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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 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 $\hat z=\alpha\cdot\hat x$.
Jun
20
comment 3D kinematic geometry problem motivated by chemistry
I'm pretty sure any angle less than $120^\circ$ produces a boat and a chair configuration. There's nothing special about $\cos^{-1}(-1/3)$.
Jun
20
comment On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
Oh, I see what you mean. Well, the very next proposition states that "Similar segments of circles on equal straight lines equal one another", and since a straight line is equal to itself, similar segments of circles on it cannot be unequal. So no.