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Response to comment 1: No changes besides those indicated in the response to comments 6 and 8 created incorrect meaning.
Response to comment 2: The zip code is correct.
Response to comment 3: The meaning is preserved and the changes are satisfactory.
Response to comment 4: This change is satisfactory.
Response to comment 5: The meaning is preserved and the changes are satisfactory.
Response to comment 6:
We are concerned about the choice to remove the dots throughout since the meaning is not preserved. The dots are not used to imply multiplication or inner vector products at any point in the manuscript, and in fact we took pains to ensure that the use was consistent (see Table I, entry for Coupling of O(n) model). The dots are meant to portray a group action, for which they are the canonical notation. See https://en.wikipedia.org/wiki/Group_action.
The removal of dots was made inconsistently. For instance, in equations (10), (11), and the inline equation on line 212 the dots are kept and not removed, presumably because they are next to a bolded roman character. These dots also represent group actions, here applied piecewise to an array of objects, and must be treated identically to dots used elsewhere in the manuscript.
If the dots may not be used to signify group action, we would like to replace them with another infix symbol rather than remove them. Replacing every \cdot with the LaTeX symbols \star (a line-centered star) or \ast (a line-centered asterisk) would be preferred to removal. Example typesetting of these options are shown in the attached image group_action.png.
If this replacement is made, we would like to use the dots to indicate inner vector products in the places that transposes are now used. This would involve the Coupling and Common Field entries for O(n) in Table I, changing "a^T b" to "a \cdot b".
Response to comment 7: The algorithm is OK as set.
Response to comment 8: In equation (5), the limits for the first product should be set as "{i, j} \in C", without the partial sign.
Response to comment 9: The figures appear satisfactory.
Response to comment 10: The definitions and acronyms used for 2D and 3D are satisfactory.
Response to comment 11: For consistency's sake, it can be set as capital "XY" without small caps.
Response to comment 12: The citation we have is Loos, Ottmar. "Symmetric spaces." (1969). A CERN library record can be found here: https://cds.cern.ch/record/106345
Response to comment 13: The information is correct.
Response to comment 14: The information is correct.
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