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Comments for the Authors
Overall, I found your article interesting and publishable. I hope you dont find the suggestions that follow discouraging, for I do believe this was a very well-written paper in general. I would like to suggest the following revisions:
p.1, bullet 1 change generate to generated
p.1, bullet 3 change generate to generated
p.3, 3 lines from the bottom put parentheses around EMBED Equation.3
p.4, Section 3.1 Your example mentions adding up the areas of the three triangles within triangle (0,1,3). This approach doesnt work in general since the center of the regular polygon might not fall within the triangle (for example consider triangle (0,1,2)). Your formula still holds and your picture might be the best to use so your readers understand why it holds. However, you may wish to at least make mention of this issue and the fact that the formula still works.
p.5, last formula on page the numerator of the last fraction should be EMBED Equation.3
p.6, third line consider writing there are EMBED Equation.3 possible triangles instead to make the dependence on n = 6 a little clearer to the reader
p.7, 2nd line I dont believe you need to use the double angle formula in the derivation of the formula for the perimeter. The formula seems to just follow from basic trigonometry as you state.
p.8, line 5 italicize k
p.8, 2nd formula k will never actually be equal to 0 since a, b, c are distinct numbers and the smallest a c can be is n 1. Having the sum start at 0 might be more convenient for your later derivation, so you may wish to keep the sum starting at 0. However, I suggest mentioning to the reader to notice that this doesnt change the sum since sin(0)=0.
p.8, Claim 1 note that the claim doesnt hold for k = 0 since p(n,0)=0.
p.8, Claim 1 Your proof of the claim is correct, although it is unclear how you use the Case I hypothesis in your proof of Case I. Also, for subcase ii, the possible values for j are EMBED Equation.3 and you need EMBED Equation.3 as well since these triples are counted in Subcase i. It is still true that you have EMBED Equation.3 choices for j.
p.8, Claim 1, Proof With the corrections above, your proof of Claim 1 is valid. However, it appears to be more difficult than necessary. Here is a 2nd possible way to prove this claim that does not rely on you breaking the proof into cases and that generalizes more easily for Theorem 5.2:
Proof of Claim 1: Lets count the number of times k occurs due to a triangle of the form (0,b,c) and where 0 is used as one of the integers forming the differences, i.e. the number of times that b 0 or 0 c + n is equal to k. In the former case, our triangle is (0,k,c) where EMBED Equation.3 , and thus there are EMBED Equation.3 such triangles. In the latter case, our triangle is (0,b,n k) where EMBED Equation.3 and there are EMBED Equation.3 . Thus, k occurs due to a triangle of the form (0,b,c) and where 0 is used exactly EMBED Equation.3 times. Due to symmetry, k occurs where any of the other EMBED Equation.3 vertices are used exactly EMBED Equation.3 . Hence, EMBED Equation.3 counts each occurrence of k twice (once for each vertex used) and thus EMBED Equation.3 .
p.9, 4th centered formula from the top an exponent is incorrect in the right hand side of the formula. The final formula should be EMBED Equation.3 .
p.9, last formula Consider keeping the negative sign with the sum instead of with the derivative. This makes the transition to the formula for the average area a little easier to see.
p.10, 3rd line Consider removing the words standard yet. I did the simplification and did agree it was quite tedious, but did not think it as being very standard.
p.12, Theorem 5.2 I think you should tell the reader from where the binomial coefficient EMBED Equation.3 comes. An argument similar to the proof of Claim 1 that I have given above will work. Also as with Claim 1, k cannot technically be equal to 0, although starting your sum there does simplify the derivation.
p.12, last formula You switched the order of the terms in the top portion of the binomial coefficient from n 1 k to n k 1 . Consider being consistent on your ordering, and continue this consistency on p.13.
p.13, 3rd line You need an additional pair of parentheses around the difference EMBED Equation.3 .
p.14, first line change EMBED Equation.3 back to EMBED Equation.3 .
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