Eric Nitardy
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 Feb21 accepted Linear Algebra of Symmetric Sums Feb20 comment Linear Algebra of Symmetric Sums I played around with induction and got stumped by the inductive step. With the insight provided by the answers below, I now see where the inductive step was headed. Feb20 asked Linear Algebra of Symmetric Sums Feb18 accepted For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero Feb18 awarded Quorum Feb12 comment Unique characterization of convex polygons @Don: You may want to look at the second and third section of my answer. Section two is a very simple approach to selecting a distinguished vertex. Section three is an more complete (but not quite finished) version of the centroid approach to selecting a first vertex. You are right the centroid approach is a bit unstable. It is also very complicated to apply to all cases. Feb11 comment For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero @ Christian: Cool! Your box function may provide a way to explore the second half of my question. Feb11 revised For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero added 16 characters in body Feb11 comment For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero @Christian: Interesting. However, your counter-example is consistent with the pseudo-proof, so I'm not worried yet. Assume for the lemma that the polygon is convex. I've edited the question ... but for non-convex polygons, maybe I should look at moments instead. Feb11 revised For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero added 9 characters in body Feb11 awarded Supporter Feb11 asked For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero Feb11 comment Unique characterization of convex polygons @Don: If you add the feature point as a vertex, it is arbitrary between which two vertices you put it. So you must choose a second vertex. If you treat the feature point separately, you must choose a first vertex for your polygon. Unfortunately, cyclic re-orderings of vertices change your $C(P)$. For example $P=(\sqrt{2},e^{i\frac{3\pi}{4}},e^{i\frac{5\pi}{4}})$ and $P'=(e^{i\frac{3\pi}{4}},e^{i\frac{5\pi}{4}},\sqrt{2})$ are the same polygon, but $C(P)\neq C(P')$. So either way, you need to choose a first(second) vertex in a way that you always choose equivalent vertices in congruent polygons. Feb10 comment Unique characterization of convex polygons Given that the OP has apparently allowed the feature point to be any interior point in the polygon, if you add it to the polygon as a distinguished vertex, you will end up with a non-convex polygon. As a result, there may not be a unique counter clock-wise ordering of the vertices. Won't that be a problem? Alternatively, if you deal the featured point separately from the polygon, you have to choose a distinguished vertex of the polygon in such a way that you always choose equivalent vertices in congruent polygons. Feb4 revised $\biggl\lfloor{\frac{x}{1!}\biggr\rfloor} + \biggl\lfloor{\frac{x}{2!}\biggr\rfloor} + \cdots \biggl\lfloor{\frac{x}{10!}\biggr\rfloor}=1001$ added 30 characters in body Feb4 answered $\biggl\lfloor{\frac{x}{1!}\biggr\rfloor} + \biggl\lfloor{\frac{x}{2!}\biggr\rfloor} + \cdots \biggl\lfloor{\frac{x}{10!}\biggr\rfloor}=1001$ Jan28 awarded Commentator Jan28 comment In neutral geometry, can a family of parallel lines leave holes in the plane? Epilogue: Now, suppose we have family of parallel lines all at angle $\alpha (≠ 90)$ to line $l$. Do they pass through every point on the plane? With the angle sum theorem this is easy. Without it, ... hmm? I've got a limiting process which looks like it will work. If you like this sort of stuff, see what you can do. Jan28 comment In neutral geometry, can a family of parallel lines leave holes in the plane? Thank you. I'll try to find that. I'm working with an unpublished manuscript by Jack Lee. It's very clear and readable. It should published someday. Jan28 revised In neutral geometry, can a family of parallel lines leave holes in the plane? added 4 characters in body