Stuff you never asked for

My name Kay | mid 20s applied maths and stats graduate | if you have questions, send asks or comment on the posts | apologies if I don't follow back | No DMs

[a-lesson-you-never-asked-for]

Feel free to ask any maths questions. I'm still learning so I apologise if make any mistakes. If you do find any errata, please let me know.

I mostly focused on computational mathematics, marine architecture and dynamical systems. I've only done honours so I'm still a long way from being classed as an expert in any of this stuff, but I hope in teaching you, I can also push myself to become better at teaching.

[a-lesson-you-never-asked-for]

lmao i won best talk at this conference - how the fuck did i do that?

[a-lesson-you-never-asked-for]

Have you ever published a paper on something so clever but also so stupid that you wonder why hasn't anyone done this before?

[fforsythiaaa]

Once when I was in undergrad, someone described something as “problematic” in class and our professor was like, “That’s cool, but ‘problematic’ doesn’t really mean anything. It means that the thing you’re describing has a problem, and in and of itself that’s not bad. Art, especially, should always have problems, or else it’s not interesting and not art, either. It sounds like you’re trying to say that this is bad, but you don’t want to say ‘bad.’ Is that right?”

So from then on whenever one of us called something problematic, he would make us talk it out until we could name the “bad” thing we were hinting at. In this particular class, 7/10 it was some type of oppression, and the remainder was like, “I’m uncomfortable because this is very new/confusing/pushing boundaries that made me feel safe.”

Once we stopped calling things “problematic” and stopping at that, class got way more interesting and... we all had to say, like, “that’s racist” or “that’s misogynistic” or “ew capitalism gross” out loud, which a lot of us had never done in a classroom before. Or we had to be like, “Uhhh... I’m not sure what’s so bad?” and confront our own beliefs and that was maybe even more useful.

Anyway. Whenever I see the word problematic, I can’t help but think of this professor being like, “Good starting point, now let’s get specific.” I think when we have to commit to saying “that’s ___” it requires a lot more careful thought about the truth and impact and complexities of whatever we’re claiming. Sometimes there really is some bullshit afoot, and also sometimes it’s art, and it should be full of problems, because that’s what art is.

[hypokeimena]

#'this is present in the text' is often a good first step #but those second and third ones (naming it; describing its function) are vital (via @elucubrare)

[cauchyfeelings]

Love how pure maths is just like:

but what if structures related to structures structurally

the category of categories contains functors between categories that preserves the categorical structure categorically

and relationships between objects themselves form objects whose relationships can also be studied

It’s giving she sells seashells by the seashore vibes

[algebraic-dumbass]

today's episode: Am I Skill Issuing Or Is The Paper Just That Bad At Explaining?

[a-lesson-you-never-asked-for]

almost definitely the latter

[jijisarchive]

Me core

[a-lesson-you-never-asked-for]

It's been a fat minute. Hope you're all well! I'm back for a bit on this blog, in the mean time, I'd like to share I have made a really cool discovery. I am suffering from a lot of stress but I am getting by :))

In the mean time, if you specialize in algebraic geometry, specifically on Kahler type objects, can you let me know? I need some help on some of the things I'm working on with regards to tropical geometries and analytical geometry. I am an idiot of the highest order in this particularly because I promised myself I would never go anywhere near the cursed objects I find within algebraic geometry. But unfortunately, I have stumbled into a really interesting application. Please and thank you :D

sincerely coming from an overworked, sleep deprived maths student

[a-lesson-you-never-asked-for]

Hi all, been a fat minute. Will start posting lessons again soon. Work and research been a bastard so havent had any self care time

[a-lesson-you-never-asked-for]

Exam soon… The only thing I’ll be calculating is the tension in the rope I’ll be hanging myself with.

Someone who once studied functional analysis

[i-am-intentionally-dense]

evaluate the sum from 0 to infinity of arctan(1/n^2) :)

[a-lesson-you-never-asked-for]

some bloke sent me a really annoying maths question that genuinely stumped me. Finally figured it out after constantly making sign errors, that happens a lot with complex numbers but meh, i finally figured it out.

There's a really neat trick here where you can convert sums of sequences into a problem of residues. I think a lot of people smarter than me can use this trick with great utility.

[a-lesson-you-never-asked-for]

Black to move, find the forced mating sequence

[boycritter]

[a-lesson-you-never-asked-for]

I had this position as white. I feel bad.

[associativeglassdesert]

can tumblr give fucking LaTeX support as a feature already instead of ai data scraping or whatever

[a-lesson-you-never-asked-for]

The closest we'll ever get is through markdown :(

LaTeX embeddings would be so nice. Messenger used to support it too and then that fucker Zucker got rid of it

[a-lesson-you-never-asked-for]

Dynamical Systems

When we talk about dynamical systems, we broadly refer to the way a particular system evolves under certain conditions. For example, if you have a particle and you give it some terrain to roll around on, what path will it follow? If you have an electron and subject it to a magnetic field, what trajectory would it follow? Or if you impart forces on a block of water, how would the shape of the boundary change?

These are all types of questions that can be answered by dynamical systems, and involve a significant depth of analysis to truly understand their mechanics. (But truthfully, I've only ever been in it for the pictures).

The Tinkerbell map The Lorenz Attractor

As usual, if anyone has any feedback or errata to point out, please do shoot me a message :). I'm still getting back into the groove of things with this so I might be missing out on stuff.

Some basic definitions

We have 2 different types of trajectories, discrete and continuous.

discrete time dynamics - this means that you have observations at discrete points in time. You take a snapshot at t=1, t=2, etc. Take the Tinkerbell map above for example. You start with a point, then apply some sort of rule, and you end up at the next point. Things happen in steps.

continuous time dynamics - this means that at each point, you prescribe a velocity (a speed and direction) in which a particle should move. So if a particle is located at (0,0) for example, in an arbitrarily short period of time, it would move in a particular direction. A good example of this is the Lorenz map, as shown in the right hand image.

The Tinkerbell Map - equations of motion

The Lorenz Map - equations of motion

NOTE: In the first case, we talk about where the next point explicitly, whereas in the second case, we talk about in how fast in a particular direction we will end up moving.

A brief discussion on derivatives

Here's a visual to explain - suppose that the blue line represents how far away you are from some particular location. The average velocity would be the slope of the green line. But the instantaneous velocity would be equal to the the slope of the yellow line.

The way we denote this derivative is as follows

The numerator describes the change in the y-values, the denominator describes the change in the x-values, and the ratio of this is the gradient (or the slope).

So in a nutshell, the average velocity represents a change in distance over a period of time, and instantaneous velocity represents a change in distance of an arbitrarily short period of time.

Where to from here?

From here, we'll introduce some broad notation around maps, orbits, and some terminology that underpin dynamical systems. If there's enough time, I'll also describe some other concepts regarding different types of orbits, and thereafter we'll get into some really cool stuff.

Anyway, that's it for now! Adios :)

[a-lesson-you-never-asked-for]