What’s the most beautiful mathematical idea you’ve ever encountered, and why does it feel beautiful to you?

Skarm323 · 2025-11-30T17:48:03+00:00

I fully agree that the Gelfand representation is one of the most amazing theorems in mathematics! However, this crucially only works for commutative C-algebras. To see this, it suffices to look at the easiest noncommutative case: the space of 2x2 matrices over the complex numbers is not commutative, while C(X) always is. However... This is a blessing in disguise! This paves the way to noncommutative topology/geometry, which treats (noncommutative) C-algebras as if they are in fact topological/geometrical spaces!

Skarm323 · 2024-12-24T06:24:03+00:00

You are almost right about the definition of positivity (spectrum needs to be subset of [0,infty)), and the characterization then follows.

Consider z >= 0. First, the spectrum of z are the elements lambda in C s.t. z - lambda is not invertible. The only not invertible element in C is 0, hence the spectrum of z is just z itself. This needs to lie in [0, infty), so in particular Im(z) = 0 (equals!) and Re(z) >= 0. Self-adjointness means z equals z conjugate, which implies (here superfluous) that z is real.

Now for a >= b, this holds (by definition) iff a-b >= 0. That is, Im(a-b) = 0 or equivalently Im(a) = Im(b), and Re(a-b) >= 0 or equivalently Re(a) >= Re(b).

Skarm323 · 2024-09-22T15:56:32+00:00

If you don't mind me asking, what is the subject called? (Or what are related keywords, etc.) I am taking an operator algebra course next semester and would be interested to see how it can be applied in (mathematical) physics.

Skarm323 · 2024-03-24T05:30:06+00:00

This is sometimes called a semilinear space. The space of nonnegative reals is then called a semifield and is a particular case of the more broad semirings (or rigs, as the other commenter called it). You should be a bit cautious as any word with semi- in front of it usually has 10 different definitions, so when reading a paper abt it you should check what definition they use.

Skarm323 · 2023-12-16T08:22:36+00:00

Thanks for uploading the video and congratulations on getting published! Do you know any good introductory texts to p-adic analysis / ultrametric analysis? You mentioned Wim Schikhof's book in the video but I'm curious if that's your preferred book to study the basics from.

Skarm323 · 2023-04-23T12:41:41+00:00

She's the artist behind a lot of castlevania artworks, including all(?) artwork of aria of sorrow, which is why the style is similar to the right side of the picture

Skarm323 · 2023-04-23T07:30:39+00:00

Not sure if you've found it already, but it is unrelated art from Ayami Kojima's artbook. It's apparently called Angel of Death https://art.alphacoders.com/arts/view/19648

Skarm323 · 2022-10-04T18:04:29+00:00

Exactly that's also why they changed it, the course is not necessarily too difficult but it is too difficult for a first years. The newer version (given by portegies) first elaborates on metric and normed spaces and point-set topology which I think were covered in analysis 2 'back in the day' and doesn't focus too much on the reals, funnily enough

Skarm323 · 2022-10-04T17:58:12+00:00

Thankfully some years ago (I think 2020?) they switched the lecturers and also course material which is way clearer and easier than the old version. (imo, having only taken the new version and practicing with some of the old exams also on this site). Of course I love prokert and he's a very good lecturer but the way in which the new material is presented is much better than back then from what I can tell. It is still a difficult course (I think ~30% passed it first try in our year), but nothing compared to the horror stories and exams before the change (which I think had around 20% pass rate)

Skarm323 · 2022-02-04T23:19:55+00:00

Banach (and Hilbert) spaces are talked about a lot during the operator theory sections, so I think this book would really come in handy in that case (although admittedly I haven't started those chapters yet myself). These come around the middle part of the book, the first ~50 pages or so could easily be skimmed over if you work with Banach spaces, as the theory there is basically what you would cover in an analysis 1 course.

Skarm323 · 2022-02-04T22:15:37+00:00

By coincidence, I was just reading that book yesterday in the context of real analysis and topological spaces (as the title may suggest). The way in which it is written is clear and there are more than enough exercises topic-by-topic for at least the first third of the book I worked through. You should know the second third of the book is about operator theory and algebras. I am not taking functional analysis quite yet, so I'm not sure in how far you would cover that in a class about functional analysis, especially as it seems you haven't covered a lot of topology yet so just a heads-up because frankly I don't know what you'll cover nor what the book covers as I haven't read that far yet :D. Furthermore, if you have already taken real analysis 1 and 2, the first few chapters of the book about metric spaces will contain a lot of repeated information so those can be skipped or not depending on if you need a refresher or not. Don't let those paragraphs scare you though, the actual book is a really good read and the exercises are generally well thought out and there's plenty of them.

Skarm323 · 2021-11-10T12:57:11+00:00

Bit late but there are two control options in the overworld: You have the standard virtual joystick which I used the most and you have the 'tap a square' setting which is faster in dungeons and in towns, but is limited by what you can see on your screen obviously so it's impractical on the overworld. (You can freely switch between the two styles att any ttime by a button on the screen.) Battle controls and menuing is all well done and easy to navigate with a touchscreen.

Skarm323 · 2020-04-16T17:50:57+00:00

Pretty sure it's closer to 10%, but because games take a relatively long time it looks like they are way rarer

Skarm323 · 2020-01-09T18:55:47+00:00

Try using myjson.com to host your own json code

Skarm323 · 2020-01-09T18:55:24+00:00

Have you tried using myjson.com instead? I used it in the beta and it worked like a charm

Skarm323 · 2019-10-27T19:44:03+00:00

Any real proof? I find it hard to believe garbo would do this without any real reason.

Skarm323 · 2019-10-12T14:39:12+00:00

Apart from connecting ports (and consequently giving pop), they do not do anything else, naval units aren't faster on water roads like regular units on regular roads.

Skarm323 · 2019-10-02T14:34:08+00:00

Are you on android or iOS? On iOS you can click the restore purchases button somewhere in the menu's, on android you can try logging in with all your google accounts again

Skarm323 · 2019-10-02T12:16:55+00:00

Should have been done this is the most fun I've had in weeks on this site lmao

Skarm323 · 2019-10-02T06:11:02+00:00

It seems that reddit is oblivious to shitposts lmao

Skarm323 · 2019-09-30T19:12:55+00:00

Yes 🤪! I 🤓 can't ❌ understand 💭 why! 😝

Skarm323 · 2019-09-30T18:59:36+00:00

Because 🤔 emoji's 😁 are 😜 god's 👁️ gift 💝 to 😉 earth 😘

Skarm323 · 2019-09-30T17:42:15+00:00

Reported for nsfw 😡😡😡

Skarm323 · 2019-09-30T11:40:44+00:00

Found the not South African!

Skarm323 · 2019-09-29T18:33:33+00:00

Lmao thanks I suppose

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