/mg/

[Anonymous No.12112291]

Maths, generally
>infinitesimals edition
Malk taths

[Anonymous No.12112299]

Sorry forgot
Alte:
>>12111249

[Anonymous No.12112315]

Hmm on second thought I could've named it "layers of mutual disconnectedness" edition but oh well

[Anonymous No.12112331]

>>12112291
>Elliptic curves

[Anonymous No.12112341]

>>12112315
faggot

[Anonymous No.12112372]

>>12112331
based

[Anonymous No.12112481]

https://youtu.be/uqBhy1c4MZc

I'm currently into constructively well-pointed Cartesian closed Heyting pretopoi with a natural numbers object.
Here's a Shulman text clarifying some notions
https://arxiv.org/abs/1808.05204

The main question of course is whether I should say pretoposes instead.

[Anonymous No.12112503]

Is it possible to capture morality in a mathematical model?

I don't mean stupid utilitarian ethics. I want to describe morality in all its beauty and complexity mathematically.

[Anonymous No.12112515]

>>12112503

[Anonymous No.12112530]

Reminder that there is this seminar starting next Monday.
http://dept.math.lsa.umich.edu/seminars_events/events.php?eventdefid=106&dt_begin=2020-07-01&dt_end=2021-06-30
>The first meeting of the University of Michigan algebraic topology seminar will occur on Monday, September 14 at 4 - 5pm EST via Zoom. Dan Isaksen will be speaking about "Deformations of stable homotopy theory". The title and abstract of his talk, as well as the upcoming seminar schedule can be found on the seminar homepage

>>12112329
Yes d(x, y)=|x-y|.

>>12112503
Categorical imperative.

[Anonymous No.12112550]

>>12112503
You might want to look into deontic logic.

[Anonymous No.12112553]

>>12112515
Unironically this.
Read Ethics.

[Anonymous No.12112604]

>>12112291
literally the worst, least accurate way to represent infinitesimals

[Anonymous No.12112636]

>>12112604
It literally isn't.

[Anonymous No.12112757]

>>12112636
>hey guys if you approach zero but never hit it youll get to the infinitesimals
how is that not what the picture is implying

[Anonymous No.12112917]

>>12112530
>using the most boring metric on Q

[Anonymous No.12112944]

>>12112917
What's the most exciting metric on Q?

[Anonymous No.12112956]

>>12112917
Sure, there may be more interesting metrics on the rationals, but the most boring one shows that there is at least one. The person asking was asking if it is a metric space, and now they can take the pair (Q, d) for any metric on Q, and it will make sense because of the example I provided.

[Anonymous No.12112976]

>>12112530
>advertising seminars
>posting anime
gmmg tranny?

[Anonymous No.12112986]

>>12112976
Some people call me that. Any interesting insights, my friends?

[Anonymous No.12113034]

>>12112917
Just use the discrete metric and have a clusterfucked useless space

[Anonymous No.12113108]

>>12112956
A non-discrete metric I meant.
/gnmg/

[Anonymous No.12113140]

>>12113108
Night, lad.

[Anonymous No.12113182]

>>12112291
how can infinitesimals be a thing if 1 = .999...?

[Anonymous No.12113186]

>>12112291
>infinitesimals
not maths

[Anonymous No.12113212]

>>12113182
R doesn't include infinitesimals

[Anonymous No.12113216]

>>12113186
>tfw she advisor-mogs everyone in this thread
is this fair bros?

[Anonymous No.12113230]

>write long as fuck technical paper over course of 2 years
>finally done
>time for final, forth, recheck
>nothing works suddenly
>kill_me(2).tiff
after a few hours of suffering I found that shit is still in order, just fucked up some exponent in maple when checking some symbolic calcs h-haha...

[Anonymous No.12113247]

>>12112481
Since you're posting your vids regularly, do you still work in academia? Asking because I had to and will most likely next semester again have to teach via video conferencing or pre-prepared vids. Any general recs/preferences for how to structure these kind of videos in general? Level of courses is likely at most beginner grad courses.

[Anonymous No.12113305]

>pi is proven to be transcendental
>e is proven to be transcendental
>we don't even know if pi+e is rational or irrational

[Anonymous No.12113334]

>>12113247
No, I'm not in academia.
Mhm, I don't know exactly what info would help you but I can brainstorm some best practices for me.
Firstly, I use OBS for recording, which probably many or most people do (obsproject.com). I don't think you need more than 1 screen including a camera window corner as I do it, but of course you can be more fancy with your window management.
I got a USB connection micro with a stand (see pic) and you can check more youtube for videos on what mic to get. I mean you'll end up getting something from Amazon I suppose and I didn't bother actually looking into what distinguishes those things. I do think a mic is better than your common laptop mic, but I must also say it gets in the way of one hand while you operate the screen.
I found having a quite brought light 1m away from my face helps with the lighting. It's less annoying than it sounds.
As doing videos and trying to be regular is whole lotta work, I eventually ended up doing those one-takes and I think it still sort of works. I get complaints that the videos are far too long, but to Quote Mark Twain
"I didn't have time to write a short letter, so I wrote a long one instead."
Have some water or something to drink on the side. With lectures, you'll sit there for an hour I guess, and I take at least 15 minutes beforehand and then the whole upload and doing title and all that semi-automated things take another 30 minutes afterwards.
As for scripting, once you make your slides or write the text you go through, the process of this does most of the scripting for you and anything beyond that that you'd write down, you might end up forgetting in the moment anyway.
I use MathJax, but I assume you have some pictures to show, so in that case you'll need slides (I think that's a more showable format than LaTeX compiled files). I see a bunch of math profs in videos now doing tablet life writing, but I personally find that annoying, the "10 ugly words per slide before they pass on" pages.

[Anonymous No.12113353]

>>12113247
>>12113334
If you don't want to go for a tablet or slides, you can conveniently render TeX with just one line importing the MathJax javascript, save it as a html and drop it into your browser - that has the advantage that you don't need much markup or editors, but for picture you'll need some more html.
Described with more features here here https://gist.github.com/Nikolaj-K/00e425a2857cd4a0991f504d14276c30
The line (let's see if this lets me post it) is

<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
\(a^2+b^2=c^2\)

[Anonymous No.12113358]

>>12113353
I don't mean to pry but have you done graduate study / are your personal study subjects relevant to your day job?

[Anonymous No.12113366]

>>12113334
>>12113353
Thanks a lot for the detailed reply!
I do have in fact experience with most of the stuff you mentioned. A good mic was the first thing I bought and obs worked well (at least with a wm that renders every workspace simultaniously like mutter). Light is still kind of an issue, will try to pester the department to buy me some professional shit.
>a lot of work
At least 3x as much as "normal" lectures from my experience... I want to try a tablet, but the one that was provided to me by my university is crap. And I don't think they'll buy me another one. From my experience, your reply, and student feedback I think a cool format would be beamer slides visible on one hand of the screen and "live" work on a black board simultaniously on the other side of the screen, would require a clicker and a pretty good camera though.

I never used mathjax (except for my homepage of course) but I'll check out your link!

[Anonymous No.12113391]

>>12113358
My Master thesis was on non-commutative emergent gravity, my PhD was in plasma chemistry, my work is in computer vision and privately I'm stuck in a foundations hellhole for 2 years now.
So I guess kinda no.

https://youtu.be/Vi2TIV3rYzc

But let's be optimistic and say it's all connected, so yes.

>>12113366
>I think a cool format would be beamer slides visible on one hand of the screen and "live" work on a black board simultaniously on the other side of the screen
Blackboard work and slides at the same time?
Clicker as in you remove move the slides while you go in depth on your blackboard. Well no wonder you take 3 times longer ;P

I donno, the Burger has a fantastic board presentation. I think he writes with Edding on some sort of plastic - would actually be interesting to know
https://youtu.be/K-JnFp6HwWE
I like this guys blackboard format, maybe one can steal from him
https://youtu.be/TpLEdu18iok

>I want to try a tablet, but the one that was provided to me by my university is crap.
The company I work with bought me an iPad and a pen mid Corona for conference calls with some drawing software but I haven't even opened it yet. Just don't do a 4 by 4 grid of 16 words and diagrams in random places.

>>12113366
>I never used mathjax
What the hell is water?!

[Anonymous No.12113419]

>>12113391
>https://youtu.be/TpLEdu18iok
This is nice, but for what I'm teaching not enough blackboard.
And what IS water, huh? Brb drilling a hole in my head and see if I find an answer...

[Anonymous No.12113427]

>>12112944
The 17-adic metric!

[Anonymous No.12113447]

I'm taking complex analysis and I missed two classes this week. We don't have a textbook and my teacher's posted notes aren't very helpful. So, I have a question about this problem. I understand how complex functions work as mapping some value from one complex plane to another complex plane(the x,y to u,v business). It's easy enough to figure that out. But I'm a bit confused when we start thinking about "strips".
Here is what I figure: the first strip is just saying that the magnitude of the real part of z cannot exceed one, so its all points between -1 and 1 on the real axis, and all on the imaginary. Similarly, the second strip is just a horizontal strip between 1 and 2 on the imaginary axis where x can be any value. But how do we show or describe the mappings?
So, for the first complex function (w = 2z +i), when we sub in (x+iy) for z, we find the real part becomes scaled by two in the image. So then, could we say that the image is defined by the magnitude of the real part being less than 2? For the second strip given, by similar logic, could we say that image is defined by the imaginary part being between 3 and 6(because v = 2y+1)? That's the only thing that makes sense to me, let me know if am on to something because the next questions are way harder.

[Anonymous No.12113457]

>>12113447
A typo, I mean between 3 and 5 for the second strip..

[Anonymous No.12113512]

>>12113447
for the second complex function: w = (1+i)z + 1, I find that u = (x - y + 1) and v = (x + y). So, in this case: wouldn't the image of both strips simply be the entirety of the complex plane, because both real and imaginary parts are defined by both variables, and in both cases, one of the variables has no limitations..

[Anonymous No.12113541]

>>12113447
take an arbitrary point in the strip and apply the transformation; what happens?

[Anonymous No.12113564]

>>12113541
It gets mapped to another complex plane...yeah?
I think i realize where I might be going wrong.
Like do I test two boundary positions given by the strip? like get an equation for u and v, test two cases, the upper and lower bounds for x, then describe what region that entails? If so, I think i was right for the mapping of the first function...
but not here:>>12113512
If i use x =-1, and 1, you get an linear equation in terms of u and v...the image should be the region in between those boundaries...I think that's what is going on here. If i am fucking up here, lemme know yet agian.

[Anonymous No.12113826]

I love math so much, bros. I really hope going for this BS doesn't end in a disaster job search-wise.

[Anonymous No.12113849]

>>12113826
As long as you have decent programming skills, it won't.

[Anonymous No.12113853]

>>12113849
Yeah, I'm going for a CS minor and do some side projects. Hopefully it will be aight.

[Anonymous No.12114068]

If I have a cake and a cherry on it and I take a random proportion of this cake (uniformly), I found that if my proportion contain the cherry then on average my proportion is 2/3 (using conditional expected value). Now I would like to find the density function of the proportion, knowing that I have the cherry. Numerically using python I found that the density is [math]f(x)=2\chi_{[0,1]}(x)[/math]. How would i find this result?

[Anonymous No.12114086]

>>12114068
Nevermind I found the proof myself. It was fun to guess with a program and prove it after.

[Anonymous No.12114160]

After finishing my leftover exams, it is time I return to topology. Today I am finishing connectedness and will start reading up on compactness.

Wish me luck mg!

[Anonymous No.12114183]

>>12114160
Have a good time studying, lad

[Anonymous No.12114316]

im studying a masters in applied math at an absolute garbage of a school
what courses should be included in such masters guys?

[Anonymous No.12114320]

Do mathematicians with experience in the relevant fields understand QM better than physicists?

[Anonymous No.12114350]

>>12114160
what are you reading

[Anonymous No.12114358]

And a good day to, folk of /mg/!

>>12114160
Good luck! Compactness is an important thing, please do make sure you learn to appreciate it.

[Anonymous No.12114364]

>>12114358
Afternoon, lad

[Anonymous No.12114371]

>>12112503
https://plato.stanford.edu/entries/logic-deontic/

[Anonymous No.12114427]

>>12114350
Munkres 2nd edition.

>>12114183
>>12114358
Thank you anons.

[Anonymous No.12114465]

>>12113419
https://youtu.be/8CrOL-ydFMI

[Anonymous No.12114604]

>>12114316
https://www.imo.universite-paris-saclay.fr/-programme-104- I’m doing this master this year, I don’t know how good the curriculum is but it’s one of the best master in applied maths from my country. Maybe try to focus learning the fundamentals aspects first like measure theory and topology until you can understand wikipedia pages on applied maths topics and not be totally lost.

[Anonymous No.12114642]

>>12112291
I know this is asked a lot but bare with me.
I was in special education (don't laugh) for math classes because math is hard for me to understand and remember. I failed my math exam in school 6 times and somehow graduated.
Is there a method for a literal retard like me to understand and remember math?

[Anonymous No.12114649]

>>12114642
khanacademy will have decent resources for your purposes
don't give up!

[Anonymous No.12114658]

Found this is a graduate-level textbook, but I think it might still be too advanced for me.

[Anonymous No.12114689]

>>12114658
they are the first degree grandetrigonometric functions

[Anonymous No.12114758]

What does /mg/ prefer:
>State Theorem abc
>proof of Theorem abc:
>blablabla
>State Lemma xy
>prove Lemma xy
>apply Lemma xy
>blablabla
>end of proof
or
>State Lemma xy
>prove Lemma xy
>Theorem abc
>proof of Theorem abc:
>blabla
>apply Lemma xy
>blabla
>end of proof

[Anonymous No.12114766]

>>12114758
The latter. Don't fuse lemmas into the middle of the proof of another result.

[Anonymous No.12114775]

>>12114758
if the lemma is used elswhere, then put them first
if it is solely for the proof, then in the middle is fine

[Anonymous No.12114781]

>>12114775
Why even call it a lemma then? It's just a step in the proof at that point.

[Anonymous No.12114818]

>>12114781
all lemmas are though. Every single lemma and theorem in every single field of maths is merely a step in the ultimate proof of mathematical unity which has its origins the fundamentals of set theory

[Anonymous No.12114831]

>>12114818
I'm all for the unification of apparently disparate areas of math via nifty results connecting them, but what you just described sounds delusional and lame.

[Anonymous No.12114832]

Anyone got any good intuitionings on trace? All I can see is that it is an invariant that can relate different matrices back to their common operator, seems like an algebraic tool mostly. Is there anything cooler?

[Anonymous No.12114889]

>>12114832
Graph theory is your friend. The trace is the number of loops, or closed walks of a given length.

[Anonymous No.12114902]

>>12114758
The second.
But I want to remark that you made the post more complicated than it should have to be, you could have used intent or something to figure out what you're asking.

>>12114775
I don't like this attutude towards a text and life in general. If the placing of the lemma is conditioned on wether it's relevant in the rest of the text, this implies the value of the lemma is only for getting over with the book, or learning it for a text.
Every lemma is a mathematical theorem that, if it's a strong nice relevant one, you might want to know. Whether it's used somewhere else _in the text_ shouldn't be the determining factor. It's status should be determined by its use in the historical context of math and its applications overall.

[Anonymous No.12115148]

>Robert B Ash, author of many math textbooks, died whilst on one of his usual afternoon walks
https://www.news-gazette.com/news/update-pedestrian-died-after-low-speed-collision-in-urbana/article_2cd9a534-0673-5b73-8c73-49ce091a1b66.html
/mg/ please be careful when you go on your walks. Don't spend all that time with your head in the clouds and watch out for actual cars from time to time!

[Anonymous No.12115168]

>>12115148
F

[Anonymous No.12115174]

>>12115148
dying isn't bad

[Anonymous No.12115185]

>>12115174
Okay, you first.

[Anonymous No.12115189]

>>12115185
okay, brb

[Anonymous No.12115630]

>>12115189
It's been over 2 hours. Did he do it?

[Anonymous No.12115683]

>>12115630
Hey guys, I'm back. Sorry it took so long. As you can see dying is no big deal. I just did it and here I am, totally fine.

[Anonymous No.12115794]

my only classmate in my program is a dude who says I'm wrong because he doesn't understand my proofs, on top he's very passive aggressive about it
how do I cope?

[Anonymous No.12115811]

Does a/b always equal a(b^-1) in the complex numbers? I can't remember.

[Anonymous No.12115826]

>>12115794
Are your proofs correct?

[Anonymous No.12115856]

>>12112291
>Next semester profs. Announced
>Get an ultra communist, known for being a jackass dude in analysis
>Fucking kill me dude, what should I do?

[Anonymous No.12115883]

I feel like this is an easy problem, but I'm stuck.

[Anonymous No.12115898]

>>12115826
yeah
one day he even messaged me "I'm not convinced, but sometimes that happens and turns out you are right."

[Anonymous No.12115982]

how do the inverse trig functions work? is it just a hard mapping, or is there a function that performs a number of operations to go from a sine input value in radians to a discrete pair of triangle side lengths

[Anonymous No.12116003]

>>12115883
Just fuck with a measure zero set.

[Anonymous No.12116008]

>>12115898
Why are you still doing math with him?

[Anonymous No.12116014]

>>12115982
Given the interval [math][0, \pi][/math], there is bijection between the interval and the interval [-1, 1] given by the cosine function. We may thus send y=cos(x) to a unique element of the first interval and call that [math]\cos ^{-1} (y)[/math]. Similar stuff for sines.

[Anonymous No.12116092]

>>12116003
kinky

[Anonymous No.12116243]

>>12115883
define f(x) as
when x in IQ and then x=p/q so f(x=p/q) = q
when x in IR just f(x) = x

IQ is measure zero so w/e

[Anonymous No.12116247]

>>12116243
should be IR - IQ, whoops

[Anonymous No.12116325]

>>12116243
This except map irrationals to 0 in case your version of integrable means over all of R simultaneously.

[Anonymous No.12116329]

>>12115883
>integrable
>unbounded on every open interval
how?

[Anonymous No.12116375]

why didn't anyone tell me that solving a linear congruence equation was the same as solving a linear diophantine equation

[Anonymous No.12116382]

>>12116329
Do you know measure theory?

[Anonymous No.12116398]

>>12116375
But I did.

[Anonymous No.12116410]

>>12116243
please learn LaTeX or don't post

[Anonymous No.12116445]

>>12116014
where can i get redpilled on trigfunctions? theres more going on here than anyone is telling me

[Anonymous No.12116512]

>>12116445
'Advance Trig' by Durell/Robson is good.

[Anonymous No.12116530]

[math] x + x\,y\,A + y\,B = x\,A + xy\,B + y [/math]

Does this remind you of something?

[Anonymous No.12116547]

>>12116530
No.

[Anonymous No.12116601]

>>12116530
Have you tried grouping terms?

[Anonymous No.12116645]

>>12114465
kek

[Anonymous No.12116703]

>>12114758
On that note, what's /mg/'s favorite lemma? For me, it's the "every sequence contains a monotone subsequence" lemma, which easily proves the Bolzano-Weierstrass theorem.

[Anonymous No.12116714]

>>12116703
https://en.wikipedia.org/wiki/Snake_lemma

[Anonymous No.12116728]

>>12116703
Really depends on the theory, but let's say Baker-Campell-Hausdorff.
And in case you really want to only consider those ZF provable statements that happen to carry the name "lemma", then I'd say Ito or Yoneda.

[Anonymous No.12116735]

I'm trying to construct a pdf that player 1 wins a 2 player game as follows:
We choose a starting point in the range [0,1]. Then each players select a number randomly from [0,1] with uniform probability. This number is added to the starting point, and then it is the next players turn. The winner is the first person to raise the starting point above 1.

[Anonymous No.12116754]

>>12116735
cont. so really its a family of pdf's parameterized by the starting point

[Anonymous No.12116824]

what's a brainlet friendly book for lebesgue measure?

[Anonymous No.12116840]

>>12116735
>>12116754
Bre, you manage to be very confusing in a few sentences.
Both start at the same starting point? Then why not start at 0 and play it to a random height?
You say
>then it is the next players turn.
without having before mentioned any action only taken by the first player.

And in any case, with those small bounds, it sounds like the game with pretty surely be over before say the fourth turn.
After n jumps, then expected distance will be [math] \sum_{k=1}^n k \int_0^1 x\x dx = \frac{n}{2} [/math], wouldn't it.
The density for the first step is uniform, a box.
I might be wrong but I think the density for the second step is just the convolution with the box again, e.g. giving 0 chance of being at 0 and at 2, but higher inbetween.
And then you keep doing convolutions, with the peak at n/2.

Easy to test in simulation if that's correct.

[Anonymous No.12116845]

fucked up the sum, ignore the k and the \x


[math] \sum_{k=1}^n \int_0^1 x\, dx = \frac{n}{2} [/math]

[Anonymous No.12116863]

>>12116840
it might be simpler to change the 'starting point' parameter to a 'smaller goal'. Basically they alternate randomly selecting uniform numbers from [0, 1] and adding them until one of them, breaks past the 'goal'. That player is the winner.
I need to solve this exactly
I'm sure there is a trick to this, much like a game where both players take turns flipping coins and the first person to get heads is the winner.
But in this case the equality that arises from analagous considerations will have an integral w.r.t the parameter.

[Anonymous No.12116868]

>>12116824
Bartle, learn analysis first or don’t bother.

[Anonymous No.12116870]

What's a proof-heavy, rigorous book on differential equations?

[Anonymous No.12116877]

>>12116863
Yeah I figured, but I think I proposed a solution: The chance for a player being at distance x after n steps is given by the symmetric function centered at n/2 given by the n-times convolution of a box with itself. You can do the convolution analytically, but I don't have it in my head. The expectation will be at the center. The player you is one step ahead is expected to be 1/2 ahead.

[Anonymous No.12116882]

>>12116870
Birkhoff and Rota, Arnold.

[Anonymous No.12116914]

>>12116870
Coddington/Levinson

[Anonymous No.12116991]

>>12116870
Brezis for PDEs.

[Anonymous No.12117005]

How do I secure funding? A pel grants don't count toward income calculations with SSI

[Anonymous No.12117013]

I'm here to beg to anyone who will listen, please increase funding for education and research. I don't care how. And I don't care where (unless you live in a real first world country)

[Anonymous No.12117031]

>>12117013
wrong. Funding for education and research must be cut to zero for anything that is not STEM-related, and maxxed for anything STEM-related. Funding to social sciences is money down the drain. I know because I know what the social science research output is like.

[Anonymous No.12117049]

I'm learning cryptography, and there's something my professor said which I don't quite understand.
My professor:
>In asymmetric cryptography, to have secure communication between n parties, you need n * (n - 1) keys. One key pair for each branch.
My calculations:
>In asymmetric cryptography, to have secure communication between n parties, you need 2n keys. Each person needs one private key and one public key.
Can anyone explain to me why I'm wrong?

[Anonymous No.12117055]

>>12117005
Solve the Riemann hypothesis.

[Anonymous No.12117103]

>>12116877
thanks for the thought anon. I think i may be able to proceed

[Anonymous No.12117230]

>>12117049
I don't really know anything about cryptography, but maybe using the same keys for everyone makes you susceptible to some sort of distributed attack?

[Anonymous No.12117248]

What are some of the coolest isomorphisms that you know of?

[Anonymous No.12117253]

>>12117248
Integers and surfaces.

[Anonymous No.12117254]

>>12117049
>>12117230

Triangle network: vertices are connected by three lines. Two keys per line -> 3*2=6 keys
Square network: vertices are connected by 6 lines. Two keys per line: 6*2=(4*3)=12 keys
Pentagon network: vertices are connected by 10 lines. Two keys per line: 10*2=(5*4)=20 keys

etc

[El Arcón No.12117256]

>>12117055
This doesn't work. Can confirm.

[Anonymous No.12117266]

>>12117254
More formally, in a network there are n agents, total connections are nC2, simply show that 2*nC2=n*(n-1). Indeed

2*nC2=n!/(n-2)!=n*(n-1)

[Anonymous No.12117286]

>>12117266
bro me and the question asker both know basic combinatorics. WHY should you need a unique key pair for each key edge on not per person?

[Anonymous No.12117307]

>>12117286
well browno I don't know shit but if we have a triangle network with agents A, B ,C then we will have publicA, publicB, publicC and privA, privB, privC. For A to see his shared crypted info with C, he will need to combine privA and publicC so a unique pair per person is necessary to have the network secured

[Anonymous No.12117320]

>>12117307
>browno
holy shit you're a fucking retard
you just justified why we at least need a key pair per person.
BUT the OP asked why his prof claimed you need MORE keys, one pair per EDGE
Fucking psueds always shitting these threads up with worthless comments

[Anonymous No.12117358]

>>12117320
>you're a fucking retard

[Anonymous No.12117360]

>>12113305
It’s an integer

[Anonymous No.12117364]

>>12117320
sorry brownio I'm tired and I know shit about cryptography but you don't either so be patient

[Anonymous No.12117388]

>>12117360
Proof?

[Anonymous No.12117420]

>>12117248
https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
https://en.wikipedia.org/wiki/Poincar%C3%A9_duality
The Algebra-Geometry correspondence in algebraic geometry is really cool too.

>>12117253
Explain

[Anonymous No.12117460]

How to get good? Do you just work through textbooks left to right doing every exercise? Seems like that will take a very long time.

[Anonymous No.12117472]

>>12117460
Ask questions, do problems, work smarter not harder.

[Anonymous No.12117483]

>>12117460
When you read a proof ask yourself:
>what is the key idea? How did they come up with it?
>Is the idea similar to something I've seen before? How else could this idea be applied?
>what goes wrong if we change X?
>can I make a different version work if we change X? Why/why not?

[Anonymous No.12117506]

Does anyone know of any books that go over counterexamples in algebra? I have ones for analysis and topology and I really liked going through them.

[Anonymous No.12117521]

Can someone please explain what the red line means?

[Anonymous No.12117528]

>>12117460
>Do you just work through textbooks left to right doing every exercise
Yes.
>t.scored 800 on the SAT of the math section by doing that.

[Xavi No.12117669]

I think for complex values and what is greater than what it should be like
z<z+1
z<z+i
z+i=z+1
this leads to a line of slope negative one through z that is equal in measure to z
everything above that line is greater than z
and everything below that line is less than z
so examples 1<1+i
2+i > 1+i because of z+1>z
5-5i=0
I maybe need a new notation for the inequality
so what do you guys think?
I don't know shit so go hard on me

[Anonymous No.12117724]

>>12112291
What prerequisites do I need to understand this?

https://arxiv.org/abs/1303.5113

Currently I'm first year masters mechanical engineering. So I know some linear algebra, differential equations, and a lot of fluid physics which is my focus. I know the basics of random processes, but don't know how to begin linking that with PDE. So where do I go from here?

[Anonymous No.12117725]

Is it possible to catch up to math grads with self-study?
I'm not going to make it bros...

[Anonymous No.12117726]

>>12117725
Yes

[Anonymous No.12117758]

>>12117420
>Explain
Take the closed real surfaces under the operation connect sum. The sphere is 0, the torus is 1, and the projective plane is -1.

[Anonymous No.12117840]

Let f(x) = x^4 +1, by the Eisenstein criterion this is irreducible over Q hence F : =Q[X]/<f(x)> is a field and [F:Q] = 4 but isnt F isomorphic to Q(i) which has degree two? Am I missing something?

[Anonymous No.12117853]

>>12117840
ignore this im either stupid or extremely sleep deprived

[Anonymous No.12117861]

>>12117840
Check the roots of x^4+1, you'll find the F is isomorphic to Q(i, sqrt(2)).

[Anonymous No.12117863]

>>12117840
>Am I missing something?
LaTeX.

[Anonymous No.12117879]

>>12117388
left to the reader

[Anonymous No.12117907]

>>12117840
u can factor f as a difference of squares in Q[i], where you can then see it needs the square root of i to split into linear terms.
Perhaps u think its splitting field has degree 4 since f has degree 4. But keep in mind that is not true in general. In general, the degree of a splitting field divides n! where n is the degree of the poly.

[Anonymous No.12117999]

>>12117256
kek

[Anonymous No.12118086]

>>12117669
You can't order the complex numbers in a reasonable way (in the sense that the order is compatible with + and *).
With your order we have that i>0, but then i^3 = -i must also be greater than 0. Thus 0 = -i + i >0

[Anonymous No.12118106]

>>12113186
Not considering the paper in your pic rigorous and correct is colonial bias and white supremacy.

[Anonymous No.12118131]

>>12116703
for me it’s undergrads should kill themselves
>>12117528
go back
>>12117669
>>12117724
>>12117840
>>12117861
>>12117907
>>12118086
Lurk moar or go back, learn to use LaTeX. Stop shitting up the thread you fucking monkeys

[Anonymous No.12118204]

>>12118131
(your prospects as a mathematician)**2 = -1

[Anonymous No.12118252]

>>12118131
>go back
To?

[Anonymous No.12118275]

>midwit gatekeeping over formatting
fucking lel

[Anonymous No.12118277]

>>12118131
>Lurk moar or go back, learn to use LaTeX. Stop shitting up the thread you fucking monkeys
summerfag

[Anonymous No.12118311]

Question:
If you have two different files with colliding hashes and split them into pieces, should those pieces not have different hashes?

I'm trying to figure out what's going on here:

https://shattered.io/

These two PDFs have the same SHA-1 hash. If you make a torrent of them (which splits them into pieces each with its own SHA-1 hash), the torrent you make will check to *either* PDF.
This shouldn't happen, am I right? Somewhere there's got to be a 1 in one that's a zero in the other and the piece with the difference shouldn't check to the other outside of the complete file where the hashes collide, right?

[Anonymous No.12118315]

>>12118311
*if you make a torrent out of one, it checks to the other

[Anonymous No.12118731]

I just realized that sigma algebras are kind of the same idea as topological spaces, only the axioms are changed.
A function [math] f: (X, \tau) \longrightarrow (Y, \pi) [/math] between topological spaces is continuous if [math] f^{-1} : \pi \longrightarrow \tau [/math] is a well-defined function.
A function [math] f: (X, \mathcal{F}) \longrightarrow (Y, \mathcal{G}) [/math] is measurable, if [math] f^{-1}: \mathcal{G} \longrightarrow \mathcal{F} [/math] is a well-defined function.
If we can find so much interesting structure on topological spaces, why can't we do that on measurable spaces too?

[Anonymous No.12118751]

>>12118731
Homeomorphisms are measurable functions with measurable inverse
Paths are continuous functions [math] f: \mathcal{B}([0,1]) \longrightarrow X[/math].
Homotopies are measurable functions [math] f: \mathcal{B}([0,1])^2 \longrightarrow X [/math].
This must be retarded, right?

[Anonymous No.12118876]

>>12117266
What >>12117286 said. I understand the combinatorics but there's no reason to have a unique key pair for each connection. Ie if A wants to communicate with B, A can encrypt with B's public key then sign with A's private key. That way we have both confidentiality and authentication. As long as both parties have a public and a private key you can double-encrypt the message. There's no reason to have a separate key pair for every line of communication that I can spot unless you're prioritizing time in the time vs space tradeoff, in which case you should just use symmetric encryption.

[Anonymous No.12118898]

Ahh yes, math, this is easy. Let me solve it for you.

[Anonymous No.12118905]

>>12118731
don't think of topologies and sigma algebras as being similar. just don't.

[Anonymous No.12118907]

>>12118731
I understood until I saw the arrow

[Anonymous No.12118979]

>>12118905
Well I don't think they are similar, but I am wondering if measurable spaces might be similarily rich in structure.

[Anonymous No.12119029]

>>12118979
What you are observing is the fact that the inverse image function induced by a continuous map is a morphism of lattices (order the topology using inclusion, use union for join and intersection for meet), and that the inverse image function induced by a measurable function is (should my memory not fail me now) a morphism of sigma algebras. This gives a way to give precise meanin for continuity or measurability by defining a function to be of that sort iff the induced function is of the corresponding sort.

[Anonymous No.12119030]

>>12118979
The are actually similar. Sigma algebras are special cases of order structures called "lattices", which can be viewed as topological spaces in several ways.

[Anonymous No.12119038]

>>12119029
>(order the topology using inclusion, use union for join and intersection for meet)

This doesn't sound quite right to me. Are you thinking of a lattice as a generalized space? The result I think you're trying to quote has you actually make the lattice into a space by taking up-directed sets as your open sets.

[Anonymous No.12119044]

>>12112291
Proof that
1+2=3
x+y=z

I saw very indetail explanation of addition 15years ago on the internet but i cannot find it. Can anyone of you point me to right direction?

[Anonymous No.12119052]

>>12119044
Successor function, Peano, Russell, blah, blah, blah

[Anonymous No.12119057]

>>12119038
No I am simply saying that the topology of a space is an ordered set with arbitrary joins, finite meets, and top & bottom elements. This is actually the basis upon which one can build the theory of pointless topology. What I am also saying is that pre-images images of unions/intersections are unions/intersections of the pre-images, as well as the empty set being the pre-image of the empty set etc. My bad, I meant pre-image instead of inverse, I'm a silly ESL poster.

[Anonymous No.12119058]

>>12119044
maybe start with the references here.
https://en.m.wikipedia.org/wiki/Successor_function#:~:text=In%20mathematics%2C%20the%20successor%20function,number%20to%20the%20next%20one.&text=The%20successor%20function%20is%20one,b)%20%3D%201%20%2B%20b.

[Anonymous No.12119098]

>>12119030
>>12119029
Hey cool, thanks that's what I was looking for.

[Anonymous No.12119108]

>>12119057
Oh I see. Well that is one connection, you don't have to go sp crazy to see how lattices are like spaces whose homomorphisms are like continuous maps. You can make the elements of the lattice into the points of a space (as opposed to the open sets, which is what you were talking about) and then put a topology on them coming from the order on the lattice.

[Anonymous No.12119218]

>>12119098
No problem!

>>12119108
Yes. This reminded me of a nice fact. There is actually a correspondence between Alexandrov spaces (arbitrary intersections of open sets are open) and preorders. Given an A-space like that, there is, at least for finite spaces, the so called Kolmogorov quotient that is homotopy equivalent to the original space and its corresponding preorder is actually a partial order. All finite spaces are Alexandrov, as are all spaces with the minimal or maximal topology.

[Anonymous No.12119340]

>>12119218
I assume there's only one person who posts anime girls and describes topology they're vaguely familiar with. Are you working towards a degree or anything or is this just a hobby for you?

[Anonymous No.12119361]

>>12119340
I am trying to get a PhD done but basically I'm just a hobbyist trying to take it as far as possible.

[Anonymous No.12119434]

>>12119361
Why do you say that you're a hobbyist then? Is it just an affectation like the anime girls?

[Anonymous No.12119482]

>>12119434
Because I know my limitations and have no interest in doing any research after this. I don't want to turn this into a job, and thus I find it reasonable to call this a hobby.

[Anonymous No.12119570]

>>12119482
what job are you aiming for?

I'm basically at a crossroads where I have to decide whether to do a Phd or not after my MSc.
I also don't want to do research. I don't like the academic life.

I'd rather work in industry building cool shit. Thing is, if I want to go that way I should probably ditch math immediately and start learning useful shit like C++ and ML, but I like doing math and there is so much more math to learn.

[Anonymous No.12119575]

>>12119482
That's pretty depressing to hear as someone who is interested in doing research? If you think these results you quote are so cool why not go on to academia? Are you independently wealthy and just going to retire after grad school?

[Anonymous No.12119579]

>>12119575
Oops too many question marks lol.

[Anonymous No.12119598]

Question: Is Algebra an autistic field of math?

What would you say is the most autistic subfield, or the one that attracts the highest number of spergs?

[Anonymous No.12119608]

>>12119598
No, number theory, set theory, category theory, all more spergy.

[Anonymous No.12119638]

>>12119598
It's homotopy theory and algebraic geometry, hands down. Category theory in other applications doesn't get hit so hard, but every trendy dilettante needs their sheaves of infinity categories.

[Anonymous No.12119660]

>>12119570
>what job are you aiming for?
I have no plans, to be honest.
>I'm basically at a crossroads where I have to decide whether to do a Phd or not after my MSc.
Would you have to pay for your studies? If so, then PhD is not necessarily the best idea. However, if you can get funding for it, then I suggest you give it a try. You can always drop out, you know.
>I'd rather work in industry building cool shit. Thing is, if I want to go that way I should probably ditch math immediately and start learning useful shit like C++ and ML, but I like doing math and there is so much more math to learn.
Can't say anything about that, sorry.

>>12119575
>If you think these results you quote are so cool why not go on to academia?
Because I feel like I have lost my mathematical edge and I know I would not be able to play the necessary social game.
>Are you independently wealthy and just going to retire after grad school?
I have savings from my jobs and a minimal life style, so I will survive a year or two without any governmental subsidies. I would still rather find some nice job than rot at home.

>>12119598
Algebra has its autists but also its chads. See some of the representation boys and you will drool.

[Anonymous No.12119661]

Help with my homework please
How do I go from H_n <= ln(n)+1

to abs(H_n) <= abs(ln(n))

where H_n is the harmonic sequence?

[Anonymous No.12119672]

>>12119638
here's (you) for mentioning infinity categories, I fucking hate that shit and everyone seems to be obsessed with it

[Xavi No.12119709]

>>12118086
no that says that zero is greater than -i
because of -i+i=0
then -i <0

[Anonymous No.12119711]

>>12119672
As far as I can tell it looks like a shitshow. I'm actually in topology and put in a little time to study this stuff, which isn't directly related to my thesis work. It's ultra trendy and people sell it as profound as set thoery or category theory, but outside the motivating applications I don't really see it going anywhere.

[Anonymous No.12119717]

>>12119570
I'm scared of where you are atm.
I just finished my Bachelors and I'll do a Masters. But I don't know what I'll do after that. I am getting good grades and I didn't fail any exams, but I don't feel like one of the people who are REALLY good. And working on cool stuff in industry sounds good. At least I have experience with coding.

[Anonymous No.12119733]

>>12119709
the rules for ordering compatible with operations are

x > y implies x+z > y+z
x > 0 and y > 0 implies x*y > 0

it's not possible to order complex numbers such that these two properties hold. not possible doesn't mean "nobody has come up with the solution yet", it literally means it's impossible.

[Anonymous No.12119745]

Why is this the case
Sin from
- pi/4 to pi/4, dx=pi/2 dy=1.4
But from pi/4 to 3pi/4, dx pi/2 dy=0.6
Same distance, more than 2x the displacement

[Anonymous No.12119747]

>>12119717
You don't need super smarts to make it in academia.
From my vantage point I can say, the smartest people don't always remain in academia.
If you have good grades (say top 5% to top 10%) and some social skills, that's all that's required.

[Xavi No.12119748]

>>12119733
uh
that doesn't make sense
yeah 1*i > 0
but the properties of i being the square root of -1 makes it
so that i^3 which is i* i^2 which makes it i* -1
-1<0
so that makes -i less than zero

[Anonymous No.12119759]

>>12119748
Take an intro to proof class or something. You lack the mathematical maturity to understand the person you're responding to so you're just gonna go in circles with this.

[Xavi No.12119786]

>>12119759
so the only thing is that
the second rule doesn't imply x*y>0

[Anonymous No.12119829]

>>12118731
>If we can find so much interesting structure on topological spaces, why can't we do that on measurable spaces too?
I don't quite thank that this statement as posed is correct,
but I'll say that defining a topology is easy (you just need pullbacks and pretty much every framework let's you build arbitrary unions anyway), while the definition of a measure space involves something like fucking R.

[Xavi No.12119834]

>>12119759
>>12119733
that applies only for the real numbers
the complex numbers are different
because of i being the square root of -1
but i>0 and 2i>0 but 2i*i= 2*-1 so it's less than 0

[Anonymous No.12119890]

>>12119834
>that applies only for the real numbers
No, it doesn't. If you don't know what a relation or a ring is then please consider dropping this until you can understand what you're trying to do.

[Xavi No.12119936]

>>12119890
really?
tell me what I'm trying to do
if it's so dangerous
instead of being so vague

[Anonymous No.12119963]

>>12119834
>i>0 and 2i>0
Nope

[Anonymous No.12119974]

Just bombed my linear algebra test.
How do you find the vector equation of a line given the equation of a plane and a point?
With the line being normal to the plane of course.

[Anonymous No.12119978]

>>12119936
It's not dangerous, it's just that you yourself lack the basic mathematical tools to even formulate a specific question. I'm not going to teach you all the math you need to know in order to ask something interesting in a thread on 4chan.

[Xavi No.12120006]

>>12119963
why?

>>12119978
I believe it
How much math is that?

[Anonymous No.12120059]

>>12116703
It's a bit mainstream but Zorn's lemma is the one I've used the most. But Urysohn's lemma is quite beautiful.

[Anonymous No.12120075]

>>12113305
pi + e = 3 + 3 = 6

[Anonymous No.12120107]

>>12119709
>>12119748
>>12119786
>>12119936
>>12120006
>You will immediately cease and not continue to access the site if you are under the age of 18.

[Anonymous No.12120108]

>>12116703
Either Ito's lemma or Schur's lemma

[Xavi No.12120121]

>>12120107
i'm 23
thanks

[Anonymous No.12120165]

[Anonymous No.12120175]

>>12119044
It is defined 3 = succ(2), 2 = succ(1) and + is the operation defined recursively as such:
[math] n+0=n,\quad n+\operatorname{succ}(m) = \operatorname{succ}(n+m) [/math]
Thefore,
1 + 2 = 1 + succ(1) = succ(1+1) = succ(1+succ(0)) = succ(succ(1+0)) = succ(succ(1)) = succ(2) = 3.

>>12119748
Definition of ordered ring: if x, y > 0 then x + y > 0 and xy > 0.

If you define i > 0, then i*i = -1 > 0, then -1*i = -i > 0, then i + (-i) = 0 > 0. Contradiction.

[Xavi No.12120204]

>>12120175
i is the only number where it's greater than zero but can be multiplied to be less than zero
there is a false contradiction

[Anonymous No.12120207]

>>12120059

[Anonymous No.12120219]

>>12120121
Are you, though?

[Xavi No.12120230]

>>12120219
I know it's hard for you to accept
but yes

[Anonymous No.12120238]

>>12120204
But it doesn't respect laws of multiplication because on one hand you have that 1*1 > 0
where 1 > 0. But i*i = -1 < 0 where i > 0. Therefore it doesn't make sense as an algebraic structure. Also what about things like 1 + i, -1 + i, -1 - i or 1 - i; what of those are positive, negative and why?

>>12120207
Please send more wallpapers with theorems, anon

[Anonymous No.12120271]

>>12120238
That's the only one I have. I got it from here.

[Xavi No.12120297]

>>12120238
1+i>0 because 1>0 and i>0 so 1+i>0
-1-i >0 because -1<0 and -i<0 so -1--i<0

-1+i and 1-i are equal to 0 but not actually 0
I can't explain any better then that yet

[Anonymous No.12120304]

[math]\mathbb{C}[/math] is not an ordered field.
Simple as.

[Anonymous No.12120306]

>>12120297
>-1+i and 1-i are equal to 0 but not actually 0
You just lost completely your argument there. One of the key points of "<" is that it is a linear ordering, and one requirement is that if a <= b, and b <=a then b = a.

>>12120271
Damn:(

[Xavi No.12120329]

>>12120306
then they are equal to zero but don't have properties of zero

[Anonymous No.12120339]

>>12120329
If you cannot even define the binary relation you're trying to talk about then please just go.

[Anonymous No.12120349]

>>12120329
By definition two equal things have the same properties. Just admit your ordering doesn't work nor it makes sense.

[Xavi No.12120365]

>>12120349
that's exactly what they said about not being able to have a square equal a negative
but okay i admit I have nothing that can convince people

[Anonymous No.12120386]

>>12120365
Nope, because what they said is that there is no real number that satisfies that, because the reals are an ordered field and effectively you just showed to yourself that "i" fucks up the order in C.

Also i makes sense because C is the minimum extension of R such that it is algebraically closed. In the same way we extend Z to have a nice algebraic property (existence of product inverses). The problem is that you lack the basic formal notions to see why C can't be ordered. And that's the reason we use R instead to compare inequalities!

[Anonymous No.12120416]

>>12120365
>I have nothing
This is all you needed to say. If you actually had something then you would be able to explain when z<w for all possible complex numbers, even if your idea doesn't obey the usual rules for orders.

[Xavi No.12120430]

>>12120416
z<w if w is above a line that goes through z with a slope -1

[Anonymous No.12120463]

>>12120238

[Xavi No.12120506]

>>12120430
-1+i is equal to 0 ordinal
that's what i should've said

[Anonymous No.12120529]

>>12120430
So that's a partial order on C.

[Anonymous No.12120557]

>>12120506
>that's what i should've said
No it's not, but w/e.

>>12120430
So you like the binary relation, call it T, so that zTw when z=a+bi, w=c+di, and d+c-a+c≥0. Do you want a cookie or something? Why should we care about this relation?

[Xavi No.12120605]

>>12120557
What are you trying to say that this could've been a thesis?
I was just sharing my thoughts

[Anonymous No.12120627]

quick question pls no bully
there is no way to multiply sint and sin(3t) together right? the closer we can get is sint(sin(3t)) ?

[Anonymous No.12120630]

>>12112530
lol I don't understand a word he's saying

[Anonymous No.12120674]

>>12120627
Use the triple angle formula to expand sin(3t).

>>12120605
There isn't much thought here. You pointed out a thing which obviously exists and said nothing of value about it, although you insinuated that it was a deep observation akin to defining the imaginary number. You also failed to understand what an ordered field is, which really showed us, I guess.

[Xavi No.12120685]

>>12120674
if it wasn't that important you could've just said that in the first place
not make it out to be like it was such a big deal

[Anonymous No.12120696]

>>12120627
You can express as a linear combination of cos(2t) and cos(4t), if you like.

[Anonymous No.12120698]

>>12120685
I never did any such thing, but thanks for playing.

[Anonymous No.12120700]

>>12120238
>inb4 muh low resolution
That's my screen size. Pass it through waifu2x or something.

[Xavi No.12120713]

>>12120698
If you read the first thing i posted you would've seen that point made first
>>12117669
>is leads to a line of slope negative one through z that is equal in measure to z
> everything above that line is greater than z

[Anonymous No.12120739]

>>12120713
Yeah I didn't read that. Good effort with this. Maybe try to look up some of the new words we used and I encourage you to return when you're not obviously underage.

[Xavi No.12120750]

>>12120739
Why do you have to be so condescending?
is there a point to be made by doing that?

[Anonymous No.12120758]

>>12120750
You're reading condescension into my writing because you're insecure. You also started arguing when people told you why your relation didn't make C into an ordered field.

[Xavi No.12120763]

>>12120758
oh don't pull that bullshit at me
>obviously underage
>maybe try to look up some of the new words
this is obvious condescension

[Anonymous No.12120869]

>>12120304
W R O N G.
Every set can be easily well-ordered. Easily.

[Anonymous No.12120880]

>>12120869
Sure, as long as a pea can be chopped up and reassembled into the Sun.

[Anonymous No.12120932]

>>12120880
yes
t. arski and banach

[Anonymous No.12121002]

>>12120869
An ordered field is not just an ordered set.

[Anonymous No.12121007]

Can someone please explain how d/dt(dz/dx) and d/dt(dz/dy) became the highlighted expression

[Anonymous No.12121011]

How much is useful commutative algebra in algebraic geometry?

[Anonymous No.12121044]

>>12121007
Literally just slot (partial z/partial x) into the chain rule, that is written on the second line.

[Anonymous No.12121047]

help
thinking 5 as it has more restrictions but no clue

[Anonymous No.12121070]

>>12114758
I prefer writing the equation, demonstrating its application, and leaving the proof for the appendix

[Anonymous No.12121096]

>>12121070
hello mr stewart

[Anonymous No.12121105]

>>12121096
Engineers don't need to know proofs.

[Anonymous No.12121109]

>>12121105
they taught them to us anyways

[Anonymous No.12121111]

>>12112291
>Tensor space
>Manifold
>Spectral
>Axiom
>p-adic

You like that baby? I can go all day...

[Anonymous No.12121138]

>>12121111
Wasted quads.

[Anonymous No.12121156]

>>12121011
as useful as feet are for baseketball

>>12121002
Yeah I wasn't seriuz

[Anonymous No.12121173]

>>12121111
Blessed quads.

[Anonymous No.12121199]

>>12121156
I don't care for your jocular ways anon. 4chan is a somber place.

[Anonymous No.12121237]

>>12121047
2,3,4 are all true, F is the cougar wife
Z sounds based

[Anonymous No.12121268]

>>12121044
how?

[Anonymous No.12121271]

>>12119598
Number theory.
>>12119672
Kill yourself

[Anonymous No.12121275]

>>12121138
>>12121173
The duality of man

[Anonymous No.12121320]

>>12113427

Do the p-adic number systems extend to numbers other than integers? I was thinking about a pi-adic metric

[Anonymous No.12121354]

>>12121199
but is it a sober space?

[Anonymous No.12121367]

>>12121320
the definition of p-adic norm on Q uses the fact that Q is the quotient field of a unique factorization domain. I suspect that's the scope of obvious generalizations of 'p-adic' metrics

[Anonymous No.12121372]

>>12113564
Think in terms of transformation. Ie translation, rotation, stretch, sheer, etc.

[Anonymous No.12121378]

can someone give me an example of a groupoid which is not isomorphic to a disjoint union of groupoids?

[Anonymous No.12121381]

>>12121354
Indeed, go to 420chan for the more general case.

[Anonymous No.12121384]

>>12121378
Can you provide more context? If you're talking about categories, just take any group as an example.

[Anonymous No.12121387]

>>12121384
And I assume you mean as a disjoint union of two or more, since otherwise the question is trivial.

[Anonymous No.12121392]

>>12121384
Yeah the groupoids are the categories which have a group as an object and the morphisms as the group elements.

[Anonymous No.12121393]

>>12121268
What do you mean how? Literally replace (partial z/partial x) in for z in the chain rule.

[Anonymous No.12121397]

>>12121387
how so? if you'd got only a single groupoid, then it be iso to itself

[Anonymous No.12121402]

>>12121384
The anon is asking for a category that consists of many objects and only isomorphisms but cannot be described as a union of disjoint components. I am not sure if that could even exist.

[Anonymous No.12121408]

>>12121397
Every groupoid can be written as a disjoint union of groupoids since every groupoid is the disjoint union of that groupoid alone. It's like asking whether every set can be written as a disjoint union of sets. It's trivial unless you constrain the index set.

[Anonymous No.12121414]

>>12121402
Obviously every groupoid is the disjoint union of however many "connected components" it has.

[Anonymous No.12121449]

>>12121408
How so?
If you got a cateogry with only one object G, you say that the coproduct of G with itself is necessarily isomorphic to G?

[Anonymous No.12121460]

>>12121449
I'm pretty convinced at this point that you have no idea what a groupoid is.

[Anonymous No.12121467]

>>12121393
But how can we replace z with dz/dx
That equation is true for z because it is a function of x and y but we don't know whether dz/dx will also be a function of x and y
Do you get what I'm saying?

[Anonymous No.12121481]

>>12121414
Yes, thanks for confirming it.

[Anonymous No.12121522]

There is nothing wrong with commutative algebra.

[Anonymous No.12121537]

>>12121449
great objection bre

[Anonymous No.12121545]

>>12121460
Your negative outlook on life will not lead to happiness

[Anonymous No.12121553]

>>12121545
It's not negative to tell someone when they're lost. I'm trying to help.

[Anonymous No.12121635]

bros... how do i foster mathematical maturity...

[Anonymous No.12121649]

All this bickering and I personally just looked up a groupoid. So it seems to me that a connected groupoid is basically just a fixed group ( the morphisms between objects are groups isomorphisms between the objects endomorphism groups). What is the use of this notion?

[Anonymous No.12121758]

>>12121649
>basically just a fixed group
This is incorrect. Do you understand what a category is?

[Anonymous No.12121834]

>>12121649
ok so yeah, any f: X -> Y induces an isomorphism between Hom(X, X) and Hom(Y,Y) by conjugation. This is a functor between C and Aut(C). I don;t see how the partial group operation comes into play
>>12121758

[Anonymous No.12121876]

Where my analysis niggas at?

[Anonymous No.12121915]

What formula can I use to accomplish this? Ideally this would be sinusoidal with some kind of crazy form of amplitude modulation that I can't quite wrap my head around.

Is this even possible with just math?

[Anonymous No.12121965]

>>12121876
Algebra > analysis

[Anonymous No.12122020]

>>12121758
Nvm I'm being dumb and pissy. I see what you mean now, sorry.

[Anonymous No.12122025]

>>12121915
piecewise function?
ex. 5sint for 0<t<π
|sint| for π < t < 2π

[Anonymous No.12122028]

>>12121965
The disciplines are asymptotically the same, up to a constant.

[Anonymous No.12122080]

>>12121915
I got something close just by fucking around, though it's maybe not exactly what you were looking for.

[Anonymous No.12122090]

>>12121915
>repeats forever
Just use fourier series, if you wanna be fancy use bump functions.

[Anonymous No.12122199]

>>12115883
>>12116329
Here's a fun example which isn't just fucking with a measure zero set, but is legitimately unbounded on every open interval no matter how you change behavior on a measure zero set:
Consider a function f(x) = 1/sqrt(x) for x in (0, 1), and 0 otherwise. Then f is integrable and unbounded at 0.
Now let q_n be an enumeration of the rationals, and consider the functions f(x - q_n). These are functions which are unbounded at each rational, and each function is integrable.
Then let [math] g(x) = \sum_{n = 1}^\infty \frac{1}{2^n} f(x - q_n) [/math]. The sum converges in L^1 norm so g is integrable, and has integral 2 times that of f. Moreover, g blows up near every rational (to the right). It is easy to check not only that g is unbounded on every open set, but also that any function which is equal to g almost everywhere is unbounded on every open set.
g is very spiky.

[Anonymous No.12122216]

>>12118731
countability assumptions = terrible, terrible behavior.
topological spaces are good because operations are either finite or arbitrary. the instant you start introducing notions of countability in topology things get disgusting too.

[Anonymous No.12122221]

>>12121635
Exercises and rebuilding proofs of theorems independently of the author’s proofs. Also, be a white or jewish man w/>3 SD iq

[Anonymous No.12122262]

>>12114889
Redpill me on this

[Anonymous No.12122287]

>>12122262

[Anonymous No.12122290]

>>12121915
If you don't need the peaks to be symmetric like a sin function then you can multiply by some factor that vanishes at infinity.

An interesting question arises though. It's intuitively obvious that if we want the peaks to be symmetric, that this function cannot arise from elementary functions which are defined everywhere. How would we show this?

[Anonymous No.12122300]

>>12122020
No but you were right. the algebraic notion of a groupoid is a partial binary operation. Yes in the case of a cat. with only isomorphisms we have a functor Endo(C) that takes each object to its automorphism group and each f: X \tp Y induces an isomorphism of Aut(X) to Aut(Y). So somehow this functor/Aut group of any object does not capture the full notion of a groupoid.

[Anonymous No.12123174]

>>12122287
I understand graph theory, I just am curious about whst you specifically mentioned

[Anonymous No.12124153]

>>12122290
Suppose we have a function f which is period with period 1 in the sense that for [math] x \in [0,1] , n \in \mathbb{Z} [/math] we have [math] f(x + n) = c_n f(x) [/math] with factors [math] |c_n| \neq 1 [/math]. Furthermore, [math] f(0.5-x) = f(0.5+x)[/math], i.e. the bump is symmetric.
We can show that such a function can't be analytic at [math] n \in \mathbb{R} [/math] if [math] |c_n| \neq 1 [/math].
Let us assume that it was in fact the case that a function f is analytic at 0 and [math] f(-x) = c_n f(x) [/math] in a small neighbourhood around 0. Then
[math] \sum_{n=0}^\infty (-1)^n a_n x^n = \sum_{n=0}^\infty c_n a_n x^n. [/math].
Therefore it must be the case that
[math] c_n = (-1)^n [/math] whenever [math] a_n \neq 0 [/math]. Hence all even or odd terms must fanish and f is either symmetric or antisymmetric. In either case we would see that [math] c_n = \pm 1 [/math].