🧵 /mg/ math general
Anonymous at Mon, 17 Feb 2025 22:29:29 UTC No. 16589624
Required Reading edition
Talk mathematics, formerly >>16562509
Anonymous at Mon, 17 Feb 2025 22:30:20 UTC No. 16589625
>>16589624
First for Verbitsky and Kaledin's Trivium
>>16568704
>>16569867
>>16573937
Anonymous at Mon, 17 Feb 2025 22:33:24 UTC No. 16589632
>>16589625
This is the other document. There's no order, you can approach both of them concurrently
Anonymous at Mon, 17 Feb 2025 22:47:37 UTC No. 16589647
>>16589409
>Why are compact spaces defined in such a peculiar way (space is compact iff every open cover of space has a finite subcover)?
Hey, I was taught that compact spaces are T2 (Hausdorff) + finite subcover condition. Was my life a lie?
Anonymous at Tue, 18 Feb 2025 01:38:24 UTC No. 16589755
>>16589625
>>16589632
where's the other one?
Anonymous at Tue, 18 Feb 2025 02:23:21 UTC No. 16589789
>>16589625
>>16589632
Algebra 9, question 18 is messed up but I'll write the question here as best as I can:
Let [math] v \in R [/math] be an Artinian ring over [math] k [/math] and
[math] P(t) [/math] be its minimal polynomial. Consider the subalgebra
[math] k[v] \subset R [/math] generated by [math] v [/math] and [math] k [/math]. Prove that [math] R_v [/math] is isomorphic to the ring
[math] k[t] / P [/math] of residues modulo [math] P(t) [/math].
Are there any other errors in either of them?
Anonymous at Tue, 18 Feb 2025 06:03:19 UTC No. 16589883
>>16589755
There's only two of them.
Anonymous at Tue, 18 Feb 2025 09:34:59 UTC No. 16589962
>>16589883
Why are the documents called trivium then?
Anonymous at Tue, 18 Feb 2025 09:40:17 UTC No. 16589964
>>16589962
>trivium: singular of trivia; anything of little importance
Anonymous at Tue, 18 Feb 2025 13:09:13 UTC No. 16590143
>>16589962
They are a "must-know" set of problems. V. Arnol'd also has a trivium but for applied math/physics students:
https://physics.montana.edu/avoront
Anonymous at Tue, 18 Feb 2025 13:12:12 UTC No. 16590146
>>16589647
No, that's a common definition. FSC by itself is sometimes called quasicompact to distinguish.
Anonymous at Tue, 18 Feb 2025 14:58:35 UTC No. 16590240
Can someone explain how term proofs work in lean 4? I get tactics proof but term proofs are so opaque to me.
Anonymous at Tue, 18 Feb 2025 23:30:49 UTC No. 16590876
>>16590240
This is too broad of a question to answer very meaningfully. Every proof in lean, even with tactics, is constructing a term of a specific type, it's just that tactics are some boilerplate corresponding to common proof patterns that generate terms for you. If you haven't read it already, this is the reference you need to start understanding type theory in Lean properly: https://lean-lang.org/theorem_provi
Anonymous at Wed, 19 Feb 2025 12:34:06 UTC No. 16591373
I envy people who love math.
Anonymous at Wed, 19 Feb 2025 15:10:29 UTC No. 16591461
>>16591373
I use to love math, but I’m not sure if I do anymore. It’s been beaten out of me by my career. Hard to find anyone in the workforce who cares about abstractions
Anonymous at Wed, 19 Feb 2025 15:25:07 UTC No. 16591471
>>16591461
Math is the ideal hobby T B H.
Anonymous at Wed, 19 Feb 2025 15:29:44 UTC No. 16591474
>>16591373
It's the only thing i genuinely like in the world. I wish i liked other things too.
Anonymous at Wed, 19 Feb 2025 22:09:54 UTC No. 16591888
>>16591759
i would have used different colors for the triangles
like red, green, blue and white
or magenta, yellow, cyan and black
Anonymous at Thu, 20 Feb 2025 01:11:23 UTC No. 16592026
Does Zorn's lemma imply excluded middle?
Anonymous at Thu, 20 Feb 2025 04:18:25 UTC No. 16592141
>>16592026
Ok, apparently it does not.
Anonymous at Thu, 20 Feb 2025 17:03:04 UTC No. 16592976
What's a good pathway to go from knowing only basics of probability to taking a graduate course on probability next semester? I know the analysis pre reqs but the actual probability i'm very sketchy on. I'd like to really learn it well if possible. Any text recommendations that include exercises would be greatly appreciated, thanks!
Anonymous at Thu, 20 Feb 2025 17:04:40 UTC No. 16592980
Do you think you have to be an intellectual masochist to be a good researcher? I'm starting to think so
Anonymous at Fri, 21 Feb 2025 07:54:36 UTC No. 16594532
>>16592976
I'd say measure theory is the main difference between intro probability and graduate level probability. I'm a big fan of "Probability: Theory and Examples" from Rick Durrett. Starts with strong foundations, works up to lots of useful topics. Lots of exercises
Anonymous at Fri, 21 Feb 2025 07:57:43 UTC No. 16594534
>>16592026
Yes, if we're assuming ZF set axioms.
Under assumption of ZF, the Zorn's lemma is equivalent to axiom of choice. We then have Diaconescu's theorem (aka Goodman-Myhill theorem) which shows that AoC -> excluded middle.
Anonymous at Fri, 21 Feb 2025 08:01:10 UTC No. 16594537
>>16589632
Why is this labeled "Geometry"? Reads more like a topology book to me.
Anonymous at Fri, 21 Feb 2025 08:06:30 UTC No. 16594543
>>16589624
Some stuff to add to the list: Aluffi's "Algebra: Chapter 0" for modern algebra with a category-theoretic flavor.
Linear Algebra Done Right for more advanced linear algebra stuff.
Hatcher for Algebraic Topology
Anonymous at Fri, 21 Feb 2025 09:03:04 UTC No. 16594573
>>16594534
Zorn's lemma implying choice requires excluded middle
Anonymous at Fri, 21 Feb 2025 11:52:29 UTC No. 16594690
Is there a weaker version of Zorn’s Lemma that is equivalent to the Axiom of Dependent Choice? My intuition tells me that there has to be some restriction made, maybe every chain with an upper bound has to be countable?
Anonymous at Fri, 21 Feb 2025 16:41:49 UTC No. 16595034
Teach me non linear analysis sci
Chaotic introductions and advanced levels
Anonymous at Fri, 21 Feb 2025 20:50:37 UTC No. 16595431
>>16589789
Any other errors in the algebra and/or geometry pdf?
I only found one in algebra 9, but the others seem okay.
Anonymous at Sat, 22 Feb 2025 02:50:42 UTC No. 16595876
Why doesn't algebraic thing have property p?
>Here's the proof.
Do geometric things a,b,c, etc have property q?
>Here's the proof.
Why don't most PDEs have closed form solutions?
>They just don't okay? No we don't have any existence disproofs or impossibility theorems or any ways to measure the space of closed-form anythings, you've just got to accept that you can't do it on faith. If it was doable it would have already been done.
This is my prof, unironically. Where do I go to find the actual existence/nonexistence theorems about the big picture of PDEs?
Anonymous at Sat, 22 Feb 2025 03:30:46 UTC No. 16595895
>>16594537
>https://en.wikipedia.org/wiki/Orde
https://en.wikipedia.org/wiki/Order
>https://en.wikipedia.org/wiki/Metr
https://en.wikipedia.org/wiki/Categ
Anonymous at Sat, 22 Feb 2025 07:32:55 UTC No. 16596007
>>16595034
chaos is so 80s, gramps
just pretend everything is a harmonic oscillator and perturb away
Anonymous at Sat, 22 Feb 2025 07:34:23 UTC No. 16596008
>>16595876
You can prove existence without requiring closed-form solutions.
Anonymous at Sat, 22 Feb 2025 08:00:24 UTC No. 16596016
>>16595876
I'd like to point out that the world is even messier than your question suggests. As a matter of fact, there are proofs that most ODE's do not have closed form solutions in terms of elementary functions.
Consider that the solution [math] f [/math] to the 1st order ODE [math] \frac{df}{dx} - g(x) = 0 [/math] for some given function [math] g [/math] is equivalent to the computation of an antiderivative of [math] g [/math]. Well we all remember how hard it is to take integrals of arbitrary functions [math]\mathbb{R} \to \mathbb{R} [/math] or [math]\mathbb{C} \to \mathbb{C} [/math]. Many of us have taken multiple courses dedicated to the solution of that particular ODE. Moreover, Liouville's theorem in differential algebra essentially states that the only functions with elementary (closed form) antiderivatives are those we can write as the sum of rational functions and finitely many logs of rational functions.
Anonymous at Sat, 22 Feb 2025 12:56:11 UTC No. 16596289
>>16590876
Explain how the convert tactic works.
Anonymous at Sat, 22 Feb 2025 14:53:53 UTC No. 16596402
>>16596289
It works by taking your prompt and redirecting it into chatgpt until you have something less imbecilic to ask about.
Anonymous at Sat, 22 Feb 2025 22:21:41 UTC No. 16596819
>>16596402
I'm sorry anon.
I'm not smart like you
Anonymous at Sat, 22 Feb 2025 22:36:54 UTC No. 16596837
>>16589624
I just started reading the Mary Dolciani books and I'm surprised you all haven't shilled her books before, starts off with logic,set theory, and mixes proofs into a lot of the material.
Modern Algebra: Structure and Method Book 1
Modern Geometry: Structure and Method
Modern Algebra and Trigonometry: Structure and Method Book 2
Modern Introductory Analysis
All from the 60s-70s during the "New Math" movement, later revisions in the 90's drop the "Modern" in the title and aren't as good apparently. Throw on spivak and you have an Algebra 1 to Calculus curriculum.