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Apr
28
comment Volume of Region in 5D Space
@MJD: More generally, Monte Carlo integration chooses a large number of random points (distributed according to the integration measure) and calculates the average of the integrand at those points. The estimate for the integral is then the total volume being sampled times the calculated average. In the present case of computing the volume of a region, the integrand is the characteristic function of the region, so this simplifies to what you wrote, calculating the fraction of points inside the region. See en.wikipedia.org/wiki/Monte_Carlo_integration.
Apr
28
comment product and box topology
@AlecTeal: I don't understand the second part of your question (is $I$ meant to be $J$?), but the answer to the first part (what $f$ is) is that it's defined by the $f_\alpha$ all being the identity map, so $f$ maps $x\in\mathbb R$ to the sequence $x,x,x,\ldots$.
Apr
28
comment Rotation of a vector distribution to align with a normal vector
@ViktorSehr: I'm not sure what you mean by "rotates the same amount". $R$ is a rotation matrix that rotates $(0,0,1)$ into $N$. It represents a particular rotation, and when you apply it to any vector, the result is that vector rotated by that rotation.
Apr
28
comment How many minutes in 1 day?
@MonK: Well, given that it doesn't coincide with my answer, I'd tend to think it's wrong :-) I don't see an error in my derivation, so you'd have to explain yours for anything more to be said.
Apr
28
comment Determinant of a block lower triangular matrix
@MarcvanLeeuwen: I'd written "are perhaps easier to derive" -- I find it easier, but others may not. To get the result for the right-hand side, you just have to argue that the non-zero terms of the Laplace expansion involve all possible products of the terms of the Laplace expansions of the components, so this is the product of the Laplace expansions multiplied out. Of course you could make a similar argument about the OP, but it involves a bit more thinking about why no non-zero terms involving elements of $C$ occur.
Apr
28
comment Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann
@ArturoDonJuan: I think both viewpoints are appropriate. The fact that you can choose how to orient the $z$-axis is equally dependent on the rotational symmetry of the function being transformed.
Apr
28
comment Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann
@ArturoDonJuan: I called that angle $\theta$, not $\cos(\theta)$. Since I can call the angle whatever I want, I'm guessing that your question is more about why I can choose this to also be the angle between $\mathbf x$ and the $z$-axis. This is because the function to be transformed is rotationally invariant, so the answer depends only on $k$ and we can choose $\xi$ along the $z$-axis.
Jun
27
comment Geometric interpretation of a Taylor series like identity
Are you aware of the derivation of the formula by repeated integration by parts? That yields what might be called a geometric interpretation, though it's probably not quite as geometric as you'd like it to be: The area under $f$ is the area under $(xf)'$ minus the area under $xf'$, which in turn is the area under $(\frac12x^2f')'$ minus the area under $\frac12x^2f''$, and so on.
Jun
26
comment Can we qualitatively predict the strategy of the German and US teams in today's World Cup soccer match?
@Hayden: Yes, that's why I put it in the answer and not in the question -- perhaps someone can come up with a more comprehensive analysis.
Jun
12
comment Multiple integral over a disc
Due to the radial symmetry, you can perform the outermost angular integral to get a factor $2\pi$. I wouldn't be surprised if it turns out to be impossible to make any progress beyond that.
Sep
23
comment Proving $\sin A + \sin B + \sin C = 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
@lanzariel: I multiplied out the three parentheses; that yields eight terms; the two terms $\pm1$ cancel, and I grouped the remaining six terms conventiently for further manipulation.
Sep
19
comment What's up with Plouffe's inverter? Is there an alternative?
Why the downvote?
Aug
19
comment Left or right edge in cubic planar graph
@draks: I don't understand what you mean by "group the edges".
Aug
8
comment Sequence of powers of Gaussian integers capturing all positive integers?
@Nemes: That isn't possible, either. Consider the residues mod $2$. If $z=0+0\mathrm i\bmod2$ or $z=1+1\mathrm i\bmod2$, then $z^2=0\bmod2$, and the exponentially accumulating factors of $2$ prevent you from hitting all even integers. On the other hand, if $z=0+1\mathrm i\bmod2$ or $z=1+0\mathrm i\bmod2$, then each power has one even part and one odd part, so due to the period $4$ in the residues mod $10$ you miss at least one of the five even residues mod $10$.
Jul
27
comment Volume of a region on the sphere
@Tomás: You're right, sorry about that. I made them all $x$.
Jul
15
comment Minimizing the time to produce $T$ items with machines that run less efficiently over time
@Mirror: Hmm -- now I don't understand your calculation. From the first term, it seems that there's a single person setting up these blenders, so the time it takes to set them up is $Q$ times the time it takes to set one of them up. But then shouldn't the blenders start producing at different times, each one as soon as you're done setting it up? And why is $f$ being multiplied by $R/Q$?
Jul
15
comment Expected Value of a Negative Binomial Random Variable
@Alex: There's no assignment here; this is an equation.
Jul
10
comment A continuous random walk of length 1
There shouldn't be an $N$ in the distribution for one step?
Jul
9
comment Nodes in spherical equations and graph matching
You've written an expression, not an equation. Perhaps you intended to set this expression equal to zero?
Jul
7
comment Using induction to prove $3$ divides $\left \lfloor\left(\frac {7+\sqrt {37}}{2}\right)^n \right\rfloor$
@Thomas: Thanks -- indeed, it should; and that $x_1$ has remainder $2$ mod $3$ was even more blatantly wrong.