does it ever make sense to have $\mathbb{C}^n$? What I think I am asking here is: can I have more than one complex plane? (My intuition says, hell why not, but part of me says there can only be one $i$).

let me rephrase that last bit

Of course $\Bbb C^n$ makes sense. Those of us who work(ed) on complex manifolds consider things locally isomorphic to that but globally far more interesting. You can do $n$-dimensional vector spaces over any field, finite or infinite.

I think is true only for vector spaces.

Hey Ted!

The $i$ that lives in $\Bbb C$ is just like any other scalar. It still multiplies vectors with as many components as you wish.

Hi EM4.

how have you been?

Of course, "there can be only one $i$" needs several grains of salt. For starters, $-i$ is a perfectly fine $i$.

I was just editing my post to work that out actually

but missed the edit window

Editing in chat is hopeless.

C^n gets used for discrete fourier transform for example

Doing OK, EM4. Being scratched and bitten beyond recognition by my kitten.

But I think I got there, becasue each plane is the span of $i$ and some real varialbe, right?

Not literally correct, no, @Andrew.

Here's what you're thinking. Start with $\Bbb R^n$ and take a basis $v_1,\dots,v_n$. Then the vectors $v_1,iv_1,v_2,iv_2,\dots,v_n,iv_n$ give a basis for the complexification $\Bbb C^n$.

if you are in $\mathbb C^n$ there is no longer an $i$

each component has its own $i$

That's not literally meaningful, @yorch. The complex scalar $i$ is universal. This is a vector space, after all, with scalars $\Bbb C$.

@TedShifrin that is what I was thinking in a more precise form :P

Ok then

there is an $i$ in $\mathbb C$

I don't know what literally meaningful is unfortunately

but whatever you say sounds good

I'm paid the big bucks to say things that make a little sense. :)

@TedShifrin What is "globally far more interesting"? Is there a few sentence / image example?

Manifolds are more interesting than vector spaces, @Andrew.

There are topological and geometric things to study that just don't happen in vector spaces.

That's true over $\Bbb C$ just as it is over $\Bbb R$.

:)

cheers Ted

Cheers? That means you're departing?

Reminds me i have that book open and am only up to the third excercise

cheers means Thank you outside of America Ted.

what book you have open?

Oh, EM4, I actually didn't know that :)

The Brits use it for hello/goodbye, I thought.

Maybe, I was just having coffee thoughts, I have quite a number of chores to get done before my first day at my new job :P

it all matters how you use it.

Oh, did you tell us about the new job, @Andrew?

@EM4 I started on Teds diff geo book

Ohhh, that book.

literally started

There are lots of phrases that mean both hello and goodbye, others for both please and thanks, ... depending on the language.

@TedShifrin not sure I did. I'm training as a service engineer for medical imaging devices. So MRIs/ CTscanners and conventional xray machines

@AndrewMicallef Ted is legend.

and the trainee pay is 1.5x my previous pay as a mechanic :P

Oh, cool beans. There's real mathematics in that medical imaging X-ray tomography stuff. Google Radon transform sometime!

I actually was reading up about line integrals around that

it seams super complex, but also parsable for me at this point

There's a lot of serious analysis there, but the basic ideas are ideas of integral geometry, which is the part of mathematics I started out in.

but for the time being I think I will be at the machine level, basic cleaning and routine maintenence like changing cathode tubes

Did they keep you on as a tutor after you sent that 5 page handout?

but there is always room for growth :P

well they scheduled the next session for a months time

I wonder if the teacher ever heard that we said the problem was wrong. :P

sooo, I think that is the polite way of giving me a slow goodbye

Yeah, maybe.

I hope the feedback went back to the teacher at least. Will see

in any case, I'll be a bit bummed if I don't get rebooked, but I do have a lot on my plate otherwise, so it won't be the end of the world

I do volunteering as well, and with my new commute times being a significant chunck of time, I'm wondering if I will ever get back to figuring out QM :P

Well, it's a good learning experience for you, but perhaps the learning is at too much cost for this particular student :P

Yeah, maybe not the best value transfer in the world huh

When I was a grad student, the friend of an old family friend wanted to hire me to tutor her high school son in math. I explained that I was very overqualified and that she didn't need someone of my qualifications. She insisted. I made some good money. The kid was bored in school and we actually covered about a month of his curriculum in a few hours each time we met.

that's my life right now Ted minus the grad.

I gather the mother thought her investment was worthwhile.

You're making good money tutoring, EM4?

somewhat it all depends.

I had big argument why area under the curve can't be negative with Harvard Chem student.

which was shocking to me.

@TedShifrin I don't think the kid I was tutoring was bored in school. It seamed to me his mother just wanted to put some pressure on him to do his homework.

or maybe the mother wanted her son to go into mathematics route.

I always say signed area to be clear, EM4. High school AP curriculum in the US defines area under a curve as $\int_a^b |f(x)|\,dx$, typically.

When I taught this stuff, I was careful to define "area under the curve" only for nonnegative functions and then to distinguish in the general case.

what I love about tutoring you make people happy and brush up on your basics. And I have some look at me like I am madman.

If you say area between two curves (so area between the graph and the $x$-axis), that's always the geometric area, i.e., the integral of the absolute value of the difference.

So, the Harvard chem student may be right, in fact.

if I remember carefully the answer was like $\frac{\pi}{3}$ and he had something like $\frac{-1}{4}$

but end of the day the problem was resolved and the student was happy and smashed the exam next day.

I'm not sure who's on which side any more.

Are you doing long-distance Zoom tutoring?

Harvard is like 6 miles away from me hahah.

MIT is nearby.

I hate zoom it kills the learning in my opinion.

Oh, I didn't know you were in Cambridge/Boston.

That's my old home.

come back Ted.

LOL, I haven't been back to visit since 99.

Went to high school 16 miles west of Boston, then (with a break in the middle) 6 years at MIT.

that time I was in Europe LOL.

I'm surprised you were alive in 99.

I was 6 years old HAHAH.

I am old.

Well, not old old, but yeah.

How was Boston back in the day?

Great city.

yes, and has a lot of schools nearby as well.

you have all the UMASS branches.

Brown

etc.

Oh, there are hundreds of colleges in the Boston area.

Don't forget music and art museums.

@TedShifrin I'm channelling my new boss in my job interview: "It's so boooring there!"

I haven't went to music museum hahaha. I didn't knew that exist.

(I was trying to defend my hometown as being a great place, full of art museums)

@EM4 I think they are called conservatoriums, and they exist

@EM4, I didn't mean music museums. I meant music — concerts everywhere (pre-COVID).

Hey, I have another mathsy question. The other day when I was playing around with decay rates, and Ted, you mentioned that ODEs werent a great way to model what I was modelling. Did you have something in mind that would be better?

(I'm thinking of taking a minute to convert the rules of my loan into an ODE and try and solve for the least money I need to pay back)

(and by minute I really mean the better part of the day at my speed)

that would better be done with a fairly complex spreadsheet. ODE is overkill.

actually, if you're solving for least money under different payment terms, maybe just a programming language of your choice. lots of moving parts, but not too many, depending on the complexity of the loan agreement.

depending on the size of the loan it may be helpful to consider tax implications too, although tax cosniderations don't matter too much if the amount of your loan isn't 'too high.'

@leslietownes yeah probably right there. I made a quick and dirty numerical simulation already, looks like a decaying exponential, but now i'm curious if it is

did i do something like this for my student loans, yes, i did.

@leslietownes there are really only two relevant terms, the interest rate, and the "fuck you for early repayment term"

yeah, most of the spreadsheet would be about figuring out how much that throws off the more obvious payment strategies.

caps on tax deductions for loan payments, if any, could also mess it up

Ahh, I didn't think about caps on deductions, that will bite me!

Is it reasonable to ask the lender "How do I pay you the least money in the long run?"

I feel like they would either dodge or lie

they will lie.

i don't know what the law is in your area, but they likely would even if it was illegal

there was a big suit against one of the main student loan servicers in the US, for automatically steering some big class of students, i think military veterans?, into more profitable payment plans.

(they probably don't know, after all the agreement isn't so trivial as to be a simple mental calculation)

there's also that. what's cheapest may depend on your income if tax is a big issue.

i spent a lot of time on the tax part of my spreadsheet only to learn that i didn't make or owe enough money for it to make any real difference.

hahah

I'm counting on tax eating 100% of the interest, but now if there are caps, i really need to reconsider

@AndrewMicallef No, ODE is fine. It was your discretization that made no sense.

hmmm, I presume you mean my time steps where massive relative to the size of the parameters. Otherwise I'm not sure how I could have improved the discretization

but you probably dont mean that cause I dont think I gave you values of the paramaters...

Oh no I did

i have made most financial mistakes that are to be made. while one should be aware of tax implications, it is not usually a good idea for tax considerations to be the main driver when it comes to such things.

I have made all of the financial mistakes my budget has allowed me to

thankfully i am willing to take a big hit as necessary, but my impatience & risk profile tends to carry a lot of heart stopping, ucler inducing exposure. for example, i was all cash for 18 mos before the 2008 financial crisis (because i figured it was coming), but after so long decided i was wrong and got back in, just in time to take a whopping hit.