/mg/ - mathematics general

[Anonymous No.15177122]

Georg Landsberg edition
Discuss mathematics here

Previously
>>15159206

Gotta question for you guys. What, in your opinion, is the advantage of discussing math here instead of on some other website? Is it just that /sci/ is the least retarded place you know of?

[Anonymous No.15177128]

Perturbation theory for nonlinear differential equations: based or cringe? I'm starting to understand and on one hand they seem wicked cool (Poincare-Lindstedt method especially). But on the other hand it seems like cheating

[Anonymous No.15177144]

>>15177122
>is the advantage of discussing math here instead of on some other website?
There are no advantages at all. This website has become extremely dumb and obnoxious

[Anonymous No.15177236]

>>15177128
That depends entirely on whether it's integrable or not.

[overAnon No.15177261]

yo been using chat gpt3 to generate examples of abstract shit, just been playing around with it for now.
what do you guys think?

[overAnon No.15177265]

>>15177261
as an example i asked to give a non mathematical example of a topology

[Anonymous No.15177293]

>>15177122
OP, I can give you a couple of things for your question...

In usual academic environments everyone functions
on the idea of collegiality. Regardless of one's
positions (politics or otherwise) they are set
aside to get to the matter at hand, mathematics.
Here, we chuck collegiality in a bin and express
ourselves as we see fit to humanly get to the
matter at hand, mathematics...or maybe how
stupid someone sounds right now.

Higher mathematics, even topics at the leading
edge, are restricted to academic and research
institutions where people in the know work and
converse. This, if any amount of it, is passed on
to students if they know where to look or talk to.
Here, anyone could potentially drop this sort of
knowledge for all to see and discuss freely
regardless of background credentials...even to
have a water cooler chat on how fake and gay the
discovery is.

They got their IAS, Harvard, Oberwolfach, Cambridge,
etc. and their leading heads, and all the fixings thanks
to funding and support.
Here, we have the janitor's closet headed by the
janitor (official title: maitre concierge san salaire),
funded by a tin can with some string and full support
from no one other than ourselves...and perhaps none
for the janitor, I guess...

[Anonymous No.15177294]

Has /sci/ still ever solved this?

[Anonymous No.15177305]

>>15177122
For discussing math, reddit is about the same; here and reddit both have their advantages and disadvantages.

MSE / MO is better but the people there can be ridiculously stuffy.

[Anonymous No.15177318]

>>15177101
why not? isn't it a set by replacement?

[Anonymous No.15177352]

>>15177344

[Anonymous No.15177360]

I am so retarded I actually posted in a dying thread.

Let me reiterate my question in this new thread.

I have 100 iq and low self-esteem.

If I learn math I will feel really really smart.

Realistically speaking, how much "progress" can a avgwit like me make? Is there any subfield of mathematics an average iq anon like myself can realistically benefit from mastering? I have a degree in Computer Engineering (we are like Electrical Engineers but more retarded) so I have some knowledge of calculus, physics, and so fourth

[Anonymous No.15177372]

>>15177352
>Is there a general solution for x
Are you asking if a solution exists, or you're asking for an actual closed-form expression for the general solution?

[Anonymous No.15177374]

>>15177352
Just copypaste and repost the whole thing but idk what having [math]x \in \mathbb{R}^{n}[/math] accomplishes in combination with [math]x_{1} + x_{2} + \cdots + x_{n} [/math]. Are you adding scalars to 2-vectors to 3-vectors to n-vectors or something else?

[Anonymous No.15177382]

>>15177318
I'm not the one you replied to, but I think you can actually.

To state it in full: we propose if f is a global choice function on sets, and Z is a set, then there is a set whose elements are precisely (z,f(z)) for some z in Z .

However, you need to use the fact that if Z is a set and f is a class function with domain containing Z then {f(z) : z in Z} is also a set.

[Anonymous No.15177396]

>>15177360
Are you asking me to estimate your mathematical capabilities with no further data?

[Anonymous No.15177416]

>>15177396
I am asking you to use your big European brain to imagine yourself as a creature with a less sophisticated brain than yourself. What branch of math would be most useful to "master" in that case?

[Anonymous No.15177418]

>>15177122
recently I heard about the borsuk-ulam thing in some pop-math video and they were mentioning that meme about how there must be 2 points on earth which have the exact same pressure and temperature

do mathematicians really think the earth is that well apprxoimated as sphere to make such a bold claim? can they be that stupid?

t. aspiring and half retarded physicist

[Anonymous No.15177421]

>>15177122
>What, in your opinion, is the advantage of discussing math here instead of on some other website?
The level of effort here is very low. If I have a serious, non-stupid question that needs a serious answer I'll go to stackexchange, but here is the best place to shitpost or exchange discussion that only requires a couple lines at a time

Plus I can call people faggots here if I don't like them

[Anonymous No.15177424]

>>15177382
yes that was my line of thinking too
I believe even without replacement one could separate [math]\{ (z,f(z)) \mid z \in x \}[/math] from [math]x \times \{ f(z) \mid z \in x \}[/math],
which should be a set since [math]\{ f(z) \mid z \in x \} \subseteq (\bigcup x) \cup \{ f(\varnothing) \}[/math]

Hence why I'm confused about the exercise

[Anonymous No.15177427]

>>15177357
I mixed things up: [math]n=m[/math]. I'm looking for a closed-form expression. I imagine there is a small chance it can be solved within a geometric interpretation.
>>15177374
Sorry, I missed new thread.
>idk what having x∈Rn accomplishes
I'm not sure, I thought it were the standard notation for vector entries. It's a row vector of n columns

[Anonymous No.15177485]

>>15177427
AFAIK there are general solutions for n<6. There's probably one for variable n but I don't know it.

[Anonymous No.15177489]

>>15177427
I might be being dumb, but can't this fail for large exponents?

Like for example, we already know that single variable polynomial equations of degree ≥5 can fail to admit solutions by radicals. So any closed-form expression would have to ho outside of that, probably using some kind of special transcendental functions.

[Anonymous No.15177497]

>>15177418
But it is true, as long as you assume the pressure and temperature are continuous.

Even if they aren't, you should be able to get bounds for the failure of this in terms of bounds on the discontinuities of P and T, etc.

Now go study up some more math, it will help you.

[Anonymous No.15177504]

>>15177418
It doesn't matter that, geometrically, Earth isn't a perfect sphere. It's homeomorphic to one, which is sufficient for Borsuk-Ulam to apply

[Anonymous No.15177521]

>>15177416
Not him, and not someone who knows much about math either. I'm struggling with logarithms and exponents.
Its not much about the goal for me I guess, but the journey. At least for many things.

I wish you can find happiness

[Anonymous No.15177522]

favorite book on hyperbolics?

[Anonymous No.15177559]

>>15177344
>>15177352
>>15177427
Am I correct in thinking that you are asking about the following sequence:
[math]\forall \{x;y;a\} \in \mathbb{R}_{+}[/math]
[math]x^a=y[/math]

.

[math]\forall \{x_{1}, x_{2}; y_{1}, y_{2}; a_{1}, a_{2} \} \in \mathbb{R}_{+} [/math]
[math] {\begin{bmatrix} x_{1} & x_{1} \end{bmatrix}} ^{{\begin{bmatrix} a_{1} & a_{1} \end{bmatrix}}} + {\begin{bmatrix} x_{2} & x_{2} \end{bmatrix}} ^{{\begin{bmatrix} a_{1} & a_{1} \end{bmatrix}}} = {\begin{bmatrix} y_{1} & y_{1} \end{bmatrix}} [/math]
[math] {\begin{bmatrix} x_{1} & x_{1} \end{bmatrix}} ^{{\begin{bmatrix} a_{2} & a_{2} \end{bmatrix}}} + {\begin{bmatrix} x_{2} & x_{2} \end{bmatrix}} ^{{\begin{bmatrix} a_{2} & a_{2} \end{bmatrix}}} = {\begin{bmatrix} y_{2} & y_{2} \end{bmatrix}} [/math]

.

[math] \forall \{x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3}; a_{1}, a_{2}, a_{3} \} \in \mathbb{R}_{+} [/math]
etc?

If so there is no general solution, per Galois groups.

[Anonymous No.15177561]

enjoying my life on the complex unit disk

[Anonymous No.15177568]

>>15177559
>If so there is no general solution, per Galois groups.
No solution by radicals, you mean.

[Anonymous No.15177656]

>>15177485
Uhm, please share if you rember! n < 6 helps in my context
>>15177489
I tried asking on a math Discord and they said this problem is "cursed". Pic is the simple case [math]n=2[/math]. I assume solutions in higher dimensions are the surface intersection of lp hyperspheres.
>can't this fail for large exponents
This is an interesting point. It makes sense, large exponents are problematic as surfaces become cube-like when [math]a_i \to \infty[/math].
>>15177559
>>15177568
This is very interesting, thank you. I will have to learn more about Galois groups and radicals to make sense of the answers.

[Anonymous No.15177797]

>>15177382
>but I think you can actually
Well you are wrong.
>>15177424
You are being misinformed by the ignorant anon, don't listen to him. You simply cannot prove that {(z,f(z)) | z in x} is a set. There are no axioms that let you prove this. You cannot apply replacement, because the axiom of replacement is stated without the symbol f. You don't have the axiom of replacement with f in it.

[Anonymous No.15177901]

>>15177122
It's much less serious and stuffy here. That's pretty much it. I also like giving career advice and asking mainly career questions. I don't ask or expect to answer any actual math questions. Honestly, I wish I had known about /sci/ more when I was in undergrad, but then again, I wasn't nearly as career oriented then.

[Anonymous No.15177964]

Is there a way to draw pictures to remember hyperbolic trig functions acting on their inverses, arctanh(sinh(x)) and such?

[Anonymous No.15177991]

>>15177901
I have gotten very good math answers on this board.

[Anonymous No.15178028]

>>15177964
Yes, unit hyperbola

[Anonymous No.15178053]

How do I go about pulling tricks out of my ass like this to solve recurrence relations? I can never figure out how they got to the G(x)-3G(x) step.

[Anonymous No.15178064]

>>15178053
Similarly, the G(x)-1 step in this example. Is there a process to go about this or is this a situation where I either have the insight or don't?

[Anonymous No.15178314]

>>15178053
>>15178064
Okay never mind I got it. The first few pages of generatingfunctionology was really helpful. Fuck Rosen.

[Anonymous No.15178470]

>>15178314
i was about to write that when I looked at your post.
generatingfunctionology is one of those books which makes me wonder why there are so little good books for self-learning.

[Anonymous No.15178577]

>>15177122
Scientifically, is mathematics all just a fiction?

https://www.youtube.com/watch?v=mhIkyqLDl9M
https://www.youtube.com/watch?v=ZQOwG-hcd_k

[Anonymous No.15178873]

>>15178577
Yes unless you assume AC

[Anonymous No.15179055]

Helloooooo /mg/. It's a me again. I have another problem. I hope you guys like it. I haven't attempted it yet but it shouldn't be super difficult.
The book says that you should try a combinatorial proof, a generating function proof and an algebraic one.
Of course I'm not going to try to do all :DDDD Me, I'm going to try it combinatorially but you should feel free to do what you want. And just to clarify, [eqn]G_n[/eqn] means the set of all permutations [n].
I hope everything else is clear.
I appreciate anyone taking the time to attempt this. If you have any interesting things you'd like to point out about this problem, let me know! And if you want any clarification or the solutions found in the book, also let me know! Have a nice day!!!

[Anonymous No.15179232]

>>15177122
>What, in your opinion, is the advantage of discussing math here instead of on some other website?
I can say nigger. Nigger.
>>15177418
>there must be 2 points on earth which have the exact same pressure and temperature
2 opposite points with the same pressure and 2 opposite points with the same temperature. Not both at once. But yeah if pure mathematics has ANY real-world applications at ALL I think we can assume the earth is topologically sphere-like and temperature is continuous
>>15178314
>generatingfunctionology
Based. Love that book

[Anonymous No.15179261]

Well, /mg/?

[Anonymous No.15179287]

>>15177797
>You cannot apply replacement, because the axiom of replacement is stated without the symbol f. You don't have the axiom of replacement with f in it.
oh my god, thank you anon. I don't know how I didn't notice this, but it all makes sense now.

To summarize, any attempt at showing that the class [math]\{ (z,f(z)) \mid z \in x \}[/math] is actually a set would need to use an instance of replacement or separation with a formula mentioning our new function symbol f (i.e. a formula that is not part of our original language). However, ZF (and so also our theory T) only includes all instances of replacement/separation for our old language.
Hope I understood this correctly.

I think I also get the motivation behind the exercise now.
Hinman calls these definitions (the ones described in https://en.m.wikipedia.org/wiki/Extension_by_new_constant_and_function_names)) "pseudo-definitions", since they're weaker than extensions by "real" definitions (the ones described here https://en.m.wikipedia.org/wiki/Extension_by_definitions).).
So while T is conservative over ZF because [math]\mathsf{ZF} \vdash \forall x\exists y (x \neq \varnothing \rightarrow y \in x)[/math], this is really all we get, i.e. we can't strengthen that "exists" to an "exists unique": [math]\mathsf{ZF} \nvdash \forall x\exists! y (x \neq \varnothing \rightarrow y \in x)[/math].
Hence not every formula in our new language has an equivalent (under T) formula in the old language. If we had that we could use replacement/separation however we want even if they only include instances for formulas in the old language, but we do not have that here!

[Anonymous No.15179293]

>>15179261
By Cauchy-Schwartz
[eqn] \frac{1}{n} \sum_{k=1}^n a_k a_{n - k} \leq \frac{1}{n} \sum_{k=1}^n a_k^2[/eqn]

[math]a_n \to \pi[/math] implies [math]a_n^2 \to \pi^2[/math] because the product of two convergent sequences is convergent with the limit being the product of the limits.

The Stolz-Cesaro theorem implies that
[eqn] \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n a_k^2 = \lim_{n \to \infty} a_n^2 = \pi^2[/eqn]

[Anonymous No.15179295]

>>15179055
Solution with genfunctions. I'll try the other as well but they seem hard.
Let
[eqn]I(x) = \sum_{n=0}^\infty x^n= \sum_{n=0}^\infty n! \frac{x^n}{n!} \\
D(x) = \sum_{n=0}^\infty D_n \frac{x^n}{n!} \\
e^x = \sum_{n=0}^\infty \frac{x^n}{n!} \\
xe^x = x\sum_{n=0}^\infty \frac{x^n}{n!} = \sum_{n=0}^\infty n\frac{x^n}{n!}
[/eqn]
Where [math]D_n[/math] is the number of derangements. We will skip the computation of it (since it's trivial with sieve), we get
[eqn]D_n = \sum_{i=0}^n \binom{n}{i} (-1)^{n-i}i! \implies D(x) = I(x)e^{-x}\text{ (recall }\sum_{n=0}^\infty (-1)^n\frac{x^n}{n!}=e^{-x}\text{)}[/eqn]
Now, fixing the elements with a specific fix amount and calculating their contribution
[eqn] \sum_{w\in G_n}\text{fix}(w) = \sum_{f=0}^n f \binom{n}{f}D_{n-f} \implies \sum_{n=0}^\infty\left\{\sum_{w\in G_n}\text{fix}(w)\right\} \frac{x^n}{n!}= xe^{x}I(x)e^{-x} = \sum_{n=1}^\infty x^n =\sum_{n=1}^\infty n!\frac{x^n}{n!} [/eqn]

The result follows (for [math]n\geq1[/math]).

[Anonymous No.15179336]

Is the following implication of Choice equivalent to it:
>For each set [math]X[/math] of nonempty sets, there exists a set [math]I[/math] that for each [math]A \in X[/math] holds exactly one [math]x\in A[/math]

This is ostensibly weaker than postulating a choice function [math]f:X\to\bigcup X[/math] exists, as I'm only asking for the range [math]I=f(X)[/math] to exist as a set.

[Anonymous No.15179385]

>>15179336
Anon, that principle is not weaker, but stronger than AC.
In fact, stated like this it is way too strong and adopting it as an axiom would make your set theory inconsistent.

To see why, consider the family (of nonempty sets) [math]X = \{ \{0\}, \{1\}, \{0,1\}\}[/math].
If there was an [math]I[/math] such that for every [math]A\in X[/math] there is one and only one [math]x\in A\cap I[/math], then you may argue as follows:
Let [math]A=\{0\}\in X[/math]. There is (a unique, but we only require existence for now) [math]x\in\{0\}\cap I[/math]. But that implies [math]0\in I[/math].
A similar argument shows that [math]1\in I[/math]. So [math]\{0,1\}\subseteq I[/math].
Now let [math]A=\{0,1\}\in X[/math]. Now we use uniquess; there is at most one [math]x\in\{0,1\}\cap I[/math]. But since [math]\{0,1\}\subseteq I[/math], we have that [math]\{0,1\}\cap I=\{0,1\}[/math].
So, any two elements in [math]\{0,1\}[/math] must be equal; in particular [math]0=1[/math].

All that said, you may weaken your choice principle to only allow families of nonempty sets that are pairwise disjoint.
The resulting principle is equivalent to choice (in the "every set has a choice function" sense).

[Anonymous No.15179414]

>>15179385
>x ∈ A ∩ I
To be frank, this looks like you misread what I asked because you're bored. Of course I mean one x ∈ A. One element for each A. Should be clear by the second phrasing of I being the image of f.
*shrugs*

[Anonymous No.15179417]

As in, for each A, you pick one element from it.

[Anonymous No.15179420]

>>15179414
No I just assumed it was a typo since it makes little sense to say that there is an I such that [something holds that doesn't even mention I].
Taking your choice principle literally it reads "Every family of nonempty sets is a family of singletons".
That this is inconsistent with ZF should be even clearer.

[Anonymous No.15179498]

Uhhhhhh /mg/ explain yourselves:
>>>/wsr/1315004
>I made a math book and it needs a front cover and a back cover. For the front cover I need two maids with huge boobs

[Anonymous No.15179590]

>>15179295
Magnificent, anon. Congratulations!
Thank you a lot for taking the time to solve this problem. I've checked your solution to the best of my abilities and I believe it's correct. Very nicely done. Concise but you made every step clear.
Again, I really appreciate you taking the time to not only give a solution but give one this clear and well written.

I'll mention, the combinatorial solution is... ridiculously easy. I found it like maybe 5 minutes after posting the problem. But of course your solution is just impressive. Algebraic one I have no idea.

I'm at the library now, I'll return home soon. Let me know if you'd like me to post the algebraic solution from the book if you're interested in seeing it. Also I make these shitty anime girls holding math books edits. Let me know if I can make one for you as a small thank you for your solution. Either way, i hope you have a great day.

[Anonymous No.15179603]

/agdg/ brainlet here. I'm trying to minimize the numerical error in my orbits. To illustrate why, consider a renderer for an Earth-Sun system. The camera is located at the origin, the Sun and Earth at $r_s$ and $r_e$ respectively. To calculate $r_e$, we can use the parametric circle equation as an approximation with $r_e = (x_e, y_e), x_e(t) = r\cos(t) + x_s, y_e(t) = r\sin(t) + y_s$. Notice that when the magnitude of $r_e$ is smallest and most in need of precision (Earth is near camera), the equation becomes the sum/difference of two large numbers and therefore subject to the largest numerical error. My intuition tells me it should be possible to rearrange the equations such that $r_e$ is a function of $t$ and some proximate $r_e(t_0)$ so that I don't need to take the sum or difference of large numbers unless the $r_e$ is already going to be large. Is this actually possible? My retard-tier trig skills have gotten me nowhere and I can't think of any fancier techniques I could use. If it is possible, can the method for circular orbits be extended to Keplerian orbits? Thanks.

[Anonymous No.15179618]

>>15179420
Then again (or differently)
Is the following implication of Choice equivalent to it?:
>For each set X of nonempty sets, there exists a set I that for each A∈X holds a x∈A

If choice holds, then the image of the choice function validates this. For two different A's, the x can be the same.

[Anonymous No.15179622]

>>15179498
he likes to hang around on /sqt/, go ask him there

[Anonymous No.15179659]

>>15177122
My brain is on vacation. I cannot into understanding this.
Please help.

By intuition f(x) goes to infinity, one case is simple, the other is not so simple.

[Anonymous No.15179665]

I don't even know my times tables

[Anonymous No.15179677]

>>15179659
What don't you understand?

>>15179665
Then learn them?

[Anonymous No.15179682]

>>15179618
Assuming this is supposed to read [math]\forall X (\varnothing\notin X \rightarrow\exists I\forall A(A \in X \rightarrow \exists x x\in A))[/math], then this is just a theorem of ZF (literally boiling down to every family of nonempty sets is a family of inhabited sets).

If you're trying to state something that looks like a choice principle then the formula following [math]\exists I[/math] better contain [math]I[/math] as a free variable, otherwise you can just drop that quantifier all together ([math]\forall x \sigma[/math], [math]\exists x \sigma[/math] and [math]\sigma[/math] are all equivalent if [math]x[/math] isn't free in [math]\sigma[/math])

[Anonymous No.15179686]

>>15179677
How do I prove that f(x) = lim fn(x)? For the non-trivial case.

[Anonymous No.15179700]

>>15179677(checked)
I learn one, but then when I learn the next set, I forget the one I learned before

[Anonymous No.15179702]

>>15179686
Just see when a given point ends up in whichever A_{n,k}. Sorry but I cant be arsed with more details

[Anonymous No.15179712]

>>15179700
Skill issue.

[Anonymous No.15179716]

>>15179682
Come on man

I said
>there exists a set I that for each A∈X holds a x∈A
Okay, let me give another try:
>there exists a set I, such that for each A∈X, this set I holds a x∈A

I would argue that my very first statement was clear, especially since I get the alternative statement of it in terms of the image of a choice function.
I think the reading is perfectly clear... But I also don't want to start a fight here.

[Anonymous No.15179717]

>>15179716
Not the guy you're replying to, but
>this set holds a x
Terrible English.

[Anonymous No.15179725]

If I say
>there exists a shelf at school that for each pupil holds a book of that pupil

you wouldn't interpret it as
>there exists a shelf at school that for each pupil that pupil has a book

Especially since you're already asking where the I is in the last statement, then if you're honest you'd come to the right reading of the statement

[Anonymous No.15179732]

>>15179717
This is the extra-explicit phrasing, but I suppose whether "this container holds an element" is pretty English is also besides the point.

[Anonymous No.15179745]

>>15179716
>there exists a set I, such that for each A∈X, this set I holds a x∈A
ok, while this phrasing sounds weird in English, I'm assuming you mean something that like this (which is a reasonable thing to ask):

[math]\forall X(\varnothing\notin X\rightarrow\exists I\forall A(A\in X\rightarrow \exists x(x\in A \land x\in I)))[/math]
I.e. "For every family [math]X[/math] of nonempty sets, there is a set [math]I[/math], such that for every [math]A\in X[/math] there is an [math]x[/math] in both [math]A[/math] and [math]I[/math]."

Now first of all, this goes back to my original assumption (here >>15179385) that you meant to say [math]x\in A\cap I[/math], since that's just saying that [math]x\in A\land x\in I[/math].
Now stated like this, it's still not quite a choice principle as it's a theorem of ZF:
Take any family of nonempty sets [math]X[/math], then taking [math]I=\bigcup X[/math] will let you prove that for every [math]A\in X[/math] there is an [math]x\in A\cap I[/math]. (Instead of choosing one element at a time, we got away with just taking all of them).

If you strengthen your principle to say [math]\exists!x[/math], then we get something that looks much more like a choice principle and it's also how I interpreted the question originally. Unfortunately now it's too strong unless we also weaken our premise to only talk about pairwise disjoint families (again, this was what my original message was about).

[Anonymous No.15179750]

NERDS

[Anonymous No.15179756]

>>15179732
I meant your English was imprecise and ungainly to the point of being unreadable.

[Anonymous No.15179758]

>>15179745
k thx

[Anonymous No.15179764]

I meant your English was imprecise and ungainly to the point of being unreadable.
It would have been better to just write everything in formal set theory notation.

[Anonymous No.15179769]

>>15179732
I meant your English was imprecise and ungainly to the point of being unreadable.
It would have been better to just write everything out in formal set-theoretic notation.

[Anonymous No.15179884]

>mfw constructivists are the communists of mathematics
>mfw the finitists are the trad-LARPers of mathematics
>mfw ultra-finitists are the anarcho-primitivists of mathematics
>mfw ZFC enjoyers are the globohomo neo-liberals of mathematics
Pick your poison.

[Anonymous No.15179905]

>>15179287
That is correct.

[Anonymous No.15179927]

>>15179884
I will pick the poison that is doing math in whatever system is most convenient for me, depending on the situation.

[Anonymous No.15179932]

>>15179884
Kek. But, damn, are these the only options currently available?

[Anonymous No.15179944]

>>15179927
ZFC then
>>15179932
I think all options are subsumend into one of the 4 "ideologies"

[Anonymous No.15179954]

>>15179884
What about ZF+AD chads?

[Anonymous No.15179974]

>>15179884
All set theorists are jewish anyway so...

[Anonymous No.15179998]

>>15179954
these are just nihilist contrarians

[Anonymous No.15180002]

>>15179954
AC does much more good than bad imo, I think measurefags should just take the hit for the rest of us

[Anonymous No.15180011]

>>15179884
I would go Genghis Khan on that eskimo retard.

[Anonymous No.15180019]

>>15180011
based but also rudepilled

[Anonymous No.15180047]

>>15177418
Borsok-Ulam Thm applies to every surface homeomorphic to a sphere retard

[Anonymous No.15180048]

>>15179884
I'll take the one that pisses you off the most

[Anonymous No.15180056]

>>15180048
they all piss me off a bit desu

[Anonymous No.15180059]

>>15180056
Then I'm all of them

[Anonymous No.15180080]

>>15179884
For me, it's the Solovay model.

[Anonymous No.15180123]

>>15180080
>you can break apart a set and get more parts than elements
miss me with that shit homie.

[Anonymous No.15180131]

Has anyone tried an axiomatization of set theory which is basically ZF + all sets are countable? You could use the countable reals in place of typical reals. I think this would probably satisfy most people.

[Anonymous No.15180202]

I have a wheel with 12 numbers on it just like a clock. I spin the wheel a few times and note the lowest number on the wheel when the wheel comes to rest. How to find the average of the lowest points I got when spinning the wheel? For example if I spun the wheel three times and the lowest points were at 3,4,5 then the average lowest point is 4. But the problem is that the number sort of wrap around. So if I get 10,11,12,1 how do I calculate the lowest point?

[Anonymous No.15180280]

Is there an algebraically closed field which is not equal to (strictly larger than) the algebraic closure of any of its proper subfields?

[Anonymous No.15180380]

[Anonymous No.15180410]

Suppose that x =/= 0. Therefore x > 0. We have that for every e > 0, e > x. But then x > x, contradiction. Therefore x = 0.

[Anonymous No.15180412]

>>15180410
this is meant for
>>15180380

[Anonymous No.15180448]

>>15180202
wtf are you talking about? the lowest number on a clock is always 1. or if you pick the number at a specific location when at rest it's singular, there is no 'lowest'.

[Anonymous No.15180458]

>>15180202
Maybe I would try to find the average location when it falls between 1-6, and 7-12 separately, then combine them in the obvious way

[Anonymous No.15180465]

>>15179884
>mfw constructivists are the communists of mathematics
And both are correct.

[Anonymous No.15180475]

>>15180465
>two more weeks till the rate of profit crosses the x axis
The entire history of the west can be reduced to the "two more weeks" meme

[Anonymous No.15180482]

>>15179974
Modern set theorists don't bother with any of this pointless shit, ironically
>>15179232
>Not both at once.
It's both at once, actually. If we're talking about just the equator, then yeah not both at once

[Anonymous No.15180515]

>>15180131
ZF includes both the naturals N, as well as the powerset axiom, stating that for every set X, the class P(X) of the subsets of X is a set.
It's a basic feature, with a simple proof, that there's never a surjection from a set X to P(X).
In turn, P(N) will be uncountable.
There's a few types of set theories without naturals or without powerset.

If you drop the naturals from ZF, you get a theory for which the hereditarily finite sets are a model ("everything is finite"). That model (with its simple binary membership relation) is maybe even better understood as a collection of graphs. (There's graph models also for weird set theories and this situaiton is a very simple special case.)
I suppose that's a sort of Wildbergerian world, if he'd do formal logic formalized theories at all.
But in reality, if this theory where N is not an object, "is" really only a theory of arithmetic, or rather there's bisimulations with first-order arithmetic. E.g. in arithmetic, you can define the primitive recursive relation
[math] x\in y\iff \exists (r < 2^x). \exists (s < y). \big(2^x (2 s + 1)+ r = y\big) [/math]
and in this way every number becomes a certain ZF-Inf set. Here all sets are ordered in a line and every number's binary representation says which numbers/sets are in it (e.g. 53_{10} = 110101_{2} is a set holding four other sets)
First order arithmetic however doesn't even have sequences as terms, if that's something you'd want to work wiht.

More interesting is to drop powerset. The go to classical theory is Kripke-Platek set theory with or without Infinity (everything I mention here has nice Wikipedia articles), which is actually also more conservative in a computability sense. E.g. it demands the subset comprehension to be sort of computable. Something which also MacLane (think topos theory) adopted for his Z like set theory.
Problem is Kripke-Platek set theory, without powerset, also doesn't have the greatest function space math (which type theory has)

cont.

[Anonymous No.15180516]

>>15180515
Another interesting alley is to drop excluded middle, since then the concept of "function" is a much more restrictive one.
E.g. Peano arithmetic PA proves that every program either halts or does not halt, which makes "halts: (N^N)->{yes, no}" a functional association. Meanwhile, in the metalogic we find that there's actually no computable function for which PA would prove that it decides the halting problem, so the above "function" is not of the computable kind (a bit like how Choice postulates there's functions that we however know not to be of the computable kind). In turn, constructively you can postulate that the function spaces like N->{0,1} a.k.a. 2^N only hold computable functions, and this object then becomes "much smaller" than the elusive P(N) and you can postulate it to exist without needing to postulate powerset.
The constructive set theories like that are nice and they are close to type theory: You can postulate that Y^X exists even for countably infinite sets Y^X, while also postulating that Y^X is not bigger than N, in the sense that a subset I_{X^Y} of N surjects onto Y^X. E.g. you can list all the computable functions from N->N, which can be taken to equal N->N. And so indeed, you can here postulate that all sets A in this theory correspond to some index set I_A of S.
Although without excluded middle you have to rethink both the meaning of "function" as well as that of "subset". This theory however has no Dedekind reals and the Cauchy reals a priori have no modulus of convergence. The index sets I will be subsets of N, but they aren't countable (because for something to be countable you'd need to be able to actually count them (programmatically), which is not something classical ZF ever requires.)
Finally, if you further give up on excluded middle and countable choice, you can turn something like Dedekind reals countable.

[Anonymous No.15180520]

index set I_A of N

[Anonymous No.15180522]

index set I_A of N

Maybe as a throwin, I can also mention the small pocket set theory
https://en.wikipedia.org/wiki/Pocket_set_theory
(https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory))

The former doesn't match your request of eveything being countable to

[Anonymous No.15180651]

>>15180516
>Finally, if you further give up on excluded middle and countable choice, you can turn something like Dedekind reals countable
OK that is pretty epic. Where can I read more about this?

[Anonymous No.15180667]

>>15180516

[Anonymous No.15180673]

>>15179884
You're forgetting traditional Platonism. When you observe a particular your intellect grasps the universals that it instantiates. This is why it's useful to construct particulars, even though the mathematical objects themselves are not constructed. Also infinity exists, it's just unqualified greatness.

[Anonymous No.15180790]

>>15180651
https://www.youtube.com/watch?v=4CBFUojXoq4

>>15180667
Why does it not like 'computable'?

[Anonymous No.15180826]

statements dreamt up by the utterly deranged: the weak-* topology is weaker than the weak topology on [math]X^*[/math]

[Anonymous No.15180923]

>>15180667
It usually outputs a lot of iq with math key words.

[Anonymous No.15180934]

Please help my retardation. What am I missing with this "counterexample to Dilworth's theorem"?
Let's have a poset {a, b, c, d, e}, such that
[math]a \leq b, c \\
d \leq b \\
e \leq c.[/math]
And there are no other relations (except reflexivity), transitivity shouldn't enforce any other.
It seems like I need at least 4 chains for a chain decomposition, but the largest antichain I see has size 3. What on earth am I missing?

[Anonymous No.15181065]

>>15180934
Surely {{a},{b,d},{c,e}} is the smallest chain decomp?

[Anonymous No.15181095]

how come that there are so many logic and foundation related posts in these threads
i thought that was a niche topic

[Anonymous No.15181131]

>>15179659
I'm confused about this still.
The K makes everything super weird.

There's no calculation for k, so this must be a sum. But then nothing makes sense at all.
I don't know what to do, or what parts are involved (I guess from real analysis, we did not go through series).

[Anonymous No.15181196]

>>15179295
>Where Dn is the number of derangements.
4chan

[Anonymous No.15181537]

Well, /sci/? What shape is that?

[Anonymous No.15181554]

>>15181537
The boundary of the dog is, assuming you use some kind of vector graphics, a connected curve with somewhere between 0 and 46 first-derivative discontinuities, depending on whether the corners I counted turn into curves with a greater zoom.

[Anonymous No.15181594]

>>15179055
>>15179590
>Algebraic one I have no idea.
I assume it means using Burnside's lemma, for which this result is literally just plugging the symmetric group into the statement

I don't see the bijection right away, although with bijections there's always the distinct possibility I'm just dumb

[Anonymous No.15181614]

>>15181095
Comes in tends, based on what posters at the time care about.

When I was into category theory 6 years ago, I responded to category theory questions and there was a lot more category theory.

[Anonymous No.15181636]

>>15180934
my best guess is that you have just misread the definition of chain decomposition. the other anon’s answer is correct

[Anonymous No.15181647]

>>15177144
It always was though. Trying to get a decent mathematics discussion going here is just naive. Lucky to get a couple of decent posts before any thread just degenerates into retards flinging pieces of shit at each other, drooling with pleasure as they do it.
The only sensible people are those who come here to blow off some steam by mocking them.

[Anonymous No.15181735]

>>15181065
Thanks. I knew it was something simple as this.

[Anonymous No.15181802]

>>15178053
The trick is always to do whatever so you can use the assumption and get your solution.

[Anonymous No.15181816]

What is the best introductory discrete mathematics text

[Anonymous No.15182133]

This proof breaks even with the slightest amount of scrutiny. I do not understand how this proof made it into the book or was accepted by anyone.

This is extremely unclear what it means with k. n and k appears to be indexes, but since k is growing, it can't be considered a matrix...
So I suppose then, A_n,k is a collection of sets (I guess as it should).

But then it assumes that f(x) = lim fn(x), for x in A.
A_n,k is a subset of A.

if we fix an x, and we determine that f(x) = c, with an x in A, but not in A_n,k. Then fn(x) goes to infinity as n goes to infinity. Which is a contradiction and clearly breaks this proof.

[Anonymous No.15182206]

>>15182133
The proof in that book is correct.
>if we fix an x, and we determine that f(x) = c, with an x in A, but not in A_n,k.
For every [math]n > c[/math] you can find a [math]k[/math] with [math]x \in A_{n,k}[/math].
>Then fn(x) goes to infinity as n goes to infinity.
Certainly not unless [math]c = \infty[/math]. Every [math]f_n(x)[/math] is by construction always smaller or equal than [math]f(x)[/math].

[Anonymous No.15182220]

in my abstract algebra class it seems that there are handful of theorems that are essentially the same theorem but for different structures (think like first isomorphism theorem for groups, rings, ...)
in class we study the structures one by one. is there a way to study them "all at once"? like is there a kind of overarching first isomorphism theorem from which all the special ones (for groups, rings, ...) are derived?

[Anonymous No.15182233]

>>15182220
These isomorphism theorems generalize in universal algebra.

[Anonymous No.15182236]

>>15182206
Aaah.
So the point is that we can't find an x that is not in A_n,k?

>But clearly we can, because all x in A_n,k are rational numbers, so let c in irrational numbers.
>Then the x not in A_n,k, and then all we can do is choose integers n, and integers are rational. Therefore it doesn't hold.
But since we take numbers between k/n^2 and (k-1)/n^2 we get all the irrational numbers too.

Then why the fuck didn't the proof say so? Takes 8 words to spell that out.

[Anonymous No.15182297]

I'm running into trouble solving a limit
I've got it to this:
[math]
lim x->2
[(2-x)] / [ (x^2-4)*(sqroot(2)+sqroot(x)) ]
[/math]


It started out like this:
[math]
(Sqroot(2) - sqroot(x) ) /
( x^2 - 4 )
[/math]

Its conjugate method I guess.

[Anonymous No.15182325]

>>15182297
>fucked up latex
not helping lol.
The limit is trivial though, no weird techniques used.

[Anonymous No.15182337]

What the fuck is the point of calculus when you can use linear algebra for everything?

[Anonymous No.15182351]

>>15182325
Thanks for replying!
Well, yes. I'm still learning the simple methods.
There I've multiplied both sides of the division by the conjugate of the numerator.
I've seen that there you're meant to factor both sides, but I don't think I see how that could be done in this case.
I think I posted with a bit of haste a moment ago. Sorry for that.

Any help would be appreciated.

[Anonymous No.15182360]

>>15182337
You can? How do you take the limit of something with linear algebra?

[Anonymous No.15182372]

>>15182360
okay, ONE thing. You can't do with linear algebra. I feel like I got trolled into doing 3 semesters of calculus before taking my first linear algebra class. Half the shit in calculus III could be done with linear algebra in a tenth of the time.

[Anonymous No.15182378]

>>15182372
>Half the shit in calculus III could be done with linear algebra in a tenth of the time.
Calculus III literally IS linear algebra, what are you talking about

[Anonymous No.15182387]

>>15182378
I took calculus III without first taking linear algebra. Now I am taking linear algebra and I feel bamboozled for wasting time setting up triple integrals for shapes in 3D space, when the same could be done in a much faster manner using linear algebra. That is what I'm talking about.

[Anonymous No.15182399]

>>15182387
>I took calculus III without first taking linear algebra.
That's on your school for being retarded enough to allow you to do this. The entirety of calculus III is just mixing calculus I+II with linear algebra. You shouldn't even be allowed to register without a linear algebra course

[Anonymous No.15182407]

>>15182399
>B in calculus III because I ran out of time during tests setting up massive integrals

[Anonymous No.15182423]

>>15179261
Am i wrong to say that the sequence converges to zero. By Abel's test the series without 1/n converges to a number. So, the numerator converges, while the denominator 1/n keeps increasing. Thus the series converges to zero.

Am i wrong?

[Anonymous No.15182451]

>>15182351
I could do it

[Anonymous No.15182454]

Do you have those useful infographics about maths? Specifically on how to get started in mathematics.

[Anonymous No.15182490]

Fuck this book and fuck this author.
Why not just write the proof instead of making outrageous proposition without any backing???

If f and g are mu almost everywhere, then they are not so at some x. i.e. A is not empty. Then the integral of h is infinite, not 0. It can only become zero if A is empty, but by assumption it's not empty.

[Anonymous No.15182506]

>>15182490
ok whatever, it's because it's mu negliable and mu(A) = 0.
still a fucking asshole of an author.

[Anonymous No.15182509]

>>15177122
Unrelated but
What the fuck is that map?

[Anonymous No.15182537]

what is a straight line? i looked it up, it said it's with no curvature, but curvature was said to be how much deviation there is from a straight line, so it's a circular explanation. I looked it up more and it said straight lines are given by axioms. Is that all there is to it? Just a axiom? Is there not any definition of a straight line? What about it systems only based on the axioms of logic, what is a straight line defined as then? I am struggling, i don't know anything about anything.

[Anonymous No.15182542]

>study [math course]
>calmly study for 6+ hours uninterrupted, actually learn
>study [physics course]
>frustrated converting doofeldorfs to 15.824 (rounded to 20) smegmameters^3 per Coulomb kilo mole-joule
>start browsing 4chan, cleaning the house, finding errands to do instead

I can't be the only one. Why is this shit mandatory. I hear in the UK you can do your entire degree only taking math courses and nothing else.

[Anonymous No.15182550]

>>15182537
From an artist's point of view, a straight line would be the shortest path between to points in space in a 2d/3d support

[Anonymous No.15182555]

>>15182542
Can't you just take out bachelors by doing your own courses in the US?

I pick the courses I want and then I'll do my thesis and take out my bachelor. I'm not in any program, just studying individual courses.

[Anonymous No.15182557]

>>15179287
Do you have a key somewhere i can use to understand what the strange symbols mean and what order to operate?

[Anonymous No.15182564]

>>15182555
You have to follow a program of study in the US, with minimum mandatory courses. I am lucky that at my university I am not required to take a foreign language course, since at a neighboring one I would be.
For my uni besides the usual math courses, I am required to take 3 physics courses and 3 programming courses. The only fun things from these are that numerical methods in physics is just taking an easier and applied numerical analysis, and my third CS course was algos and data structures, which was kinda fun I guess. Also getting to use matlab in physics, whoever made matlab deserves lots of love.

We can pick some math courses that we want, and those fall under the math electives portion of the degree program. You also can't take certain courses from a department unless you are majoring in that field or study, or you get approval from the department.

[Anonymous No.15182570]

>>15182454
wikipedia for
element (mathematics)
glossary of mathematical symbols
wolfram mathematica,

etc

[Anonymous No.15182590]

>the argument is similar and has been assigned as an exercise

[Anonymous No.15182681]

>>15177122
[math]\prod_{i \in I} \sum_{j \in J} a_{ij} = \sum_{f: I \to J} \prod_{i \in I} a_{i f(i)}[/math]
[eqn]\prod_{i \in I} \sum_{j \in J} a_{ij} = \sum_{f: I \to J} \prod_{i \in I} a_{i f(i)}[/eqn]

[Anonymous No.15182771]

Consider the tensor:

SFS:= Sneed's Feed & Seed + Seed's Sneed & Feed + Feed's Seed & Sneed

It is clear that this set is invariant under cyclic permutations. Moreover we can represent this in the general form as an operator

[math] SFS(x):= (x) F(x)\otimes S(x) + S(x) (x)otimesF(x) + F(x) S(x)\otimes(x) [/math]

In this case, the value x= Sneed gives SFS as represented above. What is the value of SFS when x= Chuck?

[Anonymous No.15182839]

>>15182771
Chuck's Fuck & Suck + Suck's Chuck & Fuck + Fuck's Suck & Chuck

[Anonymous No.15182947]

how do i solve it?
[math]e^x - x > 0[/math]

[Anonymous No.15182952]

>>15182947
i know that it is true but i got stuck at
[math
x > log(x)/2
[/math]

[Anonymous No.15182955]

>>15182947
Depends on how you define it, but it's pretty easy if you use the exponential series.

[Anonymous No.15182959]

>>15182955
i'm studying a function and this is the first derivative, i just don't how to formalize the fact that this is always true

[Anonymous No.15182982]

>>15182959
You could consider the derivative [math]e^x - 1[/math] which obviously only has the zero [math]x = 0[/math] and since the second derivative [math]e^x[/math] is positive at the zero, it's a minimum. Since the function itself is positive at [math]x = 0[/math], the function is positive everywhere.

[Anonymous No.15182992]

what are some good fringe maths youtube channels

[Anonymous No.15183005]

>>15182982
>You could consider the derivative ex−1
how did you choose this?

[Anonymous No.15183008]

>>15183005
I'm just considering the derivative of the function [math]e^x - x[/math], which is [math]e^x - 1[/math].

[Anonymous No.15183017]

>>15183008
alright, so i guess the most important part is this
>Since the function itself is positive at x=0, the function is positive everywhere.
how can i apply this property to something like this?
[math]x - cosx > 0[/math]
sorry for the retarded questions

[Anonymous No.15183021]

>>15183017
wait, i didn't think about this one much.
i can use the fact that [math]-1 <= cosx <= 1[/math]?

[Anonymous No.15183024]

>>15177294
might be misremembering, but I saw that problem as "Tantalus problem" somewhere roughly 30 years ago

[Anonymous No.15183033]

>>15183017
I mean, [math]x - cos(x) > 0[/math] is false if you consider for example [math]x = -\dfrac{\pi}{2}[/math].

[Anonymous No.15183046]

>>15183033
what can i do then?

[Anonymous No.15183093]

What's this symbol called? Does it have a code in Latex?

[Anonymous No.15183147]

>>15183046
What's your goal? As far as I understand, your goal with the expression [math]e^x - x > 0[/math] was to show that it is true for all real numbers [math]x[/math]. However, I don't understand what your goal regarding the expression [math]x - cos(x) > 0[/math] is.

[Anonymous No.15183246]

>>15182454

[Anonymous No.15183500]

I love set theory bros...

[Anonymous No.15183654]

>>15183500
basado

[Anonymous No.15183661]

>>15183500
You like set theory? Give an example of well order on [math]\mathbb R[/math]

[Anonymous No.15183969]

>>15183246
this graphic is retarded

[Anonymous No.15183974]

>>15183969
I kind of like it. Seems like a good approach to deal with the fundamentals and how to approach math before just diving into the deep end.

[Anonymous No.15183980]

the whole field of diff equations is so god awful
>yeah bro they're pretty much all unsolvable so thats why we use computers to approximate them!

[Anonymous No.15183992]

>>15183974
The Smith text is an introductory philosophical logic text. No reason to start out with this as I doubt 500 pages of drawing truth trees and translating English sentences into symbols will accomplish much. Meanwhile, when the text does become a bit more mathematical (usually when proving metalogical stuff), the readers with little mathematical maturity (it is after all the first book in the list) will probably be lost. I don't think a student that has never seen inductive definitions and hasn't done many induction proofs (other than maybe some HS level ones inducting on N) will follow arguments inducting on formulas or on derivations.
So text should be deleted from the chart or replaced with a mathematical logic text and moved to the very end.

The proof books are a good suggestion and should be the starting point of the chart. Only confusing thing is why "A Transition to Advanced Mathematics" isn't grouped together with Hammack and Velleman. It's literally another intro to proofs book (and definitely not more advanced than the other two), they're all kinda interchangeable.
I will say that Velleman should be the highlighted one in a graphic that's supposed to be "A Foundational Approach". Velleman is by far the closest to being an informal introduction to logic and set theory (no surprise there since he did his phd under Kunen, so naturally his presentation will be quite foundationpilled).

The set theory books would be kinda out of place here, but since it's "A Foundational Approach" I'll give it a pass. And they're both very elementary and well written so good choices, but possibly questionable placement in the chart, especially if one is just coming from a book like Velleman which honestly already covers most of the important content in either set theory book, just in an informal manner. (+ I doubt the student will have much use for stuff like transfinite induction & recursion, cardinal arithmetic etc. at this point)

[Anonymous No.15183995]

>>15183992
The Landau text should probably be skipped (especially if one actually read all of Enderton previously). Alternatively, just move it at the end of the chart as optional reading.

I really dislike that Lang book but I guess it works for some people. In any case, why is that even so far down and not one of the very first books lol

Can't speak for any of the remaining books but they don't seem necessary at all (tbf they're marked as optional)

Finally, this chart ends at fucking calculus lol
If that was the goal from the very beginning, one could've just gone straight from Velleman (+ possibly Lang) to the calc book of your choice

[Anonymous No.15184124]

>>15183147
the same goal as before, to prove it

[Anonymous No.15184191]

can anyone help me figure out where did cosx go? thanks

[Anonymous No.15184193]

>>15182590
>spoonfeed me

[Anonymous No.15184196]

>>15183500
Me too

[Anonymous No.15184273]

>>15183661
I'm a determinancy enjoyer
>inb4 b-but then there are vector spaces with no baserinooooooos!!!!!!!!!
ok nigga and? Maybe some vector spaces are just too big to have a basis, have you ever though about that?

[Jealous No.15184296]

>>15184191
That's it next to the "dx"

[Anonymous No.15184303]

>>15179055
By induction: G_1 = {Id} so it fixes just one point.
For the induction step, consider G_{n+1}.
We split G_{n+1} into the sets A of permutations that fix (n+1) and the set B of permutations that don't.
Notice that for all [math]\sigma \in A[/math] you can restrict it to [n] to obtain a permutation, so A has n! elements and thus
[math] \displaystyle \sum_{\sigma\in A} \operatorname{fix}(\sigma) = \sum_{\sigma\in G_n} 1 + \operatorname{fix}(\sigma) = n! + n! [/math]
For B it is more complicated, but we'll try a similar approach. We can split the permutations regarding the cycle of (n+1),
thus if (n+1) is in a cycle of length m, you get a copy of G_{n+1-m} outside; we'll denote this set as C_m, thus [math]B = C_2 \cup C_3 \cup \cdots \cup C_n[/math] (we erase [math]C_{n+1}[/math] since it doesn't add anything).
Now C_m has two parts: how many m cycles starting from (n+1) there are, which is [math]n \cdot (n-1) \cdots (n - m + 2) = \frac{n!}{(n+1-m)!}[/math]; and how many elements does it fix outside which is, by strong induction, (n + 1 - m)!. So
[math] \displaystyle \sum_{m=2}^n \sum_{\sigma \in C_m} \operatorname{fix}(\sigma) = \sum_{m=2}^n n! = n!(n-1) [/math]
which completes the proof.

[Anonymous No.15184323]

>>15180123
At contrary, in the Solovay model all subsets of R are measurable and thus the Banach-Tarski paradox fails here.

>>15184191
[math] \dfrac{x \cos x}{x^2 - \alpha^2} [/math] is an odd function, so its integral from [math] (-\infty, \infty) [/math] vanishes.

[Anonymous No.15184412]

>>15184323
>is an odd function, so its integral vanishes.
i thought that i need to somehow evaluate the integral before i can remove the cosx expression, but i guess i dont have to. thanks a lot friend.

[Anonymous No.15184448]

>>15184273
>okay if it exists then what does it look like
>I DON'T KNOW BRO I JUST ASSUMED IT ASSOOOOOOOOOOOOOOMMMMMED

[Anonymous No.15184508]

>>15177122
I was reading Amann/Escher Analysis I but I got filtered by the AM-GM inequality exercise, should I KYS?

[Anonymous No.15184520]

>>15184508
>should I kill yourself

[Anonymous No.15184528]

>>15182490
read something long like Bogachev for measure theory if you want everything proven

[Anonymous No.15184530]

>>15184508
if you spill a cup, you simply pour more in. Don't give up bro. Isn't math supposed to be about understanding things?

[Anonymous No.15184536]

>>15184530
Ok now for a more serious question. I have trouble skipping over exercises in books I am reading (because of autism or something, idk), even when I'm really struggling with them. Does that make sense? Or should I just do the ones I can do and continue with the following chapters and return to the hard ones later when I have some more time?

[Anonymous No.15184554]

>>15183500
ZFC is cringe

[Anonymous No.15184570]

>>15184536
>Or should I just do the ones I can do and continue with the following chapters and return to the hard ones later when I have some more time?
this. And there's no shame in asking for hints in stackexchange.

[Anonymous No.15184686]

>>15184554
for me it's ZF + V = ultimate L

[Anonymous No.15184953]

>>15184412
Since its addition the integrals can be separated

[Anonymous No.15185034]

>>15184448
it's makes me wonder why mathematicians keep projecting their finite intuitions into infinite contexts when time and time again it has been shown how it never works, we have tons of examples from analysis then when you take the limit weird things can happen. The axiom of choice is just the latest attempt to do this, and of course, it's makes things look just like the finite case, how convinient! And it's just as wrong...

[Anonymous No.15185178]

>>15182992
fringe how

[Anonymous No.15185202]

>>15185178
fringe in that they are built on alternative foundations that are outside the mainstream

[Anonymous No.15185527]

>>15184686
Go to bed Woodin

[Anonymous No.15185570]

>>15185527
let's be honest, it would be a huge L for the mathematics community

[Anonymous No.15185731]

Does this property hold as long as there are two distinct roots?

[Anonymous No.15185737]

>>15180516
>>15180515
Adderall: the post

[Anonymous No.15186394]

>>15184303
Very clever anon. Very clever. This solution isn't even in the book so extra congratulations. You've done a bretty good job I believe. I didn't really understand one part regarding the B part of the solution however that's really on me, I'll read this again and try to understand because I think your proof is, as I said before, bretty good.
Thank you a lot for your time and effort, I appreciate it. Let me know what you thought of the problem if you wish.

>>15181594
Sorry for the late reply anon, we got hit by fucking earthquakes. Magnitude 7.8 or something. Anyway. You're correct, it's Burnside'e lemma. I looked at the solutions from the book and there it is, Burnside's lemma.
>>15181196
I'm sorry anon, are you calling 4channers deranged?
>>15179295
Also I'm really sorry anon for bothering you but I was just curious about one thing. I don't really see why in your last line we have "f times n choose f" . I would have thought it should be "f! times n choose f". This is probably just me being a retard but I wanted to ask for clarification.


Again, apologies to all of you for the late reply. Earthquake. Keep us in your prayers.

[Anonymous No.15186423]

>>15186394
Holy shit anon are you ok? Hugs from Italy
t. I was in the /mg/ server at some point

[Anonymous No.15186444]

>>15179055
Fix any particular k. The permutations of [1,n] fixing k are in bijection with the permutations of [1,n-1], of which there are (n-1)!. Repeating for all n choices of k, we get n times (n-1)!, which is n!.
>>15186394
>earthquake
Return constantinople at once.

[Anonymous No.15186526]

>>15186423
Thanks a lot italybro. I'm quite okay, my family is far from the epicenter unfortunately some relatives are right on it. They are okay though, they're coming here to stay with us.
Italy sent us some aid I believe. Thanks a lot. You guys are great.
>I was in the /mg/ server at some point
Then perhaps I know who you are :D
Good to see you.
>>15186444
I was wondering when the simplest solution would come. This is what I have done too, quite short and easy though perhaps not as impressive as the ones people here have posted. Congratulations! Nice job.
>Return constantinople at once
Ah, no? It's ours and it's called Istanbul. Come visit sometime, as a tourist it's a pretty cool city.

[Anonymous No.15186531]

>>15177122
A long time ago I went in a thread and asked somebody to tell me books about Colors, Shapes, Numbers and Counting. Somebody told me a book about Geometic Magic Squares.

I like this book a lot and I want to study more about magic squares. What else should I read? Are there other good books about this? I like it when the math book has a very high amount of drawings.

I noticed when I looked for more magic squares information that it is listed as "Recreational math" a lot of times. What makes a math recreational or not recreational? Is it just down it if people think it is fun and it is easy enough to get picked up by hobbyists?

>Gotta question for you guys. What, in your opinion, is the advantage of discussing math here instead of on some other website? Is it just that /sci/ is the least retarded place you know of?

I don't have to login to anything or give away information as a condition of being able to talk about math. Mods usually don't interfere with discussion. Also bans for posting maids are rare now, although yesterday I got a 3 day ban from /a/ for taking about math.

If I tried to do the same posts on the math stackexchange my maids will be deleted and a fat Dunning-Kruger janny will delete my posts about Maid Space and ban me.

There is also less politics, virtue signaling and performative political correctness in the talks here, which makes them more on topic and more useful.

[Anonymous No.15186542]

>>15186526
> They are okay though, they're coming here to stay with us.
Glad to hear that
>Then perhaps I know who you are :D
Maybe, can't imagine there being too many italyfags doing alg geo in the server lol

[Anonymous No.15186599]

>>15186542
I'm glad you are ok too.
>t.THAT GUY

[Anonymous No.15186607]

if preorders are like categories and monotone functions between them are like functors, then what is would be the analogous concept of a natural transformation (between two monotone functions)?

[Anonymous No.15186647]

>>15186607
A natural transformation exists precisely when [math]FX\preceq GX [/math] for all [math]X [/math] in your domain, in which case it is unique. In short, it tells you whether [math]F\preceq G [/math] or not

[Anonymous No.15186654]

>>15186607
f(x) <= g(x) for all x, because look.

With more abstract nonsense:

Posets are equivalent to (0,1)-categories, and (n,r)-categories form a (n+1,k+1)-category, so posets form a (1,2)-category.

A (n,r)-category is such that j-morphisms, if j > r, can only be equivalences, and if j > n, are unique up to equivalence if they exist.

The equivalent of natural transformations in the (1,2)-category of posets would be 2-morphisms, and since 2 > 1, they are unique up to equivalence, so we can expect a poset between any two compatible monotonic functions, just as we have a category between any two compatible functors.

[Anonymous No.15186700]

>>15186394
>I don't really see why in your last line we have "f times n choose f"
There are [math] \binom{n}{f} D_{n-f} [/math] permutations of [math]n[/math] elements and exactly [math]f[/math] fixed points, all of those contribute the value [math]f[/math] to the sum (because there are [math]f[/math] fixed points) so all permutations with [math]f[/math] fixed points contribute a total of [math]f\binom{n}{f}D_{n-f}[/math] to the sum.
>Again, apologies to all of you for the late reply. Earthquake.
Stay strong, turkbro.

[Anonymous No.15186701]

>>15185731
why don't you verify it yourself? Just do some work on the RHS and you may end up in LHS. If you did, what restrictions do you have regarding [math]r_+,r_-[/math]?

[Anonymous No.15186712]

>>15186700
I'm dum dum. Thanks, I see now. Sorry for wasting your time.....
Again, congratulations on your solution.I think it's great.
Let me know what you thought of the problem though if you don't mind. You seem bretty skilled so I was curious if this problem was entertaining for you regardless.
>Stay strong, turkbro.
Thanks! Love you guys.
>>15186599
I do not know who THAT GUY is, I'm sorry.

[Anonymous No.15186848]

>>15186647
>>15186654
thanks anons, I think I understand.
so if P and Q are preorders, their functor category (viewing P, Q as categories) has at most one arrow (= natural transformation) between functors (= monotone functions) and so is basically a preorder again. And that preorder happens to be the pointwise order.
That is very nice

[Anonymous No.15186885]

>>15186712
Do I really have to say it...

[Anonymous No.15186907]

>>15180673
So do you think cardinals and ordinals are fake and gay?

[Anonymous No.15186950]

I don't understand integrals
At least with derivatives there was a "flow chart" to get the answer. Now with integrals it's just all up in a mist.
Am I too stupid?

[Anonymous No.15186956]

>>15182454

[Anonymous No.15186972]

>>15186950
Basic integration is the inverse flowchart of derivation. It's a game of "find the differential".
Integration by parts follows from inverting the product rule: d(xy) = xdy + ydx, so ydx = d(xy) - xdy, integrate both sides and you get ∫ydx = xy - ∫xdy.
Integration by substitution is just inverse chain rule:
if you have ∫h(x)dx and you figure out that h(x) is of the form f(g(x))g'(x)dx for some f and g, then you can take out g(x) by setting u = g(x) and considering only ∫f(u)du, with the hope that f is more easily tractable.

If all fails then you either pull some domain specific trick, or give up and feed the integral to a computer.

[Anonymous No.15186985]

>>15186972
How do I read more about this? Every integral "tutorial" I've seen just shows examples instead of just showing the rules

[Anonymous No.15186992]

>>15177122
If you could tell your high school self anything about mathematics, what would it be?

[Anonymous No.15186994]

>>15186985
Any calculus textbook worth its weight covers this. You only need the two fundamental theorems of integral calculus, and you can derive the rest for yourself by playing around with differentials and Taylor series.

[Anonymous No.15187002]

>>15186992
Study logic in your own free time instead of following formulas like a monkey, it gives you the false impression of maths

[Anonymous No.15187054]

>>15187006
Since you know that either ~A or ~B holds, you'd do a proof by cases: You do two subproofs, one assuming ~A and one assuming ~B and then try to arrive at the same conclusion in each case (here the goal is C or D)
In the subproof assuming ~A you may conclude with C as the implication ~A -> C was part of your premise. Now we may weaken this to C or D (if C holds, then especially the weaker statement C or D holds).
A similar argument is applicable to the ~B case.

This natural language argument just mirrors the rules of a Natural Deduction proof system: The proof by cases stuff corresponds to disjunction-elimination, deriving Q from P, P -> Q is implication-elimination (modus ponens) and going from P to P or Q is disjunction-introduction

[Anonymous No.15187102]

>>15186992
Don't be a retard and get a Phd in Maths. Get a real degree like a medical degree or an engineering degree instead. Or fuck even trade school

[Anonymous No.15187149]

>>15186531
>Colors
https://libgen.li/ads.php?md5=7eb503a8de40359902b3839148261b2b
>Counting
https://ipfs.io/ipfs/bafykbzacecnqtizpfue3yc7hexkwq5nqtl42ljnf4m7hrlfulhnbt4zjv3ewo?filename=a-course-in-enumeration-martin-aigner-graduate-in--annas-archive--libgenrs-nf-337636.pdf
>Numbers
https://ipfs.io/ipfs/bafykbzacedzau7e3a2c2swqabyb5u7bvufjhfy5vujk6pehjkjs6eo6w6kkyg?filename=elementary-number-theory-sixth-edition-david-m-6--annas-archive--libgenrs-nf-789189.pdf
>Shapes
https://ipfs.io/ipfs/bafykbzacecot7tbiwxs5sdk6gkv6xd7dfvuafvf6ofcl7nh2dz5ptcjyvdvcw?filename=homotopical-topology-anatoly-t-fomenko-dmitry-b-in--annas-archive--libgenrs-nf-2130091.pdf

[Anonymous No.15187298]

What's the most elementary way of showing that [math]n^{0.01} = \omega(\log^{10} n)[/math]?
One that's not "look at this plot, dude".

[Anonymous No.15187402]

what did I do wrong here?
>>15187245

[Anonymous No.15187408]

>>15186950
learn the main techniques (integration by parts, substitution, partial fraction decomposition) and do some examples. All integrals you'll have to can be solved by some (even though, often not obvious) variation of these techniques

[Anonymous No.15187410]

should i read book of proof of how to prove it?

[Anonymous No.15187413]

>>15183992
>>15183995

thanks for articulating why I think that chart is dumb and might be a prank

[Anonymous No.15187704]

>>15183980
Once upon a time, there was a partial differential equation. She was unsolvable. The end.

[Anonymous No.15187715]

>>15186950
Integration _can_ get absurdly hard but if you're just taking a calculus class everything should be fairly cookbook and you should be able to recognize what technique you need pretty quickly just from how the integral looks. If you can't, you need to practice more.

[Anonymous No.15187899]

>>15184536
You should do the opposite

[Anonymous No.15188041]

Math is all just triangles and marbles lads

[Anonymous No.15188047]

>>15184303
That is true Neptune? In larbles there is the mass two and its powers.

[Anonymous No.15188423]

>>15187410
how to prove it

[Anonymous No.15188944]

Has anyone read "Contemporary Abstract Algebra" by J.A. Gallian? Is it a good book?

[Anonymous No.15189068]

>>15177122
Can someone suggest a good book for algebra, calculus etc?

[Anonymous No.15189082]

How many hours a day do i have to study to learn math from pre-algebra to calculus in 1 year with no prior knowledge? i am getting an shitty online degree in IT so i have a lot of free time to study.

[Anonymous No.15189114]

>logic class
>lector keeps making snarky logic jokes with double negatives, connectives, vacuous truths, etc

[Anonymous No.15189160]

Is it possible to learn integration and FTC in one day alongside related rates and optimization? What's the most efficient way to learn? I feel like if I have a better grasp of it intuitively practice won't be a problem.
I already have a vague idea, but if you ask me questions regarding it I won't be able to solve them.
Have a test tomorrow.

[Anonymous No.15189277]

>>15189114
Tell him LEJ Brouwer is more chad than he will ever be

[Anonymous No.15189320]

>>15188944
Algebra for absolute beginners. There are a lot of fun exercises though.

[Anonymous No.15189406]

In polar coordinates, how come this creates the bottom hemisphere of a sphere and not the top?
pi/2≤θ≤pi, 0≤φ≤2pi. Is my book munted in the head or what.

[Anonymous No.15189409]

>>15189160
>Is it possible to learn integration and FTC in one day alongside related rates and optimization?
Not from a jedi

[Anonymous No.15189432]

why do mathematicians make up axiomatic systems where it is easier to prove a certain theory, and then later show that that axiomatic system is equivalent to ZFC?

[Anonymous No.15189445]

>>15189409
there must be a way!

[Anonymous No.15189520]

So this proof is wrong apparently, I don't know why and it's bothering me. It doesn't surprise me since I am generally a retard with proofs but
Anyone care to help?

[Anonymous No.15189932]

>>15189432
>why do
You mean why don't?

I think the answer is that basically all axioms for consistent things you're interested in - unless you study set theory itself - are likely going to end up having a model in V already

[Anonymous No.15190348]

>>15189520
First of all, your definition of [math] L_P(f) [/math] and [math] U_P(f) [/math] are wrong.
The first inequality of the second to last line should be an equality, [eqn] U_P(f^2) - L_P(f^2) = \sum _{i=1} ^n [M_i(f^2)-m_i(f^2)](x_i-x_{i-1})[/eqn]

You should use the fact that the interval [math] [x_{i-1}, x_i] [/math] is closed means that there exists [math] x_{i_m} [/math] and [math] x_{i_M} [/math] that take on the values of [math] m_i(f^2) [/math] and [math] M_i(f^2) [/math] so that [eqn] U_P(f^2) - L_P(f^2) = \sum _{i=1} ^n [f(x_{i_M})^2 - f(x_{i_m})^2](x_i-x_{i-1}) \le 2M \sum _{i=1} ^n f(x_{i_M})-f(x_{i_m}) (x_i-x_{i-1}) \le 2M \sum _{i=1} ^n [M_i(f)-m_i(f)](x_i-x_{i-1}) = 2M[U_P(f) - L_P(f)][/eqn]
Where the last inequality follows from [eqn] m_i \le f(x_{i_m}) , f(x_{i_M}) \le M_i [/eqn]

The reason your equality on the 2nd to last line is wrong and why you need to do it in such a roundabout way is that you cannot assume that the values of x which obtain the maximum and minimum of [math] f^2 [/math] on an interval are necessarily the same ones as for [math] f [/math] if the function is not the same sign on the whole interval.

[Anonymous No.15190377]

>>15190348
How do I practice proofing? I always screw up somewhere along the way or assume something which I shouldn't accidentally.
I know the general advice is "practice", but I can't really practice if I don't know how to get the ball rolling. Do I just go over definitions and theorems until I intuitively understand it?

[Anonymous No.15190504]

>>15189114
>logic class
>teacher decides to politicize it for no reason
>her example of a true statement is "Donald Trump is a racist president"
>leave because she ruined my escapism with politics

[Anonymous No.15190509]

is number theory just autism or will i be able to get a job with a msc?

[Anonymous No.15190514]

>>15190509
maybe if you specialize into becoming a cryptography expert

[Anonymous No.15190585]

Does anyone have any tips for proving this(Fermat's lil' theorem I believe) in a combinatorial manner? I already proved it using induction however I'm curious as to how else it can be proved, especially using counting techniques somehow. I appreciate anyone posting either full proofs or giving hints. Thanks!

>>15186885
Yeah, no. I get who you are :D

[Anonymous No.15190608]

>>15190348
>not the same sign on the whole interval
can you elaborate a bit on this? I don't follow

[Anonymous No.15190695]

>>15190608
-2>-3 but 9>4 I think it’s what he means

[Anonymous No.15190705]

>>15190608
If the range of f went from -2 to 1 then the range of f^2 is not 1 to 4 but 0 to 4

[Anonymous No.15190777]

If i cut down infinitely tall tree, would it take an infinite aamount of time to hit the ground

[Anonymous No.15190851]

>>15190777
no

[Anonymous No.15190919]

>>15190851
Why not?

[Anonymous No.15190939]

>>15177504
>>15180047
The Earth isn't homeomorphic to a sphere. For example this tunnel is at least one hole

[Anonymous No.15190945]

Wasn't there a trick to the l'hôpital rule that says a certain type of polynomial will always give out a limit of 0?
I should probably ask this in /sqt/ but this general seems more active

[Anonymous No.15191066]

>>15187149
Thank you for telling me. Most of these books cost too much to get a paper one but maybe I will get lucky and the store will get it.

>>15186956
Who is this maid?

[Anonymous No.15191079]

How do you learn new maths? do you create a "procedure" in your head or do you just keep doing it until it clicks?

[Anonymous No.15191096]

>>15191066
>Who is this maid?
My wife, and the board mascot.

[Anonymous No.15191379]

How do I generally prove a function is integrable on a given closed interval?

[Anonymous No.15191640]

>>15190585
>Does anyone have any tips for proving this(Fermat's lil' theorem I believe) in a combinatorial manner?
Yeah. In the integers mod p, think about the sequence 1,n,n^2,… The way it hops around between nonzero elements (assuming p does not divide n).

[Anonymous No.15191691]

>>15191640
Thank you a lot anon. I think I'm close to getting it. That was really helpful.

[Anonymous No.15191832]

What do I have to read so that I can read this?

[Anonymous No.15191852]

>>15191096
A long time ago somebody who posts this maid gave me a lot of papers about cryptography.

I think I have an algorithm that beats quantum computers. I don't know what to do with such a thing though and I don't have a quantum computer to test it with. I can't experiment with hardware I can't access and I don't have a lab with computers or assistants who are maids yet. I don't even get invited to science meetings.

I don't have a science foundation and don't get one until the 2030s.

Probably if I publish it to CC0 someone else will test it, and I will find out if it works or not from if the news is angry.

Somebody on /wsr/ gave me a small maid to help find the smallest anime maid, so I am going to draw her in a Maid Space in my book and do experiments with the maid drawing. The small maid is attached.

Thank you dra/g/ons for reading my post.

[Anonymous No.15191867]

>>15191832
17 documents written by bureaucrats

[Anonymous No.15192003]

Cerium - python
Praseodymium - latex
Prometheum - real analysis
Europium - astronomy
Dysprosium - gravity
Erbium - prose
Lutetium - 4chan
Ldk

[Anonymous No.15192065]

>>15190348
What's the correct definition of [math] L_P(f) [/math] and [math] U_P(f) [/math] ?

[Anonymous No.15192098]

>>15191852
Not /mg/-related.

[Anonymous No.15192137]

>>15190777
>>15190919

[Anonymous No.15192154]

>>15192098
Maids are definitely related to Maid General and I want to know who the pointy ears maid is so I can collect images of her on purpose and maybe watch her show if it is a maid show.

[Anonymous No.15192188]

I don't understand this. The variables just confuse me. Can anyone dumb it down a notch for me?

[Anonymous No.15192207]

are there books or notes that cover aspects of finite model theory
the ebbinghaus monograph is like a dense arcane tome to me right now

[Anonymous No.15192345]

>buy recommended books for course
>barely have time to read anything since course moves at blazing fast speed.
>taking the time to understand more would hurt my GPA and future grad school chances much more than just figuring out the "game" aspect to the course to maximize grade.

Fuck me, uni is so fucking unfun. Grad school isn't like this too is it?

[Anonymous No.15192347]

>>15192345
University makes you dumber. You are a factory farmed human.

[Anonymous No.15192354]

>>15192347
Gotta feed myself somehow chief. 4 year degree and a few actuary exams is how I get to do it. At least I'm allowed to study what I want, even if it has the annoying time cram cost.

[Anonymous No.15192374]

What do you guys write with? Any pens you prefer?
>>15192345
I relate to this so badly

[Anonymous No.15192390]

>>15192345
How many hours a week do you spend on lectures, assignments and studying?

[Anonymous No.15192414]

>>15191379
Depends on what kind of integral

[Anonymous No.15192498]

>>15192390
30 to 35. I also have a full time job though.

[Anonymous No.15192510]

>>15192137
But it's infinite

[Anonymous No.15192515]

Why are brits so good at explaining math? I've had trouble with every American professor (other than Prof. Leonard) because they don't really go into the details.

[Anonymous No.15192520]

>>15192515
Brits are tedious and pedantic by nature

[Anonymous No.15192529]

>>15192414
Riemann

[Anonymous No.15192533]

How do I get better at general mathematical problem solving? As in, looking at a problem and applying what I know to solve it.

[Anonymous No.15192535]

>>15192533
do more problems until your mind starts recognizing a pattern

[Anonymous No.15192760]

number theory should be thought in primary school.
once they are learning about division and remainders, that's where it should start. fractions don't need to be taught until much later, they are probably one of the most confusing parts for children, as seen by the amount of ordinary adults who still struggle with them. So, teaching them later, when the children are older is better. Decimals shouldn't even be taught at all in primary school. They're just a added layer of confusion for little children, who are still developing.

[Anonymous No.15192817]

Solvable for t?

[Anonymous No.15192821]

>>15192817
I mean, "[...] = constant"

[Anonymous No.15192827]

>>15192533
Get a problem book with solutions. Take notes of the tricks and steps they use to solve the problems. Obviously this doesn't work for higher-level courses and subjects where there is no actual problem book. Just autistically-written "textbooks" or review papers.

[Anonymous No.15192858]

>>15192827
Test

[Anonymous No.15192902]

>>15177964
This triangle was part of the answer to a problem I had in pre-calculus yesterday. I felt pretty smart figuring it out on my own.

[Anonymous No.15192920]

New
>>15192912

[Anonymous No.15192992]

>>15192529
Then I think you just have to demonstrate that the function is continuous except for finitely many points.

[Anonymous No.15193049]

How do I solve this?
I have n people sitting at a table. I want k of them to be sitting consecutively. How many ways for them to sit up to rotation?

[Anonymous No.15193131]

>>15192345
This is on purpose. Professors know that it's a waste of time to spend too long on a subject ever. It's best for you to be given a high and quick overview and then be pushed to the next subject where you'll build mental connections to your general knowledge.

I don't know about others, but my grad school experience wasn't much different, in fact a lot of the same courses were focused on in grad school, it's just this time it wasn't an overview anymore. I had accepted what I understood of the overview and was able to come from the start (in a lot of cases) and work and understand much more of the fundamental problems.

It's really not time effective to just go deep on a subject for the first time. Even textbooks aren't written for that. You should go through a textbook at a high level. Don't think too hard just absorb what it is saying then come back another day and go a bit deeper. That's how you master mathematics.

[Anonymous No.15193149]

>>15193049
>I have n people sitting at a table. I want k of them to be sitting consecutively. How many ways for them to sit up to rotation?
Learn English.

[Anonymous No.15193552]

>>15193049
All right a hint. I assume your table is supposed to be circular. If k<n there is a canonical place to fix so the rotation goes away: namely one end of the Row.

[Anonymous No.15194824]

bump

[Anonymous No.15194956]

>>15194824
bump

[Anonymous No.15195314]

>>15194956
bump

[Anonymous No.15195330]

bump of death