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I literally live like pic related. I'm poor, don't work (welfare and in subsidized housing), mostly crazy, live in the ghetto surrounded by addicts and criminals, and take 4th year math classes at a local university (pretty big name actually). All I do is math. Doing math and not worrying about anything else is pure kino. Sometimes I have the odd thought of "What if I got a regular job?", "What if I got a girlfriend?", but it all seems so pointless and meaningless.

>>16188044
I wish I worked in a place like that

>>16188044
>I'm poor, don't work (welfare and in subsidized housing)
You are the problem.

>>16188044
>don't work
Why don't you become a research assistant at your university?

>>16188048
Code monkey.
>>16188133
Underaged.

First year math student here. I've been reading Set Theory with Applications by Lin & Lin and another Logic book by Xavier Caicedo but I find some parts too dense to understand. Could I get book recommendations for Logic and Set Theory?

>>16188177
Ebbinghaus

>>16188044
That whole scene is kino.

>>16188177
Axiomatic Set Theory by Patrick Suppes

Bump

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I want to learn statistics and probability, are there any engaging websites, videos or books I can use for that? I'm not a math major and I don't have a very solid math background, I basically know what a first-year college student in STEM would be expected to know (more or less).
I'm only interested in probability and stats and if possible I'd like to take it to a high level.

>>16189421
I heartily recommend harvard's stat110 course, accessible here: https://projects.iq.harvard.edu/stat110/home.
The book is fantastic and freely available, as well as nice accompanying lectures on youtube.
The focus is mainly on elementary probability (though it touches on statistical stuff a good amount), which you should know before starting with any amount of statistics.
Lots of exercises, some with solutions, you should do as many as you can.

>>16189427
Thank you.
It seems to me that a lot of branches in math draw upon other branches; for example, continuous probability distributions require calculus. Can I dive into this course without any advanced prerequisites (by advanced I mean anything past precalc)? Is there a point where probability theory and statistics start to intertwine with other subjects like, I don't know, geometry, topology or whatever, to the extent that I'd need to study those as well?

>>16189431
To an extent, that's the case, yes.
For this specific book, you should definitely know how to solve an integral like [math]\int_0^1 xe^x d x[/math] or what a jacobian is.
In fact, I would be surprised if there's a good course on probability or statistics which didn't require knowledge like that, so that's what you should probably learn first (or, better yet, at the same time).

For a lot of statistics (and probability as well), calculus is enough.
If you want to get mathematical, you'd need a reasonable amount of analysis, but most of the `pure math' subjects don't really intersect with probability and stats very much.

>>16189444
I see. I'd like to get pretty deep if possible, stuff like (but not limited to) bayesian statistics, time series and stochastic processes. If calculus is enough regardless of how far I take probability and stats, then I'm good, assuming the required background can be learned in parallel reasonably quickly.

>>16189475
For the surface level stuff, calculus is enough, yes.
In fact, you can get pretty far only knowing calculus, and for the topics you mention there are plenty of `application-focused' books which you could read after an introduction like stat110.
If you have some application in mind this is what I'd recommend.

If you are also interested in the mathematics behind probability and statistics, there will be a big lurch at some point, when books and courses start assuming years of analysis knowledge.
There's no real way around that at, say, a postgraduate level, but I wouldn't worry about it for at least a year, if you're just starting out.

>>16189482
>the mathematics behind probability and statistics
Sorry, I'm not exactly sure what this means. Are you referring to something like "understanding WHY probability works the way it does" as opposed to simply applying it?

>>16189509
I mean there is a certain amount of mathematical formalism required in order to define probabilistic objects and reason with them.
I wouldn't say it answers any question of `why', it is moreso that you learn in terms of what specific mathematical (non-probability-related) concepts probability is defined, and how these concepts work.
For a nice example you can look at this article: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability).
You could probably find an answer to this paradox knowing only the basics of probability, but in order to define the question in precise terms you need more mathematics.

>>16189518
I understand. My interest lies mainly in applying probabilistic and statistical methods to real-world scenarios, so I might not be exposed to this kind of formalism.

>>16189535
Yeah, you're fine, a great deal of people don't want to bother, and rightfully so.
This means there's plenty of material that brushes these formal details under the rug, so to say, and if you're doing real-world statistics you almost certainly won't be hindered by not knowing it.

>>16189536
Well, I'll start out with stat110 and see where it takes me, then. Thanks again for your advice.

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What exactly neighborhood of [math](x,y)[/math] mean here?

>>16189697
A neighbourhood of a point is a set which has an open subset containing that point.
the condition is basically that if you consider the region of [math]\mathbb{R}^2[/math] where both [math]f_x[/math] and [math]f_y[/math] are defined, [math](x_0,y_0)[/math] sits in the interior of that region (and not on its boundary or, worse yet, outside of it entirely)

>>16189697
Open ball

>>16189697
a circle centered at (x,y) of finite size whose edge is open.
for ur pic, f_x and f_y need to exist in all point inside the circle.
the point is that f_x and f_y not only exist at (x,y), but for all points in an around it, for at least one circle. This can be a big circle or a really small one - it just needs to be of finite size

>>16188047
Me too

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>>16189421

we're currently porting ...

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I'm a programmer looking to implement a linear solver using singular value decomposition. In the book that I'm reading, this lemna is mentioned and I'm a bit confused about the terminology. It is said Q and H are fully determined by the first column of Q. What exactly does it mean for a matrix to be "determined" in this case?

>>16189540
If you have a decent calc background and some linear algebra, there are a ton of good probability and statistics books you could read. For introductory books I like Bertsekas Introduction to Probability, and Sheldon Ross's Introduction to Probability and Statistics for Engineers and Scientists. Both of these books cover a decent amount of the foundations of probability and statistics and approach the topics from the of real world uses of probabilistic ideas.

Another fairly practical book (but at a little bit of a higher level of mathematics) is Murphy's Probabilistic Machine Learning an Introduction. Despite the title indicating the book to be about ML, it has a lot of useful introductory material on univariate/multivariate probability, the basics of the linear algebra needed, and the basics of frequentist/Bayesian statistics.

>>16191877
If a matrix is singular, meaning its determinant is 0, you can't invert it because there are potentially multiple entries in the pre-image of any particular vector of your output space (especially if you are looking at the pre-image of the zero vector).

I'm not 100% certain, but my intuition is that they mean that Q can be uniquely specified if one knows the first column of Q. A.k.a. Q = P q_1 where P is some non-singular projection matrix and q_1 is the first column of Q.

Going to take a Calc class either during summer or the following semester. I don't intend on reading the assigned text (no one ever bothers with it anyway.), but I will consult a Calc textbook. Forgoing one entirely would be retarded, I'm just using a supplement instead of using whatever edition of Stewart they decide on. The Maths course sequence for stemfags in my college is (generally): Calc 1, Calc 2, Multivariable Calc, Linear Algebra/Diff. Equations. I already have a copy of Morris Kline's Calculus textbook, but I also found a copy of Hass, Weir, and Thomas' Calc textbook that I can get my hands on for cheap https://books.google.com/books/about/University_Calculus.html?id=9tegBwAAQBAJ. My question is whether or not the latter one is a "good" alternative to Stewart, and how far both textbooks would take me along the course sequence. I really like the conversational tone of Kline's text, and I like that each problem set has a fully worked-out solution available; it shows me the proper way to go about solving a problem instead of just brute-forcing it like a based retard.

>>16192097
maybe try this
https://intellectualmathematics.com/dl/calculus.pdf

>>16188959
I got that book

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For [math]{\mathcal C}[/math] any collection of sets (e.g. a topology), and [math]U\colon {\mathbb N}\to {\mathcal C}[/math] a sequence of sets, i.e. [math]U_n\in C[/math] for all [math]n[/math],
is there a name for the property that

[math] \lim_{m\to\infty} f\left(\bigcup_{n=0}^m U_n\right)[/math],
exists

and similarly is there a name for
[math]f\left(\bigcup_{n=0}^\infty U_n\right) = \lim_{m\to\infty} f\left(\bigcup_{n=0}^m U_n\right)[/math],

where as usual, the infinite union is given by
[math]x\in \bigcup_{n=0}^\infty U_n\iff \exists k. x\in U_k[/math],

>>16193984
Continuity from below

>>16193995
nice, thx

>>16187402
Sorry if noob question but, what is the "analysis" equivalent to differential equations? Real analysis corresponds roughly to calculus, and measure theory to probability theory, so what do differential equations correspond to? Just more advanced real analysis?

>>16194885
Measure theory is still very much calculus. It is just the integral part of calculus. It literally gives you a theory of integration, and most measure courses end on the Lebesgue reformulation of the fundamental theorem of calculus and differentiation under the integral sign.

Differential equations are still differential equations. It's just before vs. after all the integrals become Lebesgue integrals.

>>16194895
Thanks anon. I will go over measure theory first and in details.

>>16195622
No worries. Depending on what your field of interest is, there are different levels of "difficulty" you could go for and have enough of a takeaway.

For a pretty "easy" take on measure theory, Axler's book on it is free and not too bad. Heil's Introduction to Real Analysis is also pretty easy and covers most of the standard material (though, for some reason he uses the term "exterior measure" instead of outer measure).

If you want something more rigorous and formalized, you could go for Royden, the first half of Rudin, Folland's real analysis, Wheeden etc. if you plan on being an analyst that rigor is probably necessary, if you are only planning on applying this stuff to other problems it might not be.

since mathematics works perfectly fine without the notion of infinity (matlab works fine even though it doesn't have it), then why is it necessary to have it at all?

>>16195982
Matlab has Inf. What are you talking about? They even have the ability to create arrays of all Inf values.

Freshman here. Should I be able to integrate by parts in my head if I want to be worth a damn? I see a TA who can do it and I feel like I'll never pass if that's the expectation

>>16196018
Your TA has probably been doing it for literal years. Just keep practicing the slow way and you'll eventually get the hang of it.

When I teach linear systems theory my students are always surprised by the things I can do quickly in my head while checking their work. If they had spent the last 7 years teaching the course and the same basic material they'd get quick with it too.

>>16196007
I'm talking about the mathematics that the code is built up with. That should be obvious. Computers don't use the notion of infinity, they only deal with finite computations.

>>16196067
Mathematicians too do only "finite computations" (deductions of finitely many steps).

What are the non trivial paths (non reflexive ones) in homotopy type theory ? Why can't we just assume uniqueness of identity proofs ? What does intensional equality bring on the table ?

>>16196067
Computers absolutely have to interact with the notion of infinity even if they are restricted to finite precision integers/floating points for individual operations.

Firstly, any practical computer code you build for numerical analysis must have some heuristic mechanism to determine whether it is not converging (so it halts). There's a few ways to do this, with the simplest being a maximum number of iterations.

However, in many real implementation scenarios you don't want to limit to any particular number of iterations, so you must develop heuristics to determine whether allowing further iterations would or would not produce convergence without strictly restricting the number of iterations. The study of these kinds of implementation details and how to computationally determine convergence is a huge part of both numerical analysis and the computational side of optimization theory.

Secondly, recursive search strategies also need to deal with the notion of countably infinite sets. If you are searching for some optimal path or optimal solution and the size of the set of things you are searching through grows with every iteration, you need some way of ensuring convergence as the number of iterations grows. This is itself dealing with infinite sets directly.

AG PhD dropout here (over a decade ago), wanna get back into learning shit recreationally. Any new cool books to start up again that are light on the prereqs? Preferably tending towards algebra or geometry and light on analysis. Was thinking starting with Aluffi Algebra chapter 0 since I've already read it front to back a few times, and maybe follow up with Bosch AG/CM, but I wonder if there's a newer algebra fast-paced alternative

I suck at abstract algebra so much
How do I stop sucking

>>16196788
practice
read the textbook
when you read the textbook/notes/examples you're given, make sure you actually understand every step. if something's really tripping you up, reread it, cross-reference things in your book, ask other students or your teacher, etc. Each proof, you should be able to explain it well enough that, given a little time to refamilizarize yourself, you could recreate it yourself on paper
-- then, practice. A decent book will have exercises that have you doing similar things to what you've seen in the chapter. You start with imitation and minor changes to proofs you've seen, and with more time and practice and exposure to concepts, you grow in your ability to look at a problem and understand what ideas you've seen might apply, be more creative in how they're applied, etc. This is "mathematical maturity," and there's not really a shortcut.

>>16196018
The TA has probably seen the exact questions your doing before with other students and the exact issues they had.

Dont compare your self to others. If you must dont compare yourself to the person teaching you

>>16196285
Hard to say without knowing your level and what you remember, but you could always take up some algebraic number theory if you feel like looking into something new. After a while, you can even close the loop and look into the connections with AG. The book Primes of the form x^2 + ny^2 by Cox gives a nice motivation for the topic and is written by a non-specialist (an algebraic geometer actually)

Another option that can be nice to learn on your own is the computational side of things. For AG, I hear a lot of good about the book Ideals, Varieties and Algorithms.

I feel like I'm shilling for Cox now even though he's not my favourite author, but I guess he writes nice books for first approaches into a topic.

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Has anyone read >>16197831?
I'd like to understand the topics covered in discrete math before I learn formal data structures & algorithms for computer science.
This book seems like a good fit.

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I tried to test how good the AI is in math and it amost looked like it was onto something until it pulled a nonsense answer out of nowhere.

>>16198194
you can tell it's not thinking like a mathematician from the start because it assumes that the triangles share all three vertices instead of asking for it to be made explicit whether they do or if they only share the one edge

I only suck at research because I have no access to amphetamines.

>>16196953
>...It's important to remember that...
etc, etc.

Recommend books on mathematical physics topics

>>16198015
Actually that's a great rec, I've always had that book in my backlog since loving my undergrad algebraic number theory class but never got around to it. And also reminds me I wanted to go through Eisenbud's 3264 and all that at some point in my life

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exercises that filtered you?

https://en.wikipedia.org/wiki/Bernoulli_scheme
how can i do this in my head? I'm playing through persona and it would be incredibly useful to be able to calculate that without a calculator

This is a hypothetical question that probably belongs in /sqt/, but I'd like the input of some actual mathematicians.

If somebody were to prove the Collatz conjecture is true, but in such a way that it provided very little insight or utility in any other areas of mathematics, would that person be a shoo-in for a Fields medal? From what I've heard, the Collatz conjecture is only really interesting because it's so deceptively difficult to solve, not because it's an important problem. It makes me wonder whether the pure genius required to solve it would be deemed worthy of the prize, regardless of whether the techniques in the proof itself were of any benefit elsewhere.

Anyone here fuck with categorification? It's central to what I research but I'd be elated if I could find the right class of algebraic objects that lend themselves to a nice canonical categorification that behaves well functorially. Kinda hard when you have bend over backwards just to get Z[x]

>>16199915
In principle, the proof of the Collatz conjecture would also have implications for other repeated sequences of integer operations. This could be very meaningful for all sorts of things like random discrete point processes, sorting and indexing algorithms, and solving for termination of other random arithmetic sequences (meaning potentially decryption of pseudorandom sequences given that they are generated by processes equivalent to repeated integer arithmetic).

>>16199980
Do you just find category theoretic mathematics more compelling/interesting than set-theoretic mathematics? Most people generally seem to regard category theory as schizo-babble.

>>16199980
Are you a tranny and/or some demented autist (furry/MLP-watcher/etc.)? Be honest

>>16199915
>would a proof of Collatz be good enough for a fields medal?
yes, in my opinion. look at the most recent few fields medal recipients and what they actually did
>fano theory
>perfectoid spaces
>hodge theory
nobody knows wtf any of this mumbo-jumbo means, and I honestly think it’s damaging the reputation of the Fields Medal. an award for Collatz would be like a breath of fresh air, for everyone involved

>>16200186
Fun fact, the Fano that Fano theory is named after is the brother of Robert Fano, lab-partner of Claude Shannon and one of the pioneers of information theory.

This place is much better without cult of passion posting schizo shit, I noticed today that he has been absent

>>16196788
Why do you care? It's not real math like Analysis.

It's known that 5 queens are enough to dominate a standard 8×8 chessboard, and there are many ways to do so (pic related is one way). My question is, do any of those ways allow a sixth queen to be placed on the board, such that the sixth queen can protect the other five?

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>>16200459
4chan stop eating my images

[math]\pi cot(\pi x)f(x) = \sum\limits_{n \in \mathbb{Z}} {f(n) \over x-n}[/math]

I am working on proving that if
[math]f: \mathbb{R} \rightarrow \mathbb{R}[/math] is open, then it has an interval on which it's continuous.
I came up with the idea that otherwise any open interval maps to the union of countable set of isolated points, but it looks not enough rigorous for me.

>>16188177
My recommendations are "Introduction to Mathematical Logic" by Hodel and "Basic Set Theory" by Levy. Both are Dover books and are cheap and I think there fairly approachable. The latter goes into model theory (if you don't care about that) later on, but the early parts should be fine.

>>16200573
let f(x) = x for irrational x and f(x) = x + 1 for rational x

>>16200611
not open. wtf anon

>>16200635
Yeah, I am trying to prove it from the point of view of topology, using general definition of continuity, but it goes hard.

>>16188177
I've heard very good things about Kunen's "Basic Mathematics" (has logic, set theory, some model theory and recursion theory from what I know, idk my copy is still on its way) and then "Set Theory" (covers set theory, infinitary combinatorics so I'm guessing large cardinals? something with measure theory and topology (large cardinals again?) and forcing). Also Enderton's "Elements of Set Theory (ordinals, cardinals and that kind of stuff - might be a good introduction/first book).

For a different (categoric) approach take a look at "Sets for Mathematics".

>>16200573
>>16200745
You can't prove it since it is false.
Let f(x) = sum[ (1/n)*(-1)^b(n) ] if it converges and 0 otherwise where b(n) is the nth binary digit of x-floor(x).
f((a,b)) = R for every a<b and f is continuous nowhere.

>>16200793
Good counterexample, but what if restrict [math]f[\math] to be closed as well as open.

How to learn to do integrals (both definite and indefinite) that aren't just u-substitution or other high school stuff?

>>16200859
get a big textbook like Stewart or Thomas. it can be an old edition doesn’t matter, get it used or borrow it from a library. do as many exercises as you can handle—all of them is best—and check against the back of the book

I know it’s kind of the expected advice, but it’s the one thing that works. and it’s worth it

>>16200859
You can start digging towards Lebesgue integrals and other non-standart integrals, which students don't learn in High School.
You will have to develop better understanding concept of integration beyond simple mechanical operation on symbols.

>>16200859
Your best bet is probably working towards the Lebesgue theory of integration. Generally this is taught alongside measure theory (as it provides more context for Lebesgue integration beyond integration with respect to the Lebesgue measure).

A good book on this topic that doesn't require a ton of real analysis background is Bartle's Elements of Integration.

>>16200859
The measure theory stuff >>16200893 >>16201087 won't help except with weird pathological stuff.
Learn some complex analysis if you really want to be able to do integrals.
You will also be on the path to do asymptotic expansions/approximations for the (typical) case where there is no nice closed form solution.

Lebesgue is overrated and not useful. At most look up Riemann Stieltjes integration which can handle things like d(floor(x)), d(pi(x)), etc.

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>>16188044
>Jew shilling Jewish movie about Jewish schizo discovering Jewish numerology.

>>16201124
monotone and dominated convergence are the most fricking useful theorems to have around when integrating shit

>>16201178
Proving integrals exist or finding bounds is different from evaluating them.

>>16201203
Monotone convergence theorem and dominated convergence theorem aren't just proving that the integrals exist. In fact the main use of DCT is finding ways to evaluate weird Riemann integrals.

If you can find a sequence of Riemann integrable functions which converge to the one you care about almost everywhere and are dominated by some integrable function, you can evaluate the integral by evaluating the limit. You'll have an integral evaluation that will be exactly the same except possibly on sets of measure 0.

>>16201209
Fart on my nose.

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>>16200859
First, can you actually do Calculus-I and II integrals? I'm going to assume that you can.

You can start with this anon's suggestion (>>16200893), or you can start by learning double and triple integration, which builds on your knowledge of normal integration and delves into some practical applications. This will naturally lead you to more complex species of integrals, and when you get there, volterra integro-differential equations. Going this route would provide you an arguably more stable foundation for applying calculus to your everyday, and you can always take both routes as well, it's not one or the other. Knowledge is power.

If you have any more questions feel free to ask away. Cheers.

>>16201216
Anon, pls, I gay for you. I homo like rest of Anon. Anon pls I have no name. I fagy.

>>16201214
For free? In this economy?

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>>16196788
I made a post about a really good book last thread, you can find the post here: https://warosu.org/sci/thread/16160352#p16184781

>>16196018
You don't -have- to, but it never hurts to have that ability. The calculus for dummies book has an incredible article to simplify the process for you and it's worth checking out. https://www.dummies.com/article/academics-the-arts/math/calculus/how-to-do-integration-by-parts-192235/
Article aside though, practice is key. I can do u-subs and trig subs in my head pretty quickly, and some integrals I can do by parts in my head and others I cannot. Set a goal and practice until you get to a spot where YOU feel comfortable.

>>16201209
>weird Riemann integrals
>weird
You will never run into these integrals doing anything practical. They are usually just mathematical curiosities.

Cauchy integral formula, residue theorem, stationary phase (for asymptotics), etc. are more useful in practice.

>>16201486
> You will never run into these integrals doing anything practical. They are usually just mathematical curiosities.

Do you consider probability theory and statistics practical? What about Fourier analysis on functions with mixed supports? Laplace transform solutions to non-linear constant coefficient systems of ODE's like the kind you see in control or dynamical systems?

All three of those are cases where DCT is a great way to produce integral solutions.

>>16201486
To add to >>16201533 I don't disagree that learning the basics of integration for complex functions will also be very helpful. One of the most helpful courses I took in my EE undergrad was a complex functions course where we went through the basics of complex integration/residue theorem, and Fourier and Laplace transforms are unbelievably useful in so many different domains.

I just also think that learning the basics of Lebesgue integration can be very useful in many practical fields. You don't need to go into the same level of depth as a graduate measure course, and some books (e.g., Heil's Introduction to Real Analysis or Bartle's Elements of Integration) both have a good amount of practical utility.

>>16199994
I find it more pertinent to explaining a bigger picture than set theory. I view it now as just a tool for explaining other phenomena. Category theory lends itself to schizos who want to say a lot about nothing, and so the stereotype is pretty fitting.
>>16200163
My test is over 1000 and I can squat 4 plates at 175 bw (trust me bro). But the types you describe are literally everywhere in the field. I used to think it was just a meme.

>>16199994
>Most people generally seem to regard category theory as schizo-babble
literally no one thinks this. Category theory is useful and ubiquitous in mathematics. Many fields (like module theory and algebraic topology) benefit greatly from the categorical perspective, if only for its organizational and bookkeeping capabilities (which is basically the only purpose of set theory too when you think about it). Even Lee uses categories in Intro to Smooth Manifolds, which category theory and differential geometry seem far removed. Historically category theory will also be the correct next abstraction of functions away from relations between sets, as arrows in a category correctly capture all the behaviour of functions while being perfectly agnostic about any of the underlying processes

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so did gpt-4o get it right?

>>16202236
I don't want to burst your bubble, but yes, quite a few people within both mathematics and applied mathematics view category theory as fitting the stereotype of "abstraction for abstraction's sake." See >>16200163 for an example of one in this very thread.

You could make an argument that this perception is ignorant and category theory shouldn't have that reputation, but it does. I'm not someone who knows enough about it to really know whether that reputation is earned or not, but from my ignorance and interactions with category theorists it appears about right on the average.

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>>16189421
Montecarlo sexo

>>16202555
woo, lad. no. it made shit up to get to an answer.

>>16202578
some guy posting about troons on 4chan is definitely a mathematician

Noob here learning about cardinality. If you wrote something like [math]|\mathbb{Z}| / |\mathbb{R}|[/math], is the answer undefined or zero? Does it even make any sense to write it?

>>16203250
I'm not a mathematician either. I'm an engineer who does research on the boundary between my field and applied mathematics.

I'm sure there are plenty of mathematicians who don't find category theory to be delicately walking the line of insanity, but all of the proper mathematicians I've worked with (who are mostly analysts and probabilists rather than algebraists to be fair) consider category theory to be a discipline approached with caution. I'm sure there's plenty of interesting work to be done in category theory, but it does seem to attract schizos and computer scientists (but I repeat myself).

>>16203325
There is some arithmetic which can be done with infinite cardinals (https://en.m.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic), but division is kinda meaningless. It only works if you divide a bigger cardinal by a smaller one and the result is just the bigger cardinal anyway (plus you need to use the axiom of choice).

In the example you gave there isn't a valid answer because the cardinality of the integers is less than that of the reals.

>>16200859
This anon >>16201124 is correct. Lebesgue integration is almost completely useless for computation of integrals. It's the entire reason why Riemann integration still exists. He's also correct in recommending you complex analysis. It is very useful in computing otherwise very complicated integrals. Another recommendation is manifolds and differential forms if you finally want to understand what dx means.

>>16201533
>Do you consider probability theory and statistics practical?
Applied subjects can have inapplicable curiosities e.g., Cantor distribution.

>>16202555
nope

>>16203328
such an embarrassing post. you have no idea how out of your depth you are here.

Probably a dumb question but idc. I cannot do math at all while thrown in any way: emotionally, drunk, anything. Am I just stupid?

Why are all perfect numbers even?

>>16205274
If by "out of my depth" you mean "not devoting my life to abstraction for abstraction's sake" sure.

>>16199915
>If somebody were to prove the Collatz conjecture is true, but in such a way that it provided very little insight or utility in any other areas of mathematics
If it provides absolutely 0 insight to the rest of mathematics, then the proof is probably incorrect. Even a problem so remote from any area of active research must still provide at least SOME insight into something.

>>16199915
>>16205800
I should add onto my post. I think the Collatz conjecture, as we know it currently, is not enough.

It's not enough for some rando online to post the solution to the problem and walk away. The difficulty of the problem has to be justified with at least something, such as insight into number theory of some form.

>want to do some computerized math
>have to program and use computer
>have to read books written by programmers larping as mathematicians
AHHHHHHHHHHHHHHHHHHH
WHY IS THIS SO PAINFUL

how can i get good at math idk how to do math

>>16206373
The same way you get good at literally anything else. You start simple, practice and slowly increase your skill set and competency.

Let X be the 2-sphere with n points removed, n≥3 .
Does there exist a (geodesically) complete Lorentzian metric on X ?

>>16206504
ok what do i start with

>>16206754
Depends on where you are. Whatever the highest level of math education you feel comfortable with, a good place to start is there (or a step earlier). There's no shame in going slow and doing lots of practice problems.

If you're starting in the "high school math" area then resources like Khan academy and Brilliant are unironically pretty good (especially if you can get access to them for free).

Thoughts on gassy sets?
>>16207622

I figured a way to recursively compute formulas for sums of n^r with just a couple lines of algebra. I have been working on a few different approaches to get a closed-form expression for general r, but they seem to all explode in complexity. Fucking A

>>16207635
https://en.wikipedia.org/wiki/Faulhaber's_formula

>>16207644
I know it's a solved problem, I wanted to figure it out myself. I'm pretty confident my approach is different

>>16207635
>>16207689
post approach or gtfo

>>16206754
Probability puzzles.

>>16207726
alright, give me a day to write it up in latex

Is it possible to have a Cauchy sequence of distinct rational numbers with bounded denominators?

>>16207879
Let's see consider the set of distinct rational from 0 to 1
Throw an infinite countable set of points to mark the set, call it a subset. There is a convergent subsequence of points.
Choose M in the naturals as an upper bound for the denominator.
The number n < M is also bounded by M so the set of bounded rationals is finite. M in arbitrary so we will violate either distinctness or boundedness and we already assumed distinctness.
Now extend the argument to R using arbitrary bounds instead of zero to one.
That's just my guess anyways I don't really understand math. So sorry if you fail your homework.

>>16207879
Nevermind, I managed to prove the stronger statement that limit inferior of the denominators is infinite.

>>16207906
Correct.

Pi and e are both very close to an integer when cubed. Is this just a spooky coincidence?

>>16208279
Do a series expansion and cube em. Post what you get.

>>16207627
die

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so I was wondering if you guys knew what this kind of thing is called
basically I was thinking about how shapes can fit together. When I put two circles together there is no more space to fit any over shapes that are also convex. So in this way I'm saying it's optimized. But now I am thinking about different kinds of solutions like none smooth curves that are still convex.
If I am looking at bounded shapes and I draw a circle about the origin just splitting it in half through the origin would give me two shapes that are spatial optimized in the circle. But I want to maintain the condition that the shapes only touch at the center point. So the two circles isn't the only solution. Making the angle of the arcs between the semi circles a difference greater than 0 (epsilon > 0) we see that for the conditions of being both radially optimized and spatially optimized with only N = 2 (where N is the number of shapes) the set has no largest element.
So I was thinking in general I can add a number of radii in the circle and add an epsilon of difference.
This made me ask the question how smooth can my solutions be? There is something about the curvature about a circle where ignoring spatial optimization but considering angular optimization you cannot draw a line that is "nonzero" between the two circles. So its curvature is angle of 180 degrees if that makes sense? Because there are no degrees of freedom for the line. So it's making me think what is a general way I can define this?
Like when N = 3 what is the angular optimization?
I don't know this was very interesting to me all of a sudden. Maybe I am just a retard.
So in summary I asking how can I limit the angle of freedom between the spaces (to zero) while keeping the shapes disjoint except at one common point?
Just hoping you guys could give me some resources, information, book, set up, etc.

>>16209326
I don't know any good books/resources but this reminds me a lot of sphere packing. Unequal sphere packing is looking exactly at the question of optimizing how much of the space you can fill when you allow for the spheres to have varying radii in some set (obviously excluding r = 0).

I don't know about packing the space with non-spherical convex objects.

>>16203328
> who are mostly analysts and probabilists rather than algebraists to be fair
gee, I cannot *possibly* imagine why this group of people would find category theory to be less useful than algebraists. truly, this mystery has no answer
In all seriousness: on the one hand, even the literal inventors of the field call it "abstract nonsense." Nobody's denying category theory is abstract -- that's the point -- and there's definitely a tendency for young grad students to try to wildly overapply it, news articles to report misleadingly etc. But calling it "schizo babble" and
> delicately walking the line of insanity
> a discipline approached with caution
lol, it's not like some magic SCP shit, you're not gonna turn into some raving lunatic for Comprehending That Which Mankind Ought Not
At its most basic level, a bunch of algebraists and topologists said "hey, I keep seeing the same definitions over and over and over again, just mildly different." (quotient topology / quotient group / quotient ring / quotient module / ...; pullback of a ring/module/group/...; etc etc etc etc). How can I stop writing the same proof ten times over, with mildly different details?" And it turns out that that's the sort of perspective that, on the one hand, leads to levels of unfathomably useless abstraction; but on the other hand also often leads to genuinely nicer proofs because you can wrap up a lot of information into a single concept ("such-and-such collection of functors and families of natural transformations form a generalized cohomology theory because of X, Y and Z, so now we know a million things about it that hold for cohomology theories")
But, yes, obviously most analysts aren't going to find a need for this, because (to my limited understanding) most analysts don't do that kind of work, by definition (equally difficult work, but different). Concrete examples and actual specific choices of functions/spaces might make less use of tools meant to abstract and generalize.

>>16209584
Your response made me smile, so thank you for that.

> lol, it's not like some magic SCP shit, you're not gonna turn into some raving lunatic for Comprehending That Which Mankind Ought Not.

That's not what I intended with the "approach with caution" but I sort of like that meaning better. I more was thinking along the lines of "you'll spend a lot of time on something which won't help you towards practical career goals like publications/citations/funding/applications to other domains etc." The idea that category theory is magic SCP shit is more fun though.

By the way, I like your explanation of the motivations for category theory. It's compelling and you should put some effort into writing out a script for a mini-lecture or blog post or something explaining it. You have a good written voice for that sort of thing.

> But, yes, obviously most analysts aren't going to find a need for this, because (to my limited understanding) most analysts don't do that kind of work, by definition (equally difficult work, but different). Concrete examples and actual specific choices of functions/spaces might make less use of tools meant to abstract and generalize.

I think you are correct about this. The analysts I work with are mostly focused on functional analysis with applications to optimization on abstract spaces. This particular domain involves a little bit of algebraic topology and a little bit of "abstract linear algebra" every once in a while, but for the most part is not that algebraic.

Me and some buddies put together a paper with some kind of thing we figured.
https://ditzbitz.com/The+non-Convergence+of+Combined+Subsets+of+Infinite+Positive+and+Negative+Sets+of+Fractional+Powers+under+Addition+-+The+Ditzler-Park-Wei+Invariant.pdf

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>>16207635
Not sure how well this goes in terms of computational complexity but [math]q(n) = \sum _{k = 1}^n p(k)[/math], where [math]p[/math] is a polynomial of degree [math]m[/math], is necessarily a polynomial of degree at most [math]m + 1[/math].
So you can just evaluate it at a bunch of points and interpolate it in [math]O(m)[/math] (I'd imagine.)

How can I get better at combinatorics. I’ve learned this shit like 4 times now and I can never wrap my head around it. I’m fine with whatever BS math throws my way but for whatever reason, I can’t figure out how to arrange sally Jane and Joe in chairs or the odds of pulling a straight flush.
Any tips or tricks that helped you learn?
I’m not talking any specific type of problem, more of a general mindset thing… how to think about/approach these

>>16210841
Practice until you get it. It takes time and patience for it to click.

>>16210841
try to derive the formulas. Looking at 2 parameters, whether or not repeats are allowed and whether or not order is respected, gives a 4 quadrant square: combinations, permutations, information (permutations where repeats are allowed), and uhhh k-combinations? (combinations where repeats are allowed). It took me a while to figure them out from scratch, but now all the factorials in the formulas clearly point to the different constraints in the word problems

>>16210143
looks similar to the “p-adic rationals”

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>>16209326
I think you’re asking for a situation that looks like pic related?
>shape lives entirely in the top-left
>but it touches the origin (0,0)
>so the angle it makes there is <=90 degrees?
If that’s what you want, then the shape cannot be smooth. “Smooth” can mean a few different things, but something smooth ought to at least have a unique tangent line at every point. Then when you zoom in on the tangent line, it starts to look more and more like 180 degrees. Anything else will be pointy.

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Do you guys know any base 10 number sense exercises I can do on my own? I have 2 sets of counters. One is 24 colored farm animals and the other 20 non colored wooden blocks.

Most of the exercises i've searched on google require 2 or more people. Their aimed at preschool children which is the level I am currently at.

>>16209829
I appreciate the compliment, anon! I'll now blow my own credibility apart: I'm just an early grad student, so I can't speak to any truly advanced uses of category theory nor use in research, just where it shows up in basic algebra/algebraic topology.
I was going to type up something like "I can't imagine doing algebraic topology without concepts like functors (Hom, Ext, Tor, pi_i/H_i/H^i, ...), adjoint functors in particular (eg the suspension/loopspace or hom/tensor adjunctions), or naturality (basically every useful theorem in the subject)." But... I mean, the only algebraic topology I can imagine doing is what I've gotten from Hatcher's famous book on the topic, so I truly can't comment on research or on the subject as a whole. Likewise, I can't imagine working with, say, a tensor product without knowing about the relevant universal property -- but I've also never really needed to work with a tensor product in applications before, so who am I to say?
Anyway -- I see what you mean with how going deep into category theory might take away from career goals (or, at least, how that perspective would make sense for analysts -- I know some algebraists whose careers have mostly been centered *around* category theory). There *is* probably a grain of truth to that insofar as the applied fields, at least as far as I know, tend to get *way* more funding than pure mathematics.
But on the other hand, nobody who attempts a PhD in this nonsense is in it for the (nonexistent) money or fame. (Hell, we don't even get to brag about publication records or citation counts -- not when average publication rate is maybe ~1 paper-per-year for a full professor, peer review takes half a year if you're lucky, and let's not even mention citation count.)

>>16211170
I meant smooth almost everywhere as well as disjoint almost everywhere.
The angle at the limit would be 90 degrees to be optimized.
If N = 3 the angle would be 120.

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Is this really a rigorous definition? Was the ordered pair notion of a function not well-known when Rudin was writing?

>>16211276
He just didn't want to confuse his readers with advanced concepts like ordered pairs. Spivak goes on a whole tangent (a "big lipped alligator moment") about how ordered pairs can be encoded in set theory. Calculus pedagogy at the time just didn't know how to strike a balance between these two approaches -- introduce ordered pairs but don't go on a set theory spazzout.

>>16211206
> I appreciate the compliment, anon! I'll now blow my own credibility apart: I'm just an early grad student, so I can't speak to any truly advanced uses of category theory nor use in research, just where it shows up in basic algebra/algebraic topology

That's okay. You still know more about it than me so I'll take your word for it.

> Likewise, I can't imagine working with, say, a tensor product without knowing about the relevant universal property -- but I've also never really needed to work with a tensor product in applications before, so who am I to say?

Tensors show up all over the place in 3d array-based signal processing and I assure you that those engineers have no clue what's going on in category theory (or abstract/modern algebra in general). There definitely is a way, even if it might have many more caveats than the most rigorous approach to the discipline.

> But on the other hand, nobody who attempts a PhD in this nonsense is in it for the money or fame.

Oh, I don't really care about getting rich or famous. I just mean that it's important to make sure you're well funded enough that you can support yourself and students to keep your lab healthy.

If you're spending time burning the midnight oil working on (as an unrelated example) the Collatz Conjecture or some IUTT based theory of everything, you probably aren't going to have the best of luck being well funded enough to support graduate students, which means you're neglecting one of your fundamental responsibilities as a researcher (i.e., being productive enough to justify your own salary and support those who rely on you).

>>16207635
Here's my clever approach:
[math]\sum\limits_{n=1}^{N}(n-1)(n-2) \cdots(n-k) = \begin{cases} 0,\ N \in \{0,1,\cdots ,k\}\\ k!(-1)^k,\ N=-1 \end{cases}.\\
\sum\limits_{m=1}^{k+1} (-1)^{k+1-m} \begin{bmatrix}k+1 \\ m \end{bmatrix}P_{m-1}(N) ={1 \over k+1}N(N-1)\cdots(N-k).\\
P_k(N) = \sum\limits_{m=1}^{k} (-1)^{k-m} \begin{bmatrix}k+1 \\ m \end{bmatrix}P_{m-1}(N) +{1 \over k+1}N(N-1)\cdots(N-k).[/math]

>>16211581
should be k!(-1)^(k+1) on the first line.
Everything else is fine.
The convention is P0(N) = N

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Should I follow these books if I'm a mathlet?

>>16211999
That's a picture for ants.

>>16211999
Every chart of this kind is a meme.

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Here's an interesting problem I came up with.

If you pick three uniformly random numbers between zero and ten, what is the probability that the numbers you picked can be used as side lengths for a triangle which is less than one degree off from being a right triangle? Meaning that it would have one angle that is greater than 89 degrees but smaller than 91 degrees.

>>16211999
Obviously I cannot read your chart, but just read something that interests you. If it's too hard, go back and learn whatever prerequisites you need.

>>16211999
I agree with >>16212646 but also want to add that there's no shame in going back and looking at material you already understand from a different perspective.

In my case, I've already done the typical American analysis sequence (Abbott to Baby Rudin to Royden to Rudin) but still have been getting a lot out of going through some "fundamental analysis" books with more of an algebraic/geometric perspective like Zorich/Amann/Penot.

>>16212656
>like Zorich/Amann/Penot.
Also consider Roger Godement's books (there's four). You get a very unique treatment plus borderline schizos rants about nuclear war and the MIC (not that I disagree with him).

>>16212686
I love schizo rants by STEM authors. They always make me smile.

I think I might be a raging narcissist.

In class (I'm referring to math specifically here but it's the same in everything I do) if someone is doing better than me I feel very ashamed of myself and hateful towards them. Today I failed to finish my work in class first as I usually do, a girl and 2 boys left class before me. I was quivering with rage, empty with shame.

I thought some very unpleasant things about them. I was monstrous.

How do I cope with this? Should I endeavor to change this pattern of thought or is it ok?

>>16212634
How would you even approach this problem? Any two side lengths will have variable probability for the third to complete the condition.

>>16212634
Integers or reals?

>>16212727
reals, because with integers you could just brute force it which would be boring

>>16206504
how far can one realistically go with mathematics if they are really dumb and struggle with juggling abstracts in their head?

>>16212885
> how far can one realistically go with mathematics if they are really dumb and struggle with juggling abstracts in their head?

There's only one way to find out. Go until you either don't feel you can grasp it or you don't get enjoyment out of it anymore (whichever comes first, it's just about impossible to really grasp something you have to force yourself to even try to understand).

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>>16212073
>>16212097
>>16212646
>>16212656
Tks, I'm currently reading Basic Mathematics by serge lang. The picture prolly got resized when I copy it.

>>16212726
I don't really have any idea how to approach this. That's what makes it interesting. Solving it would probably increase your IQ.

>>16212727
We're talking about real numbers obviously.

>>16212717
That's really funny. If it works for you and you study well because of it it's probably ok, but you're almost certain to meet someone a *lot* smarter than you eventually so please don't kill them.

>>16213023
>start with set theory
>emulate an undergrad degree in 5 books
>straight into (algebraic|differential) (geometry|topology) and that's all you do
yeah it's a meme

>>16213661
Mind giving me some guidance? Currently, I'm just learning to improve my calculus and trigonometry. I found math problems way more enjoyable than leetcode.

>>16216028
Unironically your best bet for self study is probably something similar to an American undergrad track. You should spend some time getting comfortable with "applied" mathematics and problem solving before you dive into pure mathematics, especially if you are self studying for a hobby rather than in university.

1) Single variable calculus (Stewart or Apostal/Spivak)
2) Multivariable calculus (Stewart is probably fine)
3) Ordinary differential equations (Tenenbaum is fine, Blanchard's book is also good and free online)
4) Applied linear algebra (Lay is good and free)
5) Probability Theory (Sheldon Ross's book is good, Roussass is good, there's a ton)
6) Intro to proofs/advanced mathematics (I like Smith's A Transition to Advanced Mathematics, but there's a ton of equivalent books in this group too)
7) intro to real analysis (Abbott or Ross or one of the standard European books like Amann/Zorich/Penot)
8) intro to abstract algebra (Pinter or Hungerford)
9) Topology (Munkres is the standard and is fairly approachable. Mendelson's Dover book is also good and a bit easier).

After that the world is basically your oyster and you can kind of pick where you want to go. I'm someone who is much more interested in analysis, so I'd say go that way, but if algebra is your thing there's no shortage of options there too.

Then you can work on the more "advanced" math. I honestly don't know why

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>>16216550
Thanks anon. I will note it down

>>16216571
I somehow dropped the end of the last sentence in
>>16216550

I meant to say "I honestly don't know why that chart pushes you solely into algebraic topology and differential geometry." Those are both cool subjects (and in particular differential geometry has a ton of different useful applications to real world topics in physics, statistics and engineering) but those certainly aren't the only directions you can go.

Please, does anyone know a good resource on basic linear algebra for machine learning?
I'm doing ML and want to go to grad school for it, but it bothers me a bit to have shoddy knowledge of the mathematical basis of the concepts used in numpy, pandas and stuff.
I don't need to be a top tier algebraist but having a pretty solid foundational knowledge of vector spaces, matrix algebra, eigenvectors, tensors and all the other stuff relevant to AI/ML/DL/etc would be cool.
I'm also assuming that such an endeavor is useful in the first place but please tell me if it isn't.

Any suggestions? Videos or websites preferably but books are fine.

>>16216612
I didn't managed to get solve 2 problems related to those topics in my gaokao exams 8 years ago, I'm a NEET and a college dropout now anyways. So i remember how good it felt after solving a math problem back in high-school so i want to feel that feeling again. I haven't done math in a really long time now so i need to relearn some stuffs but dunno where to start.

>>16216625
Kevin Murphy's Probabilistic Machine Learning Vol 1 has a whole section on linear algebra and its applications to ML. A draft copy of the book as well as the python code used throughout is available on his GitHub.

>>16216656
Much appreciated.
I wouldn't want to be able to deploy a model and then look like a clown if someone asks me how it works. Hopefully I can remedy that.

>>16212848
Ok, from Monte Carlo it appears to be equal or just under sin(1deg) ~ 0.0175

>>16216662
In general for Machine Learning/Optimization you need to be very comfortable with the computational side of linear algebra, but it's not super necessary to have a very strong foundation in the "abstract linear algebra" like you'll find in a more proof-based linear algebra book.

The most you'll really need is some familiarity with SVD, QR decomposition, Cholesky decomposition and the Riccati/Lyapunov equations (and the basics of matrix algebra and eigenvalues etc.).

>>16217262
Damn I don't even know how to invert a matrix. I have work to do then.

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How can I approximate this integral with at least 6 exact decimals using a Gauss-Laguerre quadrature?
I split it into two integrals, one from [-inf,0] and another from [0, inf],the second one already being in a Gauss-Laguerre form, but the first one wasn't. I got stuck at the first one, I applied the x = -t substitution to try and flip the interval but that left me with exp(x) which isn't Gauss-Laguerre. How else could I have approached this?

>>16217377
There's no general way to invert a square matrix unless there are some special properties about the matrix you can leverage (e.g., if you know the exact eigenvectors/eigenvalues, or you have an easy Cholesky decomposition).

>>16217595
https://en.wikipedia.org/wiki/Invertible_matrix#Gaussian_elimination

>>16217714
Gauss-Siedel is an algorithm for inverting an invertible matrix. It isn't a closed form expression for said inverse.

>>16216028
Adding onto what kind anon said here >>16216550, since you mention calculus, trig and leetcode, it might not be necessary to learn a whole degree of maths (though I would recommend it if you're interested).
It depends on what you're wanting to do with what you learn.
If you mainly want to get good at solving CS-related problems, you can look at discrete math / combinatorics.
Pretty much any resource aimed at beginners should be fine.

If you're serious about learning maths, you already have some good advice: just follow some well-established curriculum.
It should take about as long, too (3-4 years of several hours per day).
Some particular advice:
- Try to go `wide' rather than `deep'. That is, learn a decent amount about a broad range of topics, rather than focusing a lot on a few subjects only. Kind of the exact opposite of your initial picture. Statistics, convex optimization, computer algebra systems, whatever you feel like. You'll learn a lot of foundational topics this way too, through osmosis.
- If you don't like a book or don't get a section after a few hours, just skip it and read a different book / topic, and return after a week or two. No point in getting needlessly stuck if you're self-studying.
- Focus on linear algebra, it shows up pretty much everywhere and is extremely important.

>>16217578
e^-t is just 1/e^t. Your obvious first or second guess should've been to multiply by e^t twice in both the numerator and denominator.

>>16217823
What is your point?

>>16217578
I would start by pulling out the x+cos(x)+1 since the integration variable is t.
The let t = log(u)
The exact result is just x+cos(x)+1

>>16217884
My point is that there isn't a general formula for a square NxN matrix inverse as there is in the 2x2 case as people learn. You can use machinisms like Gaussian elimination to numerically evaluate this inverse, but outside of special cases like the Woodbury matrix inversion lemma or exact diagonalization, you can't in general invert the matrix in a closed form.

>>16218040
Would you consider this to be closed form?
https://en.wikipedia.org/wiki/Invertible_matrix#Analytic_solution

>>16216550
Let's just Russian pill this list into something much less redundant.
>Zorich I, II - Analysis
>Shafarevich or Shilov - Linear Algebra

Now he can go from there. If he wants something a bit more concrete, the German book Mathematics for Physicists by Altland is good too (thanks to the shill for the rec). This book will cover algebra, calculus, and even vector calculus material.

>>16218068
I forgot to add for Arnold for Differential Equations

TIL chatgpt is very close to being able to take a screenshot of a scanned textbook and turn it into latex.

Gave it Atiyah Macdonald and it was virtually flawless for even a diagram heavy page, albeit a couple minor mistakes that could be corrected quite easily.

>>16218056
I guess in a certain sense it is. Doesn't help you much that the adjoint and the determinant are also both algorithmically defined (via cofactor expansion) rather than something with a simple expression. This also assumes the matrix is itself invertible, which we don't in general know for a given square matrix and can't easily determine from inspection in most cases (though that's somewhat of a given).

>>16217578
I don't get it x isn't dependent on t right?
Just pull it out and do a u sub.

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>>16188147
what does the "mg" in mg curriculum stand for?

>>16218508
The determinant of a nxn matrix A can be written in a simple formula
[eqn] \det(A)=\sum _{\sigma \in S_{n}}\text{sgn}(\sigma )\prod _{i=1}^{n}a_{i \sigma(i)} [/eqn]

>>16218637
I've never seen this symmetric group way of writing out a determinant. Interesting.

How would you go about specifying the permutation pairs (i, sigma(I)) for a particular NxN matrix? Sorry if this is my own ignorance showing, but I've never seen this approach before.

>>16218518
>>16217944
No, it was supposed to be x instead of t but my professor is a senile retard and I am also a senile retard for forgetting to mention that
>>16217877
Yeah, someone in /sqt/ told me that exp(x) = exp(2x) * exp(-x) which would have given me the Laguerre form... I don't know how I didn't see that during the test, it was like my brain drained all my prior calc knowledge

>>16218108
Prove.

>>16218108
please god be true please this is the one thing i want from ai, being able to tex shit for me, fuck i hate writing tex. handwriting to tex can't come soon enough

>>16218619
>thread title
Is you serious?

>>16218637
You can also "compute" it geometrically with:
[math]det(A)=\bigwedge\limits_{i=1}^{n}v_i[/math]
Very useful if you don't want to remember the jacobian when doing change of variables.

>>16219311
The RHS of what you wrote won't be a number. What you mean is
[math] (\mathrm{det}A) (\wedge_{i=1}^n e_i ) = (\wedge_{i=1}^n v_i ) [/math]
where A is any square nxn real matrix with columns [math] v_1,v_2,\ldots,v_n [/math] , and [math] e_1,e_2,\ldots,e_n [/math] is the standard basis in [math] \mathbb{R}^n [/math]

>>16219467
You know what I meant though.
You could also just hodge star it to transform the n-vector into a 0-vector (scalar).

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Ok upon reflection it's actually pretty bad. It looked good initially but then you look closer and it keeps swapping variables out with each other, and completely got rid of the coker bit.

It is interesting how much of the general gist it seems to get right though.

>>16218108
Very nice...

Whoever rewrote Rudin in [math] \mathrm \LaTeX [/math], give me the [math] \texttt{.tex} [/math] file.

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>>16219148
oopsie

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>>16188044
Based

Just finished up on some exams, either my professor went full retard (wouldn't be the first time) or I just suck at this. One of the problems was this. Let [math]\mathcal{B} [/math] be the open unit ball with the euclidean norm on [math]\mathbb{R}^{n}[/math] and [math] \mathcal{S}[/math] the unit sphere. Let [math] f[/math] be a continuous function on [math] \mathcal{B}[/math] and constant on [math] \mathcal{S}[/math]. Show that there exists a point in [math] \mathcal{B}[/math] such that the partial derivatives of [math] f[/math] are all equal to zero (basically there exists a critical point). Now if this were continuous on the closed unit ball then it's easy, we use compact by a continuous function is compact etc. However in this case I can simply take the example in [math] \mathbb{R}^{2}[/math], [math] f(x,y)[/math] equal to [math] x+y[/math] for points in [math] \mathcal{B}[/math] and equal to some random constant on the sphere and we have a clear counter example right? Can someone point out if I'm missing something.

>>16200186
I know probably this is bait, but I'll bite it anyways: all of the things mentioned are useful and have quite involved ramifications. Hodge theory is a powerful cohomology theory for complex analytic spaces; it has so many uses in complex geometry that a vast set of mathematicians are trying to generalize it for characteristic [math]p > 0[/math].
Perfectoid spaces are, in a way, a bridge between characteristic 0 and characteristic [math]p > 0[/math]; while I don't understand the details, one of the basic ideas is that Anabelian Geometry predicts that a lot of information of a field (or more generally, a scheme) is contained in the absolute Galois group (or more generally, the étale-profinite fundamental group), and Fontaine showed a construction where there is an explicit canonical isomorphism between the Galois group of objects of characteristic 0 and [math]p > 0[/math], that is the "bridge".
I'm going to assume that by "Fano theory" you mean the study of Fano varieties. They are, in essence, algebraic or complex varieties that "look alike" projective spaces [math]\mathbb{P}^n[/math]. For a better understanding of the classification problem of algebraic varieties we think we should attack the two endpoints of: Fano varieties and varieties of general type. An example of an open problem in this respect is "hyperbolicity": for complex geometry, this refers to the aboundance of non-constant holomorphic maps [math]X \to \mathbb{C}[/math]; for diophantine geometry this refers to the aboundance of rational points over a finite extension (think Mordell's conjecture in higher dimension, maybe with a refinement in the asymptotics of points counted by Weil heights). Some conjectures of Green-Griffiths-Lang predict all these notions should agree, and that hyperbolic varieties should be hereditary of general type.

>>16224616
Yeah sounds like he meant it was continuous on [math]\mathcal{B}\cup\mathcal{S}[/math] and constant on [math]\mathcal{S}[/math]

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>>16224821
Why did the latex turn out like that? Does anyone know what I did wrong?

>>16224821
A little space helps...[math] B \cup S [/math]

>>16224838
>>16224919

Failed analysis for the third time. RIP

i hate mymathlab
i hate mymathlab
i hate mymathlab

>>16219138
There are already tools for this. I needed to get some pdfs into html to make a web app to keep stats and catalog different math problems for a study tool. It's called mathpix. It worked fine for pdfs, it gets almost everything right and I'm pretty sure it works for handwriting. It's not that expensive

>>16226571
Damn...

Does anyone here know anything about Deligne-Mostow lattices? Is there any way the number 34 can be related to them? Doesn't matter how arbitrary it is.
My friend studies them and he's turning 34 and I want to get him a badge that says something like "I'm (the number of even Deligne-Mostow lattices with dimension at most 5) today!"
That example was made up, but you get the idea. I think it would be cute, but I know too little about this area to make something which would be correct.

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>>16207726
here's the formula. It's a fucking mess to do by hand for R>2. It recurses and blows up, then that C term also recurses and blows up. Still working on a pretty pdf writeup of the explanation, but the crutch of the problem is predicting the coefficients of a 'factorial expansion', similar to a binomial expansion.

Binomial expansion:
(a+b)^R

Factorial expansion:
(a+R)! / a!

>>16196107
I'm not an expert, but I'll try to give you an answer.

- The statement that every path is the trivial path is called 'axiom K'. Both axiom K and Uniqueness of Identity Proofs (UID) are consistent with Martin-Löf Type Theory. For this reason, you won't be able to find any non-trivial paths unless you go through the Univalence axiom.

The Univalence axiom states that identity is equivalent to the type of equivalences. Thus, to find non-trivial paths, we need to find non-trivial equivalences (the trivial equivalence being the identity function). The simplest example is the type of equivalences of the Booleans. The Boolean type has two constructors: true and false. There are two possible equivalences: the identity and the function that swaps true and false. By Univalence, the second function gives place to a non-trivial path.

- Since UID and Univalence contradict each other, assuming UID in Homotopy Type Theory makes the theory inconsistent. The same goes for the axiom K.

- Intensional equality means that we no longer think about strict equality, and instead, we take a more categorical view: if two objects are equivalent, they are indistinguishable (i.e. equal).

One consequence is the following: suppose that A and B are isomorphic rings. If I tell you that A is 'Noetherian', even if you don't know what that means, you are probably confortable believing that B is also Noetherian. That is because, informally, we identify isomorphic objects. Despite that, in set theory, the statement 'If A and B are isomorphic and A is Noetherian, B is Noetherian' still requires a proof. In type theory the proof is trivial: since isomorphism is an equivalence (it is), A and B are equal, so if A is Noetherian, B obviously is as well.

so today I tried my theory of computing exam, can some of you guys explain how to use Rice's theorem to prove that set I (extensional property) is not recursive in this case. By first proving that it's not trivial and then that it's extensional?
[math]
\begin{document}

The function \(\sigma(x)\) has range \(\text{range}(\sigma(x))\). Prove that:
\[
I = \{ x \in \mathbb{N} \mid \text{range}(\sigma(x)) \subseteq \{0, 1, 2\} \}
\]

\end{document} [/math]

>>16228689
[eqn]

The function \(\sigma(x)\) has range \(\text{range}(\sigma(x))\). Prove that:
\[
I = \{ x \in \mathbb{N} \mid \text{range}(\sigma(x)) \subseteq \{0, 1, 2\} \}
\]
[/eqn]

I am getting a math degree for the sole purpose of wanking my ego, how retarded i am?

>>16229927
depends how good your school is
if it's a real school, or even just an engineering school, then you're fine as long as you took enough pure math courses
if you're a state school where trig is considered "college level" and Calc III is a scary optional class -- then you could maybe still be fine, but if and only if you 1) don't pretend calc is hard, 2) don't brag about how easy you think calc is, and 3) took at least one graduate-level math course in addition to a standard pure math sequence
applied mathematicians gtfo
unless by 'degree' you mean Masters, in which case yes you're a dumbfuck no matter what; or PhD, in which case you're stupid but in a whole new, secret way compared to the undergrads - in that case it's more of a "why would you spend your youth on such a pointless thing" kind of way
in any case, though, doing this shit for ego's sake isn't uncommon -- it's not like there's so many other motivations for it. There's a certain narcissistic appeal to seeing your own brain come to terms with more and more abstruse nonsense.

>>16224639
This was interesting, anon, thanks for the writeup

In physics the motion of an object experiencing multiple forces can be calculated by adding the force vectors together and pretending that the object is only experiencing a single "net force". What would have to change about the space the object is embedded in for this to no longer apply? Can forces still be added in this way in every smooth manifold? What is the most complicated space where this fails?

>>16230292
Idk. Saw a youtube vid recently. If you can't write a lagrangian then it wont work, like for a photon, H = hbar * kc = (h/L)c = pc. Force don't work here when the speed of light is constant.

He said he couldn't think of another example.

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>Learning math (easy math)
>Try to understand what makes some funcions non-linear (like for (x+Δx)^2, if you view it geometrically you have a perfect square and you have to account for Δx twice because both base and height and changing, the total change is 2xΔx + Δx^2 but when Δx tends to 0 only its derivative 2x remains, you can also write x^2 = x * x , see that Δx needs to be accounted for twice and think something sus is happening compared to linear multiplication)
>Noticed 1/x isn't linear and asked why in a discord server with frens
>No one knows why
>Came up with an analogy similar to the square but no one seems to get it
>Someone there who is supposed to know a lot more math than me treats me like a fucking retard with questions like "Do you know what a slope is"? "What is the derivative at this point on the graph"?
>We make no progress after hours discussing and it's been imposible to explain that derivatives are related to the algorithm of the function itself
Never felt so fucking frustrated. I can't talk about math with any acquaintances because this happens. I can't talk about math here because i'm still like a toddler for this general. I don't actually study math I just ask questions for no reason but this is not working.
I should get my shit together and read books, I'll try again /mg/.

Does anyone know why [math] \tfrac{1}{r\sin\theta } \tfrac{\partial x}{\partial \phi} = r\sin\theta \tfrac{\partial \phi}{\partial x} [/math] for spherical coordinates? Doesn't have to be x or phi.

Like is there some intuitive explanation involving the gradient of phi?

>>16230358
A function of x is linear if you can write it as a polynomial where every appearance of x is without an exponent (i.e. [math]x^1[/math]).
[math]y = \frac {1} {x}[/math] is not linear because it is equivalent to to [math]x^{-1}[/math].
[math]y = 3[/math] is also linear, because there are no appearances of x at all, so every instance is to the first power... kind of by default.
The derivative thing makes it a lot easier to define (a function is linear wrt a variable iff it has a constant derivative wrt that variable) but it's by no means necessary.

There's a related, but distinct, concept called a linear operator, which is considerably more advanced. Don't mix them up.

>>16230358
>nonlinear
does 1/2 + 1/3 = 1/5? If 1/x were linear, it would be. But it's not. So they're not equal. Does 2^2 + 3^2 = 5^2? If x^2 were linear, it would be. But it's not. So they're not equal.

Why not just explain it easier

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>>16230379
>A function of x is linear if you can write it as a polynomial where every appearance of x is without an exponent (i.e. x1)
Yes. When I said you account for Δx more than once I was kinda trying to state "this is what happens when you have an exponent and why its derivative increases" in a very loose way.
>Don't mix them up.
Sorry if I do. I'll look that up.
>>16230380
I know. I started ages ago comparing
x^2 + Δx^2 =/= (x+Δx)^2
I just wanted to know why (x+Δx)^2 yields a higher number. This led to 2xΔx + Δx^2 and the whole story.
>does 1/2 + 1/3 = 1/5?
No. When I started with 1/x I did the same thing as above and long story short it's all the same stuff related to "what is this function exactly telling me to do"?
>Why not just explain it easier
Sorry I try. When I ask why is 1/x non-linear, I assume the other end takes for granted that every derivate has its own logical explanation, just like you can deduce why the derivate of sin(x) is cos(x) by looking at the unit circle.
I was asking, what does 1/x actually mean to have a decreasing derivative? I did the thing algebraically and geometrically and I literally don't know what to call this anymore because every time I say derivative this happens.

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>>16230358
anon surely you learned what a slope is in school? here are some ways to think about linearity
straight line: it's in the name
constant slope: the slope of a linear function is the same everywhere on its domain (in other words, the derivative of the function is constant) This is the same thing you were arguing with the change thing for (x+Δx)^2, just in generally useful language.
it's a linear transformation: the function f satisfies f(x+y) = f(x) + f(y) and f(xy) = xf(y) (or of the form g(x) = f(x) + C for f a linear transformation and C a constant, in order to allow lines not through the origin) cf >>16230380. This might be a useful exercise to you: prove that all linear transformations from the real line to itself is of the form f(x) = mx for some constant m (hint: what is f(0), what is f(1)?)
the exponent of "variables" are 1: cf >>16230379
for all of these notions the only linear functions are the ones of the form mx+c

>>16230492
>the other end takes for granted that every derivate has its own logical explanation
the derivative does have its own logical explanation: the derivative of f at x is the slope of f at x. A function is linear if and only if its derivative is a constant function.
>what does 1/x actually mean to have a decreasing derivative?
what do you mean decreasing derivative? it's not constant anyways and that's what matters

>>16230501
>This might be a useful exercise to you: prove that all linear transformations from the real line to itself is of the form f(x) = mx for some constant m
Can't write actual proofs . f(x) = mx for some constant m is linear because it satisfies
m (x+xΔ) = mx + mΔx
Or you can do dy/dx and see that it's derivative at any point is a constant number m everywhere.
f(0) is 0, f(1) is x
Or if you consider x to be a unit vector, you can evaluate f(m) and see if it's exactly m times your unit vector.
>what do you mean decreasing derivative?
Its magnitude decreases. It is proportional to x so its magnitude changes along with any change in x. -1/x^2

>>16230557
>f(m) and see if it's exactly m times your unit vector.
I meant consider m a unit vector, then evaluating f(x) should be x times your unit vector m
I'm writting nonsense again i'm off, thanks for trying.

>>16230557
>Its magnitude decreases
this you can see in the picture >>16230501
1/x starts with very steep slope, and then it becomes less and less steep as we go on
>>16230585
if by unit vector you mean f(1), then yes.
Let's say f(1) = m, then f(x) = f(x1) = xf(1) = xm = mx.
Similarly, if we declare that the linear functions are the ones with constant derivative, then we can also conclude that all linear functions are of the form mx+c
if f'(x) = m for constant m, then f(x) is the integral mx+c for some constant c.

How do I increase my plasticity?
I was trying to solve for x a difficult equation today and I kept trying to factor one side even though it clearly wasn't going to work. My brain just couldn't consider that messing with the other part of the equation was a better idea. Someone else got to the answer before me and that made me very upset. I beat myself with a belt when I got home as punishment, but now I have to find a solution.

How do I become a more creative mathematician?

>>16230713
>I beat myself with a belt when I got home as punishment
Just keep doing this on repeat. It's clearly going to help.

>>16230713
those single unknown problems are always solved in the same way,
multiply, add, inverse and that's pretty much it.

Quite some time ago I asked about resources to learn how to use a soroban and someone linked me documents on how to use the soroban in japanese. Does anyone have something like this and if so can you link it in the thread. Thanks.

My professor said that for a finite, discrete sample space where each outcome is not equally likely, it is not true that P(A) = 0 implies A = O (but is true when outcomes are equally likely). What would be an example where P(A) = 0 and A =/= O? Is it just the trivial case where you say an event is "possible" and put it in the sample space and assign a probability 0?

>>16231003
supposed to be A = empty set

hx=g if h and g both are 0 then there's infinite amount of solutions. Would it also be infinite solutions if g is any real number and h is any real number except 0?

>>16231017
Only if you're working with modular arithmetic.

>>16231017
if h can be any real number except 0, then in particular h can equal 1, so that x=g where g is any real number.

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what anime does he watch?

>>16231003
Yes, it is. You can always do this, except when every point has equal probability, since then (because your sample space is discrete) the total probability would be 0, not 1.

If I'm planning on writing a mathematical library in a low level language, what sources on numerics should I study beforehand?
Background: I'm a math MSc. currently working a code monkey job with zero education on numerical math and something tells me that simply representing each "real number" with a double-precision float will invariably lead to mistakes in precision, or even the floats becoming NaNs. How do I find out what's the standard way of implementing these things?

>started degree at 28, posted about it in this thread when I began my studies
>frequented this general daily or nearly daily since then.
>31 years old, recently awarded MS math, MCL
>don't really feel anything
Damn. What do I do now? Not sure if this general is small enough that anyone remembers asking me to update them.

>>16232941
gratz

If you are given two statements:
[math]L1[/math]:For all [math]φ/rightarrowψ[/math] that satisfy condition [math]K[/math], [math]¬ψ\Rightarrowφ[/math] is not derivable from [math]¬ψ\Rightarrow¬φ[/math] in [math]N[/math].
[math]L2[/math]: For every wff [math]¬ψ\Rightarrow¬φ[/math] of [math]M[/math] that satisfies condition [math]Κ[/math], [math]¬ψ\Rightarrow¬φ⊢_M ¬ψ\Rightarrowφ[/math] iff for every wff [math]¬ψ\Rightarrow¬φ[/math] of [math]N[/math] that satisfies condition [math]Κ[/math], [math]¬ψ\Rightarrow¬φ⊢_N¬ψ\Rightarrowφ[/math].

Can you derive:
[math]L3[/math]:For all [math]φ\Rightarrowψ[/math] that satisfy condition [math]K[/math], [math]¬ψ\Rightarrowφ[/math] is not derivable from [math]¬ψ\Rightarrow¬φ[/math] in [math]M[/math]?

At first I thought that by [math]L2[/math], if for all [math]φ\Rightarrowψ[/math] that satisfy the condition [math]K[/math], [math]¬ψ\Rightarrowφ[/math] was derivable from [math]¬ψ\Rightarrow¬φ[/math] in [math]M[/math], then the same inference rule would be valid in [math]N[/math]. But by L1, if [math]φ\Rightarrowψ[/math] satisfies the condition [math]K[/math] then [math]¬ψ\Rightarrowφ[/math] is not derivable from [math]¬ψ\Rightarrow¬φ[/math] in [math]N[/math]. Therefore, if [math]φ\Rightarrowψ[/math] satisfies the condition [math]K[/math] then [math]¬ψ\Rightarrowφ[/math] is not derivable from [math]¬ψ\Rightarrow¬φ[/math] in [math]M[/math]. However, I don't think this is right, since the negation of the left side of the equivalence should lead to an existential statement, that there is some [math]φ\Rightarrowψ[/math] that satisfies condition [math]K[/math] such that [math]¬ψ\Rightarrowφ[/math] is not derivable from [math]¬ψ\Rightarrow¬φ[/math] in [math]M[/math]. Is this provable or not?

>>16231335
Legend of the Galactic Heroes and Fist of the North star.
>>16232454
Read some books on Numerical Analysis. Both theoretical and practical. A good start is Richard Hammings Numerical Analysis book which is available in cheap dover print.
>>16232941
Get a Ph.D? Otherwise learn to code and A.I if you just want stable employment. I'm doing the former and got my master's around your age.

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Can someone explain to me how property (2) follows from 7.18? If [math] A_f [/math] was self adjoint it would follow, but I don't see how it does in this case.
It is from Proposition 7.11 of Brian C Hall's Quantum theory for mathematicians btw which I know some people here worked through not too long ago.

>>16233280
What is property 2?

>>16233311
Norm of [math] A_f [/math] is less than or equal to the supremum of f.
See https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.math.nagoya-u.ac.jp/~richard/teaching/s2016/Hall_2013.pdf&ved=2ahUKEwis-IzmydmGAxWNIhAIHQbDDKQQFnoECA8QAQ&usg=AOvVaw3M6Fvt6mVsonwojIvc0t5x.

>>16187402
Rank the following engineering bachelors programs from least-intensive (math wise) to most intensive:
Aerospace, Electrical, Mechanical, Computer

>>16233366
Electric is the probably the most
Software the least
Mechanical can change a lot depending on the emphasis of the program

>>16233737
Dont some EE papers get published in math journals?)

how do i linear algebra

>>16232941
MS in three years? What country does that?

>>16234211
you have vectors and matrices. matrices do stuff to vectors and you can do stuff to matrices

>>16234211
Read a book
https://babel.hathitrust.org/cgi/pt?id=hvd.32044091872531

>>16233280
I think I figured it out, but the argument laid out here is in the wrong order. First you decompose [math] A_f=A+iB[/math] into two self adjoint operators A and B, then you bound its norm by 2M where M is the numerical radius, then you prove (3) which shows that all the operators obtained are normal by commutivity, and finally you use the result for normal operators that states the norm of a normal operator is equal to the numerical radius.

>>16234211
in 2D, you can see the solution of a system of 2 lineqr equations as the point of intersection of 2 lines. If you look at it from the linear algebra perspective, the solution is how much you scale the 2D vectors A and B so that when you add them together they make C

>>16234211
just be urself

This isn't really a math question as much as a /mg/ question, but do any mobile apps read latex? Kuroba doesn't.

>>16235023
chatgpt? maybe

>>16200459
>>16200460
there are less than 5000 ways to do it, so a brute force search for it shouldn't be too difficult. no idea if anybody's uploaded a library of those 4860 states (or the fundamentals, i.e. ignoring board rotations/flips, which will be much fewer and are actually all you need) somewhere for you to search through, though

Does anyone have experience implementing singular value decomposition in a program? I'm currently following the method outlined in Solving Least Squares Problems by Charles Lawson and Richard Hanson, yet I encounter numerical instability issues which prevent convergence during the iterative application of givens rotations to produce the finalized decomposition.

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Can I visualize a matrix as a vector itself and each column being its composite vectors

>>16235810
No.

>>16235810
I suppose... But it would be a covector of vectors, strictly speaking.

On the Wikipedia page for the alternating harmonic series there's a section on random harmonic series. It says that a harmonic series with random (50/50) chance for the sign of each term to be +/- will converge with probability 1.

Would I be correct in assuming that this only happens if each term is equally likely to be positive as well as negative? Does the probability of convergence drop to 0 otherwise? I know that [math] \zeta (1+ \epsilon) [/math] converges even though [math] \zeta (1) [/math] doesn't and I'm wondering if this is another example of a tipping point.

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explain this atheists

>>16237568
Hospital

>>16237501
Of course they are. You've just been told by Big Vector that they are too abstruse to be worth studying.

>>16237568
>>16237569
Or, you know, literally just the definition of a derivative.

>noooo this function is undefined at 0, even as its limit on both side are the same so it's technically continuous!!

>>16237052
Yes, and in this case you can get slightly stronger convergence as well.
Let [math]\varepsilon_k[/math] be an iid sequence of rvs, taking 1 with probability [math]p \in[0,1][/math] and -1 with probability [math]1-p[/math], and look at con- or divergence of the random series [math]S=\sum_k \frac{\varepsilon_k}{k^c}[/math] for some [math]c\geq0[/math].
The main theorem (that I know of) is Kolmogorov's three series theorem (https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem), which completely solves this question.
If [math]c>1[/math], we have [math]\mathbb E |S| \leq \sum_k \frac{1}{k^c}<\infty[/math], so [math]S<\infty[/math] almost surely.
If [math]c\in[0,1][/math] then the first series of the theorem is 0, while the second converges iff [math]p=\frac12[/math], with the third sequence converging for any [math]c>\frac12[/math].

So, convergence holds when either [math]c>1[/math], or [math]c>\frac12[/math] and [math]p=\frac12[/math], and in no other cases.

>>16232941
congratz man, any advice/things to watch out for?
I won't be able to start a math degree until I'm 26 (3 more years) because I never got a diploma
I'm studying math and programming on the side while I do that