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Data Scientist @ Explorer Media


Jul
23
comment periodicity of an interval exchange transformation(IET)
Is the number of segments in the interval exchange finite?
Jul
22
comment Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane
@DanielRust I thought I was simplifying by using $[-1,1]^3$ instead of a unit cube which would be $[-\tfrac{1}{2}, \tfrac{1}{2}]$.
Jul
22
comment Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane
Is it possible your answer can be simplified since $e_x, e_y, e_z$ are orthonormal?
Jul
22
comment Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane
@DanielRust Can you expand that into an answer below? Basically, I need to know how the projections $p(e_i)$ change with the normal vector $\mathbf{n} \in S^2$.
Jul
22
comment An inequality on sequences with each term dividing sum of two neighbouring terms
@shadow10 You can subsitute $\frac{x_{i+1}}{x_i} = k_i - \frac{1}{\frac{x_{i-1}}{x_i}} = k_i - \cfrac{1}{k_{k-1} - ...}$ and keep substituting to get some kind of continued fraction. Or maybe you can start from: $$ 2X \equiv 2(x_1 + \dots x_n) = k_1 x_1 + \dots k_n x_n $$ So there is an average of the $\mathbb{E}[k_i]=2$
Jul
22
comment Sign convention for derivatives in a $\mathbb{Z}_2$ graded space
@vkarve $\theta \cdot \theta g'(t) = 0$ since $\theta^2 = 0$. Maybe you also need $\tfrac{d\theta}{dt} = 0$ and are left with $\theta g'(t)$.
Jul
22
comment A Cauchy-Schwartz type inequality
What about setting $x_1 = \dots = x_n = 1$ then $A = \frac{n}{k}$.
Jul
9
comment Why is Volume^2 at most product of the 3 projections?
Here is a proof in the notes of Guth on the Polynomial Method the cube $\square$ seems to be a natural shape for this type of projection inequality. Notes by Noga Alon on combinatorial nullstellensatz seem similar in spirit, though I do like the geometric flavor of the original result.
Jul
7
comment Check membership in a matrix group
Are these $2\times 2$ matrices? are these $n \times n$ ?
Jul
5
comment Working out the details of example 1.13 Hatcher: $\ E_{fg}\oplus n \approx E_f \oplus E_g$
math.stackexchange.com/questions/41936/…
Jul
5
comment Why is the Plancherel measure interesting?
@blue In a sense, there are really only two groups. Permutation groups $S_n$ & unitary groups $SU(n)$ which are related by Schur-Weyl duality. All other groups can be gotten by embedding into a permutation group or matrix group. This an exaggeration. Plancherel measure is natural since its dual to uniform measure on $G$ is natural. To make this more interesting, consider group actions on sets $X$ other than $G$ itself, & induce maps $g: L^2(X) \to L^2(X)$.
Jul
5
comment Why is the Plancherel measure interesting?
@blue Do you want class functions only or all of $L^2(G)$? Using the Peter Weyl theorem we have that $$L^2(G) = \bigoplus_{V \in \mathrm{Irr}(G)} V \otimes \overline{V}$$ so any function in $L^2(G)$ can be written as the sum of "matrix elements" of the representations of $G$. Taking the trace of both sides we get $$|G| = \sum (\dim V)^2 $$ which returns Plancherel measure under normalization.
Jun
26
comment identifying a subgroup of $S_8$ generated by 4-cycles
@DerekHolt Thanks for your reply. My main question is about the Cayley graph generated by these 4-cycles, but for today I settled on this easier question I could state more precisely.
Jun
25
comment identifying a subgroup of $S_8$ generated by 4-cycles
@DerekHolt can you find me a word in these generators giving the transposition $(12)$ ?
Jun
25
comment identifying a subgroup of $S_8$ generated by 4-cycles
@tomasz group $G$ might have an action on some other vector space
Jun
25
comment identifying a subgroup of $S_8$ generated by 4-cycles
@tomasz I am asking about the subgroup generated by the six 4-cycles I have listed.
Jun
21
comment examples of polyclic groups
@NoahS I don't know, is it? Google says upper triangular is sufficient.
Jun
21
comment examples of polyclic groups
@NoahS So $U(n)$ is locally compact topological group so it is "unimodular". Then the lattice would be $SO(n,\mathbb{Z})$ or some other arithemtic group - as in these notes. However, $\mathrm{SL}(n,\mathbb{Z})$ is not polycyclic - so I am still not understanding.
Jun
21
comment examples of polyclic groups
@YCor I can't find definitions or example of the notions of the theorem. I would like to see an explicit example of a "solvable unimodular lie group" $G$ and a lattice $\Gamma$ in that group.
Jun
21
comment Evaluate $\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx$
$\lfloor kx \rfloor \approx kx$ so your integrand is basically $ \int dx \; n! x^{n-1} = (n-1)!$. Is there a reason this approximation is not enough? $$ . $$ The Farey Fractions should be helpful since you need rational nubmers with denominator up to $n$.