/sci/ - Science & Math

I'm at Calculus 1 and already hate it.

>>15343123
go compute some derivatives whitey

What's the end game of the mathematician? Surely there'll be a point where there's nothing left to discover or discoveries become so few and far between that people won't fund the research. Then what?

>>15343262
>Surely there'll be a point where there's nothing left to discover or discoveries become so few and far between that people won't fund the research
The amount of discoveries per year is increasing rather than decreasing right now so there is no reason to believe that will happen during the next 100 years.

>>15343276
Yeah I just meant at any point in time in the future. Even 1000 years. Will mathematics become more like people just learning math and then doing client based work making models for people? I guess the same thing could happen with physics too but with that there's experiments that we probably won't be able to run for many thousands of years so maybe they'll have more time

What do mathematicians even do? Like, you majored in math, what do you do? What is the point if you're not applying that to another science like Physics, Economics, Computer science, etc...
Maybe there IS something there and i'm just ignorant because of my limited knowledge of the more advanced math concepts?

>>15343370
You're probably just bad, but don't worry, niggers trannies faggots and other mutants were never successful they just liked being called a "scientist"

>>15343419
>Brings up niggers, trannies, faggots and other mutants out of nowhere.
This is what i get for asking a genuine question in /sci/?

>>15343370
>science like Economics

>>15343463
Just trolling, all geniuses do that for the keks tranny

>>15343469
What is it, then?

>>15343370
Solve math problems under a commonly accepted proof system. The area of math is immense and is sometimes built up to help answer questions from the other sciences. Sometimes your scientist needs 'new' math to push their feeling into reality and you end up with people like Shannon, von neumann, and Einstein. Alternative you have savants like Lagrange, Galois, Grothendieck, and Perelman who make large contributions to math and the contributions aren't applied to other sciences for potentially centuries. Finally you have your average mathematical Joe who doesn't end up contributing but finds a job being a numbers consultant to kids or the c-suite.

Generally all other sciences use math as a grounding to perform scientific experiments. When teaching other sciences, they will skip over the math technicalities as much as possible till graduate school, then it depends on research focus.

>>15343370
fucking isn't just about the babies

>>>/a/251145508

>>15343727
So knowing the Whatever Conjecture has no practical use if you're not the one to solve it?
>>15343786
Yeah, but who's gonna pay you to fuck all day? What does your fucking contribute to society as a whole?

>>15343871
Lim f(x) = Death
x>∞

>>15343890
This, his baby might actually contribute to society only after 18 years (obviously) while you yourself can do research that might as well contribute even more than this offspring.
It's not a one sided coin and people sometimes overlook this, some research is important in the long term and is useful

>>15343935
I still have no idea of what you fuckers do all day.

>/sfg/ launched 7 hours ago - 265 posts
>/mg/ launched 5 hours ago - 20 posts
but also theres threads like
>>15342472
>>15343896
but no /sfg/ content outside of /sfg/
if mathfags weren't so attention hungry and desperate to launch vanity threads then /mg/ could end up being as engaging as /sfg/, but instead /mg/ is slow and boring and takes over a week to hit autosage

>>15343890
If you can understand the conjecture you have have an good idea of how that area of math works which is ~probably~ more valuable than the conjecture itself. Solving/Finding a good conjecture gets your name attached to it, which is value to some people.

>>15344140
You still haven't answered my question, motherfucker. What are mathematicians even hired for? Everyone in their respective sciences knows enough math to not need anyone else to help them with it.

>>15344008
this board is science and maths. not science and one maths general.

>>15343370
maths isn't science
they're opposites.

The study of maths, like science, is ultimately just the pursuit of knowledge, the difference is the scientist is concerned with reality whereas mathematicians aren't.

>>15344234
/mg/ could be as good as /sfg/ if it wasn't for bad attitudes from you and your ilk.

>>15344255
the difference is that scientific theories lack rigour proofs and maths is ruled by rigourous proofs

I don't get how mathematicians can rest easy knowing Godel's incompleteness theorems - I can't, it's been bugging me for over 4 months.

>>15343370
It's really strange when you think about it, but it starts to make sense when you see that those that develop pure math, are really brilliant and good at it, so they continue doing it.

Only a few or those that are not so good (but they just get filtered out so they don't "apply") do actually try to apply those concepts, but that's the small middle class between those that can and succeed, and those that apply what is already applicable.

>>15344301
Why? It's irrelevant to most of math, only perfectionists like Hilbert would actually care.

>>15344449
But you'll never know if your theorems truly make sense because you can never truly know if your axioms are consistent. Even if they were, then there would be theorems you can never prove in that system, it's a lose-lose situation :(

Sorry if this is >>/adv/ but I realized if I stay on an extra semester at my university I can get a double major in statistics as well as my mathematics degree. Is this worth it?

>>15344467
If you want to do data basedience or something like that the degree will get you into interviews faster. If that's worth it for you then do it

Let x catch a fucking break

>>15344191
>Everyone in their respective sciences knows enough math to not need anyone else to help them with it.
Yeah you haven't seen research. They use mathematicians all the time.
You're hired to do math, most of the time the business objective does not require you to solve the collatz conjecture. If your looking for discrete labels of what they do, look up what a teacher, or analyst, or actuary, or any programming position that isn't gonna be suplexed by ChatGPT does.

Let [math] M_1, M_2, \dots, M_n [/math] be n given points in the plane [math]\Bbb{R^2}[/math]. When is it possible to find a closed polygonal line [math] P_0, P_1, P_2, \dots, P_n=P_0 [/math] such that [math]M_i[/math] is the midpoint between [math] P_{i-1} [/math] and [math] P_i (1 \le i \le n) [/math]? When it is possible, how many possibilities are there?

You should be able to solve this.

Hint 1: Investigate for small n.
Hint 2: Use basic linear algebra.

>>15344301
There are people who have embraced it's oddity through algorithmic information theory.
>>15344459
>Even if they were, then there would be theorems you can never prove in that system, it's a lose-lose situation
Almost as if logic has limits.

>>15344643
>There are people who have embraced it's oddity through algorithmic information theory.
Care to elaborate?

>>15344650
They upgraded Shannon's information measure to work with a computation. Since we are trying to minimally describe objects/information, it offers another proof method to solve Godel's incompleteness theorem (and exposes the limits of AIT)
https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin's_incompleteness_theorem

I guess since were using computation a better example to "rest easy knowing Gobel's incompleteness theorem" is to learn about what a undecidable statement is (Halting problem) and see that your computer works with it's 'axioms.'

I'm almost done with an applied math degree with minor in stats and I'm about to kill myself

>>15343262
Job at mcdonalds.

>>15344618
n>=3 odd 1 solution always
n even solution exist iff centroid of even points = centroid of odd points, infinite solutions parametrized by P_0
proof: solve equations in R^2

>>15345052
why anon

How do you prove it?

>>15345207
multiply through by bd

>>15344008
Woah anon, you sure got us there. Maybe if you keep on spamming this nonsense like the lifeless loser you are, the jannies of /sci/ will make a special snowflake rule just for you.
I, for one, will go back to studying now and answer some stupid questions in /sqt/ later today, once I'm finished.

Here's a small exercise for /mg/.
Can you guess what year this was asked in? It's a pretty easy question, shouldn't take anyone more than 5 MILLISECONDS! Hahah, just kidding, take as much time as you want. I appreciate anyone solving! As always, good luck and don't hesitate to ask for hints or the solution.

>>15345245
2005006

What math field should i learn to be able to program stuff without conditional statements?
Every time that i write something that can be simplified into a math equation i fill myself with joy and think "fuck, math is great".

>>15345298
Unfortunately not anon. And your answer is way bigger than the real one! A lot bigger! How did you get it? If you let me know how you got your answer, I could try giving some hints so you get the right one. Or you can try again without hints. Up to you. Regardless, thank you for your effort. I hope you can find what you did wrong and get the right answer! Good luck!

>>15345380
Alright. So the number of integer divisors is the sum of all the (powers + 1) in a prime factorization. So [math] 2004 = 2^2 \times 3 \times 167 [/math] and [math]2004^{2004} = 2^{4008} \times 3^{2004} \times 167^{2004}[/math]. So now lets say the divisor has only 1 prime factor, we have 3 ways to make our 2004 divisor divisor.([math]2^{2003},3^{2003},167^{2003}[/math])

Now lets say the divisor has two prime factors. There's 3 combinations of 2 prime factors of the form [math]a^tb^{2002-t}[/math] for a total of [math]2001 \times 3[/math] divisors of this type.

Now for 3 prime divisors we have to distribute 1998 total powers between the 3 factors. This is [math]\binom{1998 + 3 - 1}{1998} = 1999000[/math] ways. So in total [math]1999000 + 6003 + 3 = 2005006[/math]

>>15345409
Worthless just like your copy paste method which you were too stupid to think of in a correct way.

>>15345424
Who are you mad at? Why?

>>15345409
>the number of integer divisors is the sum of all the (powers + 1) in a prime factorization.
How many non-negative integer solutions does
[eqn](x+1)(y+1)(z+1) = 2004[/eqn]
have?

>>15345245
54
>>15345409
The number of integer divisors is the PRODUCT of all the (powers + 1) in a prime factorization.

So what you need is the number of non-negative integer solutions of
[eqn](x+1)(y+1)(z+1) = 2004[/eqn]

>>15345458
No need to be so mean about it :(

>>15345409
Hi anon. Thanks for writing all that. I can see now how you went about it and I think it would have been fine had your first line not been mistaken I think.
>So the number of integer divisors is the sum of all the (powers + 1) in a prime factorization
I believe this should be the product, not the sum! I think that's where your mistake is. With this in mind, I believe you can get the correct answer with a bit more work! Good luck to you if you want to continue, I suggest you do because you're rather close anyway!
>>15345424
Look, I don't know who you are or what makes you think you can talk to people like that. But be quiet if what you're going to say is both rude and useless. I won't be replying to further comments from you unless it is an apology (not to me, to the person you replied to) and I suggest others do the same as clearly you're not interested in this problem but in insulting people who are trying.
>>15345458
Great job anon! That is indeed the correct answer. I hope you found it fun. Thank you for your answer and for telling the other anon what was wrong with his/her solution.Let me know what you thought of the problem if you want to. I hope you have an amazing day!


Oh and of course I hope no one thinks just because someone got the right answer, they shouldn't try or continue. I still absolutely appreciate anyone who comes up with an answer. Who knows, maybe there's a very unique solution?!

>>15345497
Shut the fuck up worthless tranny, you posted a simple question pretending to be some math guru or whatever you tranime idiot larp as.

Post a hard one or else do your homework

>>15345497
Thank you, I have seen the error of my ways in my previous attempt. Also im sorry >>15345458
I thought you were the other annoying faggot.

>>15345245
>>15345380
>>15345497
just cut your dick off already, tranny

>>15344008
/sfg/ over /mg/ by 750 - 60
very consistent ratios.

Let [math] f : (0,1) \rightarrow \mathbb{R} [/math] be a smooth function such that [math] f [/math] and [math] f ' [/math] are bounded on [math](0,1)[/math].
Then does [math] f [/math] extend to a smooth function [math] \mathbb{R}\rightarrow \mathbb{R}[/math]?

Thoughts on mathematical statistics? What’s a good text?

>>15347040
Consider
[eqn]f(x) = \int_0^x \cos \left(\frac{1}{t} \right) dt[/eqn]

/sfg/ tacked on another 240 posts in the time it took /mg/ to add 3.

>>15347071
Do probability then statistics.

>>15347074
Thank you for your help anon.

A new question:

Let [math] f : (0,1) \rightarrow \mathbb{R} [/math] be a smooth function, such that for each natural number [math] n [/math] the [math]n[/math]th derivative of [math]f[/math] is bounded on [math](0,1)[/math].
Then does [math] f [/math] extend to a smooth function [math] \mathbb{R}\rightarrow \mathbb{R}[/math]?

>>15347365
The answer is apparently yes: see
https://math.stackexchange.com/questions/375530/continuation-of-smooth-functions-on-the-bounded-domain

>100% of students that took the numerical analysis class last year got an F
is it really that hard? even real analysis has lower fail rate (50%)

>>15345344
Functional programming

>>15345245

Do I pick pure or applied for my masters
I can't decide, I like both

>>15348647
Depends, do you want a job outside of academia?

If I have a group G acting on a topological space X, and this action is free (no fixed points), what can I say about the orbit space X/G?

I am reading arXiv:1701.02293. On p.6-7 Pedroza says M(f ; p,q) admits a natural action of R by translations, and this action is free, so the orbit space is identified as the space of trajectories that joint p to q.

But I thought "the space of trajectories that joint p to q" exactly M(f ; p,q)? Is it true that when the action is free, X can be identified (homotopy equiv, homeo?) with X/G?

If not, what is a sufficient condition for X to be identifiable with its orbit space under some action?

>>15348708
Academia is my top priority, but I also want to be able to work outside academia if necessary

>>15348825
Applied, then. It has enough of the pure to satisfy you.

>>15343370
>science
>economics
lmao

>>15344560
this is /sci/, maybe you were looking for /pol/

>>15348825
Pure is favoured by both. "Applied" is basically saying you are not good enough for pure.

>>15348749
First of all, the action having "no fixed points" is rather vague and meaningless.

To be precise, a group action is called "free" if the only group element whose action has any fixed points is the identity element.

>>15343370
They do math, duh

>>15343890
"Practical use" as in helping people? Why would I want to do that?

>>15349616
Wow, thanks for the completely useless pedantic comment concerning the only thing in my question that I had no questions about. Next time if you have nothing meaningful to add just don't bother

why are all calculators be it on my computer, phone or a purpose built device so anti-usability and retarded?

>>15347071
Whatever you do, pair a theory text with an application text in a programming language (R tends to have the better texts). It really is useful and helps understanding a lot.

What is it with people at my uni not knowing pre-calculus? Didn't you learn that in high-school? How the fuck did you get in here to begin with?

>>15348647
Just do both nigger. Literally nothing is stopping you from taking a mix of classes.

Do you guys know where can I read more about this embedding? All I've found so far is a single sentence. I Ctrl+F'd through Hatcher's book on algebraic topology and there is a lot about embeddings, but not a single mention of Whitehead's manifold.

t. First year undergraduate student. The fucking professors won't tell me anything I don't already know, but maybe /mg/ will.

>>15349990
Worthless

>>15349991
>Worthless
The embedding? Me? You're being pretty vague here, man.

>>15350007
Probably both

>>15350010
>Probably both
Cool. Still, I don't see an argument as to why.

>>15349990
If I’m not mistaken, according to the “Construction” section in the Wikipedia page, it is constructed as a proper subset of S^3 ; hence can be embedded in R^3

>>15349990
Also don’t rely on Hatcher for any real knowledge. Understandably you aren’t aware of this since you’re a 1st year undergrad. But Hatcher is good for, and only good for, learning the fundamentals

>>15350153
Yeah, but isn't S^3 a subset of R^4? That's why it's not obvious to me.

>>15350160
Exercise: show Sn with a point removed is (homeomorphic to) Rn

>>15350157
>But Hatcher is good for, and only good for, learning the fundamentals
His book is used during first courses on algebraic topology, so that's not really a problem.
>Exercise: show Sn with a point removed is (homeomorphic to) Rn
Thanks, I'll try my best.

/sfg/ currently beating /mg/ by about 1800 to 97

>>15350168
Keep us informed.

>>15350167
Hint: look up stereographic projection

>>15350164
Actually, I've done that before, it's one of the first exercises (it was on the 2nd list of exercises we were given). So, a proper subset of S^n is naturally embedded in S^n with a single point removed, correct? Then we embed S^n with a point removed in R^n, and use both of these embeddings, right?

>>15350157
>Also don’t rely on Hatcher for any real knowledge.
Could you point me to a more detailed book then? Also, could you tell me what an American (or wherever you're from) undergrad curriculum looks like? Here in Poland there is a place where you can take topology on your 2nd semester, but usually it's done on 3rd or 4th semesters.

>The persistence of v is defined to be death minus birth
who comes up with these definitions?

>>15350180
Yeah pretty much

>>15350204
I’d say it depends on what particular topics you’re interested in

>>15349991
>someone finally posts math on /mg/
>I don't understand it
>call him worthless
Insecure imbecile, with his attitude towards learning you're washed up already. Better to quit out of math before your life gets even more fucked. Actually just quit out of life, nobody wants to interact with retards like you

>>15350567
>Insecure imbecile, with his attitude towards learning you're washed up already. Better to quit out of math before your life gets even more fucked. Actually just quit out of life, nobody wants to interact with retards like you
Thanks. Luckily I have a pretty thick skin from interacting cretins on 4chan, so it doesn't bother me that much.

>>15350573
>interacting cretins
with cretins*

Are there formal proof systems that are actually "nice to use" and reflect the way mathematicians reason in their informal proofs?
I read a book on mathematical logic and it only introduced a "Hilbert system". I understand that this makes reasoning *about* the system convenient since it only has one inference rule, but the few times I tried writing down a deduction *in* the system it turned into a mess.

>>15349990
>Do you guys know where can I read more about this embedding?
Invariants at infinity seem to be what you specifically want, and embedding concerning invariants and toplogy at the ends of space/infinity. Unless I completely misunderstand what you're asking.
https://arxiv.org/abs/1210.6741
Skimmed over this example and it seems to go over the kinds of things you'd want for invariants for ends of spaces. I apologize if I misunderstood.

What am I in for?

>>15344459
>you will never prove a theorem whose statement has more symbols than atoms in the universe

>>15344459
You don't really understand Godel's incompleteness theorem. If you really understood the metalogic nuances behind the proof you would come to a very different conclusion.

Google is dogshit and I can't find what I'm looking for in my books. True or false:
Two matrices only have the same characteristic polynomial if they are similar

>>15351187
This is closer
https://en.wikipedia.org/wiki/Natural_deduction

But equality/equivalence of terms is essentially an unsolved problem, from a practical standpoint. Most of the modern type theory stuff is just about that.

>>15352357
False

>>15352357
No, consider nondiagonalizable matrices (look at the Jordan canonical form)

Is every closed subset of Euclidean space the vanishing set of some continuous real function?

I.e., for each closed subset [math] A \subset \mathbb{R}^n [/math] is there a continuous function [math] f : \mathbb{R}^n \rightarrow \mathbb{R} [/math] such that [math] f^{-1}(\{0\}) = A [/math] ?

>>15352532
Yes, it's true in every metric space. Consider
[eqn] f(x) = \inf_{y \in A} d(x,y)[/eqn]

>>15352629
Ah right , basically the distance to A , thanks anon

>>15352532
>>15352629

As a follow-up question:

Is there a *non*-metrizable topological space X such that every closed subset A of X is the zero set of some continuous real-valued function on X ?

Between any two rationals there are uncountably many irrationals and between any two irrationals there are countably many rationals.
Why can't we use this to do a bijection from QxQ to R?

>>15352698
Why does it give you hopes it would?
Your question isn't quite clear.

Defining a totally ordered set to be "complete" if every upper-bounded subset has a supremum (l.u.b.) and every lower-bounded subset has an infimum (g.l.b.) ,

Does there exist a complete totally ordered set of every possible cardinality?

Is every closed subset of Euclidean space the vanishing set of some continuous real function?

I.e., for each closed subset [math] A \subset \mathbb{R}^n [/math] is there a continuous function [math] f : \mathbb{R}^n \rightarrow \mathbb{R} [/math] such that [math] f^{-1}(\{0\}) = A [/math] ?

What kind of sorcery are these Non-monotonic Logics? Wikipedia dont say much about it
And there's no way to make sense of somekind of proof theory of it right?

>>15343107
>Golomb
QRD?

>>15343473
go back to /pol/

Defining a totally ordered set to be "complete" if every upper-bounded subset has a least upper bound and every lower-bounded subset has a greatest lower bound,

Is there a complete totally ordered set of any given cardinality?

>>15347074
>>15347365
get a differential forms book and look for the part on pullback functions for some solutions to problems like that

Dummit & Foote or Jacobson?

is there any book which explains math with irl applications or explains why or how something works?
they always telk you how to solve something but never why or how does it work

>>15353436
I don't know, but I assume it's at least consistent - given you have have the reals have almost any ordinal size.

can have*

>>15353418
Combinatorist and an engineer. Really cool lad,worked at NASA, inspired Tetris, a bunch of stuff.He passed away in 2016 unfortunately but he left a legacy. There's a book by him and Andy Liu on combinatorics that I've recently been looking at to find some cool problems and there's quite a few. I actually intended to post one this thread but I was in a hurry and ended up posting this instead >>15345245
So I made the OP image him instead. A true genius.

>>15352470
Thanks for the link anon. That was an interesting read, though I'm not sure I understood everything written there (I'm not familiar with type theory at all for example).
Do you happen to know a text that could serve as an introduction to natural deduction (or possibly a logic book that uses it as the main proof system)?
Also, am I correct to assume that the tradeoff with using natural deduction (vs a Hilbert system) would be that proving things about the system (like soundness) are more involved?

Is it a good idea to memorize a lot of formulas instead of trying to solve them?

Bros, how do I get good at calculus. I had differential calculus last semester and I passed by the skin of my teeth. I don't wanna be a brainlet anymore. Next semster, I got Integration course. I wanna do better in this course. How do I get good in calculus and math in general?

>>15354024
As a general logic rec, I like
https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf
and
https://www.cin.ufpe.br/~mlogica/livros/Logic%20and%20Structure%20-%20Van%20Dalen.pdf

I don't know to what extent those address your query, however (especially the second one might not)

As for value of the systems, I also don't know for sure. I you got a lot of axioms like in Hilberts system, then the language is well controlled in the sense that you can enumerate sentences and proofs

>>15353964
> the reals have almost any ordinal size
Wait how does that work though? Aren’t there many ordinals much larger (in size/cardinality) than the reals (or any given set)?

>>15344264
/sfg/ is a trash fire my dude

>>15354200
Name one ;)

(without referring to 2^N itself)

>>15354100
Learning math is 90% practice 10% studying. There's no way around that.
Try out 3Blue1Brown if you feel like you're not absorbing the "studying" part well. It's great for retards like me who needs to think of a graph for half a minute to understand a statement involving functions.

>>15354386
Exists with axiom of choice, no? Or are you one of those anti-choice people

Is it possible to create a discontinous function without using multiple conditions?

>>15354659
You could do something like f(x) = |x|/x , though this doesn't exist (in the strict sense) at x=0

>>15354810
I think i remember my HS teacher telling me that a function is still continuous even if one of it's points isn't defined (1/x, tanx, etc...) for some reason.
Still, now that i did some searching, that doesn't seem to be true.

>>15354659
y = |4 - x^2| + 1

>>15354828
>is still continuous
Should be, "can still be continuous", then it's correct

>>15354872
That's not discontinuous

>>15354872
*
y = 4 (x/|x|) - x + 1

>>15354579
"Exists" what?
I wouldn't adopt choice personally, but even if you do, it will be hard to find an ordinal larger than the reals (unless, again, you define an ordinal from the reals itself).
Assuming LEM, no subset of the naturals surjects onto 2^N or the Dedekind reals, any the exponentiation map
x \mapsto 2^x
makes jumps larger than what set theory can understand

See e.g.
https://en.m.wikipedia.org/wiki/Easton%27s_theorem

Is the nullspace of a (finite square real) antisymmetric matrix necessarily even-dimensional?

and*

>>15355055

Nice explanation, thank you

>>15355056
Oops I'm stupid just take the zero matrix in odd dimension

>>15355080
Btw. our of curiosity I checked what ChatGPT 3.5 would have to say about this, and it's even worse than when I tried to make coherent statements about weak forms of choice.

>>15354659
sqrt(x^2)
look up branch cuts also

>>15343262
Math isn't discovered but invented

where can i find a good exposition of the Kronecker-Weber theorem? i'm really interested in learning and understanding this result. background-wise i more or less have the basics of ANT (integers, Dedekind rings, discriminant, localisation, class number/Dirichlet, etc.)

>>15355529
This is discontinuous on the complex plane, but the usual principal branch is continuous on the real line

Arctan(any integer) is only a rational multiple of pi for -1, 0, 1. Sure, fine, the Taylor series evaluate in a particular way that forces this but is there a deeper reason?

>>15356123
https://arxiv.org/abs/2009.06583

>>15355566
There's really no *good* reference for class field theory, it's a painful subject any way you cut it. But Cassels-Frohlich is standard and not too terrible.

>>15354828
The correct way to say this would be that those functions are "continuous on their domain"

>>15349830
If you don’t know how to articulate definitions correctly, then how do you hope to understand anything beyond the definitions?

>>15356361
are there any good elementary references that don't use CFT?

why can't i digest math books formatted like novels

>>15356428
Washington's "Introduction to Cyclotomic Fields" apparently has a chapter on it at the end. I haven't personally read that chapter, but I can vouch for it being an okay textbook from its earlier chapters.

>>15356222
Holy shit, finally a motivation for Galois groups that I don't completely hate

>>15352686
[math]X = \{a,b,c\}, \tau = \{\emptyset, \{a\}, \{b, c\}, X\}[/math].

>>15357254
Ah yes I see, thank you

\binom{n}{k} = \frac{n!}{k!(n-k)!}

>>15354828
It is continuous since points of discontinuity can only exist in domain. Fun fact: all functions from naturals are continuous.

>>15356511
Humans do not have the necessary enzymes to break down cellulose.

Topic is: will the universe, if it's isotropic and infinite, will it in that case eventually repeat. I.e. also our observable volume.
I replied that yes, of course the limit of a recurrence approaches 1 as each observable volume, with its finite size, can only hold a limited number of arrangements. A series of infinite such volumes yields a probability approaching 1, that there is at least one another copy.
Of course, this got pushback from /x/ dimwits. This here is the particular choice cut "why I am wrong", from a self-proclaimed "math degree holder".
>>15356826
>I'm not going to give you a detailed response because you don't deserve it. All I will say is this: "if something is infinite it must include every possibility" is every midwit's favourite misunderstanding of infinity. The simple counterargument is this: there are infinite numbers between 1 and 2, but it doesn't include the number 3. If there is infinite space in the universe, that doesn't mean every possible combination of atoms is contained therein.

>>15354899
>>15356370
>>15357399
I see, thanks for clarifying. There's a lot of misinformation on the internet, apparently.

>>15354085
I'm assuming you want to know this for purely practical reasons and don't actually like the process of learning where a formula comes from.
You could definitely just memorize formulas, but learning why they work can not only develop your analytical skills, but also make it easier to apply your already attained knowledge in the formula's context.
Both my HS and college teachers used "d=√((x2 – x1)2 + (y2 – y1)2)" for the distance between 2 points in a cartesian system, which is just pythagoras' theorem, although they never said that. Me noticing that allowed me to, instead of using that needlessly complex formula, use my intuition and knowledge of pythagoras to solve any problem involving it. (Same thing in a 3D system).
These things are especially important in calculus, where if you don't have a deep understanding of what something is, you'll probably have a hard time later on.

Is there a standard name for the space "n^N" of unending sequences on N into a finite range {0,...,n-1}

>>15354530
how many problems do I have to do at minimum to get a good handle? I have to deal with a bunch of other courses too so I can't devote all the time to math.
Also, I feel like an absolute retard because I can't solve even the most basic problems.

>>15358191
>how many problems do I have to do at minimum to get a good handle?
I don't fucking know, that depends on the subject, your intelligence and your current knowledge. I just practice till i'm confident enough.
>Also, I feel like an absolute retard because I can't solve even the most basic problems.
As i've said before, the studying is still integral. If your gym coach teaches the exercise in a half-assed way, you'll do it half-assed and get half-assed results.
That, or you skipped a subject that is fundamental to the current one, which is a pain, but the solution is obvious.

Incest: a game the whole family can play, by Milton Bradley.

How to I calculate the average roll of 4 x sided die where you discard the lowest roll?

>>15359525
Never mind I googled it

>>15359525
Just manually go through all cases.

Consider a cube (of uniform solid density) balanced on one vertex on top of a flat horizontal table.
What is the angle between the table, and one of the cube edges touching the table?

Is there an "elegant' way to solve this without using rotation matrices?

>>15359631
https://youtu.be/SH8z9Iou0u8

>>15345245
27

How would you rate the difficulty of Polish matura exam to equivalent exams from the west like SAT? Pic rel are two last two tasks from 2018. This is meant for 18-19 year olds and is a standardized exam that replaces entry exams.

Btw google translated ate few "of" words. It should say It should say "of this triangle" at the end of the first task.
And the proper translation for subtask b) is more like:
>Prove that the circuit L of trapezoid expressed as a function with variable "a" (f(a), or L(a)), a variable which expresses the longer base of the trapezoid, can be expressed as a function ...

Full test here (use Google translate):
https://www.matemaks.pl/matura-2018-maj.html

>>15359716
Hello anon! Thanks for your reply. I'm sorry to inform you that your answer seems to be wrong but it is not too far off which makes me think you made a small error along the way. Could you check your steps again? Let me know if you can find what went wrong but if you can't or don't want to I can send the solution instead. Good luck and thanks again for your time and effort solving this problem! I hope you succeed in finding the right answer!

Why take the absolute values? Shouldn't
[math]f(x) = \begin{cases}
x\cos(1/x) & \text{if } x \neq 0 \\
0 &\text{if }x = 0 \\
\end{cases}[/math]
work as well? Since
[math]\lim\limits_{x\to 0} \dfrac{f(x) - f(0)}{x - 0} = \lim\limits_{x\to 0} \cos(1/x)[/math]

and this doesn't have one-sided limits either?

Spent the whole weekend learning derivatives and my brain is complete mush from burnout. Feels good.

>>15360161
I know high school level derivatives. Is there something more to them later on?

>>15360401
btw does USA even have derivates in high school? Not sure what the cirrcuclum is there like.

>>15360401
They don't have derivates in HS here so i dunno. It's just pre-calculus.

>>15360401
>I know high school level derivatives. Is there something more to them later on?

Yes, derivatives form the basis of physics. They form the most important form of mathematics, Differential Equations, and you also learn how objects can change rates of speed in three-dimensional space using vector calculus, usually a 2nd year college course.

>>15359746
Definitely harder than SAT, but SAT isn't taken as seriously here as foreigners seem to think it is.
>>15360411
Yes. Good students do "freshman" calculus in sophomore year of high school or before

>>15360423
Sound interesting. can't wait to learn all of this stuff. I'm going to uni this year.

>>15360433
>learn all of this stuff
>>15360433
>going to uni
kek

Reminder that IQ is a meme and the only things you need are passion and drive.

>>15360591
This is cope

>>15360879
This is dooming.

>>15360912
No you're just a tard

Anons, where can I learn about rank-1 matrix approximation/decomposition? It's the only thing I have left to learn - any resources welcome

>getting destroyed in abstract algebra along with other students
>make a study group where we completely disregard professor's notes, and just ask chat gtp to explain each term, proof, or concept with several easy concrete and applied examples.
>its ridiculously easy, start acing his quizzes and homework assignments

Do math "professors" REALLY?

>>15361066
>Do math "professors" REALLY?
yeeees. the entire field of differential equations is basically an obfuscation. it should be taught entirely in terms of for loops in c++

Which is true

Almost all of the real numbers have exactly two decimal expansions

Almost all of the real numbers have exactly one decimal expansion

>>15360879
It’s anti-cope. It’s saying if you haven’t achieved as much as you want, you probably didn’t put in enough effort

>>15361563
This is impossible

>>15361579
Oops I thought you were saying both are true, ignore this

Does there exist a (continuous) fiber bundle with total space [math] \mathbb{R}^2 [/math] and fiber [math] S^1 [/math] ?

>>15361637
Yes. However only trivial one.

>>15361673
What is this fiber bundle?

>>15361066
Now imagine what you could achieve of you instead read a book.

>>15360934
Commas, tardo. Use 'em.

>>15349990
>>15350153
>>15350164
Here's my proof.
Since [math]W[/math] is a proper subset of [math]S^{3}[/math], there exists a [math]p\in S^{3}[/math] such that [math]p\notin W[/math]. Therefore [math]W\subseteq S^{3}\setminus \lbrace p \rbrace[/math]. The function [math]f:W\rightarrow S^{3}\setminus \lbrace p \rbrace[/math], [math]f(x)=x[/math] embeds [math]W[/math] in [math]S^{3}\setminus \lbrace p \rbrace[/math]. We know that [math]S^{n}[/math] with a single point removed is homeomorfic to [math]\mathbb{R}^{n}[/math]. Let [math]g: S^{3}\setminus \lbrace p \rbrace \rightarrow \mathbb{R}^{3}[/math] be a homeomorfism. Then [math]g \upharpoonright_{f[W]}\circ f[/math] embeds [math]W[/math] in [math]\mathbb{R}^{3}[/math].

>>15349119
NTA, but what is roughly the distinction between pure and applied maths at the graduate level? Is it an american thing? We don't really have that over here (both masters can take the same courses).

>>15351874
Thanks a lot for this article. The reason why I'm interested in such counterexamples is that I stumbled upon this discussion https://math.stackexchange.com/questions/55114/are-contractible-open-sets-in-mathbbrn-homeomorphic-to-mathbb-rn
I'd like to learn more about this property of being "simply connected at infinity". Also, I'd be grateful for any recommedations on books that deal with abstract algebra, topology, axiomatic set theory or diff. geometry. I'd like to go back to my previous uni one day and do a fully 'pure' path there, and I'd like to prepare myself somehow.

>>15361546
I found numerical methods in physics allowed me to far better understand calculus topics I was previously rusty on. The math proof is unintuitive, yet once applied it all comes together.

>>15362019
We've tried a few books, but they're not so chock full of examples. Its much more enjoyable to have chatgtp talk down to us like idiots and explain softball examples. Even better when you can correct chatgtp.

Prove that the area of the shaded circle enclosed by the graph of [math]\tan\left(x^2+y^2\right)=1[/math] is [math]\dfrac{\pi^2}{4}[/math]

Prove that the area of the shaded circle enclosed by the graph of [math]\tan\left(x^2+y^2\right)=1[/math] is [math]\dfrac{\pi^2}{4}[/math]

>>15364845
Can't help you with the books recommendations. Whatever books I once used for such things are distant memories and matters of university where they were relegated to not being purchased or the fire upon completion. Still haven't gotten around to examining whether the recommendations on https://sites.google.com/site/scienceandmathguide/ are any good or not, but some anons who seemed decently reasonable thought so.

hey, sorry to bother but would any of you mind helping me with this problem in PDEs that I have been working on?
>>15365053

>>15364935
Just integrate it in polar coordinates

[eqn]\int_0^{2 \pi} \int_0^{\sqrt{\arctan(1)}} r dr d\varphi = 2 \pi \frac{\frac{\pi}{4}}{2} = \frac{\pi^2}{4}[/eqn]

>>15364935
tan(x^2 + y^2) = 1
is just
x^2 + y^2 = arctan(1)

>>15361563
Irrational numbers have a unique expansion.

>>15365163
I disagree: consider the irrational number [math] r = \sum_{n=1}^{\infty} a_n10^{-n} [/math] where [math] a_n [/math] is a sequence of 10 ones, then a zero, then 100 ones, then a zero, then 1000 ones, then a zero, and so on.

If you add 0.999... to this it should be possible without first treating it as 1.0000 , because each "block" of ones contributes all 1's to the left, but since the number of 1s in each block is a multiple of 10, then the total contibution from a block to the digits on the left of it is 0.

Hence the result of the 2nd paragraph, minus 1 , gives a new decimal expanion for [math] r[/math] different from 0.1111111111011111111111...

>>15365331
>gives a new decimal expanion for r
different from
Actually sorry wait, I haven't checked that this is actually different from the original decimal expansion

>>15355533
I fail to believe this, as whatever natural phenomenon is in observation can be explained in mathematics, there's not much to invent when it comes to math, as it's just a description of an occurrence.

>>15343871
You would have to consider how much HP regeneration he had during the time between beatings, considering humans consistently regenerate and he wasn't going through some degeneration, it would take more than 2 half beatings to kill him, unless they were done simultaneously.

>abusively short time periods on exams with proofs only

Sadists.

>>15365535
Concepts are invented, truths are discovered. Abstractions about reality are likewise invented from observed things.

>>15365718
>Abstractions invented
Kill yourself and stop replying in this thread you can't make up definitions for words not by their common definitions.

Worthless college student piece of shit

>>15365729
>you can't make up definitions for words not by their common definitions.

>>15365843
>Worthless negrotic trash misunderstands definitions

Abstraction by definition is not invented.

DO YOUR HOMEWORK WORTHLESS COLLEGE KIDDY AND STOP ARGUING WITH ME YOU ARE WRONG

You are literally wasting your time, not even sure you are in college yet, probably failed your classes dumb fucking children

Any tips on how to study math efficiently? I have a big exam coming up in a few weeks.
I tried taking supplements like l-theanine and alpha GPC and they have minimal, or maybe a placebo effect. I can't make up my mind about caffeine though.

I switched over to computer science for my master's so that I can become a code monkey. It is incredibly boring, even the theory of computation course. Don't make the same mistake bros do your homework and go to math grad school

>>15366838
>so that I can become a code monkey.
Found your problem. Try taking some applied courses like optimization, ML (check the prerequisites for real analysis or at least a probability course), program analysis, or cryptography. Concurrency/distributed systems, if you're a fan of Dijkstra (I'm not). Maybe NLP, but honestly there's not much of mathematical interest in there. Avoid AI unless it's about robotics.

>>15361637
>>15361688

Suppose such a fibration S^1 \to R^2 \to M exists for same base space M.

Then by the homotopy LES and the fact R^2 is contractible, π_n(S^1)=π_n+1(M) for all n>0. So since S^1 is a K(Z,1), that means M must be a K(Z,2). But [R^2,K(Z,2)]=H^2(R^2;Z)=0.

So no such bundle can exist.

>>15343107
Is math worthwhile to learn if I'm not into it? if so what books do you nerds recommend.

how does the usual formal logic relate to the physics-based/physical world it often is used to "model"?

Do you like when anime use tangentially math-related sounding words?
>Absolute

How do I use a new math font in LaTeX? I use Gummi editor (I'm on Linux...) and tex-gyre is installed yet
\setmainfont{TeX Gyre Schola}
\setmathfont{TeX Gyre Schola Math}
\setupbodyfont[schola]
do not work. Can anyone help please? Gummi isn't popular and I didn't find any help googling. The default math font is lame.

>>15367258
That question is a bit too broad. I suppose you're familiar with the representation of "physical space" as a power of the reals (R^3). So I'm not really sure what you're asking. You have an idea how the physical world is related to math, and math can be expressed with increasing rigor.
A good starting point is by informally labeling things by numbers and then proceed to proof theorems about the numbers and sets of numbers, their ordering, etc.

>>15367100
Why would you want to learn something you're not into?
If it's because of your carrer, then that may also vary depending on your field. Architects learn calculus but almost never use it.

Math is pretty :)

>>15367100
>Is math worthwhile to learn if I'm not into it?
Depends.
>if so what books do you nerds recommend.
It's best to start with calculus (very good book)
https://lyryx.com/wp-content/uploads/2017/06/Guichard-Calculus-EarlyTranscendentals-2017A.pdf
then complex analysis (below are some very good notes that read really well)
https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/
then DEs (you can even use some online resource for that one as it's really basic)
https://tutorial.math.lamar.edu/Classes/DE/DE.aspx..
The last two should be much, much, easier, especially DEs, so make sure you get calculus right. Then it's up to you. If you want to do the pure stuff like algebra, topology, geometry, etc, then just go straight for it ie buy specialized books and grind. Books for topology, etc, often contain topics that go from undergraduate to advanced material (eg Hatcher's algebraic topology). Honestly same goes for applied topics... ME/fluid mech/etc, content from the three classes above will be a good enough basis. You might need a few additional things but you'll just pick that up online as you go.
(If it's physics you like then you'll additionally need to go through the material of the typical first four classes ie classical physics/electromagnetism/waves/quantum mechanics and then try and find something you like eg optics/qft/etc and study that. The four classes mentioned here should be real easy if you have a good basis in calculus, complex analysis and DEs, in fact they're usually taken at the same time by early physics students.)

How would you motivate the subject of von Neumann algebras to someone who's not well-versed in operator theory?

According to wikipedia,
>Von Neumann algebras have found applications in diverse areas of mathematics like knot theory, statistical mechanics, quantum field theory, local quantum physics, free probability, noncommutative geometry, representation theory, differential geometry, and dynamical systems.
Can someone please expand or provide references regarding these applications?

>>15345245
36

Can you prove it?

>>15367570
If we look at our most up-to-date ideas/theories of how the physical world changes over time, logic-systems or math can "model" them. After all, mathematical ideas are often originally inspired by the nature of the real world.

Addition and subtraction can correspond to discrete, physical objects being added or taken away, like even particles.

But how are things like mathematical induction/infinity/reasoning about infinity, for example, related to the real world, according to our physical theories?

It's a little surprising that reasoning with infinity is often applicable to the physical world, like using number theory to predict the behavior of a (physical) computer's results. (But really, that has to be because it *is* somehow related, commensurately)

>>15367664
fuck I meant 51???

>>15367686
Under what system of axioms are you asking if I can prove that?

Tired of the dice roll on my GPA from random bullshit retarded professors drop in their courses. There is zero reason for not having a completely standardized undergraduate math coursework, exams, and class content. Homework and exam writing should be taken away from professors and even universities, and done by committee for an entire region. Complete bullshit that you can lose a letter grade or two and have your life ruined because some jackass couldn't gauge how to write a test properly.

>>15367776
It is the axiom.

>>15367735
54 final answer fuck this

>>15367664
>>15367735
fuck these are wrong???
>>15367853
duck, this is correct!!!! Nice job anon! It's indeed 54. Third time is charm, as they say. Thank you a lot for your time and effort solving it. Speaking of, how did you solve it? How did you get those other numbers? Either way, I hope you enjoyed it! Let me know if you did so I can be happy that you did. And of course I hope you have a pleasant day!

>>15367874
Nice, thank you anon. I'm new here but I'll do my best on latex.

[math]2004^{2004} = 2^{4008}*3^{2004}*167^{2004}[/math]
so an arbitrary factor of 2004^2004 is of the form
[math]2^{x}*3^{y}*167^{z}[/math]
where x, y, z are positive integers and x is at most 4008, y is at most 2004, z is at most 2004 (though this cap eventually becomes redundant)
Then we note that the number of factors of this arbitrary factor is
[math](x+1)(y+1)(z+1)[/math]
which we need to equal 2004. We need to count the number of solutions.
We can instead consider the equation
[math]abc=2004[/math]
in positive integers because there is a one-to-one correspondence.
Now it's combinatorics because if you let A=2, B=3, C=167 you can describe the solutions like
(AABC, 1, 1)
(ABC, 1, A)
(1, BC, AA)
and so on... and then I just counted the number of solutions by hand to get 54. My counting was the source of the two previous wrong answers. I finally got the counting right on my last try, and I verified it in python by just permuting the factors of 2004. I think there's a variation on stars and bars to count solutions better? Thanks again for the question anyways.

Fellas, is it possible to go through all of gallian, doing 4 or 5 chapters a week?

Ok woops. x, y, z can be zero. a, b, c are still positive integers though. Also I didn't count like that exactly I just wanted to show the format of the tuples I was writing :o

>>15368005
>>15368076
I forgot to press the reply button woops

>>15367720
>But how are things like mathematical induction/infinity/reasoning about infinity, for example, related to the real world, according to our physical theories?
I don't think most people would stipulate that a physical theory (which is a mathematically formulated model of the empirical world) is at all concerned with explicitly relating abstract mathematical statements to said empirical world.

As a rule, the working theories are rarely treated as being a valid mirror of the empirical world when it comes to very small and very large scales, so I'd not deduce statements about abstract infinity form it.

As for induction, I think it's best to view it as a join _restriction_ of the concept of natural numbers, as well as the different predicates defined on them (the different properties of those numbers).

>15368149
>mtg
faggot. Not gonna read your post. No (YOU) for you either.

>>15368157

>>15368149
I meant infinity related to the real world as our physical theories describe it (the real world). I'm supposing physical theories, like the standard model I guess, are supposed to describe "the world" instead of just one part of it.

Even though really they may not work for everything...

I was using the idea that things like arithmetic (add one apple or someother object that follows such rules, like many kinds of particles, unlike say a droplet of water which "adding" one to another can just still be one droplet, would pretty much work the same as those arithmetical rules) are based off those objects in the real world.

I guess using the theory of the standard model involves real numbers and has space have an infinite (and real) extent, so that's using those kinds of "infinite" in the reasoning, so that's a reason it'd have to be related. But I still generally dont know much about infinity and using it

Specifically, I find it weird how you can infer things about, say, the natural numbers using it. But I guess the reason/answer is gonna be something like "Induction/Axiom of infinity is literally just X, which is literally just Y, which is how natural numbers are defined".

>a join
interesting
Also with the infimum and Schnirelmann density thing, that's related

>>15368338
>Even though really they may not work for everything...
There's no unified theory
https://en.wikipedia.org/wiki/Grand_Unified_Theory
But even if there were, they wouldn't per se speak about abstract math done by humans (not any more than we can already informally reason about it)
>so that's a reason it'd have to be related
If those models are written down in some particular mathematical formal theory, where e.g. space is R^3, then the theory tautologically claims that physical space has infinite aspects. But I don't think we learn anything about math from this process in particular, given the theory is expressed in math.
>Specifically, I find it weird how you can infer things about, say, the natural numbers using it.
Using what? We don't infer things about the formal theory of numbers using the empirical world. If out theory of numbers and physical things can be matched up (and in this way we can intuit math results using things in the world), then just because we those theories which didn't have much applications were discarded as niche. Math working for real worth use-cases is selection bias in the sense that there's plenty of math but the math you get taught at school is the one that was deemed useful.

>>15368482
>Using what?
infinity

>>15367574
You can learn to like something which is something many "nerds" can't wrap their head around.

>>15343480
pseudoscience

>Gamma(x) can be turned into f(x)/g(x), f and g are holomorphic everywhere
What functions are f and g?

>>15368533
Unless you're a Platonist, there isn't really anything you "find out about THE" natural numbers using infinity. E.g. there's no infinite objects in Peano arithmetic.

All functions from natural to real, are continuous.

>>15370065
Let A be a non-open subset of the naturals.
Wouldn't the indicator function of A be a non-continuous function from the naturals to the reals?

>>15370098
No.

I have a few papers published in another field unrelated to math before I got into pure math. Should I put them in my homepage or can I pretend they don't exist?

>>15370065
All functions from the natural numbers with the discrete topology to the reals with the standard topology are continuous.

>>15370142
The set [math]\left]\frac{1}{2}, \frac{3}{2} \right[ [/math] is open. So for [math]1_A[/math] to be continuous is neccessary that [math]1_A^{-1} \left( \left]\frac{1}{2}, \frac{3}{2} \right[ \right) = A[/math] is also open.

If [math]f(x)[/math] and [math]g(x)[/math] are polynomials with computable real coefficients, is [math] \int \frac{f(x)} {g(x)}[/math] always computable?

>>15368005
>>15368076
>>15368079
Hello again anon. Sorry for my late reply, I had an exam today and then died temporarily.

Okay so regarding your solution, it's pretty good except for the last part about counting by hand. I'm happy to say that's of course not necessary and it's not hard either. Like you say, we have A=2, B=3, C=167 and we're supposed to distribute 2 As, 1 B and 1 C among 3 boxes essentially. Well then we can just do that separately for each of them. And just as you said, it can be done with stars and bars. 2 stars 2 bars for As give us 6. 1 star and 2 bars for B give us 3, same for C and multiplying those we get 54.

Again, nice job on your solution! I think it's impressive you managed to count all the 54 possibilities by hand and it's also nice you verified it in Python. I think you'd figured out how to calculate it using stars and bars since it already came to your mind.
>Thanks again for the question anyways.
Thank you more for solving it and writing your solution here. I appreciate it. And sorry again for the late reply, hope you have a nice day.

>>15370645
Computable in what sense, with respect to what integral and is this an indefinite integral? And I suppose you exclude g's and u's with g(u)=0.
The answer is probably that yes it's fine.

>>15370065
f(x) = x

>>15369503
I may well be a Platonist. What would you say to that?

>>15371195
Why I would say?
I'd say it doesn't clarify things and it's a position people take to not have to think about philosophy any further.

I want to refresh my discrete math. Is there a good textbook, website or video series with a focus on computer science that is concise? Most textbooks are unusable because they're way too long, overly complicated and contain pointless exercises. I just want a good summary of the basics that makes use of illustrations.

>>15371286
I haven't read it but many people recommend Concrete Mathematics by Knuth.

>>15370862
You seem to be relatively knowledgable if you're the usual anime weeb anon. Been looking into fuzzy logic stuff out of curiosity, seems to have varied discrete applications but can't find much in the way of general systems post 1930s that aren't the 1990s fad. Usually implemented with respect to stochastic systems like markov chains and such, "quantum logic", etc, or so claimed. Just wondering if there's been more recent formalizations that are remotely serious.

The qualifier for "remotely serious" would be such that, like a coherent state for quantum states, the logic resolves into similarly "coherent states" mappable to things like binary logic, i.e. via emergence to coherence. Or some endeavor like that. All I find are "fuzzy logic" fads of various kinds with either limited specific application or sole focus on deriving the coherent state from a bounded system.

>>15371370
never-fuckin mind apparently I was stumbling my retarded ass toward von neumann algebra in the dumbest fucking way possible by going backward instead of realizing I was obviously thinking about ergodic theory FUCK MY LIFE

>>15371421
are you saying that you accidentally found a connection you didn't know about?

>>15371460
Oh you know just enjoying the lovely experience where every intuition I have leads to 100 year old ideas because Jon Von FUCKING Neumann exists to shit on my fucking day ensuring my perpetual irrelevance. Don't mind me time for more shut up and calculate it's all I'm fucking good for apparently

Any math guys want to explain something to a math noob?

A small circle contains an infinite number of points.
A large circle contains a larger number of infinite points than the small circle.
Right so far?

If we place the smaller circle directly within the larger circle and draw a line directly out from each point on the small circle, at 90 degrees to each tangent, we have begun mapping the points of the smaller circle onto the larger circle. But since the smaller circle contains a smaller infinity of points we can never completely map every one of its points onto the large circle.

Yet is we were to do the opposite and map the points of the large circle onto the small circle we should be able to do so since a point has no physical dimensions Therefore no matter how many points the large circle has we should always be able to map to the corresponding point on the small circle. Yet the large circle is said to contain a larger infinity of points than the small circle.

Can anyone explain this apparent inconsistency?

Thank you in advance for taking the time to explain this to a dummy.

>>15371574
>But since the smaller circle contains a smaller infinity of points we can never completely map every one of its points onto the large circle.
>Yet the large circle is said to contain a larger infinity of points than the small circle.
False.
Cardinality (size, essentially) is tricky when talking about infinity. Some infinities can be said to be larger than others, yes, but it has nothing to do with the apparent size of the objects you associate with them.
As you say, you can associate every point on the larger circle with a point on the smaller circle, and as you seem to have figured out, you can naturally make the reverse mapping, even if it seems unintuitive at first. That this one-to-one mapping is possible is indicative of the fact that the cardinality of the total number of points is equal in the two circles, despite one being larger than the other.
A slightly easier to think about it might be to consider, instead of a circle, the number line. If we consider all of the even integers, and then all of the integers in general, we can make a one-to-one mapping by simply halving each even integer (or doubling each general integer), so the two sets have the same cardinality despite the fact that intuition would tell you that there should be twice as many integers as even integers.

I see. That makes it clear. Thank you.

Is it normal to understand the same equation to varying degrees depending on the day? Processing speed is about the same, it's like I can hold an equation in my head for a longer period of time in the morning than at night. Could it be something as simple as tiredness?

>>15371661
>depending on the day
the time of day*

Knew a guy who was a programmer working on some advanced shit
Would be driving home from work when he would suddenly have an epiphany about some work related shit and have to write it down just FUCKING IMMEDIATELY! becasue he knew he couldn't hold the complicated equation inside his head for too long when he was tired.
Would crash car because he was too busy writing the shit down and have to be towed away
Company would pay all the bills and deal with any fines and police stuff cos his shit was just too valuable
After the third time they forbade him from driving to and from work and got a chauffeur to always pick him up and drop him off.

>>15371704
>shit that never happened
soiboy fairy tales

How in the fuck do you deal with this frustration of all your intuitions leading to rediscovering somebody else's fucking wheels? There has got to be a term other than that for this, but I find nobody talking about the frustration either, which is making it even worse. Like I'm uniquely fucking stupid.

>>15371756
The next time you think of a nice idea, don't rush in to work on it, but do a literature review first.
More often than not, this process will itself be a source of fresh inspiration.

>>15371769
That is what I do. I then proceed, however long that takes, to discover it's already been done. Wash, rinse, repeat. That is literally the source of the frustration.

>>15343262
Schizophrenic cave emigration
t. my friend is working on his phd and we joke about this

>>15371776
Don't know what sort of ideas you have, but for me any frustration that I get from what you're describing is outweighed by the excitement from seeing all the cool gadgets and applications that they're doing with the thing I thought up (especially if they turn out to confirm my preliminary conjectures).
Plus, as I've said, usually there exists some detail in which your approach differs from the mainstream, so there will still be something you can contribute to.

But with all that said, if you're looking for general advice, all I can offer is just to read more math, especially beyond the textbooks.

>>15355533
This. Math discovery is driven by questions and applications of the day. Yeah maybe differential geometry will be tapped out, but there are countless systems to mathematicize

>>15365535
Why is enlightenment math centered around physics?
Why is Greek math studying platonic forms?
Where did all this computer science math come from?

>>15371814
I don't know how to make anyone understand. Nobody ever understands. It isn't exciting. It's like running in place no matter what you do, and I can't make you understand just how and why that would drive somebody crazy.
>But with all that said, if you're looking for general advice, all I can offer is just to read more math, especially beyond the textbooks.
WORKING ON IT. Should've done this years ago but everyone always thought me crazy and I found being a human calculator boring as fuck in school, long before any of this stuff existed online, so I never even knew any of my crazy would've mattered. I just figured the problem was me. Only the more I read the more things make sense, and realizing I was in fact making perfect sense the whole fucking time. You have no idea what that's like. Only realizing, progressively, every idea you've tried explaining in words that people just didn't fucking understand was already understood centuries ago, or a century ago, and being treated like a nutcase for thinking them. You've no idea what a frustrating late start is like, and I have no idea how to explain it. Nor explain the horror of realizing just how outmoded you are in spite of it all.

I'd scream if I could even figure out how to make the scream mean a goddamn thing. Anyway, back to fucking working on math. I want to pretend time machines could exist so I can punch Neumann in his fucking face.

>>15353997
Interesting. Thank you

>>15367079
what do these words mean

>>15371839
Get in touch with guys like Norman J Wildberger or Ian Angell. You need to associate and communicate with people who think outside the box.

If you scoff at this then you are just another parrot and cant be helped.

>>15371661
You have masturbated more during the evening and night than in the morning. Your head is less fogged with thoughts of your mother.

>>15345344
You can't even define a step function without conditional statements. Or an equation for a circle in the cartesian plane, for that matter. The square root function, I could go on.

The other anon was wrong saying "functional programming" because it's also full of conditionals. You can't get away from them really.

You will probably enjoy learning about lambda calculus and the church-turing theorem.

>>15372156
Angell doesn't seem relevant and Wildberger's notions appear, at first glance, wholly antithetical to my own. I cannot fathom what possessed you to recommend either of them, especially given Wildberger's notions on infinities. Pointless to bother explaining as you already classified any objection as "scoffing" and "parroting" like a true zealot, and I already gave far greater consideration to your position than you deserve with such an attitude.

>>15372281
have a cracker pretty boy

>>15371370
>You seem to be relatively knowledgable
I am absolutely not
>if you're the usual anime weeb anon
Maybe, maybe not? I'm not sure who the usual weeb is. I think you might be thinking of my friend who's usually at /sqt/ and who posts remilia pictures. He's pretty knowledgeable.
>>15371421
>>15371468
If it makes you feel better, Von Neumann is sussy as hell. Some stories about him don't make any sense when you think about it for longer than 5 minutes.

I've seen 3 different videos explaining the origin of the power rule and i still don't understand it.

>>15376098
Now read a book.

>>15376098
You can prove it with the Binomial Theorem. Let [math]f_n(x) = x^n [/math] then
[eqn]f'_n(x) = \lim_{h \to 0} \frac{(x+h)^n -x^n}{h} = \lim_{h \to 0} \frac{\sum_{k=1}^n {n \choose k} x^{n-k} h^k}{h} = {n \choose 1} x^{n-1} + \lim_{h \to 0} \sum_{k=2}^n {n \choose k} x^{n-k} h^{k-1} = n x^{n-1} [/eqn]
Alternatively it follows from the product rule with induction.
Induction hypothesis: [eqn]f_n'(x) = n f_{n-1}(x)[/eqn]

First check the base case [math]n=1[/math]:
[eqn] f_1'(x) = \lim_{h \to 0} \frac{f_1(x+h) - f_1(x)}{h} = \lim_{h \to 0} \frac{h}{h} = 1[/eqn]
Now assume it is true for [math]n[/math] now for [math]n+1[/math] we get
[eqn] f_{n+1}'(x) = (f_1 \cdot f_n)'(x) = f_1'(x) f_n(x) + f_1(x) f_n'(x) = f_n (x) + n f_1(x) f_{n-1}(x) = (n+1) f_n(x) [/eqn]

>>15376209
Now prove it for irrational exponents.

>>15376235
Then you also have to know the derivatives of the exponential function, logarithm and the chain rule and then use the identity
[eqn]x^a = e^{\log(x) a}[/eqn]

So the derivative is
[eqn]e^{\log(x) a} \frac{a}{x} = x^a \frac{a}{x} = a x^{a - 1}[/eqn]

>>15376274
Now prove it for matrix exponents.

>>15375983
>If it makes you feel better, Von Neumann is sussy as hell.
Considering half my intuitions ended up leading to papers and ideas he contributed to, I doubt that. If a dumbass like me ends up there I'm sure someone given all the opportunities and lucking into that life from the start could've done it. Right place right time. Just infuriating some people get to be so lucky.
>>15372156
Having looked even further into Wildberger, he is the equivalent of a mathematical flat earther in that he functionally has to deny induction to adopt constructivism on steroids like he does. Fucking hilarious and I am not surprised given the state of /sci/. At least he seems to be a nice guy and appears to be able to teach well. Shame he's got the dumb.

I almost finished reading Leinster's short introduction to basic category theory. Are there any texts that are written at a similarly introductory level but cover some of the stuff that didn't make it into Leinster? Monoidal categories in particular come to mind

>>15377638
Here's a list of link. This also has links to other lists. Many of those might be dead links but the names are there anyhow
https://gist.github.com/Nikolaj-K/282515e58c1c14de2e25222065f77a0a

>>15377741
nta thanks for the link i'm throwing it in with the rest as I go down the list of shit to learn

>>15345207
You just multiply it across, it's self evident proof, a rule of how mathematics functions. a/b * c/d = ac/bd

Does there exist any text or perhaps website that compiles techniques for understanding very large numbers? For example consider something like [math]2^{2000}[/math]

Obviously we know it's a massive number, but how can we get a grasp on just how large it is, like how it compares to say [math]5^{350}[/math] I know there is a method with logarithms to determine how many digits are in a number, but what are some other techniques for trying to comprehend or at least grapple with extremely large values?

>>15378135
What's there to understand? Sure there is Elementary Number Theory but that sounds like overkill.

>>15377767
nta?
Reddit?

>>15378135
My meme answer would be: Divide them?

>comprehend or at least grapple with
That doesn't really sound like a very specific goal.

"extremely large values" is also relative. Playing around with Mathematica for 30 seconds tells me [math]2^{2000}[/math] is smaller than [math](22^{22})^{22}[/math] and that in turn is smaller than [math]5^{(5^5)}[/math].

>>15352130
>Serre
a great time

>>15376697
There's no luck involved, you are just a retard on a larp, now take your meds

>the other one got deleted
lmao

>>15378374
I am just astonished by your lack of mental clarity.

New thread!
>>15379790

>>15379424
hopefully this one doesn't share that fate :D