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seen Nov 6 '13 at 3:42

Mar
13
comment Linear Algebra of Symmetric Sums: a converse question
I found a straightforward way to calculate the constant term from the from the powers sums, but I have not found a good way to calculate the discriminant from the power sums. Is the only way to calculate the all the coefficients of the polynomial and then find the determinant of the Sylvester matrix?
Mar
12
comment Do equal mean and equal moment imply equal distribution?
@user7815: This paper http://ramanujan.math.trinity.edu/wtrench/research/papers/RP-112.pdf by William F. Trench appears relevant.
Mar
12
revised Linear Algebra of Symmetric Sums: a converse question
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Mar
12
comment Linear Algebra of Symmetric Sums: a converse question
Thank you for your answer. I think that you have resolved the "some of $z_j$'s are zero" issue. There is an algebra error in your $n=2$ non-distinct discussion. The criteria should be $$\frac{1}{2}\mathcal{Z_1}^2=\mathcal{Z_2}.$$
Mar
12
comment Center of gravity of a self intersecting irregular polygon
@mixkat: Given your description of the problem, finding the centroid of the polygon your group forms seems like overkill. Treat each person like a point mass and find the center of mass of your collection of points as: $$\left(\frac{\sum_{i=0}^n x_i}{n}, \frac{\sum_{i=0}^n y_i}{n}\right).$$
Mar
12
comment Linear Algebra of Symmetric Sums: a converse question
My question is: given only the $\mathcal{Z_k}$'s and that $x=(1,1, \dots, 1)$ can I tell if the matrix $M$ is singular or not?
Mar
11
asked Linear Algebra of Symmetric Sums: a converse question
Mar
10
revised From baby Hartshorne: showing the exterior of a circle is segment connected
Fixed an erroneous statement in the second proof, clarified the third proof, and added an illustration.; added 6 characters in body
Mar
10
comment Equidecomposability of a Cube into 6 Trirectangular Tetrahedra
Just making sure everyone is aware: the easiest generalization of the 2-dimesional splitting of a square into two right triangles comes out of noting that you can split the square region ($0<x,y<1$) into two congruent triangular regions: $x>y$ and $y>x$. This generalizes to 3 dimensions by splitting the cubic region ($0<x,y,z<1$) into 6 congruent tetrahedral regions $x<y<z$, $y<x<z$, $x<z<y$, $z<x<y$, $y<z<x$, and $z<y<x$. This easily generalizes into higher dimensions.
Mar
9
comment From baby Hartshorne: showing the exterior of a circle is segment connected
Again the only access I have to Hartshorne is the Google excerpts, but I know that Hilbert created an arithmetic of congruent segments (Hilbert's Foundations of Geometry see pages 15-16 and 23-36). I assume he applied that arithmetic to angle measures using Archimedes' approach for successively better estimations of arc-length.
Mar
9
comment From baby Hartshorne: showing the exterior of a circle is segment connected
There is no way to cut the sequence down to one point. Since the claim is not true in hyperbolic geometry, it is not true in neutral geometry (the Hilbert plane). I do not have access to Hartshorne so I cannot tell what the problem is. But you are trying to prove something that is not true.
Mar
9
answered From baby Hartshorne: showing the exterior of a circle is segment connected
Mar
1
awarded  Self-Learner
Mar
1
comment Second-Order Linear Differential Equation
Your approach is the variation of parameters method ($y_p=\nu_1y_1+\nu_2y_2$, solve for $\nu_1'$ and $\nu_2'$ and integrate), which works for any inhomogeneous equation. Arturo's re-writing of $\cos(t)\cos(2t)$ as $\frac{1}{2}(\cos(3t) + \cos(t))$ allows one to use the much simpler method of undetermined coefficients.
Feb
25
comment What are the postulates that can be used to derive geometry?
@Muhammad: Birkhoff clearly makes use of a well defined real number system. In Hilbert's time, arithmetic was not well founded, and he probably wanted an axiomatization independent of arithmetic. Consequently, Hilbert's axioms make no assumptions regarding arithmetic. He does build as part of his geometry a sort of arithmetic of congruent segments. The number axioms he uses, however, probably has more to do with a desire to be intuitive. Tarski's axiom set, for example, makes no arithmetic assumptions but is very short.
Feb
25
revised What are the postulates that can be used to derive geometry?
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Feb
25
revised What are the postulates that can be used to derive geometry?
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Feb
25
revised What are the postulates that can be used to derive geometry?
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Feb
25
revised What are the postulates that can be used to derive geometry?
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Feb
25
comment Why is Euclidean geometry scale-invariant?
@Joseph: Put up an answer to What are the postulates that can be use to derive geometry? question with a list of jack Lee's postulates.