Why are natural transformation diagrams commutative ?

I just started learning about category theory. I have no problem understanding the concept of category and of functor, and I can also see how functors make a category, but I struggle with the definition of natural transformations. The part about choosing a morphism in Mor(F(X), G(X)) for each X in the first category makes sense to me, but why does the diagram have to commute ? Is it in order to respect something else ? Or is it just because of convenience ?

submitted by Nevin Manimala Nevin Manimala /u/lonelygenius
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Any numerical analysis course out there in the Internet?

Of all the MOOC and online learning platforms I know of, only OCW apparently was able to display a course on numerical analysis, 18.330 Introduction to Numerical Analysis. edX, Udacity and Coursera delivered little to no results.

I am an independent learner and would eagerly learn about the subject. Do you know any interesting and sound course on it? Alternatively, some textbooks to follow? I much appreciate them for their exhaustivity, with the only downside of not imposing a fixed schedule to follow.

submitted by Nevin Manimala Nevin Manimala /u/al_taken
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Where to find resources on mathematical puzzles like this?

1) 6-sided die (sides: 1, 2, 3, 4, 5, 6), roll and win $x depending on the side facing up. How much are you willing to pay to play this game?

2) If we play a game in which Player 1 picks a number 1-11, and then player 2 can add 1-11 to that (i.e. player 1 picks 5, player 2 can add to make it 6-16), what is the strategy to win this game if Player 1 wants to make 60?

Looking for a collection of similar questions or significantly challenging ones relative to these.

submitted by Nevin Manimala Nevin Manimala /u/letsgobaby
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Complex Analysis Summer Study Group?

Hey /r/math, I will be studying Gamelin’s Complex Analysis this summer in preparation for graduate school. While I enjoy studying immensely, I find that it can be a very lonely experience. Would anybody be interested in joining a study group? I was thinking we could create a google drive folder and submit our answers to problems and have a discord or whatsapp or something to discuss further. I would expect that you would know at least some real analysis, and would have a substantial amount of time to devote to solving problems. If I get at least five people interested I’ll start this up. You can comment or PM me.

submitted by Nevin Manimala Nevin Manimala /u/McTestes68
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Looking over my notes for real at the stuff I didn’t understand, Arzelà–Ascoli was the worst offender.

Wikipedia has a great article on it, but the proof they give (at least from what I can understand) isn’t general enough to deal with compact sets in abitrary metric spaces as opposed to just Rn.

Here are the two statements: Consider a sequence of real-valued continuous functions { fn }n ∈ N defined on a closed and bounded interval [a, b] of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence { fnk }k ∈ N that converges uniformly.

And the sequential version from my class: Let A be a compact set in a metric space (M, d), and suppose (fk) is a sequence of functions fk : A −→ R which is bounded and equicontinuous. Then (fk) has a uniformly convergent subsequence

Any help would really be apreciated

submitted by Nevin Manimala Nevin Manimala /u/sectandmew
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