Okay, she knows her epidemiology.  ...That probably means they're good.  She's gonna wander off, then.

The city continues to be a city. Guards chasing a pickpocket. Trendy sweets shop with a long line. Buskers. A group of wizard-style-robed young adults standing in a tight circle doing something. Someone flying overhead! Leaky fountain dripping into the road from cracked masonry. A dog, which barks at her. She's at the edge of the academic district again.

She's gonna watch the wizard-chic teenagers!  And then make her way to the lunch place.

If she creeps up close to peek between the hems of their robes, she will find a large cauldron with weird baffle-y bits sticking up out of it. Weirdly bland-smelling cooking oil glistens on them and drips slowly in. (...actually, the lunch place smells the same way, so this is probably Apprentice Oil)

...Huh.  What're the baffles for?

The students are just gossiping and complaining about difficult coursework, not conveniently discussing their oil-making chore.

Oh, is that what this is.  Okay.  ...There any questions that her semi-dubious command of both geometry/trigonometry and spell design allow her to elucidate?

They're arguing about discontinuous functions and how they are a real pain to cast because they blow up worse than most others if you get it wrong. A girl in the crowd takes the position of 'lmao git gud'. Well, different words but that's the sentiment.

...On the one hand, yeah, on the other hand, ???

"...Discontinuous like rounding is discontinuous, or discontinuous as in there is no derivative nor singular limit for a function f of x when x equals a discontinuity within the range of your function?  Because one of those has a whole damn lot more of my sympathies."

"Wh- Who's there?"

"It's the kitsune, the one that one bunny girl made friends with-"

"Oh, yeah, if anyone could make friends with a kitsune-"

"Discontinuous like rounding, anyway, because continuous functions tend to work, poorly, if you fuck up the casting a little bit and discontinuous ones don't. I think they encourage sloppy casting." The last girl declares haughtily.

"...She does have a name, you know."

She seems to consider something for a good few long moments, idly tracing something over the ground with a paw.

"...Damn.  Fourier transforms probably wouldn't help.  The problem's the physics, not the math.  But it is possible to closely approximate arbitrary discontinuous functions by summing a series of sine waves, for functions with one-to-one mapping of input to output.  That's not exactly useful when what you're doing is more like trying to arrange a bunch of permanent magnets so the iron filings you drop in make a specific pattern, but it's true nonetheless."  ...As much as her own preferred analogy to deflecting particle beams is great, she thinks, it would only help her understand anything, so she went for magnets instead.

"Oi, I don't know her personally. It's reasonable not to know strangers' names."

"No, that does sound maybe useful- The patterns can get incredibly precise for high level research spells, it's freehand casting and template casting that needs to be fault tolerant. There are- Over in Lotus Tower there's a 120 meter spell circle and all sorts of instruments for it."

"...Well, you've got me there," she returns to the person who doesn't know Weiya.  She can't believe she's that gossipworthy.  Or, well, she can, but she doesn't want to.

As for the math discussion - "...Huh.  I would have expected the utility of Fourier transforms - excuse me, that's an old Tirran technical term for doing that decomposition to sines - to be in introducing better fault tolerance by making formerly-discontinuous spells into continuous ones.  Though I suppose that better fault tolerance works just as well in any application, and when you have such small margins already, any edge you can get is much more worth the effort."

"Maybe I'm misunderstanding what this Fourier transform actually is? You said it's a method of approximation?"

"...Sort of, yes, but also kind of not.  Not that I know the math that actually goes into all of it.  The Fourier transform as a concept is just a different way of representing the same exact math.  The Fourier transform as applied in finite iterations is approximative."

"It sounds like this wants a blackboard and a professor* or two."

*Close enough, they have a bunch of different tiers of academic-ness that don't really match western university setups.

"Probably, and I am neither of sufficient mathematical qualification, nor possessed of a blackboard.  Or hands.  Absolute most annoying thing about reincarnating, not having hands."

"Can't kitsunes shapeshift?" Someone wonders.

"Once they're mature," the bored-sounding guy replies.

"I'm sure we can find someone interested if you are interested, though?"

"I think I would be, but honestly I don't want to get anyone's hopes up.  I did not study the right sorts of math for spell geometry."

"The big hats like talking about math even when it's not useful, I think you'll be fine."

"What, you're not a big hat?"

"Oh, yeah, I'm totally going into industry." The girl shrugs.

The other three seem to think this is slightly scandalous.

"Hey, applied sciences is still science.  Somebody's gotta turn theory into practice, and efficiently, if we want to make the things the big hats figure out actually useful to the commons.  ...And seriously, I barely math, I don't even know how to do multivariate calculus."

This little group of apprentices is happy to chatter on on various topics until Weiya shows up.

And Weiya continues to be an indomitably cheerful little girl who takes her studies seriously and likes fried potato snacks!

She's happy to study with Alicia over the next however long. Few days? Couple of weeks? ...Month or two?

Definitely long enough that Alicia can manage to learn to cast apprentice spells, at least! They just make sense on some kitsune-feeling level.

They even manage to get around the pet clause by moving the cat back to his old home, where Weiya grew up, if Alicia would rather just move into Weiya's dorm for the duration.

...She's not entirely sure whether to protest the definition, to be honest, but she certainly doesn't want to part Mango from his person just for marginal comfort?  She can find somewhere to den, or a convenient chair, or just prove sapience, if having a third person sleeping over won't get them in trouble over their lease.

...She spends about as much time trying to figure out why her instincts are telling her things as she does finding things for her instincts to tell her, but she's a quick study.  Her focus is primarily on kinesis spells, with a sideline of attempting to reconstruct the periodic table of elements by comparing material divinations with eachother and het knowledge of chemistry.  (And a much more clandestine attempt to put together a sufficiently strong coal-crushing spell.  Diamonds are a girl's best friend, after all.)

Weiya gets an exception to the rules about animals in the dorm for an actual kitsune, eventually. Mango tolerates a new presence with good grace. (Weiya's roommate is a terminal introvert, and not involved much.)

Megi lets her know fairly early on that she's moving on, and is fairly huggy about it and says she does like Alicia a lot, she just does not really settle down, and asks if Alicia has any last requests or questions before it's 'see you some day!'