### Table 1. QoSMT problem with 2 rates. Runtime and approximation ratios of previ- ously known algorithms and of the algorithms given in this paper. In the runtime, n and m denote the number of nodes and edges in the original graph G = (V; E), respec- tively. Approximation ratios associated with polynomial-time approximation schemes are accompanied by a + to indicate that they approach the quoted value from above and do not reach this value in polynomial time.

2003

Cited by 5

### Table 1: Approximation algorithms in this paper.

2003

"... In PAGE 4: ...Table 1: Approximation algorithms in this paper. Both the approximation factors and the time bounds depend on the properties of the regions and the set of orientations; the results are summarized in Table1 . More speci cally, in Section 4 we give a polynomial time approximation scheme (PTAS) when the graph is a tree.... ..."

Cited by 4

### Table 1 Polynomials to implement the first division and maximum N-input XOR required.

"... In PAGE 3: ... input and four-input XORs) can be used while the required processing speed is achieved. Two other polynomials to implement the above computation scheme are listed in Table1 , along with MIz3 and G*(x). Other compromises are possible if wider XORs (three- Final division According to the scheme described here, the final division must still be carried out with G(x).... In PAGE 4: ... For a single-cell message (the worst case), which is processed in 480 ns, the final division accounts for approximately 10% of the total computation time. The two-step scheme described in this paper is summarized in Figure 2 for The three polynomials given in Table1 are the best that 708 the author was able to find in an exhaustive search up to degree 128 of multiples of G(x). (Only those multiples with consecutive terms having exponents differing by 8 or more were retained.... ..."

### Table 4: Builtin Chebyshev Polynomials: Approximation of Tn(1)

1998

"... In PAGE 3: ... Tables 3{4 show the calculation times of Tn(1) in exact and approximate modes, respectively. Note that REDUCE with on rounded did not calculate accurate approximations for large n, indicated in Table4 by the symbol 3. This bug is xed by now.... ..."

### Table 5* Low Openness Cluster Nahar-Inder Tests for Output Convergence to the Leader, USA

2003

"... In PAGE 15: ... Accordingly, it is redundant to report p-values when these calculated t-statistics are negative. The low openness group results in Table5 are again based on the United States being taken as the group leader. Here we see that there are only two instances where there is significant evidence of convergence, namely between the United States and each of Canada and Japan.... ..."

### Table 5. Computational time of approximation schemes relative to optimum

1999

Cited by 4

### Table 1 shows how much we can at best hope to reduce the degree by introducing a change of parame- ter. A related question is what is the best possible approximation order for a given polynomial degree? The answer to this question for some low degrees can be read off from Table 1. For degree 2 we may get approximation order 4, whereas for degree 4 we may get order 8 (if the corresponding equations are solvable). In fact, the natural conjecture to draw from Table 1 is that for polynomials of degree n we may get approximation order 2n, just like for planar parametric curves. However, this is too optimistic. If we consider approximating surfaces of degree n with approximation order m, the number of conditions we enforce must not exceed the number of free parameters. This condition is expressed by the inequality

in Computer Aided Geometric Design 22 (2005) 838–848 On geometric interpolation of parametric surfaces

2005

"... In PAGE 4: ... Here we have so many free variables that we may even annihilate the quartic term which requires 15 conditions but does not introduce any new parameters. All these and more possible schemes are summarized in Table1 . We note that the first perfect match between parameters and... In PAGE 11: ... In fact, for a cubic change of parameter to leave the edges of a triangle fixed requires eighteen scalar conditions. Referring back to Table1 we see that in reducing a cubic to quadratic we have two extra free parameters in addition to the standard six, leaving us ten parameters short to fix the edges of a triangle. In fact, by counting degrees of freedom and parameters one finds that the first time there are enough free parameters to both keep the edges of a triangle fixed and reduce the degree by one is the reduction from degree seven to six, which is hardly practical.... ..."