### Table 1: Size of the Macaulay matrix.

"... In PAGE 2: ... Using Stirling formula, we can check that this size is (endn) where d = maxi(di). Table1 gives the number of solutions, the size of the Macaulay matrix and the size of the matrix D to invert, for a system of generic polynomial equations of degree 2 in n variables: These gures show the limitation of such approach. Even if the matrices involved in this computation are structured or sparse [?], [?], it will not be so easy to handle linear systems of size 106.... ..."

### Table 4: Values of the coefficients a1 and a2, and mean square error for the application of polynomial and rational filters once on the image pear

"... In PAGE 4: ... (8). As an example Table4 shows the values for the coefficients a1 and a2 ob- tained from the optimization of the unconstrained versions of the three different above mentioned filters on the image pear . For a comparison the derived mean square errors after a single application of these filters with the results of the same filters where the constraint of eq.... ..."

### Table 4: Values of the coefficients a1 and a2, and mean square error for the application of polynomial and rational filters once on the image pear

"... In PAGE 4: ... (8). As an example Table4 shows the values for the coefficients a1 and a2 ob- tained from the optimization of the unconstrained versions of the three different above mentioned filters on the image pear . For a comparison the derived mean square errors after a single application of these filters with the results of the same filters where the constraint of eq.... ..."

### Table 2: The first six differential invariants for nonlinear systems in the plane V = K2 with quadratic coefficients. All sums are from 1 to 2.

in The Symbolic Computation of Differential Invariants of Polynomial Vector Field Systems Using Trees ∗

1995

"... In PAGE 2: ...easily computed in terms of a few basic operations on the space of trees. Our main result is expressed in Theorem 5 and illustrated in Figure 2 and Table2 . It provides a simple and direct combinatorial means of computing dif- ferential invariants.... In PAGE 4: ... Differential invariants are naturally expressed and easily computed in terms of a few basic operations on the space of trees. Our main result is expressed in Theorem 5 and illustrated in Figure 2 and Table2 . It provides a simple and direct combinatorial means of computing dif- ferential invariants.... In PAGE 8: ... This can be done either by hand or using DIFF-INV. See [11], for example and Table2 . Given the differential polynomials, one can then compute a basis using standard symbolic packages.... In PAGE 10: ...for planar systems with quadratic coefficients listed in Table2 . Except for I2, this is the same invariant basis as in Sibursky [11].... ..."

### Table 3.2: Computed polynomial coefficients

2002

### Table 1: Computational Flow of Coefficients of Radial Polynomials

"... In PAGE 3: ...e. Bpqs, which is schematically tabulated in Table1 . According to Table 1, the computation flow starts from Bppp (=1) and proceeds to obtain Bp, p-1, p and Bp, p-2, p using equation (12).... ..."

### Table 2 Zernike Polynomial Coefficients for Corneal First Surface Topography Examination Wavefront Error N n m Coeff (waves) Sum Var (waves^2) Percent Var Radial Poly

"... In PAGE 2: ...30 diopters (D) of slightly asymmetric, with-the-rule astigmatism. In Table2 we show the results of computing the corneal first surface wavefront error with respect to a best focus reference sphere for the optics of the corneal first surface. The sum of the coefficient vari- ances are given in column 5 of this table; 94% of the corneal wavefront error variance is contained in the single term corresponding to astigmatism and the remaining 6% is distributed in the remainder of the terms.... ..."

### Table 1. Average of sum of squares for the degrees of t-nomial multiples. Primitive polynomials with degree 4, 5, 6, 7 are considered.

2005

"... In PAGE 6: ...Table1 for multiples of primitive polynomials having degree d = 4; 5; 6; 7. We take each of the primitive polynomials and then nd out the average of the square of degrees of t-nomial multiples for t = 3; 4; 5; 6; 7.... In PAGE 6: ...2 we get that, since W (f(x); d; t) = W (g(x); d; t), the statistical param- eters based on W (f(x); d; t) or W (g(x); d; t) are also same. In Table1 , it is clear that the entries corresponding to any primitive polynomial and its reciprocal are same. 4 t-nomial multiples of products of primitive polyno- mials We have already mentioned in the introduction that it is important to nd out t-nomial multiples of product of primitive polynomials instead of t-nomial multiples of just a single primitive polynomial.... ..."

Cited by 2

### Table 2 shows the mean square error obtained for the three cubic lters and for the rational lter as reported in [8]. The rational lter had shown to lead to a lower mean square error than the lters presented in [1, 5, 7] for edge preserving noise smoothing.

1998

"... In PAGE 3: ... Table2 : Mean square error for the application of polynomial lters three times on the image \air eld quot; with SNR 6dB... ..."

Cited by 2

### Table 2 shows the mean square error obtained for the three cubic lters and for the rational lter as reported in [8]. The rational lter had shown to lead to a lower mean square error than the lters presented in [1, 5, 7] for edge preserving noise smoothing.

1998

"... In PAGE 3: ... Table2 : Mean square error for the application of polynomial lters three times on the image \air eld quot; with SNR 6dB... ..."

Cited by 2