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/sci/ - Science & Math

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Previous thread: >>14559536

>what is /sqt/ for?
Questions regarding math and science. Also homework.
>where do I go for advice?
>>>/sci/scg or >>>/adv/
>where do I go for other questions and requests?
>>>/wsr/ >>>/g/sqt >>>/diy/sqt etc.
>how do I post math symbols (Latex)?
rentry.org/sci-latex-v1
>a plain google search didn't return anything, is there anything else I should try before asking the question here?
scholar.google.com
>where can I look up if the question has already been asked here?
https://warosu.org/sci
https://eientei.xyz/
>how do I optimize an image losslessly?
trimage.org
pnggauntlet.com

>where can I get:
>books?
libgen.rs
z-lib.org
stitz-zeager.com
openstax.org
>articles?
sci-hub.st
>book recs?
sites.google.com/site/scienceandmathguide
4chan-science.fandom.com/wiki//sci/_Wiki
math.ucr.edu/home/baez/physics/Administrivia/booklist.html
>charts?
imgur.com/a/pHfMGwE
imgur.com/a/ZZDVNk1
>tables, properties and material selection?
www.engineeringtoolbox.com
www.matweb.com

Tips for asking questions here:
>attach an image (animal images are ideal. Grab them from >>>/an/)
>avoid replying to yourself
>ask anonymously
>recheck the Latex before posting
>ignore shitpost replies
>avoid getting into arguments
>do not tell us where is it you came from
>do not mention how [other place] didn't answer your question so you're reposting it here
>if you need to ask for clarification fifteen times in a row, try to make the sequence easy to read through
>I'm not reading your handwriting
>I'm not flipping that sideways picture
>I'm not google translating your spanish
>don't ask to ask
>don't ask for a hint if you want a solution
>xyproblem.info
>>
frogtranny
>>
Unanswered questions:

Maths questions:
>>14559891 [Commentary in >>14559923]
>>14564143
>>14565773
>>14566420
>>14566498 [Yeah but the degree is different.]
>>14568165
>>14571945

Physics questions:
>>14578202
>>14587783

Chemistry questions:
>>14560511 [Some anon posted a wikipedia link and quoted how methanol comes from methane but I get the general impression that doesn't actually answer the question. My bad if it does.]
>>14560974
>>14566504

Engineering questions:
>>14564254 [I think it's just because ice is less dense than water.]
>>14566717
>>14574466
>>14582645
>>14587883
>>14588380

/g/ questions:
>>14567377

Biology questions:
>>14567401
>>14570332 [ https://en.wikipedia.org/wiki/Germ_theory_of_disease ]
>>14572658
>>14589366

Stupid questions:
>>14559829
>>14559925
>>14564212
>>14566443
>>14567362
>>14570001
>>14570408
>>14571521
>>14571931
>>14574381
>>14574623
>>14574645
>>14575676
>>14580625
>>14582175
>>14584651
>>14589394
>>
>>14590123
>how does a biological mechanism for methanol production explain the biological production of methanol?
>>
>>14564254
>every single beer frosted from the top down.
what does that mean?
>>14566717
your body creates a capacitance to ground that AC current can slightly travel through. you dont have to worry about your houses wiring. heres a good video explaining it in more detail https://www.youtube.com/watch?v=1uEmX5XClPY
>>14587883
100% of the energy a CPU uses it dissipated as heat. this is something ive thought about a lot.
>>14588380
https://www.youtube.com/watch?v=8aWXb-3Hrlc
>>14567362
semiconductors if youre a chad
>>
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Is geometric algebra useful for anything besides Maxwell's equations?
>>
>>14590246
Thanks, sweetheart.
>>
>>14590263
<3
>>
Why is A mod B < B?
>>
>>14590648
remainder of A divided by B will always be an integer between 0 (no remainder) to B-1.
>>
>>14590662
And why cant the remainder be equal to B?
>>
>>14590667
if the remainder equals to B you can divide it again to have no remainder. For example, 27 mod 7 = 6, 28 mod 7 = 7 mod 7 = 0.
>>
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what are the factors which determines the life span of any species? i would be lovely if you could provide some reading material along with answer
>>
is it possible to improve your eyesight naturally or without surgery?
>>
>>14590875
You want to read about life-history theory. Here's a general intro with further reading suggestions:
https://www.nature.com/scitable/knowledge/library/life-history-evolution-68245673/
>>
>>14590881
try it and report back
>>
>>14590881
>>>https://pubmed.ncbi.nlm.nih.gov/
>>
can someone please tell me where i can see the proof that the real symplectic group is connected? I have tried 4 books on Lie groups and none of them proves this
>>
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>>14590075
$\text{Why doesn't LaTeX hyphenate sometimes? I am guessing it has to do with some retarded typography thing, but how do I enable hyphenation here? Yes, I know about \-}$
>>
>>14591020
Polar decomposition.
>>
>>14591509
weird remi
>>
Hey anons, how does this work?

Proving the <complement of ~A> (its supposed to be an A with 2 bars above it, like a double negative) is equal to A

so ~~A = {x | ~(x e ~A)}
= {x | ~(~x e A)} //(Why does negation go to x?)
= {x | x e A} = A

But how do you pull the negation/complement off of A and put it onto x, so x is now ~x.

What does ~x (not x?) even mean in this case. Its the set of all x such that not x is an element of A?

If you're describing x as in "the set of all x" how can you mention ~x?
>>
>>14592058
and what does this mean?
(as in, how are they allowed to write this, it seems like a lot of important steps and things are left out)

>>14592058
A u U = U
A u U = {x | x e A v x e U}
= {x | x e A v T}
= {x | T}
= U

How is this legal/allowed? What does the set of all x such that true mean, and how does that mean it is equal to the universe set?

Like i get it because i read the stuff on logic and logical equivalencies, but it feels like you shouldnt be able to directly subsitute sets for logical stuff, again like {x|T}=U

Sorry if this doesnt make sense but i feel like they are hand waiving this away as being allowed because they dont want to explain the real rigorous stuff behind it that makes it ok
>>
>>14591020
>>14591509
My bad, I've had a headache the whole day and my previous answer was relatively unhelpful.
You can use the polar decomposition to show that the symplectic group is homotopy equivalent to U(n) and hence connected.
Proposition 2.2.4 (ii) in McDuff and Salamon if you want a source.
>>
Assuming we just use the Roman alphabet and Arabic numerals what is the minimum number of characters we need to allow every single person on Earth to have a unique ID?
>>
>>14592360
$\lceil \; \log_{36}(7735000000) \; \rceil = 7$
>>
>>14592366
>7735000000
Weird number, where did you get it from?
>>
>>14592370
bro, come on, use your brain
>>
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>>14592366
woah, logarithms. I remember those from high school. Wish I could remember how they work.

I'll look em up. Sounds like something useful I should know. Here is a picture of a cat for your help.
>>
>>14592374
Bro googling world population gives me 7,994,649,218 in https://countrymeters.info/en/World and different values on other places.
>>
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why in the holy mother of fuck am i shadowb& on my phone, i want to fucking choke the asshole thats making me type up my fucking post again

>>14592381
>Wish I could remember how they work.
theyre the opposite of exponents. $f(x)=\log_a(x)$ is the inverse of the function $f(x)=a^x$
ty for the neko fren <3
>>14592393
your post made it sound like you had no idea what that 7 billion number was supposed to be.
>>
Any bio people here? Can women tell if you’ve jerked off recently? Like are there pheromones for that? I’ve noticed I get less interest from women if I’ve beat it the night before
>>
Sorry for the stupid question, I'm not a scientist, just came to this board because I'm curious.
I've seen several "simulations" of panoramas from moons of the solar system and in them the planets' equator/rings are at an angle with the horizon of the satellites that orbit them.
Why is this? I can think of different reason but I'm not sure which would it be:
>they can be at any angle, so statistically it's just more likely it won't be exactly parallel to the horizon
>does location on the moon, time of day or time or year have a say in that? if so, then statistically you are probably more likely to see an angle in most cases
Maybe a better way to phrase it: would it be possible to see saturn/jupiter in such a way that their equators/rings are parallel to the horizon? If so, what factors would need to be just right?
>>
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>>14592426
Another example, this one's from NASA
>>
>>14592426
assuming the poles of the moon and the poles of the planet are mostly parallel (which is usually the case),
when its "sunrise/sunset" and the planet is near the horizon, the planet will always be at an angle to the horizon. if youre standing on the equator, it will be 90 degrees, and the farther north/south you go the lower the angle gets, until at the poles its always flat, but of course theres no "sunrise/sunset" at the poles.
>>
>>14592437
Thanks a lot
>>
>>14592425
I feel like this is a question that absolutely needs to be known but nobody has studied it.
>>
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>>14590075
Is there any biological reason that a creature might have for having a mouth inside of another mouth?
>>
>>14592425
theres no literature to suggest the existence of human pheromones.
>>
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I hate it when I understand the overall calculus part but get stuck at a simple algebra step. How the hell does he collect like terms in ONE step?
In a simplified form, the initial expression is
a(x+y) + b(x-y)
I would have to distribute all of them and then collecting like terms is easy. But he just knows that a(x+y) + b(x-y) = (a+b)x + (a-b)y? How?
>>
>>14592680
Sure, for parasitoids. Say the host inside of which it has evolved to lay its eggs has multiple layers of natural armor. Then it might use its primary mouth to pierce the first layer of armor, an internal mouth to chew through the second layer, etc. until finally the ovipositor gets to the gooey center.
>>
>>14592737
Well, now that you have written it in this way, I can't help but notice it's just switching variables: a<->x, b<->y :D
>>
>>14592798
that doesnt work in general. e.g., $\; \; ab^2 \; \; \not = \; \; a^2b$
>>
>>14592737
notice that we are dividing each bn term by i in the first line, but we are multiplying each bn term by i in the second line, using $\dfrac 1 i = -i$ and a collection of like terms it's simple to see that the two lines are equivalent.
>>
>>14590075
I'm writing a program that finds the delta between an input radian (defined as "-Pi <= x <= +Pi" in a C# library) and a set of radians on the perimeter of a circle but have problems when I reach a point at +Pi because on one side it is closer to the next closest point in the set rather than the point itself because "-x - +Pi" is further away from the next point in the set which is "-x - ( - 2.0943)". What can I do to remedy it?
>>
Let G be a finite group and F a field of characteristic 0. By Maschke-Artin-Wedderburn, the group algebra $F[G]$ decomposes (as an F-algebra) as a direct sum of matrix rings $\oplus_i M_{n_i} (D_i)$ where $D_i$ are f.g. division algebras over $F$, and the $n_i$ are the dimensions of the irreps of G over F.

Now, if F is algebraically closed then this forces $D_i = F$ for all $i$, and we have $|G| = \sum_i {n_i}^2$. It seems... trivial that in the general case, we should have $|G| = \sum_i {n_i}^2 \dim_F (D_i)$. However, my textbook consistently uses the formula $|G| = \sum_i \frac{{n_i}^2}{\dim_F (D_i)}$, instead. What on earth am I missing?
>>
>>14592737
> But he just knows that a(x+y) + b(x-y) = (a+b)x + (a-b)y? How?
It's not exactly rocket surgery. Terms involving x: ax+bx=(a+b)x, terms involving y: ay-by=(a-b)y. If you're collating terms with a common factor, you ignore the factor itself. So multiples of x: a+b, of y: a-b.
>>
If youre asked to prove $a\leq 0$ is it enough to just prove $a<0$ or do you have to prove the case when $a = 0$ as well?
>>
>>14593461
Yes, I know how it works, when you write things down in two steps, then it is easy, I was just wondering how he did that in one step without "showing your work" especially since it is not just x and y but complex exponentials that require additional mental concentration. But most importantly, apparently there is no shortcut, if you do this often you will probably learn how to do it quickly and in this case he's done it a million times so he didn't bother to "show his work". But there is no magic formula or a shortcut. You need to fully remove the parentheses, distribute, and combine like terms.
>>
>>14592215
thank you very much!
>>
How often do scientists misrepresent their findings? Whether accidentally, or on purpose.

If not uncommon, are there any fields it happens in more?
>>
>>14593443
I think you're correct. $|G| = |F[G]| = \sum_i {n_i}^2 \dim_F (D_i)$
Are you sure $|G|$ is being used for the group's cardinality?
>>
>>14593708
You don't have to proceed one axiom at a time.

E.g. a+b+c+d = d+c+b+a is obvious, but formally:

((a+b)+c)+d
= ((b+a)+c)+d (commutativity)
= (c+(b+a))+d (commutativity)
= d+(c+(b+a)) (commutativity)
= (d+c)+(b+a) (associativity)
= ((d+c)+b)+a (associativity)

But outside of a few specific courses, no-one is going to expect you to "show work" for changing the order of elements in a summation.

Once you get past high school, people stop assuming that you need to be spoon-fed basic algebra. Practically any algebraic rearrangement that doesn't involve a specific non-obvious step will be treated as self-evident. Distribute, re-order, factor is about as basic as it comes. Anything you should be able to do in your head is one step.
>>
I'm trying to simplify calculating the percent change when I only have the end value and the change. Is there a way to simplify this without calculating the start value first as End - Change?
$\frac{change}{end - change}$
>>
>>14594882
No.
>>
>>14594882
are you defining percent change as change over start? then no, you need to calculate start.
>>
>>14594961
>>14594929
Thanks
>are you defining percent change as change over start?
Yeah
>>
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AAAAA why do they do this to us??
No reason is given why these crossed terms cancel out. At first I tried to understand why -j.a.sin cancels out j.b.cos and j.a.sin cancels out -j.b.cos, since they are grouped together inside the parentheses. But I finally guessed that all of them are added together even though they are parts of two different summation sequences. Then it became obvious that I should be looking at like terms with the opposite signs on two different lines (not the ones next to each other in the parentheses) and group them together. Ugh.
>>
I'm starting my degree at 23. Most of my classmates are 17 or 18. I'm really insecure about my age. I graduated at 16 and wasted a good 7 years being a shut-in. I don't even know what to say when people ask what I did for all these years.

I'd appreciate some advice if you started your degree late, or know someone else who did. Thanks.
>>
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>>14595181
>I'm really insecure about my age.
well youre acting like a 14 y/o girl so they might not even notice. when youre 40 youre gonna be wondering how on earth you ever thought 23 was old.
>I don't even know what to say when people ask what I did for all these years.
tell them you were a shut-in, its not that big of a deal.
>>
>>14595181
>I don't even know what to say when people ask what I did for all these years
The way I see it you've got four options
1) You got lost in the amazon rainforest/thailand for 5 years (lie and say you graduated at 18)
2) you were edging for 7 going for the guinness world record for largest load
3) you were working and trying to figure out what you wanted to do
4) just be honest

Newsflash, no one really cares lol. I've had classes with more than a few older students and when i was working with them, i asked what circumstances brought them to school later than usual, listen to their answer, and then start doing my work. These aren't your parents, you don't have to impress them. Worst case scenario after 4 months you will never see them again
>>
>>14595181
Anon, when someone hears you say you were a shut in for 7 years they usually assume you were depressed or dealing with loss/family issues and not a sack of shit.
Unless you're like, noticeably a lazy sack of shit.
>>
How is energy produced by renewables sources measured in comparison to fuels?

Mostly wondering about solar and wind.
Since for those you measure their energy output over time, but for things like coal you measure output per amount. So how do you compare the two?
>>
>>14595263
>Auau hauu hauu auau auu-wa hauuau!
>>
>>14595372
Not that anon but some people like that just cannot work in public working fields.
I tried, it nearly killed me. Ruined my professional career path (well doomed from the start).
The stress of dealing with person to person as a client with their very personal matters was just hell.

You just gotta find a way of dealing with people less invasively in workplaces or just solo by yourself.
That's hard to do in a society (people are finding ways on the net today, but don't expect that to continue forever, I think it's potentially going to dry up because of AI anxieties for people).
I'm looking more into how I best deal with people too and all I can think of is "I get the urge to punch them in the face when I see them, how can I use that to my advantage?".
>>
>>14595572
https://www.youtube.com/watch?v=AjNPeKgLnC8
>>
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>>14595603
>haaaauuuu!
>>
>>14595263
I remember when I turned 14 I thought my life was over.
I will double that pretty soon.

... I was right btw, my life did end when I was 14.
>>
>>14595431
For solar energy you're interested in kWh mer surface of panels.
For wind energy it'll be kWh compared to the rotation speed of the turbines (or rather, the tips of the blades since it's what prevents turbines from producing energy when there's too much wibd).
If you want to compare them to other energy sources then you'll have to rely on the capacity factor, or energy produced per surface unit required.
>>
>>14594062
Yeah, I'm positive. How confusing... thanks for the reassurance tho.
>>
>>14590075
The reason that the time independent shordinger equation evolves throughout time is because there are two parts of it the imaginary and the real. When you look at them individually they are moving through time, but when you sum them they are constant. Is this correct? This is why they call it the time independent shrodinger equation.
>>
>>14595783
> Is this correct?
No. It is what you obtain if you decompose the full Schrödinger Equation into separate equations for space and time variables. The Time Independent Schrödinger Equation is the part that only depends on position and the solutions to which are always real. However the general solution is multiplied by the time component, not summed, and that solution is complex so the full wavefunction is still complex too.
>>
>>14590075
If biting red phosphorous can make it explode, would biting a match head blow my jaw off?
>>
>>14593616
a<0 implies a<=0 trivially, so proving the former is enough
>>
>>14592680
There are real world examples, but they're typically quite different to your. His second jaw is basically a mini replica off the main one. If you need a second jaw, it's usually because your normal jaw had evolved some specific shape or structure for food acquisition, and you need different equipment inside for manipulation and/or processing of the food once you've caught it.
Consider pic-related. Its jaws are used to grab snails from the substrate, and then you can see there is a kind of secondary 'jaw' in its throat with large, flat teeth which are used to crush the shells.
>>
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>>14596102
Sorry, here's the pic
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How do you compute this limit?
>t. CSfag
>>
>>14595372
>>14595276
>>14595263
Thank you, frens.
>>
>>14596493
Something something
$\int_{i=1}^n \sqrt{x}dx = \frac{2}{3} n \sqrt{n} - \frac{2}{3}$

Something something you can approximate sums with integrals. Then your error term when divided by $n \sqrt{n}$ should go to 0 as you increase n.
>>
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I'm in space and want to build the most maneuverable spaceship. Attaching my (weightless) jet engine to a lever makes the lever accelerate faster if the jet engine is at the end. So the best design is a sphere containing all my other shit with a bunch of long arms sticking out and jet engines at the end.

Is all that right?
>>
>>14595181
the whole covid thing gives you a 2-3 year free pass. don't worry about it. I'll be 27 when I graduate. Some classmates will be 23. Nobody really gives a shit
>>
>>14596493
https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula
>>14597049
imagine the levers were REALLY long, like several kilometers long. do you think that would make for a maneuverable ship?
>>
Sent an email to a professor:
>dear prof X,
>blablablah
>best regards

He answered:
>dear anon
>blahblabla
>best regards

Now I have to reply to him. Do I still open with "dear prof X"? Or do I just answer and then close with "thank you and best regards" or whatever? Why no guide on the internet mentions this? Do they all assume that you just send ONE email and you're done?
>>
>>14597565
i usually start formal and then degrade in text messages as the email chain goes on:
>dear prof
>hey prof
>prof
>hi
>(nothing)
dont worry about it too much. i once got an email from a professor that just said
>are you on campus
in the subject line. no capitalization, no punctuation, no body. completely out of the blue, no explanation.
>>
>>14597565
After the opening of the conversation, I would just write all further replies without any preamble, and end with something like "--anon".

>>on 24th june, you wrote:
>>blahblabla
>Okay yeah, that works, I'll see you tomorrow to discuss the other thing.
>--anon
>>
>>14597561
If my engines were just moving at some fixed speed it wouldn't. But since they're applying a fixed amount of force to a lever I don't see why not. Isn't a farther away force applying more acceleration to every atom in my lever?
>>
>>14597577
>>14597589
It looks to me like you're at a more advanced level, like PhD students, which I'm not. I also don't know this prof well, just took his course and now emailing to set up the exam. I'd like to know what the proper etiquette is, even at the risk of being overly formal
>>
>>14597649
What I [>>14597589] said IS the proper etiquette, as far as I'm concerned, no matter your position. It's how you communicate with business partners and customers and suppliers as well. Formality is for opening a conversation. You wouldn't repeat formalities every 90 seconds on a telephone call, and there is little point in repeating it every email either. You have done the necessary invocations of politeness, any further openings and closings of messages just serve practical logistics.
>>
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>>14597630
Netwon's 2nd law is $F=ma$. the rotational version of this is $\tau = I \alpha$. $\tau$ is torque, $I$ is moment of interia, which is basically a measure of how far away the mass of the object is from the axis of rotation (a disc would be harder to spin if its mass were concentrated near its edge), and $\alpha$ is angular acceleration. in your problem, $\tau$ is produced by the engines, $I$ is constant (if we're assuming that the levers and engines are massless), and we're trying to maximize $\alpha$. since $I$ is a constant we'll just write $\alpha \propto \tau$ (that symbol means "is proportional to"). if the engines are perpendicular to the levers, then the torque can be expressed as $\tau = Fd$, where $F$ is the force of the engines and $d$ is the length of the levers (the distance from the engines to the center of mass). so, finally, we have $\alpha \propto Fd$.
this looks pretty open and shut for having big levers attached to your craft, but theres an issue: this seems to violate conservation of energy. rocket engines work by throwing particles out the back of it, and if the engines are producing constant force, then the speed that the particles have upon ejection should be constant too, but the math here suggests that we can increase the angular acceleration (and therefore the work done on) the ship by merely moving the engines outwards.
as you can probably guess, the assumption that the engines (or, very specifically, the fuel of the engines) is massless cannot work. massless rocket engines cannot produce thrust. so in reality, moving the engines outwards would increase the ships moment of inertia, and it would increase such that the angular momentum of the ship is increasing at the same rate in both scenarios.
cont.
>>14597649
im not a PhD student, i cant even make a dentist appointment
>>
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>>14597630
think of prying open a door with a crowbar. the closer the load is to the tip of the crowbar, the easier it is to pry, but the door will only move a tiny amount.
so, if the angular momentum is "conserved" in both cases, you want to minimize the moment of inertia of your ship in order to make it maneuverable. in fact, two ships that have equal moments will be exactly as maneuverable as each other, no matter where the engines are placed.
in conclusion, put the engines closer to the center.
>>
>>14597565
>Do I still open with "dear prof X"?
Yeah, that's fine.
>>14597732
>im not a PhD student, i cant even make a dentist appointment
Do you also plan out ordering pizza?
>>
>>14597761
>Do you also plan out ordering pizza?
im not a boomer, remi, i order it online. i do have a rat trapped in my oven tho
>>
>>14590075
sneed
>>
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>>14597769
I don't like pizza websites.
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>>14597749
>in fact, two ships that have equal moments will be exactly as maneuverable as each other, no matter where the engines are placed.
im thinking more about this and actually im not so sure this is true, i have to do some math.
>>14597779
whats wrong with pizza websites?
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>>14597749
>>14597790
yeah i was right to be suspicious, it goes as r. however, moment goes as r^2 so its still beneficial to move the engines closer.
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If I have n open sets each lying in the metric space X_i and I combine the metric spaces as a Cartesian product with the metric
[eqn] d(x, y) = \underset{1 \leq i \leq n}{max} \, \, \, {d_i(x_i, y_i)} [/eqn]
is the n-fold Cartesian product of the open sets necessarily open?
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>>14597844
$\max d_i (x_i, y_i) \leq \sqrt{ \sum d_i(x_i, y_i)^2 } \leq n \max d_i (x_i, y_i)$
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>>14590075
How do i into math? I was in horrible schools whole life. Where do i start? What books should i pick up?
>>
What is the best editor I can use for writing LaTeX?
>>
>>14598006
Vim
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>>14590075
Anyone good at contour integration here?
I'm trying to integrate $\int^{\infty}_0\frac{\sin(x)}{e^x+1}\text{ d}x$ along a rectangualr contour but my shitty complex integraion skills are clearly showing here. help is appreciated
>>
>>14598006
Visual Studio
>>
>>14598006
4chan
>>
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>>14597732
>so, finally, we have α∝Fdα∝Fd\alpha \propto Fd.
>this looks pretty open and shut for having big levers attached to your craft, but theres an issue: this seems to violate conservation of energy. rocket engines work by throwing particles out the back of it, and if the engines are producing constant force, then the speed that the particles have upon ejection should be constant too, but the math here suggests that we can increase the angular acceleration (and therefore the work done on) the ship by merely moving the engines outwards.
I can see why this would be a problem. An engine flying straight shouldn't go faster just because you stuck a lever to it. But I don't see how it's resolved by
>as you can probably guess, the assumption that the engines (or, very specifically, the fuel of the engines) is massless cannot work. massless rocket engines cannot produce thrust. so in reality, moving the engines outwards would increase the ships moment of inertia, and it would increase such that the angular momentum of the ship is increasing at the same rate in both scenarios.

It's not like the force being applied is proportional to mass of the engine, or anything. And inertia isn't even proportional to the length of my lever because of my big sphere in the middle.

Another thing: If I take my lever with rockets at the end and stick rockets going the opposite direction closer in shouldn't the outside rockets overpower the inner and make it spin?
>>
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>>14598070
yeah i think im super wrong about most of this.
consider a very simplified scenario (i drew a picture in >>14597821) where you have two opposite facing "engines" at a distance $r$ from the center of mass. instead of firing propellent, these engines just fire a single particle of mass $m$ outwards in their respective directions, causing the ship to begin spinning. we'll say that neither the engines nor levers have any mass.
you can see in that picture that the system of the two particles has angular momentum $mvr$ (actually $2mvr$ since theres two particles). conservation of angular momentum would suggest that the ship must then also have an angular momentum of $2mvr$ ($m$ and $v$ are still referring to the mass/velocities of the particles, not the ship). $L=I \omega$, so $\omega = \frac{2mvr}{I_{ship}}$. moving the "engines" outward increases $r$ while $I_{ship}$ stays constant, so moving the engines outwards would increase the angular acceleration of the ship, like you originally said.

my mistake:
>the math here suggests that we can increase the angular acceleration (and therefore the work done on) the ship by merely moving the engines outwards
angular momentum is conserved, and thats all we really care about. this was tricky to think about: if you attach a rocket engine to a ship and fire it, the momentum of the ejected fuel is the same as the momentum of the ship. but what if the rocket fires its engine and the ship spins? where is the equal and opposite angular momentum? the answer is that the ejected fuel, even though travelling in a straight line, forms a system that is "spinning" and has angular momentum.
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>>14597049
If the engines actually have zero mass, then yes. The farther they are from the centre of mass, the greater the torque. Greater torque translating to greater angular acceleration depends upon the moment of inertia remaining constant, which is the case if the engines have no mass.

For a real engine with non-zero mass, you'll get diminishing returns. If you put the engines far enough from the centre of mass, eventually the engines will be the dominant factor in determining the moment of inertia, so the distance will roughly cancel (you'll get ax/(b+cx) which tends to a/c as x tends to infinity).
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>>14598150
No problem, thanks for talking me through it, anon.

>>14598173
I was wondering why I could accelerate my massless cannon by sticking a massless telescoping lever arm to it but I guess my cannon is already traveling infinitely fast. At least I can use it to spin my ship arbitrarily quickly with my massless sail.

I guess the lesson for me here is that massless objects fuck with rotation even more than normal.
>>
>>14598070
>>14598196
Where do file.png filenames come from?
>>
How do I hide my browser history without deleting it? Please be fast.
>>
>>14598718
Delete the Chrome shortcut from the desktop and lie you use Internet Explorer.
>>
Are there any ferromagnetic alloys of non-ferromagnetic metals?
>>
>>14598628
copy pasting the image instead of saving it.
>>
>e^-jw
can you see a jew there? ^- is the nose.
>>
Hey anons, im really struggling with solving recurrence relations, and i think a big part of it is because i dont understand sums.

I dont understand how to derive even the most simple sum of n, like 1+2+3+...+n

When i get kind of close to a solution for a recurrence relation, i am always missing something, like i'll end up with n(n+1) but it should be n(n+1)/2

How and why does it become n(n+1)/2?

If you write out 1+2+3+4+5+... where does the /2 come from?

Please help my smooth brain
>>
>>14599972
I'll explain the even case and leave the odd case for you to figure out.

Suppose we wanted to find the answer to 1+2+3+...+n with n even. Then we can re-order the sum to be (1+n)+(2+(n-1))+(3+(n-2)+...+(n/2+(n/2+1). Notice that each of these elements sum to n+1, so it's just a simple multiplication. Now just how many of these n+1's are we summing? Well, notice that they are indexed by their first part (i.e. "1"+n + "2"+n-1 + "3"+n-2 etc). So in fact we are adding n/2 copies of n+1, so the final answer is (n+1)n/2. Another way to look at it is to realize that we had n elements that we were summing, and we paired up each element, so only half of them remain.

Now, can you do the case where n is odd?
>>
what is the distinction between "yield" and "conversion" with respect to a chemical reaction?

imagine for example, you react sodium and chlorine, to make salt.

Na + Cl => NaCl

and someone says that the salt was obtained in 95% yield.
or that there was a 95% conversion.
what's the difference?
>>
>>14588957
How horrible of a choice is it to use books like Serge Lang's basic Mathematics in this endeavor? I'm really bad with programming but I might need those skills in the future, but from what I understand online courses and all that is a big scam. So I'd like some pointers for that too yeah.
>>
How do I find the largest prime number less than e^n where n is an integer?
>>
>>14600236
what sort of math do you need to know?
khan academy is pretty good, i got a lot of utility out of it for calculus.
id argue that algebra is without a doubt the most important math to be adept at. for really any field where math is used, you're building on fundamentals of algebra. you want to be very comfortable with it.
>>
>>14600278
I was thinking Algebra and Geometry
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>>14600324
just my experience, but i did 3 years of chem E before switching to computer science, and i don't think i used any "significant" geometry at all.
like, sinewaves and surface areas of shapes, and working with graphs and stuff, but never any "sohcahtoa" bullshit.
maybe i just don't remember what "geometry" was, but if it was used at all it was really low level stuff. like, stuff you could just look up an equation for.

but algebra was the bulk of all the higher level math i did. i did 2 semesters of calculus, and one of linear algebra, for chemical engineering. and all that is pretty much just algebra stuff. being comfortable with variables and understanding relationships between operations. like, knowing when you can use the commutative property, or the associative property.
understanding how exponents work, and how logarithms work.

comparatively, the math i did in chemical engineering was way beyond the computer science stuff. i think there was like, a single calculus course for the comp sci curriculum, but calculus is a lot of fun once you get it. it's kind of like black magic, it's nuts that it works how it does.

but, i'd definitely recommend you hammer the algebra hardest. if you're comfortable with algebra but not super comfortable with geometry, you'll be in a much better position than if it was the other way around.
>>
>>14600337
Thanks man, would you recommend any particular books, websites, or online courses (from what I understand these are memes but again, I don't know for sure if that's the case) that would help in addition to Khan academy? I'll definitely keep in mind that algebra is the thing I should focus on the most.
>>
>>14600360
yeah actually. with respect to textbooks, honestly, i didn't get much use out of them. that is, however, just the nature of how i learn things. if i was learning on my own, without a curriculum, i would certainly be more in need of a textbook. but mostly, notes i'd taken during lectures were more than sufficient, and if i was unsure i would just google the concept to learn more about it.

if you look around, reddit might be a good starting point, but there are ways to torrent or otherwise acquire practically any textbook which exists. an e-book is usually more useful than a textbook, considering you can actually search through it in an instant.

but, a really incredible resource is that many "big boy" universities actually put up their lectures online. harvard, MIT, shit like that. "edx.org" is one site that i've used. basically, all the lectures that a student would sit through, are there, and all the material the professor would provide to them, is there.
it's free. usually it's just the first year or two of courses, so like freshman biology, freshman algebra, freshman chemistry, etc. but, i mean it's the same shit that a freshman at harvard or whatever is going through. it's the actual lecture they are sitting in, filmed.
i think you can pay some money and actually get credit hours for it, but if you can just use it to learn the information for free, and not get any credit for it. super fucking useful and i've actually used it to learn about subjects i was interested in.
>>
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yo I'm bored and dropped out
now I'm a bored NEET
what should I read?
pic rel
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>>14600369
Wow thanks anon, hopefully my boomer tier website navigation skills won't inhibit my ability to use this resource. The lectures are within the free courses right? There isn't another part of the website I'm not seeing I hope?
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>>14600398
yeah. i've only used the site for literally one biology course, but my experience was that it had like a 3 hour lecture as a video, once a week, or something.
then there was access to a powerpoint or whatever that the professor used, plus quizzes and exams and such. yeah all that was included in the free one. i think the only difference with paying is that you actually get college credits which can transfer. pretty sure all the courses are free, and you can optionally pay for it to count as credit, but don't quote me on it.

example:
https://www.edx.org/course/college-algebra-and-problem-solving
which is from the university of arizona, it's their freshman algebra. you can pay 600\$ and get credit for freshman algebra. however, be careful with that because some schools only accept certain credits. that's a huge rabbit hole and you'd have to ask the school you're planning on attending. but suffice to say, that's a link for freshman algebra as taught at arizona state university, and it's the same shit that a freshman at that school would receive, and it's free.
>>
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>>14600410
You've helped me in away I cannot describe, if you'll forgive me for being overly dramatic. Thanks anon, you're a good guy.
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>>14600424
no worries mate, good luck
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>>14600266
https://oeis.org/A040016
>>
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>>14590075
This is probably a retarded pipe dream.....but fuck it, I'll ask anyway.

>Canadian
>shitty high school grades
>got into a college (not uni.); starts in the fall

Is there any possible way I can get into a Ivy League school, or am I wasting my time for trying? My plan is to do well in college for my first year, then apply to an IL school (after taking the SATs and such, of course).
If that doesn't work, then after my first year, I plan to apply/transfer to a Canadian uni, do well, complete a year there....and then apply for an IL school.

Is this feasible or am I wasting my time?
>>
Let $H \le G$ be finite groups and $(U, \tau)$ be a representation of $H$ over some field $F$. Define $\widehat{U} = \{f: G \to U : \forall h \in H, \forall x \in G, f(hx) = \tau(h) f(x)\}$ and $\widehat{\tau}(g) f(x) = f(xg)$. I want to show $F[G] \otimes_{F[H]} U \cong \widehat{U}$ as $F[G]$-modules. It seems natural to define a map by $T(\sum_{g \in G} \alpha_g g \otimes u) = (x \mapsto \alpha_x u)$. Here $\alpha_g \in F$ are scalars and the $g$s are formal basis elements.

Is my map truly well-defined, i.e. does $T(\sum_g \alpha_g g \otimes u)(hx) = \tau(h) T(\sum_g \alpha_g g \otimes u) (x)$? The LHS is just $\alpha_{hx} u$. But the RHS is $\tau(h) T(\sum_g \alpha_g g \otimes u) (x) = \alpha_x \tau(h) u = T(\sum_g \alpha_g g \otimes \tau(h)u)(x) = T(\sum_g \alpha_g gh \otimes u)(x) = \alpha_{xh^{-1}} u$. What am I doing wrong?
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>>14598025
:( any help?
>>
Are there any animals in nature that have a rock paper scissors food web?
As in
>A eats B
>B eats C
>C eats A
>>
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>>14598025
>>14600681
Let $f(z) = \frac{e^{iz}}{e^z + 1}$. Note that for real $x$, the imaginary part of $f(x)$ is $\frac{\sin x}{e^x + 1}$. So it's enough to compute $\int_0^\infty f(x) dx$.

Since $f$ has a singularity at $z = \pi i$, we can't actually integrate over the seemingly obvious choice of rectangle. One way to avoid hitting the singularity is depicted in the picture. (The semicircle has radius $\epsilon$, and we let $R \to \infty$ and $\epsilon \to 0$.) Another possible way would be to encircle the singularity, e.g. integrate over a rectangle of side lenghts $R + \epsilon$ and $2 \pi$, and then apply the residue theorem - but the first approach seems more promising IMO. I didn't actually go through the calculations myself but I hope this will help you start.
>>
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OA, OB are both straight lines, AB can be of any curve, and ∠AOB is not fixed. Given a constant area is enclosed by the shape, how can I find the min value of x?
>>
A famous Yogi called Sadhguru talked about time and space, and I just keep thinking about it for years now. Is it of any worth?

"In yogic perception, time is the basic aspect of creation. It is because of time that there is gravity and because of gravity, everything else. Gravity is a consequence of time. It is not the other way round. The physical exists only because time – which is nothingness – takes a curve."
>>
So my internal body temperature is 36 degrees Celsius. In order to maintain that temperature, my body needs to produce the heat, and in order to do that, it needs energy which comes from food and beverages.

But what if it's currently 36 degrees Celsius where I live? My body wouldn't have to produce any heat because it's already hot enough. So I wouldn't need food, right?
>>
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Is it possible for a planet shaped like a torus to exist?
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>>14601123
Your body uses energy for other purposes than thermoregulation, so yes, you'd still need to eat.
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>>14601123
You don't need food to maintain body heat, no. Which you can verify by noting that if you die, your body will stay at 36 degrees.

There are also a lot of other aspects about your body that need to be constantly maintained if you want to live, aside from the temperature. And those still need energy and therefore food to be maintained.
>>
>>14601131
Someone wrote a delightful analysis of this: http://www.aleph.se/andart/archives/2014/02/torusearth.html
>>
what are some good resources for getting into astronomy/astrophysics?
>>
I filled out a test in 2019 that provided percentages for 25 variables. I did it again just now but got wildly different scores.
How could I easily see the differences between those two variable sets rather than just calculating manually? I'm talking basic shit like which score changed the least and which one the most.
I have SPSS and I have already punched in the variables and the respective values from the two tests.
>>
Doing a biology assignment. I need to simulate the effects of dominant and recessive alleles on a phenotype.

Could somebody recommend a simulator software, free or not, that I can use to do this experiment with instead of having to get hundreds of different colored beads and carry them over between measuring cups.
>>
>>14598025
As an alternative to what that other anon said, you could integrate $\frac{e^{iz}}{1 + e^z}$ over an infinite rectangle in the complex plane whose left corners are $(\epsilon, 0)$ and $(\epsilon, 2 \pi)$ and whose sides are parallel to the x, y-axes and then let $\epsilon \rightarrow 0^+$. Doing that gives the value of your integral to be $\displaystyle \sum_{n \geq 1} \frac{4n -1}{(1 + 4n^2)(2 - 4n + 4n^2)}$ - which, wolfram alpha tells me, is equal to $\frac{1 - \pi \text{csch}(\pi)}{2}$ (WA also gives this last expression directly when your integral is given as the input btw).

>>14601106
It looks like he's just ripping off some popsci descriptions of relativity and trying to present it as his own insight by adding some pseudoscience to it. So yeah, it's worthless imo.
>>
>>14601106
>It is because of time that there is gravity
https://youtu.be/F5PfjsPdBzg?t=5m
>>
>>14600966
have you proved that x is a portion of a circle?
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>>14590075
Explain a hermitian transformation in kids language.
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>>14600966
If x is really a minimum, it's also a minimum under those restricted set of area preserving deformations that leave the straight sides fixed and only changes the curve AB.

So treat the straight line connecting AB as an interval on the x axis, and have the curve AB be represented by a function f(x). What function minimizes its arc length for a fixed area? This is an easier problem since both the area and the arc length are simple integrals in terms of f(x), and maybe you can just guess the answer right away.

Once you know the class of curves f(x) for a given straight line distance AB and area, you can consider varying the straight line sides.
>>
>>14601608
curve $\not =$ function
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>>14601123
Your thinking about this in a strange way. Think of the heat that your body produces as waste heat that comes from inefficient use of food energy to do the things your body really needs to do, like move and think and all sort of stuff on the cellular level.
>>
>>14601616
Split it up piecewise if you need to. I just gave you a clear framework to solve the problem, don't be such a smart ass
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>>14601624
1) i wasnt OP
2) its not a moot point. your post explicitly mentions that the area and arclength are "simple integrals" in terms of f(x). breaking it up piecewise doesnt help. if you have no clue what the curve looks like you are going to have a very difficult time integrating over it unless you make some pretty consequential assumptions, like if you assume that the curve can be parametrized, or that it can be expressed as a function in polar coordinates (this is probably the best bet for OPs problem).
>>
>>14600712
Every food chain is circular if you include decomposers. You would still have your normal progression from primary producers (plants) to primary consumers (herbivores) to secondary and tertiary consumers (meso- and top predators), but every step includes a loop back to the beginning where the transfer of nutrients to a state that can be used by plants is mediated by decomposers (like fungi, earthworms, bacteria, etc.).

You could also get a circular food chain if you include parasites. For example, many fish on tropical reefs have parasites (usually small crustaceans or worms) that eat some part of them. These parasites are then eaten by "cleaners" like some species of shrimp or small fish. Sometimes these cleaners are eaten by the larger fish that come to be cleaned.
In this case you would get:
Parasite A feeds on fish B
Fish B (sometimes) eats Cleaner C
Cleaner C eats Parasite A

This doesn't usually happen though, because the cleaners perform a vital service to the fish, and it would be a disadvantage to the fish if they constantly ate the cleaners.
>>
>>14601650
Consider the arc length integral. Deform f by a function delta f as usual in calculus of variation. This will give you a differential equation for f. The only thing that isn't 100% standard is the integral of delta f is constrained to be zero. It's a pretty general approach, you don't need to know the answer in advance.
>>
>>14601323
What numbers to put in the remaining boxes to make it seem like I've finished it properly?
I'm tired of counting beads.
>>
What is the subspace generated by
$W = { (1, 1, -1, 1), (2, 1, 0, 1), (1, -1, 1, 2), (2, 3, -2, 0) }$

I solved the system but it gave me $0 = w$ (of the vector $v = (x, y, z, w)$)

I checked it with an online tool and it really is a system with no solution. The fuck do I do then? I'm supposed to find the subspace from these vectors and then find a basis for that subspace.
>>
>>14590075
is physics taught terribly on purpose or is it because they want filter people due to elitism?
>>
>>14602026
>or is it because they want filter people due to elitism?
that would still be considered "on purpose".
physics is not taught terribly in general.
>>
>>14602069
So it's normal to have passed a physics class with an A/B but not have learned anything in the process?
>>
>>14600078
ty anon but how come its n/2 copies of n+1 being added?
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>>14600883
Yes this helped a lot thank you very much!
>>14601441
This also helps thank you!

My complex analysis prof requires all work to be shown so i cant just plug it into WA xd
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>>14602148
yea, thats normal. read a book if you want to learn more.
if you were required to actually learn electrodynamics in an e-mag class then basically no one would pass. and despite what you might think, plenty of very productive and employable people nonetheless struggle with mid level physics, so this system isnt as bad as it might feel to you.
>>14602153
imagine summing from 1 to 100. the first term is 1+100, the second term is 2+99, the third term is 3+98, and so on until the last term, 50+51. so theres 50 terms of 101, or in general, n/2 terms of n+1
>>
>>14602148
>>14602026
>is physics taught terribly on purpose or is it because they want filter people due to elitism?
No. Apparently the subject matter of physics 101 is too hard for normal people, so they aren't filtering you on purpose, the average person is just too dumb. You were dumb since you didn't learn anything, but not as dumb as your classmates, hence the A/B. I've graded for intro physics classes and know how terrible the grades are.
>>
>>14602069
>physics is not taught terribly in general
Sure, the goal is to spread faith, not to damage student's foundation. But the result is the same.
>>
>>14602270
what?
>>
>>14602158
ty anon much appreciated
>>
>>14600078
>>14602158
ty i understand better now
>>
>>14602026
In some cases I'd say they try to dumb it down so much that it stops making sense to people who would actually understand it. Like in highschool physics the equations for kinematics made no sense how they were taught, then the calculus teacher mentioned to us that you're just taking derivatives of position and suddenly the problems all make more sense
>>
>>14602026
the average human is too dumb to understand anything more complicated than circuit analysis, pulleys, levers, and extremely basic ideas from thermal physics. watching stem majors of all different backgrounds fail to be able to handle rotational dynamics, oscillations, waves, geometric optics explained in the clearest of terms convinced me of this.
>>
>>14602409
>the average human is too dumb to understand anything more complicated than...
You're giving the average human way too much credit. Like I said I actually graded all sorts of undergraduate physics classes at a halfway decent school. Non-calculus based classes for liberal arts majors, classes geared towards pre-meds, classes geared towards engineers. There wasn't much difference in the quality of students in any of them. There were always a handful of 'smart' students that would get the problems, and then the majority was varying degrees of terrible. If we gave an honest grade 75% of the class would fail.

The actual upper undergraduate classes for physics majors had a majority of terrible students too, but that's a different story.
>>
>>14602427
man why did your school have all that bullshit. mine was just
calc based physics - for physics, math, chemistry, engineering, and CS majors
algebra based physics - for biology, premed, and an elective for liberal arts majors.
>>
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If you were outside a space shuttle floating in space, say 10 feet away from the shuttle and the shuttle exploded powerfully, would the force of the explosion launch you away?
>>
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What the heck is happening here? there was some work truck with two dudes sitting around, just wearing comfortable labor clothes.
I figured it's related to infrastructure maintenance but still spooked the shit out of me
>>
>>14602403
>>14602225

I never took physics in highschool and jumped straight into calc based physics. I think I didnt really learn anything because we were going through chapters like crazy. I understand the math though, which is the only reason i survived
>>
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>>14602026
I don't remember a single thing about magnetism because Fuhrer Fauci moved the second half of that course to Zoom
>>
I don't mean to be prejudicial, but why do Australian aboriginals look so weird?
>>
How do I get gud at writing algebraic equations based on physical scenarios?
>>
any good books on differential geometry with the intention of getting into GR?
>>
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>>14590262
Anyone?
>>
For anyone here who knows anything about seismology
http://ds.iris.edu/ds/nodes/dmc/tools/event/10725056
Does anything about this quake strike you as odd, or does it seem uneventful?
>>
>>14590075
Can someone explain what this an x^n means. I get a1,a2..,an are real no. Then what is an-1 x^n-q . This is confusing
>>
>>14603071
an-1 x^n-1 not an-1 x^n-q
>>
>>14603071
could be 1, could be 2, could be 100,000,000, doesn't matter. is just a way of indexing the coefficients/exponents.
>>
>>14603071
For example n=3
a_3*x^3+a_2*x^2+x_1*x^1+x_0*x^0
where x is your variable
>>
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Guys, can any of you explain to me why R1, R3 and R4 are considered transitive relations? The question is just asking to pick the transitive relations, my answer was R2 and R5, but the answer key says: R1, R3, R4 and R5, I don't understand why he left out R2 but included the others.
>>
>>14603341
R2 contains tuples (2, 1) and (1, 2) but doesn't contain a tuple (2, 2) therefore it's not transitive.
>>
>>14603441
So every element has to be related by the transitive propery? I thought 1R2, 2R3, 1R3 would suffice for it to be called transitive.
>>
>>14601596
If you flip flop the coordinates and conjugate the entries you get the same matrix back.
>>14602013
You did what? I don't actually follow.
If you solved for $Wv = 0$ and found that the only solution is $v = 0$ then the subspace is actually the entire space and $W$ is a basis.
>>14602920
Most GR books explain their differential geometry, and if you want to know DG beyond that you might as well use books for mathematicians.
>>
have a hard copy of introduction to matricies and linear transformations second edition by daniel t. finkbeiner, is a bit old but you think it's still solid intro to lin alg? or should i just get something more modern.
>>
>>14603696
Should be good, problem with old books is that they usually are intended to be used as class material and not self teaching material, but you can probably overcome that with youtube and other resources
>>
>>14603709
cheers, will keep that in mind
>>
Anyone know a good or industry standard textbook for small business finance? It uses numbers so it’s technically science.

Have corm for exchange
>>
>>14590075
>eientei
I've just imported the warosu dump I found in archive.org, allowing for unprecedented e-stalking.
https://eientei.xyz/sci/search?resto=&com=&sub=&name=&trip=&filename=yukari&md5=&capcode=&image=&deleted=&op=
https://warosu.org/sci/?task=search2&ghost=&search_text=&search_subject=&search_username=&search_tripcode=&search_email=&search_filename=yukari&search_datefrom=&search_dateto=&search_op=all&search_del=dontcare&search_int=dontcare&search_ord=new&search_capcode=all&search_res=post
>inb4 why are the results so different
I don't know.
>>
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how are they getting 81/8? i keep getting 81/16
>>
>>14599972
>derive
S = 1+2+3+...+n
S = n+(n-1)+(n-2)+...+1
S+S = (1+n)+(2+n-1)+(3+n-2)+...+(n+1)
2S = n(n+1)
S = n(n+1)/2
>>
>>14603841
it just resolved the constant multiplications.
$2(-2-(\frac{7}{4})^2)=2(-\frac{32}{16}-\frac{49}{16})=2(-\frac{81}{16})=-\frac{81}{8}$
>>
>>14603892
ahh i didnt realize the whole thing is still multiplied by 2
>>
What hobby can help improve my abstract thinking skills and work out my brain? I already do math and did bit of programming(pong, music recommendation system, drew a flag with code) . I also can’t come up with fun software ideas to build. I was thinking about maybe doing cryptography, not interested in physics btw
>>
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still have no idea what ordinal ""numbers"" are and what to do with them. is the only part of elementary set theory my peanut brain cannot comprehend.
>>
>>14603908
https://youtu.be/SrU9YDoXE88?t=12m
>>
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>>14590075
I've got this physics question and I wasn't able to solve it, I tried solving it using simultaneous equations but it didn't work, is the question incomplete?
>>
Is there an urban dictionary alternative that doesn't have random faggots posting retarded definitions?
>>
$12sin^2θ-9=0$ Solve the equation on the interval 0≤θ≤2π
$12sin^2θ=9$
$sin^2θ=9/12$
$sinθ= √3/2$
Assuming I haven't fucked up so far, the solutions would seem to be $π/3,2π/3$ however when I check these answers on a calculator neither of them seem to work. Is there no solution or did I fuck up?
>>
>>14604292
Is your calculator set on degrees by any chance? Because the solutions are good.
Also don't forget that $\sin{(\theta)}^2 = 9/12$ also implies $\sin{(\theta)} = -\sqrt{3}/2$, so you have two more solutions to find.
>>
>>14604326
>Is your calculator set on degrees by any chance?
No, but it seems like the calculator was messing up the order of operations. If I do it manually I can see that they're good. Guess I put too much trust in the desmos calculator although it should have been obvious to me it was messing up. Other solutions are $4π/3$ and $5π/3$, thanks for the help anon.
>>
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If the resistor causes a voltage drop, why does the cap still charge to 9v?
>>
>>14604692
V=IR remember. The resistor only causes a voltage drop if there is current running through it. When the cap is fully charged there is no current running through the circuit
>>
>>14603970
It's complete. i+r = 90 degrees so sin r = sin 90- i = cos i. Try using that in Snell's law
>>
>>14604848
But if I add an LED after the resistor then the capacitor will only charge to 7 volts.

Why is that different?
>>
>>14604863
Probably there is a 2V drop across the LED. If you have a multimeter you can test it.
>>
>>14604872
But what does it mean for there to be a drop across it? Isn't there also one across the resistor?
>>
>>14604877
Voltage drop across the resistor -> There is current flowing in the circuit -> voltage on the capacitor is increasing (unless the resistor is broken in which case it is effectively a bad capicitor)

If you're doing a lab just play around with the multimeter. In the real world circuit elements break, and LEDs aren't like ideal resistors so it's hard for me to give a clear answer. The theory is that the sum of voltage drops around the circuit must add up to the voltage of the battery. That part is always true.
>>
>>14604892
What confuses me is why the placing an LED in the circuit only charges the cap to 7v, but placing a resistor (of any value) there instead charges the cap to the same voltage as the battery.

If both causes a voltage drop, why is the max charge level of the capacitor only affected by 1 of them?
>>
>>14604877
LEDs (and diodes in general) are nonlinear devices, ohms law doesnt apply to them
>>
I was watching a proof of the divergence theorem and in one part it said that $dydz=ds_x$, that is, dy*dz is the x component of the surface.
why? I can't wrap my head around it
>>
>>14604894
My guess is that current only flows across the LED if the voltage across the LED exceeds a certain value and once it gets down to 2V the current shuts off. LED's don't obey Ohm's law and I'm just a physicist not an electrical engineer so this is my best guess. Resistors obey Ohm's law so as long as there is a voltage drop across it current will keep flowing and the capacitor will charge all the way.
>>
>>14604900
The surface is specified by the direction it is normal to, so s_x means the surface perpendicular to the x-axis, i.e. the yz plane.
>>
>>14604909
it makes sense now. ty anon
>>
>>14604894
>If both causes a voltage drop, why is the max charge level of the capacitor only affected by 1 of them?
the current through an ideal resistor is proportional to the voltage across it. this proportionality factor is called the resistance of the resistor. this implies that if the voltage across the resistor is nonzero, then the current is also nonzero. in your circuit, this current charges the capacitor which decreases the voltage across the resistor. this feedback creates a steady-state where the capacitor is fully charged to 9 volts and current no longer flows through the resistor. the amount of time it takes for this to happen depends on the capacitance of the capacitor.
for diodes (and by extention LEDs), the current through them is not proportional to the voltage. in general, the current through a diode looks something like $I(V)=V_0 e^{V-V_f}$, where Vf is the "forward voltage", which is often approximated as the voltage required to turn the device on. once the device is on, it can (again, approximately) deliver any amount of current. so in the case of the capacitor, the diode will at first act like a closed circuit, but once the capacitor charges high enough, the diode will "open" and will no longer carry current, thereby capping the voltage of the capacitor at 7 volts.
>>
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I'm taking the general GRE in about 1 and a half months and I haven't started any preparation at all. I'm gonna start looking at practice tests and questions this week.
I'm already admitted to grad school but I have to take it to apply for a certain position.
What should I know? Am I fucked? This all seems like a pain in the ass
>>
Let's say person A starts moving towards a point 10 light years away, and person B stays here on earth with us. A accelerates to some velocity really close to c, travels some distance at that velocity, and when close to the endpoint decelerates and comes to a full stop.
From POV of B, the trip can NEVER be shorter than 10 years, right? But from point of view of A the trip should be shorter than whatever B saw?
What's more, from A's POV the trip can be arbitrarily short depending on how close to c his velocity was?
>>
Henlo frens, I'm going through Stewart's precalc to prepare for Calculus. Why does he overcomplicate things?

Ex: to factor ax^2 + bx + c, you must find 2 factors that add up to b and whose product is ac.
Pic rel is his explanation.
>>
>>14603073
>>14603075
>>14603077
Thank you, anons
>>
>>14605020
it is an intelligence test and a measure of how much technical writing and mathematics you had to produce/learn during your degree. If you do badly it's almost always because you're a stupid person.
>>
>>14605097
>Ex: to factor ax^2 + bx + c, you must find 2 factors that add up to b and whose product is ac.
thats uhhhhh not right
>>
>>14605125
I mean I did engineering and I was good at that. I didnt do much writing and I dont play the game of semantics either
I guess its over huh
>>
>>14605128
if you have 6x^2 + 7x - 5, find two factors that add up to 7 and multiply into -30. Split x using these numbers, then group.
Ah I get it now.
>>
>>14605143
No. Engis do ok on the verbal portion, and they also do ok on the AW section. Iirc Chem Eng and EE’s score comparably to CS and slightly below Marerials Science guys on Quant section. I think if you were a good student, and you’re somewhat articulate and logical in the way you present your ideas to others you’ll score fine. I was mostly fucking with you.
>>
>>14603450
R is transitive if and only if
aRb ∧ bRc => aRc
IOW
(a,b)∈R ∧ (b,c)∈R => (a,c)∈R
>>
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>>14590075
Best way to study for the NCEES FE Exam for Electrical/Computer Engineering? I've got the official study practice exam pdf but I don't know if I should buy some third-party study materials. Ideally I'd like to pass the exam the first time. Just graduate March if that matters.
>>
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I understand getting amp and period, but I have no idea where they get those numbers for ii and iii.
>>
>>14605097
Why do they still teach factoring by trial and error? Such a trashy and unelegant method.
>>
>>14605714
For ii, you can choose any point where x is not zero to substitute into p and y, and there's a point at $(\frac{\pi}{2},-\frac{1}{2})$ conveniently. For iii, you can also choose any points so you can solve iii with the point at $\frac{\pi}{2}$.
>>
>>14590075
it absolutely baffles me how roofs stay up with so much stuff placed on top of them in a multi-story building.
>>
>>14605939
you mean ceilings
>>
>>14605946
yeah
>>
>>14605939
IIRC modern building materials are just really good, and the architectural techniques are really nice.
If you look back to something like romanesque architecture the walls needed to be absurdly thick with very small windows so they wouldn't collapse.
>>
>>14605897
Thanks. So p is y intercept, put that into y = p cos q x; then sub in a given point (𝜋 / 2 , -0.5), then solve for q.

I don't understand why q * 𝜋 / 2 = 2 though. Shouldn't it equal q𝜋 / 2?
>>
>>14602514
Civil engineering student here, this is my guess

https://en.wikipedia.org/wiki/Nuclear_densitometry

Next time you see a work ute driving around with a geotechnical / civil company's logo on it, look for a radiation warning sign on the ute somewhere. It'll be carrying one of these.

The radiation is minimal. 3.6 roentgen, not great, not terrible.
>>
How does an AI like LaMDA become sentient? isnt the whole AI thing just a regression?
>>
>>14605964
he just skipped the steps solving $-\frac{1}{2}=\frac{1}{2}cos(q*\frac{\pi}{2})$.
[eqn]-\frac{1}{2}=\frac{1}{2}cos(q*\frac{\pi}{2})\\{-1}=cos(q*\frac{\pi}{2})\\\pi=q*\frac{\pi}{2}\\q=2[/eqn]
>>
>>14606012
[eqn]-\frac{1}{2}=\frac{1}{2}cos(q*\frac{\pi}{2})\\{-1}=cos(q*\frac{\pi}{2})\\\pi=q*\frac{\pi}{2}\\q=2[/eqn]
>>
>>14606012
>>14606014
Youve got this bro
>>
>>14606024
I don't know why it doesn't work, also I can't delete my own posts for some reason.
>>
>>14590075
Let $A_n = (a_{ij}) \in \mathbb{R}^{n \times n}$ with [eqn]a_{ij} = \left\{
\begin{array}{ll}
2 & i=j \\
-1 & |i-j| = 1 \\
1 & |i-j| = 2 \\
0 & \textrm{else} \\
\end{array}
\right. [/eqn]

Which matrices $A_n$ are positive definite? It really looks like a case for the sylvester criterion but there is no nice induction pattern for the determinants of the matrices in question.
>>
testingtestingtestingtestingtestingtestingtestingtestingtestingtesting
>>
>>14606024
>>14606100
I'VE JUST FUCKING FIGURED OUT THE LATEX BUG
4CHAN PEPPERS YOUR POST RANDOMLY WITH <wbr> TAGS WHEN IT GOES ON FOR TOO LONG WITHOUT A SPACE.
SEE >>14606245
>>
>>14606252
Do I fix this in eientei? Is it objectively speaking a good idea?
>>
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Ok this will sound stupid but I've passed trig and calculus with just memory and basic understanding of what I was doing, either way.
Tan(x) = ratio of opposite side and adjacent side.
That ratio somehow gives me the mirrored hypotenuse of sorts in relation to the mirrored degree angle. Why? Pic related.
I imagined it has something to with the mirrored angle (lets call it y) so that x + 90 degrees + y is 180 degrees.
tan(45 degrees) is 1 because 45 + 45 + 90 = 180.
But I'm still kind of confused about tan. As to why o/a gives me the tangent while sin and cos are basically just moving along y and x.
>>
>>14606252
detective wemi does it again
>>
>>14605196
Im still scared Im gonna blow it. I suck at doing things on the spot, even if its something Im knowledge about
>>
How do i fix my life when i have zero friends, my family hates me, i have no skills and can't hold a job?

I'm so tired of being broke and hopeless
>>
>>14606661
Solve the Riemann hypothesis
>>
>>14606669
i'm too stupid to do math, desu

it just gets jumbled in my brain and i can never understand what to do when it's not what i've done before
>>
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>>14590075
on topic of vitamin deficiencies
>doc says you lack x
>take x
>metabolizing x can cause deficiency in y
>take x and y
>y uses z

So does a deficiency cause a break in the chain because it gets metabolized first or how can you ignore the occult vitamin relationship chart ?
>>
i found a really old ziplock bag full of magnesium shavings in my grandpa's attic. it's very oxidized and almost looks like charcoal. but, it does still burn. i took some and lit it on fire with a lighter and it does what you'd expect.

will the oxide layer interfere with any sort of chemical reactions? also, does the oxide protect the deeper layers of metal from getting oxidized? should i store it in some better way, like under mineral oil or something?

i want to use it for thermite.
>>
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>>14606605
Just calculate it. Let's call the blue line c. By construction there's an angle of 90 degrees between the radius of the circle and c , so since the sum of angles of a triangle is 180° it follows that the angle between the red and blue line is theta. Let's also call the red line b. Then we have $\cos \theta = \frac{b}{c} = \frac{\sin \theta}{c}$ so by rearranging $c = \frac{\sin \theta}{\cos \theta} = \tan \theta$
>>
>>14606888
Thermite doesn't use magnesium. It's iron oxide and aluminium. The whole point of thermite is that the end product is molten iron (it's used for welding).
>>
>>14606964
You can (and usually will) use magnesium to ignite it though
>>
Pls anons how do i find partial sums? I'm going to cry.

"Sum both sides of the identity k^2 - (k - 1)^2 = 2k -1."

I dont understand why the answer is n^2.

It says i will find a formula for
SUM k=1 to k=n (2k-1)
which is the sum of the first odd natural numbers.

What is the point in doing the sum of both sides? I just end up with a long string of additions that i cant do anything with.

This book must not have been meant for brainlets because it doesnt explain how to go from the big string of sums to a closed formula.
>>
>>14607049
Can you latex this? I have a hard time understanding the question
>>
>>14607049
If you sum the right side you just get $\sum_{k=1}^n(2k-1)$.
To see what the second sum yields write out the first couple terms
$\sum_{k=1}^n(k^2 - (k - 1)^2) = 1^2 - 0^2 + 2^2 - 1^2 + 3^2 - 2^2 + ...$
Notice how the negative part of one term cancels the positive part of the previous one. This is called a telescope sum. In the end you're left with $\sum_{k=1}^n(k^2 - (k - 1)^2) = n^2 - 0^2 = n^2$
>>
Why have universities moved from using *BSD type systems and started to use windows?
>>
>>14606964
there are many types of thermites.
iron oxide, manganese oxide, chromium oxide, copper oxide, etc.
the metals can be zinc, magnesium, aluminum, titanium, etc.
it's just a very exothermic redox reaction.
>>
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>>14607072
sorry anon i will try to learn how to do that for next time

>>14607075
thank you anon i appreciate it so much. is there a good resource or book where i can learn how to do these properly?

Anything relating to sequences or sums seems to destroy me. I've been trying to get through recurrence relations too and i just cant because at some point i end up getting to a sum or this telescope thing and then i cant go any further

the book introduced iteration for solving recurrence relations but the answers always end in some sum and then skip from the sum being written out like you did with the "1^2 - 0^2 + + + ..." and it just turns into a closed formula thing like " = n(n+1)/2" on the next line without saying how it got there.

Ty again anon i appreciate your answer very much
>>
I know that the definition of a surface integral is
[eqn]\iint_S f(x,y,z)ds=\iint_D f(r(u,v))||r_u\times r_v||dA[/eqn]
but I can't understanda it, more specifically, I don't understand the right side of the equation. can someone here explain it?
>>
If you know Specific Gravity and Specific Heat is there anyway to calculate Lantent Heat of Vaporization/Fusion? Or if you know the latent heat of either vaporation/fusion can you calculate the other?

I'm assuming there is no way. It's just a bit odd that it has to be measured.

More specifically I have a proprietary substance. The manufacturer at one time gave me the specific gravity and heat however I was foolish to think they would be as helpful in the future. They refused to supply me with latent heat information at a later date.
>>
>>14607570
>$ds$
>$\| r_u \times r_v \|$
That equation is goofy, what source are you using?
>>
>>14607624
paul's online notes
>>
>>14607629
Lmao.
When you learn regular definite integrals you're given the definition where we partition the interval and compute by multiplying base times height and adding up, right?
That doesn't work quite so nicely for surface areas because of something called the cylinder area paradox. There are multiple reasonable ways of triangulating a cylinder that gets us multiple different area values.
So what we do is the following: if we parametrize the surface with $r: \mathbb{R}^2 \to \mathbb{R}^3$ then the infinitesimal "side" $du$ is mapped to $dr_u$ and the infinitesimal side $dv$ to $dr_v$, so the infinitesimal area $du \ dv$ goes to $dr_u \times dr_v = \| r_u \times r_v\| du \ dv$, which is why when we pull back the integration from the surface to the plane using $r$ the factor pops in.
>>
>>14607629
>>14607657
Basically it's all just the chain rule.
>>
>>14590075
I´m learning calc and i don´t understand what is the dx of the integral. Obviously I integrate respect x (in the case of dx) but, like the derivative, if are treated like diferentials, dy = f´(x) dx, and if we integrate both sides
[eqn]\int dy = \int f´(x)dx [/eqn]
then we got
[eqn] y = \int f´(x)dx [/eqn]
I know it isn´t correct in theory, but i saw this in Leithold Calculus (an old book) as an analogy of what the antiderivative is. My question is how to interpret the dx of the integral
>>
should i get an MBA?
engineer with 10 years work experience
>>
>>
>>14607769
i think it's n^2 btw
>>
>>14607769
$O(n)$
for large $n$, the function approximately returns $n$ (by incrementing count $n$ times).
>>
>>
>>14607780

>>14607777
checked and thx
>>
>>14607780
is this also a complexity problem?
>>14607782
np
>>
>>14607783
yeah it's related to the post i made just after it.
>>
>>14607786
oh i see im retarded
>>
also these are my other answers, would appreciate any double checking available. gotta hit the books tomorrow and get caught up.
>>
>>14607780
>>14607782
i think its the fourth answer. after checking a pair of letters, the amount of letters left to check is reduced by two. and its not $T(n-2)+n$ because the complexity of a single function call is independent of $n$ (as long as the strings dont have read complexity).
>>14607788
1 and 2 look right. i dont know what omega means.
did you know that "big O" notation is more correctly called "big omicron" notation? which is of course an oxymoron, because omicron literally means "little O".
>>
>>14607795
that's what i was thnking too, but i wasn't sure. i was aslo confused why all of them set T(1) = T(0) = 0. not that implies in this context

and no, i didn't know that about big O actually meaning little o, funny.
>>
>>14607797
>i was aslo confused why all of them set T(1) = T(0) = 0
a string of length 1 is always a palindrome, and i suppose you could say the same about a string of length 0.
>>
>>14607801
ah ok, well have a nice night anon. here's a cute cheeseburger for your time
>>
>>14607769
isnt this O(logn*logn)?

the count is halved each time so there are x=2^k elements, so k = log(x) iterations?

and then for each iteration we count from 0 to log(x)

So that makes it logx*logx?
>>
>>14607805
no, i just submitted and he was right. i thought that too initially though
>>
well he was right insofar as it matched the expected answer. i guess the answer key could be wrong.
>>
Roughly how far would an egg need to roll across flat, hard, ground before the shell gave way and fluids were released?
>>
>>14607780
the worst case should be if each letter matches it has to do the most comparisons, so i guess n/2 comparisons in total, then +1 comparison to exit?

is this n/2 + 1? i dont know what that notation means though, im new to this
>>
>>14607804
not enough lettuce for my taste, but ty
>>14607805
>and then for each iteration we count from 0 to log(x)
no, you count from 0 to $\frac{x}{2^i}$ during each iteration, which for large x roughly equals x.
[eqn]
2^0 + 2^1 + 2^2 + \cdots + 2^{n-1} = 2^n - 1
[/eqn]
>>14607814
how the fuck would anyone answer that?
>>14607818
that question is weird because its asking to represent the complexity as a recurrence relationship. so if you call the function with a string of length n, it has to do its first two checks (which i guess counts as 1 "thing"), and then it calls itself after truncating the first and last letters, hence $T(n)=T(n-2)+1$
>>
>>14607807
oh okay, do people figure these out by writing out numbers to test? i dont understand how to visualise these.
>>
>>14607828
>do people figure these out by writing out numbers to test?
thats a good strategy, yeah
>>
>>14607827
ty anon, i dont know if ill ever get these right, i struggle with series and sums and recurrence relations and cant tell between x/2^i vs 2^i

i feel like im lacking so many foundational things
>>
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>>14607836
if it makes you feel better, i thought it was something like $O(\log_2(n^2))$ at first too.
it helps to try to read each line in english
>i = floor(i/2)
"cut i in half"
>for j = 0 to i - 1
"count to the new value of i"
>while i > 1
"repeat until we run out of i"
its basically saying that you cut off part of i, and then you count that part, and then you just keep going until i is gone. its a really roundabout way of counting to i.
>>
What happened here?
>>
>>14607855
ahhhhhh i hit the back button on my mouse will i was typing ahhhhhh
[eqn]
\frac{\sqrt{3} - 1}{\sqrt{3}+1} \\
\frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3}+1)(\sqrt{3} - 1)} \\
\frac{\sqrt{3}^2 - 2\sqrt{3} + 1}{\sqrt{3}^2-1^2} \\
\frac{4 - 2\sqrt{3}}{2} \\
2 - \sqrt{3}
[/eqn]
>>
>>14607869
>ahhhhhh i hit the back button on my mouse will i was typing ahhhhhh
Thanks for the effort anon. The only problem I have is I don't know why this works or how you would know that multiplying by the numerator would simply this specific fraction.
>>
>>14607899
This is a common trick when you have some kind of root expression in the denominator. If you have some root expression $\sqrt{a} + \sqrt{b}$ in the denominator multiply by the root conjugate $\sqrt{a} - \sqrt{b}$ because by the binomial theorem $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = \sqrt{a}^2 - \sqrt{b}^2 = a - b$. Now you have gotten rid of any square roots in the denominator.
>>
>>14607899
>I don't know why this works or how you would know that multiplying by the numerator would simply this specific fraction.
no no no, you multiply by the conjugate of the denominator. this results in something called a "difference of squares"
[eqn]
(a+b)(a-b)=a^2-b^2
[/eqn]
this knocks out any square roots from the term, at the expense of putting them in the numerator. you just change one of the signs of the denominator (either one works) and multiply both the top and bottom by it. in your problem it just so happened that the numerator was identical to the factor, but it doesnt need to be.
>>
>>14607915
>by the binomial theorem
not so sure about that
>>
>>14606661
I don't know, anon, but I believe in you
https://www.youtube.com/watch?v=siHp8U_26ys
>>
>>14607918
Is that not a thing in the rest of the world? In Germany we call that the third binomial theorem.
>>
>>14607921
ive only ever seen the term "binomial theorem" refer to expressions that look like $(a+b)^n$
>>
>>14607923
Now that I think about it, I've mistranslated. What I've been referring to is called the third binomial formula here, not theorem. "Binomischer Lehrsatz" is what the actual binomial theorem is called.
>>
>>14607921
W-what’s the second binomial theorem germany?
Have you been hiding it from us
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>>14607915
>>14607916
Thanks for the help you guys.
>just so happened that the numerator was identical to the factor
This is what made it confusing to me but I understand now.
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>>14607926
Yes that's why Nasa hired all of our nazi scientists. Jokes aside, there are three binomial formulas taught in school here:

$1. (a+b)^2 = a^2 + 2ab + b^2$
$2. (a-b)^2 = a^2 - 2ab + b^2$
$3. (a+b)(a-b) = a^2 - b^2$
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>>14607929
if you add all three together you get
[eqn]
(a+b)^2 + (a+b)(a-b) + (a-b)^2 = 3a^2 + b^2
[/eqn]

in burgerstan, we call 3 "difference of squares", and 1 and 2 dont really have names and arent specifically taught beyond teaching how to distribute.
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remi if youre here post the next OP pic
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>>14607937
We even have a song about them
https://www.youtube.com/watch?v=EYbvhWEG6kE
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>>14607657
I still don't get it
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File: helb.png (71 KB, 1755x752)
71 KB PNG
I'm sorry if this post requires a high effort answer or a lot of trouble, I'm hoping someone can help me.

I have the problem where I am missing something important but i'm not sure what it is. My textbooks and even online the solutions just seem to assume you understand how to get from A to B.

I am trying to understand sequences and sums so that I can properly learn how to identify them in algorithms, and determine big O.

Does anyone have a suggestion for learning how to do:
1. Take information that isnt of the form "1 + 2 + 3 + ... + n" and turn it into this. I was thinking recurrence relations, actual algorithms, those Sum formulas with the big E.
2. Turn that into a formula only involving n, like "n(n+1)/2".

So then i can use that formula to determine big O.

But my main problem is i am failing big time on step 1, and step 2.

On step 2 i am usually missing the /2, or my n is wrong, like i say its n, but its actually n-1, or i am just completely wrong, and that may also be because i cant do step 1 properly or at all.

Can anyone help? My book just skips over these parts because they assume i know how to do it. Pic related
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>>14609155
If you have an arithmetic progression (equally-spaced values) from a to b, the mean of the first and last values is (a+b)/2. The mean of the second and second-to-last values is ((a+1)+(b-1))/2=(a+b)/2. The mean of the third and third-to-last values is ((a+2)+(b-2))/2=(a+b)/2. And so on. It shouldn't be too hard to see that the mean of the sequence is the mean of the first and last values. And the sum of a sequence is the mean multiplied by the number of values, i.e. n(a+b)/2.

More generally, if a sequence of values is described by a degree-k polynomial f(n), then the sum of the first n terms is given by a degree-(k+1) polynomial. You can find the coefficients by defining S(n) as a generic degree-(k+1) polynomial (with symbols for the coefficients), expanding out S(n)=S(n-1)+f(n), equating same-degree terms, adding S(0)=0, and solving the resulting system of linear equations.

Also: the step marked by the second red arrow in your pic is partial fraction decomposition: a/b+c/d = ad/bd + bc/bd = (ad+bc)/bd, but in reverse. So:
a/k + b/(k+1) = (a(k+1)+bk) / k(k+1) = ((a+b)k+a) / k(k+1) => a+b=0, a=1 => b=-1 => 1/k(k+1) = 1/k - 1/(k+1). A sum of terms of the form f(k+1)-f(k) is called a telescoping series, and has the property that all terms cancel except the first and last terms: (f(2)-(f(1)) + (f(3)-f(2)) + ... + (f(n)-f(n-1)) = f(n)-f(1).
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>>14609726
bless you anon i have read this and will try to work out some problems using this

i think its all starting to make more sense
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>>14607769
>>14607780
>>14607782
>>14607783
>>14607786
>>14607795
>>14607805
>>14607818
>>14607827
You're doing a retarded undergrad's homework. stop being faggots and helping morons pass their classes.
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>>14609910
theres a new thread bro

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