/sci/ - Science & Math

How does the below expression follow from the initial statement? What properties allow for this?

>>15379531
I can't fathom why you're not just looking at the statement of Theorem 2.1, but my guess is that it's probably the Fundamental Theorem of Calculus: you can visualize the shape as being chopped up into many thin rectangles, lined up from x=0 to x=2, and where the height of each rectangle is g(x)-f(x). Since this is a continuous function, the loose intuition described above works out nicely, and lets you use integration to work out the area of the shape. The rest is just routine calculation.

>>15379528
How can i prove that if f=O(g) that f compose g = O(g^2)? ive no idea what im doing

>>15380485
This is incorrect in general

>no title for the thread
>no list of unanswered questions from previous thread
>previous thread was made two weeks ago
Is this the lowest point for /sqt/ in recent history?

What is an effective way to show [math]\int f_n \to \int f[/math] for [math]f_n(x)=e^{-x\sqrt{n}}\left(1+\frac{x}{\sqrt{n}}\right)^n 1_{x\in[-\sqrt{n},\infty)}[/math] using Lebesgue's dominated convergence theorem?
I know [math]f_n(x)\to e^{-x^2/2}[/math] pointwise, but am struggling with a nice integrable bound for the [math]f_n[/math].

>>15380601
What do you mean? The problem says that. Mayby it's not f compose g and it's f * g.

>>15380763
Actually, [math]g^2[/math] probably means [math]g[/math] composed with [math]g[/math], so then it is true if [math]g[/math] is increasing.
If [math]f=O(g)[/math], then there is some constant [math]K[/math] such that for all large [math]x[/math], [math]f(x)\leq K g(x)[/math]. Then if [math]x[/math] is large enough, so is [math]g(x)[/math] so that [math]f(g(x)) \leq Kg(g(x))[/math].

Can I ask a programming question here? I can't find the answer on google or chatgpt (it's retarded):
How do I find the second smallest distance to a point, for points in an array, without looping twice?

Finding the minimum (smallest) is easy, just loop through while checking distance, but I can't think of a way without looping twice (finding minimum, removing it, finding minimum again)...

>>15380856
Loop through the array as usual, but store the two smallest elements you've seen so far (just make sure to update them properly).

>>15380856
uhhh, something like this?
current_min = Inf
second_min = Inf
for x in array:
second_min = min(distance(x, point), second_min) if second_min > current_min else second_min
current_min = min(distance(x, point), current_min)

>>15380856
for x in array { if x < min_1 then min_1 = x; else if x < min_2 then min_2 = x; }

>>15379994
Ah, yes, forgive me for being an idiot. The theorem is right here. Does it make the problem any clearer?

>>15381187
The problem was already clear. The area under the curve is the integral so the difference in areas is the difference of the two integrals.

>>15381199
Well, yes, but how does the expression simplify to
[eqn]
\int^2_0 (\frac{5}{2}x - x^2) dx
[/eqn]
from where it started?

>>15381218
wtf? That's the easy part.

[math]g(x) - f(x) = x/2 - x(x - 2) = x/2 - x^2 + 2x = 5x/2 - x^2[/math]

>>15381241
I have no justification for being this stupid, I'm just going to thank you for your patience and hope your life improves from wherever it is now for your kindness.

Can anyone recommend me a good book or source for a review of trig? I'm taking a calc course soon and haven't done either calculus or trig in years. I have a rough grasp on the basic concepts still, but definitely am a bit fuzzy on it overall. But I also dont want to spend too much time and bog myself down on every little detail.

>>15381487
Have you looked at khanacademy?

I need to find an equality for [math]\varepsilon[/math] in terms of [math]x[/math], [math]y[/math] and [math]\gamma[/math]. I'm stuck on this problem for hours and I can't make any progress, any help would be appreciated

[eqn]
\begin{cases}

|x_n-x|<\varepsilon\\
|y_n-y|<\varepsilon\\
|(x_n+y_n) - (x+y)|<\gamma\\
|(x_n-y_n) - (x-y)| < \gamma\\
\left| \frac{x_n}{y_n} - \frac{x}{y} \right| < \gamma\\
|x_ny_n-xy|<\gamma

\end{cases}
[/eqn]

>>15382352
>An inequality for [math]\varepsilon[/math]
You already have two, and it appears [math]\varepsilon[/math] is unbounded from above. What are you trying to do?

>>15382389
I need an equality, not an inequality

>>15382393
That's not possible I'm afraid.

>>15382419
it has to be possible

>>15382440
Why do you think so?

>>15382442
because otherwise my professor wouldn't have given this problem to us

>>15382456
*shrug* If [math]y_n=y=1[/math] your latter four inequalities just state [math]|x_n-x|<\gamma[/math] and that doesn't imply anything about epsilon.

I have an expression of the form [math]\frac{\partial Y}{\partial x}dx[/math]
When I've finished finding the partial derivative I'm left with that bare differential with no corresponding differential in the denominator. What am I supposed to do with it

>>15380868
Useless answer, the point is *how* to store the smallest elements in 1 loop.

>>15380869
Even after fixing this so that 'second_min > current_min' works (it wouldn't because they're both INF at the start) it fails when then min is the first element, e.g:
var arr := [3, 6, 8, 11, 15]
var x :int = 4

>>15380931
Fails if the 2nd min is before the 1st min, e.g:
var arr := [1, 6, 8, 11, 15]
var x :int = 4

Hope you guys aren't this useless for other questions.

>>15382352
>>15382456
Nobody would have any clue why your professor is doing anything without the context behind the question. Other than by guessing. Such as what class you're in, what tools or what notes you've taken relevant to the question, what your professor is wanting you to do or use?

You haven't made it very easy to help you.

>>15382352
Are these inequalities part of a system of equations
or are each of them to be taken somewhat
separately?

>>15382508
>Hope you guys aren't this useless for other questions.
You get what you pay for.

>>15382508
> var x :int = 4
wtf is that supposed to be doing?? if you set the initial value to garbage ofc you will get garbage out.

> Fails if the 2nd min is before the 1st min
No it doesn't. 1st min would be 1, 2nd min would be 6.

>>15382496
>>15382510
there are some other informations on the problem, but it basically nails down to that system of inequalities. there is:

(a) if [math]a,\;a_1,\;b,\;b_1[/math] are real numbers with [math]a,b[/math] not zero and
[eqn]
|a-a_1|<\varepsilon \quad , \quad |b-b_1|<\varepsilon
[/eqn]
then
[eqn]
|(a+b)-(a_1+b_1)|<2\varepsilon \; , \; |(a-b)-(a_1-b_1)|<2\varepsilon
\; .[/eqn]

(b) if the real numbers [math]x,y[/math] are such that [math]|x|<a,\; |y|<b[/math] and [math]\Delta x \; , \; \Delta y[/math] are both less than [math]\varepsilon[/math], with [math]\varepsilon<1[/math], then
[eqn]
|(x+\Delta x)(y+\Delta y) - xy| < (a+b+1)\varepsilon \; .
[/eqn]

(c) if the real numbers [math]x,y[/math] are such that [math]y\neq 0[/math] and if [math]\Delta x, \Delta y[/math] are such that [math]|\Delta x|<\varepsilon, \; |\Delta y|<\varepsilon[/math], with [math]\varepsilon < \frac{|y|}{2}[/math], then
[eqn]
\left| \frac{x+\Delta x}{y+ \Delta y} - \frac{x}{y} \right| < \varepsilon\frac{2(|x|+|y|)}{|y|^2} \; .
[/eqn]

now, suppose that [math](x_n)_{n\in\mathbb{N}}[/math] and [math](y_n)_{n\in\mathbb{N}}[/math] are sequences of real numbers with
[eqn]
\lim_{n\to\infty}x_n=x \; \mathrm{and} \; \lim_{n\to\infty}y_n=y
[/eqn]
i.e., for any [math]\varepsilon>0[/math] we can obtain a [math]n_0[/math] s.t. for all [math]n>n_0[/math] we have [math]|x_n-x|<\varepsilon\; \mathrm{and} \; |y_n-y|<\varepsilon[/math]. now suppose that [math]|y|>0[/math] and [math]\gamma[/math] is a strictly positive number. the task is to find an equality for [math]\varepsilon[/math] in terms of [math]x,y,\gamma[/math] such that the inequalities

[eqn]
|(x_n+y_n)-(x+y)| < \gamma,\; |(x_n-y_n)-(x-y)| < \gamma,\; \left|\frac{x_n}{y_n}-\frac{x}{y}\right|<\gamma,\; |(x_ny_n)-(xy)|<\gamma
[/eqn]
are simultaneously true

>>15382536
I don't get what you mean by taking them separately

>>15382562
Fails for arr = [1, 6, 8, 11, 15], point = 4. Gives 2 for both.

>>15382760
for x in array { d = distance(point,x); if d < min_1 then min_2 = min_1, min_1 = d; else if d < min_2 then min_2 = d; }

>>15382701
I see what you mean. You have bounded a sum, product, difference and ratio with an arbitrary epsilon, and you now want one that works for all at the same time. You can just take the minimum of whatever: [math]\varepsilon = \min\{\frac{\gamma}{2}, \frac12, \frac{\gamma}{|x|+|y|+1}, \frac{|y|}{2}, \frac{\gamma|y|^2}{2(|x|+|y|)}\}[/math].

>>15382813
Well done, thanks.

>>15382701
Oh, okay, that helps. The equality should then be like an iterated limit, something given by the likes of the moore-osgood theorem. You're doing multivariate calculus or calc 3, right? I think this kind of thing is the relevant text to help
https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/01%3A_Limits/1.02%3A_Epsilon-Delta_Definition_of_a_Limit
https://en.wikipedia.org/wiki/Iterated_limit

Or are you before you'd be doing epsilon-delta limits? I don't want to solve your problem for you so I'm trying to be more roundabout in giving suggestions to help you figure out what you have to be reading. Though the professor and class, and your class textbook, should all already be doing that? Or am I wrong in supposing that is what your'e doing? I guess you mean this due to stating you need to find an equality, and the relevant equality or one of them could be an iterated limit.

Stupid question.
Is there any sort of quick and simple chemical testing I can do to test for the presence of Adderall in drinks like milk or water?

For the Human Genome Project, they used reference genomes to piece it all together. They used different anonymous donors, both male and female, to make a mosaic genome.

Is there anywhere where a single human specimen genome can be accessed from these donations? Even if it's a mosaic, is there any way someone could read the DNA and "recreate" a simulacrum of a human being by sussing out the alleles?
>t. knows fuck about genetics

>[math] H\left(Y\mid X\right) \;\leq\; H\left(Y\right) [/math] with equality iff [math] X [/math] and [math] Y [/math] are independent
I managed to prove the first part for the general case (entropies of σ-algebras). Does the equality case generalize as well? What do I need to know to prove it?

>>15382701
>>15382536
What I mean is you take the first two questions
and use it to solve out the fourth question, for
example. And the fifth question is independent
to the fourth--it's done separately.

Intro to physics (calculus based) is making me want to kms. Any tips on how to wrap my head around this shit?

>>15383591
Too general of a question to answer.

>>15382873
but how do I get to this? do you have any tips?
>>15382930
>The equality should then be like an iterated limit
I don't get it
>You're doing multivariate calculus or calc 3, right?
the calculus division is different in my country, but I guess this is supposed to be single variable calculus
>>15383577
oh I see. that's how I first approached the problem, but I didn't get any progress

Do physicists have a single term for both energy and matter? Matter is just clumped energy, right? Ought to have one umbrella term then.

>>15384067
That would be incredibly stupid since physically they are very different. It would be like using the same term for ice, water, and steam.

How to show [math]\cos x + \frac{2p}{\pi}\sin x \geq 1-2\left(\frac{x}{\pi}\right)^p[/math] for all [math]0\leq x\leq \pi, p\in\mathbb R[/math]? I am getting filtered.

>>15384003
What I'm doing is essentially the following: if for any [math]\varepsilon > 0[/math], [math]|a| < K_1\varepsilon[/math] and [math]|b| < K_2\varepsilon[/math], then if *you* choose [math]\varepsilon = \min\{\frac{\gamma}{K_1}, \frac{\gamma}{K_2}\}[/math] then you find that [math]|a| < K_1\varepsilon \leq K_1 \frac{\gamma}{K_1} = \gamma[/math] and [math]|b| < K_2\varepsilon \leq K_2 \frac{\gamma}{K_2} = \gamma[/math] simultaneously.

If I don't trust myself to make an SDS-PA gel right the first time, is there any harm in running it with just one lane of ladder, examining it to make sure it's good, and then loading it with samples and a new ladder? This might give me confidence that I'm not wasting sample on a fucked gel.

>>15384067
Energy (of which the rest energy is a part)
When one talks about the energy of a system or object, usually only the energy considered extractable is implied.
Example: a spring-mass system. It's at some temperature, but none of the components can convert thermal energy in any significant way, so we consider the total energy of the system to include the gravitational potential energy of the mass, the elastic potential energy of the spring, and the kinetic energy of the mass, but not the thermal energy of the components, the rest energy of the mass, or anything else.

>>15384003
>the calculus division is different in my country, but I guess this is supposed to be single variable calculus
I wasn't wrong then but perhaps the calc 3 reference was too much. Here's a calc-1 reference with example problems and walkthroughs. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Seeburger)/02%3A_Limits/2.07%3A_The_Precise_Definition_of_a_Limit
>>15384316
You completely threw me off in the first place by saying "equality" instead of "true".
Your notes here explain every single step >>15382701
Do (a), then (b), then (c). I'm not sure given the complete notes where you're having trouble. Step (a)? Steb (b)? Step (c)? The steps are nearly totally linear from steps (a) to (c) in your first post >>15382352
Literally all you have to do is exactly that in that order of the information you just gave. You need to get real specific about where you get lost. What I recommend is checking the libretext page I just linked first, and solve it as shown there.

>>15384436
These are my posts:
>>15384316
>>15382873
I didn't ask the question, but re-reading my post maybe that wasn't very clear.

>>15384449
Whups clicked the wrong post but you know who I meant anyhow. My bad.

>>15384316
but what are the [math]K_1,K_2[/math]? and why can I assume that
>[math]\varepsilon = \mathrm{min}\{\frac{\gamma}{K_1},\frac{\gamma}{K_2}\}[/math]
?
>>15384436
thank you for the reference.
as far as I understand the problem, I have to take (a), (b) and (c) as truth and solve the system of inequalities, but I don't really see how they are related

>>15383016
Amphetamines show up in piss, so try a typical urine dipstick in some positive controls of various doses
Is someone making you take your meds?

>>15382507
Integrate it for some set of x, or compare it to some other expression with a differential element

I can do pretty much all problems regarding Taylor and MacLaurin series except for these, I have no idea what they're asking, the Swedish translates to "Show that [INEQUALITY] if abs(x)le1, they have some sort of theta times x in the solution and a bunch of blabber but I just don't understand what and how I'm supposed to do

>>15384731
Since [math]e^x = \sum_{n\geq0}\frac{x^n}{n!}[/math], you can write [math]\left|e^x-\sum_{n=0}^4\right| = \left|\sum_{n\geq5}\frac{x^n}{n!}\right| \leq \sum_{n\geq5}\frac{|x|^n}{n!}\leq |x|^5\sum_{n\geq5}\frac{1}{n!}=|x|^5(e-\frac{65}{24})<\frac{|x|^5}{40}[/math].

>>15384781
Why 1/40? That seems arbitrary especially since e - 65/24 ~ 1/100

Prerequisites for fluid dynamics? I have a background in math but have zero knowledge of physics and need to learn fluid dynamics for various reasons. What's the best text for this given my background and are there any good supplementary texts?

>>15384531
So I could use a urine test on the drink? I wasn't sure if they actually detected the drug or the biological biproducts.
More like a vindictive roommate is deciding to really fuck with me. Luckily I'm hell outta dodge soon.

Any hints? This is prolly not the right place to post but can't find another ECE thread so

>>15385694
See question

>>15384731
>>15384781
>>15385694
Similarly you can show for the other one that |ln(1+x)-x+x^2/2|<=(ln(256)-5)|x|^3 and ln(256)-5<8/3 by quite a lot. There are probably techniques taught in the course or book that I don't know that are easier but more 'loose'.

>>15387006
I can see that's the question, my question was why that specific value. For a rigorous math question it appears completely random.

>>15387068
I assume the instructor has some particular method in mind to bound tail sums like those, instead of just calculating them.

Suppose X is a set and C is a collection of subsets of X which is stable under finite intersections. Is there a nice characterization of the algebra (as opposed to sigma algebra) generated by C?

>>15387626
nvm, I've found what I was looking for: the algebra generated by C is the set of all finite unions of finite intersections of sets in C or their complements (and this applies to any collection C of subsets of X, not necessarily stable under finite intersections).

is it
[eqn]
\dot{\vec r}
[/eqn]
or
[eqn]
\vec{\dot r}
[/eqn]

>>15388588
The top one.

You can have the derivative of a vector function,
but not the vector of the derivative of a function...
unless you're making a gradient vector.

>>15386681
Is this a verilog/vhdl exercise for babbys first digital logic course; what have you tried so far?
>>>/diy/mcg
>>>/diy/ohm

What is a function that takes its argument k and:
yields 1 if k < 0
and
yields 0 if k >= 0
?

I'm not be interested in a piecewise function, because that would be trivial.

>>15389184
[math] \dfrac{|k|-k}{2} [/math]

>>15379528
how to prepare for physics 2 AP test?
Should i just grind more practice tests? read the textbook?

>>15389586
>>15389184

It's actually [math] |k|-k \over 2k [/math] that does
the trick.

>>15389148
>Is this a verilog/vhdl exercise for babbys first digital logic course
Nope we haven't learnt it yet.
>what have you tried so far?
I think I figured out what to do for each option.
For 1) we can just pass the input into the output.
2) Use an OR gate
3) Use an AND gate
4) Make a half adder (?)
But I just can't figure out how to use AB. Maybe a mux would do the job but what would the circuit diagram look like?

Show that if f = O(g) and
g = O(h), then f = O(h).

Studying right now, and theres lots of questions of that type. I do not know how to do these types of problems, id appreciate if someone could solve that for me. theres a few others that are more complex, but i think if i see one worked out it'll click. my attempt was some crap with the limit definition, but it was just me throwing shit at the wall.

>>15389773
Big O notation can be ambiguous if not interpreted in the right context. I'm gonna assume that [math]f,g,h[/math] are all functions of the form [math]\mathbb{R} \to [0, \infty)[/math] and that [math]f = O(g)[/math] stands for "there exists a positive constant [math]C[/math] and positive number [math]N[/math] such that for all real [math]x \ge N[/math] we have [math]f(x) \le C g(x)[/math]". Your definition may differ, in which case I'll leave it to you to make the necessary minor adjustments.

So, assume [math]f(x) = O(g(x))[/math] and [math]g(x) = O(h(x))[/math] as [math]x \to \infty[/math]. Then there exist positive constants [math]C_1, C_2[/math] and positive numbers [math]N_1 , N_2[/math] such that whenever [math]x \ge N_1[/math] we have [math]f(x) \le C_1 g(x)[/math] and whenever [math]x \ge N_2[/math] we have [math]g(x) \le C_2 h(x)[/math]. Then for any [math]x \ge \max \{N_1, N_2\}[/math] we have [math]f(x) \le C_1 g(x) \le C_1 C_2 h(x)[/math], which shows that [math]f(x) = O(h(x))[/math] as [math]x \to \infty[/math].

>>15389855
Thank you anon :) , so I need to use the other definition instead of limits

>>15389643
Thanks. That's close, but it's undefined when k = 0, because it performs division by 0.

It was easy for me to find a function that works for all numbers ≠ 0. The thing I couldn't figure out was how to make it work for all numbers including 0.

Any ideas?

To recap, this is the problem:

What is a function that takes its argument k and:
yields 1 if k < 0
and
yields 0 if k >= 0
?

I'm not be interested in a piecewise function, because that would be trivial.

>>15390116
> I'm not be interested in a piecewise function, because that would be trivial.
Why the fuck not? The ternary operator is intrinsic to most languages.

Otherwise just extract the most significant bit of the two's complement binary representation.

>>15389675
Play around with different ideas in logisim

>>15390252
>logisim
People don't just visualize the circuit I'm their head. Wtf?

>>15390281
we mostly use the Steam game actually

Hello, can someone suggest me an alternative to AOPS books? I can't really tolerate their pdfs quality being absolute sht.
>buy them
bro I'm from Arg and our economy is tanking (finally)

>>15390116
>What is a function that takes its argument k and:
>Gives piecewise definition of a function
>I'm not be interested in a piecewise function
What the fuck

omg bros I think I just got it

are gluons called gluons because the strong nuclear force GLUES the nuclear protons together?

Does anyone know some basic asymptotic analysis? In picrel, why can we just assume that [math]k=k(n)=\Theta(\sqrt{n})[/math] as some specific function, when it is a dummy variable and not technically part of [math]f[/math]?

>>15390885
Answering my own question; let [math]\ell(n)=\sqrt{n}\cdot n^{1/100}[/math]. Then, since [math]k=o(n^{2/3})[/math] implies [math]\frac{(n)_k}{n^k}\sim e^{-\frac{k^2}{2n}[/math], we can split [math]f(n)=\sum_{k=1}^{\ell(n)}\frac{(n)_k}{n^k} + \sum_{k=\ell(n)+1}^{n}\frac{(n)_k}{n^k}=:(1)+(2)[/math]. Then [math](1)\sim\sum_{k=1}^{\ell(n)}e^{-\frac{k^2}{2n}}\sim\int_0^\infty e^{-\frac{k^2}{2n}}\, d k \sim \sqrt{n\pi/2}[/math] as in the question. Finally, [math](2)\leq (n-\ell(n)-1)\frac{(n)_k}{n^k}\sim (n-\ell(n)-1) e^{-\frac12 n^{1/50}}\sim 0[/math].
I still don't get their solution but that's ok.

i need some tips on how to define languages recursively. E={a,b,c}, x is a subset of E*, x contains all words where two letters dont repeat. I found a few stack over flow threads, and they mention finding a pattern and working with that - ive wrote a few of the words but im not sure how to turn that into a defintion

>>15391784
oops, i mean x consists of all words where no two consecutive letters are the
same.

So I saw this and I'm thinking about infinity.

If .9...=1, then wouldn't .8...9... (That's .8 repeating infinitely followed by 9 repeating infinitely.) equal .8...9, meaning that it would be turning an infinite number finite. And then wouldn't this mean that it's actually subtracting an infinite amount since the end result is infinitely smaller than the starting amount? Like, can you have multiple infinities, or is it just infinity as a concept at that point? But then wouldn't that mean all infinities are equal, which would make all numbers equal if this picture is true? Where do I start learning about infinity? Is infinity, like, a waveform that collapses once it's interacted with?

>>15391953
> That's .8 repeating infinitely followed by 9 repeating infinitely
that's a nonsense statement, you can't write such a number.

>>15392026
First off, I just did. Second off, I was told that 0.99... is the same as 1-1/10^n where n = infinity Is that right?
Wouldn't .8...+((1-1/10^n)/10^n) where n=infinity equal my made-up number?

>>15392026
>>15392095
Actually wait, I think I'm getting a bit ahead of myself. What would (1-1/10^n)/10^n where n=infinity be? Would that be an infinite number of zeroes followed by an infinite number of nines?

>>15392095
Essentially what ".8 repeating infinitely followed by 9 repeating infinitely" means is that you have a number [math]x[/math] where for all [math]n < \infty[/math] the [math]n[/math]th digit is 8 and all other digits are 9. But there are no other digits so all the digits are 8.
In order for there to be a 9 in [math]x[/math] then there must be a number [math]n[/math] such that [math] n > \infty[/math] which is obviously false.
In simpler terms, because there are already an infinite number of 8's, there is no room left for the 9's.

>I was told that 0.99... is the same as 1-1/10^n where n = infinity Is that right?
That's close, but since infinity isn't really a number there is no way to evaluate this expression. What is actually done is to consider the sequence of 1-1/10^n for all n (the natural numbers) and evaluate the limit as n goes to infinity.

If I answered IQ test questions randomly, what is the expected score of my IQ?

>>15392174
>such that n>∞ which is obviously false.
Is it? If that's false, then ∞+2=∞ wouldn't it? And then we could use that to simply break math by proving all numbers equal each other. Or what about x*∞ versus X^∞? Are these numbers equal? They certainly wouldn't look equal on a graph and wouldn't be able to be used interchangeable, which already implies that there are at least different infinities. How can they be different but also equal unless there's something else going on? There are two other explanations I can think of - One is that ∞ is an 'active' variable and changes every time it's interacted with, which would make it very difficult to use in equations and proofs, and the other is that we simply don't understand infinity and shouldn't use it because of that.

What's that hotel paradox, "A hotel has infinite rooms, but an infinite number of guests are staying there. Can you get a room?" Is there an answer for that? Like Zeno's paradox wasn't meant to prove that the race was unwinnable, it was to show the absurdity of math being unable to answer it at the time.

>>15392634
If it was a real IQ test, then the test would show as inconclusive as they ask the same question multiple times and the very different answers would show that you were giving random answers. Remember that these are often given to uncooperative and developmentally disabled children.

>>15390252
do you mean something like this?

>>15392833
And using a mux

I'm still on this infinity thing. Is (1/10^n) where n=infinity be the smallest possible positive positive number? Is there a name for that? If it's not, what is the smallest number?

>>15393109
Sorry, positve nonzero, not positive positive.

>>15393109
the set of positive nonzero reals does not have a smallest element, because it verifies the Archimedean property which guarantees that for any given real [math]x[/math], there is another real [math]y[/math] such as [math]x>y[/math]

>>15393125
But there would still have to be smallest increments, right? Zero is a real number, isn't it? So (1/10^infinity) wouldn't interfere with that property at all since it would be larger than 0. And wouldn't these increments equal (1/10^n) where n=infinity? Is the issue that introducing infinity changes the property of numbers in general since it's not a number itself? The use of infinity makes this an imaginary number anyway, doesn't it? Or putting it another way, isn't .3... an impossible paradox? We know the last number is 3, but we also know it has no 'last' number. So how can we know that which doesn't exist? There's one... like "unit of infinity" itself sort of tacked onto it, right? Should infinity be a property like positivity or negativity?

Or I guess this is getting beyond the /sqt/ so where should I start if I want to learn about infinity?

>>15393149
> So (1/10^infinity) wouldn't interfere with that property at all since it would be larger than 0
No, it would be zero.

>>15392780
>Is it?
yes, by definition.
I think you have a misunderstanding of infinity. In this context it is simply an object that is larger than all other natural numbers. So for all numbers [math] n [/math] we have [math] n < \infty [/math].
Using your example we can demonstrate that infinity is not a natural (or real) number.

>what about x*∞ versus X^∞?
Again since infinity is not a number we cannot evaluate these expressions. However we can evaluate the sequences x * n and x^n as n gets larger and larger.

>They certainly wouldn't look equal on a graph
this is a good observation, however we need to be precise with what we are considering "equal" particularly because a graph will never include infinity (it can't), so we are not comparing these two "values" directly.
In some sense they are the same since they both share the property of being unbounded. That is we cannot find a real value [math] b [/math] such that [math] a(n) < b [/math] for all numbers [math]n[/math].
However if we were to consider them equal only if they were asymptotically equal (look it up). Then we have a more precise way to verify your observation.

We don't have multiple infinities, rather our initial definition of infinity was insufficient in explaining the observation that we made. This is because we needed to be more explicit in defining the property we were looking at.

>What's that hotel paradox
Hilbert's hotel is actually a great way to understand your problem. If the hotel is already filled with 8's and we want to add a 9 what room number can we give it? any room we give will already have an 8.

>>15391953
> .8...9...
> (That's .8 repeating infinitely followed by 9 repeating infinitely.)
>>15392103

In a real number, each digit must be in a specific, well-defined digit position.

(In other words, each digit in a real number corresponds to the term k*10^n in the infinite series that constructs the number, where k is an integer from 0 to 9, and n is some integer.)

The problem with your number is that the "9" is not in a specific, well-defined digit position. In other words, that "9" would require the term 9*10^n for a transfinite value of n. The fact that n is transfinite (i.e. is not an integer) violates the requirement for a real number.

tl;dr: You're free to fantasize about a digit being in a transfinite position if you want, but you can't call the result a "real number".

>>15393308
>We don't have multiple infinities

It's okay to say it that way in an elementary school classroom.

However, if you want to be more formal, and avoid potential confusion, it's better to say this:

Infinite sets can have different cardinalities. For example, the cardinality of the integers is strictly "less than" the cardinality of the reals (according to Cantor's definition of "less than"), and, hence, they are two provably different cardinalities. We call both cardinalities "infinite", and in calculus, the symbol "∞" can refer to either one. However, when studying the cardinality of transfinite sets, we don't use the "∞" symbol or the word "infinity" because of their ambiguity. Instead, we call the different transfinite cardinalities "aleph0", "aleph1", "aleph2", etc., where "alephN+1" is defined as the cardinality of the power set of "alephN", for all natural numbers N≥0, and "aleph0" is defined as the cardinality of the integers. This creates an endless sequence of different transfinite cardinalities, each one provably "less than" the next.

A student who is unaware of the need for precise, exacting terminology in this area of mathematics might say (informally) that "there are multiple infinities". That student's mental model of the situation wouldn't exactly be wrong, but they should be corrected to use the right terminology, as described in the previous paragraph.

>>15393308
>Hilbert's hotel is actually a great way to understand your problem. If the hotel is already filled with 8's and we want to add a 9 what room number can we give it? any room we give will already have an 8.

Well, to apply Hilbert's trick, you would start by putting all the 9s in the hotel first. (I.e. you build the number from right-to-left instead of left-to-right.) Then, with all the 9s in the hotel, you can now insert an 8 by moving all the 9s to their next higher room number, and put the 8 in the (now empty) room number 0. So the hotel definitely has room for both 8s and 9s.

However, you can only accommodate a finite number of 8s that way and still keep your 9s. If you stuff an infinite number of 8s into the hotel, then all of your 9s get "pushed out", so none of your 9s have a specific integer room number anymore. Every digit must be in a specific integer room number, and if none of your 9s have a specific integer room number anymore, then that means they all got evicted from the hotel. So the resulting number would simply be 0.8888...

tl;dr: The real number system cannot distinguish 0.888...999... from 0.888...

>>15392891
you're probably expected to make your own mux

>>15394021
Yeah I guess that wasn't the best way to phrase it, but what I was actually trying to say is that [math] x * \infty [/math] and [math] x ^ \infty [/math] don't represent different infinities in the traditional understanding of infinity (that they are boundless).
I should have been more clear because I was also showing how it is reasonable to say that [math] x ^ \infty [/math] is a larger infinity than [math] x * \infty [/math] as long as you are explicit that you are making an asymptotic comparison.

>>15379528
Why is this true? I can't find any reason for it to be.
[math]E_m[/math] is magnetic energy
[math]B[/math] is magnetic flux density
[math]H[/math] is magnetic field strength
[math]J[/math] is current density

>>15394953
Why what? It's just using the identity [math]\nabla\cdot\left(\mathbf{A}\times\mathbf{B}\right)= (\nabla\times\mathbf{A})\cdot \mathbf{B}-(\nabla\times\mathbf{B})\cdot \mathbf{A}[/math]

>>15394991
Oh thanks. I didn't know what to look up and when I tried proving it I forgot I can use the product rule.

Ok bros I'm getting crazy on this one. I'm in a measure theory setting so the extended real line is alright as long as you don't have [math]\infty - \infty[/math]. But suppose I have $h(x)=g(x)=\infty$. How can I justify that the distance between [math]h,g[/math] is actually zero? (I am using uniform metric on compact intervals).

>>15396358
nvm solved, this case can't happen by construction in my problem

Hi guys, I've been trying to read von Neumann's/Morgenstern's Theory of Games book and I'm a bit stumped on the second axiom of combining (see image). Why doesn't the substitution work when you try to plug in [math]\beta u + (1 - \beta) v[/math] into v in the expression [math]\alpha u + (1 - \alpha) v[/math]? Am I big dumbo?

>>15396530
Do you mean the algebra doesn't seem to work out?

>>15396530
Pretty sure with this substitution you'd get [math]\gamma=(1-\alpha)(1-\beta)[/math].

>>15396530
[math]\gamma=1-(1-\alpha)(1-\beta)[/math]

Let [math](\Omega, \mu)[/math] be a probability space. The function [math]d_\mu (A,B) = \mu(A \Delta B)[/math] is a metric on the space of equivalence classes of measurable sets modulo null sets. Is this metric complete?

Attempt: Fix a cauchy sequence [math](A_n)[/math] (strictly speaking these aren't sets, but let's ignore this technicality). Let [math]A = \bigcup_{n \ge 1} \bigcap_{k \ge n} A_k[/math] be its setwise lim inf, which seems like a plausible candidate for the [math]d_\mu[/math] limit. We have the easy bound [math]d_\mu (A_i , A) \le \mu(\bigcup_{n \ge 1} (A_i \Delta \bigcap_{k \ge n} A_k))[/math]. How can I proceed from here?

>>15396544
Yes, here are the steps I've taken:

[math] \alpha u + (1 - \alpha)(\beta u + (1 - \beta) v) [/math]
[math] \alpha u + (1 - \alpha)(\beta u + v - \beta v) [/math]
[math] \alpha u + (\beta u + v - \beta v) + (- \alpha \beta u - \alpha v + \alpha \beta v)[/math]
[math] (\alpha + \beta - \alpha \beta) u + (1 - \alpha - \beta + \alpha \beta) v[/math]

or equivalently:

[math] (\alpha + \beta - \gamma) u + (1 - \alpha - \beta + \gamma) v[/math]

And the way I see getting at the form [math] \gamma u + (1- \gamma) v[/math] is if [math] \alpha + \beta = 2 \gamma [/math] which is nonsense.

>>15396556
This was helpful. I'm guessing this means that the gamma for the compounding lottery the first way is not necessarily equal to gamma in the second way?

>>15396647
>>15396530
You're doing it in reverse, basically.

>>15396621
Some thoughts: your metric space can be embedded into [math]L^1[/math] (you are essentially looking at indicator functions only) and since that is complete you only have to show these indicator functions converge to another indicator, which shouldn't be too tough.

how much force do we experience from the centripetal force of the earth orbiting the sun

>>15396919
The force will be given by [math]mr\omega^2[/math]. r is the distance between the sun and earth, the angular velocity comes from how long it takes the earth to orbit the sun. Plug in all the numbers and you get roughly 0.006 m/s2 per kg.

>>15396844
Thanks anon. I'm sure your suggestion works, it's just that I haven't gotten to [math]L^1[/math] theory yet in my studies. If, by chance, you have a more direct, combinatorial proof in mind, I'd love to hear it. Thanks again anyhow.

Let's say I have a function described by some power series. Borel's theorem says that every power series is the Taylor series of some function. Yet, a lot of the time these power series have a limited range of convergence. I can only ever see a small fraction of the actual function. To "shift" the window over on the function requires you to know the derivatives of that function (as every point of the function corresponds to a separate Taylor series parameterized by the function's derivatives). At that point you might as well already have an expression of the function, which defeats the point.

With all that being said, is there some general method that would allow me to "shift the window of convergence" around on a power function, so that I can see more of the function it describes?

>>15396796
Well yes, but I struggle to see the physical meaning of the differences in gamma due to the rearrangement. I can see using the tree diagram the difference between the form of the compound lotteries (and this is intuitive) but don't get why the compound preference is [math]1- (1- \alpha)(1-\beta)[/math].

>>15397112
gamma is what is 'allocated' to u. On the left it's alpha*beta, when on the right we allocate (1-alpha)(1-beta) to v, so 1 minus this to u again.

>>15397025
Analytic continuation iirc

>>15397025
You got to change the center of the Taylor series.
Make the center a slider on desmos and it'll be
a little window seeing the breadth of the function.

>>15397186
Aaaah, got it. Thanks!

Can you mathfags pls help me with >>15396951
and my replies thereafter? I suddenly feel like I cannot do algebra because I'm not 100% certain of how the notation is defined.

>>15397237
that would just shift the plot left or right. The idea is that while it is a Taylor series, every single point on the function is its own Taylor series, and those Taylor series can't really be ascertained from any of the others (I suspect)

>>15397205
oh boy this will be fun

>>15397320
>>15397237
Yeah, you're right. The series keeps its shape but
shifts horizontally from the change of center.
How about shifting the function and getting its
series at the same center (at zero)?

>>15397499
That's the problem. To shift the function, we need to have an exact description of all of its derivatives. If you know all the derivatives, you know the entire function (from the definition of a Taylor series). But, we don't know the function since all we have is a power series, that corresponds to a very specific point of that function. But not the function itself.

https://www.desmos.com/calculator/d0e3ki0cby
This graph is exactly what I want to do. Notice, though, that I'm only able to make this animation because I know explicitly what function it is.
Notice that the coefficients of g(x), are [math]f'(a), f''(a), f'''(a)...[/math]. Keep in mind, these are not derivative functions. They are *constants*, from evaluating the derivative at a specific point. To know these constants, at a point, for every point, is to know the function itself.
If we want to shift the window of the power series over, we would need to find [math]f'(a+d), f''(a+d), f'''(a+d)...[/math]. These derivative *functions* must necessarily be different derivative functions from the previous ones

that's what makes this so tricky. At this point, I'm beginning to wonder if there's some predictive way to come up with a "best guess" continuation of the derivatives, until it finally diverges too much from the original function.

can you tell I've spent a lot of time thinking about this lol

>>15397592
my original original idea from years back was to fudge the function like so: a Taylor series is always exactly equal to the function, at the center of convergence (because the first term of a Taylor series is always just [math]f(a)[/math]). So, if we just scoot the Taylor series over from [math]-\infty[/math] to [math]\infty[/math], plotting the value at the center of convergence [math]f(a) \forall a[/math], we would have the function.
See here:
https://www.desmos.com/calculator/tu3vppf2om
but that's when I realized we would need to know the function, to find the function, and we're stuck

>dude science lmao sleeping 6 hrs is as bad as not sleeping.
>slept 5 hrs a night for 3 years now
>by this gay articles logic, I should be dead.

>>15397686
Sure it's a click-bait title and everyone is different but there is a large amount of recent evidence that shows not getting enough sleep is a massive debuff.

>>15398377
I refuse to sleep more than 5 hours. I work all day how am I supposed to have a life if I sleep?

>>15398399
Work smarter, not longer.

>>15397255
Yes, this is how parens work.

>>15396621
>>15396957
I didn't double check this all, but you can probably show [math](\cup\cap A_k)\Delta A_i\subseteq \cup ((\cap A_k)\Delta A_i)\subseteq \cup\cap(A_k\Delta A_i)[/math], and then a continuity of measures argument should show that [math]d_\mu(A,A_i)\leq \lim \inf \mu(A_k\Delta A_i) < \varepsilon[/math] for large enough [math]i, k[/math].
Let me know if something here doesn't make sense, it may need some tweaking.

>>15398469
How? Every well paying job is fulltime.

>>15398692
Are you working 16hr days and weekends or something?

how can I find the coordinates of A' and B' when everything else is known?
this isn't homework, Im just a dumb engineer trying to extract data from a grid and I forgot how to geometry

>>15398726
you don't because that diagram makes no sense.

>>15398784
I want to rotate line AB by a certain angle with the center of rotation being point C and I want to new coordinates of AB.

>>15398788
Ok but that pic doesn't show that (clearly) at all. Anyways, just use https://en.wikipedia.org/wiki/Rotation_matrix to move each point A & B

>>15398726
I assume [math]x_1=x_2[/math]?

>>15398810
yes, I probably should've mentioned that
I think the rotation matrix the deleted post mentioned should work, I just totally forgot about it's existence.

>>15398813
I deleted it because it doesn't work.

>>15398816
What about it doesn't work?
>>15398726
[math](x',y')=(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta)[/math] is all you need. Just make it easy on yourself by relabelling the coordinates so that C is the origin.

>>15398726
>>15398813
>>15398822
Also what I get doing it the slow way: [math]B' = ((x_1-x_0)\cos\alpha - y_2\sin\alpha, (x_1-x_0)\sin\alpha + y_2\cos\alpha)[/math].

>>15398816
>>15398822
>>15398832
I'll be honest for a 14 degree rotation it looks a little wonky but I'll trust it
either way I'll find out once the data gets to me

>>15398822
> What about it doesn't work?
The angle alpha is not the same angle A' and B' are rotated by. However It does rotate the point O that sits between A & B.

>>15398839
What are we looking at

>>15398847
me using the rotation matrix on (10,2) and (10,-2)

>>15398849
Why does it get longer?

>>15398850
probably because of what >>15398840 said

>>15398840
ah, right.
Still, shouldn't require much more to derive, then. You can figure out the lengths from O to C and A fairly easily, so you can figure out what the angle ACO is (which is also the angle A'CO').
So, given that you know ACO, A'CO', and α=O'CO...
A'CO'+O'CO=A'CO
A'CO-ACO=A'CA=θ
or am I overlooking something just as obvious

yukariposter is rolling in his grave right now

>>15398871
not present = dead loo lay

>>15379528
How do I show [math]f:\{z \in \mathbb{C} \mid |z| < 0, z \neq 0\} \rightarrow \mathbb{C} \setminus [-1, 1], z \mapsto \frac{1}{2}(z + \frac{1}{z})[/math] is a bijection? Injectivity is pretty clear and it's also easy to find an inverse image, [math]z = w + \sqrt{w^2 - 1}[/math], but how do I show that the inverse images lies in the unit disk if [math]z \in \mathbb{C} \setminus [-1, 1][/math]. And what about the other inclusion, i.e. every [math]z[/math] on the unit disk gets mapped to some [math]w \in \mathbb{C}\setminus [-1, 1][/math].

help me pls I'm not good with inequalities

>>15398840
let O'CO=α
and let ACO=A'CO'=β (since angles are preserved under rotation)
then A'CO'+O'CO=A'CO=β+α
and A'CO=A'CA+ACO
so A'CO-ACO=β+α-β=α
what am I missing? other than that it's apparently been too long since I've done basic geometry

>>15398888
You want the angles [math]ACA' = \theta[/math] and [math]BCB' = \phi[/math] to perform the rotations of the point A,B around C.

>>15399041
well, yeah
but the thing that's not quite connecting is that since we're just rotating, [math]ACO=A'CO'[/math], yes?
and just by adding up the angles, [math]A'CO'+O'CO=A'CO[/math] (so [math]O'CO=A'CO-A'CO'[/math]) and [math]A'CO=A'CA+ACO[/math]
so why wouldn't [math]A'CA=A'CO-ACO=O'CO[/math]? i.e., why wouldn't the angle of rotation of A about C be equal to that of O about C?

this rod is rotating about the point [math] O [/math] with angular velocity [math]\omega[/math]. [math]\overline{OA} = a[/math] and [math] \overline{AB} = L[/math].
For each point [math] P [/math] on the rod, its speed is given by [math]d(P)\omega[/math], where [math]d(P)[/math] is the distance from [math]O[/math] to [math] P [/math].
Let [math]\rho(P)[/math] be the linear mass density on [math] P [/math], given by
[eqn]

\rho(P) = \lim_{P_1,P_2\to P} \frac{\mu(P_1P_2)}{s(P_1P_2)}

[/eqn]
where [math]P_1P_2[/math] is a segment that contains [math] P [/math], [math]\mu(P_1P_2)[/math] is its mass and [math]s(P_1P_2)[/math] its lenght.

Find an expression for the total kinetic energy of the rod as an integral with variable [math] x = d(P) [/math] and write [math] \rho(P) [/math] as [math] \rho(x)[/math].

I'm kinda stuck on this problem. I tried doing [math]2\mathrm{d}K = v^2\mathrm{d}m = \rho(P)d^2(P)\omega^2\mathrm{d}s[/math], where [math] \mathrm{d}s = \lim_{P_1,P_2\to P}s(P_1P_2)[/math] but I don't know how to proceed. any help would be appreciated

>>15399127
What you are doing there is adding vectors not angles.

How do you do this without changing order of integration? I've been stuck for a while, been going through other problems but this one and another one keep bugging me

>>15399368
Actually I believe the book is just trying to tell me that it's a rewriting of the same double integral but over x, I'm not supposed to solve it like that

>>15399378
Change the order of integration and it should be much simpler.

>>15399368
>How do you do this without changing order of integration?
>>15399429
>Change the order of integration and it should be much simpler.
D'oh!

What am I doing wrong. The answer is in the top right and what I have doesn’t look anything like it

>>15399457
Your last equality is wrong, [math]\frac15(-x/5)^n\neq(-1)^n(\frac{x/5}{5})^n[/math].

>>15399448
Clearly I cannot read more than one post.

>>15399460
I've looked at it again and I don't see what Im doing wrong. I thought x couln't be negative so we take a negative 1 out

>>15399368
Try change to polar coordinates.

>>15399171
if γ isn't equal to (α+β)-α in picrel, how the fuck does angle addition work?

>>15399171
wait, if γ isn't equal to (α+β)-β in picrel, how the fuck does angle addition work?

I believe this thread is slow because you cant find it through cntrl f "stupid"

>>15399639
>>15399127
>>15399645
I appreciate all the though you guys put into this
since the original line was totally vertical I just did some trig and figured it out
thanks though

if you have two moles of hydrogen, and one mole of oxygen, and that takes up some volume,
and then you burn them in a closed system, making water.
is the volume taken up by that 1 mole of water vapor less than the volume taken up by the three moles of gas? or does the heat given off by the reaction increase the pressure and therefore the volume to more than the gaseous reactants?

>>15399906
I just saw you in the mathematics general. Hi

>>15399781
If I interpreted the problem correctly, it's like sliding segment [math]AB[/math] some angle [math]\alpha[/math] about a circle of radius [math]x_1[/math] while keeping it tangent.

In that case, find the new coordinate of the point of contact, you can calculate the endpoints [math]A'[/math] and [math]B'[/math] since you know slope and distances.

>>15399502
[math]\frac{1}{5}\left(-\frac{x}{5}\right)^n=(-1)^nx^n5^{-n}5^{-1}[/math]
[math]=(-1)^n\frac{x^n}{5^{n+1}}[/math]

whereas
[math](-1)^n\left(\frac{x/5}{5}\right)^n=(-1)^n\frac{x^n}{5^{2n}}[/math].

>>15399991
I'm sorry but I can't read that. How does everyone use symbols here anyway

>>15399668
I didn't even notice it in the catalog for a while. OP done effed up

>>15399153
The mass dm of a little segment is rho(x)dx. You know the velocity. Write the sum of the kinetic energy for each dm and that will be your integral.

Your professor is using awful notation

>>15400043
That's my bad, the math didn't render, see picrel. For typing math, look up LaTeX if you're interested.

If you want to fasttrack the learning process, open up an online interpreter like
https://quicklatex.com/
and a command reference sheet like
https://en.wikibooks.org/wiki/LaTeX/Mathematics
and play around with typesetting.

I'm really a retard when it comes to dealing with summation and the product one (idk how it's called in english). I need to implement some haskell exercises using recursion and whatnot, where can I learn more about the former before tackling the programming part? thanks

>>15400135
ah okay so, You move the 5 in the denominator up to get 1 * 5^-1 which is just
5^-1
then you apply n to the denominator and numerator and then move 5^n up which changes the sign. and then you move them back down and change the sign and then you add the exponents since its the same base,

I get why is it works but that solution doesn't pop out at me naturally.

>>15400078
I think I got it now. thanks

>>15400217
Can you give a sample exercise?

If we take x^2 + i = 0 in the complex plane, then the roots are x = +- sqrt(-i) which are not in the complex plane, however the fundamental theorem of algebra states that the complex plane is algebraically closed, how does this work?

>>15400804
>then the roots are x = +- sqrt(-i) which are not in the complex plane
But they are. It might not be apparent in that form, but if you try a little bit, you can indeed find two complex numbers that square to -i.

>>15400804
let giga chinaman help you my dude
https://www.youtube.com/watch?v=Z49hXoN4KWg

>>15400806
I think i see now, if we take x = +- i^1.5 = sqrt(i^3) then i^3 = i*i^2 = -i, and i^1.5 is in C

Thanks for helping me out anon!

Place four points inside a rectangular prism. The volume of the tetrahedron spanned by these four points is at most one-third of the volume of the prism. Why?

>>15400984
Some thoughts: let's first assume the prism is just the unit square. Then the volume of the tetrahedron spanned by four points [math]v_1,\dots,v_4[/math] from this cube is given by [math]V:=1/6\det([x,y,z,\iota])[/math], where [math]x=(v_1^x, v_2^x, v_3^x, v_4^x)[/math], similarly for [math]y, z[/math], and [math]\iota[/math] is a vector of ones. By Hadamard's determinant inequality, [math]V\leq 2[/math]. You can get the general result by applying a linear map that moves the vtxs defining the cube to your prism and noting its determinant is the volume of the prism.

>>15401056
By Hadamard's determinant inequality, this determinant is at most 2*

Can someone explain what exactly is going on
I know to get the radius of convergence you set x in between -1 and 1 and then simplify which changes the one and the negative one in this case they multiply by nine. But what exactly is happening to the 9^(n+1) one for example.
Is the n+1 just ignored?

>>15400690
eh, sure.

problem sumFirstOddNs (n:z) :z {
requires: {n>=1}
assures: pic related
}

I know it ain't difficult, I'm just retarded

I have a stupid question about circuits. I was looking at a tutorial online for making a delay on circuit (this image).

So my understanding is that the voltage builds up against the capacitor until it reaches a critical value that overcomes the Zener diode. That then allows current to flow across the transistor, turning the light on. The delay comes from the time it takes for the capacitor to charge.

My question is: what does the second (the lower one in the image) resistor do? Why is this necessary?

>>15401228
You now have f(x) written as a power series, you can now use the Ratio test to determine the radius of convergence.

> Is the n+1 just ignored?
No. See above, it cancels out.

I can’t figure out how to do this one. It should involve Baire category theorem but I don’t see what sets to construct

>>15401315
you need to discharge the capacitor after the switch is opened. The capacitors discharge rate will be controlled by that lower resistor thereby delaying when the circuit turns off.

Are integers whose squares are divisible by 12, necessarily divisible by 6?

>>15403254
Use Euclid's lemma.

x^2 = 12k
=> 2|x^2 AND 3|x^2
=> 2|x AND 3|x
=> 6|x

>>15403254
Yes.

Decompose the integers into primes : [math]12 = 2^2 \times 3[/math], so the square contains at least [math]2^2 \times 3[/math] in its prime decomposition. This means that the original integer contains 2 and 3 in its prime decomposition, so it must be divisible by [math]2 \times 3 = 6[/math].

>>15403269
>>15403278
Thank you.

Do textbooks go out of copyright? How long would it take for Baby Rudin?

>>15380485
if f is O(x), then there exist nonnegative c and n0 such that f(x) <= cx for n >=n0. but then f(f(x)) <= cf(x) <= (c^2)f(x). so if f is O(x) then f o f is O(x) as well. and any function that is O(x) is O(x^2) (easy to show, left as an exercise)

>>15402572
If you don't mind a little overkill, I advise looking at Egorov's theorem. If [math]f_n(x)=\sum_{m\geq n}c_m\varphi_m(x)[/math], then by assumption [math]f_n(x)\to0[/math] pointwise. By Egorov, for any [math]\varepsilon>0[/math] there is a set [math]A[/math] so that [math]\int_A d x < \varepsilon[/math] and [math]f_n\to f_infty[/math] uniformly on [math]A^c[/math]. Then, since by assumption [math]\varphi_m(x)\leq M[/math], [math]\int_{A^c}f_n(x) dx \geq \sum_{m\geq n}c_m (c-M\varepsilon)[/math]. Now choose n large enough that [math]\int_{A^c}f_n(x) dx<\infty[/math] (possible by uniform convergence) and [math]\varepsilon[/math] small enough so that [math]c-M\varepsilon > 0[/math], and you have a bound on the tail of your series.

>>15403544
[math]f_infty[/math] should read [math]0[/math].

I want to create 2 curves, that are mirror images of each other and when they are rotated their intersection is almost perpendicular to eachother. Or if the curves are thick, they would almost form a square. The intersection doesnt have to be perfect in the middle of the rotation point.
I draw pic related, it can be improved to get a more square intersection but it sort of works. What is this called?

>>15403677
I tried involutes and made a little program to check it, but the intersection is not what i want.
Im sure this is something that exists but dont know how to look for it.

Im trying to study for an exam and have some questions about the taylor series.

This is the formula I'm supposed to use right, is the third derivative a good enough stopping point. Are there scenarios where I should go further.

Suppose [math] f:\mathbb{R}\to\mathbb{R} [/math] is a continuous function such that [math] f(x)>0 \forall x\in\mathbb{R} [/math], [math] f(n) = \frac{1}{n^2} [/math] if [math] n [/math] is a strictly positive integer and [math] f(n) = n^3-\frac{1}{n} [/math] if [math] n [/math] is a strictly negative integer.
Let [math] S(x) [/math] be the area under [math] f(x) [/math], above the [math] x [/math] axis, from 0 to [math] x [/math].

Is there any way I can evaluate [math] S(2) [/math] and
[eqn]

\lim_{h\to0^+} \frac{S(2+h) - S(2)}{h}

[/eqn]
?

>>15404190
Look up the Fundamental Theorem of Calculus.

>>15403907
Okay now I'm more confused do I use the ratio test or differentiate

I'm taking my first real analysis class. The book we're using is called Introduction to Analysis by Gaughan.
The chapters are as follows:
1. Sequences
2. Limits of Functions
3. Continuity
4. Differentiation.
My final is on 3-4 and I've sort of slacked off after the first midterm so I don't remember anything from 1-2.
Can I skip 1-2 and work on 3-4 or should I do 1-2 first?

>>15403907
> This is the formula I'm supposed to use right
Yes
> is the third derivative a good enough stopping point
Depends on question. They might asked for a certain number of terms or upto some power of x, also remember that some terms might be zero.

>>15404338
[math]f^{(n)}[/math] is the formula for the n'th derivative: https://en.wikipedia.org/wiki/Taylor_series#Definition

They are then using the Ratio Test to confirm the given Taylor series converges.

>>15404488
so for the problem with the ratio test do I actually need to find the derivative of anything?

>>15404527
No, because you are told what all the derivatives are.

>>15404540
hmm I'm trying to compare the two photos and see how they are the same formula.

I can see (x-c)^n but In the second picture why is there a n! in the denominator and numerator.

I'm really not getting this. okay so The derivative of e^x is always e^x. I get that and if x = 0 then e^x would be 1 I get that.
what I don't understand is that if there is a 0 in the numerator that is a mulyiple shouldn't the answer always just be 0

>>15404590
> why is there a n! in the denominator
because of the definition of the Taylor Series. >>15403907
> numerator
because of the given definition for [math]f^{(n)}[/math]

>>15404669
The numerator always 1 tho

>>15404669
The n'th derivative of [math]e^x[/math] is still [math]e^x[/math] for any n, so [math]f^{(n)}(0)[/math] is always 1.

>>15404713
>>15404695
oh wait that 0 is supposed to be the input for e^x okay that makes sense

>>15404725
You seem to be struggling with the notation in all your questions.

[math]f^{(n)} = \dfrac{d^nf}{dx^n}[/math]

>>15404736
okay this is embarrassing but what exactly is d in this example? is it derivative? So I would take the derivative of f and then raise it to n and then multiply it by f? and then divide it by the derivate times the input raised to n?

>>15404758
>d in this example? is it derivative?
Yes.
>>15404758
>So I would take the derivative of f and then raise it to n and then multiply it by f?
No. You take f, and you apply the differentiation operator with respect to x (i.e. take its derivative with respect to x) n times.

>>15404773
Could I get a simple basic example

>>15404779
simplest nontrivial one is probably [math]\frac{d^2}{dx^2}x^2=\frac{d}{dx}\frac{d}{dx}x^2=\frac{d}{dx}2x=2[/math]

>>15404779
this >>15404786 but the reason it's use a lot in your questions is because it's the same as saying

[math]
\begin{align}
f(x) &= f^{(0)}(x) \\
f'(x) &= f^{(1)}(x) \\
f''(x) &= f^{(2)}(x) \\
... \\
\end{align}
[/math]

as in what you first posted >>15403907

>>15404786
Ok so this f(2)x^2 ?

I get taking the derivative of x^2 twice will get you 2

I get that d^2 is d * d and dx^2 is
dx*dx

ANd then you take the derivative of x^2 once which gets you 2x, and removes a d/dx and then you do it again to get 2

so in the maclaurin series I posted

the derivitave of e^x is always e^x and since the input is always 0 the result is always 1, and then, I multiply that 1 by x^n

So basically to find the answer of a mclaurin series I just need to find what f^(n)(0) is and multiply that by x^n andthen divide by n!

f{x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a unit in R.
Can someone give an example of what this means. When they say constant polynomial do they mean just a constant?
If this theorem is true and what I think the theorem is saying is true, does it mean that any polynomial with a constant c that is a unit in R is also a unit in R[x]?

>>15404835
> So basically to find the answer of a maclaurin series I just need to find what f^(n)(0) is and multiply that by x^n and then divide by n!
Yes, precisely that.

I need help on this exercise from Humphreys' reflection groups. Note that [math]\Delta[/math] is a simple system and the root system [math] \Phi [/math] is not necessarily crystallographic. [math]\mathrm{ht}(\beta)[/math] is defined as [math]\sum_{i=1}^nc_i[/math] if [math]\beta = \sum_{i=1}^nc_i\alpha_i[/math] where [math]\alpha_i \in \Delta[/math] I tried approaches using the scalar product but nothing worked and now I don't even know where to start.

>>15404853
Okay and to find the radius of convergence, the orginal function does not matter? I just use the ratio test on the taylor series and I get it, is it always the ratio test

>>15404845
That "theorem" is not true. [math]f(x) = 2x +1[/math] is a unit in [math]\left(\mathbb{Z}/4 \mathbb{Z} \right) [x][/math] but it's not a constant polynomial.

I'm still a bit confused about the problems in my book, they don't seem structured the same as in my lessons. Which is usualy
I am given a function than I find f(c) and c is already given to me

So I do not need to find the derivative right, I'm given it to start out

Then what exactly are they doing in between the problem I am given and what I'm told the answer is

>>15404917
The original function does matter because that gives you its Taylor series. You then use the Ratio Test if you want to find the Radius of Convergence, but if you only want to know if the series convergences you could use any of the usual convergence tests.

>>15404966
sry

>>15404966
You *always* need to know the derivative. So you either have to calculate them, or you are told them, like here >>15404968

>>15404967
I meant the original funtion doesn't matter if Im already given the taylor series to start for this specific problem.

In the sense that the original is only useful in that you get the taylor series from it

>>15403345
Yes, like everything else. Works from 1928 are public domain in the US now, so wait 25 years or so.

any chemfags here?
could you tell me how hard getting something like this made would be
>>15405516
>>15405508
https://sci-hub.se/https://onlinelibrary.wiley.com/doi/10.1002/(SICI)1521-3773(19980904)37:16%3C2217::AID-ANIE2217%3E3.0.CO;2-D

>>15405524
obtaining that ligand is probably the hardest part

>>15405554
thanks anon
seems like something that might allow easier access to good quality htp.
but honestly i don't want to fuck with it because 95%+ htp sounds like death

This is probably a very simple question for you guys, but I'm a bit of an idiot.
How do I calculate how many times [something] might/probably happen if the chance is X%, and I do the experiment A times?
I can find online calculators for the chance of [something] happening at least once if you do the experiment A times, but I can't find one for this kind of probability.

>>15405627
Search up Binomial Distribution Calculator

Is the only thing my teacher is doing here is the ratio test and plugging in the endpoints to check them

>>15406242

Let [math]f:[0,\infty[\to\mathbb{R}[/math] be a continuous and strictly crescent function with [math]f(0) = 0 [/math]. Let [math]a[/math] and [math]b[/math] be two arbitrary positive real numbers. Show that
[eqn]

\int_{0}^{a}f(x)\mathrm{d}x + \int_{0}^{b}f^{-1}(y)\mathrm{d}y \geq ab

[/eqn]

I tried doing integration by parts on the first integral and then doing the substitution [math]y = f(x) [/math] and then substituting back into the inequality and I got
[eqn]

af(a) + \int_{0}^{b}f^{-1}(y)\mathrm{d}y \geq ab + \int_{0}^{f(a)}f^{-1}(y)\mathrm{d}y

[/eqn]
but I have no clue on where to go from here

>>15406242
Yes, ratio test doesn't tell you whether the series converges or diverges when the ratio is [math]1[/math]. In this case, [math]\lim_{n\rightarrow\infty}|a_{n+1}/a_n|=|x/6|[/math], ratio is [math]1[/math] for [math]x=\pm6[/math], you have to check what happens at [math]6[/math] and [math]-6[/math] yourself.

>>15406337
draw a picture

>>15406349
I thought it was strange since the ratio test was on our last exam and this is a problem to study for the next one. I guess because its used in the taylor series she wanted to test us on it again

heeeeeeeeeelp, I tried to solve for x le +- 1-y but I can't get the correct answer..... I'm mega shit at finding the bounds, does anyone have any general advice or further reading?

>>15406421
If you have a circular shaped domain it's always a good idea to consider using polar coordiantes.
The bounds will be [math]\varphi \in \left[0,\frac{\pi}{2} \right][/math] and [math]r \in [0,1][/math].

>>15406421
Switch to polar, anon!

>>15406430
>>15406443
I tried that but it didn't work, the numerator should be r^3costsint right? Since for polar coordinates the differentials are rdrdt

>>15406483
Use a u = sin theta substitution

I know I've asked about this a lot but in this problem which is from organic chem tutor
>>15403907
he does not find the points of convergence is this because its not part of the problem, if you were to find it you would just use the ratio test on the taylor series as in here
>>15404338

>>15379528
What is the expected pH of cleaning vinegar with 6% acidity? Can you tell the pH just by knowing the acidity level?
I bought a soil pH tester but it's cheap so I wanted to test it and see how accurate it is. Tap water comes out at like 7.5 which sounds normal. But I tested vinegar and it's a little over 5, like 5.1 pH. That's higher than I thought it would be and the internet says vinegar has a pH of 3 or even lower.
But the bottle says it's been "diluted with water to 6% acidity." What does 6% acidity mean? What pH should it have?

Dumb question, but in abstract algebra, what does notation like Q[3sqrt(2)] mean (cube root of 2).

When you have amnesia you don't only not have memory of the events you forgot but you also don't have the sensation of experiencing those events in real time, how is that possible? How does your subjective experience know in real time that you're going to get amnesia and forget the events you're experiencing right now?

Is it possible to solve for x without using trigonometry. I feel like it should be a and have spent hours looking for a way but I always get a x=x scenario.

>>15407047
So you should hopefully know by this point that [math]\mathbb{Q}[/math] is the field of rationals, and that [math]\sqrt[3]{2}[/math] is not a rational number.
There are some contexts where it would be more convenient if it "was", though, so [math]\mathbb{Q}[\sqrt[3]{2}][/math] is saying to take the rationals and add [math]\sqrt[3]{2}[/math] to them, and then, to maintain closure, to add in everything we'd get by multiplication/addition (and, by extension, division/subtraction) of [math]\sqrt[3]{2}[/math] with rational numbers.
So in this particular example we'd be looking at [math]\mathbb{Q}[/math] taken with all numbers of the form [math]x+y\sqrt[3]{2}[/math]. with [math]x,y\in\mathbb{Q}[/math]

>>15407047
So, given that you're at the level of asking this, you should hopefully know by this point that [math]\mathbb{Q}[/math] is the field of rationals, and that [math]\sqrt[3]{2}[/math] is not in this field.
There are some contexts where it would be more convenient if it "was", though, so [math]\mathbb{Q}\sqrt[3]{2}[/math] is saying to take the rationals and add [math]\sqrt[3]{2}[/math] to them, and then, to maintain closure, to add in everything we'd get by multiplication/addition (and, by extension, division/subtraction) of [math]\sqrt[3]{2}[/math] with rational numbers; this is called a field extension.
So in this particular example we'd be looking at [math]\mathbb{Q}[/math]taken with all numbers of the form [math]x+y\sqrt[3]{2}[/math] with [math]x,y∈Q[/math]

>>15407092
>>15407106
Thanks, my follow up question would be is this different from a polynomial ring R[x] for instance, which would be: a0 + a1x + a2x^2 + ....
Or is it possible to express Q[3sqrt(2)] like a polynomial?

>>15407106
should probably correct my other mistake and note that it'd actually be [math]x+y\sqrt[3]{2}+z(\sqrt[3]{2})^2[/math], since closure would necessitate taking the square of it as well
>>15407112
The main thing separating it from your basic polynomial ring over the rationals is that polynomials with the cube root of 2 as a root can now be reduced, similar to how we are able to break down [math]x^2+1[/math] when we start working with the complex numbers (which, themselves, are a field extension of the reals, with our extension being [math]i[/math]).
Not entirely sure what you mean by representing the field itself as a polynomial, but you can express these things as quotient rings, e.g. [math]\mathbb{Q}[x]/(x^3-2[/math]

>>15407144
Thank you, last question, what's the Quotient Ring you introduced mean and did you/how did you derive it from Q[3sqrt(2)]?

>>15407162
They're analogous to the idea of a quotient group in group theory, if you have any experience with that.
[math]x^3-2[/math] is the simplest irreducible polynomial with rational coefficients with [math]\sqrt[3]{2}[/math] as a root, so it's a very natural place to start looking for our desired field extension; we call this the minimal polynomial of our extension.
When we take the quotient ring with respect to this, we're establishing an equivalence between all polynomials that differ from each other only by some multiple of [math]x^3-2[/math]; this is because if one can be written as [math]a+b\sqrt[3]{2}+c(\sqrt[3]{2})^2[/math], so can anything else in this equivalence class. This is important because it means that [math]x^3-2[/math] itself, and any multiples thereof, reduce to [math]0[/math]; or, rather, we can reduce any multiple of [math]x^3[/math] to [math]2[/math]. If we take some polynomial in our ring, then, we can then use this to forcibly reduce it to the form [math]a+b\sqrt[3]{2}+c(\sqrt[3]{2})^2[/math], establishing an isomorphism between our field extension and our quotient ring.

>>15406421
I just can't get this to work at all, please someone post a solution

>>15407366
[eqn]\frac{1}{4} - \frac{\log(2)}{4}[/eqn]

>>15407386
I'm aware of the answer, can you show me your solution to get to the answer? Thanks

>>15407396
There are a million possible way. For example you can first do the integral over y and then the integral over x

For the inner integral substitute u = 1 + y^2
[eqn] \int_0^{\sqrt{1-x^2}} \frac{xy}{(1+y^2)^2} dy \\
= \frac{x}{2} \int_{1}^{2-x^2} \frac{1}{u^2} du \\
= \frac{x}{2} \left( \frac{1}{x^2 - 2} + 1 \right) \\
= \frac{x}{2} + \frac{x}{2(x^2 - 2)}
[/eqn]

Now you do the other integrals

[eqn] \int_0^1 \frac{x}{2} dx = \frac{1}{4} [/eqn]
[eqn] \int_0^1 \frac{x}{2(x^2 - 2)} dx = \frac{1}{4} \int_{-2}^{-1} \frac{1}{u} du = \frac{-\log(2)}{4} [/eqn]

So the end result is
[eqn] \frac{1}{4} - \frac{\log(2)}{4} [/eqn]

>>15407440
I fucking tried u-substitution and polar coordinates, I couldn't get it to work, I'll have to look at my solution and compare it to yours and find what I did wrong, thanks man

>>15407440
>>15407463
Ah FUCK bro I didn't even try to change the order of integration, man that shit and finding bounds really filter me, thanks again

Really more of a medical (psychiatry) question but I'm wondering if there's a specific point I can research regarding suicidal tendencies. I don't want to look it up on google because I don't want my shit to get flagged.
Unsimply put, for the decade and a half I've had at least one suicidal thought a day but in the last few years, about once every two weeks, I end up with something far stronger. My mind loses all abilities to focus, and does what I can only describe as shaking and it leaves me with an outright compulsion to kill myself. I'd best describe the feeling as going five days without water and staring at a glass full of it while trying not to drink it. I had gotten pretty good at dealing with them, even using alcohol as medicine if its really bad, but lately they've been coming on stronger and stronger.
I'm not looking for sympathy, I'm not looking for advice, just seeing if anyone has any useful info for me to go over regarding it to see if I can get these new impulses under control before the one time I can't.

>>15406978
yeah. pH is defined as the -log() of the concentration of [H+] ions. pKa tells you Ka, which tells you how much [AcOH] dissociates into [H+] and [AcO-] solution. "6% acidity" could mean anything, I assume it's 6% v/v (by volume, so 6mL pure AcOH diluted to 100mL with water).
Do the math and you should be able to predict the pH.

>>15407570
Try private browsing

I am a [math] \mathrm \LaTeX [/math] expert.

>>15408455
Is it pronounced lateX or latech

>>15408487
It is pronounce lay-tech because it is supposed to be a play on the words latex and tech.

Is there a book that sort of just explains the theorems/proofs in Abstract Algebra clearly without all the fluff of textbooks. I know the fluff is important but I sort of just want the theorems/proofs listed in an easy to understand manner.

>>15379528
Good Morning /Sci/entists!

When I use my latex template, making a banner causes the first section of the book to get distorted. It makes a big gap between the section name text and the table of Combinators. I tried using negative vspace, but it seems to do nothing until suddenly jumping way too far and getting too close to the text.

How do I fix the spacing?

>bonus question
I am changing all the birds into maids. If you know a good anime maid that matches the letter, please tell me.

Thank you /sci/entists for reading my post.

>>15408783
Who knows? Templates are retarded and you shouldn't use them. No way to tell without reading the entire source code of the template. You should first write the material, using as little packages as possible, then think of cosmetics.

>>15408811
Material is written and I am just making it pretty now. The template is called Kaobook. I don't remember where I got it from. I was looking for templates with big margins so my book can get margin maids.

I have never tried to use LaTeX without a template. I am just going to fix the rest of the document and hope somebody tells me while I typeset other things. If that doesn't happen I will play with it a little more when I run out of other work. If I can't fix it, I will just post it as-is because it doesn't harm the quality of the writing or the appearance of the maids, just makes the first section of every chapter look silly.

>>15408847
I am assuming you are talking about margin notes. You don't need any special template for that. Latex already has it, like I have done here. If you can upload just the naked manuscript without any superfluous packages on github, I can see if I could help you later. I am very busy right now.

>>15408880
I never understood using margins like that, if you're going to write so much text, just add another paragraph in the body. If not, just add it as a footnote. Now it's like there are two pages next to each other and you have to switch between them when the author feels like it.

>>15409197
I mostly did it for experimentation, but I suppose it has mainly to do with textwidth. If you are using a small font, you can only cram in so much characters in a line. Beyond a certain limit, it just become hard to read. The max characters per line is ideally considered to be no greater 80 characters. So if the page you are using is large (like A4), you will have extremely wide margins, which will be left unused. So it is better to be used some way or the other. Hence, it makes sense to move the footnotes there.

Personally, I use the margins a different way. I overflow large headings outside the margin. This is because they use a larger font, so the size of the paper relative to them is smaller, so it makes sense to have narrower margins for them. Secondly, I also indent theorem environments outside the margin, this makes them easier to track when glancing through the page. Picrel. I also overflow tables and figures outside the margin, since the whole character per line is not really relevant then.

>>15379528
What are the AI chat engines that you use and like that don't require a login? Here are my three "go-to" chat sites:

https://deepai.org/chat
https://beta.character.ai
https://www.perplexity.ai

Are there any others you can recommend that don't require a login?

>>15409227
This looks great

Why is top left equal to top right?
Bottom is my attempt but I don't get where the minus come from.

>>15409638
Top looks like the chain rule [math]\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}[/math]. What context is this question though? The negative sign will come from some other relationship.

>>15409696
Thermodynamics. I had to prove that [math]c_p-c_v=\frac{TV\alpha^2}{k_T}[/math]. I eventually got to the point where to be truth the relation that I mentioned must be truth. I also thought it was chain rule, but I don't get where the minus come from.

>>15409706
Ahh. Use Maxwell's relationships.
[math]
\dfrac{\partial P}{\partial T}\bigg|_{V} = \dfrac{\partial S}{\partial V}\bigg|_{T}
= \dfrac{\partial S}{\partial P}\bigg|_{T} \dfrac{\partial P}{\partial V}\bigg|_{T}
= -\dfrac{\partial V}{\partial T}\bigg|_{P} \dfrac{\partial P}{\partial V}\bigg|_{T}
[/math]

>>15409735
aaaaaaaaaaaaaaaaaaa tytytyty

Anyone know if this converges?
[math]\sum_{n=0}^{\infty}\frac{cos(kna)}{2^n}[/math]

>>15410337
assuming your coefficients are real, observe that every term in the sequence is going to be bounded between 1/2^n and -1/2^n

>>15410383
Yeah, just ended up throwing it at wolfram and it converges.

>>15410337
Yes, by comparison test.

A substance has the following properties:
>at T = T0 = cte, the work done by a reversible expansion from V0 to V is [math]W = RT_0 ln(\frac{V }{V_0})[/math]
>the entropy is given by [math]S=R(\frac{V_0}{V})(\frac{T}{T_0})^a[/math], (V0, T0 and a constants.).
Calculate the Helmoltz free energy.

So the Helmoltz free energy is F=E-TS and dF=-SdT-pdV, so at T constant [math]dF=RT_0 ln(\frac{V }{V_0})[/math]. But how do I get F from here?

I want to reduce 3-SAT to the following problem: Given a simple graph that has round and square vertices, square vertices can only have an edge to round vertices and vice-versa. Assume that the degree of every square vertex is atleast 3, can we remove k of the n round vertices so that every square vertex still has degree 3 or more? The condition that every clause has to be true is easy to model, but i'm having trouble reducing the conditions for the literals

bump

bump

>>15411538
Are you sure you didn't flip V_0 and V in the entropy? Your formula says the entropy gets bigger if you put it in a smaller volume while keeping the temperature the same.

Anyway, you know dF/dT = -S so you can integrate
F= -\int S dT + C(V)
where C is a function depending on volume but not temperature.

Now set F(T_0, V)-F(T_0, V_0) = -W to solve for C(V) up to an overall constant.

Will we ever see a cure for it in our lifetimes that doesn't involve giving you erectile dysfunction

>>15417229
20 years will do that to someone, oh well