Yes, the stuff I taught in my year-long applied math course. But if you think about parametrizing $k$-dimensional manifolds in $\Bbb R^n$, then once you get past tangent planes to curvature, etc., you're at second (and sometimes higher) partial derivatives.
@sevdaicmis You're writing incorrect things. Unless that notation is the complement of the graph, which would be weird notation. Again, pictures should lead you to a valid proof.
copper: no. and i've always been surprised how little patent figures matter. i've spent years on a single patent case without ever looking at the figures.
in the beforetimes when people would hand me printouts of patents the first thing i would usually do is tear out the figures to save me the trouble of flipping past them every time.
ted: that and there are legal doctrines which basically say that the figures don't matter if they aren't supported by text. i'm kind of surprised that there are so many technical drawing shops whose only business is generating figures for patents. i think they prey on the small inventor who thinks they need a lot of them.
they are legally close to irrelevant to the scope of a patent right. sometimes it does help to have a visual aid, but if there's no supporting text for it, you wouldn't get to the point where you could use it.
so often you find yourself not looking at them until very late in a dispute.
@TedShifrin most probably. but, i've managed to derive $b_x<0$. the quadratic would be zero at $b_x=p_x/2$ and $b_x=3p_x/2$. we are looking for $b_x$'s that make the quadratic negative, that would be $3p_x/2<b_x<p_x/2$, which shows $b_x<0$.
There was a follow-up question to that Moon's orbit precession question I answered a couple of weeks ago. I initially misunderstood the new question. Oops. And then it took me a couple of hours to realise it could be solved with a simple cross-product. Even so, it was still a bit tricky. It's so easy to make a sign error when doing 3D trig. :) I expected the equations to become a hellish mess, but remarkably everything simplified right down. astronomy.stackexchange.com/a/47554/16685
The 3D diagrams were actually quite helpful in verifying that my equations were (probably) correct, and that I hadn't made some silly blunder.
Traditional navigation doesn't use them though, since there's no sane way to do them using latitude & longitude spherical coords, unless you translate back & forth to Cartesian coords, and navigators never did that.
Let $E = \{(x,y) \in \mathbb{R}^2: y = \sin(\frac{1}{x}),\ \text{for}\ 0 \lt x \leq 1\} $
Showing it's not compact. Means to show that it is either not closed, not bounded or both.
I can see visually that it is not. In particular it is not closed. I'm having an issue to do this formally. After a bunch of messing around I've settled on the idea of the ball around $(0,0)$ and showing that the complement of my set $E^c= \{y = \sin(\frac{1}{x})\} \cup \{y \neq \sin(\frac{1}{x}) \forall (x,y) \in \mathbb{R}^2$ is not open. Visually what is going to be happening is that any ball around $(0,0)$ …
@copper.hat A month or so ago on the Space Exploration stack, there was a question about the feasibility of using tidal power on the Moon. The OP thought it'd work well because the Earth's gravity is so much stronger than the Moon's. We pointed out that the Earth barely moves in the Moon's sky, and although there would be tides from the Sun, they'd run on a monthly cycle.
The Moon's motion is scarily complicated, if you want precision. Just calculating the time of the full & new moons to within 30 seconds involves series with ~15 trig terms. That's good enough for predicting eclipses. By the start of the 20th century, the full lunar theory involved around 1400 terms. And they worked out all those terms by hand, doing calculations using log & trig tables.
@LearningCHelpMeV2 I wouldn't bother memorising it. Just remember that it exists. You can derive it (and similar expressions) fairly quickly when you need them. OTOH, if you're onto doing those crazy competitions where you have to do high speed algebra, then sure, memorize it.
@leslietownes Analog computers were fairly crude back then. I doubt they could compete with 7 figure trig tables. But I guess they could be helpful in verifying that 2 trig series were (approximately) equivalent.
@PM2Ring at least there are no stupid butterflies flailing their wings and affecting the precision of computation. i mean, for god's sake, if we just eliminated the entire butterfly population on the earth we could predict the weather so much better. i'm surprised no one has thought of this.
i love analog computation. the idea of using on-the-fly reconfigured hardware is something that needs a lot more exploration i think, not just gpu silliness.
And that Moon stuff wasn't just an academic exercise. Because the Moon's motion relative to the stars is so fast (and it's large parallax) it was the best timekeeper that we had, until atomic clocks came along. Official / legal time was ultimately based on time from astronomical observations, and that meant lunar observations.
copper there is a writer calvin trillin who used to write a lot on food and other issues in the new yorker. he has a memoir of his young days at princeton where apparently he was friends with a faber of the faber pencil fortune. they called him "the pencil king."
Even now, although UTC is derived from TAI (atomic time), it's still kept in synch with the Earth's rotation, but we now use a network of extragalactic sources (quasars), as well as the Moon. We regularly observe the Moon & a bunch of satellites to get the fine details of Earth's rotation. That's the job of the IERS. iers.org/IERS/EN/Home/home_node.html
@copper.hat When I was in my teens, I had access to a pretty groovy electronic analog computer, built in the late 1960s. It was in a local science museum. maas.museum/powerhouse-museum
I was hoping to answer this question about inscribing an equilateral polygons in an ellipse. math.stackexchange.com/q/4301282/207316 Unfortunately, the OP declined to improve their question. My latest algorithm is quite fast, if the eccentricity isn't too extreme. It gets tricky with really skinny ellipses. You have to feed it a reasonable initial approximate solution, and then procede with caution. Otherwise, it converges on bogus solutions.
@TheGreatJRB Sorry, I don't think you can access rejected edits.
Let $A,B\overset{\tt iid}{\sim} \mathcal{N}(0,1)$, $X=ae^A>{0}$ and $Y=be^B>{0}$. What is the density of $Z=X/Y$?
I've tried $\texttt{change of variable}$, $\texttt{convolution}$, and $\texttt{CDF}$ I'm just lost.
I made an edit to an answer so that the one who gave the hints would get the credit...
Rejected 2 hours ago: Harish Chandra Rajpoot reviewed this 2 hours ago: Reject This edit was intended to address the author of the post and makes no sense as an edit. It should have been written as a comment or an answer. Leucippus reviewed this 2 hours ago: Approve user2715281 reviewed this 2 hours ago: Reject This edit deviates from the original intent of the post. Even edits that must make drastic changes should strive to preserve the goals of the post's owner.
Interesting. I thought I could see that stuff, but I can't.
The way I would suggest handling this is to ADD to your original question in accordance with the comments after Minus's answer, comment asking him if that edit is correct, and then accept his given answer.
I had put all kinds of code in it that I run through a c++ program that colors the text. I was going to put it in Overleaf as a solution to an exam I reviewing for classmates
@TheGreatJRB You should be able to copy at least some of it. Where there are 2 columns, they get combined, but the stuff with only 1 column should be ok. Eg,
@TheGreatJRB No. Please don't do that. But it wouldn't give me any points anyway because the system automatically reverses serial voting on a daily basis.
I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.
During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voti...
It's a necessary policy, but it can also be annoying. Eg, if you're researching some topic, and 1 or 2 experts have written a bunch of excellent answers to several questions you can only upvote a few of them per day, or they'll get reversed. Conversely, if some clueless person goes on a spree of posting a bunch of really bad answers in a short space of time, you can only safely downvote a few of them.
Ah right, obviously Leslie. I was trying to think of the simplest counter-example for this commutation problem and it didn't occur to me to pick an operator that doesn't commute with its own adjoint.
Yeah, the raising/lowering of the harmonic oscillator was the first thing that came to mind as soon as I saw you wrote U = H
But that is complicating things unnecessarily
You know what I just realized this very moment..... When you all were talking about infinite dimensional Hilbert spaces and the operator that sends $e_n \to e_{n+1}$... that's exactly the raising operator for the harmonic oscillator.
Never realized before that that "weird operator" that always gets brought up as a counter-example is also one of the most used operators in physics . . .
