MATHEMATICA Computations of March 2005 for:

Basic Global Relative Invariants for Nonlinear Differential Equations

by Roger Chalkley

Memoirs of the American Mathematical Society, Providence, Rhode Island, November 2007,  Number 888,

ISBN 978-0-8218-3991-1

QA371.C435     2007

Previously, the computations had been done with Version 3.0 of MATHEMATICA and  all of those computations can be downloaded with a click here.  To show that each of the computations included in the manuscript can be performed effectively with Version 5.1 of MATHEMATICA available during March of 2007, we have evaluated all of the preceding notebooks with Version 5.1.  Those evaluated notebooks can be downloaded by clicking the hyperactive text below.

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Page 11, Formulas (1.49) and (1.53);  Page 13, Formula (1.59)    For a check on these formulas, a MATHEMATICA  notebook can be downloaded by clicking  here.  Its evaluation shows directly that these formulas specify relative invariants.

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Page 13,  (1.60), (1.61), and (1.62)    For a check on these formulas, a MATHEMATICA notebook can be downloaded by clicking  here.  This verification that (1.49),  (1.53), and (1.59) specify relative invariants is independent of the preceding one (though it is sufficient to set m = 2 in the preceding one).

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Page 15.  Formula (1.76)    For a MATHEMATICA notebook showing that  (1.75) and (1.77) yield (1.76), click  here.

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Page 16, Formula (1.81); Page 17, Formula (1.82)     For a MATHEMATICA notebook that checks these formulas, click  here.  For the  method used to discovery these formulas along with complete computational details, see the notebook prepared for pages 168-169.  It can be downloaded by clicking  here

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Page 147, Computations for (14.17)    For a MATHEMATICA notebook that checks the computational proof of Theorem 14.8, click  here.  (Avoid Version 5.0.)

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Page 150, Computations for (14.27)    For a MATHEMATICA notebook that checks the computational proof of Proposition 14.13, click  here.   (Avoid Version 5.0.)

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Page 151, Computations for (14.33)    For a MATHEMATICA notebook that provides the computational proof of Proposition 14.14, click  here.  (Avoid Version 5.0.)

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Page 152, Computations for (14.47)    For a MATHEMATICA notebook that provides the computational proof for Formulation 14.15, click  here.   (Avoid Version 5.0.)

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Page 153, Computations for (14.48)    For a MATHEMATICA notebook in Version 3.0 that provides the computational proof for Formulation 14.16, click  here.  (Avoid Version 5.0.)  The computations of this notebook do not appear explicitly in the manuscript.  Version 5.1 fails to evaluate this notebook; e.g., click here.  The software engineers at MATHEMATICA have used the preceding notebook in Version 3.0 to correct the inefficient implementation of the  Coefficient  function in Version 5.1.  They expect the  Coefficient  function in the next Version of MATHEMATICA to be as useful as it was in Versions 3.0 and 4.1.  After the coefficients for (14.48) in Formulation 14.16 have been found, we can use Version 5.1 to check (14.48) then has the desired properties.  A notebook that does the latter can be downloaded by clicking here

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Page 162,   Computations for Example 15.7    For a MATHEMATICA notebook that verifies the assertions made in Example 15.7, click  here.

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Page 166, Formula (15.47); and pages 167-168 Section 16.1   The method used to discover (15.47) and the evaluation of the Input statements for Section 16.1 are presented in the notebook that can be downloaded by clicking  here

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Pages 168-169, Section 16.2   The discovery of (1.81) and (1.82)  of pages 16-17 is presented in the notebook that can be downloaded by clicking  here.  This notebook also contains evaluations for each of the Input statements in Section 16.2.

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Pages 169-171, Section 16.3    An excellent check for the correctness of Formulas (1.14) through (1.38) on pages 7-9 is provided by the computations in the notebook that can be downloaded by clicking  here.  The Input statements for this notebook were obtained by copy-and-paste from the corresponding Tex file of the manuscript.  Thus, we also have a check on this Tex file.

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Pages 169-171, Section 16.3   An excellent check for the correctness of Formulas (1.14) through (1.38) on pages 7-9.  As before, but with  p = 10  in place of  p = 7, click  here.

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Pages 172-177     A MATHEMATICA notebook for the computation of linearly independent relative invariants of weight  s  for  Q_{2} = 0 when  s = 2, 3, 4, 5, 6, 7, 8  can be downloaded by clicking  here.  The last part of the notebook contains a summary of the results.

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Pages 172-177     A MATHEMATICA notebook for the computation of linearly independent relative invariants of weight  s = 9  for  Q_{2} = 0 can be downloaded by clicking  here.  The last part of the notebook contains a summary of the results.

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Pages 172-177    A MATHEMATICA notebook for the computation of linearly independent relative invariants of weight  s = 10  for  Q_{2} = 0 can be downloaded by clicking  here.  The last part of the notebook contains a summary of the results.

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Pages 172-177    A MATHEMATICA notebook for the computation of linearly independent relative invariants of weight  s = 11  for  Q_{2} = 0 can be downloaded by clicking  here.  The last part of the notebook contains a summary of the results.

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Pages 190-191, Example 18.5    A MATHEMATICA notebook that contains the computations for  Example 18.5 can be downloaded by clicking  here.

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Page 198, Example 19.8    A MATHEMATICA notebook that contains the computation for Example 19.8 can be downloaded by clicking  here. (There is a slight change with respect to the constants in line b6 of page 198.)

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Pages 203--208, Proposition 19.12, Proposition 19.13, and Remark 19.17    A MATHEMATICA notebook that contains the computational proofs for Proposition 19.12, Proposition 19.13, and Remark 19.17  can be downloaded by clicking  here

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Page 209, Proposition 19.20    A MATHEMATICA notebook that contains the short computational proof of Proposition 19.20 can be downloaded by clicking  here.

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Pages 210--212, Formulas (19.93)--(19.95), Section 19.9, and Example 19.24    Click  here to download a MATHEMATICA notebook that: (i) explains how formulas (19.92)-(19.94) can be used with machine computations,  (ii) checks the formulas on the middle of page 212, and (iii) supplies computations for Example 19.24 on pages 212-213.

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Page 214 ,   A notebook for the MATHEMATICA computation of the last displayed formula on page 214 can be downloaded by clicking  here.

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Page 217,   A check on the formula in line 17 about B_{n}(z) can be downloaded by clicking  here.

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Page 217, Formula (19.125)    A notebook for the MATHEMATICA computation can be downloaded by clicking  here

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Page 218, line b2    A MATHEMATICA notebook that checks the corrected misprint about  (-1)^(2n-1)n  in place of (-1)^n  can be downloaded by clicking here.

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Page 272,   For a check on our generalized Laguerre-Forsyth identity with copy and paste, click here.

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Page 311,   For a check on formula (27.27), click  here.

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Page 313,   For a check on Theorem 27.7 for n = 1, click  here.

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Page 313,   For a check on Theorem 27.7 with n = 2, click  here.

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Page 313,   For a check on Theorem 27.7 with n = 3, click  here.

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Page 313,   For a check on Theorem 27.7 with n = 4, click  here.

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Page 313,   For a check on Theorem 27.7 with n = 5, click  here

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Page 317,   For a check that the case k = 3 of (27.46) specifies a relative invariant of weight 3, click here

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Page 330-331,    For the computations on page 329 as a modification of the ones for Section 16.4, click  here.

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Pages 347-353,    For relative invariants of (30.1), click  here

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Pages 353-356,   For the relative invariants of (30.28), click  here.

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In early 2009, we noticed that Version 6.0 refuses to evaluate various Input commands that offered no difficulty at all for earlier versions of MATHEMATICA.  In particular, Version 6.0 refuses to evaluate printM when it is inserted after its corresponding definition in those notebooks accessible above that correspond to page 172.  Shortly thereafter, we found that Version 7.0 evaluates the notebooks without difficulty.