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1d
comment Why is not parity transformation just a rotation?
For any vector $v\in\mathbb R^3$ there is some rotation that maps it to $-v$, but there is no one rotation that maps every vector $v$ to $-v$. For example, rotating by $180^\circ$ about the $x$-axis maps $(0,0,1)$ to $(0,0,-1)$ but does not map $(1,0,0)$ to $(-1,0,0)$.
Jul
1
comment How can you do algebra with rounded numbers?
If both sides of the equation $9x=5$ are rounded, you get the constraint $4.5/9.5 < x < 5.5/8.5$. This works as long as both coefficients are positive.
Jun
30
comment Advanced calculus: Solving quaternion differential equations
You could always expand out the quaternion multiplications and treat the system as a linear ODE in 8 real variables.
Jun
30
comment Formula to represent 'equality'
Anyway, I'll leave this here: en.wikipedia.org/wiki/Gini_coefficient
Jun
30
comment Formula to represent 'equality'
If you want something that's "easily readable for a person who is not familiar with statistics", just report the maximum and minimum value. "Talks lasted between 5 and 30 minutes."
Jun
29
comment Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
"they need to know how to check obvious mistakes" Again, you keep track of the order of the error term: see the second half of @2012rcampion's answer. I am the downvoter, so you don't need to go on a witch hunt.
Jun
29
comment Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
"How do we know which approximation is good enough in a given context? There is no satisfying answer to this." Yes, there is. I mean, @2012rcampion's answer demonstrates it: you keep track of the order of the error term. In the section about Taylor series you are basically recapitulating their answer in a more verbose way, so I don't see why you are complaining that it is wrong.
Jun
29
comment Is $y=mx+b$ linear?
Indeed, if a "linear equation" were linear in the form the OP expects, it would always have a trivial solution.
Jun
27
comment Convex signal reconstruction for convex generator function?
The problem can have multiple minima (e.g $f(x)=x^2$ with $y>0$) so it can't possibly be formulated as a convex optimization unless you find a way to eliminate one of the minima.
Jun
25
comment Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
See Limitations of approximating $\sin(x)=x$.
Jun
24
comment Components of a vector given three points?
I would assume the grid lines are at integer coordinates, from which you can infer the coordinates of the points.
Jun
22
comment Show that a sinusoid having a frequency larger than one corresponds to a sinusoid having a frequency less than one.
What are the values of $\cos(2\pi n)$ and $\sin(2\pi n)$ when $n$ is an integer?
Jun
21
comment Scalar derivative of ${\rm tr}~[A(x)\log A(x)]$ where $A(x)$ is a square matrix
What is the log of a rectangular matrix? Is it the entrywise log?
Jun
21
comment The likelihood of being an accountant vs being an accountant and a plumber
Don't worry, "Rahul" is a very common name in India so you should not assume that all those comments on Dr. Gelman's blog are mine!
Jun
20
comment Why doesn't -1 = 1 (spot the falacy)
en.wikipedia.org/wiki/…
Jun
20
comment The likelihood of being an accountant vs being an accountant and a plumber
But as you admit, you don't know that he's a plumber -- you only know it with high probability. There is a small chance that he might just be a talented amateur (and also an accountant). So the probability of being accountant is still greater than that of being an accountant and a plumber.
Jun
20
comment Minimum curve for the distance between two points at the plane
wolframalpha.com/input/…
Jun
20
comment Minimum curve for the distance between two points at the plane
You should get $\partial F/\partial y'=y'/\sqrt{1+y'^2}$ without the $y''$ in the numerator.
Jun
20
comment 3D kinematic geometry problem motivated by chemistry
Let the bond angle be $\theta$. In the plane, draw an equilateral hexagon $ABCDEF$ with angles $\angle A=\angle D=\theta$ and the rest equal to $\pi-\theta/2$. Then sides $BC$ and $EF$ are parallel. Now you can "pick up" vertices $A$ and $D$ and rotate them out of the plane about the $BF$ and $CE$ axes respectively. If you rotate them all the way to the inside, you get a concave planar polygon with remaining angles $\theta/2$. Because $\theta/2<\theta<\pi-\theta/2$, by the intermediate value theorem there is some rotation for which the remaining angles are also equal to $\theta$.
Jun
20
comment Proof that $\mathbb{R}^+$ is a vector space
Fun fact: This unusual vector space over $\mathbb R^+$ is just the image of $\mathbb R$ (with the usual vector space structure) under $\exp$. That is, each $x\in\mathbb R^+$ has a corresponding $\hat x\in\mathbb R$ via $x=\exp\hat x$, and the vector space operations are defined by copying them over: $z=x\oplus y$ iff $\hat z=\hat x+\hat y$, and $z=\alpha\odot x$ iff $\hat z=\alpha\cdot\hat x$.