eNP l ; where p = k div NP and l = k mod NP . In the above identity, the index of vector basis eP p is associated with the processor index on which element A(k) is located after being distributed using a BLOCK distribution.... ..."

Cited by 8

### Table 6: Number of I/O passes for the tensor product IR AV IC with various data distributions. D = 16, Bd = 512, M = 222, and N = RV C.

1996

"... In PAGE 27: ... We now show that by using an appropriate cyclic(B) data distribution, a better performance program can be synthesized for most of the cases. Several typical examples are shown in Table6 . We notice that when we increase B, we can reduce the number of passes of data access for most of the cases and the decrease in the number of passes can be as large as eight times.... ..."

Cited by 7

### Table 2. Comparison: Tensor-product B-splines, MARS, and PIMPLE Function Distr. L2 Fit L1 Fit MARS PIMPLE

"... In PAGE 12: ...ixture: .95 N(0; 2) + .05 N(0; 25 2), and Slash: N(0, 0.01)/U(0,1). Table2 gives the values of NE computed from 1000 replicates of sample size n = 100. The associated SD apos;s are standard deviations of individual NE apos;s from simulated samples.... ..."

### Table 2: Number of I/O passes for tensor productIR AV IC with various data distributions. D = 16, Bd = 512, M = 222, and N = RV C.

"... In PAGE 8: ...We now show thatbyusing anappropriate cyclic(B) datadis- tribution, abetterperformance program canbe synthesized. Sev- eral typical examples are shown in Table2 . We notice thatwhen we increase B,we canreduce the numberof passes of dataaccess formostof the cases andthe decrease inthe numberof passes can be as large as eighttimes.... ..."

### Table 3: The number of I/O passes required for the tensor product IR AV IC using a cyclic(B) distribution, where Nt (= M BdD) is the maximum number of the physical tracks in a memory-load.

1996

"... In PAGE 21: ... The method of determining the cost of a tensor product has been discussed in Section 6. The values of C0 for di erent cases can be found in Table3 and Table 4 presented in Section 8.... In PAGE 21: ...ound in Table 3 and Table 4 presented in Section 8.2. A special case of k = j needs to be further explained. When k = j, we assume that C[j + 1; j] = 0 and we use C[i; k] to represent the cost of grouping all the tensor product factors from i to j together. Because the grouped tensor product is a simple tensor product, the value of C[i; k] in this case can also be determined by using Table3 and Table 4 presented in Section 8.2.... In PAGE 22: ... 8.2 Performance of Synthesized Programs for Tensor Products The number of I/O passes required by the synthesized programs for a tensor product are summarized in Table3 and Table 4 by going through various cases using the approach presented in Appendix C. We can verify that the results presented here are more comprehensive than the results presented in [10].... In PAGE 22: ... However, for those conditions, we can have that C gt; Bd, and V C gt; M. If we further assume that C lt; BdD, then from the results in Table3 , we can synthesize a program with V C... ..."

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