/sci/ - Science & Math

I just checked https://map.worldweatheronline.com/
The map seem incoherent, why would there be such a discrepancy in the area shown in this pic?

>>15828045
why is time largely said to be one dimension when there is clearly a present future past? why not just treat them as different axes of one thing?

>>15828171
> there is clearly a present future past?
> why not just treat them as different axes of one thing?
because those are just words for a direction along a specific axis. just like we use "here, forwards, backwards".

Why doesn't the sticky work?

>>15828280
It hasn't for some time. The account that created the link no longer exists.

I don't know where I fucked up.
The info of the task I need to do is in the image. I have two forces, F1=60N with its corresponding angle of 17 degrees and F2=70N, angle is 36 degrees.

My task is to calculate the resulting force of the entire system, which means calculating the components of F1 and F2 in both the X and Y axis. The drawing is exactly redrawn as was shown in the task. My issue is that I seem to get negative values for Fr(y) and when I punch it into my calculator to get the final value, it gives me a math error (I assume because you can't get a root of negative numbers, it just turns into the complex number i, which is not what I want here) and in general I'm just confused as to where I seem to fuck up. Maybe I used the trignometric identities wrong (sin and cos) for F2? I'm really not sure.

>>15828290
One of the most important things to learn in physics is that a lot of the time, the signs don't actually matter, only what they are relative to each other.
In this case, the only important thing is that F1 and F2 have opposite signs, and you getting a negative value is just telling you that the force is in the direction you've assigned to be negative. Try again after flipping the signs for everything on the y-axis, so that up is negative and down is positive

>>15828290
I don't think that's it, I looked over it just now and realized my F2(x) and F2(y) values are inverted according to their trigonometric identities, so I swapped them and I also realized I punched the negative number into my calculator wrong.

I now got the entire resulting force for the system (Fr) to be 42.33N. Would be helpful if any anon could come by and confirm if that's really the solution since my mechanics homework doesn't have any solution at all.

>>15828290
I think that you have to be 18 to post here.

>>15828330
I'm 18, I'm a freshman in college. Want me to take a picture of my student ID and birth date for you?

So -GMm/r is negative and mgh is positive and that is just a convention that depends on the reference point? For near-surface calculations we let W=mgh and W increases with height while in a more general case W=-GMm/r and W is negative and decreases with height and approaches zero at infinity? That's all there is to it? This always confused me.

>>15828335
potential energy is a scalar, force is a vector so the sign tells you the direction it is acting.

I have to check whether +-*/ are closed or open maps from the Euclidean R^2 topology to R (also Euclidean). I know how to do it with open sets: take a ball and check if I can move an epsilon in both directions for any point in its image. But how would I do it for closed sets?

>>15828410
by definition, a set is closed iff its complement is open

Im not a mathematician, please bear with me
suppose I have two M x N matrices A and B and I concatenate them into an M x 2N matrix C. If I perform singular value decomposition on the dataset C that provides outputs 'U' (M x 2N) and 'V' (2N x 2N) with 'U' being spatial modes ordered by greatest contribution to mean values and 'V' being temporal coefficients. The decomposition is such that [math]C = UV^{T}[/math].
Is it possible to determine individual contributions of A and B to U and V?
again, I really dont know much about this stuff but I'm trying to see if it's even possible/reasonable before I bother delving deeper into it

>>15828045
>>15828490
>>15828490
>>15828490
>>15828490>>15828490
>>15828490
PLEAAAAASE

1. How reliable are climate models?
2. Is Zeke Hausfather credible as a climate scientist? Or is he on the same tier (i.e. overly optimistic) as Mann?

What is the largest fatty acid that humans can actually digest?

does [eqn]f(z)=\sum_{n\geqq 1} z^{n!}[/eqn] diverge for all [math]|z|=1[/math]?
i have proven that it converge swith R=1 and diverges for [math]z=e^{i\pi/k}[/math] where k=1,2,3 etc

>Square bottle uses less plastic than a similarly sized round bottle
WHAT!? How does this work?
I always though cylinders were the most efficient?

Is this due to the fact that it's such a small quantity that structural integrity of the container is so easy to achieve that shape doesn't matter? Or are these guys straight up lying or cheating somehow?

I mean, I've never seen stuff like this come in cylinder shaped bottles. Maybe a big jug if you buy in bulk though. Anyone ever see hydrogen peroxide in a cylinder? I haven't. I just always assumed this practice of putting medical supplies/etc in boxy bottles was for space efficiency in shelving/shipping but never thought it was about using less plastic per bottle...

If someone can explain this, I would appreciate it because I always thought cylinders were the most resource efficient way to store fluids.

I took 10mg of melatonin last night because I had to be up early today. Woke up with severe brainfo that has yet to go away. Today has been horrifying; I can't remember the last time I felt this retarded.

My heartrate goes up too much now after drinking just a a bottle of wine and it never did this before. It goes up to 110 after just a 750ml bottle of wine of a 12.5% abv. If I drink more than this it'll go up even more. What's going on, /med/? Why is my body not able to handle alcohol anymore?

Is there a way to retrieve the result of a BLAST request made using NCBIWWW in python, probably using the query_ID of a previously made request?

>>15829257
> Or are these guys straight up lying
Maybe not lying but they are certainly wrong.
> space efficiency in shelving/shipping
This is why square bottles are used. You can pack them with no wasted space and the containers are shorter.

Round bottles became the norm because that was the natural shape when glass rather than plastic was the most common material for containers.

Suicide suicide suicide

>>15828290
Something is up with the x component. You might want to start over.

Does anyone else find a life built around learning really alienating? I love learning and thinking about big problems, but as soon as I got skilled enough to develop unique approaches to things & novel answers to problems, I met super harsh and hostile disagreement from people based on my conclusion. Like I would give tons of supporting evidence for why I disagreed with the common view, but it's like 9 in 10 humans will ferociously defend whatever people already believe, just on the basis that most people believe it.

Even when I explain in detail why Y is true and not X, any positive reaction I get is because I posted this in a community of Y supporters. They will like the post, give it a thumbs up, but none will respond to my actual points, just agree with the conclusion.

And the longer I go on & deeper I get into my areas of passion, I feel that the # of people who understand what I'm even talking about continues to shrink. Shit is depressing man. Eventually I'll have to keep it all to my own head

>>15829800
> I met super harsh and hostile disagreement from people based on my conclusion
The fact you give absolutely no context or information about the topic you're talking about makes me believe you're a schizo.

>>15829808
For example, the other day I mentioned that war can be one of the greatest experiences in life. This isn't a crazy view, most of the world saw it this way before 1900, the Greeks wrote about it quite a lot, but when I said this everyone instantly misrepresented my argument & said "hurr go get blown up by an IED or drone striked or something". I explained that war went to shit around modern times, that all the excitement and glory and potential riches for men in war have been stripped away, the rewards of war have disappeared while the downsides have multiplied. But everyone stopped replying to me.

Shit like this happens all the time. If you have an unconventional view of anything, you'll get shut out on principle. It makes me wonder what's even the point of educating average people when on 99% of the things they believe, they more or less lock them in for life regardless of anything they are told.

>>15829832
> most people disagree with my minority opinion
> this shocks me
this is you.

>>15829839
I don't mind if they disagree, that's 100% to be expected. But if you have unconventional views, the most common thing you find is they are rejected on principle. And this works the other way around: If you have conventional views, you can give any justification for believing them because most people ignore the reasoning and evaluate based on the conclusion. I love when people give counter-arguments to me, because I just want to learn and know the answer, not be right. If you have unconventional views, you won't often get a good counter-response.

I appreciate the guys who debate Flat Earthers by pointing out how going up in a balloon high enough, you can see the curvature of the earth. Meanwhile have you looked at how most people interact with Flat Earthers? Seriously, they don't have an actual counter. "We've known earth is round for centuries LUL". Okay, how? How do you know? A lot of them don't know how we know, they accept it on principle, a priori. When you think about it, the fact 90%+ citizens accept basic facts about life just because everyone else does, and firmly believe in it, is a big problem. It means anyone with an unconventional view is just shouted down. Imagine how much better off we would be if there weren't hordes of people who accepted ideas on principle, and argued in their favor despite only believing in them because everyone else does. It's horrible

>>15829846
What are you expecting? What is your overall point? Most people have neither time nor inclination nor ability to somehow digest "everything" encompassed and your personal floating interests are your own. That "most people" don't have time or care to waste on crazies like flat earthers should not shock you.

Nevertheless, some do on every major website they show up on, and with not only rigorous rationale for evidence of Earth's shape but even experiments constantly demonstrating it. Flat Earther's own experiments constantly demonstrate it and not a single one of the lying shits have ever admitted it. Then you have the MAGE project, and the second MAGE project, demonstrating a curved horizon with cameras at a crazy high altitude all to be completely ignored by the same flat-tards demanding exactly that be shown.

What's baffling isn't that people don't care or think your opinions are abnormal. What's baffling is the hypocrisy you display by using FE as your example. If anything, people who push that lie have more thoroughly demonstrated every single thing you're complaining about than any other group. As a result of that paradox I haven't the first clue what in the fuck you think your point is because it seems to me to be pure self-contradiction.

>>15829980
>What are you expecting? What is your overall point? Most people have neither time nor inclination nor ability to somehow digest "everything" encompassed and your personal floating interests are your own. That "most people" don't have time or care to waste on crazies like flat earthers should not shock you.
Then why do they have such impassioned opinions about things they don't know anything about? They should just shut up instead of yelling.

>>15829982
>Then why do they have such impassioned opinions about things they don't know anything about?
It's called a social norm. If you think society could function without them you're either autistic or retarded. I am not meaning by that to say social norms are good for their own sake, only that your take on the reaction is completely detached from even the most basic perspective taking. Yeah, the vast majority of people have very strong opinions about things they know very little about and won't change their mind regardless of the evidence you give. Most of the people who will simply wouldn't care about the subject enough to have a strong feeling on it to begin with. Nothing is ever going to change that.

If you enjoy arguing things contrary to social norms plenty of groups and people love doing that. Most people just want to get the fuck on with life and rely on those very same social norms to offload the rest. Odds are you do too unless you really are autistic.

>They should just shut up instead of yelling.
Or maybe you shouldn't slap a cactus and complain that it stuck you? I'd be more sympathetic if there were anything that could be done about it. You can either live with it or choose to stop living, or I guess become the equivalent of a jackass street preacher everyone wishes would get hit by a bus.

>>15828045
Do micrscopes rely on the particle nature of photons or wave nature? I was wondering how you could increase microscope resolution beyond normal wavelength limit if one had a way of getting phase information could can we see below the half wavelength limit? Is there some theory to this? I cannot find anything except some off hand comments.

>>15830020
>Do micrscopes rely on the particle nature of photons or wave nature?
Yes.
>I was wondering how you could increase microscope resolution beyond normal wavelength limit if one had a way of getting phase information could can we see below the half wavelength limit? Is there some theory to this?
I think you mean this? https://en.wikipedia.org/wiki/Quantum_microscopy
Or perhaps quantum metrology? https://en.wikipedia.org/wiki/Quantum_metrology

There are various related terms and concepts I think map on to what you want. That is, trying to get beyond the standard quantum limit, which I assume is what you meant about the half-wavelength limit. https://en.wikipedia.org/wiki/Quantum_limit

>>15830031
I think what he's asking (let me shoehorn my own question here) is, if you had two coherent sources of light, and offset them by a phase difference, could you resolve things below the half wave limit?

e.g. if I had two 500nm lasers, offset in phase by 10nm, could you resolve things at 10-20nm or whatever?

>>15830167
I don't think so, combining two waves will never produce a new wave with a smaller wavelength.

However what you describe is similar to how LIGO works but that is looking for the phase difference, it does not resolve anything. What I don't know is if such a method could be modified to do so.

Help would be much appreciated

>>15830167
>I think what he's asking (let me shoehorn my own question here) is, if you had two coherent sources of light, and offset them by a phase difference, could you resolve things below the half wave limit?
I'm the one you asked and the answer is No but that is a thing already
https://en.wikipedia.org/wiki/Interferometry

The wiki page linked on the standard quantum limit also references interferometry. As for why there's the limit,
https://en.wikipedia.org/wiki/Squeezed_coherent_state
https://10m.aei.mpg.de/standard-quantum-limit-sql/
Or if you already know it "because Heisenberg’s uncertainty principle".

>>15830192
I'm the anon he asked and yes it's called interferometry. He was trying to ask about the ratio of the phases. Or I am pretty darn sure he is.

what dey mean with the bit at the end that the universe is flat but if the mass density was different it would be curved? so it's flat on average but it curves locally like einstein's gravity analogy? or it's flat everywhere because gauss's triangle angle sum measurement in the mountains would always show 180 degrees because the universe is flat and einstein's analogy with mass causing curvature isn't actually a thing in real life? but then what is gravitational lensing?

https://www.youtube.com/watch?v=lFlu60qs7_4

>>15828717
bumpu

>>15830351
>>15830167
>>15830020
>>15830031
>>15830192
https://www.sciencealert.com/ligo-has-surpassed-the-quantum-limit-we-can-explain

>>15830647
ah neat so basically realization of the entanglement idea assuming it actually worked and can be replicated and stuff. I haven't read or looked at the paper just saying. Thanks for sharing.

Bizarre question I’ve been thinking about. If I used a magical device to remove a one-atom thick plane of matter from a person, what would happen? I don’t think they would just fall apart into two pieces.

Very stupid, but what's the 101 on how social media algorithms work?

I mean, they all have different ways to calculate which content will be shown to whoever, though in the end it's about engagement, staying on the platform, clicking ads, etc. But what parameters are usually involved in these calculations?

Say Bob like sports and hates math, Ana likes math and hates sports, John likes both sports and math, then one day Bob likes a sports post, this post is pushed to John that also like sports, but not to Ana who hates it. Some page posts a math related thing, it shows to Ana and John who should like it, but not to Bob. Is it that simple? Now, say Bob finally likes a post about math, which he usually hates, does that count more on pushing this content forward since it has reached outside of its usual audience?

I'd like any resource on how the basics of it works.

>>15831091
Nothing would happen. Anything at the atomic scale like that wouldn't work at all like a sword. Let's change proportions, imagine I cut a mountain in half with a magic katana, sure some roots were cut out, some leaves, perhaps a lizard out of luck, but do you think I've disrupted that ecosystem in any given way? wouldn't the weight of the dirt and rocks fill the gap immediately? The proportion between a sharp sword and a mountain is far less dramatic than a molecule and a human being. You would not even be able to cut in half a single cell if you're only taking a single atom thick plane of it. A drop of water has over a sextillion atoms, anon, take a trillion of them out, you would see no difference.

Also, the question doesn't make any sense, are you disrupting some molecules by taking a single atom out of them? I don't even know how this could possibly function.

>>15829257
I'm not 100% sure about it, but I think the most efficient in terms of plastic use would be a hexagon shaped bottle. But I think the square is much much better than the round ones in terms of storage and I think they account for that to get to that conclusion. You need less plastic to wrap them up, you can transport more of them at once and save on carbon, etc

>>15829257
>Anyone ever see hydrogen peroxide in a cylinder?
https://www.laballey.com/products/hydrogen-peroxide-30-lab
i dont think ive ever seen anything higher than 3% come in a square bottle.

I think my prof's slides for Solidworks are worth $1,000, holy shit the 'textbook' is useless, the line "move your mouse cursor onto the blahblah flyout and click the left mouse button" ctrl-c ctrl-v'd thousands of times. I looked around on youtube, only saw a few good tutorials, amd they only showed the first couple vids for free then linked to their site and wanted hundreds of dollars for the rest. Anybody have good slides/vids for Solidworks?

If I listen to Mozart before a math test will I score better

Or is there something less annoying to listen to that does the trick

anyone read pic is it worth buying for someone interested in optics as a hobby? Cant find it on libgen or in the library.

>>15829257
Spheres are the most efficient (highest ratio of volume to surface area). The advantage of cuboids is that they tessellate into larger cuboids (e.g. cargo containers, trailers) without gaps. Cylinders are something of a compromise between the two.

>>15831244
Depends on what atmosphere you want as you
study. For me, classical is what I enjoy but not
study though. So, I go for some bossa nova in
low volume, like this:
https://www.youtube.com/watch?v=jsAUxuX-FOc

Maybe you want a sinister, technological vibe
as you want to hack the math test, then check this:
https://www.youtube.com/watch?v=Z-VfaG9ZN_U

Or maybe a blend of outer space and earthly soul
uplifted to intelligently placed drum and bass.
See your math test from a third eye,
from-the-outside-in, nature-is-yours, perspective:
https://www.youtube.com/watch?v=yvsRrpIPCoE
https://www.youtube.com/watch?v=UHRTQCF7oKw

if [math] f\colon \mathbb{R} \rightarrow \mathbb{R}, f(x)=\lfloor x\rfloor\colon=max\{ m\in\mathbb{Z}: m\leq x \} ,f^{-1}(\{0\})?[/math] I know it's [0,1), but what's the correct way of proving it

How can we use Lagrange and Hamilton to solve some tough classical mechanics problem in few lines ?

>>15831863
If for some x in \bR you have that f(x)=0, then max\{m\in\bZ\mid m\leq x\}=0 and so 0\leq x as 0 is element of \{m\in\bZ\mid m\leq x\}, if 1\leq x, it follows that 1 is in the set and so 0 cannot be the maximum, contradicting that f(x)=0, thus x<1. This shows that [0,1) is subset to the fiber of f at 0, the converse is straightforward.

>>15831869
Bump
The tough competition level problems

>>15831869
That shit is useful only for simple artificial physics problems involving finitely many point masses.
In the real world problems where object deforms as forces act of them with hysteresis effects all that fancy shit is useless. That's why they don't teach Hamiltonian mechanics to mechanical engineers. Computer simulations based on Newtonian mechanics is still the way to go.

If oil on earth comes from algae and plankton, then how did it get to Titan? How is there any oil or hydrocarbons in space?
>https://www.space.com/4968-titan-oil-earth.html

>>15832216
What makes you think carbon only exists in life? Oil is mostly just carbon and hydrogen. There's plenty of both in the universe.

Theoretically speaking, would it be possible to engineer a virus that inserts my DNA into other people’s cells, totally replacing their original DNA with my own?

>>15833053
Theoretically speaking anything that isn't impossible is possible. However realistically speaking the answer is absolutely not. It would require a virus large enough to carry a payload of your entire DNA and a mechanism to swap it with a target cells, and a method to replicate itself - that is no longer a virus, it's something else entirely. You would have gone passed possible and into the realm of pure science fiction.

>>15830584
bump

there are contradicting statements like einstein's theory says it gets curved but they haven't measured a single case of space actually being curved, not even locally?

>In a universe with zero curvature, the local geometry is flat.
https://en.wikipedia.org/wiki/Shape_of_the_universe

Are there any good book translated from Russian on the topics of general, organic and inorganic chemistry? I have some English books on the topic, they're somewhat adequate but most are structured around exams like GED or SAT or whatever. Plus the constant mentioning of imperial units annoys me. I also have some German books, but German authors tend to complicate things unnecessarily imho.

I'd like some books that are about chemistry for the sake of chemistry, not for the sake of passing exams.

>>15833395
>chemistry for the sake of chemistry, not for the sake of passing exams.
Never gonna find that in a (((book)))
Read the methodology sections of papers

>>15833404
You're meaning nobody took the time to compile a book from methodology sections of papers?

>>15833408
They are all dumbed down to the point of uselessness. There may be a few handbooks on common preparations, but "textbooks" are universally garbage.

Can the Pi number be written in the Roman numerals

>>15833837
No. They had no method to write arbitrary decimals.

id like to thank everyone that voted for my nomination of witten as the new OP image.

I'm curious if anyone here has had someone proselytize them on campus

Had someone try to proselytize me on campus a long time ago; here's my shitty experience:

He claimed that being lustful, lying, drunkenness and jealousy are all sins and one can look to Christ for forgiveness. I told him these are all normal human emotions and no one should be ashamed of them and we should look to our loved ones to keep those feelings in check. The dude was so baffled that all he said was " Ive been doing this for 30 years". Then he left, packed up and went home.

How retarded would I be for doing an B.S in MechE if I'm currently on the fence on whether I just want to do bioengineering or become an MD? Should I do ChemE instead?

>>15834272
chemE is the hardest engineering discipline

mechE is hard but chill and you get to be around cool people.

>bioengineering or become an MD

depends on how long you want to stay in school for.

>>15828045
When i heat an iron rod on one end in a way of a sinusoidal temperature change, how fast travels that sinus wave through the rod?

>turn stove off
>pan still hot
>but egg not cooking
Please help me make it make sense.

>>15834634
Is it that this specific image in Starfield art book that I inserted depicting lost dimensions near perfectly or perfectly and if 'shimmied' into opens a gateway to thinking?

Am I doing it now successfully?

-----
Take note of all these take time to note them.

Training 1

Training Extended

Objective

Means

Other

Everything I need to tell you

>>15834634
It's in the weapons section. It is the first image of a single gun that's got something remincent of something triangular.

>>15833837
XXII ÷ VII

>>15828045
Can I take Faraday's Law and do the following:
[math]\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \Rightarrow \vec{B} = -\int \nabla \times \vec{E} dt[/math]?

I want to find the magnetic field at a point caused by an electric field from an electric point charge that moves over time.

>>15834750
Sorry, let me clarify the last line:
There is a point charge that is moving -> It creates an electric field that changes with time because the point charge is moving -> The electric field creates a magnetic field

>>15834750
>I want to find the magnetic field at a point caused by an electric field from an electric point charge that moves over time.
I admit I am not sure but maybe my bungling will help you if you explain why I'm wrong like you would a rubber duck.
My thinking is that it would be 0 since,
https://en.wikipedia.org/wiki/Gauss%27s_law_for_magnetism
Divergence theorem. Magnetic field B has divergence = 0, https://en.wikipedia.org/wiki/Solenoidal_vector_field
So you're inadvertently trying to calculate a magnetic monopole? https://en.wikipedia.org/wiki/Magnetic_monopole

I hope my idiocy helps you though either way. If I happen to be coincidentally correct in my reading what you're doing trust me it's only coincidence

What can I use to solve college math problems and show me step by step solutions? I am in an online college math class and the teacher wants pics of me working out each problem
I am an old man, please someone help me with this.

>>15834901
how bout you just work the problems yourself?

fentanyl citrate (p. 558) https://web.archive.org/web/20170908140647/https://books.google.com/books?id=7uk0GJckmy0C&pg=PA557#v=onepage&q&f=false

>>15834996
It's beyond my comprehension, I don't have the time to learn all this with a family and work. I'm not trying to get a degree in anything math related, it's just a general requirement.
I know no one is going to help me here, unfortunately I'm not savvy with the AI stuff, but thought I'd ask either way.

>>15835588
>beyond my comprehension

this is not true at all

>have the time to learn all this with a family and work

i have met someone who became an engineer even though still had to deal with this. Stop making excuses.

https://www.youtube.com/@ProfessorLeonard/playlists

this is the best youtube channel for undergrad math and lower.

If I have to rewrite [math]y=f(x)[/math] and it so happens that I need to divide by [math]y[/math] can I just ignore [math]y=0[/math] because [math]y[/math] is a dependant variable?

>>15835702
doesn't work like that, no
your new function will have a hole at any x such that y=0, and will be undefined there

>>15835642
Thank you, I appreciate it

How to test for statistical significance of this?

if A is a column with one of these values (-1,0,+1), and B is a column with values in range of (100-130), C is a column with values in range of (100-130)

Hypothesis: [C-D] is less when A is +1 compared to when A is -1 or 0.

The data is all on google sheets.

if we have [math]y\in f(\bigcup_{\alpha}A_{\alpha})\equiv\cdots\equiv \exists \alpha(x\in\ A_{\alpha})\equiv\exists \alpha(f(x)\in f(A_{\alpha})[/math] how exactly do we get to the fx from the last equiv. I know it intuivitely, but can't come up with a correct explanation.

>>15835880
what definition exactly

let [math]A=\{1,2,...,n\}[/math]Show that exists
[math]\sum_{i=1}^{n-1}C^{n-1}_{i}\cdot i^{n-1-i}[/math] images of[math]f:A\rightarrow A[/math] so that [math]f^{2}(A)=\{1\}[/math]

what the fuck is the meaningful difference between these two null hypothesis?
it's the same fucking question but with different numbers, and yet the first is represented with less than or equal to, but the second is an equal.
why why why

Can someone tell me how do arbitrary constants work? if I try to solve this:

[math]
\ln(y^{2} + 2) = ln(x^2 + 4) + C \\
y^{2}+2 = e^{C}(x^2 + 4)
[/math]

Why is [math]e^{C}[/math] multiplying ????? please link a video or website if you can't explain because I can't find anything

>>15836142
[math]e^{a+b}=e^{a}e^{b}[/math]
this is a basic property of exponentials

>>15836142
[eqn]\begin{align}
\ln(y^{2} + 2) &= ln(x^2 + 4) + C \\
e^{\ln(y^{2} + 2)} &= e^{(ln(x^2 + 4) + C)} \\
y^{2} + 2 &= e^{ln(x^2 + 4)}\ e^{C} \\
y^{2} + 2 &= (x^2 + 4)\ e^{C}
\end{align}[/eqn]

>>15836146
>>15836149
thanks, forgot some basic stuff.

>>15828045
If I fill up a perrier bottle up with normal tap water and use it to train my grip strength (by gripping the bottle at the widest point and squeezing as hard as I can), what are the chances it will fucking explode in my hand?

>>15828045
i didcovered that if you add 3 to an even number, it will become odd, and when you add 3 to an odd number, it will become even.

eg: 3+4=7 and 3+7=10.

can this be used for anything?

>>15836637
you can generalize '3' to all odd numbers. What about for multiplication and exponentiation?

>>15836735
>What about for multiplication and exponentiation?
Think about it.

Just... Just think about it for a minute.

>>15836747
I'm prompting the first guy to figure it out

>>15836772
ah my bad

How come it seems like maths people really don't care very much about the idea of about proofs study but non-math stemtards will pick up a proofs book and act like they god just blessed them and they're geniuses now?
I'm a mathematician and my proofs class didn't have a book, I've never read a proofs book in my entire life, but most of the books I read are books full of proofs. Math is being proven.
I see these CS people holding up their proofs book acting like it's the Bible with the word of god.
I don't get it is there something fucking magic in there?

>>15836779
Ego. It's always ego. Look into grigori perlman and why he left mathmatics

>>15830584
BUMP

THIS IS INFURIATING

ALL THESE PEOPLE TALKING CONFIDENTLY ABOUT GRAVITY BECAUSE CURVED SPACETIME BUT WE HAVEN'T ACTUALLY FOUND EVIDENCE THAT THE UNIVERSE IS ANYTHING BUT FLAT

https://www.reddit.com/r/explainlikeimfive/comments/17mf1u4/eli5_gravity_isnt_a_force/

FUCK YOU

>>15837009
sigh alright i'll bite
The geometry of the total universe [math]\ne[/math] the geometry of any given point or area of space.

So were you just dumb, and now realize you're dumb, or are you trolling?

>>15836914
>due to feeling disappointed over the ethical standards in the field
the fuck does that mean?

>>15837012
veritasium said there was no evidence of curvature even in the context of gauss's triangle measurement in the mountains

see also >>15833387
>In a universe with zero curvature, the local geometry is flat.

>>15837019
Reading between the lines: he hates how political academia is. It doesn't help geniuses like him thrive since he just wants to focus on the work

He put it in those words because it's not worth being slanderous but people in the know can see what he is saying behind the rhetoric

>>15837051
[math]geometry_{universe} \ne curvature_{spacetime}[/math]
>no evidence of curvature
[math]geometry_{universe}[/math]
>the local geometry is flat
[math]geometry_{universe} \ne curvature_{spacetime}[/math]

Anything else?

>>15837066
wait wtf academia is super political?
should I avoid going to do a PhD and just get a masters in whatever is hot?

>>15837116
you either accept it for what it is or quit. Too many profs will gate keep you from reaching your best self

>>15828045
is it possible to have spontaneous emission from the zeeman effect? Say a external magnetic field splits the energy level of a electron in a atom so once we take away the external magnetic field were does the energy go that excited the electron to the higher energy level (if the spin was aligned)?
Wondering if its possible to build lasers which pump electrons via magnetic excitation.

[eqn]\int_1^\infty \frac{dx}{x^x}[/eqn]How can it be done?

>>15837303
The upper limit will vanish to zero. The lower limit you could probably approximate using some kind of series expansion. There's no closed form solution.

>>15837303
Yes.
[eqn]\int_1^\infty \frac{dx}{x^x} = \int_1^2 \frac{dx}{x^x} + \int_2^\infty \frac{dx}{x^x} \leq \int_1^2 \frac{dx}{x^x} + \int_2^\infty \frac{dx}{x^2} < \infty[/eqn]

[math]\int_1^2 \frac{dx}{x^x}[/math] is finite because you are integrating a continuous function over a compact set and [math]\int_2^\infty \frac{dx}{x^2} = \frac{1}{2}[/math] is finite too.

>>15837316
>There's no closed form solution.
How'd you know that? Is there a proof for it?

>>15837317
I know the limit exists. I'm interested in what the limit is.

>>15837331
> How'd you know that? Is there a proof for it?
Ok. It may be more correct to say no known closed form solution exists.

> I'm interested in what the limit is.
You can integrate it numerically to get ~ 0.70417

>>15837390
pretty sure you could prove it definitively with Liouville, but I can't be assed to go through the motions
https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)

>>15837397
I know that the antiderivative can't be expressed in terms of elementary functions. But that's not my question. My question is if you can express the value of the improper integral in terms of an elementary expression (with a square root, logarithm or in terms of the transcendent numbers pi, e and so on). Both are no equivalent, because there are non-elementary antiderivatives with elementary values of their improper integrals. The sine cardinal or the Gauss function, for example.

>You can integrate it numerically to get ~ 0.70417
I know but that's sort of boring, don't you think. Thing is
[eqn]\int_0^1 \frac{dx}{x^x} = \sum_{k = 0}^\infty \frac{1}{k^k}.[/eqn]So I think there must be a relation to a discrete sum for the integral from 1 to infinity. Also I want to prove that there can't be an elementary expression of the sum(s), which is a different problem.

>>15837536
That's like saying since [math]\sum_{n=1}^{\infty} \dfrac{1}{n^2}[/math] and [math]\sum_{n=1}^{\infty} \dfrac{1}{n^4}[/math] have closed form answers then surely that must be the case for [math]\sum_{n=1}^{\infty} \dfrac{1}{n^3}[/math] as well.

>>15828045
Is there anything different in the preparation of injectable vs insufflation (nasal spray) peptides?

I.e. Is there anything stopping me from putting injectable bottles into a nasal spray bottle and using it that wayy?

How do I get something like this to work on google sheets?

>>15837718
Ask on /g/. We don't use proprietary software here.

>>15837725
Thanks

>>15837136
What if I just want to teach while I make video games as a hobby?
I can't into finance because I don't fucking understand finance and I don't know what trading is, I don't own a single stock and I doubt I have the money to even buy one.

>>15828045
In the book I'm reading it says that the elements of a finite group G can be represented by matrices as follows. A column matrix u (called a basis vector) is chosen with components u_i (called basis functions). An element X of the group is selected and its effect on each u_i (i=1, ..., n) is determined so that you get u'_i = X u_i. This generates a new column matrix u' so that there must be an n*n matrix M(X) such that u' = M(X) u.

This is where my confusion comes in. Apparently M(X) isn't the representation of X for the following reason: "because we have already chosen the convention whereby Z = X Y [where Z, X, and Y are group elements] implies that the effect of applying element Z is the same as that of first applying Y and then applying X to the result, one further step has to be taken. So that the representative matrices D(X) may follow the same convention, i.e. D(Z) = D(X) D(Y), and at the same time respect the normal rules of matrix multiplication, it is necessary to take the *transpose* of M(X) as the representative matrix D(X). Explicitly, D(X) = [M(X)]^T ."

Why is it the case that, given Z = X Y, D(Z) = D(X) D(Y), but M(Z) =/= M(X) M(Y)? How do you prove it?

Normal modes for pic related (2D motion only), but I'm only interested in the zero frequency normal modes. I've found that they're 4-fold degenerate, but I can only intuit 3 of them:
1) x-translation
2) y-translation
3) rotation
What is the 4th zero frequency mode? Also, if the spring system is modified slightly to have springs on the diagonal, the number of zero frequency normal modes drops to 3; which one drops out?

If a block is slipping around a slope which cause the slope to move backs. According this statement, both the bodies will have displacement then how can we use total momentum conservation theorem ? Centre of mass is also displaced..?

>>15838796
Why don't you just write out the differential equations and calculate the eigenvectors of the matrix?

>>15838824
The total horizontal momentum and the x-coordinate of the centre of mass is constant.
It's not conserved vertically since you have an outside force (gravity).

>>15838796
Are the angles of the springs to each other fixed? If not it can be squashed into a parallelogram.

>>15835905
Count possibilities for [math]f^{-1} (\{ 1 \})[/math] and then count functions mapping [math]A \setminus f^{-1} (\{ 1 \}) \to f^{-1} (\{ 1 \})[/math]

Last year I did a classical mechanics and electrodynamics course in which we used resnick halliday Krane for reference and aced it..This year I am taking an classical mechanics class which included Lagrange and Hamilton's theorems and a little bit advanced calculus in mechanics

Not able to understand things well, we are using Taylor classical mechanics, are their any particular pre requite to it ? And can i see the solution solved examples or do I have to try those first ?
How should I study ?

The explanation in Taylor classical mechanics are not as vivid as in resnick Halliday

T.2nd year mechanical engineering student

>>15839106
Best way to study is to try to get the syllabus of the class ahead of time and try to follow the chapters in the syllabus. Profs are usually more than happy to give you the most current syllabus.

>>15828045
Hi, /lit/ here, hope this is the right thread.
I would love some advice for a sci-fi short story I'm developing, as I haven't dealt with any maths/physics since I graduated high-school over a decade ago.
The story has a plot device that involves matter expanding concentrically/spherically at a rapid pace. I want to express that the rate of expansion declines as a function of area (already) covered, where the initial speed/rate is significant (~1% c) but the decline also 'speeds up' so that by the time the total area covered a approaches 10km the sped drops very rapidly down to speed of sound levels and then remains constant (or rather any further decline becomes so miniscule that it seems practically constant by human standards).
In the story this is supposed to function like a law of nature, so I want to express it as a formula. Any help would be much appreciated. As I said I'm not familiar with math/physics, so please excuse the poor explanation...

Consider the ring R[[x]] of power series with coefficients in another ring R where x is placeholder for an element in yet another a non-commutative ring, e.g. x could be a matrix. Now consider the power series of exp and log. How can you show that the series of the logarithm is the inverse to the series of the exponential function?

I looked it up online, but all proofs rely on either differentiation, commutativity of the argument x or analytical properties of the exp or log.

I know these are dumb questions but I'm gonna ask anyway:

>first

The problem was to convert [math]x^2+y^2=6y[/math] into polar form. This is how they did it:
[eqn]
x^2+y^2=6y \\
r^2(\sin^2{\theta}+\cos^2{\theta})=6r\sin{\theta} \\
r^2=6r\sin{\theta} \\
r=6\sin{\theta}
[/eqn]
Why were they justified in dividing by [math]r[/math] in the last step when [math]r[/math] could be [math]0[/math]? I know [math]\theta=k\pi,k\in\mathbb{Z}[/math] will make [math]r=0[/math] and that doing the case [math]r=0[/math] separately just gives me [math]0=0[/math] (which is not quite useful) but I still don't understand when I can too ignore variable being [math]0[/math] and just divide. I think it has something to do with [math]r[/math] being dependant variable.

>second

Now the problem was to solve matrix equation [math]AX-2X=M^T[/math] for unknown [math]X[/math]. All was pretty clear to me except for the second step which was to isolate [math]X[/math] like this: [math]X(A-2I)=M^T[/math]. However, there is no proof (that I see) that this is valid. For [math]AX[/math] to be valid, [math]X[/math] must have [math]m[/math] rows and [math]k[/math] columns, and [math]A[/math] must have [math]k[/math] rows and [math]n[/math] columns. But for [math]AX-2X[/math] to be valid, [math]AX[/math] and [math]X[/math] must have the same number of rows and columns. Therefore, [math]k=n[/math]. But there is no proof that also [math]m=n[/math], so [math]A[/math] which has [math]n[/math] rows might not be compatible with [math]I=XX^{-1}[/math] for addition/subtraction.

>>15839261
Sorry for fucked up formatting but Idk

>>15828045
are we (humans) technically anthropomorphic animals (furries)??

our ancestors come from the plesiadapis, which were squirrel-like. if that means we are furries, then what are some body parts that still resemble the ones from the plesiadapis?

>>15839269
I forget the details but there's a line of thought in anthropology/evolutionary biology theorizing that human intellect and culture are a result of (self-) domestication i.e. changing the phisiology in such a manner that the potential for aggression and violence are reduced produces higher intelligence as a secondary biproduct.

Or something like that. Not sure if this would click with your line of thought.

Anyone know what forums or IRC channels or whatever are available for discussion of distributed control systems? I'm particularly interested in Honeywell's Experion PKS, I have knowledge I'd like to share

>>15839261
There may be some constrained so r is never zero, but in general you don't have to divide by r to get the same result.

[math]r^2=6r\sin{\theta} \implies r(r - 6\sin{\theta}) = 0[/math]. So either [math]r = 0[/math] or [math]r = 6\sin{\theta}[/math].

>>15839243
Wdym "commutativity of the argument x"? What do you think x shouldn't commute with, elements of [math]R[/math]?

>>15828045
ok /sci/, I've been hitting my head against the desk for a while trying to prove the following

Starting from the descending power series solution to Legendre equation

[eqn](1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+n(n+1)y=0[/eqn]

we arrive to the following expression for one of the solutions:

[eqn]y=a_0\left[ x^n-\frac{n(n-1)}{2\cdot (2n-1)}x^{n-2} + \frac{n(n-1)(n-2)(n-3)}{2\cdot 4 \cdot (2n-1)(2n-3)}x^{n-4} - \frac{n(n-1)(n-2)(n-3)(n-4)(n-5)}{2\cdot 4 \cdot 6 \cdot (2n-1)(2n-3)(2n-5)}x^{n-6} + \cdots \right][/eqn]

where [math]a_0[/math] is an arbitrary constant, and the general term can be expressed as
[eqn](-1)^r \frac{n(n-1)(n-2)(n-3)(n-4)(n-5)\cdots(n-2r+1)}{2\cdot 4 \cdot 6 \cdots (2r) \cdot (2n-1)(2n-3)(2n-5) \cdots (2n-2r+1)}x^{n-2r}[/eqn]
where [math]r[/math] goes from [math]0[/math] to [math]\frac{n}{2}[/math] if [math]n[/math] is even
or [math]r[/math] goes from [math]0[/math] to [math]\frac{n-1}{2}[/math] if [math]n[/math] is odd

Now, the thing is to prove that the election of [math]a_0=\frac{(2n-1) \cdots 5\cdot 3 \cdot 1}{n!}=\frac{(2n)!}{(n!)^2 2^n}[/math] makes the previous expression for [math]y[/math] takes the value [math]1[/math] when [math]x=1[/math], i.e.:

[eqn]\frac{(2n-1) \cdots 5\cdot 3 \cdot 1}{n!} \left[ 1-\frac{n(n-1)}{2\cdot (2n-1)} + \frac{n(n-1)(n-2)(n-3)}{2\cdot 4 \cdot (2n-1)(2n-3)} - \frac{n(n-1)(n-2)(n-3)(n-4)(n-5)}{2\cdot 4 \cdot 6 \cdot (2n-1)(2n-3)(2n-5)} + \cdots \right] = 1 [/eqn]

What I'm trying to do is to prove this using the compact form

[eqn]y(1)=a_0 \sum_{r=0}^{N} (-1)^r \frac{(n!)^2(2n-2r)!}{r!(2n)!(n-r)!(n-2r)!}[/eqn]
where [math]N=\frac{n}{2}[/math] if [math]n[/math] is even, or [math]N=\frac{n-1}{2}[/math] if [math]n[/math] is odd.
Without having to use Rodrigues formula or any other recurrence relations. Just the finite sum thing.

Another way to put this is to prove that

[eqn] \sum_{r=0}^{N} (-1)^r \frac{(2n-2r)!}{r!(n-r)!(n-2r)!} = 2^n[/eqn]

Any help will be appreciated.

>>15839477
ITT: yous youing yous

>>15839480
lol I was about to (You) myself mentioning this is not homework, I'm just an enthusiast

>>15838901.
I would do this but I'm a lazy bum.
>>15838927
Yeah, this seems to be it, thanks.

Observation 1: Atoms share Electrons
Observation 2: Sharing is communist
Is the universe therefore made up by a bunch of communists?!

What would the potential difference between A and B be and why? I tried just pretending A and B formed a loop, I tried using the outer loop and just adding the voltage up along the paths between A and B.

Why is my computer telling me this? The answer in the of the book says there is a solution. And when I plot the lines, they clearly intersect. The linear system is

x1 - 4x2 = 1
2x1 - x2 = -3
-x1 - 3x2 = 4

>>15839823
Disregard. Apparently that function from numpy needs a square matrix as input.

>>15839058
>>15839403
hey look its remi, feels like i havent seen you in ages

>>15839477
>>15839477
As >>15839480 said, here I am youing myself to note the following.

The last expression can be put as

[eqn]\sum_{r=0}^{N} (-1)^r {n \choose r}{2n-2r \choose n} = 2^n[/eqn]

I think this can be somehoe linked to the identity:

[eqn]\sum_{r=0}^{n} {n \choose r} = 2^n[/eqn]

But I haven't figured out how yet (even if possible)

>>15839830
this is why its important to actually read the errors (or catch only specific errors)

>fall into black hole
>1.6 trillion G's pulling on me
>perfectly fine
>stand on a platform above a black hole
>1.6 trillion G's pulling on me
>instantly turn into mush
please explain

>>15840165
The weak equivalence principle. You experience no force when in free-fall.

>>15840561
that's seems .... wrong, there is still a force pulling you right?

>>15840165
you get squished in the second because the gravity is counteracted by the normal force the platform exerts on you
there is nothing pushing back against you in the first scenario, so you end up as healthy as someone getting sucked into a black hole can be, which is probably not extremely but it's less certain than getting pulled flat.

A ball is tied to a thread of length(inextensible) length 2l
Currently the ball is kept at a distance l with the slacked thread.

Now velocity u is applied to the ball of mass m and it stretches a distance 2l with an angle of 60° and final velocity V

How can I calculate the tension ? As per I know the it will have two components vsin60 and vcos60
Vsin60 will be zero as the tension won't tell the ball moves forward

Now what should be done ?

>>15840750
No. You are moving in a straight line but one that happens to be along a curved space time. You are not experiencing any acceleration so you feel no force on you. It's one of the core principles of General Relativity and it's why astronauts on the international space station feel weightless even though they are constantly falling towards the earth.

If you were falling into the black-hole feet first you would however eventually experience tidal forces due to the difference in gravity (curvature) between your feet and your head. You'd be stretched out like spaghetti and eventually die (it's called spaghettification).

>>15839261
2nd question, anyone? Am I just retarded?

>>15839261
>Why were they justified in dividing by r in the last step when r could be 0?
[math]r = 0[/math] is a degenerate case for polar coordinates and has no angle, so you just have to keep track of it separately.
>second
Your insight seems correct. Usually you'd isolate it like [math](A - 2I)X = M^T[/math], which doesn't really change much.

>>15840796
>Your insight seems correct.
So what's the deal here? Why do we do it if it's not valid? And why would you undistribute [math]X[/math] on the right side when it's on the left of [math]A[/math] and matrix multiplication is not commutative?

>>15840796
>Your insight seems correct.
So why do we do it if it is not valid.

havent really done any physics in a while so sorry if this is a braindead question

lets say i have a container with oil and water. then i put an object in it until it sinks into the water portion.

if i were to calculate the absolute/gauge pressure would the equation just be p_0 + p1gh + p2gh or would stil just be p_0+p1gh

p1= density of oil

p2 = density of water

>>15840811
We don't.
There could be some special circumstance that allows for that switcheroo but I'd need the original material to tell.

I'm sure the answer is out there but I can't figure out what to google to find it.
Python prints the top numbers, but I want them to be printed like the bottom numbers. What do I google? "round" obvisouly doesn't work because they are on different orders of magnitude.

>>15840860
I don't think there's anything, it's just an introductory workbook with some definitions and theorems in it. Anything specific that would help?

>>15840879
https://stackoverflow.com/questions/34032855/rounding-scientific-notation-in-python
?

What did the science mean by this?

>>15840879
You can use `"{:.2e}.format(number)"` I guess. It returns a string so wrap it in float() to get a number if you need to.

Credits go to ChatGPT.

>>15840879
You can use `"{:.2e}".format(number)` I guess. It returns a string so wrap it in float() to get a number if you need to.

Credits go to ChatGPT.

>>15839477
>>15840028

Alright, I think I got this
I ended up needing to prove that
[eqn]\sum_{r=0}^{N} (-1)^r {n \choose r}{2n-2r \choose n} = 2^n[/eqn]
To achieve this, we start from the following binomial expression:
[eqn][(1+x)^2-1]^n= \sum_{r=0}^{n} {n \choose r} (1+x)^{2n-2r}(-1)^r = \sum_{r=0}^{n} {n \choose r} \left[ \sum_{k=0}^{2n-2r} {2n-2r \choose k } x^k \right](-1)^r[/eqn]
Here, we find that the [math]x^n[/math] coefficient happens when [math]k=n[/math], which leads to
[eqn]\sum_{r=0}^{n} (-1)^r {n \choose r} {2n-2r \choose n } [/eqn]
On the other hand, we have that
[eqn][(1+x)^2-1]^n = [1+2x+x^2-1]^n=[2x+x^2]^n=[x(2+x)]^n=x^n(2+x)^n = x^n \left[ \sum_{r=0}^{n} {n \choose r} 2^{n-r}x^r \right][/eqn]
And here, the whole coefficient for [math]x^n[/math] happens when [math]r=0[/math], so then:
[eqn] x^n {n \choose 0} 2^{n}x^0 = {n \choose 0} 2^n x^n = 2^n x^n[/eqn]
Then, equating both coefficient of [math]x^n[/math] in both results we have that:
[eqn]\sum_{r=0}^{n} (-1)^r {n \choose r} {2n-2r \choose n } = 2^n [/eqn]
Buuuuut, the binomial coefficients of the form [math]{2n-2r \choose n}[/math] will be positive and non zero only if
[eqn]2n-2r \geq n[/eqn]
Or
[eqn]\frac{n}{2} \geq r[/eqn]
This means, the last finite sum runs from [math]r=0[/math] to [math]r=\frac{n}{2}[/math] at most, i.e.:
[eqn]\sum_{r=0}^{n/2} (-1)^r {n \choose r} {2n-2r \choose n } = 2^n [/eqn]

Then proved.

I still have to check what happens when the sum runs from [math]r=0[/math] to [math]r=\frac{n-1}{2}[/math] (for the [math]n[/math] odd case) but I think I made it good enough to this point c:

>>15840881
>>15840895
Thank you!

>>15841023
You're welcome anon.

>>15841071

[math]1. \int x\sqrt{4+5x^2}dx\\
2. \int x^2\sqrt{4+5x^2}dx\\
3. \int x^3\sqrt{4+5x^2}dx[/math]

Why are 1 and 3 easily solvable with a simple sub but for 2 I have to use trig/hyperbolic subs? Can I take it as a general rule that if x^n is even I have to do the trig sub?

hi i was supposed to calculate mass displacement from this spring model equation but I'm not sure what my mistake was

>>15841180
you're solving
[eqn]0=y'(t)[/eqn] for [math]t[/math]?

>>15841224
isn't that the extreme point?

>>15841180
>>15841234
yeah, i'm just not familiar with displacement stuff (thought this was some mechanics thing)

anyway, seems like last two lines was where you messed up, check denominators, yo multiplied by 10 again when it was not necessary

What's the deal with the new sticky? It's got good links but it's all over the place.

>The Feynman lectures on physics
>https://www.feynmanlectures.caltech.edu/
>Landau and Lifshitz
>https://archive.org/details/landau-and-lifshitz-physics-textbooks-series
>string theory FAQ
>https://ncatlab.org/nlab/show/string+theory+FAQ

>>15841369
Nvm found the thread where mods were asking for suggestions before half-assedly throwing something together.

The following is a hypothetical example of a log(kobs) versus pH profile for the hydrolysis of an ester with an added acid
such as acetic acid . Examine this curve and state which form of acid or base catalysis is occurring at the various pH regions.
Write a rate law that would describe this curve.

Pls, I know there is some MIT chad out there that can answer.

>>15841875
This is specific and general acid/base catalysis.

How would you solve matrix equation [math]XA+BX=C[/math]?

>>15834634
essentially, the heat of the pan is not going into your food. it’s energy travelling from the element, through the pan, and into your food. When you turn it off, this flow ceases and you’re left with just the energy still dissipating from.

>>15842134
[eqn]X = \sum_{k=0}^\infty (-1)^k B^k C A^{-(k+1)} [/eqn]
assuming A is invertible and that series converges.

>>15842397
Interesting. It's more complex than I've expected.

Why I should afraid the death?
Expect genetics

>>15842568
>It's more complex than I've expected.
The closed form is but you can also solve it as a big ass linear equation like an idiot.

Can a Universal Turing Machine find a formula for the roots of any nth degree polynomial?

>>15842731
No. It's well-established that there is no general formula to find the roots of any polynomial above degree 4. It's not that we haven't found it yet; it's that it outright cannot exist.

>>15842734
Functions that involve infinite sums can solve a general 5th,6th and 7th degree polynomial. What I meant to ask is if there is a UTM that can find such general formulas any nth degree polynomial.

let [math]P=\mathbb{R}\backslash \{0\}, f: P\times P \to P\times P, f((x,y))=(\frac{x}{y},\frac{y}{x})[/math] How do I Prove that f is surjective?

let [math] P=\mathbb{R}\backslash \{0\}, f: P\times P \to P\times P, f((x,y))=(\frac{x}{y},\frac{y}{x}) [/math] How do I Prove that f is surjective?

>>15842810
You don't.

>>15842810
It's not?
[math](1,2)[/math] is very clearly in the codomain, but f doesn't map to it

>>15842819
but if we have a random [math](l,m)\in P\times P(codomain), [/math] and we construct an equality [math](l,m)=(\frac{x}{y},\frac{y}{x})[/math] doesn't this prove that it's surjective since x/y and y/x will always give a real number other than 0 so the equality holds since l,m have to be real numbers \ {0}?

>>15842819
>>15842821
nvm I'm retarded

>>15842826
It would, if you could always construct that equality. But you can't.

>>15842826
It doesn't map to the whole codomain. It only maps to the hyperbola where the product of both components is 1.

why do they say there are discrete packets of energy? is it just a measurement problem, insufficient resolution, or do they know these packets of energy exist

>>15842863
The Franck–Hertz experiment showed this.

>>15841117
If you try to make a sub in 2, you would get another sqrt like in the beginning, and the easiest way is to perform tan sub, so I think that's why if it's even you get the sqrt. Then you have to do trig sub again

>>15841117
whats the sub for 3?

What's the easiest way to create a histogram that will display the mean, the median, quartiles, and standard deviation on the graph? I'm a TA and I want to send students summary statistics on midterm performance without it taking too much of my time.

>>15842940
matlab, matlab plots (python), excel

>>15842959
>matlab plots (python)
excuse me, Matplotlib

>>15842926
It looks like you need
5x^2 = 4 sinh^2(u)
that is
x = (2/sqrt(5)) sinh(u)
then
dx = (2/sqrt(5)) cosh(u) du
and
sqrt(4 + 5 x^2) = 2 cosh(u)

You will be left with something like
(some constant) integral sinh^3(u) cosh^2(u) du
= (some constant) (integral sinh^3(u) du - integral sinh^5(u) du)

For integrals of powers of hyperbolic sines there are some formulas here.
https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions
First use the recursive formula there to reduce those integrals to
integral sinh(u) du = cosh(u)
Then do back substitutions
sinh (u) = (sqrt(5)/2) x
cosh (u) = sqrt(1 + sinh^2(u)) = sqrt(1 + (5/4) x^2) = sqrt(4 + 5x^2)/2

>>15842926
What's inside the sqrt. Apply linearity.

How can I proof the completeness axiom of R?

I found several proofs but I don't understand a shit.

Also, I don't understand how to proof the Inverse Property of Multiplication with Cauchy sequences of rational numbers.

Any help would be thankful. Have a nice day everybody :)

>>15842999
>completeness axiom
IIRC you just replace the sequences with sequences that converge sufficiently fast and then do a diagonal slash.
Like you've got the sequence of sequences [math](a_n) _m[/math] and you do [math]n \to (a_n) _n[/math].
What's the inverse property of multiplication btw?

If I put a 360 degree 9g servo hooked to a pi, what's the easiest way to create some sort of motorized or belt driven slider than can operate on it?

>>15828045
Is it worth paying for connected papers? Google seems trash in finding the things I wanted and 5 graphs is not much.

How can I divide a division into parts based on their decimal?
I know I can divide a multiplication this way:
a.b * c.d = a * c + a * d + b * c + b * d
And a division so far:
a / c.d + b / c.d

>>15842999
You can prove the completeness axiom, because it's an axiom. You usually introduce the completeness in R using the Dedekind cuts or converging sequences in combination with Cauchy sequences, as this anon >>15843765 said.

Is Leibniz notation truly better? I find Newton notation like f'(x) far more natural to work with than dy/dx

The book im working out of says many people find Leibniz notation easier, especially for related rates. Also it says chain rule is "easier" using this notation.

But for me it doesn't feel that way. Is there any tricks or memorization techniques for converting between the two notations?

>>15844244
>Is Leibniz notation truly better?
Better than Newton's at least but Euler is better than both.
>f'(x)
That's Lagrange's notation.

>>15844143
>You can't prove the completeness axiom, because it's an axiom.
I assumed he wanted to prove it from the Cauchy sequence construction since he brought it up in the next line.
>>15844244
Lagrange notation immediately becomes a mess once you reach the chain rule tbqh.
Mathematicians are in this eternal loop where young people are taught Lagrange/Newton notation because they might come across it in textbooks, so they occasionally use it because it's convenient and the burden is passed on to the next generation.
>>15844301
Based.

>>15844244
>>15844378
To Newton he formed calculus as a tool to do physics. It's why he almost exclusively used the dot notation in his Principa because it all to do with classical mechanics and orbits, so everything was a change w.r.t time. In that context it is a more natural and easier to use notation. However in the more general mathematical framework Leibniz is clearly the better notation.

Smart anons tell me
Would a magnet hitting the top of a tube it is inside of over and over again make it fly?

Is
[eqn]
\begin{pmatrix}
x&y
\end{pmatrix}^\top
[/eqn]
an acceptable notation for a vector [math](x,y)[/math]?

>>15844686
as the magnet approaches the top on the tube just before collision, it also pulls the tube downward. if the whole system is airborne, then the tube will move down as the magnet moves upwards, and specifically in such a way that the center of mass of the system remains stationary both before and after collision (conservation of momentum). so no, no flighty.

>>15844724
If you have a matrix to go with it, the notation
would be that of a row vector. The usual (x,y)
notation would be acceptable, or even the
vertical orientation at least.

I don't really get the Monty Hall problem. Let's say [math]D_i[/math] for i = 1, 2 ,3 describes "the i-th door having the car" with [math]P(D_i)=1/3[/math] for all doors. And let [math]D_i^c[/math] be the complementary event for each.
After the player chooses door 1 and the game master opens door 3 revealing a goat, the probability of door 2 having the car would be [math]P(D_2|D_3^c)=\frac{P(D_2\cap D_3^c)}{P(D_3^c)}=\frac{P(D_2)}{P(D_3^c)}=\frac{1/3}{2/3}=1/2.[/math] same for door 1.
can someone tell me where I'm off?

>>15844896
you are not calculating the right probabilities
you are supposed to show that by selecting another door your odds of getting the car is higher

>>15844896
The easiest way to intuitively understand it is to just write it out. Let's use C to represent a car and G to represent a goat, and assume that you pick the first door.

The initial possibilities are
C G G
G C G
G G C

When you pick the first door, Monty gets to pick which of the remaining two doors to open, and will always pick one with a goat. Our scenarios thus reduce to

C G
G C
G C

so in 2/3 of the possible scenarios, switching is the right move

>>15844896
The game master knows the contents behind each door and deliberately chooses a door with the goat so the information you get from
>the game master opens door 3 revealing a goat
is not just that door 3 didn't contain a goat but also that the game master chose to open this door over door 2. So why did the game master not open door 2? The reason is simple: The car is behind door 2.

>>15844896
Stop calculating things for a second.
The probability you picked the door with the car is one third. After the game master opens a door with the goat, the probability you picked the door with the car *remains* one third. The car needs to be behind one of the doors yet to be opened, so the other one has a 1-1/3=2/3 chance of having it.

was talking to my friend about evolution
he claims it wasn't real because it makes no sense for animals to have non-working developmental organs/limbs if natural selection is a thing.
Like "lol animals went from having no arms to having arms, but wouldn't they have not survived natural selection in between having no arms and having arms"
what do I think about this

>>15845597
you think you need a friend who isn't a retard.

>>15845597
If you think about it, organisms are self-replicating machines. We think of this as the pinnacle of the future and human creation, but we live with it every day. Wouldn't it take some kind of creator to develop the engineering required of life and its delicate environments?
I feel like there are "parameters" of evolution that don't allow adaptations/mutations to stray from the organism's original design. All I know is, there's a lot of assumptions that people stand firmly on as dogma and it PISSES me off

>>15845597
how is having weird snubbed arms worse than having no arms?

>>15845597
>Like "lol animals went from having no arms to having arms, but wouldn't they have not survived natural selection in between having no arms and having arms"
> humans develop arms
> then build airplanes to fly
> birds just develop wings
bigthink

P_1 = (4, pi/4, 2). Are these calculations correct? Looks false desu. Online calculators are totally useless.

>>15846276
doesn't look totally useless to me

>>15846277
What did you enter as input? Is the transformation from spherical to Cartesian correct?

>>15846286
*cylindrical to Cartesian, sorry

>>15846286
"convert Cartesian to spherical" is sufficient and will bring up a menu where you can input your vector
and, yes, the cylindrical to Cartesian map is also correct

>>15846300
thanks

>I have two kids, one of them is a boy.
What is the probability both are boys?
>I have two kids, the older one is a boy.
What is the probability both are boys?

>>15846406
https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

>>15846438
Not an argument.

Anyone know how to make this in metapost? I can do the circles, but what exactly is the optimal way of doing the C set?

>>15846725

Consider the gravitational force [math]\mathbf{F}_g = -mg \mathbf{e}_z[/math]. I want to express the force in terms of cylindrical coordinantes. How can I go about doing this?

>>15847001
Nvm I just realized, it's one and the same thing.

How can I elegantly show that an algebraic structure is a field? Is there a way to do it without checking the field axioms? Maybe by constructing an isomorphism?

>>15847032
>Maybe by constructing an isomorphism?
There's https://math.stackexchange.com/questions/44374/intermediate-ring-between-a-field-and-an-algebraic-extension , which means a possible proof strat along those lines is:
>embed your ring into a field
>show it contains a subfield
Can't think of anything other than that, the axioms and some random theorems (every finite integral domain is a field, for example).

Say we have the proposition: there exist irrational numbers x and y such that x^y is a rational number.
Do I understand this correctly that the only times where x and y are irrational that x^y is rational is only when :
x = sqrt of a rational number where the sqrt makes it irrational(so sqrt4 for example wont fit), to the power of some number(lets call this number 1)
y= some number such that when multiplied with 1 gives us an even number

>>15847071
no
consider [math]x=e[/math] and [math]y=\ln(a)[/math] for [math]a \in \mathbb{Q}[/math]

What is the difference between spherical unit vectors, [math] \hat{r}[/math], [math] \hat{\theta}[/math], [math] \hat{\phi}[/math], and a spherical basis, [math]\mathbf{e}_\pm = \mp\frac{1}{\sqrt{2}}\left(\mathbf{e}_x \pm i\mathbf{e}_y\right) [/math], and [math]\mathbf{e}_0 = \mathbf{e}_z [/math]?

>>15847270
They aren't related.

What is the angle between two lines with (positive) reciprocal slopes?

>>15847085
the point here is that e is transcendental so e^a is still irrational for every a in Q

>>15847892
That's not what's being asked, though. In fact, it's almost exactly the opposite.
The question was about x^y being rational when both x and y are irrational. e^a is irrational if a is in [math]\mathbb{Q}[/math], yes, but the condition was precisely that a isn't. But [math]\ln{a}[/math] would be for any rational a

>>15847896
yah e and ln(a) are both irrational (the latter bc transcendental) hence each such e^ln(a) is a counterexample. Sorry if it was unclear
>>15847888
Are you asking for a specific formula? Well, when slope=0, the angle is 90 deg, and when slope=1 the angle is 0 deg. I would consider using the arctan function as part of your formula

Convince me that I don't have CJD. Please.
Over the past 4 months, my vision has gotten fuzzy and blurry, my memory has gotten significantly worse, it's been unable to learn anything new. I find myself talking out loud to nobody. My voice has become soft and my words slur. I can hardly visualize in my mind eye, and my spatial awareness has completely disappated (if I close my eyes, I don't "know" where I am). I don't remember opening this thread, but I know I did. Please help

>>15828045
>warosu
It is down. Does anyone know more? Do we need to mirror it asap?

>>15848057
No, I don’t know, but it’s been down at least six hours.

How to solve complex quadratic equations? The only way I know is using the quadratic formula: find the determinant of the equation and then its roots, and finally plug them into the quadratic equation. Is there some other good way? I've noticed that trying to use systems of equations is usually way too complex. Maybe something involving polar forms?

>>15848066
I made the deal with the devil.

KLUMOSA
KLUMD...

>>15848079
It has to be tesseract internet aligned. If that worked.

It could be the answer was under our/my nose.

>>15848057
Just use eientei lmao.

[eqn]
\begin{vmatrix} 212 & 106 & a \\ 424 & 212 & 106 \\ 636 & 317 & 106 \end{vmatrix}
[/eqn]
I tried to simplify this determinant and came up with:
[eqn]
\begin{vmatrix} 212 & 106 & a \\ 0 & 0 & 106-2a \\ 0 & -1 & 106-3a \end{vmatrix}
[/eqn]
But these two determinants are not the same. Why?

Fuck me but I cant.
[eqn]
4a=2b+c
3b-2c=4
2a+bc-b=?
[/eqn]
How to solve this shit?

How to solve this shit?
[eqn]
4a=2b+c \\
3b-2c=4 \\
2a+bc-b=?
[/eqn]

>>15847888
Build 2 unit vectors then dot product them. I had worked this out in order to use calculus to show the focal property of parabolas, took several pages, while the given geometric proof is like 5 lines lol

I'm tasked to find the order of the n cycle [math](1\, 2\, \ldots\, n)[/math] of the symmetric group [math]S_n[/math]. Idgi, isn't that just the identity with order 1?

>>15848410
that it is. Question might just be checking to see if you understand that fact

>>15848420
hm okay, thanks

>>15848450
Cool. It's ok. It's still going to be a problem rawly. There is a raw nightmare present that drags me down but I gather all the good logics.

>>15848203
If you solve one equation for one unknown then you can use this to eliminate that unknown from the other equations.

>>15848410
The symmetric group S_n is the set of maps from {1,2,...,n} to {1,2,...,n} together with composition as multiplication.

The n-cycle (1 2 ... n) is the map
1 -> 2
2 -> 3
...
n -> 1
It's not the identity of the group. It has order n.

>>15848484
>The symmetric group S_n is the set of maps from {1,2,...,n} to {1,2,...,n}
I meant bijective maps from {1,2,...,n} to {1,2,...,n} of course.

>>15848484
ohhhh I see, thanks for the save

if f maps from natural numbers to integers, how do I prove by contradiction that f is not surjective if f(n)=(-1)^n times n
I got (-1)^n times n=-2, but don't know to show a contradiction. I already showed it's injective if that is needed here.

Anyone know the best library to use for linear algebra in Python? I'm just learning and my textbook starts with solving linear systems, so that's what I'm doing right now. I was using Numpy and Scipy and their linalg.solve() function only accepts square matrices. They have linalg.lstsq() which can accept non square matrices, but it tries to approximate (the least squares) solutions to linear systems, which means that if there is no solution, it returns the closest approximation. This is a problem if I'm trying to check my work, since sometimes the goal is to show that a linear system is inconsistent. So you can see that neither of those two functions really work well for my purposes.

Bonus question: what do you think is a good order of material to learn linear algebra? My text book does linear systems, matrix algebra, determinants, vector spaces, eigenstuff, least squares, in that order. I watched through the 3blue1brown series on linear algebra and he does transformations first, then algebra, then determinants, dot, cross, change of basis, eigenstuff, etc., and really hammers home the geometrical understanding. He doesn't talk about linear systems or elementary row operations in the entire series. So the order is similar but the starting material is different. What do you think?

>>15848534
If [math]f(n)=(n)(-1)^n[/math] maps surjectively to the integers, it must map to 1. What can you say about the absolute value of an integer x that f maps to 1? There's at most one natural number of a given absolute value; is this a valid solution?

>>15848534
Think about what it means to be surjective.

>>15828045
Is there a book about reading scientific papers you could recommend?

I'm tired of watching youtube videos with "le funny memes" every 10 seconds that talk about those studies and of reading books made by hacks who pull stuff out of their asses. I want to get to the source. But I have no academic education and some terminology often used in those studies is hard to digest. Is there a book that could help me out here?

>>15848622
Nvm, chat gpt does a job well enough for me.

Is there a better explanation of why science is so broken?
https://www.youtube.com/watch?v=Y_Yy-wXNe2I

>>15848556
>So the order is similar but the starting material is different. What do you think?
Linear algebra is really useful in a wide variety of fields, and depending on their field, a teacher might emphasize different things.
>start with vector spaces and linear transformations
This makes sense if your goal is pure mathematics—emphasis on concepts and theory, possibly proving theorems
>start with matrix multiplication and row operations
This makes sense for any kind of applied goal, from engineering to computer graphics. Pure math also often begins here. A big advantage from the teacher’s perspective is that it’s easy to make up 100 homework and exam problems; the theory stuff needs a lot more careful planning

But all roads lead to the same place

Recommend books on cryptography

Exept Bruce Schneier

These two formulas are driving me insane. Am I retarded?
If [math]\textbf{r}_1[/math] and [math]\textbf{r}_2[/math] are INTEGERS, why are they in bold as if they were vectors? And why are they summing over [math]\textbf{r}_1[/math] and [math]\textbf{r}_2[/math] if they don't even appear inside the summation argument?
The very same goes for [math]\textbf{k}_1[/math] and [math]\textbf{k}_2[/math]

Test

>>15850066
[math]\textbf{r}[/math] is a vector.
[math]\textbf{r}_1[/math] is the first component of the vector [math]\textbf{r}[/math].
[math]\textbf{r}_2[/math] is the second component of the vector [math]\textbf{r}[/math].
Those components are both integers here. You're summing over the pairs
[math]\textbf{r} = (0,0) \\
\textbf{r} = (0,1) \\
\vdots \\
\textbf{r} = (0,N-1) \\
\textbf{r} = (1,0) \\
\vdots \\
\textbf{r} = (N-1,N-1) [/math]

which are [math]N^2[/math] many pairs in total.

Have any of you heard of Thomas Kuhn?

>>15850090
Yes he LARP's the same boring philosophical discussions for decades with zero progress.

>>15850084
Ok, that makes sense, thank you. So I could write the summation more explicitly as pic related?
I still don't get why they are bold though.

>>15828045
Can anyone debunk my stupidity in response to the quickly deleted Monte Carlo thread?

>>15850099
>valid method when you're simulating something that is random in the first place

Lol. Lmao even.
>graphing calculator
>simulate dice
>need over 9000 simulations to approximate equal distribution of 1/6 chance per side
Is this real life or Fanta sea? Let's find out.
>play Yahtzee for real
>keep rolling the same numbers

Statistics is truly a lie and it's yet again a miracle if the sun comes up tomorrow.

>>15850128
Not that anon but I agree that the boldness is an unusual choice, if I write a vectors coordinates I usually do it without making them bold, like [math]\mathbf{r}=(r_1,r_2)^\top[/math].

>>15848556
> sometimes the goal is to show that a linear system is inconsistent
Remove rows to form a square matrix, solve the reduced system, then evaluate Ax-b with the remaining rows to obtain the error.

def solve2(a, b):
....c = a.shape[1]
....a1,a2 = a[:c],a[c:]
....b1,b2 = b[:c],b[c:]
....x = np.linalg.solve(a1,b1)
....return x, a2 @ x - b2

Also: bear in mind that when you're using floating-point, almost every square matrix is full-rank. The determinant will almost never be exactly zero. E.g. np.linalg.rank performs SVD then counts the number of singular values which are smaller than the estimated rounding error.

How do I do this? I've yet only seen weak forms of different PDE:s that have all been on a much simpler domain. I can't even formulate the correct integrals

>>15850257
You start with the first equation. Multiply it with a test function, integrate over Omega and then do integration by parts to move the derivatives to the test function. The boundary terms you get can be replaced with the other equations.

>>15850280
Inner product notation, but do I integrate again to fully get rid of the derivatives of the trial function? Also, I should split the contour integral since I have two BC:s?

>>15850125
I see
He was mentioned alongside Popper in this talk I found, I'm not very familiar with philosophy of science but he used their names like he was swearing unpleasantly
https://www.youtube.com/watch?v=eBq42B3K3lk

>>15849670
Right on. To give some context, I'm not actually in a class right now. I did take a linear algebra class like three years ago, but don't remember a whole lot. I'm definitely more interested in the applied goal, I'm trying to learn this to help me be a better data scientist. Thanks for responding.
>>15850245
Will play around with your function. Not quite sure what rank and Singular Value Decomposition are yet, but I'll get there. Still reacquainting myself with the super beginning stuff, pic related. Thanks!

>>15828941
Your mom desu

I had my first Math exam at uni from Limits. Only solved 2 out of 8 problems. Of course, I spent only 4 hours "studying"(copy-pasting web notes to the notebook which I could use during the exam) and started at 3 am the same day of the exam. I was able to solve all problems during lectures, but I am still depressed and fucking embarrassed. Had plenty of time, but spent it instead walking from corner to corner in my room. Horw to cope?

I want to learn about superconductivity, like its theories, practical stuff about making it, and some other stuff I might not know about. I have a background in chemical engineering with a good grasp on chemistry and math.
What are the prerequisites for learning about superconductivity? Any good resources anons can recommend?

>>15850806
you didn't study and only did the easy problems in less than an hour
legit skill issue

I have a matrix of values and I'm going to "extract" data along 30 and 60 degree lines by taking data from every row and every other column (and vice versa).
I'm a little worried about loss of resolution due to skipped points. Is there a recommended interpolation method for this?

>>15851121
Why do you need to interpolate any data when you have the data right there?

Can digestive acid do this to clothing that fast? Explain

Why is:
[math]\left( \frac{\partial u_i}{\partial x_k} dx_k \right)^2= \frac{\partial u_i}{\partial x_k}\frac{\partial u_i}{\partial x_l} dx_k dx_l[/math]
? Where does the index [math]l[/math] come from?

>>15851640
Einstein summation innit?

>>15851689
Yes, a sum is implied on the repeated indices. But how does that explain [math]l[/math]?

>>15851886
[math]\displaystyle \left( \sum_{k = 1}^n a_k \right)^2 = \sum_{k = 1}^n \sum_{l = 1}^n a_k a_l[/math]

>>15851891
Damn that was stupidly easy. Thanks

I must be missing something fundamental. I'm doing this in the vectorspace of all polynoms of second degree or less with x in [0,2]. Why does the norm induced by the given scalar product not work to produce vectors that fed into a scalar product give somethign different than 1?

>>15852006
Why does the induced norm not norm the vector such that it, fed into the scalar product, only produces 1?*

>>15852006
[math]\frac{\sqrt{8}}{\sqrt{3}}\neq\sqrt{8}\sqrt{3}[/math]

>>15852016
Yes but x/sqrt(8)/sqrt(3) = x/(sqrt(8)sqrt(3)). I couldn't find an faq or anything, how can I do fancy math symbols?

>>15851891
why do you post anime pictures with your solutions?

>>15852026
[math] tags and LaTeX formatting, more-or-less

>>15852016
I am actually just super retarded and didn't know the order of operations mattered here, thank you anon

>>15852054
Tradition.

Anyone know where you can download Magma?

https://www.sciencedirect.com/science/article/abs/pii/S0940960204800025
I thought the plica semilunaris was supposed to be vestigial? wtf?

Brushing up on tensors and I've learned that implicit summation is only valid when indices are in covariant/contravariant pairs, e.g.:
[math] {A^i}_i, A^iB_i,{{T^i}_i}^j[/math]
are all valid (contractions), but something like
[math]{A_i}_i[/math]
makes no sense.
But when I take a look at something like my old emag textbook, I find that all indices are in the down position, and summations just occur over any repeated index. So, when is it OK for tensors to be expressed and summed over when all indices are down?

>>15852543
It's just a convention and will depend purely on the author of the textbook and the material in question.

>>15852543
>So, when is it OK for tensors to be expressed and summed over when all indices are down?
Never lmao.
An index showing up as many times lowered as it does raised is what ensures the coordinate invariance.
Summing over repeated lowered indices is, as you've already brought up, only fine if half of those lowered indices are actually supposed to be raised but the author didn't bother.

>>15852589
*ensures the coordinate invariance of the sum, to be clear.
Feel free to take a minute to check this.

>>15852589
think i just found the answer here

If an equilatateral triangle has side length [math]\sqrt{3}[/math], then its center must be [math]0.75[/math] away from any vertice (i.e. half of median = half of height, which is [math]1.5[/math]), correct?

>>15852615
That's exactly the same as implicitly adding the metric tensor to raise and lower indexes so the sums work out.

>>15852406
tap a volcano vent