š§µ /mg/ maths general
Anonymous at Fri, 24 Jan 2025 16:51:03 UTC No. 16562509
[math]/\mathfrak{mg}/[/math]
Deutsche Mathematik edition
Talk maths, formerly >>16534183
Anonymous at Fri, 24 Jan 2025 17:43:52 UTC No. 16562601
the most beautiful math ever produced is by weierstrass and his analysis school and 30 students who revolutionized all mathematics
Anonymous at Sat, 25 Jan 2025 01:48:44 UTC No. 16563103
anyone here do math in cursive? I find it very difficult to take nice looking notes because different lecturers force you to take different notes and often they're very non linear and you often have to go back and cross something out because they made a mistake or you did
i can't really figure out how to do proper spacing either, especially when there's multiple exponents and stacked fractions
i dont know why but it really bothers me, i dont have ocd but it make me anxious somehow
i've been using graph ruled a4 notebooks and gel pens but they're pretty bad for the angle i'm using and i really dont enjoy pencils so i'll probably switch to a fountain pen
i'm curious if there's anyone who has a legit though out formatting style with proper spacing etc.
Anonymous at Sat, 25 Jan 2025 01:59:36 UTC No. 16563111
>>16563103
I do have ocd, I just write whatever without much thought and then turn it into LaTeX, improving your tools won't change anything, you actually have to put in some work studying penmanship for things to look decent (at least 6-7 months), but is it really worth it?
Anonymous at Sat, 25 Jan 2025 02:03:31 UTC No. 16563116
>>16563111
i think you write posts without much thought either but since you appear to have not read my post i was strictly referring to formatting
>improving your tools won't change anything, you actually have to put in some work studying penmanship for things to look decent (at least 6-7 months)
that's just your dogshit opinion you can be entitled to as a flaming retard with OCD rotten brain but it's wholly unrelated to my post
Anonymous at Sat, 25 Jan 2025 02:08:48 UTC No. 16563123
>>16563116
Yeah you're right I don't really put much thought when answering to retards in this shithole and also yeah you nailed it I barely read your post. I thought you were whinning about not being able to format your notes properly, my opinion is that that is a waste of time, but whatever you do you.
Anonymous at Sat, 25 Jan 2025 03:33:33 UTC No. 16563178
How do I stop being bad at math? I can barely understand high school level math and everything beyond that is incomprehensible to me.
Anonymous at Sat, 25 Jan 2025 03:53:55 UTC No. 16563195
>>16563178
what about it do you not understand?
Anonymous at Sat, 25 Jan 2025 04:00:23 UTC No. 16563201
>>16563103
i dont write in cursive i write in retarded half-cursive because i never learned real cursive
i like using graph ruled notebooks and my fountain pen, occasionally i fuck up and smear the fountain pen but the tactile feedback from writing with a decent one is unparalleled, good experience. gel pens are decent
my formatting changes based on whether or not im loaded on stimulants. on stimulants i get the ability to write even better organized and spaced notes on completely unlined paper. i usually measure spacing in terms of writing eq'ns with the line of the graph going through them and then writing exponents or subscript shite on the lines above and below where i wrote the equation, if that makes sense. pretty hard to commit to specific spacing for notes tho
the formatting pisses me off regularly but ultimately it doesnt matter that much; i seldom even go back and look at my own notes, i just take the notes to keep myself busy while listening to a lecture
Anonymous at Sat, 25 Jan 2025 05:13:04 UTC No. 16563236
>>16563195
I can grasp the basics of those concepts but I struggle with developing a deeper understanding of them. I alao have a hard time utilizing them when I try to solve math problems
Anonymous at Sat, 25 Jan 2025 06:20:43 UTC No. 16563262
How do I fraud my way through a math PhD
I don't know how I got here or why the hell I'm here, but somehow I am a fully funded math PhD student.
I don't even know what a Lagrange multiplier is I had to ask deepseek
Anonymous at Sat, 25 Jan 2025 07:10:16 UTC No. 16563290
>>16563236
BHAHAHAHHAA
Tell me more
Anonymous at Sat, 25 Jan 2025 15:08:36 UTC No. 16563538
>>16563262
What does your position require you to research?
Anonymous at Sat, 25 Jan 2025 15:10:28 UTC No. 16563543
>>16563236
What concepts?
Anonymous at Sun, 26 Jan 2025 00:27:11 UTC No. 16564330
After studying Analysis and Topology, how much Category Theory do I need to learn to study the dual notions of Oralysis and Bottomology?
Anonymous at Sun, 26 Jan 2025 15:14:07 UTC No. 16564924
>>16562601
>beautiful
>math
when will this meme end?
Anonymous at Sun, 26 Jan 2025 15:17:06 UTC No. 16564932
>>16564924
what are you doing here, midwit?
Anonymous at Sun, 26 Jan 2025 15:18:45 UTC No. 16564935
>>16564932
>midwit
midwits latch onto the idea that math is "beautiful" because that's the same word every other normie uses. then you look at what they think is "beautiful" and find out their taste is garbage.
Anonymous at Sun, 26 Jan 2025 15:23:24 UTC No. 16564942
>>16564924
>>16564935
holy shit you are ELITE and may I say BASED
Anonymous at Sun, 26 Jan 2025 15:27:47 UTC No. 16564950
>>16564935
sublime
you happy?
Anonymous at Sun, 26 Jan 2025 15:32:27 UTC No. 16564956
>>16564950
even worse.
how about "interesting." that's all it is. no more circlejerking over math
Anonymous at Sun, 26 Jan 2025 15:36:27 UTC No. 16564962
>>16564956
There are many interesting things out there that carry no aesthetic value. Female hyenas having massive clitori is interesting, but itās neither beautiful nor sublime.
Anonymous at Sun, 26 Jan 2025 15:41:10 UTC No. 16564969
>>16564935
Personally I find all math that I don't understand but want to understand beautiful. Then when I properly understand it it's no longer beautiful and I move on to the next thing.
Anonymous at Sun, 26 Jan 2025 16:05:47 UTC No. 16565001
Why do computer scientists prefer to talk about directed sets rather than chains when it comes to partial orders/preorders?
I'm not well-versed in CS but from the little bit I've seen in introductory texts to program semantics it would seem that most of the given examples for DCPOs would also apply to chain complete orders and I feel like the theory of those is way easier? Maybe that's naive of me but for example I still don't fully comprehend the fixed point theorem for DCPOs and would have no idea how to come up with that on my own while the one for chain complete orders is quite intuitive and straightforward.
Anonymous at Sun, 26 Jan 2025 16:11:08 UTC No. 16565009
>>16563543
simple stuff like calculus
Anonymous at Sun, 26 Jan 2025 16:12:07 UTC No. 16565013
>>16565009
Name a specific concept you lack the intuition/deeper understanding of.
Anonymous at Sun, 26 Jan 2025 16:22:23 UTC No. 16565029
>>16565013
Differential
Anonymous at Sun, 26 Jan 2025 16:40:28 UTC No. 16565068
>>16565029
Pretty sure that's not a concept in calculus. Don't you mean the derivative?
Anonymous at Sun, 26 Jan 2025 16:44:18 UTC No. 16565076
With roughly one month and a half at my disposal to study differential geometry and Riemannian geometry, what are some good textbooks I could use if I want to study everything in this time frame?
Also, should I skip topological manifolds?
Anonymous at Sun, 26 Jan 2025 17:44:10 UTC No. 16565167
>>16565001
Whatās a directed set? If thatās the same as a word (a function from a subset of the naturals to some set), then itās simply because thatās how computer memory is set up.
Anonymous at Sun, 26 Jan 2025 18:04:45 UTC No. 16565189
>>16565076
Perhaps Differential Geometry and Its Applications by John Oprea?
>>16565167
NTA, but I understand that directed set means the set with preorder (relation that's both reflexive and transitive) where any two elements have an upper bound: https://en.wikipedia.org/wiki/Direc
(It's generalization of a sequence.)
Anonymous at Sun, 26 Jan 2025 18:17:55 UTC No. 16565209
>>16565189
>(It's generalization of a sequence.)
A net (a function from any directed set) is a generalization of a sequence (a function from the naturals directed by the usual order).
Anonymous at Sun, 26 Jan 2025 21:49:30 UTC No. 16565437
Greetings mathematicians!
Can anyone give to me a homeomorphism between [math](-\infty, a][/math] and [math](a, b][/math]
and a homeomorphism between [math](a, b][/math] and [math][a, b)[/math]?
Anonymous at Sun, 26 Jan 2025 21:58:21 UTC No. 16565454
>>16565437
Dear Anonymous
I'm sorry, but your questions (especially the second) are far too trivial. This thread is not about solving your homework. (Have you ever heard about exponential function and 1-x function)?
Anonymous at Sun, 26 Jan 2025 22:44:29 UTC No. 16565552
>>16565189
Ah, so directed sets are natural generalizations of chains from tosets to posets. My guess is that CS guys would need this for various data structures like trees?
Anonymous at Mon, 27 Jan 2025 01:56:02 UTC No. 16565745
The FUCK do I do with a BS in math. I only did this shit cause it was easy.
Now my lame ass parents are taking about how I need a job so I can get a wife.
Women don't even like asian guys, I just want to play video games.
Anonymous at Mon, 27 Jan 2025 01:59:07 UTC No. 16565747
image:
17-gon
source(image):
https://www.wolframalpha.com/input?
quote:
"If you had to choose a few words or symbols to encapsulate your legacy, what would you pick? Johann Carl Friedrich Gauss (1777ā1855) left behind a trophy case stocked with mathematical achievements to choose from, but above all, he wanted a "regular heptadecagon" etched on his headstone. The highly symmetrical 17-sided shape starred in a proof that Gauss considered one of his greatest contributions to math."
source(quote):
https://www.scientificamerican.com/
Anonymous at Mon, 27 Jan 2025 03:15:16 UTC No. 16565902
>>16565437
First, create the homeomorphism between (a,b] and [a,b). That should be easy enough: create a continuous bijection within [a, b] such that f(a) = b and f(b) = a.
Then, ask yourself which property of this set could be exploited to create a homeomorphism to an infinite interval. Tip: use the distance between x and something, and put it in the denominator.
Anonymous at Mon, 27 Jan 2025 12:10:36 UTC No. 16566257
>>16565745
based
Anonymous at Mon, 27 Jan 2025 19:17:06 UTC No. 16566546
I'm struggling to understand why unsoundness doesn't imply inconsistency.
For some statement P which is independent of PA but proven by True Arithmetic, isn't the proof of P contained in the standard natural numbers which appear in all models of arithmetic, including non-standard ones?
Anonymous at Mon, 27 Jan 2025 20:51:47 UTC No. 16566646
>>16565747
image:
2.125-gon
or 17/8-gon
(unfortunately, the tips are missing)
source(image):
https://www.wolframalpha.com/input?
Anonymous at Mon, 27 Jan 2025 21:08:25 UTC No. 16566665
image:
22/7-gon
(unfortunately, the tips are missing)
source(image):
https://www.wolframalpha.com/input?
note:
22/7 ~ pi
Anonymous at Mon, 27 Jan 2025 21:37:18 UTC No. 16566731
>>16566714
image:
pentagram
or 5/2-gon
source(image):
https://www.wolframalpha.com/input?
Anonymous at Tue, 28 Jan 2025 08:22:24 UTC No. 16567239
What is stratification
Anonymous at Tue, 28 Jan 2025 13:53:56 UTC No. 16567429
>>16566714
Does D*v Z*kh**m still work at The Pentagon?
Anonymous at Tue, 28 Jan 2025 14:28:53 UTC No. 16567447
>>16565068
>the derivative
is a ratio
of two differentials
Anonymous at Tue, 28 Jan 2025 14:35:38 UTC No. 16567454
>>16566731
a pentagram?
why that reminds me of
Satan's chosen jeople
Anonymous at Tue, 28 Jan 2025 15:27:04 UTC No. 16567491
omg
a 6/5-gon
or 1.2-gon
https://www.wolframalpha.com/input?
note the central hexagram
Anonymous at Tue, 28 Jan 2025 18:29:31 UTC No. 16567696
>>16566665
image:
1/7 of 22/7-gon
source(image):
https://www.wolframalpha.com/input?
Anonymous at Tue, 28 Jan 2025 18:45:35 UTC No. 16567712
>>16566646
image:
1/8 of 17/8-gon
source(image):
https://www.wolframalpha.com/input?
Anonymous at Tue, 28 Jan 2025 21:00:00 UTC No. 16567904
>>16567696
image:
3 revolutions of a pi-gon
source(image):
https://www.wolframalpha.com/input?
Anonymous at Tue, 28 Jan 2025 21:33:53 UTC No. 16567964
image:
19/7-gon
(unfortunately, the tips are missing)
source(image):
https://www.wolframalpha.com/input?
note:
19/7 ~ e
Anonymous at Wed, 29 Jan 2025 00:07:20 UTC No. 16568140
Guys, PLEASE help me.
I have the opportunity to enter grad school. I was a good student overall, but only God knows how I got my C in Abstract Algebra.
I'm undecided between going for Riemannian Geometry and Model Theory. What should I do? Which of these is least likely to require me to understand fields and Galois theory?
Anonymous at Wed, 29 Jan 2025 00:18:54 UTC No. 16568152
>>16568140
Learn fields and galois theory.
Anonymous at Wed, 29 Jan 2025 12:01:54 UTC No. 16568626
What are some statements independent of ZFC that are resolved by positive set theory (in particular, [math]\mathsf{GPK}^+_\infty[/math])
Anonymous at Wed, 29 Jan 2025 13:43:46 UTC No. 16568690
i'm moving out
of TFBT = this f*ck*ng bait thread
Anonymous at Wed, 29 Jan 2025 14:02:13 UTC No. 16568704
Does anyone have the trivium by Verbitsky?
I can't seem to find the combined pdf of the questions.
Anonymous at Wed, 29 Jan 2025 16:12:36 UTC No. 16568817
>>16568152
Yeah, I'll obviously have to learn it at some point if not just so I can feel less like a fraud. But I'm fucked if it shows up right as I start my program.
Anonymous at Wed, 29 Jan 2025 16:14:33 UTC No. 16568818
>>16568140
NOT Model Theory.
Anonymous at Wed, 29 Jan 2025 17:34:19 UTC No. 16568886
>>16568140
>Riemannian geometry
modern Riemannian geometry requires good knowledge of Lie theory and understanding of category theory, so good luck with that if you struggled with finite groups.
Anonymous at Wed, 29 Jan 2025 19:19:34 UTC No. 16568999
>>16568886
I didn't struggle with groups. My problem was when halfway through the year the professor had to go home due to health issues and we got a new one to teach us the fields, extensions and Galois theory part.
Anonymous at Wed, 29 Jan 2025 21:13:16 UTC No. 16569273
What are some good, non-introductory, logic textbooks that, preferably, also talk about Gƶdel incompleteness theorems?
Anonymous at Wed, 29 Jan 2025 22:04:13 UTC No. 16569408
>>16568999
If you do go to grad school, youāll realize that if you want to actually learn any subject, you do it yourself. Professors are all lazy cunts anyways, even if there are some needles in the shitstack.
Anonymous at Wed, 29 Jan 2025 22:07:10 UTC No. 16569416
>>16569408
Big true. The only grad level math courses I've taken are electives in analysis and measure (information theorist EE here).
Even then, there was an absolute fuckton of self-teaching required to understand what was going on in the course (and that's just a reality of the material, my professor was pretty great). The biggest thing courses do is provide you a clear structure for your self-teaching.
Anonymous at Wed, 29 Jan 2025 23:57:04 UTC No. 16569576
Redpill me on obscure categories and their relation with other categories. I'll start. The category of pointed sets consists of sets with a single distinguished element and a morphism between such sets, ie a function that satisfies f(e_1) = e_2, where e_1 and e_2 are the respective distinguished elements.
This category lets one talk of kernels and related objects such as exact sequences in an algebra-free context. Examples of non-algebraic structures where this might show up are manifolds on which a frame has been fixed, so that the diffeomorphisms are restricted to those that are "inertial with respect to the frame".
Anonymous at Thu, 30 Jan 2025 00:11:46 UTC No. 16569590
I might add that such āpointed manifoldsā correspond to the choice of the Levi-Civita connection, which makes the torsion vanish.
Anonymous at Thu, 30 Jan 2025 02:35:42 UTC No. 16569760
whats the type of math that will allow me to make cool stuff with computer graphics?
t. retard
Anonymous at Thu, 30 Jan 2025 04:12:59 UTC No. 16569867
>>16568704
>>16562509
Anyone has the PDF?
Anonymous at Thu, 30 Jan 2025 05:09:10 UTC No. 16569925
>>16569760
Linear algebra
Anonymous at Thu, 30 Jan 2025 13:25:12 UTC No. 16570178
>>16569273
Ebbinghaus-Flum-Thomas
>>16562704
More like: Abstract erosion from a load crossing over your moms pussy
Anonymous at Thu, 30 Jan 2025 13:26:15 UTC No. 16570180
>>16570178
*traversing
Anonymous at Thu, 30 Jan 2025 14:10:07 UTC No. 16570218
>>16569760
3d vectors and matrix, are involved, doing calculations with these, if you boil it down, cgi is a bunch of points in 3d space affected by other points and vectors , not sure what the matrix are for, I guess resterizing the whole mess
t not a /sci/ chad
Anonymous at Thu, 30 Jan 2025 14:25:04 UTC No. 16570234
>>16570218
>not sure what the matrix are for, I guess resterizing the whole mess
How did you even find your way to this thread?
Anonymous at Thu, 30 Jan 2025 16:56:14 UTC No. 16570377
>>16563103
>anyone here do math in cursive?
Your red pen reminds me of my TWSBI clear plastic converter pen, and those do come with italic nibs. I really miss that pen, I broke it a long time ago and I never bothered to buy another.
Cursive should be avoided in general. It's just a sloppier version of Chancery Italic, which is made anachronistic by the fact that we do not write with utensils with italic nibs (i.e. you are not an italian scribe employed by the vatican writing with a quill carved out of a feather). Our "nibs" are a fine hair-width now, in the form of a pencil tip or a piece of chalk, etc.
To expound more on cursive being retarded. Handwriting where the ideal is the nib never leaving the page is like typing where your fingers never leave the keyboard. Silly. If "swipe type" didn't automatically typeset your writing properly, then it would look similar to this retard shit:
>iuytdfghjikooiuytrerfghn'uytb nmnbvcxdsadrftgrerfv bvgfrde
This is the same exact problem, I am not joking, with cursive.
And then even in isolation, the letters turn into random glyphs of their own with no rhyme or reason discernable, especially if you're a second grader. Imagine torturing a kid with memorizing picrel and being told to write in this ugly script instead of exposing him to the actual roots of European culture and the actual method of handwriting as it was developed. Couldn't be me.
It's illegible and is self-defeating in purpose. Also, two things: 1) it makes you gay. Imagine what it does to your personality if the way you write might as well look like you dot your "i"s with hearts and punctuate your periods by drawing little daisies. Writing in cursive is the equivalent of speaking with a gay lisp. 2) if the goal is "economy of movement" and faster writing to capture lecture notes, then you should actually be using the journalism technique of "shorthand". We don't do this because this is a COPE. It is NOT done because of speed. It's done because people are GAY
Anonymous at Thu, 30 Jan 2025 17:07:50 UTC No. 16570392
>>16570377
>Cursive should be avoided in general
stopped reading past this
Anonymous at Thu, 30 Jan 2025 17:09:40 UTC No. 16570395
>>16570377
However, if just for the sake of better style, the beauty of Chancery Italic is that it can be learned in 6 or 7 minutes unlike the other anon's suggestion of 6-7 months of practice. All that's left to do is to write normally, in print, but when you would write in symbols, make them special by using the italic form. Even without an italic nib, it's distinguishable and looks correct.
Anonymous at Thu, 30 Jan 2025 17:15:34 UTC No. 16570404
>>16570392
Cursive is just a late degenerate form of traditional penmanship.
Anonymous at Thu, 30 Jan 2025 17:19:37 UTC No. 16570410
>>16570395
>autistic enough to write simplified chinese
>too autistic to make it legible or make sense
Anonymous at Thu, 30 Jan 2025 17:29:35 UTC No. 16570423
>>16570410
As far as writing with a mouse in ms paint is concerned, it serves its function as sample text, Would you have preferred Brazilian Portuguese?
šļø Anonymous at Thu, 30 Jan 2025 17:36:51 UTC No. 16570434
>>16570395
>png file with transparency
Dear Anon, this is how your image looks like on my browser. You had one job.
Anonymous at Thu, 30 Jan 2025 17:47:47 UTC No. 16570446
>>16570395
>png file with transparency
Dear Anon, this is how your image looks like on my browser. Sincerely, you had one job.
Anonymous at Thu, 30 Jan 2025 17:48:35 UTC No. 16570447
>>16570446
Maybe stop complaining to others if the problem is with your own theming?
Anonymous at Thu, 30 Jan 2025 17:52:31 UTC No. 16570450
also ich find ja unvollstaendigkeit muss unvollstaendig sein, sonst waer sie ja universell und somit widerspruechlich; die metawerkzeug ausrede glaub ich nich, weil's kein system gibt das von aussen vollstaendig analysiert werden kann. sprich, entweder unvollstaendigkeit funktioniert nur so wie das axiom der wahl, oder unvollstaendigkeit muss wissen wenn wir mit unvollstaendigeit ueber unvollstaendigeit reden, und das klingt alles nich so als waer das vollstaendig. also, find ich jetz so.
Anonymous at Thu, 30 Jan 2025 18:32:17 UTC No. 16570487
>>16570377
>GAY
Anonymous at Thu, 30 Jan 2025 18:47:46 UTC No. 16570501
Math anons please help, I am doing an independent study in a grad school level topic in my undergrad because it was offered to me by my favorite professor. It seemed like a good idea because if he is offering it it means he thinks Iām capable. Every single problem from the book that I do I either donāt know where to start or I just completely miss the mark. Is this normal for this level content? I feel like I am learning a lot, I know much more about the topic than I used to, but arenāt I supposed to be getting some questions right first try? Or is it that with this high level stuff you just learn by being wrong all the time? I imagine my professor is frustrated that Iām a midwit.
Anonymous at Thu, 30 Jan 2025 19:55:23 UTC No. 16570563
>>16570501
>arenāt I supposed to be getting some questions right first try?
not if you're jumping years ahead and aren't absolutely hardcore engaged out of interested, no.
as in, i think excepting "just because" was a mistake
Anonymous at Thu, 30 Jan 2025 20:19:29 UTC No. 16570588
>>16567727
No I will not "formalize it in lean" and no, I will not try to make the paper "more rigorous". Frankly I'm done thinking about this dumbfuck problem, so do it yourself.
Anonymous at Thu, 30 Jan 2025 22:06:17 UTC No. 16570694
>>16570501
>I know a lot more about the topic than I used to
Sounds like it's a success.
What topic?
Anonymous at Thu, 30 Jan 2025 22:18:04 UTC No. 16570706
>>16570563
So isnāt that kind of fucked then? Professor must have way overshot my abilities. Itās like going up to a primary schooler and asking if they want to take a bench pressing class for adults, then when the primary schooler gets their chest caved in they āshouldnāt have just accepted the classā. I thought, with my professor telling me over and over that he thinks I should do it, that I should therefore do it. If youāre like āheh, obviously you shouldnāt have done thatā is my professor just fucking with me? Also FWIW I think I miscommunicated, to me that sentence did not mean I did it ājust causeā I do have a very good interest in what we are talking about and thought I could do it. I gave it deep consideration. I just chose wrong anyways I guess.
Anonymous at Thu, 30 Jan 2025 22:24:22 UTC No. 16570721
>>16570694
Eh, I was avoiding ādoxxingā myself if my professor lurks here but it probably doesnāt matter, itās Lie Algebras. Iām comfortable enough with all the definitions and concepts itās built on, but then when it comes to the proofs Iām just dead in the water.
Anonymous at Thu, 30 Jan 2025 23:00:15 UTC No. 16570760
>>16570721
>if my professor lurks here but it probably doesnāt matter, itās Lie Algebras
you realize that this is not some superduper niche thing, right?
Anonymous at Fri, 31 Jan 2025 00:17:07 UTC No. 16570825
>>16570760
Being specifically someone posting at the times I am, with the exact problem I am, with specifically this exact independent study that I started just now, in this semester. Tell me how many people you think fit that description reasonably, right this second. Also, nothing I said even slightly implied that I think itās some āsuper duper niche thingā. I should have expected /sci to be illiterate.
Anonymous at Fri, 31 Jan 2025 01:02:11 UTC No. 16570844
Does anyone have that meme that's like "This is nonsense made up by lunatics. We've been had!" It's just some graphs (can't remember what they were off the top of my head since this is like three or four years ago now), and it appears as if it was made in MS Paint.
Anonymous at Fri, 31 Jan 2025 06:11:13 UTC No. 16571040
>>16570825
Youāre paranoid. Plenty of undergrads doing research projects. And expecting a professor to browse 4chinzā¦ Thatās a tall order.
Anonymous at Fri, 31 Jan 2025 10:30:34 UTC No. 16571169
>>16570234
amazing and informative insight your knowledge shines through
Anonymous at Fri, 31 Jan 2025 15:05:19 UTC No. 16571323
>>16570392
Me too.
Anonymous at Fri, 31 Jan 2025 17:52:23 UTC No. 16571437
>>16569576
>non-algebraic structures
which immediately get algebraized
You can also do galois theory with them.
Start with monic polynomial.
vectorize coeffs.
let this be your base point in C^n
closed loops starting and ending at base point induce permutations on roots
Anonymous at Fri, 31 Jan 2025 18:18:06 UTC No. 16571452
How are people like Tao and Schulz supposedly among the smartest humans who've ever lived, yet they have not amassed wealth and resources or other markers of success in our society. Maybe this is the wrong thread to ask in, I'd just like to know what is their metric, or possibly optimized/parabolized breakpoint for material success vs intellectual pursuit as a function of FUN maximization. I only thought of this because Tao's wife is unattractive, but there are smart sexy girls like the Deepseek engineer, so he doesn't need to stoop so low.
Anonymous at Fri, 31 Jan 2025 19:45:13 UTC No. 16571517
>>16571452
maybe because they're real people who live in the real world and not a set of numbers in some imaginary 5 dimensional metric space that exists only in your incel fantasies?
Anonymous at Fri, 31 Jan 2025 23:51:52 UTC No. 16571809
>>16571437
This is something akin to roots of semisimple Lie algebras or am I thinking in the wrong direction?
Anonymous at Sat, 1 Feb 2025 01:09:51 UTC No. 16571844
Anonymous at Sat, 1 Feb 2025 01:42:50 UTC No. 16571859
>>16571844
Genuinely funny
Anonymous at Sat, 1 Feb 2025 03:10:46 UTC No. 16571920
>>16571809
I'll give a small example
p(x) = x^2-b
r1 = sqrt(b)
r2 = -sqrt(b)
base point (0,-b) in C^2 corresponds to our p(x)
vary vector (0,-b) over some path in C^2 back to (0,-b)
If b winds around 0 an odd number of times then r1 and r2 swap due to the sqrt branch
performing one loop then another is just composition of induced permutations
We can guarantee no winding in innermost radicals (in the formulas for the roots) by restricting to composing commutators of loops.
We can guarantee no winding in second innermost radicals (in the formulas for the roots) by restricting to composing commutators of commutators of loops.
...
https://en.wikipedia.org/wiki/Commu
S1 through S4 will eventually terminate (meaning the formulas for the roots have finitely many nested radicals)
We see the group S5 does not terminate. It gets stuck at A5 in general. This is pretty much Abel Ruffini.
Anonymous at Sat, 1 Feb 2025 09:51:03 UTC No. 16572085
>>16570844
>>16571844
>>16571859
You are sub 80 IQ monkeys. Go back to plebbit faggots.
Anonymous at Sat, 1 Feb 2025 15:09:34 UTC No. 16572245
>>16571844
Here u go
Anonymous at Sat, 1 Feb 2025 20:18:20 UTC No. 16572536
>>16572465
Substitute [math]x = ny [/math]
[eqn]n^{-s} \int_0^\infty x^{s-1} e^{-x} dx\\
= n^{-s} \int_0^\infty n^{s-1} y^{s-1} e^{-ny} n dy \\
= \int_0^\infty y^{s-1} e^{-ny} dy [/eqn]
Anonymous at Sat, 1 Feb 2025 20:37:13 UTC No. 16572542
>>16572536
I thought there might be a substitution necessary but I suck at math so I didn't see it. How do you even see substitutions like that? there should be a book purely on substitutions honestly, I would read that.
Anyway, thanks a zillion.
Anonymous at Sat, 1 Feb 2025 20:43:15 UTC No. 16572545
>>16563178
you literally just need to have a working memory and the ability to recognize patterns which is innate in all humans
how can you be bad @ math
Anonymous at Sat, 1 Feb 2025 22:16:02 UTC No. 16572611
There was a person here a while back who asked this exact question. I'm reading Category Theory In Context by Riehl and noticed this and thought they might find it interesting if they read the thread.
Anonymous at Sat, 1 Feb 2025 22:17:03 UTC No. 16572614
>>16572611
Here's the lemma.
Anonymous at Sat, 1 Feb 2025 23:00:19 UTC No. 16572657
what math do I need to know to understand neural networks? i'm a college freshman who has taken up to integral calculus
Anonymous at Sat, 1 Feb 2025 23:07:15 UTC No. 16572665
>>16572657
Some basic calculus based probability theory, a good bit of linear algebra, some basics of graph theory. That's pretty much it for the simpler parts of neural networks.
Anonymous at Sat, 1 Feb 2025 23:14:52 UTC No. 16572671
>>16571517
I don't understand your answer.
Anonymous at Sat, 1 Feb 2025 23:28:18 UTC No. 16572675
>>16572611
Wait, I don't get the fifth one. Isn't the order-reversal morphism (going from leq to geq) an isomorphism too? Or is that only true for tosets?
Anonymous at Sat, 1 Feb 2025 23:55:15 UTC No. 16572693
>>16572671
your ideas about "success" or "fun maximization" have nothing to do with the real world or real every day people's motivations.
Anonymous at Sun, 2 Feb 2025 03:26:22 UTC No. 16572860
>>16572675
Example (v) isn't talking about isomorphisms in the category of posets, but about the isomorphisms in a fixed poset (viewed as a thin category)
Anonymous at Sun, 2 Feb 2025 05:58:56 UTC No. 16572953
>>16571452
They have a blast doing math, my guy. That's the simple answer.
Anonymous at Sun, 2 Feb 2025 12:02:11 UTC No. 16573146
>ywn intuitively understand stationary sets
Anonymous at Sun, 2 Feb 2025 17:07:28 UTC No. 16573331
>>16573146
A stationary set is a sponge.
Anonymous at Sun, 2 Feb 2025 23:10:36 UTC No. 16573652
Can a Maths senpai help a kouhai out?
>>16573621
>>16573643
Anonymous at Sun, 2 Feb 2025 23:22:45 UTC No. 16573657
>>16573318
Can you describe what parts have gotten you so far? Also, what is your stats/math background? Is it the general concepts that are tripping you up or the mathematical mechanics?
Anonymous at Mon, 3 Feb 2025 01:49:25 UTC No. 16573749
What's a strategy for showing that an element does not belong to an inductively defined set?
For example if one were to define the even naturals inductively as 0 is even and if n is even then so is n + 2, how would one show that 1 isn't even?
Anonymous at Mon, 3 Feb 2025 02:13:26 UTC No. 16573762
>>16562509
Is it too informal/juvenile to use tcolorboxes in an undergrad thesis to denote between Definitions/Theorems/Lemmas etc...
Anonymous at Mon, 3 Feb 2025 02:42:00 UTC No. 16573786
>>16573749
your definition isn't strong enough to prove that.
you need
"0 is even and for n > 0, n is even if and only if there exists even m >= 0 such that n = m + 2."
your definition isn't strong enough because it doesn't say that ONLY 0, 0 + 2, 0 + 2 + 2,... are even.
Anonymous at Mon, 3 Feb 2025 03:40:33 UTC No. 16573852
how can i self learn a higher lvl math text book by myself most effectively? ?
Anonymous at Mon, 3 Feb 2025 05:37:50 UTC No. 16573937
>>16562509
Does anyone have the trivium PDF by Verbitsky?
The website by him is down.
Anonymous at Mon, 3 Feb 2025 06:41:37 UTC No. 16573953
is structure and interpretation of classical mechanics a good into to calculus based CM if I already have a trig based overview of basic physics?
Anonymous at Mon, 3 Feb 2025 06:53:12 UTC No. 16573959
>>16573318
They were never good at writing a rigorous mathy book which just makes it that much worse when they start using more abstract math concepts. If you are struggling you need to put it away and just study more pure math for a bit. Least Squares Regression uses elements of calculus and statistics but it is primarily an application of Linear Algebra. Hence the frequent use of matrix equations.
Anonymous at Mon, 3 Feb 2025 07:33:44 UTC No. 16573978
Any good books on three dimensional projective geometry?
Anonymous at Mon, 3 Feb 2025 07:34:48 UTC No. 16573980
>>16573318
Read the og: The Nature of Statistical Learning by Vapnik.
Anonymous at Mon, 3 Feb 2025 12:45:17 UTC No. 16574175
>>16573953
sir, this is the mathematics general
Anonymous at Mon, 3 Feb 2025 13:25:13 UTC No. 16574198
Is there a proof of four-colour theorem that doesn't require a shit-ton of computer calculations? (Or a proof that's impossible to prove a four-colour theorem without a shit-ton of computer calculations?)
Anonymous at Mon, 3 Feb 2025 13:29:03 UTC No. 16574200
>>16574198
Yeah, you do the calculations yourself.
Anonymous at Mon, 3 Feb 2025 13:33:06 UTC No. 16574201
>>16574198
go ahead and find it. I believe in you, anon.
Anonymous at Mon, 3 Feb 2025 16:16:44 UTC No. 16574371
>>16574175
but it has lots of math in it tho
Anonymous at Mon, 3 Feb 2025 18:46:06 UTC No. 16574503
>>16574371
Your mom has lots of BBC in her but she isnāt BBC herself.
Anonymous at Mon, 3 Feb 2025 20:06:38 UTC No. 16574609
>>16574503
Mutt's law remains undefeated.
Anonymous at Mon, 3 Feb 2025 22:12:44 UTC No. 16574787
It's been years since I've done some math. What are some fun topics one can dive into? I have a degree in math so any suggestion is welcomed, I don't care how advanced.
Anonymous at Mon, 3 Feb 2025 22:19:29 UTC No. 16574798
>>16574787
Dessins are pretty entertaining as a new challenge.
Anonymous at Mon, 3 Feb 2025 22:25:29 UTC No. 16574809
>>16574798
https://en.wikipedia.org/wiki/Dessi
this what you mean?
Anonymous at Tue, 4 Feb 2025 00:45:13 UTC No. 16574940
>>16573749
>>16573786
To define a set inductively usually means to define it as the *least* set closed under some rules.
The even naturals are indeed the least subset of N that contains 0 and is closed under addition by 2.
But many other subsets of N are closed in that sense and a lot of them will also contain 1 (for example N itself is closed under both rules and contains 1).
So you absolutely have to use the fact that the evens are not just closed under your two rules but are actually the least closed set if you wish to prove that 1 is not even.
A common way to proceed is to prove the equivalent statement that every even number is not equal to 1. This can be proven inductively (not using induction on the naturals, but using the induction principle arising from the inductive definition of the evens): Since the evens are the least subset of N containing 0 and closed under addition by 2, it is sufficient to prove that the set of naturals not equal to 1 is also closed under both these rules. Then the evens are a subset of the naturals not equal to 1, hence every even number is not equal to 1, so 1 is not even.
But the set of naturals not equal to 1 immediately contains 0 since 0 is not equal to 1 and it is also immediately closed under addition by 2 since n + 2 cannot be equal to 1, so we are done here.
Anonymous at Tue, 4 Feb 2025 00:46:14 UTC No. 16574941
>>16574940
Something somewhat related to what the other anon said: Defined this way you can also prove (again inductively) that every even number is either 0 or arises from addition by 2 of some other even number. The converse of that statement is immediate, so the even numbers are *exactly* those naturals that are either 0 or of the form e + 2 with e again being an even number.
The upshot to all this is that this is essentially a corollary of Lambek's lemma applied to the posetal category of subsets of N with endofunctor/monotone function given by taking subsets of N to their image under addition by 2 and also appending a 0 to that image. The even numbers are then the initial algebra/prefixed point of this functor/function, which by Lambek's lemma turns out to be an actual fixed point, which here ends up being the statement that n is even iff it's equal to 0 or of the form e + 2 with e again being even.
Anonymous at Tue, 4 Feb 2025 00:48:13 UTC No. 16574944
>>16574809
Yep. Absolutely magnificent little thingies. Their scope is frankly unbelievable.
Anonymous at Tue, 4 Feb 2025 03:02:05 UTC No. 16575016
>>16574787
If you like probability, PGMs are super cool and can handle a ton of different topics in both probability theory and statistics.
Anonymous at Tue, 4 Feb 2025 06:11:07 UTC No. 16575096
>>16574198
Nope, I'm working on it.
Gotta be more general than tait's conjecture.
https://en.wikipedia.org/wiki/Cauch
https://en.wikipedia.org/wiki/Kirch
https://en.wikipedia.org/wiki/FKT_a
Anonymous at Tue, 4 Feb 2025 16:29:09 UTC No. 16575448
>>16573852
Hereās how I read my textbooks. Iām an engineer.
>Start by reading the book, actually reading it.
>When you encounter something you donāt know, take the time to understand what its saying and understand what it means.
>if they show proofs, try doing the proofs yourself.
>re-read the part you originally got stuck on and refine your understanding on that part (sentence, definition, example, etc) until you are confident you understand the concept.
>repeat until youāve read the whole book.
Anonymous at Tue, 4 Feb 2025 22:21:11 UTC No. 16575788
>>16575770
By the way, the (x,y) points are ((Ļ/2)-1,Ļ/2) and (Ļ-(Ļ/2)+1,Ļ-(Ļ/2)) where the curve touches the line.
Anonymous at Tue, 4 Feb 2025 22:28:41 UTC No. 16575792
Forcing feels like witchcraft. how did that jew cohen even come up with it...
Anonymous at Tue, 4 Feb 2025 22:58:08 UTC No. 16575812
anyone using AI/LLMs to help with math? Or are they not good enough at the upper level math subjects? I know wolfram exists but never used it.
Anonymous at Tue, 4 Feb 2025 23:24:03 UTC No. 16575848
>>16575812
LLM's don't follow logical rules. They are interpolation systems, not directly constrained by mathematical logic. If you need "close but you'll never know if it's right," an LLM is probably not a bad first pass. If you actually need to be certain it is correct and logically consistent, LLM's will not help you. They can't even know how many of each particular letter are in any given word without having to conjure up some human written Python code to check.
šļø Anonymous at Tue, 4 Feb 2025 23:28:35 UTC No. 16575856
>>16575770
Some partial but loose approximations I've found for small parts: [math]\sqrt[3]{6x}; \left(\frac{\pi}{2}\right)^{-1} + \sqrt{2x - \pi + 3}; \frac{3\pi}{2} + 1 - \sqrt{3 + 3\pi - 2x}[/math]
Anonymous at Tue, 4 Feb 2025 23:35:05 UTC No. 16575871
>>16575770
Some partial but loose approximations I've found for small parts:
[math]\sqrt[3]{6x}; \frac{\pi}{2} - 1 + \sqrt{2x - \pi + 3}; \frac{3\pi}{2} + 1 - \sqrt{3 + 3\pi - 2x}[/math]
Had to delete my previous post because of a LaTeX conversion error, AI mistaked -1 for ^{-1}.
>>16575812
They usually suck, which is why I still need human assistance.
Anonymous at Wed, 5 Feb 2025 04:49:54 UTC No. 16576089
>>16574503
leave mom and jamal out of this and just answer my question nerd
Anonymous at Wed, 5 Feb 2025 05:49:21 UTC No. 16576115
>>16575812
It is interesting to see deepseek think.
You can learn from the AI. At the very least it definitely can solve book problems and be your answer key when self studying.
>>16575848
retard
deepseek is solving putnam problems in 10 minutes
I trust its logic more than any random undergrad.
Anonymous at Wed, 5 Feb 2025 07:07:39 UTC No. 16576166
>>16575812
Trying to get an LLM to talk about math is easily the most disappointing thing I've ever done with one. They're dumb as fuck and the only other place I've noticed the hallucination problem as badly as mathematics is in translation
They're not even reliable at regurgitating standard things that are probably in their training data several dozen times in different books and posts
Anonymous at Wed, 5 Feb 2025 07:20:53 UTC No. 16576172
>>16576166
Example problem to give LLM so I can see what you mean?
Anonymous at Wed, 5 Feb 2025 07:37:17 UTC No. 16576185
>>16571452
Terence Tao makes well over half a million dollars a year just in base salary from UCLA. Sure, he could make a lot more by quitting and diddling derivatives in some quant shop, but there's no way you can say he isn't a financially successful person
Anonymous at Wed, 5 Feb 2025 07:57:11 UTC No. 16576192
>>16575770
[math]f^{-1}(x)-a= \lim\limits_{n \rightarrow \infty} \dfrac{nD_{z=a}^n[log(f(z)-x)]}{D_{
Anonymous at Wed, 5 Feb 2025 07:59:08 UTC No. 16576195
>>16576192
should be n+1 on the bottom D
Anonymous at Wed, 5 Feb 2025 08:35:52 UTC No. 16576225
>>16571452
you sound ugly
Anonymous at Wed, 5 Feb 2025 08:55:32 UTC No. 16576233
>>16576115
saaar sukdeep will solve everything ples invest
Anonymous at Wed, 5 Feb 2025 10:58:17 UTC No. 16576291
>>16576172
Prove R(3,4)<=9. A basic problem in graph theory with widely published solutions that are definitely in their training data but none of the scamgpt models are able to solve it. Deepseek r1 was able to solve it though.
Anonymous at Wed, 5 Feb 2025 10:59:39 UTC No. 16576293
>>16576172
Ask any of the llms to produce or reproduce any proof that every vector space having a basis implies the axiom of choice. Its a standard proof yet none of the llms are able to reproduce it.
Anonymous at Wed, 5 Feb 2025 20:27:40 UTC No. 16576729
>>16576192
is this some kind of troll where the limit approaches 0?
Anonymous at Wed, 5 Feb 2025 21:20:45 UTC No. 16576768
>>16576729
No. It converges.
Suppose you wish to solve f(a)=x
Taylor expand f(a) around the solution z
f(a)-x=f(z+(a-z))-x=f(z) +(a-z)f'(z)+...-x
f(z) = x by definition of z.
log(f(a)-x) ~ log((a-z)f'(z) +...) = log(a-z) + log(f'(z) + (...)/(a-z))
Differentiate n times wrt a to get LHS^(n) ~ -(n-1)!(-1/(a-z))^n for a near z
... n+1 times ... LHS^(n+1) ~ -(n)!(-1/(a-z))^(n+1)
LHS^(n) / LHS^(n+1) ~ (z-a)/n for a near z.
"a near z" essentially means a is closer to z than any other solution.
This is basically the householder method. Great way to get pade expansions for inverses.
https://en.wikipedia.org/wiki/House
Instead of doing many newton iterations, just do 1 infinite order householder step.
You still need an initial point a (hopefully one that is convenient).
Anonymous at Wed, 5 Feb 2025 21:30:48 UTC No. 16576777
>>16575871
slightly better approximation
[math]\sqrt[3]{6x}\left(1+\frac{1}{
Anonymous at Thu, 6 Feb 2025 04:00:20 UTC No. 16577045
>>16576768
I see, logarithmic derivatives that are part of fractional calculus. But derivatives up to the ā-order isn't practically computational yet theoretically more precise perhaps, but I probably don't need that many orders anyway, 3 to 8 might be enough. Are you simply doing that raise because of the vertical tangents around at 0 and Ļ of [math]f^{-1}(x)[/math] where there needs to be more precision? I've also been thinking about using Taylor series again.
Does /sci/ even support [code]code blocks[/code]?
Anonymous at Thu, 6 Feb 2025 04:57:54 UTC No. 16577073
In what class and what context do most mathematics students first learn about Galois connections?
Anonymous at Thu, 6 Feb 2025 05:30:12 UTC No. 16577091
>>16577073
looks like a generalization of a standard result from intro grad school algebra
Anonymous at Thu, 6 Feb 2025 16:44:05 UTC No. 16577576
bump function
Anonymous at Thu, 6 Feb 2025 18:36:57 UTC No. 16577705
>>16577576
Here you go...
Anonymous at Thu, 6 Feb 2025 21:32:33 UTC No. 16577839
>working on textbook for weeks
>stuck on basic fucking problem for hours
>ask online
>some faggot solves it in 15 seconds
Why even bother lmao. jesus
Anonymous at Thu, 6 Feb 2025 21:35:10 UTC No. 16577841
>>16577839
forgot the gif
Anonymous at Thu, 6 Feb 2025 22:16:40 UTC No. 16577888
>>16577839
What was the problem?
Anonymous at Thu, 6 Feb 2025 22:51:41 UTC No. 16577934
>>16563262
I did my math PhD. The trick I learned is to copy the textbook over and over until I understood it. Good luck
Anonymous at Thu, 6 Feb 2025 22:53:22 UTC No. 16577937
>>16577934
Sorry just realized I am referring to US, where a PhD starts with a few years of classes before the thesis work begins. When you get to the thesis work, literally just pester the fuck out of your advisor and do whatever he tells you and go to him as soon as you have any problem
Anonymous at Thu, 6 Feb 2025 22:54:08 UTC No. 16577940
>>16577888
Those problems were tough.
Anonymous at Thu, 6 Feb 2025 23:07:19 UTC No. 16577957
>>16577940
>muh number theory
meme for faggots stuck in the 17th century
Anonymous at Thu, 6 Feb 2025 23:13:21 UTC No. 16577965
>>16577940
Why the fuck are you posting open problems? I asked you for the problem you weren't able to solve and someone else solved it in 15 seconds.
Anonymous at Thu, 6 Feb 2025 23:45:23 UTC No. 16577995
>>16577965
How could they be open if dude solved them in 15 seconds?
Anonymous at Fri, 7 Feb 2025 01:06:57 UTC No. 16578042
I am getting brutally raped by this proposition in Serre's course in arithmetic. anyone know how to prove that the ideals he's talking about are a basis for neighborhoods of zero? I feel like it should be obvious but I can't prove it for some reason.
Anonymous at Fri, 7 Feb 2025 01:19:46 UTC No. 16578050
Is PDE field for autists? It's just brutal computation, right?
Anonymous at Fri, 7 Feb 2025 02:52:39 UTC No. 16578102
I fucking love Time Series Analysis by James Hamilton
Anonymous at Fri, 7 Feb 2025 10:17:52 UTC No. 16578297
>>16578050
Wait till you find out about concentration inequalities.
>>16578102
Read a REAL book on Time Series like Brockwell & Davis' methods.
šļø Anonymous at Fri, 7 Feb 2025 18:20:50 UTC No. 16578624
>>16578042
He said to give [math] A_n = \mathbb{F}_{p^n},[/math] i.e. field of p^n elements, the discrete topology and to give [math] Z_p [/math] (the ring of p-adic integers using projective limit definition) the topology inherited from the product topology of the [math] A_n. [/math]
But then if you consider a neighborhood of 0 of [math] Z_p, [/math] [math] B = \epsilon_2^{-1}(\{ 1, 0\}) [/math] where [math] \epsilon [/math] is the projection onto the 2nd coordinate, B contains the element (..., 1, ... 1, 1) which is not contained in any of those ideals except [math] p^0\mathbb{Z}_p = \mathbb{Z}_p [/math] because every other ideal has 0 in the last coordinate. But [math] \mathbb{Z}_p [/math] obviously isn't a subset of [math] B [\math] so how can the ideals generate all neighborhoods of 0? what the fuck is he talking about?
Anonymous at Fri, 7 Feb 2025 18:31:13 UTC No. 16578634
>>16578042
He said to give [math] A_n = \mathbb{Z}/p\mathbb{Z} [/math], the discrete topology and to give [math] Z_p [/math] (the ring of p-adic integers using projective limit definition) the topology inherited from the product topology of the [math] A_n. [/math]
But then if you consider a neighborhood of 0 of [math] Z_p [/math], [math] B =\epsilon_2^{-1}(\{1,0\}) [/math] where Ļµ is the projection onto the 2nd coordinate, B contains the element (..., 1, ... 1, 1) which is not contained in any of those ideals except [math] p^0\mathbb{Z}_p= \mathbb{Z}_p [/math] because every other ideal has 0 in the last coordinate. But [math] \mathbb{Z}_p [/math] obviously isn't a subset of [math] B [/math] so how can the ideals generate all neighborhoods of 0? what the fuck is he talking about?
Anonymous at Fri, 7 Feb 2025 18:33:47 UTC No. 16578639
>>16578634
>An=a/pa
I meant mod p^n not p
Anonymous at Fri, 7 Feb 2025 21:52:46 UTC No. 16578835
>>16564969
yeah the beauty is supposed to be in the proofs, not the results
Anonymous at Fri, 7 Feb 2025 22:17:00 UTC No. 16578867
>>16578050
It depends. Solving particular BVP problems are for midwits aka engineers. Itās brute force calculations as you said.
Actually studying (linear) PDEs is an autistic topic that involves group theory, functional analysis and representation theory. If you want to get an idea of how a professional mathematician approaches PDEs, see Harmonic Analysis by Folland.
Anonymous at Sat, 8 Feb 2025 13:20:29 UTC No. 16579388
What are the most interesting finite sequences in mathematics?
Anonymous at Sat, 8 Feb 2025 13:25:28 UTC No. 16579393
>>16579388
I think the Heegner numbers are pretty neat
Anonymous at Sat, 8 Feb 2025 14:42:30 UTC No. 16579431
>Cleo debunked
/mg/ on suicide watch
Anonymous at Sat, 8 Feb 2025 14:48:37 UTC No. 16579433
>>16579431
What happened?
Anonymous at Sat, 8 Feb 2025 14:58:18 UTC No. 16579437
>>16578835
That's not it. The best proofs are the boring ones. The beauty is in the understanding, which makes all the proofs seem natural and makes a whole subject click.
Anonymous at Sat, 8 Feb 2025 15:01:45 UTC No. 16579438
>>16579431
Yeah it's over now. Last time the Cleo fraud allegations were posted here people got really salty, like little children being told Santa Claus isn't real.
Anonymous at Sat, 8 Feb 2025 15:06:02 UTC No. 16579442
>>16578042
>>16578634
>anyone know how to prove that the ideals he's talking about are a basis for neighborhoods of zero?
First convince yourself that they form basis of neighborhoods of zero.
First note that the set
{0}x{0}x...{0}xA_(n+1)xA_(n+2)x....
is open in the product topology.
Its restriction to Z_p (which is a subset of the prod_k A_k)
is exactly p^nZ_p. Thus each p^nZ_p is a neighborhood of zero.
Conversely, if I is a neighborhood of zero, it's the restriction to Z_p of an open set of the form V x A_(n+1) x A_(n+2) x..... As it contains 0, V contains (0,0,...,0) so the open set contains p^nZ_p.
Anonymous at Sat, 8 Feb 2025 15:08:40 UTC No. 16579445
>>16578634
It seems like you're confused about what a basis of neighborhoods is. A basis of neighborhoods at a point x is a collection of open sets U_n, all containing x, such that for every open set U containing x there is an n such that U_n lies inside of U.
Anonymous at Sat, 8 Feb 2025 17:23:58 UTC No. 16579528
>>16578297
That's a pretty laborious proof of Azuma-Hoeffding vs a martingale argument + induction.
Also, I'd stick to Hamilton.
Anonymous at Sat, 8 Feb 2025 17:42:33 UTC No. 16579542
>>16579528
Hamilton what?
Anonymous at Sat, 8 Feb 2025 18:12:04 UTC No. 16579570
>Going through how to prove it with lean
>Get through first 2 set of exercise pretty easy, at most take an hour or two
>get to one exercise I can't solve no matter what
>been on it for 2 days
>There's no solution set anyway
ov3r
Anonymous at Sat, 8 Feb 2025 19:52:40 UTC No. 16579635
>>16579437
wrong
Anonymous at Sat, 8 Feb 2025 20:20:56 UTC No. 16579654
>>16579570
Lean is not easy so it's pretty undestandable. With easier proofs you can just go with the flow and it will eventually work. For harder ones I recommend planning ahead, maybe go step by step on paper, and think about which tactics you'll want to use. Or, post the problem here to give you some pointers.
šļø Anonymous at Sat, 8 Feb 2025 20:24:58 UTC No. 16579662
>>16579445
> such that for every open set U containing x there is an n such that U_n lies inside of U
yes... and I showed that every point containing X does not lie inside U
>>16579442
>Conversely, if I is a neighborhood of zero, it's the restriction to Z_p of an open set of the form V x A_(n+1) x A_(n+2) x..... As it contains 0, V contains (0,0,...,0) so the open set contains p^nZ_p.
This doesn't work because you didn't show that there is a p^nZ_p containing any point of I that subsets I. I can contain points that have no non-zero coordinates and there is no p^nZ_p that contains points with non-zero coordinates except Z_p itself which does not necessarily subset I.
I'm pretty sure that he is using a definition for "basis of neighborhoods of 0" that is based on topological rings (https://en.wikipedia.org/wiki/Topo
Anonymous at Sat, 8 Feb 2025 20:26:55 UTC No. 16579667
>>16579445
> such that for every open set U containing x there is an n such that U_n lies inside of U
yes... and literally showed that there a point x in u such that there is no p^nZ_p containing it that subsets U.
>>16579442
>Conversely, if I is a neighborhood of zero, it's the restriction to Z_p of an open set of the form V x A_(n+1) x A_(n+2) x..... As it contains 0, V contains (0,0,...,0) so the open set contains p^nZ_p.
This doesn't work because you didn't show that there is a p^nZ_p containing any point of I that subsets I. I can contain points that have no non-zero coordinates and there is no p^nZ_p that contains points with non-zero coordinates except Z_p itself which does not necessarily subset I.
I'm pretty sure that he is using a definition for "basis of neighborhoods of 0" that is based on topological rings (https://en.wikipedia.org/wiki/Topo
Anonymous at Sat, 8 Feb 2025 20:54:05 UTC No. 16579705
>>16579654
Nevermind I cracked it
Holy moley
with lean it feels like I'm tinkering with hieroglyphics because all the tactics are like a grab bag of different techniques instead of a cohesive whole. A lot of the time there's something I want to do but I don't know how to translate it into lean code.
Anonymous at Sat, 8 Feb 2025 21:05:22 UTC No. 16579716
>>16566546
Every true statement of arithmetic is an axiom of true arithmetic. So the proof is just a oneliner, "this is an axiom". This doesn't actually help you distinguish between true and false statements.
Anonymous at Sat, 8 Feb 2025 21:29:10 UTC No. 16579737
My pde professor gives us homework that uses m, n, o, p, r, mu, nu, omicron, pi and rho as separate symbols mixed together and we're supposed to mix them perfectly without mistakes. We have complained and he has suggested that we have an audiobook about the history of different alphabets play in headphones that we wear while we sleep to better internalise the greek alphabet. We asked him if he could use different symbols and he says it looks nice and symmetric.
Anonymous at Sat, 8 Feb 2025 22:17:35 UTC No. 16579770
>>16579737
all of those letters are easily distinguishable except o and omicron which are literally identical. I have literally never seen anyone use omicron as a math symbol so I kind of don't believe you when you include omicron.
Anonymous at Sat, 8 Feb 2025 23:12:48 UTC No. 16579809
>>16579667
>This doesn't work because you didn't show that there is a p^nZ_p containing any point of I that subsets I
That's not how a basis of neighborhoods works.
See
>>16579445
Anonymous at Sat, 8 Feb 2025 23:22:41 UTC No. 16579811
>>16579809
That is clearly wrong since there are open sets in the topology that can't be generated by the sets [math] p^n\mathbb{Z}_p [/math] hence it would be meaningless to refer to them as the "basis of the neighborhoods of 0" as they do not generate those neighborhoods. Everything in the proposition followed trivially when I substituted the alternate definition he was obviously using so I am clearly right, whereas according to your definition the first sentence of Serre's proof of the proposition has nothing at all to do with the actual proposition, hence I am clearly correct. You seem to have simply invented your definition out of thin air as it makes no sense based on what the word "basis" means. faggot.
Anonymous at Sat, 8 Feb 2025 23:32:55 UTC No. 16579817
>>16579811
>>16579809
I now see that you are correct. sry for calling you a faggot
Anonymous at Sat, 8 Feb 2025 23:50:14 UTC No. 16579827
There is a general notion of a subobject and a quotient object in category theory via monomprhisms and epimorphisms equivalence classes respectively. Subobjects and quotient objects are duals. Here's something I'm struggling to figure out: is there a general notion of isomorphism theorems for generic categories?
In the category Grp, the first isomorphism theorem can be thought of as a universal epimorphism [math](G,\phi:G\twoheadrightarrow H)[/math], which states that H is a quotient object of G up to equivalence. I'm struggling with translating the other three isomorphism theorems into the language of category theory.
Anonymous at Sun, 9 Feb 2025 03:05:34 UTC No. 16579944
>>16579827
The closest theorem I'm familiar with, that applies to a generic category and which somewhat resembles the first isomorphism theorem would be that the coequalizer of the kernel pair of a morphism is its regular coimage.
Arguably that's not quite the first iso theorem since it speaks about the regular coimage (i.e. the greatest regular epi out of the domain that admits a factorization) rather than the image (the least mono into the codomain admitting a factorization). These two notions coincide in a regular category, and one could argue that this fact is the category theory version of the first iso theorem.
Anonymous at Sun, 9 Feb 2025 06:32:11 UTC No. 16580029
>>16579827
>first isomorphism theorem
Probably somewhere here
https://en.wikipedia.org/wiki/Isomo
Anonymous at Sun, 9 Feb 2025 10:51:17 UTC No. 16580170
>>16579827
A good example to keep in mind is that the map the ring inclusion
Z -> Q
is both a monomorphism and an epimorphism in the category of rings.
Would you call Q a subobject of Z?
Anonymous at Sun, 9 Feb 2025 10:52:18 UTC No. 16580171
>>16580170
Oops meant to say a quotient object
Anonymous at Sun, 9 Feb 2025 11:36:10 UTC No. 16580195
Quadratic forms represent ellipsoids. Is there a similar visualisation for bilinear forms.
Anonymous at Sun, 9 Feb 2025 12:13:35 UTC No. 16580229
>>16580029
The article only talks about algebraic objects. Iām talking about generic categories since subobjects and quotient objects exist for non-algebraic categories. The simplest example is subsets and quotient sets.
>>16580170
I fail to see how this is an epimorphism. Itās an inclusion, as you said, so you miss every fraction. You can certainly construct a set epimorphism, but all that tells you is that |Q| /leq |Z|. You can construct a dual epimorphism, so |Q|=|Z| as expected. So Q is the same set as Z up to equivalence but a different ring from Z. In general, Z is the initial object in the category Ring, so it will always be a subobject of some ring, even if improper.
Anonymous at Sun, 9 Feb 2025 12:57:06 UTC No. 16580276
>>16580229
Plain sets are still very much algebras in the universal algebra sense, they're the algebras of the empty signature
Anonymous at Sun, 9 Feb 2025 13:03:33 UTC No. 16580283
>>16580276
Sure, I guess you can think of them as some weird vacuous algebra. But I want some pure, distilled arrow autism without any reference to universal algebras. This anon >>16579944 suggested the categorial notion of kernel, which looks like something that can work for what Iām looking for. After all, normal subgroups and ring ideals are just kernels.
Anonymous at Sun, 9 Feb 2025 15:15:15 UTC No. 16580382
>Exercise says _ -> For all X blah blah blah blah
>Then says put something in _ to prove the theorem
>Put in 1=0
is this cheating?
Anonymous at Sun, 9 Feb 2025 16:34:38 UTC No. 16580414
>>16580229
>I fail to see how this is an epimorphism.
It's an epimorphism because a homomorphism from the rationals is determined by the values on the integers.
Anonymous at Sun, 9 Feb 2025 16:44:35 UTC No. 16580421
>>16580414
>from the rationals
the original post clearly stated that the source is the integers. Sober up, anon.
Anonymous at Sun, 9 Feb 2025 16:48:01 UTC No. 16580422
>>16580421
Yes, the map from the integers to the rationals is an epimorphism and I just explained why.
Anonymous at Sun, 9 Feb 2025 18:55:37 UTC No. 16580524
Let R be a commutative ring and S a commutative R-algebra. Is there a name for the property that for every x,y in S, if x ā y then there is an algebra homomorphism f:S ->R with f(x) ā f(y)? Or something equivalent for their dual schemes?
Anonymous at Mon, 10 Feb 2025 12:13:36 UTC No. 16581199
>>16580422
You said the map is an inclusion, ie we have n |-> n. Every fraction doesnāt have a fiber under this map. Are you retarded?
Anonymous at Mon, 10 Feb 2025 12:40:22 UTC No. 16581222
>>16581199
>You said the map is an inclusion, ie we have n |-> n.
Yes.
>Every fraction doesnāt have a fiber under this map.
Correct.
>Are you retarded?
No. Are you?
Anonymous at Mon, 10 Feb 2025 14:35:47 UTC No. 16581312
>>16581199
What do you think the definition of an epimorphism is?
Anonymous at Mon, 10 Feb 2025 16:24:43 UTC No. 16581457
When dealing with an homotopy, say, [math]H:X\times [0,1] \to Y[/math], can I write [math]H:[0,1]\times X \to Y[/math] instead?
I like interpreting the parameter as time, and just to agree with most notations in physics, for me it'd be preferable to have the "time" argument listed first.
Is it just a matter of notation and convetion or is there a reason for writing [math]X[/math] first in the cartesian product?
Anonymous at Mon, 10 Feb 2025 17:33:10 UTC No. 16581598
>>16581457
>Is it just a matter of notation and convetion or is there a reason for writing X first in the cartesian product?
The former. No reason
Anonymous at Mon, 10 Feb 2025 19:51:22 UTC No. 16581779
>>16581222
>>16581312
Ok, so I looked into it and itās an epimorphism in the category theory sense (the arrow definition), but not an surjective epimorphism as one might expect. Apparently, this is not a āregularā epimorphism because itās not coequalizer of the two arrows used in the category-theoretic definition. But then you could have just mentioned that, asshole. Nobody cares about āirregularā category-theoretic epis or monos as far as I know.
Anonymous at Mon, 10 Feb 2025 20:09:21 UTC No. 16581805
>>16581779
>because itās not coequalizer of the two arrows used in the category-theoretic definition
What do you mean by that? What two arrows?
Anonymous at Mon, 10 Feb 2025 21:07:32 UTC No. 16581859
>>16563123
based oldfag
Anonymous at Mon, 10 Feb 2025 22:30:09 UTC No. 16581935
>>16576185
The ultra rich take risks and it pays off. For every wealthy person there's a lot more that failed despite being smart. It's better to just do what you love and Tao gets paid well enough I'm sure to be comfy. Why risk it all? For what? To run some startup and exploit people for financial gain and make more products the world doesn't need?
Anonymous at Mon, 10 Feb 2025 22:46:41 UTC No. 16581957
>>16581805
The two arrows that need to be equal for a morphism to be an epi (or dually a mono). The same arrows used in the equalizer/coequalizer definitions.
Anonymous at Tue, 11 Feb 2025 00:40:22 UTC No. 16582023
>>16569760
Geometric algebra
Linear algebra
Calculus of manifolds, numeric methods of
Anonymous at Tue, 11 Feb 2025 00:42:09 UTC No. 16582026
>>16570825
I reported you to DEIA, Brian.
Anonymous at Tue, 11 Feb 2025 04:56:32 UTC No. 16582167
Somewhat related to the ongoing discussion:
Apparently monos, injections and regular monos (arrows that are the equalizer of some pair of parallel arrows) all coincide in the category of sets (and similarly for epis, surjections and regular epis).
I'm familiar with the proofs that regular monos are indeed monic (in any category) and that in the category of sets, the monos are precisely the injections.
Combining the former theorem with one direction of the latter theorem tells me that a regular mono in Set is an injection.
I'm struggling with the converse (and would appreciate a hint): Given an injection I would like to show that it's an equalizer of some diagram, but I can't quite up with another set and two functions that make this work...
Anonymous at Tue, 11 Feb 2025 13:22:17 UTC No. 16582445
>>16582167
>but I can't quite up with another set and two functions that make this work...
Because that statement is false. A regular morphism is a stronger notion. Regular monos/epis are universal properties ie there's a notion of uniqueness to them. Some random injection from one set to another doesn't have anything "special" about it.
Generally speaking, if you want counterexamples to your "set-like" intuition about categories, then always look at topological spaces. They fuck up your set-theoretic intuition. It's the simplest example of a category where a morphism that is both epi and mono isn't an iso (we require an explicit inverse continuous map for homeomorphisms). For regular vs "usual" monos/epis, Top has topological embeddings vs just injective continuous maps. The latter isn't a very useful notion in Top, while the former is gives the usual notion of a topological subspace. This is why it makes more sense to define a subobject of an object as an object with a regular monomoprhism to its superobject rather than a generic monomorphism. This anon >>16580170 provided a different example in the category of rings.
Anonymous at Tue, 11 Feb 2025 13:37:06 UTC No. 16582462
>>16582445
Every mono/epi is regular in Set
Anonymous at Tue, 11 Feb 2025 13:49:39 UTC No. 16582477
>>16582167
Given a mono f:X -> Y take the disjoint union of two copies of Y and quotient by the relation identifying the parts of the copies that are in the image of f. The two functions are the ones taking Y to its first copy and its second copy.
Anonymous at Tue, 11 Feb 2025 14:33:34 UTC No. 16582507
>bidirectional statement
>prove a implies b
>half an a4 page, an hour of work, but it all comes together in the end and flows logically
>feel a warm and fuzzy feeling in my heart
>now prove b implies a
>1 line, a few equivalences, 5 minutes of work
>huh? wtf
>the line of thinking works for a implies b
>fuck
Anonymous at Tue, 11 Feb 2025 14:35:54 UTC No. 16582509
>>16582507
>>huh? wtf
Huh yourself? Equivalences with one trivial and one nontrivial direction are very common.
Anonymous at Tue, 11 Feb 2025 14:39:34 UTC No. 16582514
>>16582509
He said that both directions are trivial but it wasn't obvious until he started doing the second direction. Nice reading comprehension.
Anonymous at Tue, 11 Feb 2025 15:26:30 UTC No. 16582567
>>16582514
I got that, but he hadn't realized that yet at that part of the greentext. So the "huh wtf" was not warranted. Nice reading comprehension.
Anonymous at Tue, 11 Feb 2025 17:25:41 UTC No. 16582704
>>16582509
just say haha nice meme man
not everything is an argument
Anonymous at Wed, 12 Feb 2025 00:12:43 UTC No. 16583232
>>16582567
>he hadn't realized that yet at that part of the greentext.
nuclear grade autism
Anonymous at Wed, 12 Feb 2025 09:16:40 UTC No. 16583483
How do I learn to plot an Euler / clothoid curve (using Python)? I canāt understand the math behind it and every other resources seems to have a different definition of how x and y coordinates are defined for the bend.
Anonymous at Wed, 12 Feb 2025 10:38:00 UTC No. 16583516
How do i become not shit at math? I can't get past multivariable calculus. I don't even need it but i took anyways cause
>just finish all le math you already started em anyways.
Anonymous at Wed, 12 Feb 2025 13:24:55 UTC No. 16583598
What is the most important constant between 1 and 2? sqrt(2)? Golden ratio? Zeta(2)? Zeta(3)? Something entirely different?
Anonymous at Wed, 12 Feb 2025 13:44:25 UTC No. 16583623
>>16583598
When you grow up, you realize that what is āimportantā or isnāt depends on the context. Is zeta(2) important? To someone working with Bose-Einstein condensates it is. It shows up in some niche electrodynamics BVPs. But to someone working with finite groups (combinatorics) it isnāt, because it never shows up.
Anonymous at Wed, 12 Feb 2025 13:52:38 UTC No. 16583636
>>16583598
Zeta(3) for the troll.
Anonymous at Wed, 12 Feb 2025 14:06:11 UTC No. 16583647
>>16583623
>depends on the context
Would you kindly fuck off with postmodernism? Regardless of context the most important constant between 3 and 4 is pi. Regardless of context the most important constant between 2 and 3 is e. And now, between 1 and 2...
PS: Formula for a critical temperature od Bose-Einstein condensate features Zeta(3/2).
Anonymous at Wed, 12 Feb 2025 14:15:03 UTC No. 16583666
>>16583647
>muh postmodernism
you must be 18+ to post, kiddo. Clean your room.
Anonymous at Wed, 12 Feb 2025 14:26:03 UTC No. 16583671
>>16583598
probably pi/2
Anonymous at Wed, 12 Feb 2025 15:52:42 UTC No. 16583721
>>16583647
>Regardless of context the most important constant between 3 and 4 is pi.
what about 1.4e
>Regardless of context the most important constant between 2 and 3 is e.
what about pi/1.4
Anonymous at Wed, 12 Feb 2025 19:04:15 UTC No. 16583884
>>16583721
Neither of those is very important
Anonymous at Wed, 12 Feb 2025 19:34:00 UTC No. 16583899
>>16583884
Then pi is not important either.
Anonymous at Wed, 12 Feb 2025 19:52:34 UTC No. 16583914
>>16562509
I'm gonna get massively (justifiably) shit on for this but I didn't want to make an entire thread. I need input from people who are experts in this area before I dedicate thousands of hours to it.
I have a day job making good money, not at all related to STEM. I'm not necessarily looking for a career change. However, I enjoyed math in school though I never had to do anything proof based or rigorous. I did calculus sequence (1-3?) and got A's easily, but never had to do beyond that.
I need you to be completely real with me. Is math worth self studying to a deep level as a hobby? Could applied maths or pure maths knowledge / Masters have enough translational benefit to REALLY enhance my life or even become an alternative career eg data scientist, actuary?
I'm having a bit of an identity crisis and am looking for some hobby to be stimulating. However, if I can't hope to catch up to asian olympiad kids with private tutoring or possibly make money doing it, I don't want to turn my small interest in math puzzles and math youtube videos into 5,000 hours of wasted time just to be mediocre/no career or financial benefit.
TLDR I like math as a layperson but don't LOVE it. Is math education worth pursuing, or its saturated with H1b geniuses already so quantitative careers are a non-starter for adult learners.
If you guys love advanced/pure math just for sake of mental masturbation and having a sense of superiority over 'midwits', more power to you. But I need something with this much mental effort to produce real RESULTS, at least over a 5+ year time frame.
Anonymous at Wed, 12 Feb 2025 20:19:51 UTC No. 16583946
>>16583914
>I need you to be completely real with me. Is math worth self studying to a deep level as a hobby?
Only if you enjoy it. I have a job and self study it as a hobby but I have no illusions about it ever becoming useful to anything.
>Could applied maths or pure maths knowledge / Masters have enough translational benefit to REALLY enhance my life or even become an alternative career eg data scientist, actuary?
Being a data scientist and an actuary requires very little math afaik.
>I'm having a bit of an identity crisis and am looking for some hobby to be stimulating
Math is a great hobby for that.
Basically, if you enjoy pure math, go for it, but have no illusions about it ever becoming useful. Very little math is required for for quantitative careers. If you want to do those just learn the requirements directly.
Anonymous at Wed, 12 Feb 2025 20:24:48 UTC No. 16583952
>>16583946
Thank you for your honest input.
I think my problem is FOMO with time usage. Say the 10k hours I put into math I instead put into, idk, martial arts? I'm unlikely to ever get into a serious fight, but it has translational benefits eg health and fitness, confidence etc. Maybe math has enough cognitive benefits (brain aging) to justify the time even if nothing else? Personal satisfaction? I don't know. Maybe I'm just depressed. Arrival fallacy is a bitch. I have my 'shit together', and now I'm stuck with contemplation and 'is this it?'
Anonymous at Wed, 12 Feb 2025 23:51:45 UTC No. 16584119
>>16583914
>Is math worth self-studying to a deep level as a hobby?
>Could math [...] become an alternative career?
These are very different questions. If you go to a serious job interview and explain that you learnt math in your spare time you will probably be looked at funny.
Also, there's enough applicable and applied math available to study to fill a lifetime.
>So is it worth it?
How should I know? It depends on what you want, and you seem to be worried about some strange things.
>if I can't hope to catch up to "asian olympiad kids"
You can't.
>or make money doing it
Probably not, if you have no idea where to look for it.
>I don't want my small interest in math to turn into 5000 hours of wasted time
So, don't? If you have enough of an interest to do some puzzles, maybe just buy one book on a topic which seems interesting, and work through it. That seems like a decent middle ground, no? I recommend probability.
Anonymous at Thu, 13 Feb 2025 00:55:13 UTC No. 16584169
https://en.wikipedia.org/wiki/Knast
>For a, b in L we write [a, b] for the closed interval with bounds a and b: {x ā L | a ā¤ x ā¤ b}. If a ā¤ b, then āØ[a, b], ā¤ā© is a complete lattice.
why?
Anonymous at Thu, 13 Feb 2025 01:52:13 UTC No. 16584215
>>16584169
nevermind i found a proof
Anonymous at Thu, 13 Feb 2025 09:13:02 UTC No. 16584474
>>16584119
>that you learnt math in your spare time you will probably be looked at funny.
Why?
Anonymous at Thu, 13 Feb 2025 16:57:45 UTC No. 16584886
I think I stumbled onto a way to derive the normal distribution, or at least "inspire" it, that I haven't seen before.
I've been making a stat tracker for a game I play, and I've noticed some obvious outliers from when I first started as opposed to the current trend.
Obviously I could pick them out but I was bored so I decided to try and weight the data based upon how well it fit the mean.
I figured taking each instance of the data and finding it's squared distance from the mean would be useful, and I decided to normalize it with an arbitrary constant.
I then figured that for this weighting function, I'd want the limit toward 0 to be 1, and toward infinity to be 0.
I ended up using a negative exponential with my squared distance as an argument, and I realized that I had a gaussian.
Nothing groundbreaking obviously, but I had fun
Anonymous at Thu, 13 Feb 2025 22:35:51 UTC No. 16585200
Should I study something "fun" rather than something more focused to my career?
Thinking robotics over embedded IoT
Anonymous at Fri, 14 Feb 2025 01:08:36 UTC No. 16585345
Iām trying get kicked off the phone
Anonymous at Fri, 14 Feb 2025 06:17:05 UTC No. 16585501
>>16585111
Tits or gtfo
Anonymous at Fri, 14 Feb 2025 10:59:30 UTC No. 16585656
>>16584474
Because very often, when someone says they learnt a subject on their own, there's something the size of texas missing. If mathematics is an important part of the job, whoever's hiring won't want to verify your knowledge himself, and just expect a university to have already done that.
If maths isn't important to the job, you're just "the guy with the weird hobby".
Anonymous at Fri, 14 Feb 2025 18:02:09 UTC No. 16585883
>>16585501
You prove the relationship between integral and derivative from memory
Anonymous at Fri, 14 Feb 2025 18:08:19 UTC No. 16585885
>>16581882
Yeah, it's a bit messy. The only things I've found useful is knowing what a Hausdorff space is and how Urysohn's lemma works.
Anonymous at Sat, 15 Feb 2025 08:35:34 UTC No. 16586452
>>16583914
>I enjoyed math in school though I never had to do anything proof based or rigorous
repliers missed the important part of the post
math is completely different after you introduce proof
you don't even know if you like math until you try proof
Anonymous at Sat, 15 Feb 2025 14:15:37 UTC No. 16586659
>>16586452
Not just that. There's also research level math which is again completely different from the undergraduate proof-based math.
Anonymous at Sat, 15 Feb 2025 14:23:15 UTC No. 16586667
>>16586659
it's not as big a change though, the end of undergrad can be pretty close to research
high school math however is just calculation
Anonymous at Sat, 15 Feb 2025 14:39:03 UTC No. 16586681
>>16562509
2+2=4
Quick Maths
Anonymous at Sat, 15 Feb 2025 14:55:49 UTC No. 16586712
>>16586667
Not really. Undergrad math is very elegant. At the end of undergrad, you'd be doing what? Stuff that has largely been done in early 20th century. But current math is just extremely ugly, and disorganised. You would simply have to take results from subjects you know nothing about at face values. What you'd be doing would seem to have absolutely no correlation with real life. You'd be reading 10 page long proofs on results that you have no idea why anyone would care about. I mean even stuff like functional analysis, differential geometry and whatever people learn at the late undergrad level is still fairly motivated and organised, but bleeding edge math is just absurdly soul crushing. That is, unless you are number theory autist.
Anonymous at Sun, 16 Feb 2025 00:34:46 UTC No. 16587405
Theorem: there exists an injective function from A to B if and only if there exists a surjective function from B to A.
Now, there is a rather trivial proof of this theorem if we go by the usual fiber definitions of surjective and injective. However, that proof relies on the Axiom of Choice. Is there a proof that doesn't rely on the choice function?
Anonymous at Sun, 16 Feb 2025 06:24:56 UTC No. 16587634
>>16587405
If A is empty then there's always an injection from A to B but there's only a surjection from B to A if B is also empty. So you need an exception for that. With that exception, it's provably equivalent to the axiom of choice.
Anonymous at Sun, 16 Feb 2025 07:04:50 UTC No. 16587644
>>16587405
The forward direction of that theorem is simply true in classical mathematics (for inhabited A that is, otherwise the empty function into an inhabited codomain is an injection but there are no functions at all going the other way). The backwards direction is known as the partition principle (PP).
PP requires choice in the sense that it's not provable in ZF, whether PP actually implies AC is still an open question.
Funnily enough, the forward direction in its full generality is constructively equivalent to excluded middle, so neither direction is quite as innocent as it looks
Anonymous at Sun, 16 Feb 2025 14:01:48 UTC No. 16587892
>>16587634
>If A is empty
I HATE VACUOUS STATEMENTS
I HATE VACUOUS STATEMENTS
Anonymous at Sun, 16 Feb 2025 14:09:54 UTC No. 16587902
>>16587892
based actually
reject pedantry
Anonymous at Sun, 16 Feb 2025 14:11:34 UTC No. 16587906
>>16587644
>The forward direction of that theorem is simply true in classical mathematics
It still requires the choice function, I think. You need to specify where the elements outside of the range go. So you need to pick some distinguished element of A where you send all elements outside the range.
Anonymous at Sun, 16 Feb 2025 14:26:53 UTC No. 16587916
>>16587906
You need to pick some element of A to send those elements that lie outside the image to, yes.
But that's only making one singular choice and essentially boils down to eliminating an existential quantifier (namely from the assumption that A is inhabited) rather than invoking choice.
More generally, making only finitely many choices doesn't require AC for the same reason.
Anonymous at Sun, 16 Feb 2025 14:33:37 UTC No. 16587918
>>16587916
>essentially boils down to eliminating an existential quantifier (namely from the assumption that A is inhabited) rather than invoking choice
>making only finitely many choices doesn't require AC for the same reason.
Please, elaborate. In my limited understanding, the difference between the choice function and quantifiers is that the latter cannot specify a unique element where the elements of B will be sent. Even with unique existential quantifiers, itās simply a shorthand for āif there exist a,b then a=bā, but it never actually specifies what those elements are.
Anonymous at Sun, 16 Feb 2025 14:39:51 UTC No. 16587923
Ok, I think I got it. We can phrase the forward direction as āsince A is nonempty, there exists a unique a in Aā. And then we send everything outside the range to a.
But in the backwards direction, we have to pick individual elements from every fiber. But then the same trick can be done as aboveā¦ My head hurts. I donāt get it.
Anonymous at Sun, 16 Feb 2025 15:06:52 UTC No. 16587954
>>16587918
>>16587923
We don't need uniqueness to justify eliminating an existential quantifier.
Whenever you have a premise of the form [math]\exists x\varphi(x)[/math] you'd usually proceed by introducing a fresh variable, let's say [math]y[/math] (one can also choose [math]x[/math] itself if it doesn't occur free in any other active assumption). You'd then add [math]\varphi(y)[/math] to your list of assumptions and continue proving, at some point ending up at a conclusion [math]\psi[/math] that doesn't contain [math]y[/math]. Existential elimination then says that you may conclude with [math]\psi[/math] altogether.
Think about how you'd prove something elementary like "the successor of an even number is odd" and how you treat the existential quantifier in the definition of even.
What is a little more complicated is when you have not just one set you're supposed to pick an element from, or two, or three... (which would all just boil down to n many existential eliminations), but you're given a general (possibly infinite) family of sets (all of which are inhabited of course) and you're required to pick one element out of each one.
If you know that all these sets are actually singletons, i.e. they contain exactly one element, then this doesn't require choice (this is sometimes called principle of unique choice/non-choice or function comprehension principle). In classical set theory this is simply true by definition (because functions are identified with functional relations). This is the reason behind why the theorem "every bijection has an inverse" doesn't require AC (because every fiber of a bijection is a singleton). Indeed that principle is equivalent to every bijection being a set isomorphism (similarly to how AC is equivalent to every surjection being right-invertible).
Anonymous at Sun, 16 Feb 2025 15:11:06 UTC No. 16587958
>>16587954
I kinda lost the plot here and forgot to actually answer the last question, sorry.
The reason why the backwards direction of the original theorem requires choice is because the fibers of the surjection are just a plain family of inhabited sets. We don't know that this family is finite so we can't just repeatedly apply existential elimination a finite amount of times and since we're given a mere surjection rather than a bijection the fibers aren't necessarily singletons, so there are no canonical choices to make.
So all we're left is to invoke AC
Anonymous at Sun, 16 Feb 2025 16:13:28 UTC No. 16588013
>>16587958
>The reason why the backwards direction of the original theorem requires choice
I understand that (we have a possibly infinite family of fibers), but the forward direction is still murky to me. You say
>Think about how you'd prove something elementary like "the successor of an even number is odd" and how you treat the existential quantifier in the definition of even.
but this isn't a constructive statement. We invoke the axiom of schema of specification to specify that we're talking about even numbers and not all naturals. Whereas in the forward direction of the original statement, the element of A that we send all the elements of B outside the range is non-constructive. I cannot specify that element for an arbitrary set using a logic formula, so I'm forced to pick via AC. Am I missing something here?
Anonymous at Sun, 16 Feb 2025 16:14:29 UTC No. 16588016
>>16588013
*but this is a constructive statement (via axiom of schema of specification)
Anonymous at Sun, 16 Feb 2025 16:55:10 UTC No. 16588057
>>16588013
I'm afraid we're talking past each other a little bit, or at least I'm not quite following what you're trying to say.
For example, I'm not sure what your usage of the terms constructive and non-constructive are referring to here, as it doesn't agree with how these would usually be used (what is a constructive statement? what do you mean when you say that an element is non-constructive?)
Discussing constructive validity (in the precise sense) of the forward direction of your theorem isn't completely uninteresting as it's not provable in intuitionistic set theories (like mentioned before, it is indeed equivalent to excluded middle). But that has very little to do with picking an element out of A (rather, it's because the image of f isn't generally a decidable subset of B, so a piecewise "function" defined based on being inside the image isn't necessarily total and hence not necessarily a function at all).
Similarly, Specification doesn't really have anything to do with this.
If I understand correctly you're confused as to why one can just pick an element out of A without AC and then proceed through the rest of the proof.
Rather than going on about why this works (again, because that's exactly what existential elimination allows us to do), my question would be how you've tackled other theorems in the past that also contain an existential statement in its premise?
For another example, how would you go about proving something like "If [math]X[/math] is inhabited then [math]\bigcap X\subseteq\bigcup X[/math]"?
Just like in your theorem, one would pick an [math]x\in X[/math] (since [math]X[/math] was assumed to be inhabited) and then go on with the proof. Would you say that this requires choice too?
Anonymous at Sun, 16 Feb 2025 17:06:54 UTC No. 16588072
>>16588057
Ok, so I think there is a lot of semantics going on in this discussion, which is why most mathematicians stray away from ZFC and these foundational results. It's very confusing for someone who doesn't work in predicate logic.
>Just like in your theorem, one would pick an xāX (since X was assumed to be inhabited) and then go on with the proof
My understanding is that in these kinds of proofs we are implicitly saying "[math]\forall{x}\in{X}[/math]" when we say "let [math]{x}\in{X}[/math]". At least this is the kind of intuition one has when doing analysis proofs. There is a hidden universal quantifier.
However, in the forward direction, this would be incorrect. We cannot define a function g by saying "this element maps to some arbitrary element" or, more formally, [math]\forall{b}\in{B}\forall{a}\in
So my suggestion is to stay on topic instead of going on these tangents. How would you explicitly construct a surjective function from B to A given an injective function from A to B using existential quantifiers only?
Anonymous at Sun, 16 Feb 2025 17:08:55 UTC No. 16588076
>>16588072
to make it clear, by
>hidden universal quantifier
I mean
>implicit universal quantifier
Anonymous at Sun, 16 Feb 2025 17:22:55 UTC No. 16588093
>>16588057
Ok, I think I'm starting to get it now. Here's a sketch of a proof:
Let f:A->B be an injective function. Since A is nonempty, there exists a unique a in A. Define a function from B to A as follows. Send be to an element of its fiber under f if that fiber is nonempty. Otherwise, send that element to a. This function is surjective by construction.
I think I got it now. Thanks for these hints!
Anonymous at Sun, 16 Feb 2025 17:35:25 UTC No. 16588108
>>16588072
You already know the textbook proof of your theorem and I'd just present it to you again, so that alone probably won't clear this up.
You can kinda sorta think of the existential quantifier as a generalized disjunction (this is a bit of a lie but it works well enough here). Now the way to use a disjunction that's given as a premise is to write to subproofs, both of them ending with the same conclusion.
I.e. if we know that [math]P_1\lor P_2[/math] holds and we know that [math]P_1[/math] entails [math]R[/math] and also that [math]P_2[/math] entails [math]R[/math], then we can conclude with [math]R[/math] altogether.
For example if your theorem instead read something like "If [math]0\in A[/math] or [math]1\in A[/math] then if there is an injection [math]A\to B[/math] there mist be a surjection [math]B\to A[/math]", then the way to prove it would be to supply to subproofs, one assuming [math]0\in A[/math] and one assuming [math]1\in A[/math], but both of these concluding with the "if there is an injection then there is a surjection" stuff.
You probably already know what these subproofs would look like concretely. In both of them the surjection is constructed by mapping a [math]b[/math] in the image of [math]f[/math] to its unique preimage and those outside are mapped to [math]0[/math] in the first subproof and to [math]1[/math] in the second.
Anonymous at Sun, 16 Feb 2025 17:41:29 UTC No. 16588122
>>16588108
Now to generalize from disjunction to existential quantifier: When we were given a binary disjunction we provided two subproofs, both ending with the same conclusion. For a ternary disjunction we'd provide three subproofs and so on. The real way to generalize this to simply provide only one subproof that works for all cases simultaneously.
So when given an existential statement like [math]\exists P(x)[/math], we should look for a proof that [math]P(x)[/math] entails [math]Q[/math]. To ensure that this one subproof works for all cases, the conclusion [math]Q[/math] should not mention the variable [math]x[/math] (just like the conclusion in each subproof for a disjunction remained the same!).
Then existential elimination says that [math]Q[/math] holda altogether.
Now for your problem let [math]P(a)[/math] be the statement [math]a\in A[/math] and [math]Q[/math] the statement "if there is an injection then there is a surjection".
You're given that [math]A[/math] is j inhabited, i.e. that [math]\exists a P(a)[/math] holds. You want to prove [math]Q[/math] and notice that [math]Q[/math] does not contain the variable [math]a[/math].
So if we now provide one of those "universal subproofs" that [math]P(a)\to Q[/math] then we are done.
So you prove that there is surjection given that there is an injection and that [math]a\in A[/math]. And the subproof will look just like the previous ones except that all the [math]b[/math] outside the image are mapped to our one fixed [math]a[/math] (whatever [math]a[/math] may be).
Anonymous at Sun, 16 Feb 2025 17:52:45 UTC No. 16588141
>>16588093
This is mostly correct except for two details
>Since A is nonempty, there exists a unique a in A
No, there exists at least one a in A but there may be many others. But that's not a problem and the rest of the proof goes through since it works with any one a in A.
>Send b to an element of its fiber if that fiber is nonempty
This is where we need to use the fact that f is injective. Since f is injective the fibers of f contain at most one element, so the fibers that are nonempty contain at least one element and at most one element, hence exactly one element.
So we map those b to that *unique* element in its fiber. If we didn't know that f was injective then there could possibly be more than one element in b's fiber and we'd actually have to invoke choice here to choose one element out of each fiber. But this is not necessary here.
Anonymous at Sun, 16 Feb 2025 18:13:23 UTC No. 16588170
>>16588141
>No, there exists at least one a in A but there may be many others.
True, uniqueness would make A a singleton. I was just trying to make sure that the function maps to a unique element, but I see why it would. Existence of a implies {a} also exists and we can make the uniqueness statement in the definition of a function via the uniqueness of a in the singleton {a}.
Anonymous at Mon, 17 Feb 2025 05:20:11 UTC No. 16588819
>>16562509
Alright so like , you know how in math stuff can get simplyfied.
I am trying to do the oposite for a personal project and try to bluid a system to complexify in the context of boolean algebra and to write it by hand
But I am having troubles defining what it is to complexify.
Origininally I wanted to like just see it as putting all posible configurations if the equation were to actualy happen and be fine.
Like f=y+x'+yx'
But like f= x'+y+i could be like f=x'+y+i+iy+x'i+x'y+x'yi
But then I arealised adding yix' whould make the ecuation more complex.
And I was introduced to the idea of rounding up from myself but then if everythibg that will make the page larger must count then even something as simple as 2+2 maximised whould be
(2+2) +(*1*0)+(*2*0)+(*3*0)+(*4*0).......
And now I am sad because this was suppused to be the last problem of a personal project that I really needed to complete and not only I can't define maximising as a concept but I think I forgot how to do bolean algebra because of it.
Can any of you anons save me how can I define anti simplification in a logical way but still posible to write with a pencil way.
Anonymous at Mon, 17 Feb 2025 05:28:23 UTC No. 16588825
>>16588819
I am going crazy over this , I was so close to finnaly get on my mission.
My plan. But I need to deal with this tecnicallity.
Is just the restriction by hand is very bad for this , although I might be able to salvege it with my new writing number system regardless maybe?
Does anyone get me or do I sound like a drugie right now.
Anonymous at Mon, 17 Feb 2025 08:07:20 UTC No. 16588935
>>16588819
I don't think there's a useful way to do this, sorry. Could there be another way to finish your project? What problems are you having with boolean algebra?
Anonymous at Mon, 17 Feb 2025 12:15:19 UTC No. 16589143
>[eqn]\underset{\boldsymbol {\beta}}{\arg\min}L(Y,\hat Y)[/eqn]
here is your ai bro
Anonymous at Mon, 17 Feb 2025 18:26:53 UTC No. 16589409
Why are compact spaces defined in such a peculiar way (space is compact iff every open cover of space has a finite subcover)?
Anonymous at Mon, 17 Feb 2025 19:09:18 UTC No. 16589444
>>16589409
https://arxiv.org/abs/1006.4131
Anonymous at Mon, 17 Feb 2025 20:40:01 UTC No. 16589520
>>16581882
You don't study separation axioms in a vacuum by themselves, you learn about them and their uses in some specific context (e.g. locally compact Hausdorff spaces or CW-complexes).
Anonymous at Mon, 17 Feb 2025 22:08:01 UTC No. 16589592
Where is the new thread?
Anonymous at Mon, 17 Feb 2025 22:31:34 UTC No. 16589629
>>16589592
Here:
>>16589624