Anonymous at Tue, 17 Dec 2024 04:04:59 UTC No. 16515492

First for Hugh A. Thurston's Differentiation and Integration
https://archive.org/details/differentiationi0000thur

Anonymous at Tue, 17 Dec 2024 04:19:57 UTC No. 16515514

>inb4 Verbitsky fag
His Trivium is more useful than that late 20th century string theorist curriculum:
>>16493957
>>16493959

Anonymous at Tue, 17 Dec 2024 04:40:17 UTC No. 16515541

any grad school or beyond chads here?

could you guys recommend the fastrack way to get up to speed on bachelor's level math? Kind of like the pareto-principle of math curriculum? I am a graduated EE fag peasant but want to get into grad school for statistics PhD. Heard they require a lot of background in pure math.

Anonymous at Tue, 17 Dec 2024 04:44:29 UTC No. 16515545

>>16515541
Follow anon's suggestion. Feel free to skip (only) Lang:
>>16509496

Anonymous at Tue, 17 Dec 2024 05:37:58 UTC No. 16515588

>>16515545
>fudding lang

Anonymous at Tue, 17 Dec 2024 05:42:37 UTC No. 16515589

>>16515541
you don't need most of the shit in a math bachelor. I assume you already did calculus and linear algebra. So if you can top that with some real analysis and measure theory you should have all you need.

Anonymous at Tue, 17 Dec 2024 11:50:55 UTC No. 16515771

>>16502194
What is their twitter? I am curious.

Anonymous at Tue, 17 Dec 2024 13:00:14 UTC No. 16515813

>>16515589
I think the basics of bachelor's level group theory end up being quite helpful in statistics later. As someone who did an EE bachelor's and is doing a statistics oriented PhD (technically still EE but the program I am going through requires real analysis, measure, smooth manifolds and convex analysis), the main thing I wish I had more background in from undergrad is abstract algebra. I don't need it enough to spend time taking a grad level course on it, but having some basics of group theory, rings fields and modules would be helpful as well.

Anonymous at Tue, 17 Dec 2024 14:22:21 UTC No. 16515848

Three funny names of international authors a found recently:
>Dingding Dong
http://math.uchicago.edu/~may/REU2017/REUPapers/Dong.pdf
>Semën S. Kutateladze
https://ncatlab.org/nlab/show/Sem%C3%ABn+S.+Kutateladze
>Arjum Nigam
http://math.uchicago.edu/~may/REU2022/REUPapers/Nigam.pdf

Anonymous at Tue, 17 Dec 2024 19:25:13 UTC No. 16516083

>>16515813
Are you doing a PhD in statistical signal processing by any chance?

What core pure math curriculum courses would you recommend that is obviously not covered in EE undergrad curriculum?

Anonymous at Tue, 17 Dec 2024 19:26:15 UTC No. 16516085

>>16515545
>>16515589
Thanks for the suggestions anons.

Anonymous at Tue, 17 Dec 2024 19:29:42 UTC No. 16516088

>>16515541
>Follow-up question
Is it advisable to learn and practice International Math Olympiad level math and problem solving skills for someone already graduated? Does it help to excel in grad school?

Anonymous at Tue, 17 Dec 2024 21:02:22 UTC No. 16516184

norman wildberger's youtube is working again

Anonymous at Tue, 17 Dec 2024 22:50:04 UTC No. 16516324

>>16516184
thanks norm

Anonymous at Wed, 18 Dec 2024 01:27:53 UTC No. 16516458

>>16516184
Why was it down?

Anonymous at Wed, 18 Dec 2024 01:29:09 UTC No. 16516460

>>16516458
He got hackerd

Anonymous at Wed, 18 Dec 2024 01:35:13 UTC No. 16516466

>>16516083
Yes statistical signal processing and non-linear optimal estimation.

Ultimately, a good 90% of statistical signal processing is answering two questions "Can you find the eigenvalues/eigenfunctions?" and "Can you figure out how good/bad your estimates/decisions are?" Those are deceptively simple concepts in question, but can very quickly lead you into deep waters when your convergence requires bounding random complex eigenvalues. If you get into sensor array design, you basically need to act as a translator between graduate pure-math manifolds textbooks and your applied tasks in R^n.

At an undergrad level? Take a proper real analysis course, an introductory functions of complex variables/analytic functions course, and a proper mathematics department probability theory course. If you have the ability to take one, I'd also recommend a course in measure theory, because convergence of statistical functionals will make a lot more sense of you understand the basics of measure.

I'm assuming you've also done a linear algebra, ODE's, and some kind of signals and systems/Fourier analysis course. If you haven't, you should do that.

Anonymous at Wed, 18 Dec 2024 10:02:55 UTC No. 16516888

For some mesurable space [math](X,T,\mu) [/math] I'm considering the sigma algebra induced by the set [math]N [/math] of all elements of [math]T [/math] whose measure is 0. I've found that if the measure of [math]X [/math] is finite then the induced sigma algebra is the union of [math]N [/math] and the set of complements of the elements of [math]N [/math].
Things get messy if the measure of [math]X [/math] is not finite however, does my union in the finite case still work as the induced sigma algebra in the infinite case?
For [math]A_{1},A_{1}\in N [/math], [math] X\A_{1}\cup X\A_{2}=X\A_{1}\cup (X\A_{2}\X\A_{1})[/math]. The measure of the left hand side is the measure of X, the measure of X\A is also the measure of X but the last term could be anything. Any pointers?

Anonymous at Wed, 18 Dec 2024 11:11:57 UTC No. 16516921

If I want to visually represent a complete graph with n nodes in such a way that all edges are of the same euclidean length, can I only do so in n-1 dimensions? What's this called and where can I read about it?

Anonymous at Wed, 18 Dec 2024 12:33:25 UTC No. 16516955

Is there a name for this type of curve?

It's simlar to an ellipse: You have two points and a given angle t, and you want all points that when you connect them to these two points, they make the angle t.
For a right angle, the curve would just be a circle (with the two points just being antipodal points on it).

Anonymous at Wed, 18 Dec 2024 12:37:46 UTC No. 16516956

Nevermind >>16516955, I'm retarded, it's just a circle (sort of)
https://en.wikipedia.org/wiki/Inscribed_angle

Anonymous at Wed, 18 Dec 2024 12:43:21 UTC No. 16516962

>>16516955
It's still a circle just with the two points forming a chord.

Anonymous at Wed, 18 Dec 2024 17:43:46 UTC No. 16517215

>>16516466
Isn't statistical signal processing being quickly phased out in lieu of machine learning? Basically spamming compute power in Bayesian statistics instead of wasting brain power in classical statistics?

>At an undergrad level? Take a proper real analysis course, an introductory functions of complex variables/analytic functions course, and a proper mathematics department probability theory course. If you have the ability to take one, I'd also recommend a course in measure theory, because convergence of statistical functionals will make a lot more sense of you understand the basics of measure.
I am not an undergrad anymore but going on to do my master's in signal processing. I was aiming to do courses similar to picrel (marked ones). Am I covered?
I have done signals and systems, linear algebra, and ODEs yes.

Anonymous at Wed, 18 Dec 2024 17:45:13 UTC No. 16517217

>>16517215
forgot to include picrel

Anonymous at Wed, 18 Dec 2024 18:20:18 UTC No. 16517251

>need to get a perfect score on my final exam to get an A in the course
>would have to get a 0 to fail the course

Anonymous at Wed, 18 Dec 2024 18:56:57 UTC No. 16517282

>>16517251

Anonymous at Wed, 18 Dec 2024 19:13:31 UTC No. 16517295

>>16515848
>Dingding Dong
https://youtube.com/watch?v=vzkp10zbgMA

Anonymous at Wed, 18 Dec 2024 19:14:53 UTC No. 16517298

im a undergrad first year and fucked up first quartile (Linear Algebra 1, Calculus, Set Theory)
any tips man?
In 4 weeks Ill do resits aswell as Analysis, linear algebra 2 and Programming. Probably wont pass LinAlg2.

Anonymous at Wed, 18 Dec 2024 19:26:18 UTC No. 16517305

>>16516088
>Is it advisable to learn and practice International Math Olympiad
“mostly no”, in the sense that the benefits are not that direct, and you are probably busy. instead, learn pure-mathematics-style problem-solving through material such as real analysis, group theory, and the other suggested core pure math courses

but if you enjoy em, by all means do olympiad problems too

Anonymous at Wed, 18 Dec 2024 19:34:43 UTC No. 16517313

>>16517305
I mean, I do enjoy them but it's more of a regret that I was too poor to know about these competitions when I was in HS. I would only like to seriously do them now if it would actually help with problem-solving skills in advanced math courses.

Anonymous at Wed, 18 Dec 2024 20:09:30 UTC No. 16517344

>>16516888
You have [math]N = \{A \in T:\mu(A)=0\}[/math], and [math]S=\{A \in T:\mu(A)=0\text{ or }\mu(A^c)=0\}[/math].
Since [math]N \subseteq S[/math], and [math]S[/math] is a sigma algebra, also [math]\sigma(N) \subseteq S[/math].
The other inclusion is pretty easy as well.
What are you worried about, precisely?
Maybe you should try to show [math]S[/math] is indeed a sigma-algebra.

Anonymous at Wed, 18 Dec 2024 22:55:38 UTC No. 16517485

How many shots can you do before you start fucking problems up?

Anonymous at Thu, 19 Dec 2024 01:50:02 UTC No. 16517599

>>16517215
> Isn't statistical signal processing being quickly phased out in lieu of machine learning? Basically spamming compute power in Bayesian statistics instead of wasting brain power in classical statistics?

There are people who approach problems this way. In general I think it's a bit more nuanced than that because there are many aspects of classical signal processing that Machine Learning just doesn't easily accomplish.

As an example, if you're doing multi-target tracking, you need some process for determination of track formation and track deletion. There have been some papers that have tried to use things like CNN's or some transformer sequence model, but they all suck and are super unreliable. What ends up actually moving the field forward are (generally adaptive) graphical models, with classical statistics based bounds for determination of the probability of rare events.

Also, a lot of the people who are doing the best research in the mathematical theory behind Machine Learning are statistical signal processing types. It's not like these are separate disciplines in a meaningful sense. Signal detection is fundamentally a classification task. Estimation is fundamentally a regression task. Whether you use a linear filter or a Bayesian kernel approximation doesn't really change the fundamentals of the problem to be solved.

The course list looks good. I think you'll be fine. If you need more, you'll do more.

Anonymous at Thu, 19 Dec 2024 02:59:09 UTC No. 16517635

>>16517298
Just grind at practice problems. If you have to, try and get problems from other books (e.g., Schaum's outlines or any of the relevant springer problem books) and you'll be alright.

Anonymous at Thu, 19 Dec 2024 05:03:53 UTC No. 16517720

I keep fucking up problems because I forget to put negatives where they belong or stupid shit like saying 5 x 5 is 20 then the entire problem is wrong
Do I need practice?

Anonymous at Thu, 19 Dec 2024 06:31:46 UTC No. 16517776

>>16517344
That was the issue, wasn't able to show it was one, but funnily enough, the way you wrote the set helped me figure it out, cheers.

Anonymous at Thu, 19 Dec 2024 08:09:47 UTC No. 16517812

>>16517720
Learn your times tables by heart.

Anonymous at Thu, 19 Dec 2024 10:22:46 UTC No. 16517885

>>16517720
Visit neurologist. Seriously. I began making such stupid mistakes (like dropping signs, writing that cos(0)=0) and I was diagnosed with sclerosis multiplex.

Anonymous at Thu, 19 Dec 2024 10:28:51 UTC No. 16517891

>>16517485
0, as someone whose studying measure theory

Anonymous at Thu, 19 Dec 2024 13:55:46 UTC No. 16517994

>watch video about Crash Team Racing shortcuts
>suddenly the homotopy definition pops up

Anonymous at Thu, 19 Dec 2024 14:36:54 UTC No. 16518025

>>16517994
I mean, it is an optimal feasible path planning problem, which is a discrete geometry and geometric graph theory problem.

Anonymous at Thu, 19 Dec 2024 14:37:58 UTC No. 16518026

>>16517891
Hell, I need a shot before I even open a measure theory book. Probably doesn't help with the comprehension of the material but it does help with the existential dread.

Anonymous at Thu, 19 Dec 2024 15:19:13 UTC No. 16518068

>>16517599
>a lot of the people who are doing the best research in the mathematical theory behind Machine Learning are statistical signal processing
I thought they were theoretical statistics people who did that.
How different would a statistics PhD be from a PhD in (statistical) signal processing?

Anonymous at Thu, 19 Dec 2024 16:12:24 UTC No. 16518089

>>16518068
It depends on the school you go to and the focus of the program. For my program you're required to do a "minor" (meaning 4 grad level courses before your defense) in either math, statistics, physics, computer science or another engineering discipline. Of the 12 courses you're required to do before defense, 4 of them could be statistics (which, at my school, means you're doing about half of the same lecture material as a stats PhD).

Personally, I did mine in math focusing on analysis (class 1: baby rudin, class 2: measure, class 3: measure theoretic probability theory and stochastic processes, class 4: linear functional analysis and spectral theory). However, one of my lab-mates did his focusing on Bayesian statistics and actually has a stats professor on his advisory committee, so I'd say his degree is pretty similar.

Anonymous at Thu, 19 Dec 2024 16:34:23 UTC No. 16518098

How much of topology can I learn if I don't know about complex numbers?

Anonymous at Thu, 19 Dec 2024 16:46:00 UTC No. 16518108

>>16518068
Oh, I forgot, there definitely are theoretical statistics researchers working on the frontiers of ML, but they aren't alone. Take a look at the major topics for IEEE's Fusion conference and you'll see dozens of engineering PhDs from different universities and national labs working on both the development of ML-integrated statistical signal processing systems and the statistical characterization/modeling of ML-systems in the context of signal processing tasks.

If you look at the most cited recent papers in IEEE Transactions on Signal Processing, you'll find quite a lot of focus on trying to catch the theory up with the practice. That's not to discount the work by guys like Trevor Hastie. He just isn't doing this alone in a vacuum.

Anonymous at Thu, 19 Dec 2024 17:45:51 UTC No. 16518165

Is it ok to hand out Walmart job applications to dumb people in your uni

Anonymous at Thu, 19 Dec 2024 19:26:14 UTC No. 16518248

>>16518089
why did you choose to go with math? wouldn't statistics make more sense for you?

>>16518108
So basically EEs with signal processing background can choose to go into theoretical ML research, like statisticians can? I knew SP folks did a lot of work in ML, but I reckoned they only specialized in the niche of wireless communications at that.

Anonymous at Thu, 19 Dec 2024 20:29:32 UTC No. 16518321

>>16518248
> Why did you choose to go with math? Wouldn't statistics make more sense for you?

I figured that if I had a strong analysis background and probability background, that would allow a much easier path towards self-teaching from theoretical stats books like Keener/Shao, than the other way around. My signal detection/estimation research also relies a lot on constrained convex optimization, and having taken analysis was a godsend for that.

> So basically EEs with signal processing background can choose to go into theoretical ML research, like statisticians can?

It depends on the researcher, the money they have access to, and their personal research interests. I'd say more SSP researchers in the ML space have a focus on "how do we apply ML to target tracking/signal detection/multi-user communications" etc., but a surprising number of those more "applied" tasks require you to get down in the mud with some fundamentals of probability theory/statistics.

It's not an accident that the Covariance inequality for Bayesian estimation (which applies in either a classical or ML-based approx. inference case) was produced by an EE (van Trees). Variations on that same covariance inequality is the one used for performance bounding every single ML based regression system out there.

Anonymous at Thu, 19 Dec 2024 21:10:58 UTC No. 16518371

i'm trying out MathJaX for the first time sorry if this is not allowed

[ math ] /sqrt{2*2} + /sqrt{2} [ /math ]

Anonymous at Thu, 19 Dec 2024 21:12:42 UTC No. 16518373

[ eqn ] /sqrt{2*2} + /sqrt{2} [ /eqn ]

Anonymous at Thu, 19 Dec 2024 21:19:14 UTC No. 16518380

>>16518373
>>16518371
Don't. Put. Any. Fucking. Spaces. Within. Tags.

Anonymous at Thu, 19 Dec 2024 21:19:54 UTC No. 16518381

>>16518380
>Reddit-speak

Anonymous at Thu, 19 Dec 2024 21:23:45 UTC No. 16518383

>>16518381
>autocaps
phonefag

Anonymous at Thu, 19 Dec 2024 22:11:31 UTC No. 16518431

>>16518098
What's there to know about complex numbers you can't brush up in a day? Do you mean complex analysis? If it's the latter, you only need elementary real analysis mainly because of maturity and concrete examples.

Anonymous at Thu, 19 Dec 2024 22:17:53 UTC No. 16518435

>>16518383
Nice catch, he's a Redditor AND a phonefag.

Anonymous at Thu, 19 Dec 2024 22:30:46 UTC No. 16518448

>>16518431
I mean the latter. I know the arithmetics of complex numbers, their polar forms, de Moivre's formula etc., but statements like "bounded entire function must be constant" are almost black magic to me (at least I understand every word.)

Anonymous at Thu, 19 Dec 2024 22:35:52 UTC No. 16518452

>>16518448
>how much x do i need to know before learning y
JUST LEARN y AND LEARN THE BITS OF x WHEN YOU NEED
Complex analysis gives some useful intuition for things like homotopy/homology but basically none of what most people learn in AT has anything to do with even complex numbers. Why? Because topological spaces are not contingent on some ground field, and even then when ppl speak of manifolds it is usually real manifolds. Stop procrastinating and just do the math you want to do.

Anonymous at Fri, 20 Dec 2024 01:02:57 UTC No. 16518530

What's a good scientific calculator for uni?
Can't have graphing for cheater reasons

Anonymous at Fri, 20 Dec 2024 01:37:16 UTC No. 16518548

>>16518530
>Can't have graphing for cheater reasons
ngmi

Anonymous at Fri, 20 Dec 2024 02:28:16 UTC No. 16518575

I’m studying basic arithmetic and I’m already running into shit that has me stumped. The product of 65 x 26 is less than 64 x 27. That’s extremely counterintuitive to me. Why does the +1 increase from 26 to 27 have more weight than the decrease from 65 to 64?

Anonymous at Fri, 20 Dec 2024 02:35:57 UTC No. 16518590

>>16518575
because the +1 for 26->27 is being multiplied by 64, while the +1 for 64->65 is only being multiplied by 26. it becomes a bit more apparent if you break it down slightly
[math]65 \times 26 = (64+1) \times 26 = 64 \times 26 + 26[/math]
[math]64 \times 27 = 64 \times (26 + 1) = 64 \times 26 + 64[/math]

Anonymous at Fri, 20 Dec 2024 02:54:44 UTC No. 16518610

>>16518530
I have a Casio fx-991EX that I got for like $15 during undergrad and it's been my main calculator since. I still use my inspire sometimes if I really need it, but that Casio is definitely worth the basically nothing that it costs.

Anonymous at Fri, 20 Dec 2024 05:08:25 UTC No. 16518757

>>16517298
You should take a group theory course before linear algebra 2

Anonymous at Fri, 20 Dec 2024 15:56:07 UTC No. 16519179

>>16515464
LaTeX is honestly my favourite software ever written. It allows me to do all my work from the comfort of my bed :)

What editor do you guys use? I am still stuck on overleaf since I'm too lazy to configure nvim for it. Are any of the simpler all in one GUI programs worth using?

>>16515541
Get a book that goes over proving things. I get the impression that you're in the US where they make you sit exams during your PhD. I'd imagine the concepts on something like Real Analysis are not going to be too tough if you got through EE, however, actually going from a blank page to a proof is a daunting task. If you're looking to go into stats probably measure theory/probability stuff might be relevant.

>>16517298
Do not give up anon. I fucked up my entire first year and 1/4 of my second but am now on track to do quite well. Depending on where you are in the world and what university you're at, please make sure you know exactly how your university weights classes for you average. In my country first years (tend) to not actually contribute to your final average at all.

>>16518530
Check if your university recommends one. If you aren't taking a class with a calculator this semester, try check other classes you might be taking. Then, get the recommended one. Only one class in my entire degree used a calculator and honestly, I barely remember actually needing it in the exam. Honestly a bit strange that it was even required. Worse yet, my first one broke, I then lost my second and then found it after having just sat the exam and having bought a third. Whoops.

Anonymous at Fri, 20 Dec 2024 17:20:13 UTC No. 16519269

>>16519179
>I'm too lazy to configure nvim for it
nvim with snippets is worth it

Anonymous at Fri, 20 Dec 2024 19:02:45 UTC No. 16519329

>>16515464
How conditioning on an "event" defined using measure theory?

Anonymous at Fri, 20 Dec 2024 19:53:34 UTC No. 16519360

>>16519179
I use TeXStudio

Anonymous at Fri, 20 Dec 2024 20:04:37 UTC No. 16519369

>>16519329
You have a probability space [math](\Omega, \mathscr{A},P)[/math]. Take any set [math]B \in \mathscr{A}[/math] with [math]P(B)>0[/math] then you can consider the probability space [math](B, \mathscr{A}', P')[/math] with
[math]\mathscr{A}' = \{ B \cap A | A \in \mathscr{A} \}[/math]
[math]P'(A) = \frac{P(A \cap B)}{P(B)} [/math]

Anonymous at Fri, 20 Dec 2024 20:06:16 UTC No. 16519370

>>16518098
There is no real obstruction to learning all of (algebraic) topology without knowing anything about complex numbers as far as I know, though you would miss out on a few (beautiful) examples. The double cover of S^1 has a very simple description in complex polar coordinates, for example

Anonymous at Fri, 20 Dec 2024 20:26:02 UTC No. 16519384

>>16519269
Thanks I will get this set in one of these days

>>16519360
Thank you as well, I will use this for now.

Anonymous at Fri, 20 Dec 2024 23:53:40 UTC No. 16519539

>>16518321
>It depends on the researcher, the money they have access to, and their personal research interests. I'd say more SSP researchers in the ML space have a focus on "how do we apply ML to target tracking/signal detection/multi-user communications" etc., but a surprising number of those more "applied" tasks require you to get down in the mud with some fundamentals of probability theory/statistics.
That's good to hear. I was afraid I would pigeonhole myself into wireless communication if I go the SSP route.

>>16519179
>What editor do you guys use?
I use VSCode. Effortless git backups.

Anonymous at Sat, 21 Dec 2024 00:43:18 UTC No. 16519567

I thought if I went to uni for math it wouldn't be just hammering roots and logarithms to shit out some limit for 80% of the time but that i'm gonna learn cool proofs and learn tons of definitions and try to apply them but nope
I'm less than 3 months in and I'm basically guaranteed to fail at this point.
Also my fucking lecturer is such a senile retard i legitimately have no clue when he's rambling and when he's actually talking about shit i need to know he just doesnt make it clear.
I'm just too tired to try to memorize 12 algorithms to solve textbook problems and then lose half of the points for not doing the problem explicitly the same way it was shown to me before.
I don't even know what to do at this point I'm 1 year in the dump or maybe I should just give it up for good but I have no other besides hobbyist IT stuff. It's so humiliating but I just can't fucking work on this shit my i just can't handle 4 hours of solving boring problems that are identical to the shit i've been doing in high school

Anonymous at Sat, 21 Dec 2024 00:44:24 UTC No. 16519568

And it's not like I could've ever caught up because I'd have to start from the beginning

Anonymous at Sat, 21 Dec 2024 00:48:17 UTC No. 16519575

Let C be some closed solid regular cube in [math] \mathbb{R}^3 [/math].

Is there a (2-)plane P in [math] \mathbb{R}^3 [/math] satisfying the following conditions?
(1) P passes through at least 2 different vertices of C ; and
(2) the intersection of P with any given edge e of C is a set of vertices of e . In other words, P does not touch the "interior" part of any edge of C .

I'm pretty sure the answer is "no", but what's an efficient/clever way to show this?

Anonymous at Sat, 21 Dec 2024 00:53:44 UTC No. 16519581

>>16519539
git backups are very helpful and I'm making the most of my univeristy's overleaf premium by making as many backups as possible

>>16519567
I understand this feeling anon, I nearly failed my first year. It was pretty boring and I only even went to campus maybe 3 or 4 times a month. Just stick through the boring shit and try keeping yourself stimulated with more interesting work. By the time the good stuff finally comes around you will find it easy since you've been self teaching. It isn't the best strategy for grade-maxxing but it's personally gratifying.

Anonymous at Sat, 21 Dec 2024 00:56:26 UTC No. 16519583

>>16519568
You'll get your f***ing hands dirty like the rest of us.

Anonymous at Sat, 21 Dec 2024 01:04:51 UTC No. 16519592

>>16519581
buddy i'm already on my way out i'd have to ace 2 tests and suck lecturer's cock to let me rewrite one of the old tests and maybe i could make it when the best i've done was to reach like 30% even if i tried i wouldnt get more than 60%
linear algebra is palatable but calculus is 100% memorization and I'm pretty sure I've fucked my brain to a point where I could be diagnosed with "dyscalculia" i just cant focus on it i am too tired i cam barely manage to wake up in the morning i am already thinking of which minimum wage job i can get

Anonymous at Sat, 21 Dec 2024 03:08:23 UTC No. 16519680

>>16519592
>calculus is 100% memorization
Agreed, honestly why even take calculus, Wolframalpha does all that for you anyway. It's basically just to exercise your algebraic and analytic thinking, but I've literally never needed to use it myself and I'm in my final year of grad school in math.

Anonymous at Sat, 21 Dec 2024 03:11:27 UTC No. 16519682

>>16519370
>The double cover of S^1 has a very simple description in complex polar coordinates
More than that, literally any finite connected covering of S^1

Anonymous at Sat, 21 Dec 2024 07:01:19 UTC No. 16519830

>>16519369
What if event is of probability 0?

Anonymous at Sat, 21 Dec 2024 07:07:34 UTC No. 16519836

>>16519539
> That's good to hear. I was afraid I would pigeonhole myself into wireless communication if I go the SSP route.

It can happen, but it doesn't need to. It all just depends on your research interests and the funding available to you. My recent research is technically in a subset of sonar, but really it's about random matrix theory and convergence of non-linear programs with random variable Hessians.

Anonymous at Sat, 21 Dec 2024 09:02:14 UTC No. 16519882

Thoughts on OpenAI automating math research?

Terry Tao did not expect progress to be made on FrontierMath for years, and now look at o3...

Anonymous at Sat, 21 Dec 2024 09:16:12 UTC No. 16519888

>>16519882
I think, if `AI' will be able to generate code in some proof assisting language like lean, essentially formalizing its current sometimes-schizo ramblings, it's over.
That's a big if, though, and they'd have to make the generating multiple orders of magnitude cheaper than it currently is for the impact to be significant.
It's certainly interesting.

Anonymous at Sat, 21 Dec 2024 12:48:27 UTC No. 16520005

>>16519583
if by getting my hands dirty you mean working construction for the foreseeable future than you're right
>>16519680
it's mandatory
truth be told i'm not doing too well with linear algebra either but failing it doesn't make me illegible to continue studying
i like logic a lot which probably makes sense since i have some CS background

Anonymous at Sat, 21 Dec 2024 15:08:16 UTC No. 16520063

>>16519888
The fact is, there's no breakthroughs needed anymore for that to happen. Just engineering + getting better / more gpus.

I think a lot of mathematicians, especially the truly autistic ones, might kill themselves once they're made obsolete

Anonymous at Sat, 21 Dec 2024 15:20:19 UTC No. 16520074

>>16520063
nta, but you won't get orders of magnitude cheaper with some basic engineering + better gpus. That would already require a pretty serious breakthrough. Moreover, I don't think the quality of the ai is there yet either. Not saying it's impossible, but it's also not in the state of "no breakthroughs needed".
>I think a lot of mathematicians, especially the truly autistic ones, might kill themselves once they're made obsolete
do you have some weird investment in this? Are you the same tard that made the ai post in /scg/? Be honest, are you an undergrad/underaged/NEET?

Anonymous at Sat, 21 Dec 2024 15:21:32 UTC No. 16520075

what can i use the frobenius norm of a matrix for

Anonymous at Sat, 21 Dec 2024 15:25:57 UTC No. 16520078

TIL m comes *before* n in the alphabet

Anonymous at Sat, 21 Dec 2024 15:55:02 UTC No. 16520094

>>16520074

o3 is just doing the same thing o1 does (generating tons of chain of thought solutions + having another model selecting the best one) and the leap between them is immense. I don't see why they can't just push the same for o4 etc, until the distilled and cheap models will be good too

>do you have some weird investment in this? Are you the same tard that made the ai post in /scg/? Be honest, are you an undergrad/underaged/NEET?

No idea of what scg is. Also I'm a postdoc but not in math. Math will fall first and other fields will follow

Anonymous at Sat, 21 Dec 2024 19:48:31 UTC No. 16520213

Anonymous at Sat, 21 Dec 2024 19:53:51 UTC No. 16520220

>>16520213
Stop posting this meme.

Anonymous at Sat, 21 Dec 2024 19:54:52 UTC No. 16520221

>>16520078
Yeah, math people prefer to use n before m, and only introduce m as a second variable, meanwhile a is preferred to b, making it seem like the reason is alphabetical order, while in fact it's preferred because n stands for "number".

Anonymous at Sat, 21 Dec 2024 20:03:54 UTC No. 16520225

>>16520213
See:
>>16515514
String theory is dead. Update your curriculum.

Anonymous at Sat, 21 Dec 2024 21:20:06 UTC No. 16520265

>>16520063
Surely formal verification is still a human issue though?

Anonymous at Sun, 22 Dec 2024 05:53:42 UTC No. 16520595

Math teachers should be curved graded against each other and the bottom 10% move down a grade level every semester

Anonymous at Sun, 22 Dec 2024 06:20:37 UTC No. 16520603

Why do they hit calc students with Delta epsilon rigor that isn't matched until literally 5 semesters later?

Anonymous at Sun, 22 Dec 2024 06:30:47 UTC No. 16520607

What the fuck is algebra

Anonymous at Sun, 22 Dec 2024 07:03:57 UTC No. 16520615

>>16520603
>Why do they hit calc students with Delta epsilon rigor that isn't matched until literally 5 semesters later?
Europe exist, they dont delay rigour there

Anonymous at Sun, 22 Dec 2024 08:05:42 UTC No. 16520646

>>16520607
In (modern) algebra not only they use "x" and "y" with variable meaning but "+" (plus) and "*" (times) as well. For example, they study commutative and non-commutative operations and in the latter case they accept the entities being multiplied may not be real/integer numbers like usual.

Anonymous at Sun, 22 Dec 2024 08:06:14 UTC No. 16520647

>>16520615
European universities confuse me. They clearly start out ahead of US universities in terms of math education, but seem to completely drop the ball. As someone who did his undergrad/master's in the US and then worked in Italy/Switzerland for a bit, their master's degrees in my field barely even covered what we did in my bachelor's program. Granted, the Bologna process is a 3+2 master's and the average US bachelor's length for a good stem program is 4.5-5 years now.

Anonymous at Sun, 22 Dec 2024 08:51:23 UTC No. 16520674

>>16520647
Well don't for profit universities get most of their profit from undergrads?

Anonymous at Sun, 22 Dec 2024 16:58:54 UTC No. 16520941

>>16520646
I told my shrink about non real entities multiplying and I got forced on meds

Anonymous at Sun, 22 Dec 2024 17:04:23 UTC No. 16520947

How come bijective homomorphisms between groups/rings/fields/modules are always isomorphisms,

but e.g. bijective continuous maps between topological spaces aren't necessarily homeomorphisms?

I understand how to prove these facts with symbols, but what's an "overarching" reason for this?

Anonymous at Sun, 22 Dec 2024 17:07:26 UTC No. 16520950

>>16520646
For background, the non-commutative case can cover situations like e.g. matrix multiplication, which is of utility and interest in physics and engineering to say the very least

>>16520941
Sucks 2 suck

Anonymous at Sun, 22 Dec 2024 17:24:26 UTC No. 16520976

>>16520947
>How come bijective homomorphisms between groups/rings/fields/modules are always isomorphisms,
Perhaps it's because in algrebra isomorphism is *defined* as bijective homomorphism?
>but e.g. bijective continuous maps between topological spaces aren't necessarily homeomorphisms?
Because in topology there are continuous and bijective functions whose inverse is not continuous. Classical sample is an interval [0; 1) and a circle. You may easily turn an interval into a circle, but to turn a circle into interval you must tear it somewhere and thus introduce discontinuity.

Anonymous at Sun, 22 Dec 2024 17:26:48 UTC No. 16520983

>>16520976
>Perhaps it's because in algrebra isomorphism is *defined* as bijective homomorphism?
No, an isomorphism is a bijective homomorphism whose inverse is also a homomorphism. It's just that being a bijective homomorphism turns out to be sufficient.

Anonymous at Sun, 22 Dec 2024 17:36:09 UTC No. 16520992

>>16520976
What I'm asking is, why are morphisms in the category of groups/rings/fields/modules whose underlying set-functions are bijective necessarily isomorphisms,
but this is not the case for other concrete categories like the category of topological spaces?
>Because in topology there are continuous and bijective functions whose inverse is not continuous.
Like I already said, I know how to prove these at a basic level, so you repeating this is not helpful to me. But kudos to you for coming up with a proof yourself.
Btw, another nice example is: on any given set X, the "identity" function is continuous from the discrete to codiscrete topology on X and is bijective, but its inverse isn't continuous.

>>16520983
Yes, this aligns with what I've been taught.

Anonymous at Sun, 22 Dec 2024 17:41:27 UTC No. 16520996

>>16520674
> Well don't for profit universities get most of their profit from undergrads?

This is kind of true but also kind of not. On the margin, the most profit is from international students who pay cash (largely because most of them are ineligible for subsidized loans/financial aid). This is a big reason why so many schools (even public ones) have been orienting themselves so much towards the luxury resort model. They want to attract rich Chinese and Indian people with money to spend (launder) by sending their kids to the US for a good education.

My personal opinion is that US universities do a better job at providing a broad range of coverage, and if you go to a good university that coverage ends up fairly deep as well. This gets meme'd on by people who are pointing at the required humanities courses for accredited programs, but it also end up with a much broader coverage of whatever your major is.

My bachelors was in EE, and by the time I'd finished it I'd done all of the basic calculus sequence, linear algebra, real analysis and an introductory analytic functions course, as well as probability theory, stochastic processes, linear dynamical systems and control theory. The Italian/Swiss colleagues I'd worked with had a depth of knowledge in a few particular niches that my education did not provide, but they entirely excluded really fundamental skills in favor of waiting until an elective Master's course to talk about them at all.

One of the aerospace engineers I'd worked with did his entire master's degree at a fairly good Italian university and had never taken a formal probability course. You couldn't even take third year undergrad courses at my university without a probability prereq.

Anonymous at Sun, 22 Dec 2024 17:42:49 UTC No. 16520999

>>16520947
I can't quickly find a reference, but iirc it is because groups/rings/fields/modules are all algebras (in the universal algebra sense) and in this general setting bijections are good enough. There are also some categories where some property + mono + epi = iso, but this is probably not the best line of thought to enter, since for example ring morphisms can be epi without being surjective.
>>16520976
isomorphism is *defined* as invertible morphism

Anonymous at Sun, 22 Dec 2024 18:14:21 UTC No. 16521017

>>16520947
Because groups/rings/fields/modules are models of first order theories, and homomorphisms are simply maps between models that preserve truth claims in the first order theory. Any such bijection turns out to be an isomorphism of models.
Meanwhile topology is a second order theory.

Anonymous at Sun, 22 Dec 2024 18:17:32 UTC No. 16521022

>>16520999
>I can't quickly find a reference, but iirc it is because groups/rings/fields/modules are all algebras (in the universal algebra sense)
The statement holds for all first order theories, not just algebras.
For example graphs and partial orders also have the property that morphisms which are bijective are isomorphisms. As do all structures which are modelled by first order axioms.

Anonymous at Sun, 22 Dec 2024 18:25:32 UTC No. 16521027

>>16520947
Another way to think about it is that with topology the structure is not just the underlying set X but also the topology t. So in a sense a "true" notion of a morphism would be a map (X,t) -> (Y, t') where t' is the topology of Y, considered as a pair of maps on the underlying spaces X->Y and the map from open sets to open sets t -> t' which respect each other. This would be the model theoretic way of looking at morphisms between models. A morphism is a bijection between models then if it's a bijection X-> Y together with a bijection t -> t'. In such a case, the map is a homeomorphism. However, this model theoretic notion of morphism is not very useful in topology, and instead another type of morphism is chosen, that of a continuous map, which is more complicated and doesn't respect the model theoretic structure in a nice way.

Anonymous at Sun, 22 Dec 2024 18:29:09 UTC No. 16521032

>>16521017
>>16521022
Awesome, this is exactly the kind of thing I was looking for! So, what distinguishes groups/rings/fields from topologies is whether they use 1st order or 2nd order logic. Thanks anon, I'll look into this further.

Anonymous at Sun, 22 Dec 2024 18:38:02 UTC No. 16521038

>>16521032
>whether they use 1st order or 2nd order logic
Or I guess to be more precise, whether they are *models* of these.

>>16521027
Thanks, this is another interesting perspective.

🗑️ Anonymous at Sun, 22 Dec 2024 18:51:30 UTC No. 16521049

>>16520947
You need the inverse map to be continuous?

🗑️ Anonymous at Sun, 22 Dec 2024 18:53:47 UTC No. 16521051

>>16521049
*to be a homeomorphism

Anonymous at Sun, 22 Dec 2024 19:15:53 UTC No. 16521070

How quickly should I be going through a math textbook? When is it time to say you're done and move on to the next one?

Anonymous at Sun, 22 Dec 2024 19:39:41 UTC No. 16521095

>>16521070
when u do the hw problems and feel you got a good understanding of it

Anonymous at Sun, 22 Dec 2024 19:48:07 UTC No. 16521098

>>16521032
Nta, but no need to talk about 2nd order logic. It is enough to think that algebraic structures only involve algebraic operations in the structure signature, while other structures like topologies involve families of subsets in the signature. Yes, there's no way to talk about "subsets" in the pure first order logic theory taking operations as primitives, but we usually already have set theory when we want to study algebra and topology.

Anonymous at Mon, 23 Dec 2024 06:59:11 UTC No. 16521530

Any good but short calculus (single and vector) books you can recommend?
I already have some proof experience, worked through most of Hammack and the first few chapters of Tao Analysis 1 and half of Axler.
But I never had Calc classes.

Anonymous at Mon, 23 Dec 2024 07:58:47 UTC No. 16521563

>>16521530
>Good but short single variable calculus
>Proof/elementary analysis experience
Consider:
>>16515492

Anonymous at Mon, 23 Dec 2024 12:21:08 UTC No. 16521684

>>16521530
Get a classic "differential and integral calculus" book like Piskunov.

Anonymous at Mon, 23 Dec 2024 16:18:46 UTC No. 16521832

would learning math help a software engineer? what opportunities could it unlock?

Anonymous at Mon, 23 Dec 2024 17:14:41 UTC No. 16521865

>>16521832
Software Engineer can mean a lot of things. Most programmers nowadays are essentially just library and framework plumbers.
So it depends on what you actually do.

Anonymous at Mon, 23 Dec 2024 17:14:43 UTC No. 16521866

>>16518869
>Other sciences seek to discover the laws that God has chosen; mathematics seeks to discover the laws which God has to obey.
J. P. Serre
>From a metaphysical standpoint, I would say that mathematics is exactly what God didn't have to create, because it was all there from the beginning and He couldn't but take it as it is. (Maybe it may be said that mathematics is just part of God's own nature, namely the part of it to which human reason has access just by its own feeble means ...) God had an infinite choice about how to build a universe, with its laws (spiritual, physical, biological ...), and maybe there are many or even infinitely many such, of which we only know (ever so little) one. But whatever way he thinks up his Universe, He's got to use the same mathematics, with 2 + 2 = 4, and not 3 or 5. It is not in His power to change this, any more than to change his own nature – and surely He never had any wish to do so!
A. Grothendieck

Anonymous at Mon, 23 Dec 2024 18:34:26 UTC No. 16521955

I wish proof assistants were more user friendly. Solo learning math I never know for sure if my proofs are valid unless I ask and I don't want to ask for every exercise.

Anonymous at Mon, 23 Dec 2024 19:58:38 UTC No. 16522036

>>16521955
play the natural number game
very fun

Anonymous at Mon, 23 Dec 2024 20:04:35 UTC No. 16522042

>>16521866
Allegedly there's a prohibitively long proof of the fact that 2+2=4. Where can I read (about) it?

Anonymous at Mon, 23 Dec 2024 21:23:50 UTC No. 16522114

ayo nigga we intergrating arrows

Anonymous at Mon, 23 Dec 2024 21:45:04 UTC No. 16522145

>>16522042
I guess you are thinking about Alfred North Whitehead and Bertrand Russell''s [math]\textit{Principia Matematica}[/math]. The following is a contemporary equivalent:
>Inspired by Whitehead and Russell's monumental Principia Mathematica, the Metamath Proof Explorer has over 26,000 completely worked out proofs in its main sections, starting from the very foundation that mathematics is built on and eventually arriving at familiar mathematical facts and beyond. Each proof is pieced together with razor-sharp precision using a simple substitution rule that practically anyone (with lots of patience) can follow, not just mathematicians. Every step can be drilled down deeper and deeper into the labyrinth until axioms of logic and set theory—the starting point for all of mathematics—will ultimately be found at the bottom. You could spend literally days exploring the astonishing tangle of logic leading, say, from the seemingly mundane theorem [math]2+2=4[/math] back to these axioms.
Let [math]2,4\in\mathbb{C}[/math]
https://us.metamath.org/mpeuni/mmset.html#trivia
>The complete proof of [math]2+2=4[/math] involves [math]2,913[/math] subtheorems
>These have a total of [math]26,323[/math] steps—this is how many steps you would have to examine if you wanted to verify the proof by hand in complete detail all the way back to the axioms of [math]\textrm{ZFC}[/math] set theory.

Anonymous at Mon, 23 Dec 2024 23:15:13 UTC No. 16522227

Fun generalization of mobius transformations. Can also use split quaternions to shorten things.
[math]\pmatrix{a & b & c\\ \bar b & \bar a & \bar c\\ d & \bar d & r}\sim \dfrac{az+\bar az+c}{dz+\bar d \bar z +r},r\in \mathbb{R}.\\
\text{Introduce}\ j:j^2 = 1\ and\ \forall z \in \mathbb{C},\ jz=\bar zj.\ \text{This gives the split-quaternions (the usual quaternions have }j^2=-1).\\
\text{The matrix representing the transformation can be reduced to a 2x2 matrix of split quaternions}:\\
\pmatrix{a+bj & c (\frac{1+j}{2})\\ (1+j)d & r (\frac{1+j}{2})},\ r\in\mathbb{R},\ a,b,c,d\in\mathbb{C}. \text{Care must be taken when multiplying since the j does not commute.}[/math]

Anonymous at Mon, 23 Dec 2024 23:32:43 UTC No. 16522244

>>16521832
The ability to graduate with a cs degree with a good gpa and stand out from mathlets

Anonymous at Tue, 24 Dec 2024 00:22:41 UTC No. 16522281

>>16522227
should be az + bz* + c on top of the fraction

Anonymous at Tue, 24 Dec 2024 04:26:44 UTC No. 16522431

Is this problem solved:
Let [math]\omega(m)[/math] denote the number of distinct prime factors of [math]m[/math]. Does there for every natural number [math]k[/math] exist infinitely many natural numbers [math]n[/math] such that [math]\omega(n)<\omega(n+1)<\dots<\omega(n+k)[/math]?

Anonymous at Tue, 24 Dec 2024 04:30:43 UTC No. 16522435

>>16515464
get the fuck in here >>>/x/39491447

Anonymous at Tue, 24 Dec 2024 05:37:02 UTC No. 16522492

1 != 0 is an artificial restriction on what fields are, and there SHOULD be a field with 1 element, which is the zero ring.

If your reasoning for not allowing it is "it causes too many exceptions in results" then that same reasoning can be used to say 2 isn't a prime.

Anonymous at Tue, 24 Dec 2024 06:25:30 UTC No. 16522512

>>16522492
I want my fields to be integral domains.

Anonymous at Tue, 24 Dec 2024 07:04:42 UTC No. 16522526

>>16522512
THE ZERO RING IS AN INTEGRAL DOMAIN VACUOUSLY YOU NEGRO

Anonymous at Tue, 24 Dec 2024 08:48:01 UTC No. 16522560

>>16522526
but it doesn't have a field of fractions
inb4 it's its own field of fractions, you need to change the construction to fit this particular example

Anonymous at Tue, 24 Dec 2024 08:49:56 UTC No. 16522561

>>16521032
>>16521098
adding on to this, if you have relations in your language then you can have bijective homomorphisms that aren't isomorphisms
For example, there are bijective homomorphisms of partial orders that aren't isomorphisms

Anonymous at Tue, 24 Dec 2024 10:14:14 UTC No. 16522593

>>16522492
>then that same reasoning can be used to say 2 isn't a prime.
The reasoning is in fact why 1 isn't considered a prime.

Anonymous at Tue, 24 Dec 2024 10:15:46 UTC No. 16522594

Mathematicians are so stupid. There are only countably many formulae in FOL. So, just enumerate through them and you'll eventually prove everything. What's the big fuss?

Anonymous at Tue, 24 Dec 2024 10:19:10 UTC No. 16522595

>>16522594
Why are plebs so consistently filtered by infinity?

Anonymous at Tue, 24 Dec 2024 11:11:49 UTC No. 16522613

>>16522594
See:
>>16522145
If a simple fact like that takes so many steps in fully formal language, how much time would it take to enumerate all theorems up to that fact and verify it? (You can think of every step as new theorem which is the concatenation of all the preceding ones). Let's see your results

Anonymous at Tue, 24 Dec 2024 11:31:14 UTC No. 16522627

>>16522613
>le it's too much heckin work
Lmao. If you give a simple MBA grad the work of proving theorems, he would prove all the mathematics by the end of the year.

Anonymous at Tue, 24 Dec 2024 11:38:02 UTC No. 16522630

>>16522627
How much time would it take her to yield and single out the shortest Godel sentence in your hypothetical system?

Anonymous at Tue, 24 Dec 2024 11:59:43 UTC No. 16522643

>>16515464
I got one, what's the level of energy required to infer infinity?

Anonymous at Tue, 24 Dec 2024 12:03:29 UTC No. 16522644

>>16522643
Like from a calculator battery for e.g. haha pfft

Anonymous at Tue, 24 Dec 2024 14:24:39 UTC No. 16522729

Question for my /mg/ nerds, but how do you guys feel about the new ai math news? Do you feel like it's overhyped?
The one time I used chatgpt and told it to do a word count it couldn't even do it properly, and it would be different answers whenever I would ask it to redo it

Anonymous at Tue, 24 Dec 2024 14:44:44 UTC No. 16522739

>>16522729
Yes and no. Currently available AIs are genuinely extremely impressive but they aren't actually PhD level and are very inconsistent which is a big problem.

Anonymous at Tue, 24 Dec 2024 14:50:57 UTC No. 16522743

>>16522593
No it isn't.

>>16522560
It IS its own field of fractions, whether you like it or not.
The field of fractions of an integral domain is the localization at the maximum set such that the map x -> x/1 is injective. Works for the zero FIELD.

Anonymous at Tue, 24 Dec 2024 15:16:34 UTC No. 16522754

>>16522743
>The field of fractions of an integral domain is the localization at the maximum set such that the map x -> x/1 is injective
Where can i find this characterization in the literature?

Anonymous at Tue, 24 Dec 2024 16:31:51 UTC No. 16522801

>>16522754
It's called "using your head", algebralet

Anonymous at Tue, 24 Dec 2024 16:45:57 UTC No. 16522812

>>16522801
Touché, but i still want to read something similar

Anonymous at Tue, 24 Dec 2024 16:52:19 UTC No. 16522816

>>16522812
Read Atiyah and Macdonald, chapter 3.

Anonymous at Tue, 24 Dec 2024 17:19:30 UTC No. 16522828

Atiyah is an Arabic name meaning "gift", because Michael Atiyah is a gift to the world.

Anonymous at Tue, 24 Dec 2024 17:45:10 UTC No. 16522861

>>16522743
>No it isn't.
Yes it is.

Anonymous at Tue, 24 Dec 2024 18:20:10 UTC No. 16522899

41^2 = 1681
42^2 = 1764
43^2 = 1849
44^2 = 1936
and so on
each goes up a hundred and down a perfect square

Anonymous at Tue, 24 Dec 2024 18:24:43 UTC No. 16522906

>>16522899
I'll go up your ass and down your mom

Anonymous at Tue, 24 Dec 2024 18:29:19 UTC No. 16522912

>>16522906
There is no need for such vulgarity.

Anonymous at Tue, 24 Dec 2024 19:10:50 UTC No. 16522954

>>16522729
Impressive, but still way overhyped. Personally I still don't see in it more than a glorified documentation search and/or autocomplete+. That can be genuinely useful of course, but is way different from the "ai will solve it" narrative.

The thing is: all current llm fail miserably on very simple, original logical questions and calculations. They get some impressive things right too.
Now if you work on anything that isn't monkey work or library plumbing then you will have to spend as much time verifying the AI's potential answer (if it isn't bogus right away) as you would have spent coming up with it in the first place.

And I currently don't see anything changing in that regard except huge marketing buzzword babble.

Anonymous at Tue, 24 Dec 2024 19:23:17 UTC No. 16522969

Grant's making shitposts again

Anonymous at Tue, 24 Dec 2024 19:36:40 UTC No. 16522978

Has anyone read pic related, or the apparently similar book "Guesstimation" or something else that is similar? Are they any good? I find getting better at estimating arbitrary problems interesting. But the official descriptions sound cheesy.

Anonymous at Tue, 24 Dec 2024 20:05:55 UTC No. 16522998

>>16522899
That's nice but check this

469^2 = 219961
470^2 = 220900
471^2 = 221841
...
499^2 = 249001
500^2 = 250000


Each goes up a thousand and down a perfect square.

Anonymous at Wed, 25 Dec 2024 00:49:02 UTC No. 16523183

Sup, /mg/. What's your take on that book?
>>>/g/103614046

Anonymous at Wed, 25 Dec 2024 01:11:28 UTC No. 16523201

>>16522978
the wiki mentions a neat book called dead reckoning, check it out

Anonymous at Wed, 25 Dec 2024 02:09:25 UTC No. 16523243

>>16515464
Can someone explain this post from /x/:
>>>/x/39492043
So, that was the foundational theory. Eventually, a mathematical proof was developed to prove it:

Wave Function Expansion
Psi(x,t) = A exp(i k x - i ω t)
= A exp(i k x) exp(-i ω t)
= A [cos(kx) + i sin(kx)] [cos(ωt) - i sin(ωt)]

Probability Density
|Psi(x,t)|2 = Psi*(x,t) Psi(x,t)
= A2 [cos2(kx) + sin2(kx)]
= A2
This shows the wave-like nature remains intact until a measurement occurs.

Measurement Process
For an operator M acting on the state |Psi>, the possible outcomes are its eigenvalues mᵢ, and the probability amplitude for each outcome is ⟨ϕᵢ|Psi⟩, where |ϕᵢ⟩ are the eigenstates.

Collapse Mathematics
After measurement, the state collapses into |ϕⱼ⟩ with probability |⟨ϕⱼ|Psi⟩|2.

Wave-Point Transformation
Psi(x,t) can transform into a delta function delta(x - x0), normalized so that integrating its squared magnitude over all x gives 1.

Uncertainty Principle Connection
Δx Δp ≥ ℏ / 2
This sets the fundamental limit on positional and momentum precision.

Quantum Entanglement
For an entangled state like (1/√2)(|0⟩1|1⟩2 - |1⟩1|0⟩2), measuring one particle collapses the overall state instantaneously.

Conservation Laws
∫|Psi(x,t)|2 dx = 1 (total probability),
E = ℏ ω (energy),
p = ℏ k (momentum).

Complete Transformation
Psi(x,t) = A exp(i k x - i ω t)
delta(x - x0) exp(-i ω t)
delta(x - x0)

Mathematical Identity
As the width σ goes to zero, a normalized Gaussian becomes delta(x - x0), showing how wave packets can become point-like.

This expanded proof demonstrates that wave functions hold complete information, and measurement collapses them. Position and momentum remain complementary. Probability is conserved. The transformation can switch from wave-like to point-like while preserving all conservation laws. In essence, both descriptions—waves and points—are two faces of the same coin, just like mass and energy.

Anonymous at Wed, 25 Dec 2024 04:18:05 UTC No. 16523314

>>16522729
> ai now solves these mega hard problems real mathematicians have to spend hours on. The solutions totally can't be found online or in its training data!
> no you may not see or evaluate any of the problems, or the ai output, just trust us
Convenient

Anonymous at Wed, 25 Dec 2024 04:56:16 UTC No. 16523345

Can someone help me understand some of the points of this question?

>Let [math]A[/math] be an arbitrary subset of the set [math]\{1,2, ...,2n\}[/math] of size [math]n+1[/math]
>Prove that there exists two distinct elements [math]a[/math] and [math]b[/math] such that[math]a[/math] divides [math]b[/math]

[math]A[/math] is a set of size [math]n+1[/math]; [math]A:=\{1,2, ..., n+1\}[/math]?
[math]B:=\{1,2, ..., 2n\}[/math]
[math]A \subseteq B[/math], therefore [math]a,b \in A,B[/math]?

Are the above assumptions correct?

Anonymous at Wed, 25 Dec 2024 07:04:49 UTC No. 16523405

>>16523345

A is an arbitrary subset of B, so A is not necessarily {1, 2, ... , n+1}. A could be {n-1, n, n+1,..., 2n-1, 2n}, or any selection of n+1 elements of B.

Think of it like this: the statement to prove is that at most half of the elements of B can be chosen such that none of the chosen subset A divide any of the elements in B which aren't in A, nor do any of the elements of A divide any of the other elements in A. Any subset of more than half of B (aka the subset A of size n+1) must contain an element that divides one of the elements of B that is not in A, or one of the other elements of A. (an element of A is still an element of B, as its a subset. Which is why dividing one of the other elements of the chosen subset is relevant.)

Anonymous at Wed, 25 Dec 2024 11:31:13 UTC No. 16523497

>>16523183
I like it, it's a fun read.

Anonymous at Wed, 25 Dec 2024 11:54:30 UTC No. 16523508

>>16523345
>>16523405
Jumping into this conversation to say that it's not clear where "a" and "b" are from in the statement of the question.
Most likely it's that both are in A, because otherwise the statement is either trivial or wrong.

I'm guessing this is a question about the Pigeonhole Principle, and since A has n+1 elements, you need to define n "pigeonholes" and a rule to place the "pigeons" (elements of A) in them.

Anonymous at Wed, 25 Dec 2024 14:12:00 UTC No. 16523604

why are category theorists so annoying ;_;

Anonymous at Wed, 25 Dec 2024 14:18:42 UTC No. 16523611

>>16523604
a coconut is just a nut

Anonymous at Wed, 25 Dec 2024 16:24:31 UTC No. 16523692

>>16522036
My progress so far

Anonymous at Wed, 25 Dec 2024 21:07:16 UTC No. 16523881

>>16523692
Became more frustrating than fun in the advanced multiplication world. At one point I had to use a previous theorem but it was not listed anywhere on the right panel which pissed me off.

Anonymous at Thu, 26 Dec 2024 00:19:08 UTC No. 16523947

>>16515464
How do I get into mathematical proofs as a hobby? I have a book called Calculus by Michael Spivak, but it's a good bit over my head, not going to lie.

Anonymous at Thu, 26 Dec 2024 01:33:44 UTC No. 16523971

>>16523947
>Textbooks
Journey into Mathematics: An Introduction to Proofs - Joseph J. Rotman
Proof, Logic, and Conjecture: The Mathematician's Toolbox - Robert S. Wolf
Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics - John Baylis, Rod Haggarty
How to Read and Do Proofs - Daniel Solow
>Lecture Notes
A Primer for Logic and Proof - Holly P. Hirst and Jeffry L. Hirst
https://www.appstate.edu/~hirstjl/primer/hirst.pdf
Modicum Mathematicum: A Swath Through The Basic Language Of Abstract Math - Paolo Aluffi
https://math.hawaii.edu/~pavel/Aluffi_notes_321_Modicum.pdf
Proof, Sets, and Logic - M. Randall Holmes
https://randall-holmes.github.io/proofsetslogic.pdf
Basic Concepts of Mathematics - Elias Zakon
https://www.trillia.com/zakon1.html

Anonymous at Thu, 26 Dec 2024 02:01:53 UTC No. 16523984

>>16523692
>>16522036
Are you doing the latest, dumbed down version? Because the superior version is this:
https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/index2.html

Anonymous at Thu, 26 Dec 2024 02:53:44 UTC No. 16524011

>>16523984
Looking at "Advanced Addition World" it's the exact same things I proved in the later version's advanced addition world. How is it dumbed down?

Anonymous at Thu, 26 Dec 2024 11:32:49 UTC No. 16524229

>>16524011
Idk i was just quoting some anon from the lambdaplusjs /math/ board

Anonymous at Fri, 27 Dec 2024 07:07:31 UTC No. 16524941

Is the corestriction of a morphism onto its regular image always an epimorphism? I feel like this should be true but I'm completely blanking on a proof right now

Anonymous at Fri, 27 Dec 2024 20:36:35 UTC No. 16525479

I read some post by Joel David Hamkins somewhere arguing that the concept of a "standard" model of arithmetic is far more vague and poorly defined than most people seem to believe, and recently I've started to see things the same way. How do you know "when to stop" with induction? Isn't the concept of finiteness a little circular?
Obviously unsoundness can make "non-standard" models undesirable, but hypernaturals and the like seem natural enough (pun not intended).

Anonymous at Fri, 27 Dec 2024 22:48:05 UTC No. 16525654

>>16525479
first order logic problems

Anonymous at Fri, 27 Dec 2024 23:46:02 UTC No. 16525729

>>16525654
The true theory of second-order arithmetic isn't definable in second order logic either, and the issue with higher order interpretation of first order theories is that consistency results become useless.

Anonymous at Sat, 28 Dec 2024 01:00:08 UTC No. 16525788

Is it too autistic too draw a black square after each exercise? I draw a black square for easy exercises and write QED for hard exercises.

🗑️ Anonymous at Sat, 28 Dec 2024 01:21:02 UTC No. 16525811

>>16525729
>The true theory

Yeah but I'll take my categorical model, my ability to define what "finite" vs "infinite" means, and all the other benefits.

>muh no completeness theorem
No one cares

Anonymous at Sat, 28 Dec 2024 06:47:50 UTC No. 16526067

>>16525788
Exercises do not merit the tombstone symbol by merit of being an exercise. It is intended to be a punctuation mark at the end of a proof. The symbol plays a formal grammatical role, and is interchangeable with QED. Length and difficulty have nothing to do with it. You wouldn't drop your periods with proper grammar, would you?

Anonymous at Sat, 28 Dec 2024 07:02:57 UTC No. 16526076

Is there an uncountable subset [math]A[/math] of [math][0, 1][/math] such that [math]\forall a\in(0, 1], \{x\in A : a\le x\}[/math] is at most countable?
That is, I want all uncountability to be concentrated near zero.

I feel that maybe Cantor set can be modified to satisfy this. Like, instead of removing open intervals from the middles, remove them from the rightmost part:
- remove [math](\frac23, 1)[/math]
- then remove [math](\frac49, \frac59)[/math] and [math](\frac59, \frac23)[/math]
- then [math](\frac8{27}, \frac9{27}), (\frac9{27}, \frac{10}{27}), (\frac{10}{27}, \frac{11}{27})[/math] and [math](\frac{11}{27}, \frac49)[/math]
- etc.
It should still be uncountable (proof that Cantor set is uncountable should work for this set as well? And it feels like there is an obvious bijection between those two sets). And if I choose arbitrary [math]a\in(0,1][/math], there will be only a finite number of points to the right of [math]a[/math], and therefore uncountable to the left.

Is this correct?

Anonymous at Sat, 28 Dec 2024 07:13:32 UTC No. 16526083

>>16526076
Pretty sure that the Archimedean property would guarantee this is impossible. Exists definitionally for the hyperreals though. Note of caution, I only thought about this for 45 seconds, so might be wrong. Try assuming A exists and then contradict Archimedean property for a fixed a.

Anonymous at Sat, 28 Dec 2024 07:33:27 UTC No. 16526094

>>16515464
best algebra book if i want to get foolishly good at algebra, or just things you'd recommend reading after one semester of abstract algebra? i like algebra

Anonymous at Sat, 28 Dec 2024 08:14:20 UTC No. 16526103

Either I'm retarded or my professor is, I've got a function [math] f(x,y)=\frac{y^{2}x}{1+x^{6}} [/math] which can easily be shown to not be lebesgue integrable with respect to [math] \lambda_{\mathbb{R}^{2}} [/math]. The partial function for some fixed y, [math] f^{y}=f(\cdot,y)[/math] is certainly lebesgue integrable with respect to [math] \lambda_{\mathbb{R}} [/math], however my professor justifies this by saying that this partial function is continuous and [math] |f^{y}|\leq y^{2}\frac{1}{|x|^{5}} [/math] and states that the right hand side is lebesgue integrable in x over the reals.
That really doesn't seem to be the case though, is the right hand side truly lebesgue integrable?

Anonymous at Sat, 28 Dec 2024 08:58:28 UTC No. 16526115

>>16525479
Its not vague and poorly defined. Its the smallest inductive set, end of.

Anonymous at Sat, 28 Dec 2024 10:17:24 UTC No. 16526149

>>16523345
Something something pigeonhole principle.

Anonymous at Sat, 28 Dec 2024 10:33:59 UTC No. 16526155

>>16526103
If you want to be autistic then you can bound it like this

[eqn]|f^y(x)| \leq \begin{cases} y^2 & |x| \leq 1 \\ \frac{y^2}{|x|^5} & |x| > 1 \end{cases} [/eqn]

Anonymous at Sat, 28 Dec 2024 10:59:48 UTC No. 16526181

>>16523692
>>16523984
I remember doing this a few years ago. Got filtered by one double induction question where I could not possibly know how to do it because the technique needed to prove it wasn't actually explained as possible. That was very frustrating and a major bad part of the natural number game.

Anonymous at Sat, 28 Dec 2024 10:59:48 UTC No. 16526182

>>16526155
NTA, but that's a good one. Then you can just use DCT to show that it's Lebesgue integrable almost everywhere. I need to spend more time brushing up on measure. There's so many neat party tricks in analysis.

Anonymous at Sat, 28 Dec 2024 11:01:34 UTC No. 16526185

>>16526094
I really liked Algebra: Chapter 0. It's actually the book that changed my mind about category theory being exclusively for CS-troons and schizos. Now I'm aware there's a third category of category theorists, galaxy brained algebraists who use category theory as a mechanism to solve algebra problems.

Anonymous at Sat, 28 Dec 2024 11:04:04 UTC No. 16526188

>>16526185
Huh? How do you use category theory to solve algebra problems?

Anonymous at Sat, 28 Dec 2024 11:05:24 UTC No. 16526190

>>16526076
No, since A can be written as countable union of at most countable sets.

Anonymous at Sat, 28 Dec 2024 11:08:26 UTC No. 16526193

>>16526115
What do you mean by "smallest"? From the perspective of any non-standard model, it is working in [math]\omega[/math].

Anonymous at Sat, 28 Dec 2024 11:10:11 UTC No. 16526195

>>16526155
Cheers, I think another way would be [math] \int |f^{y}|d\lambda=y^{2}\int \frac{|x|}{1+x^{6}}d\lambda(x)= y^{2}(\int_{[-1,1]} \frac{|x|}{1+x^{6}}d\lambda(x)+\int_{|x|>1} \frac{|x|}{1+x^{6}}d\lambda(x))\leq y^{2}(\int_{[-1,1]} |x|d\lambda(x)+\int_{|x|>1} \frac{1}{|x|^{5}}d\lambda(x))[/math]. The last integrals being obviously finite. What do you think?

Anonymous at Sat, 28 Dec 2024 14:05:09 UTC No. 16526297

>>16526094
>foolishly good at algebra
>just things you'd recommend reading after one semester of abstract algebra
Jacobson for both.

Anonymous at Sat, 28 Dec 2024 14:19:58 UTC No. 16526308

>There is no set [math]X[/math] such that [math]2^{X} = |\mathbb{N}| [/math].
What if there was? Are there any abstract generalisations of power sets that enable us to do some kind of inverse power set operation for sets that don't normally have an inverse power set?

Anonymous at Sat, 28 Dec 2024 15:29:19 UTC No. 16526339

How long would it take for me to brush up on Precalc/Trig and Calculus 1 stuff? What's the best way of going about that?

Anonymous at Sat, 28 Dec 2024 16:51:53 UTC No. 16526391

>>16526308
No. It's impossible for a power set to be a limit cardinal, by definition.

Anonymous at Sat, 28 Dec 2024 17:30:01 UTC No. 16526422

Does there exist a Lebesgue integrable function that does not almost equal a Riemann integrable function?

Anonymous at Sat, 28 Dec 2024 19:15:55 UTC No. 16526500

>>16526391
Okay, is there a way to construct N so that it isn't a limit cardinal?

Anonymous at Sat, 28 Dec 2024 19:26:26 UTC No. 16526512

>>16526422
Did you already browse >Counterexamples in Measure and Integration - Franziska Kühn and René L. Schilling

Anonymous at Sat, 28 Dec 2024 19:32:37 UTC No. 16526515

>>16526512
No.

Anonymous at Sat, 28 Dec 2024 19:48:48 UTC No. 16526525

I have a BA in Math from a pretty chill school. I have been a public school teacher for the past few years but want to go back to grad school to escape the terrible pay and working conditions. My city has good programs in Financial Mathematics.
The furthest I have gone is a few independent study courses in Graph Theory, Topology, and Category Theory. My background in Prob and Stat is severely lacking. I am now 27. Is it worth it, or am I just fucked? If the rest of my life is just staying up into the wee hours developing lessons to teach Algebra I and Geometry to kids who can't read or count, I'll be really mad.

Anonymous at Sat, 28 Dec 2024 19:50:12 UTC No. 16526528

Are there functions which are continuous in one direction and non continuous in another?

Anonymous at Sat, 28 Dec 2024 19:53:32 UTC No. 16526534

>>16526525
Probability is just real analysis and linear algebra. You'll be alright. Just get a good "transition" probability textbook (or two, one that is all calculus/linear algebra but is a bit more advanced and one that is properly measure). If you self study them, you'll probably be fine.

Anonymous at Sat, 28 Dec 2024 19:55:47 UTC No. 16526538

>>16526534
Thank you. Not to bother, but do you have any specific recommendations?

Anonymous at Sat, 28 Dec 2024 19:59:11 UTC No. 16526545

>>16526528
"Continuity" in a classical sense doesn't really work that way. For something to be continuous, you have to be able to arbitrarily approach it in any direction.

There are "smooth" maps in lower spaces that are continuous in that lower space, but that requires a bit of topology and doesn't always work. (Take, as an example, the function
[math]
f(x,y) = 1_{x^2+y^2=1}(x,y)
[/math]
This function is not continuous in [math]\mathbb{R}^2[/math] but is continuous in angle [math]\theta[/math] if you convert to polar.

So, you can't really have a function that is properly continuous in some directions but not in others, but you can have a non-continuous function in one space that is continuous in a lower dimensionality projection.

Anonymous at Sat, 28 Dec 2024 20:00:24 UTC No. 16526547

>>16526534
>Probability is just real analysis and linear algebra.
If you are intention is to never go beyond baby's first measure theoretic probability course, then yes, that is an accurate description.

Anonymous at Sat, 28 Dec 2024 20:01:36 UTC No. 16526550

>>16526538
Do you want something more applied or more analysis heavy? Casella and Berger's Statistical Inference covers a lot of probability and statistics from a primarily calculus/linear algebra perspective.

If you want something more "pure math" there's no shortage of good textbooks. My favorites ATM are probably Le'Gall's Measure Theory, Probability and Stochastic Processes, as well as Bremaud's Probability Theory and Stochastic Processes.

Anonymous at Sat, 28 Dec 2024 20:04:00 UTC No. 16526554

>>16526547
The kid is asking about entry level probability theory. Your autistic screeds about concentration inequalities for random eigenfunctions can wait.

Anonymous at Sat, 28 Dec 2024 20:07:28 UTC No. 16526559

>>16526550
I have never actually taken a single applied course. My background, meager as it is, is entirely pure. Something tells me dipping my toes into applied would be good for me, but realistically, an analysis approach would be far smoother at this point.
I actually have a copy of Bremaud already so I supposed I'll start there. Thank you.

Anonymous at Sat, 28 Dec 2024 20:10:17 UTC No. 16526563

>>16526559
No problem, you've got this. Remember, there's always a deeper rabbit hole. Don't let yourself get chased into losing the forest for the trees. The basics will work for most use cases (read, statistics) and when you need something more sophisticated, you'll figure it out when you get there.

Anonymous at Sun, 29 Dec 2024 00:56:25 UTC No. 16526936

Anyone have a good rec for a calculus of variations resource/textbook? It can rely on some basic functional analysis/measure theory, but ideally it would be more applied than functional analytic in nature. Thanks in advance

Anonymous at Sun, 29 Dec 2024 01:49:19 UTC No. 16527037

>>16526500
No. You can represent the power set of k as the set of all functions from k onto {0,1} - think of the 1s of a function as the elements present in some subset, and the 0s as their complement in k. The set of all finite binary sequences is countable, so for there to be a k such that P(k) = |N|, k would have to be larger than any finite set but smaller than N, which immediately produces countless obvious contradictions.

Anonymous at Sun, 29 Dec 2024 01:57:10 UTC No. 16527054

>>16526936
Israel M. Gelfand's maybe, but havent read it

Anonymous at Sun, 29 Dec 2024 01:59:29 UTC No. 16527063

>>16527054
I ended up ordering that one on Amazon for cheap. I keep running into problems in information theory that require calculus of variations and I only really understand it in a super heuristic way. I'm hoping by spending some time learning calculus of variations I'll be more comfortable with maximum entropy/minimum distortion distribution problems (which seems to require CoV by default).

Anonymous at Sun, 29 Dec 2024 06:29:49 UTC No. 16527357

>>16526195
[math] \int |f^{y}|d\lambda=y^{2}\int \frac{|x|}{1+x^{6}}d\lambda(x)= y^{2}(\int_{[-1,1]} \frac{|x|}{1+x^{6}}d\lambda(x)+\int_{|x|>1} \frac{|x|}{1+x^{6}}d\lambda(x))\leq y^{2}(\int_{[-1,1]} |x|d\lambda(x)+\int_{|x|>1} \frac{1}{|x|^{5}}d\lambda(x)) [/math]

Anonymous at Sun, 29 Dec 2024 07:01:20 UTC No. 16527373

>>16526185
thanks! i wish this book were cheaper, it has been recommended to me a few times. pdf will have to do for now

>>16526188
i guess you may have to read the book to find out, huh

>>16526297
thanks, i assume you mean Jacobson Basic Algebra I and II, right? any comments on what you like particularly about Jacobson? looks like used copies of this aren't too expensive


>>16526339
how much do you need to brush up on, and to what end? how thorough do you want to be? how much time do you have to dedicate to working on it?

the best way of going about that would be to pick an appropriate textbook for precalc and for calc, and then to work through the exercises in relevant chapters

if you're reviewing you can probably get pretty sharp at this stuff in about a month

Anonymous at Sun, 29 Dec 2024 15:42:04 UTC No. 16527696

>>16527373
>how much do you need to brush up on, and to what end? how thorough do you want to be? how much time do you have to dedicate to working on it?
Enough to take a physics course. I'm about to enter my next semester for school in a week. I may only end up taking a couple of classes.
>the best way of going about that would be to pick an appropriate textbook for precalc and for calc, and then to work through the exercises in relevant chapters
Would Khan Academy work as well?
>if you're reviewing you can probably get pretty sharp at this stuff in about a month
Damn, I definitely should've started earlier then. lol

Anonymous at Sun, 29 Dec 2024 16:05:49 UTC No. 16527716

>>16527373
>thanks, i assume you mean Jacobson Basic Algebra I and II, right? any comments on what you like particularly about Jacobson? looks like used copies of this aren't too expensive
It treats a lot of advanced topics and is very comprehensive without being very slow like Aluffi. Aluffi has this patronising tone which I really dislike—treating the reader like a kid who needs jokes to be in their book. However, it's also a nice book if you don't mind the slow pace and its use of category theory from the beginning.

Jacobson gets through the basic stuff quite quickly and doesn't bog you down with loads of superfluous exercises though Aluffi does mark the important exercises. This is not to say, it is terse. It has a lot of examples to motivate the definitions. In particular, it explains in detail the relationship between dihedral groups and regular polygons, which surprisingly some books like Herstein don't bother with at all. Clearly, one can't describe it as a terse book. Neither is it dry like Dummitt & Foote.

Maybe this is pedantic, but it's the only basic book I have read that actually formalises the notion of polynomial expressions. I really like its treatment of polynomial rings as the ring generated by appending an additional element to a subring. I haven't seen it anywhere else.

I can't say much beyond the first few chapters. I never got beyond PID since I don't care much for algebra but it's the only algebra book that made the subject interesting for me. That said, I have heard it becomes really difficult when you get to the more advanced topics especially the ones in the second volume.

>looks like used copies of this aren't too expensive
Make sure you are getting the second edition. The first one doesn't talk about Chinese Remainder Theorem for some reason. Who knows what other basic stuff is missing.

Anonymous at Sun, 29 Dec 2024 16:31:33 UTC No. 16527735

>>16527373
>i guess you may have to read the book to find out, huh
Can you not give an example?

Anonymous at Sun, 29 Dec 2024 17:13:14 UTC No. 16527792

>>16527696
>Would Khan Academy work as well?
Nta, but at the basic level the best exercise sheet you can find is on Khan Academy, because you get instant feedback (they are interactive). And the videos are optional, it is better to look up things as you need them, from internet or your textbook of choice, instead of listening to the jeet.

Anonymous at Sun, 29 Dec 2024 17:41:42 UTC No. 16527834

>>16516921
You can assume that one of the nodes is [math] v_0 = 0 [/math]. Let [math] v_1, v_2, \ldots, v_k \in \mathbb{R}^n [/math] be the other nodes. Then [math] v_i^{\top}v_i = 1 [/math] for all [math] 1 \leq i \leq k [/math]. On the other hand, the angle between different nodes must be [math] \frac{\pi}{3} [/math] which implies [math] v_i^{\top} v_j = \frac{1}{2} [/math] for [math] i \neq j [/math].

We now need to show that the vectors [math] v_1,\ldots,v_k [/math] are linearly independent, as this implies that [math] n \leq k [/math] (i.e. there are at most [math] n+1 [/math] vectors). Consider the matrix
[eqn] M = (v_1, v_2, \ldots, v_k) [/eqn]
By considering the inner products see that
[eqn] A = M^{\top} M = \begin{pmatrix}
1 & 1/2 & \ldots & 1/2 \\
1/2 & 1 & \ldots & 1/2 \\
\vdots & \vdots & \ldots & \vdots \\
1/2 & 1/2 & \ldots & 1
\end{pmatrix}. [/eqn]
But you can show that [math] \det(A) = \frac{n+1}{2^n} \neq 0 [/math] which implies that [math] A [/math] has full rank [math] k [/math], so [math] M [/math] has full rank either.

Anonymous at Sun, 29 Dec 2024 17:53:29 UTC No. 16527851

>>16527834
The determinant should of course be [math] \frac{k+1}{2^k} [/math], not [math] \frac{n+1}{2^n} [/math]. As this implies that the rank of [math] M^{top}M [/math] is [math] k [/math], the rank of [math] M [/math], having [math] k [/math] columns, must also be [math] k [/math].

Anonymous at Sun, 29 Dec 2024 18:08:42 UTC No. 16527877

>>16516921
>>16527834
It's called a simplex. It's pretty easy to prove that such a graph must be a simplex. Prove the following simultaneously by induction
1) The n-simplex is the only complete equidistant graph with n+1 nodes
2) The midpoint of the simplex is the only point that's equidistant to all the vertices in an n-simplex inside the affine plane spanned by the simplex, and its distance from some vertex is <1

Base case n=1 easy.
Inductive step:
Take a n+2 complete equidistant graph. Remove a vertex v. By induction, what's left is a simplex. The vertex we removed cannot lie in the plane of the n-simplex. d(v,w) = sqrt(d^2 + a^2), where w is any vertex other than v, d is the orthogonal distance to the plane of the n-simplex, a is the distance from v' = v orthogonally projected onto the plane and w. Since all d(v,w)'s are equal, and d is constant, that means the orthogonal projection is the midpoint, so v is somewhere on a line orthogonal to the midpoint of the n-simplex. There are two unique points on this line satisfying the distances requirement, and both lie outside the plane of the n-simplex. Therefore we get a n+1 simplex.
The midpoint of the n+1 simplex is also somewhere on that line so its distance to one of the vertices is <1.

This has gotten a bit verbose but tl;dr you're forced to follow the inductive construction of the simplex as taking n-simplex, extruding its center to an orthogonal new dimension a bit to get a n+1 simplex.

Anonymous at Sun, 29 Dec 2024 22:54:56 UTC No. 16528395

>>16527735
of course i can't! i haven't read the book yet!

>>16527696
just to brush up on what you'll need for physics, i don't think you need a full month. if you're able to lock in and really focus this week, you can make good progress.

khan academy is alright for working on exercises--if you're just going in to take calculus based physics 1, i think it's fine to start with khan academy. but in the long run, the better way to study will be with textbooks.

check out Paul's Math Notes and use khan academy for exercises like >>16527792 said

a week is plenty of time, make sure you start yesterday though

>>16527716
but what if i am a kid who needs jokes to be in their book? jk
sounds like a good book, thank you for the rec

Anonymous at Sun, 29 Dec 2024 22:55:30 UTC No. 16528397

>>16528395
>cant read the book yet
IQ? Very important for math

Anonymous at Sun, 29 Dec 2024 23:07:27 UTC No. 16528421

>>16528397
yh lol theres basically no point to math unless you have a tested iq of 135+

Anonymous at Sun, 29 Dec 2024 23:09:31 UTC No. 16528426

>>16528421
>yh lol theres basically no point to math unless you have a tested iq of 135+
I have a 118 IQ and I can do math just fine.

Anonymous at Sun, 29 Dec 2024 23:11:12 UTC No. 16528429

>>16528426
No you can't
if ur anything short of answering research questions in low dimensional topology stfu

Anonymous at Sun, 29 Dec 2024 23:12:48 UTC No. 16528434

>>16528426
Idk, my IQ is 131 and my mathematical intuition is rather limited. The cutoff being slightly higher makes sense to me.

Anonymous at Sun, 29 Dec 2024 23:14:46 UTC No. 16528439

>>16528429
Don't be ridiculous.
>>16528434
Cutoff for what?

Anonymous at Sun, 29 Dec 2024 23:19:23 UTC No. 16528446

>>16528439
>ridiculous
being below par for a doctoral student is not ""just fine""

Anonymous at Sun, 29 Dec 2024 23:42:16 UTC No. 16528484

>>16528446
Not specializing in finite dimensional topology doesn't make one below par for a doctoral student.

Anonymous at Mon, 30 Dec 2024 00:37:04 UTC No. 16528597

>>16528397
nice bait, comprehensionlet

i can't provide an example [because] i have yet to read the book, the fuck?

my iq is at least 69 so i think i'm equipped just fine

Anonymous at Mon, 30 Dec 2024 00:44:04 UTC No. 16528617

>>16527716
Personally, I kind of like when there's some jokes/levity in the book. It gives the text a bit more personality than being just being "Springer Graduate Texts in Mathematics: Introduction to Categories for People Who Already Know Categories." If anything, I'd say the more difficult the topic, the more necessary it is to sometimes include something a bit less serious and, dare I say, fun, so it isn't just a dry slog the whole time.

Maybe that's just my dumb engineer brain talking, and I don't appreciate the "rigor" of three line elegant but incredibly opaque proofs with no instructional quality.

Anonymous at Mon, 30 Dec 2024 00:47:10 UTC No. 16528619

>>16528617
one must develop the nebulous concept of "mathematical maturity" to really appreciate how hysterical the elegant but incredibly opaque three line proofs can be

Anonymous at Mon, 30 Dec 2024 00:57:31 UTC No. 16528638

>>16528619
I guess if I didn't learn it in my applied math undergrad or the math electives I took in my PhD program, it might be beyond my capacity.

Anonymous at Mon, 30 Dec 2024 01:03:58 UTC No. 16528649

>>16528638
i have yet to find a non-insane way (i.e. laughing my ass off when something is just completely over my head) to appreciate it in my pure math undergrad, so who's to say whose capacity it is really beyond

Anonymous at Mon, 30 Dec 2024 02:24:53 UTC No. 16528847

>>16528649
My favorite is when they define a dozen pages worth of Lemmas so that the proof for their 2 line theorem can be 3 lines long instead of just having a longer theorem and longer proof.

My spectral theory textbook does open with some joke like "This entire textbook is supporting lemmas for the spectral theorem" if I'm remembering like that.

Anonymous at Mon, 30 Dec 2024 02:25:55 UTC No. 16528853

>>16528847
Remembering correctly.* Sorry, I need sleep.

Anonymous at Mon, 30 Dec 2024 02:30:06 UTC No. 16528869

Anybody else become a complete shell of themselves after learning so much maths?

I can barely get two words out these days

Anonymous at Mon, 30 Dec 2024 02:36:30 UTC No. 16528891

>>16528869
Grad school just did that to me in general.

Anonymous at Mon, 30 Dec 2024 03:30:40 UTC No. 16529146

Use your big quantitative brains to write an equation for x = distance from object where a magnifying glass flips an image

Anonymous at Mon, 30 Dec 2024 03:36:36 UTC No. 16529166

>>16529146
You'd need to know how the lensing works.

Anonymous at Mon, 30 Dec 2024 03:41:48 UTC No. 16529180

>>16529166
I don't know and you don't figure it out

Anonymous at Mon, 30 Dec 2024 03:48:51 UTC No. 16529209

>>16529180
You'd need to know the lensing effects to start the problem (how is the lens curved and how does it change the image projection). Without having a formula for the way that light curves when passing through the lens, you'd have no way to measure projected distance.

Anonymous at Mon, 30 Dec 2024 04:10:18 UTC No. 16529298

>>16529209
I wear glasses i know they are different
I'm asking what variables in the lens affect the output. No this isn't my homework.

Anonymous at Mon, 30 Dec 2024 04:29:54 UTC No. 16529349

>>16529298
I'm not an optical physicist, so my answer is probably lacking some nuance. My understanding is there's basically three components that matter for lensing, thickness, curvature, and spectral properties (does it impact all colors/wavelengths uniformly or does it attenuate/distort some more than others?). Curvature will tell you both where the focal point is, and how steep the tangent lines are at any given point approaching the focal point.

If you have a very sharp curvature, you can expect very large distortions at the edges (meaning you'll get a lot of non-linearity and stretching in the image). Beyond that, I'd recommend you look at an optical physics reference. It's been a while since I've opened it up, but I remembered liking the bit of optics coverage in Serway and Jewett's university physics book.

Anonymous at Mon, 30 Dec 2024 05:06:12 UTC No. 16529399

>>16528617
I think it depends on the joke. There are some subtle deadpan jokes you can add it academic literature that may cause a nice chuckle. For instance, an actually funny joke is the statistical mechanics suicide one, or the "better known for other work" paper.

But then you have shit like Jay Cummings which literally describes itself as a "meme filled book," which just makes cringe. Aluffi is not that bad, but it has an entire theorem environment dedicated to a joke, and then a paragraph having to explain the joke. That's not funny and just wastes my time.

>the more necessary it is to sometimes include something a bit less serious and, dare I say, fun, so it isn't just a dry slog the whole time.
A good book wouldn't need humour to be fun or not-dry. The exposition of the topics itself ought to be the primary aspect making the book fun.

Anonymous at Mon, 30 Dec 2024 05:34:47 UTC No. 16529443

>>16529146
Everyone who takes an optics class learns this. Any introductory book will teach it. Optics is one of the original important uses for Electrodynamics. They start out with a certain reasonable approximation, talk about the index of refraction, then you go into basic 1 sided lens, then you do a 1 sided mirror, then you go into 2 sided lenses and 2 sided mirrors. The derivation is basic enough that I'd imagine you can just google the answer instead of looking in a textbook. On the Physics GRE there's like one or two questions on it, funny enough.

🗑️ Anonymous at Mon, 30 Dec 2024 05:35:48 UTC No. 16529445

>>16529443
ignore 2 sided mirror

Anonymous at Mon, 30 Dec 2024 05:37:14 UTC No. 16529449

>>16529443
*2 mirrors

Anonymous at Mon, 30 Dec 2024 08:59:35 UTC No. 16529579

>>16528847
lmao that rules

>>16529399
yeah, there's an art to it, and deadpan humor plays very well with things that are presented more formally

the statistical mechanics suicide one is a hilarious example

maybe math is ruining my sense of humor because lately the funniest thing in the world to me is just stating a tautology. normal people laugh at this too though

Anonymous at Mon, 30 Dec 2024 14:05:54 UTC No. 16529805

These typesetting mistakes bother me a lot.

Anonymous at Mon, 30 Dec 2024 16:20:07 UTC No. 16529946

I sucked at math in school and always had a suspicion that the people were good at it were better at mentally framing things. For instance when multiplying 20 x 40, you could either think of these numbers as 2 tens and 4 tens, or you could think of them as distinct entities. If you frame it in the first sense then you’d simply have to multiply 2 x 4 and add two zeros to get your answer. Frame it in the second sense and it becomes a seemingly bigger problem. Are there books that teach you how to fundamentally hone such problem solving skills?

Anonymous at Mon, 30 Dec 2024 18:04:08 UTC No. 16530029

>>16529946
>20x40
40 x 10 = 400, then double it.

Anonymous at Mon, 30 Dec 2024 18:57:54 UTC No. 16530072

>>16527037
Is there no way to take a couple axioms away from ZFC so that P(finite set) isn't always guaranteed to be finite?
What about the other option - is there a set of axioms where there is no smallest infinite set?

Anonymous at Mon, 30 Dec 2024 19:21:46 UTC No. 16530086

>>16530072
You would have to take away most of the axioms in zfc or nbg to get either of those, idk more about logic than that.

Anonymous at Mon, 30 Dec 2024 19:26:58 UTC No. 16530089

>>16530072
>Is there no way to take a couple axioms away from ZFC so that P(finite set) isn't always guaranteed to be finite?
Technically depends on what you mean by finite, since ZF without C can have Dedekind-finite infinite sets (that is, infinite sets which aren't in bijection with any proper subset of themselves), and for any infinite set X, P(N) injects into P(P(X)).
By the usual meaning of finite (in bijection with some Von Neumann natural), not in any sane set theory, no.
>What about the other option - is there a set of axioms where there is no smallest infinite set?
Sure, it's consistent with ZF (assuming the consistency of a measurable cardinal) that there is an infinitely decreasing chain of cardinals. Of course, those are incomparable with N so you can't meaningfully say whether they're bigger or smaller than it.
As far as the well-orderable infinite sets go, no.

Anonymous at Mon, 30 Dec 2024 22:18:05 UTC No. 16530226

>>16529805
Where is the mistake here?

Anonymous at Mon, 30 Dec 2024 22:19:29 UTC No. 16530227

>>16529805
>>16530226
Nevermind, this can indeed be confusing.

Anonymous at Mon, 30 Dec 2024 22:50:36 UTC No. 16530248

>try to learn lean from book
>get question with stuff like "assume a - 5b = 4" and "b + 2 = 3" prove a = 9
>Hmm I think I will conclude b = 1 and then substitute
>Try to write that
>can't do it
>check book
>there's no way to introduce that conclusion
>totally confused
>Check book again, must have missed out to do this
>"So the way you do this problem is to write a - 5b + 5b, then 4 + 5b, then subtract 10 from 4 and then add 2 to b so you have -6 + 5(b+2) :). The rest is trivial"
>??????????????????????
>delete lean

Anonymous at Mon, 30 Dec 2024 22:54:38 UTC No. 16530253

>>16530248
Which book, which chapter, which exercise?

Anonymous at Mon, 30 Dec 2024 23:04:24 UTC No. 16530268

>>16530253
Book is "The mechanics of Proof"

🗑️ Anonymous at Tue, 31 Dec 2024 00:45:48 UTC No. 16530345

>>16530248
In general, for "simple" algebra like that your strategies should be:
1) Try to reduce the equations
[math]
b+2=3 \Rightarrow b= 3-2 = 1
[/math]
Now you have b isolated and not including any other variables.
2) Substitute and solve:
[math]
a-5b= 4 \Rightarrow a-5(1) = 4 \Rightarrow a= 4+5 = 9
[/math]

Just go slow, give yourself a bit of patience and if you don't get the answer you're looking for, give yourself a few minutes and try it again but use a different strategy until something works.

Anonymous at Tue, 31 Dec 2024 00:48:47 UTC No. 16530349

>>16530248
The mathlib theorem you are looking for to cancel out subtraction like that is eq_sub_of_add_eq: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Basic.html#eq_sub_of_add_eq. You have to get good at mathlib-fu to learn Lean.

Anonymous at Tue, 31 Dec 2024 00:50:49 UTC No. 16530350

>>16530248
>>16530349
Want to add also, I'm not sure what this retarded book you're using is, but if you're already familiar with how to make proofs on paper this "Mechanics of Proof" book doesn't look great. I recommend "Mathematics in Lean" by Avigad and Massot if you already have the 'mathematics' part down and are just trying to learn Lean: https://leanprover-community.github.io/mathematics_in_lean/mathematics_in_lean.pdf

Anonymous at Tue, 31 Dec 2024 01:24:12 UTC No. 16530365

>>16530349
>>16530350
The book is called "Mechanics of proof" by Heather MacBeath

I jumped into it because I finished the Natural Numbers Game without too much trouble and thought a proper textbook would be the next step. My mathematical skills are fairly below average but I didn't think The NNG was difficult.

Anonymous at Tue, 31 Dec 2024 01:40:23 UTC No. 16530376

>>16530365
Well I recommend "Mathematics in Lean" to you as a serious sequel to the NNG. The problem of subtraction cancellation that you are having even is an exercise in that book, so I'm sure you'll find out how to solve your problems from that one.

Anonymous at Tue, 31 Dec 2024 09:17:39 UTC No. 16530622

Bros what is it like to think about math? Is it fun?

Anonymous at Tue, 31 Dec 2024 09:28:03 UTC No. 16530634

>>16530376
NTA but going through that one myself. It's very nice.

Anonymous at Tue, 31 Dec 2024 10:46:07 UTC No. 16530726

>none at present

Anonymous at Tue, 31 Dec 2024 10:55:29 UTC No. 16530734

>>16530726
Why are my tax dollars going to fund homological algebras if they have no practical applications?

Anonymous at Tue, 31 Dec 2024 11:18:20 UTC No. 16530749

What the fuck are subtraction and division?

Anonymous at Tue, 31 Dec 2024 11:34:20 UTC No. 16530758

>>16530749
Mathematicians are still trying to figure that out.

lowercase sage !!IaxlA1xvEP/ at Tue, 31 Dec 2024 11:51:50 UTC No. 16530770

>>16530749
>What the fuck are subtraction
Addition of additive inverse.
>and division?
Multiplication by multiplicative inverse.

Anonymous at Tue, 31 Dec 2024 12:13:13 UTC No. 16530785

>>16530770
Fuck you

Anonymous at Tue, 31 Dec 2024 17:25:19 UTC No. 16531010

>>16530376
I am that anon and I am working through it as well and I find it much easier than Mechanics of Proof, which is odd because Mechanics of Proof is advertised as the easier of the two.

Anonymous at Tue, 31 Dec 2024 19:13:29 UTC No. 16531079

I used to fantasize about being given an abel prize or a fields medal but recently I've found out they call you in advance to ask you if you're going to accept it to avoid another Perelman's situation (a mathematician who has a field's medal has revealed this on numberphile). That means it's possible that there have been mathematicians who were about to be awarded a fields medal or an abel prize but who rejected it but nobody knows about it except themselves and the prize committee. Which is an interesting thought.

Anonymous at Tue, 31 Dec 2024 19:42:36 UTC No. 16531092

>>16530758
How long will it take?

Anonymous at Tue, 31 Dec 2024 20:06:05 UTC No. 16531098

>>16530785
Fuck you, he gave the correct answers.

Anonymous at Tue, 31 Dec 2024 20:45:35 UTC No. 16531120

>>16531092
Depends. They're still debating if Thrembo exists. Could be years.

Anonymous at Tue, 31 Dec 2024 21:57:54 UTC No. 16531206

>type "simp"
>proof clears
>No clue why
>move on to the next question

Anonymous at Tue, 31 Dec 2024 21:59:15 UTC No. 16531207

>>16531079
No, because anyone who seeks so much attention that's they reject such prizes would publicly reject it.

Anonymous at Tue, 31 Dec 2024 22:06:05 UTC No. 16531209

>>16531206
That's fine, the goal of mathematics is not personal understanding but knowing that it's been rigorously checked to hold.

Anonymous at Tue, 31 Dec 2024 22:10:31 UTC No. 16531211

>>16531209
It's amazing how many problems I solve without knowing exactly what I'm doing. It's totally opposite from how I think of math normally. Almost like I'm feeling my way through the problem instead of thinking.

Anonymous at Tue, 31 Dec 2024 22:18:07 UTC No. 16531216

>plug hw into wolfram
>never learn anything
>fail exam

Anonymous at Tue, 31 Dec 2024 22:22:00 UTC No. 16531219

>>16531216
Not wolfram, chudcel. it's a PROOF ASSISTANT. A little out of your league I bet huh? heh...well we can't all be geniuses.

Anonymous at Wed, 1 Jan 2025 04:28:07 UTC No. 16531513

>>16531206
It works because someone else already proved the theorem in the standard library or in mathlib and the simplifier found and applied that proof. That's proof by "its in the textbook."

Anonymous at Wed, 1 Jan 2025 07:19:26 UTC No. 16531569

>>16530734
you have a problem with constructing a giant pair of tits you can walk into??
what are you, a fag?!?!

Anonymous at Wed, 1 Jan 2025 10:41:48 UTC No. 16531644

>Tits "buildings"

Anonymous at Wed, 1 Jan 2025 11:06:42 UTC No. 16531646

>>16530726
source?

Anonymous at Wed, 1 Jan 2025 11:39:22 UTC No. 16531651

>>16531646

Anonymous at Wed, 1 Jan 2025 12:06:37 UTC No. 16531658

>>16531513
Is there a way to see exactly which lemmas `simp` is applying?

Anonymous at Wed, 1 Jan 2025 12:07:08 UTC No. 16531659

>>16531658
'simp?' with a question mark

Anonymous at Wed, 1 Jan 2025 12:15:38 UTC No. 16531663

>>16531659
Thanks!

Anonymous at Wed, 1 Jan 2025 15:27:07 UTC No. 16531750

The question of "what is subtraction" has interesting answers but I'll only say one here. It's obvious that any two integers can be subtracted from each other but it isn't obvious how two group homomorphisms can be subtracted from each other.
There are many different operations out there that reduce to ordinary subtraction in the case of the integers but are inequivalent for other "types" of input. I don't think it's possible to get a good bigger answer as to which one is "true subtraction" unless we have a way to map the space that contains all of these different operations.

Anonymous at Wed, 1 Jan 2025 16:24:10 UTC No. 16531784

>>16531750
Also, if category theory could map spaces like that it would have already been done by now. Type theory can probably do it, though.

Anonymous at Wed, 1 Jan 2025 16:26:20 UTC No. 16531787

>>16531750
Now do division.

Anonymous at Wed, 1 Jan 2025 17:15:09 UTC No. 16531806

>>16531750
Even understanding adding 1 to a number is beyond current mathematics. Division is like light years away.

Anonymous at Thu, 2 Jan 2025 02:56:09 UTC No. 16532231

Need cleo gf

Anonymous at Thu, 2 Jan 2025 07:00:02 UTC No. 16532336

>>16532231
you post here? what the fuck?

Anonymous at Thu, 2 Jan 2025 20:49:42 UTC No. 16533107

This is literally me omg

Anonymous at Fri, 3 Jan 2025 05:58:47 UTC No. 16533585

Is a person here able to suggest a quality university catered to mathematical foundation, an alternate request would be for a particularly fine public course's syllabus.

Anonymous at Fri, 3 Jan 2025 17:38:43 UTC No. 16534123

Bros, How can I get a scholarship for a master's degree in mathematics?
My university has an agreement with the University of Paris Saclay, but since my country is garbage, they only let me apply for the master's degree in applied mathematics, but I would like to do pure mathematics.

I have been searching on different websites for scholarships but the only thing I have been able to find have been scholarships that only include tuition fees.
What I'm looking for are scholarships like the Sohpie Germain foundation, (https://www.fondation-hadamard.fr/en/our-programs/transversal-programs/graduate-program/apply-for-a-sophie-germain-scholarship/) , which give you money while you study so you can live.

Anonymous at Fri, 3 Jan 2025 18:30:29 UTC No. 16534161

I hate lean's syntax so much.

Wondering if Metamath might be better to go into 2bh

Anonymous at Fri, 3 Jan 2025 20:31:18 UTC No. 16534295

>>16534183