/sci/ - Science & Math

mechanics: find v avg of a biker hitting 50km/h in the first part of the travel and 75km/h in the second

why the book impplies that the answer is 60km/h if the results show 75km/h?

Δt = s/v avg > s(first part)/50
s(second part)/75

travel's total:
s(total, or 2, the travel's two parts)/3750 = 75km/h

what happened here?

>>14515175
If you want someone to do your homework for you go to >>>/wsr/

>>14515185
theres no one there plus the thread says also homework

>>14515192
>the thread says also homework
OP is retarded and can't read.

>>5942502
>Reminder: /sci/ is for discussing topics pertaining to science and mathematics, not for helping you with your homework or helping you figure out your career path.
>If you want advice regarding college/university or your career path, go to /adv/ - Advice.
>If you want help with your homework, go to /wsr/ - Worksafe Requests.

>>14515175
We call the total distance [math]2s[/math], in kilometers;
The first half was done in [math]s / 50[/math] hours. The second half was done in [math]s / 75[/math] hours.
The total time is [math]s/50 + s/75 = s/30[/math].
Velocity is total distance over total time, [math]2s / (s/30) = 60[/math]
>>14515197
>NOO NOT THE RULERINOS IN THE STICKERINO

>>14515197
last thread has been used mostly for homework

>>14515204
thanks bud

>>14515204
>>14515208
>>14515212
Anon just got an A on his engineering homework and he is clearly retarded. When the Boeing 797 crashes I know who to blame.

>>14515149
I've heard frequently about the vibrating nature of atoms, and the promise of nuclear reactions in the alchemy of new elements. Sometimes, i hear prime numbers spoken of as the atoms of number. If such an analogy was to be extended, would there be a way to 'harmonically simulate' as if by nuclear decay a prime with a combination of other primes? could perhaps a collection of musical ratios vibrate together to form a ratio not of the prime factors one began with

>>14515215
"I will crash every plane on the sky for a single (You)." - Anonymous

Unanswered questions:

Maths questions:
>>14497694 [/mg/]
>>14504953
>>14505901 [An idea has been posted, more would probably be good.]
>>14508023 [Follow up in >>14508034]
>>14508690
>>14509850
>>14509901
>>14510522
>>14512822 [It is.]


Astrobiology questions:
>>14490272

Physics questions:
>>14496597
>>14500263 [Ass question.]
>>14505682
>>14506284 [His dimensions are totally fucked but there might actually be something broadly of the sort, so I'm not putting it in stupid.]

/g/ questions:
>>14492773
>>14496443
>>14505825
>>14508688
>>14512188

Chemistry questions:
>>14505678
>>14508361

Engineering questions:
>>14497613
>>14500919

Biology questions:
>>14513242

Stupid questions:
>>14492818
>>14494309
>>14494972
>>14495861
>>14497183
>>14496210 [Follow up in >>14497230]
>>14499823
>>14502494
>>14502502
>>14506052
>>14506152
>>14508700
>>14510176
>>14513579
>>14513881

>>14515307
>I will crash every plane on the sky for a single
There's enough retards that can't do their homework here to kill half the planet.

whats the difference between /math embed and /eqn embed

>>14504953
The notation P(A|B) means the probability of A happening if we know that B happened; in other words, A conditioned on B.
Assuming B has nonzero probability, we define this as [math]P(A|B) = \frac{P(A\text{ and } B)}{P(B)}[/math]. For example, P(Y = 0 | Y ≥ 1) = 0 in your problem.

>>14508023
Yes. Generally, if B and C are two bases of the same space, there is always a linear transformation taking one to the other -- the very definition of B being a basis means that C can be written as a linear transformation of B's vectors, and this transformation X must be invertible for C to be a basis.
Then, you can "lift" this transformation up to the full M-space by setting the orthogonal complement of ker(M) to not be changed by X, which makes sure it's still invertible.

>>14500919
If the current source is DC, then a capacitor acts like a break in the circuit -- no DC current can travel through an ideal capacitor. A capacitor only conducts when the voltage varies.

If [math]\displaystyle \sum_{k=1}^\infty |a_k|[/math] converges, does [math]\displaystyle \sum_{k=1}^\infty a_k[/math] converge as well? Why? If not what are the conditions to make it converge?

>>14515454
yes always, it is trivial to prove with the triangle inequality

>>14515519
Under what condition does triangle inequality work for infinite sum?

>>14515526
There is no condition, it's just always true.
You can easily extend the triangle inequality to a countably infinite sum through induction.
If you're concerned about having an uncountable infinite set, in order for it to have a convergent series it can only have countably many non-zero terms so we can again use the triangle inequality with induction.

im taking calculus 3 as a spring course. We literally started 3 weeks ago and I have a midterm worth 35% of my grade tomorrow. The TA starts every lab session with "I gotta start writing now or we won't get through all the examples". -> proceeds to write like a crack addict for 1 hour mumbling under his breath.

spring courses are a meme.

>>14515307
I answered the gamma matrix question

>>14506284
>>14515718

Hello good peoples of sci
I was thinking about CERN conspiracy theories.
People think that 1 or 2 particles hitting each other are somehow going to open a portal to the hell vampire dimension?

but that made me think. Aren't there billions of particles constantly smashing each other near the speed of light around earth? CERN is just doing it on a much smaller scale in a controlled environment.

Really stupid fucking question, I'm just asking for clarification here
So the magnetic quantum number is equal to the range of l, i.e. (-1, 0, +1). E.g. if l = 2, then ml = -(2, -1, 0, +1, +2)
But then my textbook throws this curveball at me,
>The magnetic quantum number is also equal to (2l + 1). So if l = 1, then ml = 3, i.e. (-1, 0, +1)
But how does that even make sense? What does 3 tell me here? How is it equal to the range (-1, 0, +1). It doesn't elaborate any further and I'm left confused

>>14515149
Is the stress tensor covariant or contravariant?

>>14515779
>but that made me think. Aren't there billions of particles constantly smashing each other near the speed of light around earth? CERN is just doing it on a much smaller scale in a controlled environment.
thats right
portal to the hell vampire dimension open all the time
now ask yourself, 'why aren't vampires from the hell vampire dimension bleeding me dry?'
the answer may surprise you :)

>>14515149
Why not:
1/0=∞
2/0=2∞

>inb4 you absolute braindead retard
I knew that already

>>14515960
It means that if you know l, then there would be 2l + 1 possible choices for ml. ml is related to the schrodinger's equation for the wavefunction, I can't really explain it but you can only use integers for ml and they have to be smaller than l for the equation to work. Physically, ml describes the number of atomic orbitals in a subshell so there are 3 orbitals when l = 1, i.e. p subshell.

>>14515326
math is inline, eqn adds a line break.
>>14516205
https://en.wikipedia.org/wiki/Cauchy_stress_tensor#Cauchy%E2%80%99s_stress_theorem%E2%80%94stress_tensor
>It can be shown that the stress tensor is a contravariant second order tensor
>>14516393
We can do that. It just doesn't have all of the properties we want.

So when I simplified a boolean expression, I got 1 as the answer. How do I represent this in a logic circuit?
Like what should the input and output be? How should I draw it. Pls and ty

>>14516732
> We can do that. It just doesn't have all of the properties we want.
Interesting. Do you or someone else care to elaborate?

>>14516843
Please help. /g/ is too retarded to answer this.

I'm not sure if this counts as a stupid question or not but here we go.

You know the whole issue with measuring the one way speed of light, because of the clock synchronization? What if you had two synced atomic clocks, let them run, and then agreed on a point in the future in which the light would be shot from one clock to the other? Then you could just measure when it hits on the other to get the one way speed. Is this possible? If not, why not? It seems like a simple solution to the issue so there must be something I'm overlooking here.

Why do physicists look down on engineers?
Physics in 2022 is a joke. WOO QUANTUM RESEARCHERS SAY MUTIVERSE COULD EXIST!!! OMG THE CAT IS IMMORTAL IN THE BOX???

It is more akin to (bad abstract) art than physics now.

>>14516914
Sure.
We want [math]a/a = 1[/math], because that's the base definition of division. So [math]0/0 = 0 \infty = 1[/math].
We want [math]a - a = 0[/math], because that's the definition of subtraction.
We want [math](b + c)a = ba + ca[/math], that's distributivity and essentially half the definition of mulitplication for natural numbers (the other half being [math]1a = a[/math]).
And we have [math]0 = a \infty - a \infty = (a - a) \infty = 0 \infty = 1[/math].

>>14516965
Because engineers say ignorant things like your post

>>14516969
Ahh that made a lot of things click. Thank your very much

>>14516989
Ignorant how?
you haven't really criticized my views except in that you've applied labels, you have not analyzed them in any way and attacked them logically.

>>14517003
I have no responsibility to do that

>>14517015
Opinion -> Disregarded
Well done buddy

I am sorry if my question is not worded right. I am trying to create a framework for understanding how to prove general problems.

Like the question "Show that if 3n+2 is even, then n is even"

So this question's "theme" is odd/even, so it will require knowledge and use of the definition of odd/even.

But also in terms of If/Then, we assume the "if" part. So we assume 3n+2 is even. and then we use that assumption to prove the "then" part, by using the definition of an even integer.

Is there a guide or resource that puts this into words that my smooth brain can apply to even harder proofs that are not so clear-cut?

Thank you

>>14517067
damn idk what so say anon, what are you doing right now that requires this breaking down. Because I don't see how it gets anymore complicated then how you already described it.

>>14516940
sure, how do we synchronize?

>>14517370
My assumption was that atomic clocks are supposed to be synchronized and that's the concept of the clock, is that not the case? It makes a lot more sense if you can't sync them I suppose

>>14517370
>>14517403
Actually would you even need to synchronize? If the clock can determine its current point in time you can just account for the time difference afterwards.

>>14516919
What's the Boolean expression?

how do i do this? I got the mass of the cube and then did some stuff with avogadro's numbers to get my answer. but it was marked wrong

>>14515149
You know how certain chemicals and shit like cum or bleach is visible in ultraviolet light? Is anything uniquely visible in infrared only? I'm autistically considering an invisible ink you can only see with NODs on.

>>14517403
they work on radiation emission rates of a certain element so yeah they automatically sync up on account of subparticle wacko jacko

>>14517603


# of atoms = N * (density) * volume / (Molecular Weight)

>>14517888
too late retard

https://en.vrijdag.nl/security_print/infrared_security_ink#:~:text=What%20is%20infrared%20ink%3F,equipment%20to%20see%20infrared%20printing.

I want to make a fantasy weapon called an Ion Grenade, but I don't know if that would work.
It's basically a flash grenade but magic, and I don't know if having a whole lot of 'clustered' pos/neg Ions would really do anything because I know literally nothing about Ions beyond Destroy All Humans having an Ion Detonator that launched a plasma Ion Bomb that disintegrated shit in its explosion radius, and the Low Orbit Ion Cannon that still gets used in raids to fire ebin lazors for DDoSing people.
How do Ion theoretically make flash gooder? It's easy to stretch and bullshit things a little as this is fantasy, I just need some groundwork.

are there any non-trivial functions whose iterates are invariant w.r.t. to the original function

I've found that [math]\sqrt{a-x^2}[/math] is, but only for even a. This makes me think e or some trig function abomination is going to wind up being the answer, if there's a closed form solution at all
https://www.desmos.com/calculator/dtte85yy3b

>>14518107
in less retarded words, are there nontrivial functions [math]f(x)[/math] such that
[math]f(x) = f(f(x)) = f(f(f(x))) = f(f(f(f(x)))) = f^{n}(x)[/math]

>>14518115
Suppose you have such a function on a set X, that is f: X -> X. Then the image of X under f is some (not necessarily strict) subset S of X, and f, restricted to S, is the identity map.
A function of this form can be constructed, then, by choosing S, letting f(x) = x for x in S, and letting f(x) be any arbitrary value in S for x not in S.
Examples on the real line include:
f(x) = x for all x.
f(x) = |x|.
f(x) = x modulo 10.

Can you plate something with any alloy? If im not clear enough are there certain alloys that you cant plate steel with? Particularly can you plate stirling silver on steel or does it have to be pure silver?

I have this course in uni named "Models and Simulations" but the teacher and his notes are terrible. Content so far is stuff like monte carlo simulations, markov chains and multi-state models.
What books would you reccomend to learn this?

>>14518107
>but only for even a
ITYM only for an even number of iterations.

As has been pointed out, f(f(x))=f(x) => f(y)=y => f is the identity map on its domain.

If you only need it to hold for an even number of iterations, then f(f(x))=x => f=f^-1, i.e. f is its own inverse. This means that the graph of y=f(x) must be symmetric about y=x. In order for f to be a function, it must be monotonic decreasing.

The example of √(a-x^2) is a quarter-circle of radius √a, centred at the origin. y=√(a-x^2) => x^2+y^2=a. Any equation which is symmetric in x and y is inherently symmetric about y=x and so will work for some domain.

For the case where f^(kn)(x)=x for all n (i.e. whenever the number of iterations is a multiple of k), for odd k my gut feeling is that you're looking at a rotation of 2π/k about some point in the complex plane (i.e. for real x, f(x) will be complex). For even k, you can throw a reflection in there.

>>14518115
I don't think so because they all have to cancel out to f(x)=x or f(x)=constant. Even if they look complicated.

>>14515307
>>14508690
[math]\max_{(x,y)}f(x,y) = \max_x \{ max_y \{f(x,y) \} \}[/math]
Makes some sense to me given some very strong assumptions, for examply f is continuous and youre optimizing over a compact set and so on. Do you have more info on the assumptions for f?
Otherwise, ive actually never seen this approach for optimization, more often than not some version of gradient descent is used, which to work also makes some strong assumptions.
If you want to discuss more let me know, I can suggest some good reading material.

let [math]\omega[/math] be a p-th primitive root of unity where p is an odd prime. I know [math]G = Gal(\mathbb{Q}(\omega) / \mathbb{Q})[/math] is cyclic of order p-1. a cyclic group of order n has a unique order d subgroup for every divisor d of n, so by the fundamental theorem of galois theory, there exists a unique field [math]\mathbb{Q} \subset M \subset \mathbb{Q}(\omega)[/math] such that [math][M : \mathbb{Q}] = (p-1)/2[/math]. how to show that [math]M = \mathbb{Q}(cos((2\pi)/p))[/math]?

if I'm not mistaken, M is the field fixed by the unique order 2 subgroup of G, but how to use this to explicitly describe M?

>>14518292
>Makes some sense to me given some very strong assumptions
Don't be stupid, it makes sense on anything.
The proof is so awkward but simple that it just doesn't merit posting.

>>14515149
do i need to be born into money to pursue a doctorate in math?

I do not really see any point into going into debt if i do love the subject

>>14519087
you'll make 300k starting

>>14519099
im sure that's if you pursue a stats route

what would i make if i went towards either pure math or applied?

X and Y are written as 52 and 252 respectively in base A, and are written as 44 and 206 in base B

I'm retarded so I don't know how to find X, Y, A and B but I solved a different equation and found that A and B satisfy the equation
5A−4B=2

>>14519087
You realize you are paid as a graduate student right?

how do i do this? idk how to work with matrices and linear independence.

>>14519263
only if you teach right? but what if i just want to focus on my studies instead of that?

>>14519280
If you become an RA then you get paid without teaching, but even so the TA workload is not that heavy.

>>14519286
i see, thanks for letting me know.

>>14519274
2x2 symmetric matrices has a natural basis of [math]\left\{\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&1\\1&0\end{bmatrix},\begin{bmatrix}0&0\\0&1\end{bmatrix}\right\}[/math].
In this natural basis, the B matrices have the coordinates [math]\left\{\begin{pmatrix}1\\-2\\1\end{pmatrix},\begin{pmatrix}2\\1\\3\end{pmatrix},\begin{pmatrix}4\\-1\\-5\end{pmatrix}\right\}[/math]
To show they are a basis, show that these coordinate vectors are linearly independent.
In the natural basis, the last matrix has coordinates [math]\begin{pmatrix}4\\-11\\-7\end{pmatrix}[/math] -- now you have to change coordinates to B.

How I solve this?
[eqn]\int_{1}^{2}\int_{-2}^{2}\frac{y\sin(x)}{(1+x^2+y^2)^3}\ dxdy[/eqn]
The solution manual says this is equal to 0, but it doesn't explain how.
The integral calculator online gives me an answer that would take me two hours to write by hand and doesn't either explain why the definite integral is 0.

>>14519367
I forgot to add that I'm not allowed to switch to polar coordinates, otherwise I would have done so.

>>14519367
Symmetric on [math]x[/math].

>>14519389
Wow, it was that easy.
Welp.

>>14519367
Yes, that does look odd.

>>14519367
If anything, the limits of integration from -2 to 2
should be a dead giveaway to check on odd
or even functions. The integrand has an odd
function for a numerator and an even function
for a denominator with any y constant.

say I can be randomly chosen as one of x winners out of y players in some lottery, are the chances of me being among those x just x/y? isn't that just the percentage of winners out of total players and shouldn't I calculate the chances of me being among those x instead?

>>14517905
dude i must be retarded, i have tried several different formulas and it keeps marking them all wrong. what are you getting?

>>14519692
NTA but I got [math]7.38e22[/math] using [math]\frac{1.08^3}{10.5}\div107.87*6.02e23[/math]. Did you not use 1.08^3?

>>14519744
no i didnt. thanks for pointing that out...

I always struggle with absolute values. I have [math]D=\{(x,y):|x|\leq y, 0\leq y \leq 1\}[/math] and I'm integrating over this region in space.
What are my boundaries then?
Any tips/videos on how to deal with absolute values better? I always struggle with them.

>>14519762
If you know the graph of the absolute value, it's
a V-shape from the origin upwards. Or, it's the lines
of y=x and y=-x put together. The y being between
0 and 1 inclusive is a horizontal strip going left
to right. Putting the absolute value and the strip
together, it is a triangle whose coordinates are
(0,0), (-1,1) and (1,1).

>>14519762
The region is the triangle bounded by y≥-x, y≥x, y≤1. 0≤y is redundant here as it's implied by |x|≤y.

Note that |x|≤y => -y≤x≤y. For integration, it would be better to integrate dx dy with -y≤x≤y, 0≤y≤1. If you integrate dy dx, the bounds are |x|≤y≤1, -1≤x≤1. The |x| in the y bounds would mean that the integral has to be split into two, one with -x≤y≤1, -1≤x≤0 and the other with x≤y≤1, 0≤x≤1.

Putting aside questions about the existence of f(x), what is
[eqn] (f(x)^{n})' [/eqn]
in terms of f(x)?

>>14519939
>>14519862
Thanks.

>>14519957
n f^{n-1} f'

this DOES deserve its own thread,but im rangebanned
>will sleeping standing up make me (nofap nosex)invulnerable to wet dreams?

I'll start here instead of making a thread.

I have some generic, speculative questions regarding a topic that I know nothing about whatsoever. I'm pretty much retarded, I just like to think about things, and lately I've been thinking about a story that takes place in a setting where essentially matter scanning and replication has been invented.

It's very new, it only works with certain materials, so it's mostly used to replace industrial shipping - instead of shipping a bunch of big things across the globe you download the plans and print them out with your smart matter printer.

You can't replicate anything super mechanically complex, and certainly not an animal or person, both because the smart matter printing process isn't yet flexible enough to produce all the materials required, and because on the scanning end you wouldn't be able to get a precise enough scan on something fragile/moving/alive. Everybody also knows that you wouldn't want to clone a person this way because it gets you into all sorts of philosophical issues and the fidelity isn't 100% so they'll probably break down and go crazy or something.

The story, of course, follows from a breakthrough in materials and maybe some error correcting algorithm on the scanning end that lets someone stick a person in.

Is this concept sound? Like, internally consistent enough for a story? What would be the ramifications? What else would such technology affect societally? I'd like to have a solid core mechanic, given the smart-matter unobtanium-style hand-wave, that can be explored and followed to its conclusions in an interesting way.

Please tear my idea to shreds thank you scientists

Given the region in space [math]D = \{ (x,y): x\geq 0, y\geq 0, x^2+y^2 \leq 1\}[/math] and the following integral:
[eqn]\iint_{D} \frac{xy}{(1+y^2)^2} \ dxdy[/eqn]
I'm regardless of the order of integration I'm getting zero as a result of integrating, while the solution manual lists something else as an answer.
What am I doing wrong?

Whats the relationship between superclusters such as laniakea and the cosmic web?
What are the leading theories on whats at the centre of these clusters, if anything?

>>14520237
First off, D isn't the upper half of the unit circle, it's the quarter of the unit circle in Quadrant I. You should have *both* x and y positive.

Second, use Paul's Notes section on Cartesian to polar double integrals to make it easy.
https://tutorial.math.lamar.edu/classes/calciii/dipolarcoords.aspx

Just remember that dx.dy is r.dr.d\theta, not just dr.d\theta. (The infinitesimal rectangle would not have d\theta as a side length but rather r.d\theta.)

You should get 0.07671 if I'm not mistaken.

>>14520276
>You should have *both* x and y positive.
Damn, this completely went over my head. I need to focus more and stop skipping so many steps.
>Second, use Paul's Notes section on Cartesian to polar double integrals to make it easy.
I'm not allowed for these questions, in the next section I'm about to start with I can finally do that.
Thanks for the answer lad.

>>14520281
>I'm not allowed for these questions
Sort of surprising since it's so awkward without it. Makes for a bit of an irritating exercise.

But anyway, if that's the case, just consider the double integral over (x,y) as an ordinary integral over x that's nested in an integral over y. That is, rather than explicitly integrating over an area, you're holding y constant and integrating over a horizontal line, and then putting those horizontals together into the area by integrating over y.

So, hold y constant and deduce what the boundaries of the region must be to do the integral over x. That should be pretty easy. You've basically done it in your picrel.

>>14517067
Try working mod 2.

>>14520053
For people yes it's consistent, check out the game SOMA which has a similar story device (except it only works with people)
The big question for transporting people is whether you actually transport a person, or if you create a new person and murder the person entering the "portal"
Otherwise you can just go the star trek "whatever it just works" route which is also consistent enough

Replacing industrial shipping, on the other hand, is just ridiculous logistically, and probably the worst candidate for 3d printing -- it's by its nature extremely inefficient, while just putting it on a boat is both efficient, simple and reliable.
Just think about how much raw matter you would have dig up from the ground to provide the printing stuff, and that's not to speak of the energy costs for converting dirt and rock into a complete wood/concrete house

What, biochemically, determines whether a sperm has XX or XY chromosomes?

>>14520727
God

>>14516969
1/0 = 0^-1 = (e^iPI + 1)^-1
1/0^2 = 0^-2 = (e^iPI + 1)^-2 = 1 / (1 -2 + 1) = 0

>>14519762
|x| = (x^2)/x

Can I safely drink water out of a clean, clear, uncolored glass vase? Looked into it and was overwhelmed by the different types of clear glass and how drinkware is apparently distinct.

>>14515149
>The protons have a positive electric charge, the electrons have a negative electric charge, and the neutrons have no electric charge.
so if energy is porportional to mass how does something have no charge but mass

For my bioinformatics assignment I have to Identify all the amino acid mutations in this fasta file. However, there are like 2000. Is there a software where I can input this file which then points out all the irregularities and the line where it occurs? I need to make a table listing every single mutation - there has to be something out there right?

>>14521303
dont wozza im retarded just needed to convert it into a protein

can i get help for c) ?

>>14520727
A sperm only has one set of chromosomes, so it's either X or Y, not XX or XY.

>>14521523
Ok, so then what determines whether it has an X or a Y?

I've been meeting with a scientist from one of my school's lab once a week during the past semester as part of a sort of internship. Next week is our last session for the year what sort of gift could I get him?

>>14518735
Not writing everything out because it's 5 am and I'm hungover, but:
$[M:Q]=(p-1)/2\Rightarrow |Stab(M)|=2\Rightarrow Stab(M)=\{id, conjugation\}$

$w+\bar{w}=2\cos((2\pi)/p)$

>>14518735
Not writing everything out because it's 5 am and I'm hungover, but:
[M:Q]=(p-1)/2\Rightarrow |Stab(M)|=2\Rightarrow Stab(M)=\{id, conjugation\}

w+\bar{w}=2\cos((2\pi)/p)

>>14521169
>if energy is porportional to mass how does something have no charge but mass
It sounds like you think energy should be an implication of having a charge?

>>14521552
A process called meiosis, in which for every X gamete there is a Y (if you are a male). So at production the ratio of x,y sperm is 1:1

Why does my interest in anime and games plummet as I age?

>>14515149
Is it normal to not have actually learned anything after taking calc based physics 1 and 2 even tho i got an A? i feel like those classes are there only to filter people, not teach people

>>14522012
depression

How is it possible to use Fermi golden rule on a system with the following interaction hamiltonian?
[eqn]H_I = K e^{i(kx -\omega t)} \hat{e}\cdot \vec{p} + K^* e^{-i(kx -\omega t)} \hat{e}\cdot \vec{p}[/eqn] As far as I understand, the rule can only be used if the system has an interaction hamiltonian of the form:
[eqn]H_I = V_0 e^{-i\omega t}[/eqn] And in this case it's impossible to rewrite the hamiltonian as a constant times a SINGLE complex exponential of time

Hi anons, retard here. I'm having trouble with limits at infinity proof. Pic related. I can't understand how the conclusion is inferred from the last step(s). I underlined the problem parts. I'm new to calculus so maybe I'm out of my depth. I'm not even sure how to ask this question really. I think I don't quite understand the 𝜀 in the inequality and how the conclusion follows. If someone could go into a bit more detail or point me to a book or something that explains this and similar stuff in more detail I'd much appreciate it.

would it be beneficial for me as an engineer to a class on mathematical proofs?

>>14522157
I think you gotta look at the frequencies of the diagonal of the Hamiltonian and the rule has those frequencies as exponentials, basically exp(-tH).
The thing of the form exp(itw) is probabily only stemming from the unitary evolutions that make some constant operator time dependent, but those w's shouldn't distract you from the frequencies given by the energy that the Hamiltonian represents.

PS capitalize the person's name. Hamiltonian.

(or let's say the time evolution has the frequencies <H> as exponentials, which aren't equal to w's but they depend on said w's)

Why do japanese policewomen wear funny hats?
>>14521334
[math]W[/math] is generated by [math]e_1, \ldots, e_5[/math], [math]X[/math] is generated by [math]e_3, \ldots, e_7[/math].
>>14522186
If [math]c > 0[/math] and [math]x > 0[/math] then [math]\displaystyle \dfrac{c}{x^r} = \left | \dfrac{c}{x^r} \right | = \left | \dfrac{c}{x^r} - 0 \right |[/math]
The conclusion is just the definition of a limit.

>>14521867
how does something have energy but no charge

>>14522198
I don't think so, no.
Learning how to prove things is just learning how to form logical arguments that help you reach your desired result.

>>14522219
Wtf are you talking about lol

>>14522157
>As far as I understand, the rule can only be used if the system has an interaction hamiltonian of the form:
That hamiltonian is not even hermitian so you're misunderstanding something. Reread your notes/textbook or see
https://en.wikipedia.org/wiki/Fermi%27s_golden_rule#The_rate_and_its_derivation

What is the scalar equivalent of acceleration? In other words, velocity is to acceleration as speed is to what?

>>14522375
https://physics.stackexchange.com/questions/477427/is-there-a-scalar-acceleration

had this on my graph theory exam
>prove that there is a cycle of length 30 in a complete graph with 30 vertices, with every two vertices having one directed edge between them
what's the correct answer here? I answered that it's possible to have a vertex with outdegree zero and thus no cycle of length 30, but apparently that's wrong.

had this on my graph theory exam
>prove that there is a maximum cycle of length 30 in a complete graph with 30 vertices, with every two vertices having one directed edge between them
what's the correct answer here? I answered that it's possible to have a vertex with outdegree zero and thus no cycle of length 30, but apparently that's wrong.

>>14522219
I don't really understand what you are saying. The exp(iwt) appear because this is a hamiltonian for a charged particle in an E.M. field

>>14522299
>That hamiltonian is not even hermitian so you're misunderstanding something
Sorry if it's a stupid question, but why should the interaction hamiltonian be hermitian? It's just a term in the total hamiltoniant, which is:
[eqn]H=H_0+H_I[/eqn] this should be hermitian, right?
By the way pic related is the thing I was referring to

>>14522415
Sounds correct to me.
Explicitly, we can choose the vertices to be [math]\{ 1, 2, \ldots, 30 \}[/math] and for there to be a vertex from [math]a[/math] to [math]b[/math] if and only if [math]a < b[/math]. Fits everything as far as you posted and has no cycles at all.
Have you double checked the definition of complete directed graph? Maybe your explanation for why it doesn't work is too unspecific and you should have constructed a complete counterexample?

>>14522449
>for there to be a vertex from
An edge, obviously.

>>14522449
Maybe it's the difference between "cycle" and "directed cycle".

>>14522493
That's trivial tho. Also, the definitions in whatever book anon's using probably don't even allow for talking about undirected cycles in directed graphs.
Maybe he meant path instead of cycle? Then it's non-trivial, but the length drops to 29. Unless you decide to be schizophrenic and count lengths by vertices in the path, in which case it's 30 again.

>>14515149
Medical friends, do you know if it will ever be possible to do a procedure on the brain that will fix serotonin, dopamine, or whatever levels permanently?
Basically would replace the need for pills and it wouldn't create tolerance overtime like pills do.

>>14521552
Sperm are produced by taking one chromosome from each pair. Each normal cell (with chromosome pairs) produces two sperm; if the original cell is XY, it will produce one X sperm and one Y sperm.

To find the vertex of this equation, I have to convert it into standard form and use -b/2a, right?

This question has been driving me nuts.
Can someone help me. This is how far I got.

>>14523310
Forgot pic.

>>14523266
what happens when 2x-5 = 0?
>>14523317
did you miscopy the problem because f(x-1) = 10/16 works exactly

>>14523365
Youu're right, anon. I'm still stuck here tho.

>>14523506
Take logarithms on both sides.

>want to answer questions
>too stupid to answer any of them because I don't study maths or physics
its over, isn't it?

>>14523933
What do you study?
Most of the questions are inevitably going to be maths and physics because of the maths/physics/engineering subject overlap.

>>14523933
>>14524005
homosexuals cringelarping as anime girls

>>14524007
I've cropped your image so you can stop replying to me with reddit screenshots.

>>14524005
engineering

>>14522415
knock a vertex out and prove there is a maximum cycle of length 29 on the remaining graph

>>14524034
Surely you can answer a lot of questions in these threads then.
There's basic algebra like >>14523266 and >>14523317 all the time.
You can probably even answer >>14508690 if you put half an hour into it. The proof is basic, anons are goofing because they're looking at the theorem and incorrectly thinking it looks non-trivial.

Can someone motivate me to learn set theory? I'm trying to finish the first 5 chapters of Enderton but somehow I just cant get it stated...

>>14523933
this is a math homework board, no one is answering the science questions

>>14515307
hey just because I don't know the proper terminology doesn't make it a stupid question
(t. >>14496210 and >>14497230)

- given the stated goals, is a normal distribution w/ a weighted mean and standard deviation an appropriate model for the task, or are there better options?
- since it needs to tread more carefully when its data is old, is there a more effective way to accomplish that than artificially lowering the mean and increasing the standard deviation based on the time since the last sample?
- using the standard deviation as a proxy for uncertainty in the underlying value, is there a better way to handle exploration vs exploitation than poking around a few percentile from the mean?
- when the process is certain enough of a user's strength, it should adopt a bias for weight recommendations slightly above the mean (since the goal of exercise is to get stronger, and that happens by progressively using heavier weights to provide a challenge). what would make a good condition for it to be "certain enough" to start increasing the recommended weight?

>>14523726
Thanks fren. I was tired yesterday and messed up something so trivial.

>>14522436
[math] H_0 [/math] should also be hermitian since it's the unperturbed hamitonian, which means [math] H_I = H - H_0 [/math] is also hermitian.
>pic
What they actually mean is that the perturbation can be decomposed into components with negative and positive frequencies and these allow transitions from higher to lower states or vice versa, respectively. as an example, if the system is an electron in an atom and the perturbation is some electric field of a single frequency [math] \hbar \omega = E_{out} - E_{in} > 0 [/math], then [math] V(t) = V_0 e^{i \omega t} + V_0^\dagger e^{-i \omega t} [/math], where V_0 is some time-independent operator (V(t) is hermitian but V_0 need not be). The negative frequencyy part [math] V_0 e^{i \omega t} [/math] causes transitions [math] |out \rangle \rightarrow |in \rangle [/math] (since this is an electric field, this transition is an example of stimulated emission of a photon) and the positive frequency part causes transitions [math] | in \rangle \rightarrow | out \rangle [/math] (absorption of a photon). The rates of both these transitions are the same because [math] |\langle out | V_0 | in \rangle |^2 = |\langle in | V_0^\dagger | out \rangle |^2 [/math]. If the perturbation were not hermitian, then the operator corresponding to positive frequency would be completely different from [math] V_0^\dagger [/math], so you couldn't couldn't say that these rates would have to be equal

>>14522725
Finally, an actual answer. Thanks.

I'm lost, how did Rudin come up with the [math]-m_2\leq m \leq m_1[/math] such that the last inequality is true?

Let [math]G[/math] be a metrizable group. Let [math]H[/math] be a closed subgroup (which need not be normal) such that the quotient space [math]G/H[/math] also happens to be metrizable. Suppose that [math]g_\ast g_n H \to g_\ast gH[/math] in [math]G/H[/math]. Does this imply [math]g_n H \to g H[/math]?

Since [math]H[/math] need not be normal, coset multiplication is not well defined, which got me confused

How to solve this?

I just can't into questions like this that say given <log equation> show that <log equation>. My professor got mad that my class couldn't solve questions like this and told us that he would include a lot of questions like this in the finals. How can I git gud and solve this kinds of questions? I think he's trying to fail us. Pls help.

>>14524611
What's [math]g_*[/math]?

>>14524770
Use logarithm properties to write the first equation as [math]\log a = \log b[/math] and then do [math]a = b[/math]

How do I pay attention during lectures in class? I can only focus for a minute or two before I unconsciously drift into thought. I can't miss these lectures.

>>14524416
Thank you very much for the answer anon. So, if I have an interaction hamiltonian:
[eqn]H_I = K e^{i(kx -\omega t)} \hat{e}\cdot \vec{p} + K^* e^{-i(kx -\omega t)} \hat{e}\cdot \vec{p}[/eqn] Then the Fermi golden rule tells me that the absorption rate is:
[eqn]\Gamma_{in \space out}=\frac{2\pi}{\hbar}K^2|\langle out | e^{i(kx -\omega t)} \hat{e}\cdot \vec{p} + e^{-i(kx -\omega t)} \hat{e}\cdot \vec{p} | in \rangle |^2 \delta(E_{out} - E_{in} - \hbar \omega)[/eqn] The problem is that I'm supposed to show that:
[eqn]\Gamma_{in \space out}=\frac{2\pi}{\hbar}K^2|\langle out | e^{ikx} | in \rangle |^2 \delta(E_{out} - E_{in} - \hbar \omega)[/eqn] And I have no idea how to do that. I know that [math]e^{ikx}[/math] can be brought out of an integral in [math]dx[/math] and that its square modulus is one, but I cannot do that in this case. I also don't know how the operator [math]\hat{e}\cdot \vec{p}[/math] can disappear like that, when it should at least turn the [math]| in \rangle[/math] state into its first derivative

>>14524416
Thank you very much for the answer anon. So, if I have an interaction hamiltonian:
[eqn]H_I = K e^{i(kx -\omega t)} \hat{e}\cdot \vec{p} + K^* e^{-i(kx -\omega t)} \hat{e}\cdot \vec{p}[/eqn] Then the Fermi golden rule tells me that the absorption rate is:
[eqn]\Gamma_{in \space out}=\frac{2\pi}{\hbar}K^2|\langle out | e^{i(kx -\omega t)} \hat{e}\cdot \vec{p} + e^{-i(kx -\omega t)} \hat{e}\cdot \vec{p} | in \rangle |^2 \delta(E_{out} - E_{in} - \hbar \omega)[/eqn] The problem is that I'm supposed to show that:
[eqn]\Gamma_{in \space out}=\frac{2\pi}{\hbar}K^2|\langle out | e^{ikx} | in \rangle |^2 \delta(E_{out} - E_{in} - \hbar \omega)[/eqn] And I have no idea how to do that. I know that [math]e^{i \omega t}[/math] can be brought out of an integral in [math]dx[/math] and that its square modulus is one, but I cannot do that in this case. I also don't know how the operator [math]\hat{e}\cdot \vec{p}[/math] can disappear like that, when it should at least turn the [math]| in \rangle[/math] state into its first derivative

>>14524770
uh the rhs goes to log xy from property log m + log n = log mn
now use the properly alogx = log (x^a) to change rhs to xy^1/2
then you go (x-y)^2 = 9xy
and you’re done.

Can I get some recommendations of places for online video lessons? Specifically chemistry, biology, philosophy, literature, gender studies, theology and data science.

I can teach you biology and gender studies in my discord server

>>14524565
Let m be the smallest integer greater than nx (test: how do we know there is a smallest such integer?). m-1 therefore cannot be greater than nx.

>>14524990
Since you're only looking at the absorption rate, the contribution of the negative frequency part will be negligible in comparison to the positive part. See the proof of the rule in the wiki article I linked earlier or in this article
http://staff.ustc.edu.cn/~yuanzs/teaching/Fermi-Golden-Rule-No-II.pdf
>I also don't know how the operator e^⋅p⃗ can disappear like that
It won't, your book probably has a typo or something

>>14525067
But why did he bother with m_1 and m_2 in previous step? Couldnt he just set m = floor(nx) + 1 and be done with it?

>>14525078
It's hard to guess why without context, but it looks like he's proving that floor(nx) exists, essentially. Otherwise, you're right.

>>14525045
Hilarious but I just want stuff to listen to while playing vidya.

consider an automobile moving at 90kmph, the driver gets distracted for 2 seconds. what is the distance travelled during this time?

uniform motion mechanics question i'm legit self-learning physics from the ground up i'm not trolling nor too lazy to do my homework cause this is in fact not even homework, its just a question on my book that i couldn't figure out why it says the answer is 200m

tried the s=so vt formula but it got fucked

>>14525127
First you convert.
https://www.wolframalpha.com/input?i=90+kilometers+per+hour+in+meters+per+second
Then you use [math]\Delta s = vt = 25 m/s \times 2 s = 50 m[/math].
No idea where 200m came from, your book is drunk.

>>14525134
my goofy ass was looking at the wrong answer, sorry and thanks anon

>>14524781
Any fixed element of [math]G[/math], sorry, I should have made that clear.

So why are there no electric dipoles or magnetic quadrupoles in a nucleus? (And ideally why of the 2^k-pole moments it's only electrics for odd k and magnetics for even k?)

>>14525230
Right.
You have a topological group [math]G[/math]. [math]H[/math] is a closed subgroup of [math]G[/math]. [math]G/H[/math] is the set of left cosets of [math]H[/math] equipped with the strongest topology such that [math]p: g \to gH[/math] is continuous.

Your result is true, but the proof is ugly.

>>14515149
hey do any of you have access to the proceedings of the london mathematical society, and if so can you get me this pdf
https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-23.3.532

>>14525319
learn to pirate before you graduate man
https://sci-hub.st/10.1112/plms/s3-23.3.532

>>14525339
tyvm I appreciate it

>>14525068
I'm probably retarded, I still don't get it. In the proof, in case of absorption, the term with the denominator [math]\omega_{fi}+w[/math] can be neglected when compared to the term with the denominator [math]\omega_{fi}-w[/math], and I understand it. But I have no terms with those denominators here.
>It won't, your book probably has a typo or something
Sorry you're right. Then what I'm trying to understand is how: [eqn]|\langle out | e^{i(kx -\omega t)} \hat{e}\cdot \vec{p} + e^{-i(kx -\omega t)} \hat{e}\cdot \vec{p} | in \rangle |^2 \delta(E_{out} - E_{in} - \hbar \omega) =|\langle out | e^{ikx} \hat{e}\cdot \vec{p} | in \rangle |^2 \delta(E_{out} - E_{in} - \hbar \omega)[/eqn]

>>14524863
Ty anon
>>14525001
Thanks, anon. I know the rules and can do 'solve for x' type log questions, but this type of given <>, show <> type of questions are a bit hard. I'll practice more.

>>14525306
Fair enough. Thanks, anon.

Is it possible to learn physics(uni lvl) by myself?

Is it a typo in my book that says "if x1>x2 then since f is strictly increasing, we can conclude f(x1) > f(x2)"

Wouldn't this be strictly decreasing? It was about using contraposition to prove that f is one to one if f is strictly increasing. The other case was if x1 < x2 which makes sense to me that f is strictly increasing there

>>14525119
listen to sissy hypno or something

How do you use contraposition to prove if A is a subset of B, then B' is a subset of A'?

I tried to look it up online but the answers are confusing, and seem to be taking the contraposition of just A subset B, by its definition "if x is an element of A, then x is an element of B".

But that doesnt match the usual contraposition where you turn the negation of the hypothesis into the new conclusion and the negation of the conclusion into the new hypothesis.

And also taking the negative of the definition of a subset seems useless. What good is turning Vx (x e A -> x e B) into Ex(x e A & x e B')? Now you just have "for some x" and that doesnt seem useful at all.

Im sure im misunderstanding something fundamental so thank you for your patience

>>14525632
Of course it is. We don't know what level you're at though.

>>14525647
It's correct. There's no typo.
>>14525663
Why on earth would I?

>>14525776
>It's correct. There's no typo.
okay thank you

>>14525744
Zero. I forget all what was in school

>>14515149
is there anything that indicates that gravity pulls things in like a spiral (whirlpool, tornado)?

>>14525712
>Vx (x e A -> x e B) into Ex(x e A & x e B')
This is a really good start and is the correct negation, you should write out your other statement like this too. It's hard to give help for this problem without just revealing the answer, but writing out the correct contrapositive is basically the whole proof.

>>14522198
No, physics would be far more useful. I found that physics classes were good for learning to solve problems of many types and you can gain deeper knowledge of stuff that will apply directly to whatever engineering field you're in.

>>14525911
i passed calc based physics 1 and 2 with an A so far but i genuinely have no real understanding of anything. It makes no sense to me.

>>14525776
because its hot

>>14525939
Not really.

if all aerospace and terrestrial motion ceased would it be possible for the air currents around the earth to reach some form of steady flow after a very long time? (provided you freeze time so its just stays as one day with constant unchanging sunlight/moonlight at any given point)
>>14525993
you'll realize eventually

>>14525871
I'm not sure what to do with it.

The contrapositive ends up with one being Vx and the other being Ex. Am i doing something wrong?

so "Vx(x e B or x e A') or Ex(x e A and x e B')"

This just has no meaning to me :(

>>14526307
>The contrapositive ends up with one being Vx and the other being Ex. Am i doing something wrong?
Yes, they should both be using the existential operator

>so "Vx(x e B or x e A') or Ex(x e A and x e B')"
When proving conditional statements you don't want to write it in this form. You should write it out in the [math]p\Rightarrow q[/math] form. The basic Idea is that you assume that the antecedent is true and then show that the consequent must be true. Is this an assingment for a class? or are you working out of a book? There should be some demonstration of how to prove conditional statement that you should review.

This is a tautological statement so there is technically a way to make sense of it in the form you wrote, but there is a much easier and more useful way to prove this.

>>14526376
I've just been trying to work through the book on my own, and i'm a bit past the material on logic and unions, but i find i still struggle with proofs like this.

But what use does the existential operator have for proving anything if its only talking about some x and not all x?

I think it can be written as Ex(x e B' & x e a --> x e A & x e B'). But i'm not sure what this means with Ex in there. It could be like Ex(p->p) and p->p is the same as ~p or p, which is just p.

But still i just cant see any meaning in it, especially with only the existential operator involved :(

Sorry i have such a smooth brain

>>14526474
>>14526376

Im also confused why i have to prove it this way using quantifier stuff. The other proofs are all like "assume x e A, then by definition of <something>, x e B, which is the desired conclusion" or something like that.

Can the proof be written as something like "Suppose x is not an element of B, and is an element of A...

[math]e=\lim_{n\to\infty}(1+\frac{1}{n})^n[/math]
[math]e^x=\lim_{n\to\infty}(1+\frac{x}{n})^n[/math]
How does the x get in there?

>>14526542
The x was always there, it was the 1 that got substituted in.

>>14526550
So how do you derive [math]e^x[/math]? I found [math]e[/math] by thinking about compound interest where P=1, r=1, and t=1. I just don't see how you can make the jump.

>>14526607
There are many ways (including different definitions of e), but here's one:
If we have shown that the first limit exists and define its limit as e, then you can use essentially the same argument to prove that the lower limit exists for all positive integer x.
Next, we take a positive rational r=a/b and do a little bit of variable substitution in the limit:
[math]\displaystyle e^a=\lim_{n\to\infty} \bigl(1+\frac{a}{n}\bigr)^n=\lim_{n\to\infty} \bigl(1+\frac{a}{bn}\bigr)^{bn} = \Bigl(\lim_{n\to\infty} \bigl(1+\frac{r}{n}\bigr)^{n}\Bigr)^b[/math].
So we've proven that [math]e^r = \lim\limits_{n\to\infty} \bigl(1+\frac{r}{n}\bigr)^{n}[/math] for positive, rational r. Substituting [math]n\mapsto -n[/math] shows that this works for negative r as well.
Extending from the rationals to the reals is trickier, because real-valued exponents are usually defined in terms of the exponential function, which would be circular here.
We'd like to just define [math]e^x = \lim\limits_{r\to x, r\in\mathbb Q} e^r[/math], but you have to prove this limit exists, and don't know a simple way to prove that.
The only way I know of is to take a detour to prove that e^r is equal to the usual Taylor series, which gives us enough continuity to safely take that limit above.

what happened to that educators thread? i was gonna post in it :c

>>14526542
[eqn]
e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) ^n \\
e^x = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) ^{nx} \\
e^x = \lim_{n \to \infty} \left( 1 + \frac{x}{nx} \right) ^{nx}
[/eqn]
the key thing to notice is that n is going to infinity, but both instances of n here are being multiplied by x, so both n's are going to infinity "at the same rate", but it doesnt matter how quickly or slowly n goes to infinity, so you can cancel out the x's and change both the nx terms to just n.
another way to think of it: you might imagine n being incremented like 1, 2, 3, 4, ... all the way to infinity. but it doesnt need to go like that, it can be increased at any rate you want so long as it goes to infinity. it can go like 2, 4, 6, 8, ... or like 1, 10, 100, 1000, ... or it can go like [math]\frac{1}{x},\frac{2}{x},\frac{3}{x},\frac{4}{x},\cdots[/math] which means you could cancel out the x's and it would go like 1, 2, 3, 4, ... instead. but its all the same process.

>>14526783
Oh! lim of n = lim of nx = infinity! I got it. I'm familiar with calculus enough to get that.
The one thing I was never sure about though was the whole x^2 goes to infinity faster than x or x! to infinity faster than x thing. I can see that, I really do, but how do we objectively determine which functions increase faster than others? obviously, it can't just be a matter of slope as 2x and x are just as fast as each other essentially, right? so what is it?
>>14526686
I'm going to be honest, the other guy explained it more simply and clearly than you did imho but I still really appreciate the effort, pal. Thanks a lot, anon.

>>14526474
>>14526497
>I think it can be written as Ex(x e B' & x e A --> x e A & x e B')
This is wrong, quantifiers do not distribute like that, the correct form should be [math] (\exists x(x \in B' \wedge x \notin A')) \Rightarrow (\exists x(x \in A \wedge x \notin B)) [/math]
How you want to approach the proof is to assume that [math] \exists x(x \in B' \wedge x \notin A') [/math] is true and to derive [math] \exists x(x \in A \wedge x \notin B) [/math] using the substitutions [math] x \in A \Leftrightarrow x \notin A', x \in B' \Leftrightarrow x \notin B [/math].
For this proof you don't need to pay attention to the existential quantifier at all. When writing proofs, you should also limit the use of formal notation whenever possible

>But what use does the existential operator have for proving anything if its only talking about some x and not all x?
it's generally a lot easier to prove properties of a single element rather than to prove properties of many elements (potentially infinite)
Although it is unnecessary for this proof, noticing how to change a universal statement into an existential statement through the use of the contrapositive is extremely helpful, but that probably wasn't what you were asking.

confused about
>Velocity at splat time: sqrt( 2 * g * h )
in a free fall calculator: how long it takes to fall from height h and what's the splat speed.
isn't it simply g*time?
what is sqrt( 2 * g * h ) anyway? how did they arrive at this formula?

>>14526815
>obviously, it can't just be a matter of slope as 2x and x are just as fast as each other essentially, right? so what is it?
pretty sure 2x is twice as fast as x. i think what you mean to say is something like [math]\frac{2x}{x}[/math] is a finite number, where as [math]\frac{x^2}{x}[/math] can be arbitrarily large.
>I'm going to be honest, the other guy explained it more simply and clearly than you did
i was trying to provide an intuitive explanation, but the other guy (remi?) was actually rigorous. i appreciate that you said you liked mine more, though, probably pissed him off.

>>14526821
Using the equations [math] y = h - \frac{gt^2} 2 [/math] the object hits the ground when y is 0.
We can solve for this time to find [math] 0 = h - \frac{gt_0^2} 2 \Rightarrow t_0 = \sqrt{\frac{2h}g}[/math].
You are correct in pointing out that [math] v = gt [/math] but since we know how long it takes for the object to hit the ground we can solve [math] v_0 = gt_0 = g\sqrt{\frac{2h}g} = \sqrt{2hg}[/math]

>>14526833
> i think what you mean to say is something like 2x/x is a finite number, where as x^2/x can be arbitrarily large.
Yes, I mispoke, my bad. How do we establish the speeds of e^x vs. x! vs. x^n? I'm sorry, does my question make sense?
> i was trying to provide an intuitive explanation
Which part of your explanation was unrigorous? doesn't it just follow from the definition of the limit (which should be rigorous)?
> but the other guy (remi?) was actually rigorous.
You know his name? haha, that's pretty cool.
> i appreciate that you said you liked mine more, though, probably pissed him off.
lol, really didn't mean nothing by it but thanks again :)

>>14526859
>How do we establish the speeds of e^x vs. x! vs. x^n?
i think you can do something like l'hopital, maybe, iunno. not sure how that'd work for the factorial (gamma) function tho
>Which part of your explanation was unrigorous?
the biggest jump is pulling the exponential inside of the limit
>You know his name?
dont tell anyone but i immediately assume any analysis post that uses capitalization/punctuation is remi. i estimate that this has a 5% false positive rate and a 0% false negative rate

>>14526816
thank you anon i appreciate your help. is the quantifier ignored/removed by using existential instantiation? or is it just not necessary to consider so it isnt mentioned

>noticing how to change a universal statement into an existential statement through the use of the contrapositive is extremely helpful,
tyvm anon i will keep that in mind and try and look out for that in the future

the two subset terms in the hypothesis and conclusion (if thats the right terms) look so similar i'm just confused why contraposition is required or thought of?

and also i thought you were supposed to use contrapositive proofs to get the simpler term as your hypothesis and then prove the more complicated term as the conclusion

i would never have thought B' subset A' was the simpler of the two, but maybe it just doesnt look simpler, or maybe my thought was just niave or wrong

>>14526912
>is the quantifier ignored/removed by using existential instantiation?
you could do that, but you don't need to since you are just going to reintroduce the quantifier on (essentially) the exact same statement, the real "work" in the proof is just rewriting the antecedent into the consequent using the substitutions I described.

>i'm just confused why contraposition is required or thought of?
In this proof, using contraposition on the conditional outside the quantifiers isn't super useful. But as you probably saw when you looked up solutions online, taking the contrapositive on the conditional inside of the quantifiers makes the proof trivial. (both methods are very simple though)

I assume that this is just a busywork exercise to get you to engage with topics presented, but now you know a pretty neat fact about sets.

Does anyone have any good books about radio waves and frequency modulation and stuff? I really wanna learn more about it

>>14526542
[eqn]\lim_{n\to\infty}(1+\frac{a}{n})^n = \lim_{n\to\infty} e^{\ln{(1+\frac{a}{n})^n}}=\lim_{n\to\infty} e^{\frac{\ln{(1+\frac{a}{n})}}{\frac{1}{n}}} [/eqn]
We can then drag the limit into the [math]e[/math] and use L'hôpital's rule:
[eqn]\lim_{n\to\infty} \frac{\ln{(1+\frac{a}{n})}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{\frac{1}{(1+\frac{a}{n})}\cdot \frac{-a}{n^2}}{\frac{-1}{n^2}} = \lim_{n\to\infty} \frac{a}{(1+\frac{a}{n})} = a\\
\Rightarrow \lim_{n\to\infty}(1+\frac{a}{n})^n = \lim_{n\to\infty} e^{\frac{\ln{(1+\frac{a}{n})}}{\frac{1}{n}}} = e^{a}[/eqn]

Asking this here. I tried /g/, but they doesn't read.
What do I need to read to understand what this means?

>>14515149
i know you couldn't use entanglement to send long-distance messages, but if sets of entangled photons were separated very far away, could that be used to get random noise -- but precisely equal random noise? Would that sharing of information be faster than light?

>>14527042
bless you anon ty for your help

this question was in a section of the book that was summing up different proof types. they introduced contrapositive with a normal example and then this was the next one that they "left to the reader." pretty sneaky!

why are all famous scientists from the past such disgusting creatures who always shit on each other, accused each other of plagiarism, bully each other, and were just overall despicable human beings? shouldn't smart people be free of jealousy?

How do I solve this:
ARHHH

>>14528003

>>14528008

>>14525405
Looks like your LHS is wrong because you're not applying the rule correctly. I don't think you're even supposed to prove anything here. You're taking the absolute square of the entire interaction hamiltionian's matrix elements when in fact you should only be taking the absolute square of the matrix elements of the component of H_I which corresponds to the frequency which causes the transition because that's what the rule says you should do (and the derivation of this is given in the pdf posted earlier).

I am heating ethanol to 150 F for a lab application in open containers.

I am close to the boiling point and I'm concerned there will be vapors that can be ignited with a static spark.

What kind of anti-static mat can protect me? Is it really necessary?

>>14516393
The limit of 1/x as x approaches 0 in infinity. However simply put, division isn't defined for dividing by 0. There are plenty of videos and stuff you can read that go into detail about it. Just google it. It's not complicated.

>>14528008
>>14528011
here's your homework solution:
[eqn]
\log (x^4) = 4 \log x = 4 \log (2 ^k) = 4k \log 2 = 4k \\
\log (y^4) = 4 \log y = 4 \log (20^k) = 4 \log \left((2^2 \times 5)^k \right) = 4 k \left(
\log 2^2 + \log 5 \right) = 4 k (2 + t) \\
\log (x^4) + \log (y^4) = 4k + 4 k (2 + t) = 4k (3 + t)
[/eqn]
(all logs log base 2)
just apply log rules and use prime factorization of 20
brush up on log rules if you've forgotten

>>14528073
im not a chemist, but can't you just use a fume hood?

>>14528171
That's a good idea. However I do not have one ... but I can build one.. hmm. But I still would like to know if these mats can protect against that sort of thing.

Does anyone have any idea why this is giving me garbled output? Is it just the editor?

still waiting on this --> >>14522255

>>14522255
You know that energy in physics isn't just literal wall outlet electric energy, right?

>>14528054
>You're taking the absolute square of the entire interaction hamiltionian's matrix elements when in fact you should only be taking the absolute square of the matrix elements of the component of H_I which corresponds to the frequency which causes the transition
Then the rule it's incorrectly stated...? Pic related are equations from the PDF, and in the Fermi golden rule there appears the entire interaction hamiltonian matrix element [math]W_{fi}[/math].

>>14528054
>You're taking the absolute square of the entire interaction hamiltionian's matrix elements when in fact you should only be taking the absolute square of the matrix elements of the component of H_I which corresponds to the frequency which causes the transition
Then the rule it's incorrectly stated...? Pic related are equations from the PDF, and in the Fermi golden rule there appears the entire interaction hamiltonian matrix element [math]W_{fi}[/math].

why does centripetal acceleration equal V^2/r. how is it derived? and what happens if the object itself moves with acceleration around the circle? will the centrepedal acceleration remain V^2/r and V will change ?

can the two outer layers of corrugated cardboard be considered decoupled?

>>14529349
Do you have a middle school understanding of vectors? That's the bare minimum I'll need.
We imagine a particle in uniform counter-clockwise circular motion whose position is the vector [math]\vec{r} (t)[/math]. The particle's velocity vector is obtained by starting from [math]r[/math], turning it a quarter counter-clockwise, dividing it by the radius and multiplying it by the velocity.
If you take a second to look at things, you'll notice that the velocity vector is also in uniform circular motion. Because it completes turns at the same time as the particle, those two have the same angular speed, hence the speed of the velocity vector is [math]V^2 / r[/math] (I could write out the ins and outs of the computation but that would probably just confuse you more).

>>14529414
Something like this?

>dividing it by the radius
why?

>>14529448
>Something like this?
Yeah.

>>dividing it by the radius
>why?
The position vector has modulus (norm? Not sure what terms you're used to) equal to the radius. The velocity vector has norm equal to the speed. So to get from the former to the latter we turn it in the right direction, divide by the radius and multiply by the speed.

Could I use a ham radio transceiver (~150W) to drive a small RF plasma chamber? If so, how would I go about matching the impedance? I am very new to RF.

If I have the absolute entropy of a substance at a given temperature and pressure, how does this entropy vary with a change in pressure at the same temperature?
I've been doing this exercise for four hours now and I'm almost done but I can't get this.

If I have the absolute entropy of a substance at a given temperature and pressure, how does this entropy vary with a change in pressure at the same temperature?
I've been doing this exercise for four hours now and I'm almost done but I can't get this.

>>14529616
Come on anon. How bad are your google skills when every search gives you an answer - https://en.wikipedia.org/wiki/Entropy#Isothermal_expansion_or_compression_of_an_ideal_gas

>>14529662
You don't get it, that gives you the change in entropy when you vary the pressure, it does not give you the absolute entropy of the substance at this new pressure, which is what I need.
Imagine you walk 100 meters to the right, and then 2 meters up. You've walked a total of 102 meters, but the distance to your original point is more, it's something like that.

>>14529465
i suggest you ask on /diy/ there is /ohm/ and also /ham/ generals there.

>>14529662
You don't get it, that gives you the change in entropy when you vary the pressure, it does not give you the absolute entropy of the substance at this new pressure, which is what I need.
Imagine you walk 100 meters to the right, and then 2 meters up. You've walked a total of 102 meters, but the distance to your original point is different, it's something like that.

>>14529699
Okay thanks anon.

>>14529326
They're probably just calculating the W_{fi} in the last line using only the time-independent part of W(t) i.e. W. In any case the notation is terrible. This pdf has a more rigorous explanation than the one posted earlier. Try reading it maybe
https://courses.cit.cornell.edu/mse5470/2015_ece4070_mse5470_primernotes_fermigoldenrule_dj.pdf

>>14529349
> why does centripetal acceleration equal V^2/r. how is it derived?
x(t) = (r cos(θ), r sin(θ))
x'(t) = (-r θ' sin(θ), r θ' cos(θ)) (by the chain rule)
x''(t) = (-r θ'^2 cos(θ), -r θ'^2 sin(θ)) (assuming θ''=0, by the chain rule and product rule)
= -θ'^2 x(t)
v=rθ' => ||x''(t)|| = rθ'^2 = r(v/r)^2 = v^2/r

> and what happens if the object itself moves with acceleration around the circle?

x''(t) = (-r θ'^2 cos(θ) - r θ'' sin(θ), -r θ'^2 sin(θ) + r θ'' cos(θ))
= -θ'^2 x(t) + θ'' (x'(t)/θ')

IOW, the acceleration has a centripetal component proportional to the square of the angular velocity and a tangential component proportional to angular acceleration.

For constant angular velocity θ''=0, which simplifies to the previous form.

If i flip a coin infinitely many times, is the chance of getting infinitely many heads in a row 1? Or is it 0

We can brute force adaptive camo on a sphere or cylinder quite easily with small devices that serve the dual purpose of camera and projector. Each pixel on each sensor-lcd needs to be mapped to another pixel on another sensor-lcd. The devices have some sort of duty cycle between image capture and projection. Difficult but not impossible.

When it comes to clothing however this approach seems like a fools errand because the orientation of all of the devices is constantly changing.

Question 1. Is there a device accurate enough to determine the orientation for each sensor with respect to a reference plane?
Question 2. How much calculation would it take to convey the sensor pixel to the lcd pixel?

>>14526607

>>14530230
The probability is zero. Think about it this way. You can consider any sequence of coin flips to be represented by a string of H’s and T’s. An infinite sequence of flips is then HHTHTTTTH… going on forever. Your question is then, what is the probability that this infinite string ends in HHHH… forever. The probability is zero because the probability of having such an infinite sequence is zero, and we’re only talking about one infinite string, so we don’t have an infinity*0 issue to work out. Your problem is analogous to drawing a number from the real number line between 0 and 1 and asking what the probability is that the number you get ends in 0000… which is to say that it comes from a particular subset of the rational numbers. The rational numbers are countably infinite, while the real numbers between 0 and 1 are uncountably infinite, so the probability is zero.

>>14530230
In case >>14530615's reply went over your head.
Imagine the infinite binary tree with H,T as the symbols. The lengths of the branches are countably infinite; the number of branches is uncountable. A wishy washy proof that the number of branches is uncountable is that you can't count to the end of the first branch, i.e., HHHH..., so you can't start counting the number of branches. Your question asks: how often will we select a particular branch if we select a branch at random? >>14530615
gave you the answer: 1 over uncountable infinity, which for all intents and purposes is equal to zero.

I am redoing calculus studies and find it really slow and hard. I remember the first time I did it I got more than 1 x the work done in 20 times less time. Is it because I've aged a few years? What's going on? Or is it the change of books? Maybe I forgot to check my answers the first round?

I studied Anton Howard, now studying Stewart.

>>14529873
I see... I have written my own, extended and corrected version of the Fermi golden rule, with two separate equations, one for absorption and one for emission. Although I have to say it bothers me a lot that I can't find it stated like this anywhere, and until then I'll always suspect that what I've written is wrong.
As it stands, it seems that you have to know the 3 pages of proof to "apply the rule" correctly, which essentially means "just redo the proof for the situation at hand". Then why call it "rule" if it can't be readily applied as it's stated? It should have been called Fermi golden procedure.

Anyway thank you very much for your help

>>14530498
So you prove e by introducing e and ln with the base of e in the second step? Isn't circular logic?

can somebody enlighten me on career opportunities for an applied mathematics degree? I got baited by Good Will Hunting into studying mathematics, I don't mind teaching math or actuary but I wouldn't want that as my first option.

say an algorithm has a time complexity of floor(n/2) + ceil(n/2) where n is an integer, is it acceptable to consider it as having a total time complexity of n given that the floor of half an integer plus the ceiling of half that integer is always that integer?

Can someone figure this out please I'm a dumb ass

MAX(1000, 702000 / 10, 653 * 100, 1000 + (MAX(11 - 25, 0) * 1000))

hopefully that makes sense.

>>14531145
> is it acceptable to consider it as having a total time complexity of n given that the floor of half an integer plus the ceiling of half that integer is always that integer?
Yes, except it's not for that reason, but because [math]\lfloor n / 2 \rfloor + \lceil n /2 \rceil = n[/math]
Wolfram can actually prove it on its own: https://www.wolframalpha.com/input?i=floor%28n%2F2%29+%2B+ceil%28n%2F2%29+%3D+n

>>14531168
how about a recursive relation like t(n) = some constant + t(floor(n/2)) + t(ceil(n/2))? in this case you can't simplify it to t(n) or it would make the recursion infinite, right?

What is this field of logic called? I like doing these kinds of problems, and I can occasionally find an image like this that has problems similar to ones I've done before. However I can't get a definitive answer on the name of the field or types of problems. Propositional logic?

>>14531196
Right.
But in that case it might be easier (assuming t is monotonic increasing) to do [math]t(n) \leq \text{some constant} + 2 t(\lceil n / 2 \rceil)[/math]

[math]\int^{\infty}_{-\infty}cos(\alpha x) \, U(x) \, dx[/math]

Where [math]U(x)[/math] is a generic periodic function. Is it true that this integral is different from zero only if [math]\alpha = 0[/math]?
If so why?

>>14531027
no

>>14531224
Yes that's it. You can find a lot of those problems and more in Kleene's books.

When proving a conditional statement like If P then Q, is it correct to say that what you're actually stating is

"Q is true when P is true"? (and specifically not stating "When P is true, Q is true") or does the order not matter?

I am trying to rationalize contrapositive proofs where you say "Suppose Q is not true... therefore P is not true." This is stating "P is not true when Q is not true" right? Or is it stating "When P is not true, Q is not true"

Sorry I am confused

Let F be a field of characteristic 0. My book states that the nth cyclotomic polynomial over F is irreducible if and only if [math]Gal(F(\omega)/F) = (\mathbf{Z}/n)^\times[/math] where omega is a primitive nth root of unity. Why, though?

(not looking for a rigorous proof, just a short explanation)

Can someone explain how !A + AB = !AB

A calc I used said by using A + !AC = A+C rule, but my value is !A + AB, not A + !AC

>>14515149

What does the beta symbol mean here in this equation?

I left some of the text around it for context.
This is the paper
https://arxiv.org/pdf/2005.07513.pdf

Thank you

>>14531403
Not quite.
If [math]U(x)[/math] and [math]\cos (\alpha x)[/math] have the same period ([math]2 \alpha \pi x[/math]) then we can break up
[math]\displaystyle \int_{- \infty}^{\infty} \cos (\alpha x) U(x) \ dx = \sum_{k \in \mathbb{Z}} \int_{k 2 \alpha \pi}^{(k + 1)2 \alpha \pi} \cos (\alpha x) U(x) \ dx = \sum_{k \in \mathbb{Z}} \int_{0}^{2 \alpha \pi} \cos (\alpha x) U(x) \ dx[/math]
, so the integral either zeroes or diverges.

>>14515149
do isogametic species produce hormones? Do those that produce motile isogametes produce testosterone?

I feel dumb for not being good at math

Please rate these linear algebra textbooks:
Strang
Lang
Lay
Axler done right
Bretscher

Which one is the best for a beginner?

>>14531786
A -> B ≡ ¬A ∨ B
≡ B ∨ ¬A
≡ ¬¬B ∨ ¬A
≡ ¬B -> ¬A

>>14532504
radical. i never thought of the idea of adding a double negation

ty

Is store-bought hot chocolate mildly disgusting because they add weird shit to it everywhere or just where I live?

>>14531888
it aint, do a truth table
S = !A + AB
!S = !(!A + AB) = A(!(AB)) = A(!A+!B) = A!B
S = !(A!B) = !A + B

>>14532838
>store-bought hot chocolate
that could be a lot of things, you talkin the powder? or the much more economical and just as tasty method of hot milk + choco syrup?
try adding coffee creamer to it, shit is amazing.

>>14532852
>that could be a lot of things, you talkin the powder?
No, I mean the styrofoam cup beverage.

>>14532861
like, buying it prepared from a cafe or something?

>>14532871
Yeah but not cafe-cafes, store-cafes.

>>14532877
>store-cafes
like a grocery store? what country?

>>14532901
>like a grocery store?
Like gas station stores which sort of double as cafes but can also sometimes be found in the wilds detached from their gas stations.

>>14532908
i think youre thinking of "convenience store", and they dont sell hot chocolate in the states, or at least ive never seen them.

>>14532913
>i think youre thinking of "convenience store"
Oh yeah, that's the term.
>and they dont sell hot chocolate in the states, or at least ive never seen them.
Queer.
So they just sell coffee and/or have coffee vending machines?

>>14532930
ive never gotten coffee at a gas station before, but maybe the machines do have hot chocolate. this picture i found on google seems to suggest that.
also, an american wouldnt really call it a vending machine, it would be "coffee machine" or "coffee dispenser" or something. a vending machine is something you give money to.

is there an electric or electromagnetic equivalent/analog of the navier-stokes equations?

>>14532973
maxwells equations

is there a fluid dynamics equivalent/analog of the maxwells equations?

>>14533009
ohm's law

Where am I supposed to find a quiet spot to study and learn in peace if my own home isn't an option?

>>14533258
starbucks

>>14533278
I'm not a youth trying to use the bathroom.

>>14533291
you sure are picky for someone with home issues

>>14531888
~A+(A&B)/~(A&B)
https://programming.dojo.net.nz/study/truth-table-generator/index

>>14515149
I have a .tex files with 100s of chapters. I want some way to split the the chapters to different .tex files, and replace it with \include. How can I do it?

>>14515149
What does the asterisk represent in second order differential equations? I keep seeing things like y*(x) = (v_1)(y_1) + (v_2)(y_2), and my professor has even written "c_1*" where c_1 is one of the constants in the solution of a homogeneous differential function. Looking at the first equation I assume this has something to do with the connection between the constant terms in a homogeneous solution and it complementary u_1 or v_1 values in a non homogeneous equations solution. Is this right? Can't find anything online about it.

>>14533875
Forgot pic, sry.

Is a poor man's speedball ((dex)amphetamine + benzos) mid/long-term neurotoxic if you only do it once every couple months?
If you doubt that infrequent number: I do not enjoy this shit, especially not recreationally. It only happens incidentally during exam periods.

>>14532952
>also, an american wouldnt really call it a vending machine, it would be "coffee machine" or "coffee dispenser" or something. a vending machine is something you give money to.
I meant a coffee machine you give money to like in the picture.
Plastic/styrofoam ups are usually piled up to the side.

>>14532184
Nice, thank you

>>14532349
Same

How do you find good problem sets? I'm almost finished with prof Leonard's precalc, and starting Calculus soon. Where can I get good, hard problem sets? I find paul's math notes, while good, to have relatively easy problem sets. The questions my professor gives are harder. Thanks.

>>14533875
Whenever I've encountered it, it refers to the particular solution of an inhomogeneous ODE. For a linear ODE (or linear system of ODEs), the particular solution is the steady-state value when all derivatives are zero.

>>14533779
I'm not saying that there is no ready made solution.
But making a (Python) script is probably easiest.
Read the original file line by line, check if the line regex matches /^\\chapter/, break it all up into multiple files.

a car travelling at 70kmph breaks and stops after travelling 37,8m. what distance it would travel if its velocity was 60kmph?

not homework i just can't figure out im sorry again guys

>>14534571
Question makes no sense, I am newfag so maybe this is bait, but assuming the car at both speeds stops at 37.8m... then both cars will travel that distance. The slower one will take longer though.

>>14534571
Using [math]{v_f}^2-{v_0}^2 = 2as [/math] where v_f and v_0 are final and initial velocity respectively, a is the deceleration and s is the displacement, you can find out that a = -5.001 mps. Using the same equation again with initial velocity at 60 kmph will give 27.78 m.

>>14534650
cars dont undergo constant deceleration, its a function of velocity (overdamped oscillator, looks something like [math]v(t)=v_0 e^{-t}[/math])

>>14534718
Except that can't possibly be what they intended for the problem, because it says that the car stops.

>>14534727
he said it wasnt homework, i assumed it was a real world question

>>14535435
I've always read "not homework btw" as "this is a book exercise, yes, but I'm not being graded on it, hence asking for help isn't cheating".
I was going to point out that if it was a real problem anon should know the braking speed (parameter thingy) but I can't actually find that shit on google so apparently not.

What are the rules for proof by contradiction?

When there is a P and a Q, am i getting my "contradiction" from proving ~P & P, or is it from proving ~Q & Q?

The book says ~P --> (Q & ~Q) is logically equivalent to P. But i thought the point was to prove the conclusion Q.

So i thought a proof by contradiction was "P-->~Q", which is ~(~P v Q), which is P & ~Q. I took this to mean i was supposed to assume the hypothesis was true AND that ~Q is true.

And then by making both of these assumptions, i will prove ~Q is true, which contradicts the original Q of the original statement.

Do i have this all wrong? I'm so confused.

>>14535783
What, precisely, are you trying to prove?
>there is a P and a Q
isn't enough of a description.

>>14535810
There are a bunch of questions along the lines of "If P, then Q". But there are also questions that are just "P"

For instance i think a question like "prove sqrt(2) is irrational" would be a "P"

But then theres "If x^2 is even, then x is even." Which definitely seems "P->Q"

And proof by contradiction is used for both of them.

>>14535839
Oh and i forgot to ask, where does the Q & ~Q come from in "prove sqrt(2) is irrational"? If P is "sqrt(2) is irrational" then ~P must be "sqrt(2) is rational." But there's no Q & ~Q to prove.

>>14535435
Real world cars actually come to a stop.

Real-world deceleration comprises a mix of braking force (which results in roughly constant deceleration) and aerodynamic drag (which is proportional to the square of the speed). But unless you're driving a box truck with fucked brakes at 150 mph, braking force is significantly higher than aerodynamic drag.

>>14535875
>which results in roughly constant deceleration
how do you figure? the torque caused by the brake pads should be proportional to the velocity and the force of the brakes.
>Real world cars actually come to a stop.
dont be a smart ass

>>14535839
If you want to prove "P implies Q", then assume "P and not Q", and derive "Q or not P". That might be deriving "not P" or deriving "Q", either of which would imply "Q or not P", but not necessarily deriving either of them by themselves.

>>14535783
> The book says ~P --> (Q & ~Q) is logically equivalent to P. But i thought the point was to prove the conclusion Q.
No, the point is to prove the conclusion P.

(¬P -> _|_) -> P

There's no actual need to bring a second variable into it. (Q∧¬Q)≡_|_.

>>14535839
>But there are also questions that are just "P"
>For instance i think a question like "prove sqrt(2) is irrational" would be a "P"

Essentially, a proof has the form A∧B∧C |- X where A,B,C are premises and X is the conclusion. If there are no premises then X is a tautology. Proof by contradiction doesn't care /per se/ whether you have any premises. If you do, you can use them in proving ¬P->_|_.

P->Q is just an alternative way of writing (¬P)∨Q, but it's often used because it mimics P|-Q. At the meta level, (P|-Q)|-(P->Q), i.e. if you can prove Q from P then P->Q is a tautology. And conversely (P->Q)|-(P|-Q).

>>14535913
Thank you anon. Just so i fully understand what you mean, what is _|_ and |- mean?

I'm not sure i have seen it before

>>14535899
>the torque caused by the brake pads should be proportional to the velocity
That's not how friction works. Friction is largely independent of speed, beyond the fact kinetic friction (when speed is >0) is usually less than static friction (when speed = 0).

If it was actually proportional to speed, you really would get an exponential decay with the speed never actually reaching zero.

>>14535927
>what is _|_ and |- mean?

_|_ is "bottom" or "false". |- is "proof", X|-Y = "assuming X, Y can be proven".

>>14535929
>If it was actually proportional to speed, you really would get an exponential decay with the speed never actually reaching zero.
i disagree. friction is a macroscopic model of a microscopic phenomenon. saying that the car would never stop is like saying a spring would oscillate forever because its a decaying sinusoid.

so this is a dumb question but I would like clarification. Also I can't into Latex because I am dumb so I will use U for union and n for intersect. so P(M)=probability of owning microwave, P(C)= Probability of owning CD player. So P(M)=.75, P(C)=.25, P(MnC)=.16, the problem is asking for probability they have microwave or CD, but not both so I do P(M)+P(C)-2*P(MnC). My professor is telling me not to use the 2, but we are not considering the P(MnC) so we need to subtract off two of it. He told me "or" has to mean one or the other but I told him in the formula we have been given P(AUB)=P(A)+P(B)-P(AnB), the AUB is not an exclusive or, to which he told me he had never heard of exclusive or in his life. Am I wrong in including the 2? I feel like the fact that they are asking for not owning both means we need to subtract P(MnC) twice.

>>14536039
>friction is a macroscopic model of a microscopic phenomenon.
Doesn't change the fact that friction is much closer to constant than it is to being proportional to speed. There isn't a "standard" model of friction that's any better than F=μN (F = friction force, μ = coefficient of friction, N = normal force). Any more accurate model will be based upon empirical analysis of a specific system.